Epidemic modelling an introduction

CAMBRIDGE STUDIESIN MATHEMATICAL BIOLOGY: 15EditorsC. CANNINGSUniversity of Sheffield, UKF. C. HOPPENSTEADTArizona State University, Tempe, USAL. A. SEGELWeizmann Institute of Science, Rehovot, Israel

EPIDEMIC MODELLING: AN INTRODUCTION

This is a general introduction to the ideas and techniques required to understandthe mathematical modelling of diseases. It begins with an historical outline of somedisease statistics dating from Daniel Bernoulli's smallpox data of 1760. The authorsthen describe simple deterministic and stochastic models in continuous and discretetime for epidemics taking place in either homogeneous or stratified(non-homogeneous) populations. A range of techniques for constructing andanalysing models is provided, mostly in the context of viral and bacterial diseases ofhuman populations. These models are contrasted with models for rumours andvector-borne diseases like malaria. Questions of fitting data to models, and the useof models to understand methods for controlling the spread of infection arediscussed. Exercises and complementary results at the end of each chapter extendthe scope of the text, which will be useful for students taking courses inmathematical biology who have some basic knowledge of probability and statistics.

CAMBRIDGE STUDIESIN MA THEM A TICAL BIOLOGY

2 Stephen Childress Mechanics of Swimming and Flying3 C. Cannings and E. A. Thompson Genealogical and Genetic Structure4 Prank C. Hoppensteadt Mathematical Methods of Population Biology5 G. Dunn and B. S. Everitt An Introduction to Mathematical Taxonomy6 Prank C. Hoppensteadt An Introduction to the Mathematics of Neurons

(2nd edn)7 Jane Cronin Mathematical Aspects of Hodgkin-Huxley Neural Theory8 Henry C. Tuckwell Introduction to Theoretical Neurobiology

Volume 1 Linear Cable Theory and Dendritic StructuresVolume 2 Non-linear and Stochastic Theories

9 N. MacDonald Biological Delay Systems10 Anthony G. Pakes and R. A. Miller Mathematical Ecology of Plant Species

Competition11 Eric Renshaw Modelling Biological Populations in Space and Time12 Lee A. Segel Biological Kinetics13 Hal L. Smith and Paul Waltman The Theory of the Chemostat14 Brian Charlesworth Evolution in Age-Structured Populations (2nd edn)15 D. J. Daley and J. Gani Epidemic Modelling: An Introduction

D. J. DALEY and J. GANIAustralian National University

Epidemic Modelling:An Introduction

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

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This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First published 1999First paperback edition 2001Reprinted 2005

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Library of Congress Cataloguing in Publication dataDaley, Daryl J.

Epidemic modelling : an introduction / D.J. Daley and J.M. Gani.p. cm. - (Cambridge studies in mathematical biology ; 14)

Includes bibliographical references and index.ISBN 0 521 64079 21. Epidemiology-Mathematical models. 2. Epidemiology-

Statistical methods. I. Gani, J. M. (Joseph Mark) II. Title.III. Series.RA652.2.M3D34 1999614.4'01'5-dc21 98-44051 CIP

ISBN 978-0-521-64079-4 hardbackISBN 978-0-521-01467-0 paperback

Transferred to digital printing 2007

To Nola, for constant support and understanding [DJD]

To my late wife Ruth, who first directed my interestto biological problems [JMG]

. . . The history of malaria contains a great lesson for humanity—that weshould be more scientific in our habits of thought, and more practical inour habits of government. The neglect of this lesson has already cost manycountries an immense loss in life and in prosperity.

Ronald Ross,The Prevention of Malaria (1911)

.. . It follows that epidemic theory should certainly continue to search for newinsights into the mechanisms of the population dynamics of infectious dis-eases, especially those of high priority in the world today, but that increasedattention should be paid to formulating applied models that are sufficientlyrealistic to contribute directly to broad programs of intervention and control.

Norman T. J. Bailey,The Mathematical Theory of Infectious Diseases (1975, p. 27)

. . . The level of economic development of communities generally determinesthe level of health services. The higher the level of economic development,the more effectively did surveillance and containment principles apply andthe earlier was variole major [smallpox], in particular, eliminated from thec o u n t ry- F. J. Fenner,

Smallpox and its Eradication (1988)

.. . Statistical science has made important contributions to our understand-ing of AIDS. Statistical methods were used in the earliest studies of the eti-ology of AIDS, and evidence for sexual transmission came from case-controlstudies among gay men, in which AIDS cases were compared to matched con-trols. It was found that high numbers of sexual contacts were a risk factorfor AIDS. R Brookmeyer and M. H. Gail,

in Chance 3(4), 9-14, 1990

Contents

Preface xi

1 Some History 1

1.1 An empirical approach 11.2 A deterministic model 2

1.2.1 The Law of Mass Action 61.3 Prom curve-fitting to homogeneous mixing models 71.4 Stochastic modelling 111.5 Model fitting and prediction 131.6 Some general observations and summary 15

1.6.1 Methods and models 161.6.2 Some terminology 16

1.7 Exercises and Complements to Chapter 1 17

2 Deterministic Models 20

2.1 The simple epidemic in continuous time 202.2 The simple epidemic in interacting groups 232.3 The general epidemic in a homogeneous population 272.4 The general epidemic in a stratified population 352.5 Generation-wise evolution of epidemics 382.6 Carrier models 452.7 Endemicity of a vector-borne disease 462.8 Discrete time deterministic models 482.9 Exercises and Complements to Chapter 2 53

vn

viii Contents

3 Stochastic Models in Continuous Time 56

3.1 The simple stochastic epidemic in continuous time 573.1.1 Analysis of the Markov chain 583.1.2 A simplifying device 613.1.3 Distribution of the duration time 62

3.2 Probability generating function methods for Markov chains 633.3 The general stochastic epidemic 66

3.3.1 Solution by the p.g.f. method 683.3.2 Whittle's threshold theorem for the general stochastic

epidemic 733.4 The ultimate size of the general stochastic epidemic 77

3.4.1 The total size distribution using the p.g.f. method 773.4.2 Embedded jump processes 793.4.3 The total size distribution using the embedded jump chain 813.4.4 Behaviour of the general stochastic epidemic model: a

composite picture 833.5 The general stochastic epidemic in a stratified population 853.6 The carrier-borne epidemic 943.7 Exercises and Complements to Chapter 3 100

4 Stochastic Models in Discrete Time 105

4.1 The Greenwood and Reed-Frost Models 1054.1.1 P.g.f. methods for the Greenwood model 108

4.2 Further properties of the Reed-Frost model 1114.3 Chains with infection probability varying between households 1154.4 Chain binomial models with replacement 1184.5 Final size of epidemic with arbitrary infectious period 1234.6 A pairs-at-parties model: exchangeable but not

homogeneous mixing 1264.7 Exercises and Complements to Chapter 4 131

Contents ix

5 Rumours: Modelling Spread and its Cessation 133

5.1 Rumour models 1335.2 Deterministic analysis of rumour models 1385.3 Embedded random walks for rumour models 1415.4 A diffusion approximation 1455.5 P.g.f. solutions of rumour models 1495.6 Exercises and Complements to Chapter 5 151

6 Fitting Epidemic Data 154

6.1 Influenza epidemics: a discrete time deterministic model 1556.2 Extrapolation forecasting for AIDS: a continuous time model 1586.3 Measles epidemics in households: chain binomial models 162

6.3.1 Final number of cases infected 1626.3.2 Cases infected for different types of chain 163

6.4 Variable infectivity in chain binomial models 1646.5 Incubation period of AIDS and the back-calculation method 168

6.5.1 The distribution of the AIDS incubation period 1726.6 Exercises and Complements to Chapter 6 174

7 The Control of Epidemics 175

7.1 Control by education 1767.2 Control by immunization 1797.3 Control by screening and quarantine 184

7.3.1 The single prison model 1867.3.2 Interaction of a prison with the outside world 1877.3.3 A quarantine policy in prison 189

7.4 Exercises and Complements to Chapter 7 192

References and Author Index 194

Subject Index 205

Preface

This monograph is designed to introduce probabilists and statisticians tothe diverse models describing the spread of epidemics and rumours in apopulation. Not all epidemic type processes have been included. Withminor exceptions, we have restricted ourselves to the spread of viral andbacterial infections, or to the propagation of rumours, by direct contactbetween infective and susceptible individuals. Host-vector and parasiticinfections have been mentioned only very briefly.

Throughout the book, the emphasis is on the mathematical modelling ofepidemics and rumours, and the evolution of this modelling over the pastthree centuries.

Chapter 1 is a historical introduction to the subject, with illustrationsof the most common approaches to modelling. This is followed in Chapter2 by an account of deterministic models, in both discrete and continuoustime. Chapter 3 analyses stochastic models in continuous time, and in-cludes detailed studies of the simple, general and carrier-borne epidemics.In Chapter 4, the main stochastic models in discrete time, namely the chainbinomial models are studied, and a pairs-at-parties and related models out-lined. Chapter 5 considers models for the propagation and cessation ofrumours, and exploits some of the techniques introduced earlier to analysethem; the results highlight differences between these and the classical epi-demic models. Chapter 6, which is essentially statistical, is concerned withthe fit of various models to observed epidemic data. The book ends withChapter 7, which describes three main methods of controlling epidemics. Alist of references that also incorporates an author index, and a subject indexare provided at the end.

While the monograph cannot claim to be comprehensive, our hope is thatreaders who master its contents should have little difficulty in reading thecurrent literature on epidemic modelling. The two main treatises on the

xii Preface

subject are Bailey (1975) and Anderson and May (1991); both are oftenreferred to in the text. The former considers most of the classical epidemicmodels, both deterministic and stochastic, while the latter concentratesessentially on deterministic results. We believe that both types of modelshave a role to play in describing the spread of infections in large and smallpopulations, and have attempted to give due weight to each.

Current epidemic modelling relies on a great variety of mathematicalmethods; we have endeavoured to emphasize the intrinsic interest of suchmethods, as well as demonstrate their practical usefulness. Exercises andcomplements have been provided at the end of each Chapter, with the dualintentions of extending the text and of providing opportunities for readersto practise their skills in modelling and the analysis of models. The exercisesare not of uniform difficulty.

For those who wish to reach the forefront of current research, the vol-umes of papers edited by Mollison (1995) and Isham and Medley (1996),both arising from a six-month research programme on Epidemic Models atthe Newton Institute in Cambridge in 1993, provide further illustrations ofepidemic modelling and many challenging problems in the field.

Finally we thank all of our colleagues who have collaborated with usover many years in both this area of applied probability and others; theircontributions are too many to mention individually, save that DJD paystribute to David Kendall who first introduced him to the topic of this book,and JMG expresses his gratitude to Norman Bailey and Maurice Bart let t,pioneers of stochastic epidemic modelling.

We also thank David Tranah, and the Copy Editor and others at Cam-bridge University Press for their cooperation in producing this book withoutinvoking LaTJ X.

Daryl Daley, Joe GaniCanberra, October 1998

Some History

The mathematical study of diseases and their dissemination is at most justover three centuries old. To give a full account of the history of the subjectwould require a book in itself. The interested reader may refer to Burnetand White (1972) for a natural history of diseases, to Fenner ct al. (1988)for an account of smallpox and its eradication, and to Bailey (1975) andAnderson and May (1991) for an outline of the development of mathemat-ical theories for the spread of epidemics. We shall be concerned with themore modest task of placing some of the recent epidemic models in per-spective. We therefore present a selective account of historical highlightsto illustrate the developments of the subject between the seventeenth andearly twentieth centuries. Creighton (1894) gives a descriptive account ofepidemics in Britain to the end of the nineteenth century, and Razzell (1977)for smallpox.

1.1 An empirical approach

The quantitative study of human diseases and deaths ensuing from them canbe traced back to the book by John Graunt (b. 1620, d. 1674) Natural andPolitical Observations made upon the Bills of Mortality (1662). These Billswere weekly records of London parishes, listing the numbers and causes ofdeaths in the parishes. In his book, Graunt discussed various demographicproblems of seventeenth century Britain. Four of his twelve chapters dealwith the causes of death of individuals whose diseases were recorded inthe Bills. These death records, kept irregularly from about 1592 onwardsand continuously from 1603, provided the data on which Graunt based hisobservations.

1. Some History

Table 1.1. Numbers of deaths due to eight causes, and related risks

Causes Deaths Risk1. Thrush, Convulsion, Rickets, Teeth and Worms; Abortives,

Chrysomes, Infants, Liver-grown and overlaid 71,124 0.3102. Chronical Diseases: Consumptions, Ague and Fever 68,271 0.2983. Acute Diseases, and Miscellaneous 49,505 0.2164. Plague 16,384 0.0715. Small-pox, Swine-pox, Measles and Worms without

Convulsions 12,210 0.0536. Notorious Diseases: Apoplex, Gowt, Leprosy, Palsy,

Stone and Strangury, Sodainly, etc. 5,547 0.0247. Cancers, Fistulae, Sores, Ulcers, Impostume, Itch, King's Evil,

Scal'd-head, Wens 3,320 0.0148. Casualties: Drowned, Killed by Accidents, Murthured 2,889 0.013Source: Graunt (1662). The figures for Groups 1, 2, 3, 6 and 8 are quoted directly inGraunt's text, while those for Groups 4, 5 and 7 are obtained from his complete tableof casualties appended after his comments in 'The Conclusion'.

In the 20 years 1629-36 and 1647-58, there were 229 250 deaths recordedfrom 81 different causes. Table 1.1 consolidates these data into eight maingroups. The relative risks of death from each of the eight causes are indi-cated in the column furthest to the right in Table 1.1.

The main killers were Groups 1, 2 and 3; Graunt was led to observe that

whereas many persons live in great fear, and apprehension of someof the more formidable, and notorious diseases following [Group 6]; Ishall onely [sic] set down how many died of each: that the respectivenumbers, being compared with the Total of 229,250, those persons maythe better understand the hazards they are in.

The notorious diseases were further broken up by Graunt into the sub-categories of Table 1.2. Among these, apoplexy appears to have been thelargest killer. Graunt's analysis of the various causes of death provided thefirst systematic method for estimating the comparative risks of dying fromthe plague, as against the chronical or other diseases, for example.

These observations may well be considered to be the first approach tothe theory of competing risks, a theory that is now well established amongmodern epidemiologists.

1.2 A deterministic model

A more theoretical approach to the effects of a disease, namely smallpox,was taken by Daniel Bernoulli (b. 1700, d. 1782) almost a century later.Smallpox was then widespread in many parts of Europe where it affected

1.2. A deterministic model

Table 1.2. Deaths due to notorious diseases

Causes

1. Apoplex2. Cut of the stone3. Falling sickness4. Dead in the streets5. Gowt6. Head-ach7. Jaundice8. Lethargy9. Leprosy

10. Lunatique11. Overlaid, and starved12. Palsy13. Rupture14. Stone and strangury15. Sciatica16. Sodainly

Total

Deaths

13063874

24313451

99867

6155529423201863

5454

5547

Risk xlO~3

5.6970.1660.3231.0600.5850.2224.3530.2920.0260.6762.3081.8450.8773.7640.0221.980

24.196Source: Graunt (1662).

a large proportion of the population, being responsible for around 10% ofthe mortality of minors (cf. Bernoulli's model-based estimate in the lastcolumn of Table 1.3) while those who survived were immune to furtherattack but left scarred for life. In 1760 Bernoulli read his paper 'Essaid'une nouvelle analyse de la mortalite causee par la petite verole et desavantages de l'inoculation pour la prevenir' to the French Royal Academyof Sciences in Paris. His intention was to demonstrate that variolation, i.e.inoculation with live virus obtained directly from a patient with a mild caseof smallpox, a procedure that usually conferred immunity, would reduce thedeath rate and increase the population of France. Bernoulli's argument isreadily recognized as the following problem in competing risks.

Suppose first that a cohort of individuals born in a particular year hasan age-specific per capita death rate ji{t) at age t. Then given an initialpopulation size £(0) = £0> its size £(£) at age t satisfies the equation

so

where M(i) is the cumulative hazard. We shall use £(•) below.Consider another cohort subject to both the general per capita death

rate /x(t) as above and a further hazard (infection) like smallpox with aconstant infection rate (3 per individual per unit time. Individuals succumb

= £(0) exp ( - ft n(u) du) = £(0)e~M«) = C(t), (1.2.2)

1. Some History

Table 1.3. Age profile of population afflicted with smallpox (Bernoulli)

Age (yrs)t

0123456789101112131415161718192021222324

Total&(*)

1,3001,000855798760732710692680670661653646640634628622616610604598592586579572

Age cohortImmune

z(t)

0104170227275316351381408433453471486500511520528533538541542543543542540

Suscept.x(t)

1,300896685571485416359311272237208182160140123108948372635648.542.53732.4

SmallpoxIncidence

13799786656484236322824.421.418.716.614.412.611.09.78.47.46.55.65.04.4

CumulativeDeaths

17.129.539.247.554.560.565.770.274.277.780.783.485.787.889.691.292.693.894.895.796.597.297.898.3

AnnualTotal

30014557382822181210987666666666677

MortalitySmallpox

17.112.49.78.37.06.05.24.54.03.53.02.72.32.11.81.61.41.21.00.90.80.70.60.5

Source: Bernoulli (1760). Note that Halley's table (column 1) starts at t = 1; Bernoulligives reasons for choosing cohort size 1,300 for t = 0. Bernoulli used a = ft = 1/8, andobtained his figures by smoothing to the mid-point of the previous year, so his figure 17.1for t = l , coming from 1017.1, differs from 1014.9 = 8 x 1000/[7 + exp(-l/8)] as followsfrom (1.2.6) (cf. Gani, 1978).

only once, the result of such infection being either death in a fraction a ofcases or immunity for the remainder of life in the complementary fractionI — a. Denote the number of individuals still susceptible to the diseaseat age t by x(t), and the total number of the surviving cohort of age t,whether immune or not, by £p(t) as shown in Figure 1.1. To simplify themathematical model, the infectious state is assumed to be instantaneous,so that as soon as an infection occurs, the infective individual either dies orrecovers immediately. Then for the x(t) susceptibles and z(t) = £@{t) —x(t)immunes in this cohort,

x(t) =-(n(t) + 0)x(t) (1.2.3a)and

z(t) = -v{t)z{t) + (1 - a)/3x(t). (1.2.3b)

Cohortsize

1200-

1000-

8 0 0 -

6 0 0 -

4 0 0 -

2 0 0 -

00

1.2. A deterministic model

10I

15I

20

-+- Age (years)

Figure 1.1. Survivors £/? ( ) and immunes z ( ) in cohort ofinitial size 1300 (data from Table 1.3). The susceptibles of age t in thecohort are x(t) = £#(£) — z(t).

These equations are solved via integrating factors. Using M(t) and x(0)£ ( 0 ) = £(0) = £0 a s before, we have

whence

and

x(t) =

Integrating on (0, t) and simplifying,

(1.2.4)

- e""')(1.2.5)

using (1.2.2); observe that £(£) = £0(£) when the infection rate (3 = 0.Equation (1.2.5) relates the sizes of the surviving cohorts of age t in pop-

ulations with ({3 > 0) and without (/? = 0) smallpox, respectively. Bernoulliused it in the form

(1.2.6)1 - a + ae -#

to estimate the size of a surviving cohort £(£) in a 'state without smallpox'on the basis of Halley's (1693) data from Breslau, now Wroclaw. This esti-mation required parameters a and /?; on reviewing what evidence he could

1. Some History

from a number of areas, Bernoulli fixed on a = j3 = 0.125. Use of theseestimates in (1.2.6) yields the data in the last column of Table 1.3; entriesin the other columns are derived from this column and Halley's data. Ob-serve that, granted the validity of Bernoulli's assumptions, smallpox causedbetween 10 and 40% of deaths between ages 2 and 23.

Suppose Bernoulli had had available observations of the form of his £(•)at (1.2.6), for a 'state without smallpox' with a death rate similar to that/JL(-) prevailing in the areas from which Halley's data were drawn (column 2of Table 1.3). Then taking differences of (1.2.5) with itself for times t = t'and t = t' 4- 1 yields

so thatIn A(fr(f)/C(O) = "(<» + 0 0 , (1-2-7)

where a = — ln[a(l — e~^)]. This is the simplest way of expressing theresult (1.2.5) for the purpose of estimating f3 and a conditional on suchextended data being available. All that Bernoulli could do was to presentthe advantage of variolation (i.e. absence of deaths due to smallpox) on thebasis of his model-based calculations. Note too that the population riskof death from smallpox (cf. Tables 1.1-2) as implied by Table 1.1 is about100/1300 « 7.7%, higher than in Table 1.1 because the London populationfrom which Graunt drew his data, included more immigrants than Breslau.In Halley's day Breslau had rather few immigrants, and hence, propor-tionately more infant and childhood deaths, smallpox being more prevalentamongst children than adults.

1.2.1 The Law of Mass ActionThe Law of Mass Action has found wide applicability in many areas ofscience. In chemistry, the idea that a reaction is influenced by the quantitiesof the reactant materials goes back at least to Boyle (c. 1674). Around 1800,C. L. Berthollet emphasized the importance of mass or concentration of asubstance on a chemical reaction, but this was not generally accepted for halfa century. Ultimately, Guldberg and Waage (1864-1867) postulated thatfor a homogeneous system, the rate of a chemical reaction is proportionalto the active masses of the reacting substances (Glasstone (1948, p. 816)).

Applied to population processes, if the individuals in a population mixhomogeneously, the rate of interaction between two different subsets of the

1.3. From curve-fitting to homogeneous mixing models

population is proportional to the product of the numbers in each of thesubsets concerned. In any population it is possible for several processes tooccur concurrently, in which case the effects on the numbers in any givensubset of the population from these various processes are assumed to beadditive. Thus, in the case of epidemic modelling, the Law is applied torates of transition of individuals between two interacting categories of thepopulation, such as susceptibles who, as a result of contact with infectives,themselves become infectives; a second simultaneous process is that of theinfectives who become removals. These two processes underlie equations(1.3.2a) and (1.3.2c) respectively: when more than one process is involved,as for the numbers of infectives in equation (1.3.2b), the effects are additive.

Application of the Law to transitions that occur in discrete time is not sostraightforward, but, subject to certain constraints on the size of the changeinvolved (see e.g. Section 2.8 below), it remains valid.

The Law also has a stochastic version when the process concerned is as-sumed to be Markovian, and the rate is then interpreted as the infinitesimaltransition probability.

Implicit in the 'proportionality' aspect of the Law, is an assumption thatthe quantities concerned in inducing the transition are subject to homoge-neous mixing with each other. The Law can then be seen as the result ofsuperposing all possible contributions of the individual components to theinteraction, these individuals being regarded as equally likely to interactwith each other in a given (small) interval of time.

1.3 From curve-fitting to homogeneous mixing models

First issued in 1837, each Annual Report of the Registrar-General ofBirthsyDeaths and Marriages in England included tables of causes of death andcommentaries. The Report for 1840 includes a contribution from WilliamFarr1 entitled 'Progress of epidemics', in which Farr attempted to char-acterize mathematically the smoothed quarterly data for smallpox deaths.Some 26 years later, in a letter to the London Daily News of 17 February

xFarr was appointed compiler of abstracts to the General Register Office in 1839 andremained there until retirement in 1879. Early volumes of the Annual Reports containpapers of Farr prefaced by a 'Letter to the Registrar-General'; they cover a variety ofissues pertaining to the data in the Reports. Thus, in the Sixth Annual Report (1842)Farr noted that the annual small-pox death-rates per 106 live individuals for the years1838-42 were 1101, 604, 679, 408 and 172 respectively, and remarked that 'The reductionin the mortality from small-pox since 1840 was probably the result, at least in part, ofthe Vaccination Act' [of 1840]. Later he gave the 1850 death-rate as 263.

1. Some History

Table 1.4. Deaths from smallpox in consecutive quarters 1837-39

Observed deathsDeaths averaged overtwo consecutive quartersPercentage change

Sum.18372513

Aut.18373289

Win.18384242

Spr.18384484

Sum.18383685

Aut.18383851

Win.18392982

Spr.18392505

Sum.18391533

Aut.18391730

2901 3766 4365 4087 3767 3416 2743 2019 1637+30 +16 -6 -8 -9 -20 -26 -19

Source: Farr (1840).

1866 (quoted by Brownlee, 1915), he attempted to predict the spread ofrinderpest among cattle by a similar method.

Table 1.4 gives the observed deaths in the smallpox epidemic of 183739 drawn from Farr (1840), together with the average values of consecutivequarters, for 10 quarters in all. Farr concluded that as the epidemic declined,he could detect an approximately steady rate of deceleration in the numberof deaths per quarter. Brownlee (1906) later carried out work of a similartype: he fitted Pearson curves to epidemic data for several diseases, and forseveral different locations.

But these pragmatic approaches were essentially limited, so long as therewas not an appropriate theory to explain the mechanism by which epidemicsspread. By the beginning of the twentieth century, the idea of passing ona bacterial disease through contact between susceptibles and infectives hadbecome familiar, and Hamer (1906) first foreshadowed the simple 'massaction' principle for a deterministic epidemic model in discrete time. Thisprinciple, which incorporates the principle of homogeneous mixing, has beenthe basis of most subsequent developments in epidemic theory (see Section1.2.1 above and Anderson and May (1991, p. 7) for discussion).

Hamer, noticing the rise and fall of infectives in the course of a largerange of epidemics, argued against variable infectivity. Specifically, he wrotethat to explain the eventual decline of an epidemic, 'the assumption ofloss of virulence or infecting power on the part of the organism is quiteunnecessary'. He also put forward a numerical argument about the initialincrease and eventual decline of the number of infectives in a population; thisindicates that he was aware that both susceptibles and infectives affected thenumber of new infectives listed in the weekly reports of measles in London:

Now the outbreak will take much longer to decline to extinction than ittook to rise, for those especially exposed have in large part been alreadyattacked and the disease must spread, in the main, among personswhose manner of life brings them comparatively little into contact withtheir fellows.

1.3. From curve-fitting to homogeneous mixing models

Let xt, yt be the numbers of susceptibles and infectives respectively attimes t = 0,1,2, . . . . Hamer's idea was equivalent to expressing the newnumber of infectives at time t 4- 1 by Ayt such that

Ayt=Pxtyt (4 = 0,1,2,...) (1.3.1)

where the constant (3 is such that (3xtyt < xt for all t, i.e. (3 < l/(maxi>i yi).These new infectives are a proportion (3 of the number xtyt of contacts be-tween susceptibles and infectives, where /? is known as the infection param-eter. Because of the constraint on /?, it follows that in a closed populationin which yt — N — xu we have (3xt(N — xt) < xt or f3(N - xt) < 1; this iscertainly satisfied if /3 < 1/N.

Continuous time versions of epidemic equations were used by Ross (1916)and Ross and Hudson (1917) in their studies of populations subject to in-fection. But the form of equations most commonly used to characterize thetypical general epidemic with susceptibles x(t), infectives y(t) and immunesz(t) (such as a measles epidemic) is due to Kermack and McKendrick (1927).They assumed a fixed population size N = x(t) H- y(t) 4- z(t), and using thehomogeneous mixing principle for continuous time t > 0 derived the (now)classical equations

d c-77 = -0xy, (1.3.2a)d£^ = 0xy - 7i/, (1.3.2b)

^ = 72/, (l-3.2c)

subject to the initial conditions (x(0),2/(0),z(0)) = (xo,2/o>0)- Here (3 is2

the infection parameter, similar to that in (1.3.1), and 7 is the removalparameter giving the rate at which infectives become immune. In caseswhere death or isolation may occur, z(t) represents all removals from thepopulation, including immunes, deaths and isolates.

Dividing equation (1.3.2a) by (1.3.2c) gives

dx (3 x , 7— = x — with P=~3>dz 7 p (3

the parameter p being the relative removal rate. The solution of this equa-tion is

x = xoz-z/p,2Some authors write (3 = (3'/x(0) so (1.3.2a) becomes x = -0'(x/xo)y.

10 1. Some History

\w=N-z

0 2oo

(a) XQ p

Figure 1.2. z^ as the point of intersection of w = N — z and w = xo e"

so that

Hence

with the parametric solution

= N-z- xoe~z/p.

•/o(0<t< oo). (1.3.3)

Kermack and McKendrick obtained two basic results, referred to as theirThreshold Theorem. The first is a criticality statement, and comes fromequation (1.3.2b). Writing this equation as

shows that if the epidemic is ever to grow, then we must have dy/dt\t_0 > 0,or xo > p, i.e. the initial number of susceptibles must exceed a thresholdvalue p. The second, deduced from (1.3.3), states that as t —> oo, z(i) —>Zoo < N. Now in the limit t —• oo, N — z^ — x0e~ZoG^p = 0 (see the graphsin Figure 1.2).

Suppose now that x0 is close to N; then z^ will be the approximatesolution of

0 = N - Zoo - N - - f + p > <»*•>

1.4. Stochastic modelling 11

so that if N = p -\- v with i / « p , then

Zoo ~ T ^ T - ~ 21/. (1.3.5)1 + I//P

Thus, if x0 « A/" = p -f i/ with */ > 0, then since 2/oo = 0, Xoo + oo = A/" gives

(see Exercise 1.1 for the next term in the approximation).Kermack and McKendrick's paper also includes the observation that, ac-

cording to the model, some susceptibles survive the epidemic free from in-fection. At the time this was a significant result. We may view these threeresults, stated formally later at Theorem 2.1, as typical of the qualitativeinsights which mathematical models of epidemics attempt to achieve.

1.4 Stochastic modelling

The spread of an infectious disease is a random process; in a small group ofindividuals, one of whom has a cold, some will catch the infection while oth-ers will not. When the number of individuals is very large, it is customary torepresent the infection process deterministically as, for example, Andersonand May (1991) mostly do. However, deterministic models are unsuitablefor small populations, while in larger populations, the mean number of in-fectives in a stochastic model may not always be approximated satisfactorilyby the equivalent deterministic model.

One of the earliest of stochastic models is due to McKendrick (1926),but the most used may well be the chain binomial model of Reed andFrost, put forward in their class lectures3 at Johns Hopkins University in1928, and based on one originally suggested by Soper (1929). Reed andFrost never published their work; Helen Abbey (1952) later gave a detailedaccount of it (see also Wilson and Burke (1942, 1943)). It should however bepointed out that En'ko (1889) anticipated some aspects of Reed and Frost'smodel by nearly 40 years, fitting data on measles epidemics recorded inSt. Petersburg to a discrete time model similar to theirs (see En'ko (1889)and Dietz (1988)).

3E. B. Wilson records these dates as February 2-3, 1928, and refers to correspondencewith Dr Frost shortly thereafter: 'I strongly urged Dr Frost to publish his theory of theepidemic curve, but he thought it too slight a contribution' (Wilson and Burke, 1942,note 3).

12 1. Some History

xt

5

4

3

2

1 : = X3 = 1, ^3=0

JT3 = X4 = 0, = 00 1 2 3 4

Figure 1.3. Two sample paths of a Reed-Frost epidemic, ending att = 3 ( ) and £ = 4 ( ) respectively.

The model is based on the assumption that, in a group of Xt susceptiblesand Yt infectives at times t = 0,1,2,..., where the time unit is the averagelength of the serial interval (see Figure 1.3), infection is passed on by 'ad-equate' contact of an infective with a susceptible in a relatively short timeinterval (£, t + t) at the beginning of the period (indeed, instantaneously att + 0). The newly infected individuals Yt+\ will themselves become infec-tious in (£ + 1, £-hl-he), while the current infectives Yt will then be removed.Each susceptible is assumed to have the same probability 0 < q < 1 of notmaking adequate contact with any given infective, or qYt of not makingcontact with any of the Yt independent infectives during (£, t -h e). Thus,for each susceptible, the probability of infection will be 1 — qYt; assumingthe independence of each susceptible, the probability that there will be Yt+\infectives at t + 1 can therefore be taken to be the binomial probability

XtXt,Yt} = (1.4.1)

where Xt = Xt+\ + lt+i. Figure 1.3 depicts two possible paths of anepidemic starting from (Xo, Yo) = (4,1); for one path, X\ > X2 = X3 = 1and y3 = 0, so that the epidemic terminates at t = 3, while for the other,X2 > X3 = Xt = 0 and Y4 = 0 and it terminates at t = 4.

Because of the structure of (1.4.1), it is easily seen that the probabilityof an epidemic such as that in Figure 1.3 would be

L = P{XuY1\X0,Y0}P{X2,Y2\X1,Yl}P{X2,0\X2,Y2}

x2(1.4.2)

1.5. Model fitting and prediction 13

i.e. the product of a chain of binomials. Hence the model is referred to as achain binomial model; such models will be discussed in much greater detailin Chapters 4 and 6 below.

Since the later 1940s, when Bartlett (1949) formulated the model forthe general stochastic epidemic by analogy with the Kermack-McKendrickdeterministic model, stochastic models for epidemic processes have prolifer-ated. Most have relied on discrete or continuous time Markov chain struc-tures, and we shall consider some of these in subsequent chapters. At thisstage all that needs to be said is that reviews of the literature of epidemicmodels (see e.g. Dietz and Schenzle (1985), Hethcote (1994)) indicate thattheir number has grown very rapidly in the past 50 years.

1.5 Model fitting and prediction

Epidemic modelling has three main aims. The first is to understand betterthe mechanisms by which diseases spread; for this, a mathematical structureis important. For example, the simple insight provided by Kermack andMcKendrick's model that the initial number of susceptibles XQ must exceedthe relative removal rate p for an epidemic to grow, could not have beenreached without their mathematical equations (1.3.2).

The second aim is to predict the future course of the epidemic. Againusing Kermack and McKendrick's general epidemic model as an example,we learn that if xQ = p + v is the number of susceptibles at the start ofthe epidemic and v is somewhat smaller than p, then we can expect theirnumber to be about Xoo = p — v at the end. Thus, we could predict thatthe total number of individuals affected by the epidemic would be about 2vif we wished to estimate the medical costs of the epidemic, or to assess thepossible impact of any outbreak of the disease.

The third aim is to understand how we may control the spread of the epi-demic. Of the several methods for achieving this, education, immunizationand isolation are those most often used. If one were able, for example, toreduce the number of susceptibles XQ in the Kermack-McKendrick modelby immunization to a level below p, the epidemic would be much reducedin size.

In order to make reasonable predictions and develop methods of control,we must be confident that our model captures the essential features of thecourse of an epidemic. Thus, it becomes important to validate models,whether deterministic or stochastic, by checking whether they fit the ob-served data. We now outline an example of such model fitting in the case of

14 1. Some History

Table 1.5. The Ay cock measles epidemic

t

xtYt

01111

11089

28622

32561

41213

5120

Source: Abbey (1952).

a measles epidemic. If the model is validated, it can then be used to predictthe course of the epidemic in time.

Following the pioneering study of measles and scarlet fever by Wilson etal. (1939), Abbey (1952) was among the first to use a stochastic epidemicmodel for the estimation of a 'contact' parameter, and for testing the validityof the model. Among her many sets of data was one for a particular measlesepidemic in 1934 studied by Aycock (1942); to this data set she decided tofit a Reed-Frost model for t = 0 ,1 , . . . , 5, the unit of time being a 12-dayperiod. Using the same notation as in the previous section, Table 1.5 recordsthe progress of the Aycock epidemic.

From the Reed-Frost model, the probabilities of these results in eachindividual time interval can be worked out respectively as

__ ^9\22/^9\86

(1.5.1)Abbey obtained estimates of the probabilities qi of no contact in each sep-arate interval i = 1,2,3,4 by the Maximum Likelihood method as

=0.9454, fc=(_) =0.988.

If one assumes that the probability of no contact has the same value qthroughout all intervals, then the probability of the epidemic is given by

L(q) = Li(q)L2(q)L^(q)L/i(q)1 (1.5.3)

with functions Li as in (1.5.1). The Maximum Likelihood estimator q of qsatisfies the relation

dlnL 9 198<f 1342o21 793<f° 2320= - 1 ~ ~ i ^ ~ i Too - -,—"W + —^- = °> (1-5.4)

q=q 1 - 9 1 - r l-q22 1-q61 qdq

which can be solved numerically to yield q = 0.9685.

1.6. Some general observations and summary 15

The fit of this model to the data turns out to be far from perfect: on agoodness-of-fit test Abbey reported %2 = 53.1 on 4 degrees of freedom, andnoted a significantly improved fit from estimating (rather than counting)the number of susceptibles. Abbey (1952) found the same true of othermeasles, cHickenpox and German measles data, and investigated variationof the contact rate either with time or between individuals, or both, asother possible reasons for the inadequate fit of the model to the data (seeChapter 4).

1.6 Some general observations and summaryMathematical techniques and models used in the study of epidemics forma major part of this book. They usually encompass, for any given model, aset of assumptions which can roughly be described as belonging one to eachof the categories listed below.

The 'epidemic process' can be characterized as the evolution of someinfectious disease phenomenon within a given population of individuals.The properties of the process fall naturally into three categories:(1) assumptions about the population of individuals within which the dis-

ease first manifests itself, and then spreads;(2) assumptions about the disease mechanism: how it is spread, and the

mechanism of recovery or removal, if such occurs; and(3) mathematical modelling assumptions that allow the specification of the

two preceding properties.So far as the population is concerned, we make assumptions about(a) its general structure: it may be a single homogeneous group of individu-

als (apart from (c) below), or a collection of several homogeneous strataor subgroups, or else generally heterogeneous so that each individual isdifferent;

(b) the population dynamics which specify whether the population is closedso that it is a constant collection of the same set of individuals forall time, or open, allowing individuals to give birth and die, and toemigrate or immigrate or both; and

(c) a mutually exclusive and exhaustive classification of individuals ac-cording to their disease status-, thus, at any given time, an individual iseither susceptible to the disease, or incubating it, or infectious with it,or possibly an infectious carrier without any symptoms of the disease,or a 'removed case.' A removal has been infectious or an infectiouscarrier but is so no longer, whether by acquired immunity or isolationor death.

16 1. Some History

Given the population, we next turn to a mathematical description of themechanism(s) which specify how the disease is spread and how, if at all,individuals may ultimately recover temporarily or permanently from thedisease. We mostly restrict ourselves to the assumption that the diseaseis spread by a contagious mechanism, viral or bacterial, so that contactbetween an infectious individual and a susceptible is necessary. After aninfectious contact, the infectious individual or carrier succeeds in changingthe susceptible's disease status: there follows an incubation period duringpart of which the disease is latent within the newly infected susceptible.After this the susceptible itself becomes an infective (see Figure 1.4 andSection 1.6.2).

1.6.1 Methods and modelsThis monograph is as much about mathematical methods as about theepidemic models themselves. So, unlike Bailey's (1975) classical treatise, itis primarily organized around the various mathematical techniques used tostudy epidemic models. Consequently some models recur in several places.

Chapter 2 outlines a few deterministic models, after which Chapter 3describes some stochastic models in continuous time, Chapter 4 others indiscrete time, and Chapter 5 models for rumours. Chapter 6 discusses thefitting of models to data. In Chapter 7, three examples are given of howepidemic modelling may help us to control the spread of epidemics.

Our hope is that the reader who masters the methods outlined here willbe well prepared to tackle the more comprehensive treatises of Bailey (1975)and Anderson and May (1991), and papers in the recently published volumesedited by Mollison (1995) and Isham and Medley (1996).

1.6.2 Some terminologyIt may be useful to clarify some terms commonly employed in epidemictheory: these are illustrated in Figure 1.4 (cf. also Anderson and May, 1991,§§3.1.1 and 3.2.4). We assume that there is an instant at which infectionoccurs for an individual; this is the start of a latent period during whichthis individual is not infectious. There then follows an infectious periodwithin which symptoms will appear; the incubation period is the time fromfirst infection to the appearance of symptoms, and this is necessarily greaterthan or equal to the latent period. The serial interval is the time betweenfirst infection and the infection of a second individual; this is again largerthan or equal to the latent period, but can be either smaller or larger than

1.7. Exercises and Complements to Chapter 1 17

Individual'sdisease state: Susceptible Latent < Infectious > Immune/Removed

1 1 1 1 1 > timeEpoch: tA tB tc t£> tE

<— Incubation period —>< Serial interval •

tj\'. Infection occurs £#: Latency to infectious transitiontc- Symptoms appear t^: First transmission to another susceptibletE- Individual no longer infectious to susceptibles (recovery or removal)Note: tp is constrained to lie in the interval (iB» i?)» s o t& > tc (as shown) andtD < tc are both possible.

Figure 1.4. Diagrammatic representation of progress of a disease in anindividual.

the incubation period. Often the serial interval and latent period are usedinterchangeably although their meanings are different. The fundamentalquantity in the process of infection is the serial interval, but the latent periodis often used in the literature because it is assumed that a second infectionwill occur as soon as the first infective becomes infectious. Anderson andMay (1991, §3.2.4) use the term generation time of the disease agent; interms of the notation in Figure 1.4 it is the expectation of (£# — t^) +\(tE — £#)> i-e. the mean latent period plus half the mean infectious period.

1.7 Exercises and Complements to Chapter 1

1.1 Show that for positive n, e and p there exists a unique root of the equationn +e = z + ne~z/p

satisfying 0 < z — e < n, z = (, say. Expand the exponential term to thirdorder and deduce that in the limit e [ 0, Q « 2v — \v21 p where 0 < v =n - p = O(pi) (cf. Daley and Gani, 1994, §4.1). (The notation here and ofequation (1.3.4) is related by z = Zoo, n = xo and n + e = N.)

1.2 In the general epidemic model sketched in Section 1.3, the quantity Ro =n/p = /3n/j = (initial no. of susceptibles)/(relative removal rate) coincideswith the Basic Reproduction Ratio in Section 3.5 below. Show that forRo = 2 in this model starting from t/o < ô, about 20% of the populationsurvive the epidemic. Bartholomew (1973, p. 346) gives 2% for Ro — 4.

1.3 The data in Table 1.6 come from bar charts labelled a, . . . , j in En'ko (transl.1989). En'ko extracted the daily numbers of measles cases for several yearsfrom records at the St Petersburg Alexander Institute, where the date of ameasles case was determined by the appearance of a rash on the face, andat the Educational College for the Daughters of the Nobility, where the datewas determined by the date of transfer into the infirmary. En'ko allocatedcases recorded 0-7, 8-17, 18-29, 30-41, 42-53, 54-65, .. . days following the

18 1. Some History

Table 1.6. Numbers of cases in successive generations of several measles epidemics

1stday

09203244

0821

091937

0112535

0122437

0

11612

153

2311

1211

1111

1 2 3

1865,

2540

2320

1875,

00

22

1879,

5163

1882,

006

031

1884,

100

100

b

4421

i

30

c

50

e

031

f

000

4

4300

21

02

131

130

5

031

52

10

10

6

3

22

11

11

7

0

12

00

8 9

1

10 1

11

1stday

010223446

091935

081832

011202951

08193144606985

0

12224

1111

2221

11231

137524111

1

14032

2250

153

011

2

00213

18111

14

110

3

00144

74

47

011

4 5

1870, h00232

22141

1874,g

123

75

1875,a

206

72

1888, d

120

Combined counts2101830

0342175

0431843

0391884

111

6

1040

175

50

20

7

101

16

51

11

8

21

11

20

2

> of 10 epidemics

2251764

01812133

1131861

61070

9

0

2

12

2

. J

21351

10

1

1

10

1

1321

11

1

1

22

Source: From bar charts in En'ko (transl. 1989). See Exercise 1.3 for more explanation.

initial case on day 0, to generation 0, 1, . . . respectively. The table gives foreach epidemic the first day of observing a case for each generation, and thenumber of cases reported on the various days within the generation, for aslong as there were any such cases.(a) Investigate whether the spread of the dates of recording cases within anepidemic shows any systematic trend from one generation to another (if not,then periodicity is strong, and identifying a case is variable).(b) Repeat the analysis of (a) on the combined data: what sort of additionalvariability does such pooling of data introduce?(c) The construction of analogues of qt and q as in section 1.5 entails theestimation of the population size as well.(d) If a model is fitted as in (c), then a \2 goodness of fit test can beperformed.


1.4 Wilson and Burke (1943) list the monthly numbers of measles cases in Prov-idence RI for the years 1917-1940 as in Table 1.7. Plot out the course of thenine epidemics for which the epidemic curves peak around May 1918, Mar.1921, Mar. 1923, Jan. 1926, Apr. 1928, Jan. 1932, May 1935, Mar. 1937,Mar. 1940. Observe that there is a marked seasonality effect if the datasetis treated as a whole.(a) Assuming a mean serial interval of 0.5 months and a closed populationfor the course of the epidemic, investigate how estimates of any or all of JV,/o, /3, 7 might be constructed.(b) Repeat this analysis for the record as a whole assuming instead animmigration of 2130 new susceptibles each September (thus, Aryear varies aseach year changes).(c) Repeat the analysis of (b) but assuming now a steady immigration of178 new susceptibles per month.

Table 1.7. Measles cases by months in Providence RI 1917-1940Year

191719181919192019211922192319241925192619271928192919301931193219331934193519361937193819391940Total

Jan.

33551

12532989680513

20575458421

279904131194222335697485

Feb.

47984

1275854

1228611

13602

11218902

203701157748115354957300

Mar.

623734

1366653

147036

6481

42226114957402134392

11844405306890

Apr.

1091232

42793902668711153481

10813994

158199318

1351767112

1184627684

May1191299

54042662538316181962

8832762345681329

1953834720

3175437852

June

367804

288992211730301052

80011146358116

1061279171290

2863724934

July132613

1462823291558486

5083822179254424111310

1571211989

Aug.

72333810196250827748990225174436420495

Sep.28145173213101831220484001201

165

Oct.1625321610181093620

1910051020890

507

Nov.

851

1907

13175

417073602

3370110930

2671

1435

Dec.

5533

1912665272

12244236100

1548017487733

4461

4385

Total492414335

202224081017462798

1936477560

40791367109

34005703252795307562377220

1872311351221

Source: Wilson and Burke (1943).

Deterministic Models

In deterministic models, population sizes of susceptibles, infectives and re-movals are assumed to be functions of discrete time t = 0,1,2,... or dif-ferent iable functions of continuous time t > 0. Such approximations to thetrue, integer-valued numbers of individuals involved in an epidemic, allowus to derive sets of difference or differential equations governing the process.The evolution of this epidemic process is deterministic in the sense that norandomness is allowed for; the system develops according to laws similar tothose for dynamical systems. It is usual to regard the results of a determin-istic process as giving an approximation to the mean of a random process:there are examples related to this in the next chapter (see equations (3.2.4)and (3.3.6), and Exercises 3.1, 3.2, 3.5 and 3.11).

2.1 The simple epidemic in continuous time

A simple epidemic is one where the population consists only of susceptiblesand infectives; once a susceptible is infected, it becomes an infective andremains in that state indefinitely. A simple epidemic may be thought of asone where(a) the disease is highly infectious but not serious, so that infectives remain

in contact with the susceptibles for all time t > 0;(b) the infectives continue to spread their infection until the end of the

epidemic (see equation (2.1.2) and below for interpretations of the 'end'of the epidemic).

An infection which may approximate these conditions is the common coldover a period of a few days. This simple epidemic model is the same as thelogistic model of population growth, attributed1 to Verhulst (1838).

1 Miner (1933) remarks that 'Verhulst's work was generally forgotten until after the

20

2.1. The simple epidemic in continuous time 21

We suppose that the total population is closed, i.e.

x(t) 4- y(t) = N (all t > 0)

where, as throughout this chapter, x(t) and y(t) denote the numbers ofsusceptibles and infectives at time t, with initial conditions (x(Q),y(0)) =(%o-> Vo) with 7/o > 1. Then assuming that the individuals of the populationmix homogeneously, we can write

%=0xy = l3v(Ny), (2.1.1)at

where /? is the pairwise rate of infection (i.e. infection parameter) and, incontrast to the discrete time case (cf. (1.3.1)), the condition (3 < l/N isno longer needed. This differential equation, the so-called logistic growthequation, is readily solved, since

y(N-y) \y N-yJ N

so integrating on (0, £),

l n ^ l n ^N - y(t) N-y0

Hence

As t —> oo, equation (2.1.2) shows that y(t) —> N, so that according to themodel all individuals in the population eventually become infected, thuscausing the end of the epidemic (in the mathematical sense).

In this model we have both x(t) > 0 and y(t) > 0 for all finite positive£, so the question arises as to when we may consider the epidemic to haveterminated in practical terms. Realistically, we could define the 'end' of theepidemic to occur at Ti = inf{t : y(t) > N - 1}, i.e. when the number ofinfectives is within 1 of its final value. Since the function y(-) has a positive

independent rediscovery of the logistic curve by Pearl and Reed in 1920', and that to hisknowledge 'the only reference to the work of Verhulst in modern times prior to [1920] is[a paper in 1918 by Du Pasquier]'. Bailey (1975) gives no account of its emergence inepidemic theory; Bailey (1955) attributes the stochastic version of the model to Bartlett's1946 lecture notes (see Bartlett, 1947).

22 2. Deterministic Models

derivative for finite t, T\ is determined by y(Ti) = N - 1. It follows from(2.1.2) that

2/oiV __ KT 1

SO

Table 2.1 illustrates the values of T\ for various values of yo when N24,50,100,1000 for the simple case where /? = I/AT.

Table 2.1. Ti determined from y{T\) = N -1 when 0 = I/AT

2/0

110

N = 24

6.27103.39793.1355

50

7.78365.27813.8918

1009.19026.79234.5951

1000

13.813511.50196.9068

Observe that as yo increases from 1 to |7V, the time T\ taken to reachAT—1 is halved, as follows from the symmetry about y = ^N of the derivativeat (2.1.1). Also, as N increases from 24 to 1000, T\ increases rather slowlyfor, as (2.1.3) shows, Tx = O((]nN)/0N).

Thus, if the unit of time is the day, in a classroom of 50 schoolchildren ofwhom one has a cold initially, the infection spreads among the whole classin fewer than eight days if (3N = O(l).

Sometimes epidemiologists are more interested in the epidemic curve,which is the rate of occurrence of new infectives, here dy/dt. We see from(2.1.1) that

dy _ pN2y0(N - yo)e^Nt _ (3yo(N - y0)dt [yoePNt + (N - yo)}2 [ c o s h / 3 N t + (l- 2yo/N) s i n h 0Nt]2 '

(2.1.4)It has a maximum when

At this time we have x(t) = y(t) = |JV, and (dy/dt) = /3(|JV)2. The dashedcurve in Figure 2.1 illustrates equation (2.1.4), i.e. the epidemic curve forthe deterministic model of a simple epidemic (cf. also Figure 2.12 below).

2.2. The simple epidemic in interacting groups 23

y(t)

dy/dt

Figure 2.1. y(t) ( ) and dy/dt ( ) for the simple epidemic.

2.2 The simple epidemic in interacting groups

We suppose in this section that a closed population now consists of m groupsof sizes Ni,..., ATm, in each of which a simple epidemic may break out.Assume that these groups interact with each other as follows. In place ofthe single pairwise infection rate /? as in (2.1.1), suppose that susceptiblesin the j th group are subject to infection from infectives in the i th group atrate fiij per interacting pair; for i = j we set /?j = jijj (j = 1 , . . . , ra). Figure2.2 illustrates the model, which is due to Rushton and Mautner (1955).

For ij = l , . . . ,m:

Figure 2.2. Infection rates in interacting communities i,j = 1,.. . ,m.

Let Xj(t), yj(t) denote the numbers of susceptibles and infectives in eachof the groups j = 1 , . . . ,m respectively. Then we readily see that wheninfection is transmitted both within and between groups, equation (2.1.1)can be generalized to the set of equations

dt + (j = 1 , . . . , ra), (2.2.1)

subject to the initial conditions Vj(O) = yj0 and Xj(0) = Nj — yjO, as followsfrom Xj = Nj — yj.


While these equations may be solved numerically, explicit algebraic resultsare obtainable only if the parameters and initial values have a relativelysimple structure. For example, we might set f3j = (3 (all j) and fyj = /3K forsome K 1 for infection between different groups. Then (2.2.1) becomes

(2.2.2)dt

A further simplification is to set Nj = N (all j). Then (2.2.2) becomes

(j = l , . . . ,m) . (2.2.3)

If all the initial values t/jo — Vo are the same, this set of equations basicallyreduces to the single equation

- y)y[l + (m - l)/c] =

where /?' = /3[1 -j- (m — 1)K] and

VoN

as is consistent with (2.1.2) when m — 1.We now show that if the yjo are different but N3 = N for all j , there

exist explicit parametric solutions of (2.2.3) for the yj(t). Write r = fit,a = 1 -f (m — 1)K and Xj = N — yj. Then (2.2.3) becomes

— (j = l , . . . ,m) . (2.2.4)

The further transformations Uj = eaNT(xj/N) and v = (1 - e~aNT)/a leadto

— lnLT,- = C + K (j = 1, . . . ,m), (2.2.5)

or in terms of m-vectors U and InU and the mxm matrix B = (6^) definedby

I/2lnU =

f lnC/i 1lnC/2

Un(7mJ

' 1

2.2. The simple epidemic in interacting groups 25

where In U involves an abuse of notation,

lnU BU. (2.2.6)dv

Note that for t = 0, r = 0, v = 0 and xj0 = N - yj0 = NUj(0).This matrix equation can be solved as follows. First use the inverse

B 1 = (bij) of B (assuming |B| ^ 0) to give

dvi.e.

d m

2 = 1

Define X by setting In X = B - 1 In U, so that In U = B In X. These relationsare equivalent to

from which it follows that

d- In Xj = Uj=

i=l

Then for all j = 1 , . . . , m,

V ^ = (flrf = F(v) say. (2.2.8)i= l

Hence on integration,

X«-\v) - X«-\0) = (K - 1) T F(u) du = G(v),Jo

orr ] vc^-i) ii/(«-i)

F(ii)duj = [Xp^+Giv)]

where X,(0) = ni l i^f l(0) = UZo (*Jo/Nf\ Now from (2.2.7), for allj = l , . . . , m ,


But from (2.2.8),

,K/( ,C-1)

K - 1 dv '

whence

v =n. K. — 1 /o,

so that the time is given parametrically in terms of G. We can now find thesolutions for Uj(v) and hence for the original yj — N — Ne~aNPtUj.

These computations simplify as follows in the special case where yio — 1and yjo — 0 for j = 2,..., ra, so the y3,(t) for j = 2 , . . . , m are identical forall t > 0, and equations (2.2.4) reduce to the two equations

Further transformations as in (2.2.5) lead to

= xi + (m — 1)«£2 — Na,( 2 - 2 - n )

[1 + (m - 2)«]a;2 - iVa.

lnC/2 J - { K 1 + ( m - 2)K J I f/2 J - B I C/2We note that

! _ 1 (l + (m-2)K - ( m - l ) « l- ^ I 1 JB

where

X = |B| = 1-f (m - 2)« - (m - 1)K2 = (1 - k)[l + (m

so that

It follows that when £ = 0, v = 0 and

Xt(0) = u[1+{m-2)K]/K(0)U^{m~1)/K(0) = (1 - # - i

X2(0) = UiK/K(0)Ul/K{0) = (1 - AT-1)""7^.

2.3. The general epidemic in a homogeneous population 27

///

J

>']1/

fhk \' \ \

\ \\ \ww

o t o(a) K = 0.1

Figure 2.3. Numbers of infectives y\ and 2/2 (and y2 ( ), for a Rushton-Mautner simple epidemic spreading in twocommunities, with #i(0) = £2(0) = N, yi(0) = 1 and 2/2(0) = 0, and K asshown.

(b) « =), and infection rates y\

Hence

Xi(v) = ((1

X2(v) - ((1

where

v = — 1

K ~ l Jo' d 5 . (2.2.13)

Hence, for any time v, G(v) is known by (2.2.13), and thus also X\{v) andX2(v). Prom these Ux{v) and U2(v) are obtained as Ui = ^X^7 7 1"1^ andC/2 = X ^ 1 " ^ ^ " 2 ^ , and thus z, and Vj = N - Xj (j = 1,2).

Figure 2.3 depicts the spread of infection in a population of two equallysized strata. For larger K as in (a), the outbreaks largely overlap and re-inforce each other, whereas in (b) the epidemics occur more slowly andapproximately in sequence.

2.3 The general epidemic in a homogeneous population

In the classical model for a general epidemic that we now describe, the sizeof the population N is assumed to be fixed as for the simple epidemic of


Section 2.1, but infectives may die, be isolated, or recover and become im-mune. Individuals in the population are counted according to their diseasestatus, numbering x(t) susceptibles, y(t) infectives and z(t) removals (dead,isolated or immune), so that x(t) is non-increasing, z(t) non-decreasing andthe sum x(t) + y(t) -b z(t) = JV, for all t > 0. The differential equationgoverning x(t) is

d c— = -(3xy, (2.3.1)

where /? > 0 is the pairwise rate of infection as before, and (x,y, z)(0) =(#o>2/o?2o) with yo > I, zo = 0. The number of infectives simultaneouslyincreases at the same rate as the number of susceptibles decreases, anddecreases through removal (by death, isolation or immunity) at a per capitarate 7 > 0, so that

^ L i y . (2.3.2)

Finally the number of removals increases at exactly the same rate as theloss of infectives, so that

ft=iv. (2.3.3)Observe that (d/dt)(x(t) + y(t) + z(t)) = 0, as is consistent with the totalpopulation size remaining fixed at TV.

In their first paper entitled 'A contribution to the mathematical theory ofepidemics', Kermack and McKendrick (1927) proposed these equations asa simple model describing the course of an epidemic. We can write (2.3.1)and (2.3.3) as

l . d * = _ 0 d z = _ l dzx dt 7 dt p dt' K }

where p = 7//? is the relative removal rate. Integrating this differentialequation directly, and using the initial values xo and z0 — 0 as above, weobtain

x(t) = zoe-2 ( t ) / p . (2.3.5)

A second integral is also readily obtained: equations (2.3.1-2) imply thatx(t) and y(t) satisfy

ax xso

x(t) + y(t) - plnx(t) =xo + yo- plnx0. (2.3.6)

Within the region where x, y and z are non-negative, equation (2.3.2)yields the inequality y > -7?/, which in turn implies that y(t) > yoe~l1 > 0


(all 0 < t < oo). Similarly, x > -px(xo+yo) so that x(t) >0 (all 0 < t < oo). However, from (2.3.1), x(t) is strictly decreasing for allsuch t. Consequently, x(t) and z(t), and hence y(t) as well, converge tofinite limits XQO, Z^ and ^ as t -> oo, with y^ — 0 as we would havel i m ^ o o i > 0 otherwise. Further, from (2.3.5), XQO = x0e~Zoo/p. Becausez<x> < #o + 2/o < oo, XQO > 0, and equation (2.3.2) then implies that y isultimately monotonic decreasing; it is in fact monotonic decreasing for allt > 0 if and only if x0 < 7//? = p.

Kermack and McKendrick's results constitute a benchmark for a range ofepidemic models, so we state them formally for later reference.

Theorem 2.1 (Kermack-McKendrick). A general epidemic evolves ac-cording to the differential equations (2.3.1-3) from initial values (xo,2/o?O),where x0 + Vo = N.(i) (Survival and Total Size). When infection ultimately ceases spreading, apositive number XOQ of susceptibles remain uninfected, and the total numberZoo of individuals ultimately infected and removed equals xo + yo — x<x> andis the unique root of the equation

N-zoo = x0 + y0-zoo= xoe-Zo°/p, (2.3.7)

where yo < Zoo < #o + 2/o? P — i/P being the relative removal rate.(ii) (Threshold Theorem). A major outbreak occurs if and only if y(0) > 0;this happens only if the initial number of susceptibles Xo > p.(iii) (Second Threshold Theorem). If XQ exceeds p by a small quantity u,and if the initial number of infectives yo is small relative to v, then the finalnumber of susceptibles left in the population is approximately p — v, andZoo « 2V.

The major significance of these statements at the time of their first publi-cation was a mathematical demonstration that even with a major outbreakof a disease satisfying the simple model, not all susceptibles would neces-sarily be infected. Conditions were given for a major outbreak to occur,namely that the number of susceptibles at the start of the epidemic shouldbe sufficiently high; this would happen, for example, in a city with a largepopulation. These conclusions were consistent with observation, such ashad been noted by Hamer in his 1906 lectures.

It remains to demonstrate part (iii) of the theorem. Kermack and McK-endrick did so by first finding an approximation to x(t) as an explicit func-tion of t. To this end, observe that substituting from (2.3.5) into (2.3.3)


together with the constraint on the population size yields

dzi ^ . ~\vj •"u^ / • yZ.o.o)

This differential equation does not have an explicit solution for z in terms oft. However, using the expansion e~u = 1 — u + \v? + O(u3) and neglectingthe last term, yields the approximate relation

which can be solved. First express the right-hand side as in

\ 2 x ° i f x ° A 2 fx° 2

Now setting

a=\JQ(N-xo)+(—--l\ (2.3.10)

this reduces to

dz p2

Substituteatanh^^f.-p

p2 L xo\ p

where at time t = 0, z0 = 0, so that atanht;0 = — [(#o/p) ~ !]• Then wecan readily see with this substitution in (2.3.11) that

Hence

d^ P2J , 2 2 , U2 \ P2 u2 d v

-r « -— a - a tanh v) = —asech v—.dt 2x0

v f x0 dt

dv T ,— « f 7a, so v « ^7^ -h

andi (£2 ) ^ ! tanh (I7erf - ^) (2.3.12)()XQ

" 1with </? = tanh"1 [ ( l /a)((x o /p) - l ) ] .


Equation (2.3.12) yields an approximation to Zoo = limôo z(t),namely

( Y (2-3.13)Xo \ p

Now from equation (2.3.10) for a, when 2XQ(N — XQ) <<C (ZO — p)2 andx0 > p,

(2.3.14)

or, writing XQ = p + ^ for some positive z/,

~

Equivalently, XQQ ~p-\-v — 2v = p — V. Note that this result is obtainedfrom the approximation (2.3.9) to the differential equation (2.3.8).

Another route to part (iii) of Theorem 2.1 is to analyse equation (2.3.5)more directly. Observe that the function f(z) — x^~zlp is convex mono-tonic non-increasing for z > 0, so it intersects the line g(z) = N — z =#o 4- 2/o — z at most twice. In fact, since #(0) > /(0), there is exactly onepoint of intersection z^ say, in z > 0 unless t/o = 0, in which case z — 0is also a point of intersection. Now if /'(0) = —xo/p ^ —1? the point ofintersection in z > 0 is necessarily close to the origin; conversely, z^ is muchlarger than zero if /'(0) < —1. This effectively substantiates (ii). For (iii),again use an expansion of the exponential function, this time in (2.3.7), sothat for ZQO > 0 and yo ~ 0,

which in fact is the same as (2.3.14).We now analyse this model more carefully using Kendall's (1956) meth-

ods. Kendall noted that Kermack and McKendrick's approximate resultswould in fact be exact if the infection parameter (3 were not constant butrather a function of z, namely

0{z) = d z) W( zvl (°<z<p'p=fy- (2-3-15)p) \ p

Note that z cannot be allowed to be equal to or larger than p, otherwise /3(z)becomes zero or negative. We see that /?(0) = /?, and that as z increases,(3(z) decreases monotonically as shown in Figure 2.4, so that the per capitainfection rate decreases as the number of removals increases. For /?(z) toremain within 20% of the initial value /?, it is enough that z < \p.


0 9Figure 2.4. Kendall's modified infection parameter f3(z) ( ) and /3 ( ).

The solution of equation (2.3.4) with (3{z) in place of /? is

This equation is precisely the case t —• oc of equation (2.3.12), obtainedfrom (2.3.7) using the expansion of the exponential function to the secondorder. Thus, the approximate solution (2.3.12) for z underestimates thenumber of removals, since the infection parameter /3(z) is always less thanits initial value (3 as in (2.3.1-3).

Returning to equation (2.3.7) for constant /?, Kendall (1956) viewed theepidemic rather more generally, first from time t = 0 to t = oo, with

t=- [* — r (0 < z < Zoo = 2(00)), (2.3.17)

where t —> 00 as z | z^. We have already noted in (ii) of Theorem 2.1that Zoo is a positive root of (2.3.6); this is illustrated in Figure 2.5 (see alsoFigure 1.2).

Notice that there is a second root Z-00 < 0. Now we can imagine theepidemic as starting at time close to t = — 00 with a very small number e ofinfectives and N + |z-oo| — c susceptibles, and evolving to 0 infectives andN — Zoo susceptibles at time t = 00. The total number of removals wouldthen be Zoo + 12-001 fr°m a total population N' = N + |^-oo|-

In order to consider the evolution of the equations (2.3.1-3) on the wholereal line as time interval, it is convenient to take as the time origin the epoch£1 corresponding to x(t\) — 9 susceptibles because the peak of the epidemiccurve occurs at this instant. Indeed, we can see directly from (2.3.2) thaty = 0 for x = p, so that y(t) is at a maximum there, as asserted. Notethat from (2.3.5) the corresponding value of z is z{t\) = /oln(xo/p) = zp.


= XQ -f 2/0 - z

Z-ooO

Figure 2.5. Zoo and z-oo as points of intersection ofw = xo -f 2/0 — z and w = xoe~z^p.

We remark that in plotting the evolution of several epidemics of a givendisease in different time periods, a useful common origin is exactly such atime where the epidemic curve is believed to have peaked. For example,Wilson and Burke's (1943) data given in Exercise 1.4 could be plotted inthis way.

In terms of the functions #, y and z satisfying (2.3.1-3), x{t\) = p corre-sponds from (2.3.17) to the time

tl = ->pln(xo/p) dw

N -w- xoe-w/f> '

we now define (xf(u), yf(u), zf(u)) = (x(ti+u),y(t\+u),z(ti+u)--Zp). Suchfunctions satisfy the equations (2.3.1-3) with (x, y, z) replaced by (xf, y1, z'),and initial conditions

(4>Vo,zo) = - p - - p - pln(xo/p),0),

except that we consider them as being defined for all — oo < t < oo, andsatisfying (xf+ y' + z')(u) = N — zp (all u). The limiting values at u = —oo,oo are (7V',0,-Iz^l) and (Nf - \ZLQOI - ^ , 0 , ^ ) respectively, wherezLoo = z-oo - zp, z'oo = Zoo- zp. Note that I^LÎ + z ^ = |z_oo| + zôo.These quantities are illustrated in Figure 2.6.

This information can be interpreted conveniently in terms of the intensityof the epidemic, defined by

i =\Z-c

N1 (2.3.18)


w = N -zp-z'

Figure 2.6. Some relations between (xo,yo;N), (xoo,Zoo,z-oo;N) and

using x' = (xrz'/p, Nf = pe^~^/p, and TV' - K J -z'oo = pe~ *'<*>'p. Hence

TV'

or

Further, since N' = pe|2:-<~l/p, or \ZLQOI = pln(N'/p), then

W-c—N'i

ln(N'/p)

(2.3.19)

(2.3.20)

Table 2.2 lists various indicators of the characteristics of an epidemic interms of the intensity. Note that an epidemic of zero intensity representsthe limiting case where i | 0.

We can see from Table 2.2 that for all i in 0 < i < 1, or equivalently,for 0 < p < Nf < oo, there are more removals after u = 0 than before.For example, if N' = 1000 and p = 896, so N'/p = 1.116 and i = 0.2,then about 20% of the population become infected and thus about 800susceptibles remain at the end of the epidemic. On the other hand, withthe same N' but now p = 390 so that N'/p = 2.564 and i = 0.9, about90% of the population are affected by the disease and only 100 remainsusceptible at the end of the epidemic. Broadly speaking, while a smallmajor outbreak occurs when the population size parameter N' is in the

2.4. The general epidemic in a stratified population 35

Table 2.2. Characteristics of a general epidemic in terms of the intensity i

Intensityi

00.20.40.60.80.90.99

Relative sizeN'/P

11.11571.27711.52722.01182.55844.6517

Peak incidencey'(o)/N'

00.00560.02540.06790.15560.24180.4546

Severity before peak4o/(l*'-<J+4o)

(0.5000)0.50940.52120.53790.56570.59210.6662

Notes. For given i, N'/p = | ln(l - i)\/i, y'(0)/N' = 1 - (p/N')[l + ln(N'/p)), andl*'-ool/(*~ + K o J ) = (P/N'i) WN'/P) = [ln(N'/p)]/| In(l - i)|. See text.

region of the critical threshold size p, most of the population is affected (i.e.a large major outbreak occurs) as soon as N' is 3 or more times p.

Table 2.2 can also be used to relate the measures (#o,2/o) m t n e originaltime scale t to the 'standardized' measures of the table. For, supposingthat (a?o, Vo)i P a n d N are given, then we can solve equation (2.3.5) to findthe value zp for which xo — p, namely z p = p\n(xo/p)y and hence obtainz' = z — zp. Then z-co and z^ are the two roots of

N - z - xoe~z/p = 0,

and finally N' = N 4- |^-oo|- All the quantities of Table 2.2 can now befound.

For example, (a?0, yo) = (800,100), N = 900 and p = 390 gives zp = 280.2,z-oo = -78.6, z^ = 796.1, so AT' = 978.6 and z = (78.6 4- 796.1)/978.6= 0.8938, i.e. close to 90% of the population are infected by the epidemic.

Note that the epidemic is skewed about the central value zp\ in the ex-ample just given, z^ = z^ - zp = 796.1 - 280.2 = 515.9, while -z^^ =\z-oo ~ zp\ = 78.6 4- 280.2 = 358.8, which is about two-thirds of 515.9. Thecorresponding values of the susceptibles x' are x'^ — 103.9 and x!_oo —978.6. The value v discussed after (2.3.14) could be estimated as either800-390 = 410 or 390-103.4 = 286.6, again differing appreciably. In otherwords, in terms of Xo = p+v, the rough approximation x^ « p—v — x$ — 2vholds at best for a small range of intensities i

2.4 The general epidemic in a stratified populationA major aim of this section is to indicate how Kermack and McKendrick'sresults as stated in Theorem 2.1 extend to the more general setting of Sec-tion 2.2 in which an epidemic spreads in a stratified population (hence,


non-homogeneous). We do not need to specify here the basis of the stratifi-cation: it may be spatial (i.e. geographical), behavioural, cultural or socio-economic, for example. In this section, in addition to the pairwise infectiouscontact rates 0ij for an infective in the ith sub-population or stratum toinfect a susceptible in the jth stratum, for i, j = 1, . . . , m, we also supposethat there are per capita removal rates jj for the removal of infectives fromthe jth stratum; Zj(t) denotes the cumulative total of such removals bytime t. Then by analogy with the basic equations (2.4.1-3), again by theLaw of Mass Action, we have the differential equations (d.e.s)

Vj = + • • • +(2.4.1)(2.4.2)(2.4.3)

for each j = l , . . . ,ra, with initial conditions (XJ,yj,Zj)(0) = (xjo,yjo,fy-These equations are expressed more compactly using vector notation similarto that of Section 2.2, extended to include the vectors

x =X2

y =

2/12/2 Z2

7 =

7i72

We indulge in the same sort of abuse of notation for lnx as with lnU in(2.2.6), and extend it to the use of diag7-1 = diag(7f 1,7^"1,. . . ,7^*) forthe diagonal matrix whose elements are the reciprocals of the elements ofa vector like 7. In this vector notation, with B = (A?)? the differentialequations (d.e.s) (2.4.1-3) are expressible as

x = - diag(x)B;y, y = diag(x)B'y-<iiag(7)y, z = diag(7)y. (2.4.4)

Writing B 7 = B'diag(7~1), the first and third of these give (cf. (2.3.4))

dlnx . , . 1x—r— = -B /diag(7"1)z = -B 7 z .

Integration on (0, t) coupled with the initial conditions leads to

j - 1, •.., m), (2.4.5)i=\

2A. The general epidemic in a stratified population 37

provided that this solution curve or trajectory lies in the region X definedby

Xj, Vjy Zj > 0, Xj + yj + zj = Xjo + yjo (j = 1, • • •, m). (2.4.6)

It is not difficult to check that the trajectory does indeed lie in X: it fol-lows from (2.4.1) that trajectories in X are monotonic in each Xj, andyj(t) > 0 for 0 < t < co assuming the matrix B is primitive (i.e. forsufficiently large n, all components of B n are strictly positive). Thenusing the boundedness as well, it follows that the component vectors oflini£_>Oo(x,y,z) = ( x 0 0 ^ 0 0 ^ 0 0 ) exist and satisfy

y~=0, x ^ x o - f v o - z ~ ,x? = xj0 exp ( - [B7zoo]i) (j = 1,..., m).

Thus, the d.e. system (2.4.1-3) yields a unique limit point ( x 0 0 ^ 0 0 ^ 0 0 ) .Further analysis completes the proof of part (i) of the following statement,which is an analogue of the Kermack-McKendrick Theorem 2.1.

Theorem 2.2. A general epidemic evolves in a stratified population ac-cording to the differential equations (2.4.1-3) in the region X, starting frominitial values (xo,yo,0), where the transmission matrix B is primitive andthe removal rate vector 7 has all components positive.(i) (Survival and Total Size). When infection ultimately ceases spread-ing, positive numbers of susceptiblcs Xj° remain in each of the strata j =1 , . . . , m. The numbers of removals Zj° constitute the unique solution in Xof the equations (2.4.7). If Xo is replaced by any XQ that is componentwiselarger than XQ, i.e. XQ >: xo, then z°° is replaced by z/oc >z z°°.(ii) (Threshold Theorem). A major outbreak occurs if and only if the eigen-value Amax of the non-negative matrix diag(xo)B7 with largest modulus,lies strictly outside the unit circle, where B 7 = B/diag(7~1).(iii) (Second Threshold Theorem). If x0 = f -h Ax, where the componentsof Ax are non-negative and sufficiently small and Amax(diag(£)B7) = 1 <Amax(diag(xo)B7), then

( 2 A 8 )2=1

~ 1 )i.e. z°° w 2[l 'diag(v~1)(x0 - £)]v, where v is the right eigenvector of theeigenvalue 1 of the matrix diag(£)B7.

The detailed proof of this theorem can be found in Daley and Gani (1994);we confine ourselves here to some indicative comments. Watson (1972)treats special cases of the model.


Equations (2.4.7) can be solved iteratively via the transformation T :(n) *—> z(n+1) defined componentwise by

- expf-CB^W),]), (2.4.9)

starting from any convenient z 0) e X such as z 0^ = yo. Thus, the vectorz°° is a fixed point for T, i.e. Tz°° = z°°, and under the stated conditionsit is the only fixed point in X.

The criticality statement (ii) is motivated in part by recalling that for ageneral epidemic in a homogeneous population, the initial point (xo,yo) isclose to the singular point (#o, 0) for the pair of d.e.s (2.3.1-2) describing theevolution of the process. The linear d.e. system approximating (2.3.1-2) atthat singular point either diverges when XQ > p or converges when x0 < p. Alinear approximation to (2.4.1-3) in the neighbourhood of (xo,yo) ~ (xo, 0)leads to a similar divergent/convergent dichotomy, determined here by thedominant eigenvalue of the matrix analogous to the product xo/^T"1 of theKermack-McKendrick Threshold Theorem. We shall meet Amax again inSection 3.5 in the context of the Basic Reproduction Ratio.

The approximation in (2.4.8) is derived by a power series expansion inthe solution equations (2.4.7), much along the lines of the argument below(2.3.14) based on (2.3.5).

The description we have given for an epidemic in a stratified popula-tion is not the only possibility. For example, Ball and Clancy (1993) andClancy (1994) allow individuals to move between strata, hence offering amore general cross-infection mechanism than our formulation with the ma-trix B allows. Exercise 2.4 sketches details of a special case of Theorem 2.2,and Exercise 3.9 evaluates Amax for a special case of B.

Exercise 2.5 discusses briefly the question of stratified population ana-logues of the 'other' root Z-OQ (see Figure 2.5) and the standardized in-tensity measure i at (2.3.18). Certainly, B" 1 is well-defined when B 7 isprimitive.

2.5 Generation-wise evolution of epidemics

One can envisage an infection in a population as spreading along a sequenceof links from any given individual to a number of others. If for examplethe infection is 'psychological' like a rumour or item of important news(cf. Chapter 5), then there may be a strong interest in knowing how manyindividuals are infected direct from the initial infective(s): this is the number

2.5. Generation-wise evolution of epidemics 39

of first-generation infectives. Individuals infected first-hand from one ofthese first-generation infectives are then second-generation infectives, andso on through to the j th generation. The accuracy of the 'infection' maybe affected by the closeness or otherwise of the source of an individual'sinfection to the initial infective(s). In the case of the simple and generalepidemic models of Sections 2.1 and 2.3 the proportion of j th-generationinfectives can be derived algebraically. To the extent that this involvesconsidering sub-divisions of the population, there is some overlap with theanalysis in Sections 2.2 and 2.4.

For clarity below and consistency with all of our later exposition, we setN = x0 and / = yo in notation used up to this point, and N' = N +1.

Consider first the simple epidemic model of Section 2.1. Use yj(t) todenote the number of infectives at time t who have acquired their infectionby contact with some (j — l)th infective, i.e. with an infective countedamongst the yj-i(u) infectives of the (jf — l)th generation at some earliertime u < t, the instant of contact for the individual concerned. So, ina simple epidemic in a population of size N', there is a constant numberyo = I of initial infectives, yi (t) who by time t have been infected first-handfrom these y0, and so on up to yj(t) j th-generation infectives who by timet have been infected from the ?/j_i(£).

We assume homogeneous mixing of the population, with the infectionspread from one individual to another. Then, when the size N of the closedpopulation is large, the equations describing the spread are

j^t) 0 = 1,2,...), (2.5.1)

withoo

J2 N + I-x(t) = N'-x(t)- (2.5.2)

note that the number of generations j must in fact be finite. In addition tothe equations at (2.5.1) we also have the relation y = (3xy = /3(N'—y(t))y(t)as at (2.1.1), from which we know that y(t) = IN'/(I + Ne-W*) (see(2.1.2)).

To solve the d.e.s (2.5.1), rewrite them as

(2.5.3)

so

/Jo

/ ^ ^ d u j (j = 0 , 1 , . . . ) ,o Jo


150

100

1 2 3 4 0 5(a) (b)

Figure 2.7. Growth of 'generations' 0, 1, 2, 4, 6, 8 of infectives in asimple epidemic with (AT, /) = (1000,1), 0 = 0.0001, on 0 < 0N't < 10;part (a) is an enlargement of the start of (b). The number at the right-hand end of the curve shows the generation number; generation 6 is themodal generation for large t.

where W(Uj) = yiu^/yiuj) = (d/dt) lny( t ) | t = u / Thus,

yo(uo)W(uo)W(u1)...W(uj)duodu1...dur

0<uo<ui<-"<u3<t

Here, yo(u) = / (all u > 0), and the (j + l)-fold integral is symmetric in itsarguments, so with

lnU(t)= /V(«)d«= / 'Jo Jo du

we have

(2.5.4)

Equation (2.5.4) shows that the sequence {y3\t)} is proportional to the termsof a Poisson distribution with increasing mean In U(t) (cf. Daley, 1967a).For small t (meaning, 0N't = o(l)), lnU(t) « (1 - I/N')0Nrt. Figure 2.7illustrates the growth of the 'generation sizes' in the case (JV, /) = (1000,1),0 = 0.0001 and /?AT't < 10. Part (a) of the figure shows the behaviour forsmaller t.


To study the generation-wise evolution for the general epidemic model ofSection 2.3, we continue to use the functions yj(t) as defined above (2.5.1),and introduce thf function Zj(t) to denote the number of j th generationinfectives removed by time t. In place of the d.e.s (2.5.1), we define t/_i = 0and now have for j = 0 ,1 , . . . ,

x = -/3xy,Vj = fayj-i -IVj, (2.5.5)

where y = YlJLo Vj as before, and y = (f3x — ^)y. Thus, (3x = 7 -I- y/y, andthe second of the equations at (2.5.5) becomes

or equivalent ly,

Here we have written Wj(t) = eytyj(t) (j = 0,1,...) and it;(t) = efor which u;0(t) = / (0 < t < 00) and Wj(0) = 0 (j = 1,2,...). SettingVF(t) = dlnu7(£)/d£, we see that the functional form of these equations isexactly the same as in (2.5.1), with solution

, j ( t ) _

J '

*similar to (2.5.4). Then, since /0* W{u) du = \n[w(t)/w(0)] = jt+ln[y(t)/I],

Thus, as for (2.5.4), the relative frequencies of the generation types amongthe infectives at time t are proportional to the terms of a Poisson distribu-tion. However, since from (2.5.5),

.,(.)-7 / * * ( . ) * . -7 / / ' . - ' • ' 1 t t l a | ; | l ) / " ' ' < . , (2.5.7)0 Jo J'

we see that for the removals at time t, the numbers in the various generationsare proportional to the terms of a mixed Poisson distribution. The mixing


15 2010(a) (b)

Average generation number among removals in a general), deaths in a linear birth-and-death process (—•—•—)

Figure 2.8.epidemic (and 'aged' infectives in a simple epidemic ( ) with (AT,/) =(1000,1), (3N't < 20 and (a) 7 = 0.01 (hence, p = 100, Ro = N/p = 10),and (b) 7 = 0.05 (hence, p = 500, Ro = 2). See Section 3.5 for discussionconcerning the Basic Reproduction Ratio RQ.

factor 7e 1U du arises from the distribution of the time an individual, onceinfected, remains so until it is removed.

As a measure of the average number of generations represented amongthe removals we use

Tm(t) = (2.5.8)

The denominator here is z(t) = N +1 - x(t) - y(t) = p]n[N/x(t)], whilethe summation in the numerator yields, on simplification,

0ln[N/ x(t)} Jo

*) du. (2.5.9)

This function is conveniently studied numerically by means of the three d.e.s

rh = m (2.5.10)

The function m{t) is plotted in Figure 2.8 together with the correspondingfunctions from the simple epidemic and linear birth-and-death processes(see Exercises 2.6 and 3.5 respectively).

We see that the ultimate 'average' generation number of those infectedin a general epidemic is larger than the corresponding 'average' in a simpleepidemic. This occurs because in the former the number of infectives isreduced by removal, thereby slowing down the rate of spread of infection


(so, the pool of susceptibles remains larger). Thus the difference in theultimate averages reflects the later occurrence of infection in the generalepidemic as against the simple epidemic. These interpretations are furtherborne out by the comparison of the two sets of averages for the same initialconditions and pairwise infection rate, but increased removal rate of (b)over (a) in Figure 2.8.

It follows from (2.5.7) that Zj° = limôo Zj(t) equals the limit as t —• ooof the given integral. Recall from (2.3.6) that x(t) and y(t) are related by

y(t) = C + plnx(t)-x(t) where C = N + / - pin AT. (2.5.11)

Also, from Kendall's argument referred to following (2.3.17), the monotonicfunction x(t) has inverse r(x), given by r(N) = 0 and for N > x > XQQ,

Ndx x_ fN ^__ fN

i\~L 0xy"Jx

Writing v{x) = /Qr W(u) du, the limit of the integral (2.5.7) after a change

of variable from t to x gives

ao \nu-u) ' JXoo 3\(2.5.13)

where

N

Dunstan (1982) used Daley's (1967b) result (2.5.7) to show that for fixed/ and j and large iV, with N ^$> p,

(2.5.15)

irrespective of the infection and removal rates /3 and 7. This right-hand sideis the same as limt_>oo Vj(t) in the simple epidemic (see (2.5.4)), and henceis a limit property of {z?°} when p I 0. To explain this coincidence, observefrom the d.e.s at (2.5.1-2) and (2.5.5) that the simple and general epidemicmodels with the same initial conditions and pairwise infection rate /?, have


100

30

Figure 2.9. Growth of odd-numbered 'generations' in a general epidemicon 0N't < 30 with (iV,/) = (1000,/), 0 = 0.0001 and (a) 7 = 0.01(hence, p = 100, ft, = 10), (b) 7 = 0.05 (hence, p = 500, Ro = 2).Generation number shown at right.

30

approximately the same behaviour for 0 < t < 0(7 l/N) =Then for sufficiently small 7, at the end of this 'initial' phase, the populationin the general epidemic will consist almost entirely of infectives for whichthe generation-type will be the same as under the simple epidemic. Thesimple epidemic by this time has essentially reached its final state, whilein the general epidemic model each j th-generation infective subsequentlybecomes a j th-generation removal.

Let us now review both these results and the properties of birth-and-death processes (see Exercise 3.5), in conjunction with the fact that for adiscrete time Galton-Watson branching process the mean sizes of successivegenerations form a geometric rather than Poisson sequence. We see thatthe Poisson nature is attributable to the modelling assumption that anyinfective, irrespective of 'age', is as likely as any other to infect a susceptible.The mixed Poisson characteristic in the proportions of removals, comes fromthe delay between an individual's being counted first as an infective andsubsequently as a removal in the general epidemic, or a death in the birth-and-death process. The fmiteness of the population reduces the averagegeneration number from a linear function, as in the linear birth-and-deathprocess, to either In U(t) or /0* W(u) du, both of which have finite limits ast —• 0 0 .

Figure 2.9 shows how the odd-numbered Zj(t) evolve in general epidemicsfor the same models and initial conditions as in Figure 2.8.

Exercise 2.7 sketches another model for the generation-wise evolution ofan epidemic in which successive generations are non-overlapping.

2.6. Carrier models 45

2.6 Carrier models

A major complication with certain diseases such as typhoid, tuberculosisand poliomyelitis is that the infectives who are the source of infection in thecommunity, may be individuals who do not display any symptoms and areapparently healthy. It may take the detection of a geographical pattern ofinfection, or a mass screening programme before such infectives are iden-tified and removed from contact with susceptibles. We call such infectivescarriers to distinguish them from susceptibles who, on being infected, maybe quickly recognized by their symptoms and removed from the population.

This type of situation requires a somewhat different model. The infectionof a susceptible through contact with a carrier, in the simplest model, nowresults in the removal of the infected susceptible while the number of carriersremains unaltered. Carriers are distinct, and their numbers are diminishedby an independent removal process. Applying the Law of Mass Action,the numbers of carriers w(t) and susceptibles x(t) follow the differentialequations

x = —(3xw and w = —yw. (2.6.1)

These are easily solved, given initial conditions (x(0),u>(0)) = (xo,wo), as

w(t) = woe-^ and x(t) = xoexp ( - (f3wo/j)(l - e~7*)). (2.6.2)

In practice, when a carrier-borne disease is recognized in a communitywhere it is normally absent, measures are quickly taken to locate the sourceor sources of infection. Suppose then that the introduction into the com-munity of wo carriers occurs at time t = 0, and that the identification of thedisease through some susceptibles who have developed symptoms occurs atsome time to later, at which point the removal rate 7 increases to 7' say.The result of such activity is that the d.e. for w is no longer as at (2.6.1)but

-yw (0<t< to),(t > to),

while the d.e. for x is unchanged. Then the solution is now that in (2.6.2)for 0 < t < to, while for t > t0 it is

- = { _ ; , : . :::::'"" (2.6.3)

w(t) = 0 \x(t) = x(t0) exp ( - (f3w(to)h')(l - e - ^ - 1 0 ) ) ) .

Carrier models have a much simpler structure than the general epidemicmodel, due to the fact that the process w(t) in the model is independent ofx(t); x(t), however, depends on the evolution of w(t). See also Exercise 2.8.


2.7 Endemicity of a vector-borne diseaseThe development of epidemic modelling owes much to the work of malariolo-gists Ronald Ross (1857-1932) and George Macdonald (1903-1967); see e.g.Macdonald (1973). Malaria, unlike viral diseases such as measles, mumpsand HIV which spread by contact between infectives and susceptibles, iscaused by a parasite whose life-cycle is spent partly in an intermediatehost, the Anopheles mosquito. The mechanism of infection in this case isdifferent from that which we have so far studied; we must now track thedisease-status of both human and mosquito populations. Our aim is tomodel the progress of an outbreak of malaria, and thereby describe condi-tions for its endemicity. We outline briefly a simple deterministic model,noting that it is capable of considerable refinement, in both deterministicand stochastic settings (see e.g. Anderson and May (1991, Chapter 14) and,particularly, Bailey (1982)).

The life-span of malarial infection is relatively short compared with thehuman life-span, and a mosquito population is typically unaffected by theprevalence or absence of malaria. Thus, it suffices to consider two popu-lations of constant sizes iVi humans and AT2 mosquitoes, in both of whichindividuals are either infected or not, the numbers of susceptibles and in-fectives being Xk(t) and Yk(t) respectively, Xk{t) + **(*) = Nk (A; = 1,2).A human infective with malaria loses infectivity (and is again susceptible)at rate / / j ; similarly a mosquito infective dies at rate /i2 and is immedi-ately replaced by a new-born susceptible. A mosquito depends on bitinga human for a blood-meal during which it can ingest infected blood if theblood-source is malarial; in its turn it will infect a human susceptible whenit is infective (see the cited references for more biological detail). Then thehuman malaria-infected population, neglecting any latent period, changesat rate

and similarly, for the I2W malaria-carrying mosquitoes,

where j3 denotes the feeding rate per mosquito and 7^ the infection trans-mission rate from an infective of type i to a susceptible of type j (i, j = 1,2).

In terms of proportions xk = Xk/Nk = 1 — yk (k = 1,2),jV2V\ = 0l2ij^V2Xi - /Ziyi = fii(a1y2x1 - 2/1),

2/2 = /? ( )

2.7. Endemicity of a vector-borne disease 47

0.5(a) threshold just exceeded

0 0.5 1(b) threshold well exceeded

Figure 2.10. Level curves 2/1 = 0 and 2/2 = 0 in the 2/1-2/2 phase-plane.

where ax = /?72i(N2/Wi)/Mi, a2 = £712/A*2- By inspection, 2/1 = y2 = 0 isa stationary point, as is any solution of the equations 2/1 = 0 = 2/2 > namely

Thus

2/1 = a i 2 / 2 ( l - 2 / i ) and 2/2 = <*22/i(l - 2/2).

2/2 = (2.7.2)

is a possible equilibrium value; it always lies in (—00,1), and in (0,1) if andonly if aic*2 > 1, i.e.

£-* < ^rN>- (2-7-3)£>721 V>2

Thus, according to the model, if malaria is to remain endemic a thresholdcondition must be satisfied, namely that the relative recovery rate of humansfrom malarial infection must be smaller than the relative infection rate of thenew-born mosquito population. Further, as also noted by Ross (1916), toeradicate malaria, it suffices to reduce the mosquito population sufficiently.

Macdonald (1952) distinguished two types of behaviour of infected popu-lations where malaria was endemic: in some populations malarial infectionwould ebb and flow quite markedly ('unstable' endemic), while in othersit would remain at an approximately constant level. Sketching the curvesT/i = 0 and 7/2 = 0 from (2.7.1) in the 2/1-2/2 plane, and supposing thatthe system lies above the threshold as at (2.7.3), these curves cross at thestationary point at an angle lying between zero and a right angle. The firstextreme, where the curves cross at a small angle, corresponds to a situationwhere the infection and transmission levels are only just sufficient to yieldan endemic situation: small changes in the level of one of 2/1 and 2/2 (in Fig-ure 2.10 it is 2/2, i.e. the fraction of the mosquito population that is infected)


determine large changes in the equilibrium level of the other. Under thesecircumstances the disease, while endemic, is still subject to large changesin infectivity resulting from relatively small changes in the infestation levelof the mosquito population. The second extreme corresponds to the casewhere a significant change in the level of either y\ or 1/2? determines a sig-nificant change in the other; this corresponds to an endemicity whose levelis sustained quite steadily.

2.8 Discrete time deterministic models

All data available for epidemics are gathered at discrete time intervals, as forexample the numbers of infectives recorded in a population on consecutivedays, or in consecutive serial intervals. It is therefore proper to examine theprinciples of deterministic modelling in discrete time, whether as skeletonsof continuous processes or as discrete processes in their own right.

Gani (1978) studied a discrete time equivalent of the simple epidemic,where observations are made at the times t = 0 ,1 , . . . . Let a closed popu-lation of size N consist of xt susceptibles and yt infectives at time t where%t + Vt = N and yo is given, with 0 < y0 < N. One way of describing theprogress of a simple epidemic, appealing to the Law of Mass Action, is bythe difference equation

y t+i = mm(yt + (3yt{N - yt), N) (t = 0,1,...), (2.8.1)where /3 represents the pairwise infection rate over the discrete time unitconcerned; this parameter is usually different from that for the analogousmodel in continuous time. Omitting the min(-) operator and expressing therelation in the form yt+\ — yt = /3yt(N — yt) stresses the analogy with thelogistic equation (2.1.1) for population growth.

We study the process {yt} through the function

f(y) = / ^ ( / r 1 + N - y) = y + f3y(N - y) (2.8.2)which is concave, /(/3"1) = N = f(N), and has 0 and N as fixed points. Weuse this function because, at least for 0 < yt < min(iV, Z?"1), yt+i = f(yt)-Define a sequence {yf} by y% = y0, yt*+1 = f(y$) so that yt = yt providedthe right-hand side lies in the range (0, N). There are two cases to consider,according as f3N < 1 or (3N > 1. In the former case, this discrete timeepidemic parallels the behaviour of the simple epidemic in continuous timeas presented in Section 2.1 above. To see this, note that f(y) is a parabolaas shown in Figure 2.11 (a), and that for y < N, y < f(y) < N. Thus,starting from positive y0 < N, y$ < N < 1//3 for alH = 1,2,... so yt = y*,and yt | N as t —• 00.

2.8. Discrete time deterministic models 49

N(a) PN < 1 (b) 0N > 1

Figure 2.11. Discrete time logistic epidemic.

When (3N > 1, it follows from (2.8.2) (cf. Figure 2.11(b)) and the con-cavity of / that if (3~l < y* < N, then f(y*) > N, while if 0 < y* < /3'1then (f3N)y* < f(y*). Consequently, for t such that maxo<s<t{y*} < (3~l,

< 2/t+1, and therefore for some t sufficiently large, t = T say, wehave > (3~l > ?/f, and thus

Vs =

> N. This means that for such

(2.8.3)

Thus, for /?iV > 1, the entire population contracts the disease in a finitetime T-hi .

This type of behaviour is expressed formally in the theorem below: thereis no such analogous behaviour of continuous time deterministic simple epi-demics. The result is a consequence of the discretization of time.

Theorem 2.3 (Threshold Theorem for Discrete Time Logistic Equation).For an epidemic described by the logistic growth process in discrete time asin equation (2.8.1) with an initial number of infectives yo in (0, N), either(a) (3 < N~l, and at any time t = 0 ,1 , . . . after the initial infection, apositive number of susceptibles remains in the population (i.e. yt < N); or(b) j3 > N~x and no susceptibles remain after some finite time T -f-1.

Spicer (1979) used a discrete time deterministic model for predicting thecourse of influenza epidemics in England and Wales. It is an analogue of thecontinuous time general epidemic model of Section 2.3. Let xt, yt denotethe numbers of susceptibles and new infectives on day £, and (3 the pairwiseinfection rate as defined at (2.8.1). Removal of infectives occurs daily, withthe proportion ^ of the new infectives on any given day t remaining in thepopulation j days later. Assume that 0 < ipj < 1 (all j = 0,1,...).


30-

20-

10-

0-0 8 10 12

Figure 2.12. Weekly deaths from influenza and influenzal pneumo-nia in Greater London, 1971-72. (Data from Spicer, 1979.)

Suppose the epidemic starts with XQ susceptibles and yo new infectives onday 0. Then using the same Mass Action principle as before, the sequence{(xt, yt)} evolves according to the law, for t = 0 ,1 , . . . ,

yt+i = f3xt =xtyt+\, (2.8.4)

while the number zt+i of infectives removed on day t is given by

=Yt- {Yt+l -i=o

Spicer used the estimated proportions ô , . . . ,^5 — 1-0, 0.9, 0.55, 0.3, 0.15and 0.05, with xjjj = 0 for j = 6,7,. . . , to predict the progress of influenzaepidemics in Greater London in 1958-73, and hence to estimate the numberof deaths (as a proportion of the number of infectives) from influenza andinfluenzal pneumonia in that period. The important factor was to obtainan estimate of /?#o; this was found to range between 4.5 and 7.7 in theperiod 1958-73. Figure 2.12 illustrates the quality of the fit of the model tothe data (see also Section 6.1 and Exercise 6.1 below). If we assume thatdeaths are proportional to the number of new infections in any week, thenthe plotted points and fitted curve of Figure 2.12 are proportional to theepidemic curve defined around (2.1.4).

Saunders (1980) used a slightly different approach to study the spreadof myxomatosis among rabbits in Australia. After infection, a rabbit goesthrough a latent period of about 7 days, followed by an infectious period ofabout 9 days during which symptoms can be observed. Nearly all infectives

2.8. Discrete time deterministic models 51

die, so that removals in this case are due to death. The model postulatesthat a rabbit infected on day t enters and remains in the latent period untilday t + t—1 (£ = 7). It then becomes infectious on day t + £ and remainsso until day t+£+k-l (fc = 9), after which it dies.

Let ?/i (t) denote the number of rabbits first observed to be infectious onday t; these were infected on day t — £, and remain infectious and in thepopulation until day t + k — 1. Hence on day £, the total number of infectivesis

fc-iy(t) = Y,yi(t-j), (2.8.5)

i=osince any rabbits becoming infectious before day t — k -f 1 have died by dayt. Starting with XQ susceptibles at time t = 0, the number of remainingsusceptibles at time t is

t+t

iO'), (2.8.6)

while the number of those in the latent state is

i(t + J). (2-8.7)

The total number of rabbits alive on day t is

Saunders used the homogeneous mixing model

£) = (3y(t)x(t) (2.8.8)

with an estimated f3 = 0.0031, £ = 7, k = 9, but the fit of the model provedunsatisfactory (see Figure 2.13(a)). An improved fit results from using apopulation dependent model in which

i.e. (3 has been replaced by a/p(x(t)), on the assumption that each infectedrabbit makes contact with a constant number a of rabbits each day. Anestimate of a gave a = 0.34, and this change with p(n) = y/n, resultedin better agreement with data, as can be seen from Figure 2.13(b). Saun-ders also used a goodness-of-fit criterion to discriminate between possibledensity-dependence functions p(n) = c, y/n and n, where c is a constant.


120-100-8 0 -

6 0 -

4 0 -

2 0 -

U

\ \

1 1 1 1 10 10 20 30 40 50 0 10 50

(a) (b)Figure 2.13. Rabbit population affected by myxomatosis: observeddata ( ) and fitted models ( ): (a) homogeneous mixing, and(b) population dependent model. (Data from Saunders, 1980.)

It is possible to develop a discrete time model equivalent to the generalepidemic model of Section 2.3 in which for t = 0 , 1 , . . . ,

xt+i -xt= fixtyt, -yt= /3xtyt - - zt = ~iyt, (2.8.10)

with Xt + yt -f- zt — TV, and /?, 7 are the infection and removal parameterswhich may differ from those in the continuous time model. More correctly,to avoid negative values of xu one should rewrite the first two of theserelations as

xt+1 = max(0, xt - (3ytxt), yt+1 = yt - >yyt + mm{xu f3xtyt) (2.8.11)

and restrict the value of 7 to (0,1). This model was studied in detail byde Hoog, Gani and Gates (1979) who derived an analogue of the classi-cal Kermack-McKendrick theorem, but it is somewhat more complex thanTheorem 2.3; we refer the reader to the 1979 paper for details.

Enough has been said to indicate that while it is mathematically moreconvenient and elegant to deal with deterministic models in continuous time,their discrete time analogues are used fairly often when it comes to analysingdata collected daily, or at other serial intervals. The examples sketchedabove are intended to indicate both the usefulness and the complexity ofdiscrete time deterministic models.



2.1 Mimic the solution of Section 2.2 in the two special cases:(a) For the simple epidemic model in a homogeneously mixing population ofSection 2.1, put m = 1 and K, = 0 in Section 2.2, and deduce that F(v) = 1.(b) Suppose m — 2, fin = #22 > P12 > 0 = #21; investigate the solution.

2.2 Suppose that the classification of a population by its disease state (suscepti-ble, infective, removal) is extended by interpolating a latency state betweensusceptible and infectious, with the numbers w(t) in this class at time t sat-isfying the differential equations x = —fixy, w = fixy — 6wy y = 8w — 72/, andz = jy. Show that the final size in such an epidemic model coincides withthe general epidemic model at equation (2.3.6), and is independent of 6.

2.3 Use a latency state as in Exercise 2.2 in a model for a simple epidemic, so thatthe equations of the model consist of the first two equations (for x and w)and y = 6w in place of the other two equations. Investigate whether anysolution, parametric or otherwise, is available to describe the time evolutionof such a model.

2.4 In the general epidemic in a stratified population described by the d.e.s at(2.4.1)—(2.4.3), suppose that the rates fi tJ — fi 3 and 7j = 7 (all j). Show thatthe solution curve of the d.e.s satisfies

(0)]1/f t = Mt ) /x , (0 ) ] 1 / f t (all t),

and hence that

where Z = X/"Li ZT ls ^ n e u n iQu e positive root of

Prove that the dominant eigenvalue of Theorem 2.2(ii) is given by Amax =£7=1*,o/V7.[Ball (1985) calls such a process a Gart epidemic after the work of Gart (1968,1972).]

2.5 Show that the quantities Zoo and z-00 illustrated in Figure 2.5 can be deter-mined iteratively as limn_+oo z^ and linin-ôo z^~n~^ respectively, where

s<"+1> = y0 + Xo[l - exp(-s(n )/p)],

and z(0) = 0 and n = 0 , 1 , . . . . For a stratified population, the first of theseequations has the analogue (2.4.9). Investigate the relations

*<--*> = - [ B - 1 ln[l + (y0 - B<->)/XO] ]„


where ln[] here denotes a vector with components ln[l + (yj0 - z^ n ) ) /x j 0] ,much as below (2.4.4), as a possible analogue of the equation for ^~ ' n - 1 \When zj~oo) = limn-oo z{rn) exists and differs from ^(oo) = limw.->oo^n),define analogues of i at (2.5.18) by ij = [ôo) + l^'Ô/ATj where N'j =Xjoexp(—[B 7z^~oo^]J). As an analogue of zp consider the vector B" 1 ln[p]where ln[p] is the vector with components \n[yj/0jj].

2.6 For a simple epidemic model, the analogues of the j th generation removalsZj(t) in a general epidemic model (cf. (2.5.7)), are the jth generation in-fectives zs

3(t) who have been in this state for an exponentially distributedtime with mean I/7, i.e. 'aged infectives'. These zs

3(t) satisfy the d.e. Zj =7(yj ~ zj) where quantities with superscript s refer to the simple epidemic.Show that the analogue ms of the function m at (2.5.8) equals Ms/zs, whereMs = ^2 • 3zj> illustrated in Figure 2.8, satisfies the d.e.

lny.= 7 M + l n y

2.7 The following is a prescription for using continuous time methods to trackthe generation-wise evolution of an epidemic process when the latency periodis large relative to the infectious period, and successive generations of infec-tives are non-overlapping. Introduce functions {Xn(), Yn() : n = 0,1,...}satisfying the d.e.s

Xn(tn) = -0Xn(tn)Yn(tn) and Yn(tn) = -^Yn{tn) (t > 0),

together with boundary conditions Xo(0) = N, Yo(0) = I and, for n =0 , 1 , . . . ,

Xn+l(0) = Xn(00), IWi(O) = Xn(0) - Xn(00) = Xn(0) - Xn + l(0).

In this formulation, 'time' tn runs from 0 to oo while the number of nthgeneration infectives declines from Yn(0) to zero; at the same time, the firstgeneration latent infectives are increasing, and the number of susceptiblesdeclines from Xn(0) to Xn(oo) through the growth of such infectives. Atin = ooa new time axis tn+1 starts at 0 and the process iterates. The d.e.for Yn has solution Yn(tn) = yn(0)e~7*n and so

ln[Xn(t«)/Xn(0)] = -[0Kn(O)/7](l - e~7t") -> -/?Fn(0)/7 (tn - oo).

(a) Show that Xc» = limn_>oo Xn(tn) exists and satisfies equation (2.3.7).(b) Prove that, if / is allowed to vary, then limjio[Yi(0)/>o(0)] = 0N/-y = ROas in Figure 2.8 and in Section 3.5.(c) Compare Fo(0)H \-Yn(0) numerically with Z0(oo)H \-Zn(oo), usingthe solution of the generation-wise evolution results in Section 2.5, for somesuitable 0/*y, N and /.


(d) Interpret the sequence {Zn(oo)} in the context of a discrete time generalepidemic model (cf. discussion at end of Section 2.8 and Chapter 4).

2.8 In the carrier model described by the d.e.s at (2.6.1), suppose that a (small)proportion a of the susceptibles infected become carriers, so that the secondof the equations is now

dw „ 7 d In x dx

with x and w related by w = wo -f a(xo — x) — p\n(xo/x). Deduce that thenumber of susceptibles #oo surviving the epidemic is the unique solution ofthe equation

a(x0 - Zoo) + wo = plnOro/xoo), ô > Xoo > 0.

[Observe that with this modification, the carrier model is similar to the gen-eral epidemic model.]

2.9 Suppose that outbreaks of an epidemic take place in a given region (or village)each year. Consider how to combine data from several years so as to displayfeatures of the evolution of the disease; how could these data be used toestimate parameters of a suitable model? As a possible model, assume thesame population size for different years and the same parameters j3 and 7 inthe case of the general epidemic of Section 2.3 or its discrete analogue (cf.En'ko's work in Dietz (1988), En'ko (transl. 1989), and Exercise 1.3 above).

Stochastic Models in Continuous Time

In this chapter and the next, perhaps the most important of this book,we describe the evolution of an epidemic as a stochastic process. Such astochastic approach is essential when the number of members of the popula-tion subject to the epidemic is relatively small. While deterministic methodsmay be adequate to characterize the spread of an epidemic in a large popu-lation, they are not satisfactory for smaller populations, in particular thoseof family size.

We shall use the tools of non-negative integer-valued Markov processes incontinuous or discrete time, retaining much of the approach of the previouschapter in the dissection of the population. However, we no longer usecontinuous functions to approximate the evolution of the epidemic; rather,we consider a closed population of individuals, each of whom is classifiedas either a susceptible, an infective or (except for the 'simple' epidemic ofSection 3.1) a removal. This S-I-R classification1 is possible because weregard the epochs where individuals change their classification as randomlydetermined points in continuous time, or as fixed points in discrete time. Formathematical convenience but with sufficient simplistic realism, we assumethat the processes involved are Markovian. The reader is presumed to havesome familiarity with the elements of Markov processes on countable statespace, as for example in Bailey (1964), Cox & Miller (1965), Karlin & Taylor

1Waltman (see Waltman and Hoppensteadt (1970, 1971)) introduced this notation todesignate both the possible disease states of an individual, and the possible progressionof disease in an individual; here, it denotes susceptible —> infected —> removed. A simpleepidemic as in Chapter 2 or the first two sections below is an S-I epidemic. For an S-I-Sdisease like malaria, a simple model envisages that an individual is either susceptible,infectious or (after recovery) susceptible again. Superinfection is not allowed in thissimplest version in which individuals are either infected to the same extent, or susceptible;a variant of this model would allow superinfection. Hethcote (1994) surveys deterministicmodels more extensively.

56

3.1. The simple stochastic epidemic in continuous time 57

Xt

N

0 tx tjv-l t^-2 ts £2 ^1Figure 3.1. The death process {X(t) : t > 0}.

(1975, 1984), Bartlett (1978) or Ross (1983). In this chapter we concentrateon Markov models of epidemics in continuous time.

3.1 The simple stochastic epidemic in continuous timeA simple stochastic (Markovian) epidemic model in continuous time is onein which the population consists only of susceptibles and infectives; oncea susceptible is infected, it becomes an infective and remains in that stateindefinitely. Thus, the main features of the simple deterministic epidemicin continuous time also hold for the simple stochastic epidemic we nowdescribe.

We suppose that the total population is closed, so that

X(t) + Y(t) = N (all t > 0) (3.1.1)

where X(t) and Y(t) denote the numbers of susceptibles and infectives attime t, with initial conditions (X, Y)(0) = (N, I) where / > 1. We assumethat {(X, Y)(t) : t > 0} is a homogeneous Markov chain in continuoustime, with state space the non-negative integers in Z2 satisfying (3.1.1)and N > X(t) > 0. The only non-zero transition probabilities occur for0 0 is the pairwiseinfection parameter.

Because of (3.1.1), such a Markov chain is in fact one-dimensional on thatpart of Z satisfying N > X(t) > 0, or equivalently, / < Y(t) < N + /, the

58 3. Stochastic Models in Continuous Time

former being a death process and the latter a birth process on the integersconcerned. This enables us to write down certain properties of the modelfairly easily. For example, concentrating on {X(t) : t > 0} as a pure deathprocess and referring to Figure 3.1, it is clear that X(-) evolves by unitdecrements at the epochs ts, £/v-i, • • • ,t\ say, where

NTi ( 1 < J < J V ) , (3.1-3)

and the Tj are independent exponentially distributed random variables forwhich ETj = l/[/3i(N + I - i)]. Consequently

1 N 1 1 [ N 1 N~j

1=3 v

/ N(N 4- I -I)

= 1 , 7 = 1 ) , (3.1.4)

l n / • _ 1 ) / / _ 1 ) + 7/v,/,j otherwise.

x <

Here 7/v,/j = 7iv — 7/- i + 1N+I-J — lj-i with the sequence {77} definedby

3 17o = 0, 7 j = 5 3 T - l n j ( j - 1 , 2 , . . . ) , (3.1.5)

t=i z

and 7j converges as j —> 00 to Euler's constant 7 ^ = 0.577216... Equation(3.1.4) shows that the mean time Etj until the number of susceptibles isreduced to j — 1 (j = N,..., 1) can be approximated by the inverse of thelogistic rate (see Exercise 3.1); Williams (1965) and Bartholomew (1973,Chapter 9) give further comparisons. See Exercise 3.2 for varti when / = 1.

3.1.1 Analysis of the Markov chain

We now analyse the Markov chain {X(t) : t > 0} by the usual methods,given that the only non-zero transition probabilities, for 1 < i < N, are

Pr{X(t + St) = i - 1 I X(t) = i}= /3i(N 4- / - i)St 4- o(6t),Pi{X(t + ft) = t I X(t) = i} = 1 - p%(N + / - i)St - o(6t),


with X(t) = 0 being the absorbing state. Then the forward Kolmogorovequations for the state probabilities pi(t) = Pr{X(t) = i \ X(0) — AT},(0<i< iV), are

pN = -/3NIpN(t),Pi = -0i(N + / -

(o.l.o)

where p = —. In matrix terms, for P(t) — (PTV(£), - • • ,Po(t))', P = —-Tr-ot dt

equals( NI

-NI (N •-(JV-1)(I + 1) (AT-2)(7

-1) Oj

PN-l(t)PN-2(t)

Po(t)

so thatP = -0AP(t). (3.1.7)

Here the square matrix A of order N + 1 is the difference of two matrices,one diagonal and the other subdiagonal:

A = diag(JV7, (N - 1)(7 + 1) , . . . , 1(7 + N - 1), 0)- subdiag(AT7, (N - 1)(7 4-1),. . . , 1(7 + N - 1)).

Note that both the diagonal and subdiagonal matrices may have repeatedelements, as in Example 3.1.1 worked out shortly.

The solution of this matrix equation (3.1.7) is known to be

P(t) = e - ^ P ( O ) , P(0) = (1,0, • • •, 0)' = eN+1, (3.1.7')

but explicit results may not be easy to derive by this method. A muchsimpler approach is through the Laplace transforms pi(0) = /o°° e~etpi(t) dt,Re(0) > 0, 0 < i < N which, applied to equations (3.1.7), yield

fJoIJo

- p dt = 0pi(O) = -0i(N + 7 - i)pi(0)

4- f3{i + \){N -f 7 - i - l)pi+i(6), (1 < i < N - 1),f°° _0tdpo

Jo dt °


We can express equations (3.1.8) in matrix terms as

0A)P = eN+x (3.1.9)

where //v+i is the unit matrix of order N -f 1, ejv+i is the (N + l)-vectordefined at (3.1.7') and P is the vector of Laplace transforms pt(0) of theelements pi(t) of P. Equivalently,

PN(0) =0NV

We can solve for Pi{6) recursively, starting with PN(6) to obtain the result

N

where some factors Q-\-j3j{N+I—j) appear twice for pi(6) with z < ^(

Example 3.1.1. To illustrate the procedure, suppose that (N,I) = (4,1),13=1. Then

4 ' iy"v ; (0 + 4)(0 -f 6) ' Fzy > ~ (6 + 4)(0 -f 6)2 '

(fl + 4)2(0 + 6)20 '

Note the double factors in pj{6) for j = 0,1,2. Inverting these Laplacetransforms shows that for all 0 < t < oo,

6 ( e e 2 t e ) ,= 36( - e"4< + te~4t + e"6i + te"6'), ( j

= 1 + 27e~4t - 36ê~4t - 28e~6t -

For details of similar calculations see Bailey (1963; 1975, Chapter 5).


3.1.2 A simplifying deviceIt will be obvious that the fact that some of the eigenvalues A = j(N+I—j)(0 < j < N) of the matrix A in (3.1.7) are repeated, increases the complexityof the solutions for Pj(t). In the case of Example 3.1.1 where

A = diag(4,6,6,4,0) - subdiag(4,6,6,4)

the solutions Pj(t) involve both e~4t, e~m and £e~4t, te~6t. Suppose theinitial number of susceptibles is not an integer N but the quantity N€ = N+cwhere 0 < e < 1; make X(t) = e an absorbing state for the termination ofthe epidemic. Then the matrix A becomes

Ae = diag(4-fe,6 + 2e,6 + 3e,4-h4€,0)-subdiag(4H-e,6-|-26,6 + 36,4-h4€).(3.1.12)

The eigenvalues are now all distinct, so there is a solution of the form

p)(t) = aje-W + bje-^2t + Cje-(2+e^ + d3e~Â\where for simplicity j denotes the state X(t) = j -f- e. Elementary calcu-lations starting from Pe(t) = —A ePe{t), with Pe(-) much as at (3.1.7), leadto

e - < 4 + * p%{t) =

(6 + 2c)(4 + e) ( ) J '(6 + 3c)(6 4- 2c)(4 + c)

Pi W ^-7-: (2 + 2e)3e (2-f2c)(2-c)e-(6+2c)t _ e -

(3.1.13)

Recovery of the solutions at (3.1.11) for e = 0 is a straightforward matterof taking limits. By inspection, as e —• 0, p\(t) —•> e~4* and p|(£) —>2(e~4i — e~6t). In the case of p|W> taking limits for the last term yields

lim e~6t(e~3€t — e~2et)/e = —te~ 6*

so that- e ) - te~6t).

Similarly we can check that p\(t) —> pi(t) as at (3.1.11), and then appeal tope

0(t) as at (3.1.13) to complete this derivation of the full solution (3.1.11).We use this simplifying device later when considering the probability

generating function method in Section 3.2. The solution (3.1.13) for small6 can be considered as a close approximation to the exact solution (3.1.11).


3.1.3 Distribution of the duration timeA quantity of some importance mentioned earlier is the duration time t\ ofthe epidemic (see Figure 3.1); it equals the sum

N

Tj (3.1.14)

of N independent exponential variables, each with its own parameter

The Laplace-Stieltjes transform2 of Tj equals

so that if t\ has the p.d.f. /i(£), then

jff .-AM* =This p.d.f. is related to the state probabilities pj(-) at (3.1.7) by

fx(t) dt = px(t) /3(N + / - 1) d«, (3.1.16)

so for example when AT = 4, I = 1 and ^ = 1 as in Example 3.1.1,

/ i ( t ) = 4 x 36[-e~4 t + te"4 t + (1 + *)e~6'],

and thus t\ has the distribution function

Fx{t) = 1 + e~4*(27 - 36t) - e"6t(28 + 24t). (3.1.17)

Kendall (1957) used the transform at (3.1.15) to obtain the limit distri-bution of î in the case /?= 1, / = 1, N —> oo. Using the transformation

W = (AT + 1)*! - 2 In N, (3.1.18)

2We distinguish between the Laplace-Stieltjes transform E(e~0X) — J°° e~0x dF(x)of a random variable X with distribution function F(-), and the Laplace transformJo e~^tp(t)di of an integrable function p(-). They coincide when F has a p.d.f. /(•).

3.2. Probability generating function methods for Markov chains 63

it follows from (3.1.15) that, with /(•) denoting the p.d.f. for W,

• - « - ) * - " ""II ( > ^ "- jf

}2ft (l + ") "1 [T(l + 0)}2 (N - oo); (3.1.20)

here Euler's formula r ( H - z ) = zT{z) = Iim7v—oo N z f l ^ i W + {zlJ)] l (e&-Whittaker and Watson, 1927, p. 237) justifies taking the limit. Inversion ofthe Laplace transform [F(l 4- 6))2 leads to

f(w) = 2e~wK0{2e-w/2), (3.1.21)

where Ko(-) is the modified Bessel function of the second kind of order zero(see Exercise 3.3).

This limit distribution for t\ is interesting as an example of a sum of in-dependent non-identically distributed r.v.s having a non-Gaussian limit dis-tribution. This is not surprising, for the components in the sum at (3.1.14)are certainly not uniformly asymptotically negligible. These componentscan be grouped into three phases, so that the sum t\ consists of a startingphase of the epidemic where the first few Ti,T2,... have means that areO(l/iV), then a middle phase consisting of most of the N r.v.s Tt, when iis not close to either 1 or AT, with means that are O(l/iV2), and an endingphase with the last few T/v,T/v_i,... when the means are again O(l/N).Daley, Gani and Yakowitz (2000) use this three-phase decomposition of t\.An analogue for the general epidemic is indicated in Exercise 3.13.

3.2 Probability generating function methods for Markov chainsAn alternative way of studying a Markov chain {X(t) : t > 0} on the non-negative integers as state space, is through the analysis of the probabilitygenerating function (p.g.f.)

N

<p(z,t) = E(zxM) = Y^ P f { * ( 0 = 3 I X{fy}zj (M < 1), (3.2.1)


where 0 < X(t) < N. It is usually possible to obtain a partial differentialequation (p.d.e.) for (p(z,t) from the Kolmogorov equations for the prob-abilities for X(t), at least when the transition rates are simple polynomialfunctions of the non-negative integers as is often the case in models of pop-ulation processes. In some cases the p.d.e. admits an explicit solution. Weconsider the Markov chain for the simple epidemic of Section 3.1 to illustratethe method; we shall use the technique again in Sections 3.3 and 3.6.

We use the forward Kolmogorov equations (3.1.6) for the probabilities ofX(t) taking values in 0, . . . , AT, but first simplify them by changing the timescale so that the new time t' = (3t. Then since

equations (3.1.6) are replaced by the same relations with (3 = 1 and t'in place of t. For convenience below, we write t = tf. Now multiply theequation for dpj/dt by zj and sum over j . Writing

£(«)* ' , with v>(z,0) = ^ and ^ M = £ &&j=0

this gives

j=o

where PJV+I = 0. Simplifying the sums yields

(3.2.3)

If we differentiate (3.2.3) with respect to z and let z | 1, we deduce that

^ ^ = E(X(t)[(X(t) - 1]) - (N + 1 - l)EX(t), (3.2.4)

so with & = EX(t),

Recall from (2.1.1) that for the deterministic model of a simple epidemic,with the change of time scale used here, the function x(t) denoting thenumber of susceptibles at time t satisfies

x = —x{N + I — x).

3.2. Probability generating function methods for Markov chains 65

Consequently, since x(0) = £o = ^(0)> EX(t) « #(£) so long as vanX(t) issmall relative tox(N+I — x). But, it should be noted that the deterministicresults do not necessarily approximate the mean of the stochastic model wellfor all t.

We can solve the second-order p.d.e. at (3.2.3) by the standard methodof separation of variables. This means writing <p(z, t) = Z(z)T(t) for somefunctions Z and T, in which case (3.2.3) becomes

-I-l)— = -X say, (3.2.5)

where= dZ(£) „ = d2Z(z) . = dT(t)

dz ' dz2 ' dtand A is a constant. The simpler relation at (3.2.5) gives T = —AT, sothat T = e~At. However we know from Section 3.1.2 that this result will becorrect only if the initial number of susceptibles N is replaced by Ne = N+e(0 < e < 1) in the eigenvalues of the matrix A of (3.1.7). Assume this isdone: then from the rest of (3.2.5),

z{\ - z)Z" - (1 - z)(Ne + / - \)Z' -\Z = 0. (3.2.6)

This is a hypergeometric differential equation of the general form

z(\ - z)Z" + [c - (a + 6 + l)z]Z' - a&Z = 0,

where c = 1 — (ATe + / ) , a + 6 = — (iVe -f / ) , a& = A. A suitable solution of(3.2.6) is known to be the hypergeometric function

Ziz) -[Z) ^ r(a)r(b)T(c+k)k\

The eigenvalues Xj must be such that F(a, 6; c; 2) is a polynomial in z ofdegree not greater than iV, which means that a or 6 must be a negativeinteger such that a + b = —(N f + /) is satisfied. Thus the Aj must be of theform

A, = j(Ne + / - j) (0 0M) = ^ a i e ~ J ' ( N e + 7 " J ) ^ ( - ^ i - AT, - / ; 1 - iVe - / ; z), (3.2.8)


where the coefficients a7 can be obtained from the initial condition

N

(p(z,0) = zN = J2aJF(~JJ - Ne - 7; 1 - N€ - 7; z)3=0

and certain orthogonality conditions satisfied by the functions i?(-,-; *; z).If we now let e —» 0, so Ne —> iV, we recover the exact solution to (3.2.4).Bailey (1963) gives full details of this solution in the case 7 = 1.

Clearly, while this result may be satisfying analytically, it is not easyto use in practice. Nevertheless p.g.f. methods can be useful, especially incondensing equations from which summary information like moments canbe deduced.

3.3 The general stochastic epidemic

We consider next an S-I-R model in which the total population N + Iis subdivided into X(t) susceptibles, Y(t) infectives and Z(t) immunes orremovals, with (X,F,Z)(0) = (AT, 7,0) and

X(t) + Y(t) + Z(t) = N + I (all t > 0). (3.3.1)

We assume that {(X, Y)(t) : t > 0} is a bivariate Markov process; (3.3.1)ensures that Z(t) = N + I — X(t) — Y(t) is known when (X, Y){i) is known.We again assume that there is homogeneous mixing so that in the timeinterval (£, t + 6t) infections occur at rate /3ij6t and removals at rateyielding for the infinitesimal transition probabilities

Pr{(X, Y)(t + 6t) = (i - 1, j + 1) | (X, Y)(t) = (ij)} = pijSt + o(6t),)(t + St) = (ij - 1) | (X,Y)(t) =

, y)(t + « ) = (i, j) | (

where /? is the (pairwise) infection parameter and 7 the removal parameter.Then if we write the state probabilities as

Pij(t) = Pr{{X,Y)(t) = (i,j) I (X,Y)(0) = (AT,/)}, (3.3.2)

we can readily derive the Kolmogorov forward equations in the form

(0 < i + j < N + 7,0 < i < N,0 < j < N +1),

3.3. The general stochastic epidemic 67

subject to the initial conditions PNI(0) = 1, Pij(O) = 0 otherwise. Note forlater use that the distribution {Pn} of the ultimate size of the epidemic,meaning, the distribution of the number of initial susceptibles ultimatelyinfected, is given by

Pn = Pr{ lim (X, Y)(t) = (JV - n, 0) | (X, Y)(0) = (N,I)} = P7v-n,o(oo).t—+00

This distribution is computed most efficiently, in terms of time, numericalprecision and range of values of N + / , via an embedded jump process: seeSection 3.4.2 and Figure 3.2.

Some slight simplification is achieved if we define tf — /3t so that d/dt =/3d/dt' much as before at (3.2.2). With p = 7//? as the relative removalrate, the equations (3.3.3) become (again writing t = t1 for convenience)

^ | W = {i + 1){j _ i ) p s + 1 J . 1 _ j{i + p)pij

(0 )It. Sucha procedure is tedious (see Exercise 3.4), and it is preferable to use a sys-tematic approach such as one based on a mixture of Laplace transform andp.g.f. methods. Such solutions were offered around the same time by Gani(1965, 1967), Siskind (1965) and Sakino (1968): we outline Gani's solutionof 1967.

It is worth remarking that these solutions are possible here, essentiallybecause the processes concerned are Markov chains with well-ordered samplepaths. The states (i,j) for {(X, Y)(t) : t > 0} have an hierarchical structure,with the value of X declining by single units, and the value of Y increasingor decreasing by single units also. The process is 'strictly evolutionary':once a state is visited and left, it cannot be revisited, so, each state (i,j) isentered either once or not at all. The forward Kolmogorov equations (3.3.3)relate the derivative of any given state probability, to state probabilities forthat state and the states immediately preceding it along any well-orderedsample path. See also Section 3.4.2 and Exercises 3.6 and 3.7.


3.3.1 Solution by the p.g.f. methodMultiply the respective equations in (3.3.4) by zlw^ and sum them over i andj . Then for the p.g.f. (p(z, w; t) of the bivariate Markov process {(X, Y)(t) :t > 0} denned by the sum

£ (x, r)(o) =0<i<N

we obtain the p.d.e.

with <p(z,w;0) = zNw1. Differentiation of (3.3.5) with respect to z or wfollowed by putting z = w = 1 yields (in the original time scale)

-0cov(X(t),Y(t)),

Pcov(X(t),Y(t)),

respectively. Neglecting the covariance terms and putting (x(t),y(t)) =(EX(t), EY(t)) yields the d.e.s. of the deterministic model of Section 2.3.

Equation (3.3.5) is more simply studied in terms of its Laplace transform/•OO

(p(z, w;0)= / e~etip{z, w; t) dt (Re(8) > 0)(3 3 7)

7(M)= E E0<i<N 0<j<N+I-i 0<i<N

where

Taking Laplace transforms of (3.3.5) yields

0<p(z,w;O) - 2 W =

, Pn(0)= e-etPij(t)dt.

Substituting for <p(z,w;9) from (3.3.7), we deduce the set of equations

p)w~p}^,dw , (3.3.8)


These equations can expressed succinctly in matrix form as

A — + 0F = u/e7v+i, (3.3.9)

where F = (f^ / /v-i • • • /o) ' and e^+i is the (AT -f l)-row vector with 1 inits first row and zeros elsewhere, as at (3.1.9). With IJV+I denoting the unitmatrix of order N + 1, the square matrix A = A(w) of order N -f 1 equals

—p(l — W)IN+I +wdia.g(N,N — 1 , . . . , 1,0) — u>2 subdiag(AT, N — 1 , . . . , 1)= -pljv+i 4- wA'(O) + \w2A"(ti). (3.3.10)

Since each component of the vector F is a finite series in w, F has a finiteTaylor series expansion

N+I jF(w;0) = Y^ — FiJ\O;0), (3.3.11)

where F^^(0; 0) is the jth partial derivative of F{w; 0) with respect to w atw = 0. Then setting w = 0 in (3.3.9) we have for / > 1

orF'(O;0) = (0/P)F(O;0). (3.3.12)

Differentiating (3.3.9) with respect to w yields

[A'(w) + dlAr+^F'^jfl) 4- AHP"(ti ; ;f l) - Iti/êAr+r, (3.3.13)

setting t/; = 0 and rearranging gives

F"(0; 9) = - ([A'(0) + 91N+1}F'(0; 6) - I6ueN+1), (3.3.14)

where ^1 / is the Kronecker delta and A'(0) = diag(A^, TV - 1 , . . . , 0)+pIN+1.Differentiating (3.3.13) leads to

A"(w)F'(ti;; 9) + [2A'(w) + 9IN+1]F"(w; 9) + A(w)F'"(w; 9)' 2 (3.3.15)


so setting w = 0 and rearranging as before, we have

F'"(0; 0) = - [[2A'(0) + 0I,v+i]F"(O;0) + A"(0)F'(0;0) - .I1*2'pi (1 — 2)1P1

(3.3.16)In general, further differentiation of (3.3.15) with respect to w followed

by setting w = 0 and rearranging, shows that F(J'+1)(0; 0) equals

P\(3.3.17)

These equations yield a first-order recurrence relation when expressed inthe matrix form

f Fk+1)(0;6>n9 { F<»>(O;0) J

A'(O) + «Af+i (J2)A"(0)^j f FO)(0 ;e) j I\6jt (eN+1)

plN+i 0 J lpW-D(0;^)J ( / - j ) ! I 0 J'(3.3.18)

this relation is valid for j = 0 , 1 , . . . , N + I when we define F<0)(0; 9) =F(O;0), F<- 1 ) (0 ;«)=0.

We see that for j = 0 , . . . , / - 1,

where for k < j we define the product Jl^fc ^* = EjEj-i • • • Bfe+iBfc. Forj = / , however, we have

and for j = 1 + 1 , . . . , / + AT,

r F C + ^ O ^ ) 1 _ r r B . f F(o;*) | /! r ew+1)I F(')(0;») J ~ l = l B l I 0 J 7 1 0 J ' (3-3-19)

i—0

F(O;0) | /! A f e w + 1 )0 J ~ 7 11 Bi [ 0 J (3-320)


Since

(FF

Hence,

p(JV+7+l)(()

AT+7+l)(0;e)^N+/)(0;6>)

i=0

)

0

= 0, it follows that

)) \ / ' N+I (

[ iV+7 -i y, r iV+7 -I

n B i j 1F ( O ; 0 ) = 7 [ n Bij

where [*]JV+I denotes the truncated (N + l)-square matrix consisting of thefirst TV + 1 rows and columns of its argument. Now from (3.3.18), becausejA'(Q) -f 0IJV+I is a diagonal matrix and A"(0) is a subdiagonal matrix,both with non-zero elements, any product n i = o ^ ls a l ° w e r triangularmatrix with non-zero eigenvalues for Re(0) > 0. Hence its inverse exists,and thus

(3-3-21)

Thus, a complete solution of the general stochastic epidemic, in the sense ofdescribing the Laplace transforms of the state probabilities of (X(t),Y(t))for finite t > 0, is given by

w,9) ) _N¥f1w* ( FW(O;0) \«; 9)dvj- j ^ J ( PW-D(O; 9) J

F(w,9) (3-3"22)

WJ In*-1* 1 fF(0; e)) T ' W J n [n^1 B 1 f ejv+!Z . f P J [ o J .

with Bi as defined at (3.3.18) and F(O;0) given by (3.3.21).Example 3.3.1. Gani (1967) illustrates this solution for the simplest case(iV, /) = (1,1), from which it is clear that the solution is not easily calculatedin practice for larger N or / . First we have

J),


The matrices B; are given by

( 9 0 0 00 0 0 0p 0 0 00 p 0 OJ

so the required products are

(6(1

P0

0p60

f l + p + 00900

0P

-2p0

p2BiB0 =

p0(l0

According to (3.3.21) this yields

P(O;0) = [B2B1B0]^1[p-1B2]2e iV+i

0p0

0 0 0p + 0 0 0

0 0 0p 0 0J

0 0- 2 00 00 0

0 00 00 00 0

) 0 00 00 00 0J

p + 6){2 + 2p + 6){p + 0){2p + 6)6{p + 0){2P + 6) 0

2p6 0(l + pP

) ( 2 + 2p + 0^0)) { 0 J1 . (3.3.23)

The full solution to the 2-person epidemic may now be obtained from(3.3.22) as the upper left 2 x 2 matrix in

so F(w; 9) equals

0

8(1 +p + 6)2

)(3.3.24)


This Laplace transform solution is more than enough to indicate thatthe form of the time-dependent solution to the problem of describing theevolution of a general stochastic epidemic is algebraically formidable, atleast with pen and paper (see also Exercise 3.4). Fortunately a compositepicture of such stochastic development is possible; we concentrate on theultimate behaviour of the epidemic process, first with the analogue of theKermack-McKendrick threshold theorem, and then with a description ofthe ultimate size of the epidemic.

3.3.2 Whittle's threshold theorem for the general stochasticepidemic

One may well ask whether there is a stochastic threshold theorem similar innature to the Kermack-McKendrick theorem for the deterministic generalepidemic (part (ii) of Theorem 2.1). We start by considering the case, firstpublished by Bartlett in 1955 (around equations (19) and (20) of §4.4 ofthe first edition of Bartlett, 1978), in which the susceptible population N isvery large, so that

Pr{(X, Y)(t + St) = (i - 1, j + 1) | (X, Y)(t) = (i, j)} « NjSt + o(St)instead of the exact ijSt -f o(6t), and

Pr{(X, Y)(t + St) = (ij - 1) | (X, Y)(t) = (ij)} = pjSt + o(St).We write Y(t) for the marginal process of the number of infectives thatsatisfies these approximate relations exactly, observing that such a {Y(t) :t > 0}, with Y(0) — / , is a birth-and-death process with rate parametersN for births and p for deaths. The p.g.f. of Y is given by

^ pe^-P^jz - 1) - (Nz - p) V-l)-(NZ-P)\ 1 *» (3_3_25)

1 - pt(z - 1)Note that the result for N = p can be readily derived from that forN^pby setting N = p + e in this case and letting e —> 0. We are interestedin the probability of extinction of Y(i), i.e. in limôo Pr{Y(t) = 0} =limôo </?(0, t). Prom the first of equations (3.3.25) we see that the be-haviour differs between N > p and N < p, namely

< * < " > ")>r=f-i]imip(0,t)= ' " *~ ~ ' N


When N = p the second equation in (3.3.25) yields the limit 1, which isequal to the limit of either of the expressions in (3.3.26). Thus, for N < p,\imtôo "Pr{Y(t) = 0} = 1 so that the epidemic outbreak is likely to besmall. On the other hand, for N > p, the limit is positive but < 1, sothat the outbreak may be either small or large. Whittle (1955) gave a moreprecise analysis of Bartlett's approximation; the essential step is to boundthe number of infectives Y(t) in the actual epidemic process between twobirth-and-death processes like Y(t).

For any £ in (0,1) we can ask whether the intensity of the epidemic,meaning the proportion of susceptibles who are ever infected, exceeds £. Tothis end define

TT(C) = Hm Pr{X(O) - X(t) < NQ = 2 ^ Pn (3.3.27)

with Pn the probability that the final size of the epidemic is n, as givenbelow (3.3.3). We can bound the component Y{t) in the epidemic process{(X, Y){t) : t > 0} by F(f) in two bivariate processes

{Y(t), U(t) = I -f X(0) - X(t) : * > 0},

where Y(-) is a birth-and-death process with birth parameter either Ai = N,as in (3.3.25), or A2 = N(l — C), and with death parameter \x\ — /i2 = p.Note that £/(£) counts all the individuals who have ever been infected up totime t, including the initial / infectives. Write

Pjk(t) = Pr{(Y,U)(t) = (j,k) I (Y,U)(Q) = (1,1)},

and

Now the forward Kolmogorov equations for the birth-and-death process withgeneral birth parameter A are

-£jT = XU ~ l)Pj-i,*-i - (A + p)jPjk H- p(j 4- l)Pj+i,fc,(0 < 3 < N + /, / < k < N + / ) ,

defining pjk = 0 when (j, k) lies outside the permissible range. This leadsto the p.d.e.

^ = [Xz2w -f (A H- p)z + p] ^ , (3.3.28)


with the initial condition (f(z,w]0) = zNw1. Equation (3.3.28) can besolved by classical methods. The characteristic equations for Lagrange'smethod yield

d(f _ dt __ dz _ dw~0~ ~ T ~ Xz2w + (\ + p)z + p ~ ~0~'

Denote the two roots of the quadratic equation Xz2w + (A + p)z + p = 0 inz by 771 (w), f/2(w)> then

where for 0 < w < 1, rji > 772 > 0. The general solution is thus of the form

<p(z,w;t) =

where g is a function which can be found from the initial condition

Write ^ = 2—^-, so that z = ^ f^ 2 , implying thatT)\ — Z £ 4" 1

It then follows that

^,»; *) = -7 [^j-^tr 1""2u4"^ '^ '• (3-3-30)V y L {z - r)2)e-Xz{m-m)t + (m -z) J v ;

As t —> 00, the asymptotic distribution of the total number of individualsultimately infected is given by

lim <p(l,w;t) = w*4(w) = [ ^ ^ ] ( 1 - \/T=~^)/, (3.3.31)t->oo

where K = 4Xp/(X -f p)2. We now expand this relation, using Lagrange'sexpansion for the function tp(s) = s7, where s(w) = 1 — y/1 — KW canconveniently be given as the root S(K) of s2 — 2s + KW = 0 that satisfieslim^ô s(w) = 0. Note that

s(2 -s) sw = L — — .


thus using the formula

for Lagrange's expansion, we see that this leads to

PJX) = lim Pr{U(t) = I + n}= lim Pr{X(0) - X(t) = n}t—*oo t—*oo

r(2n + J - l ) ! Xnpn+I <M-1\n\(n + I)\ (A + p)2«+/ ^ n ^ v V> ( 3 3 3 2 )

where {Pn(X)} is the probability distribution for the total number of initialsusceptibles ultimately infected, for a specific birth parameter A.

Now we know from (3.3.31) that for N —> oo,

lim < ,1; t) = V Pn(A) = i ( A ) (P < A)' (3.3.33)U (p>A),

so that

f ; [ ( ^ ) ] / (3.3.34)n=0

Thus for TV large enough, in the case of an epidemic of intensity £, withbounding values X\ = N > X(t) > A2 = N(l — C) for the birth parametersof the birth-and-death processes involved, we have approximately

n=0 n=0 n=0

or equivalently

(3.3.35)

We can draw the following conclusions from this result.

Theorem 3.1 (Whittle's Threshold Theorem). Consider a general epi-demic process with initial numbers of susceptibles N and infectives I, andrelative removal rate p. For any C in (0,1), let TT(C) denote the probabilitythat at most [NQ of the susceptibles are ultimately infected i.e. that theintensity of the epidemic does not exceed £.

3.4. The ultimate size of the general stochastic epidemic 77

(i) Ifp < N(l - <), then

(ii) IfiV(l-C) <P<N, then

In both these cases there is a probability approximately equal to 1 — (p/N)1

that the epidemic achieves an intensity > C for small (.(iii) If p > N, then TT(Q = 1, and the probability that the epidemic achievesan intensity greater than any predetermined £ in (0,1), is zero.

3.4 The ultimate size of the general stochastic epidemicVarious approaches have been used to find the distribution {Pn} of the ulti-mate size of an epidemic, where in the notation used in and below (3.3.2-3),

Pn=limGPN_nj0(t) (3.4.1)

is the probability that n of the initial susceptibles become infected at somestage during the epidemic. When there are no more infectives, there aretherefore n + / removals including the / initial infectives. In this sectionwe indicate two methods of finding this distribution, one as an illustrationof the p.g.f. technique of Section 3.3, the other using an embedded randomwalk technique that is much superior numerically, in terms of speed, numer-ical accuracy and population size that can be conveniently handled. Thismethod is essentially due to Foster (1955). We give a third method, basedon another embedded Markov chain, in Section 4.5, while other methodshave been used by Bailey (1953) and Williams (1971), as outlined in Bailey(1975).

3.4.1 The total size distribution using the p.g.f. methodFor a Markov process on countable state space in continuous time, thelimiting probability TT that the process is in state i can be computed fromthe Laplace transform Pi{9) of the probability pi(i) that the process is in iat time t. The Abelian property

ni = lim pi(t) = lim 0pi(6),t—•oo 0—»0


where pi(6) = /0°° e~etpi(t)dt, yields this probability TT,. In terms of thegenerating function fi(w,8) of transforms defined at (3.3.7), this meansthat, since for all i, \imt-+ooPij(t) = 0 for j =£ 0. then

Pn = lim 0pN-n,o(0) = lim 6fN-n(0,0).B—*0 0—>0

Thus

Pi

{PN)

= lim 6

f IN (0,9)

I fo(O,0) )

= lim0F(O,6>), (3.4.2)

where F(0,0) is given by (3.3.21). From that equation, recalling that

Bo=

we see thatiV+I / + Nn BA 6F(0,0) =I\\ T\ Bi

whence, taking the limit as 0 —> 0, we find

AT+I

1 / lim 0F(0,0)

nN+l

0

(3.4.3)

liV+l

Thus, P = lim^_,o "(0^ 0) equals the right-hand side here pre-multipliedby the inverse of the matrix product on the left-hand side, where as notedabove (3.3.21), this matrix product is non-singular, with a unique inverse.The following example indicates what is involved in using this matrix resultin practice.Example 3.4.1. We compute the distribution of the ultimate size of theepidemic in the case (AT, /) = (3,1) with p = 1. Setting

= i4 =

1 J

A"(Q)-2A = -2

(03 0

2 01 0)


the products appearing in (3.4.3) on the right- and left-hand sides are,respectively,

(AA -12A) (3A -6A] (2I / 0 J I / 0 J { I

2A -2A0

= 24(A3 -AA- AA),

3A -6A0

\ ( 4A -12A ) f 3[ I / 0 J { I

= 24(,44 -

Hence, after some calculation,

W 2A -2A \ ( A 0 ^J ( I 0 J [I Oj

- AA2 - A2A + A2).

256-11160

81-382

• >

16-7 1,

— x f 64 1-210

> 0 >

( 0.2500 >0.08330.1042

w 0.5625 >

p =

Observe the presence here of both positive and negative terms. In largermatrices this is a source of numerical instability, assuming that there is thecapacity to handle the larger size (e.g. square matrices that are O(100)).

3.4.2 Embedded jump processesWe commented below (3.4.3) that properties like the size3 of the generalstochastic epidemic are most readily studied from a practical point of viewby using an embedded jump chain technique. Aspects of this technique canbe found for example in Todorovic (1992, §8.7); we rehearse here, for thesake of notation and general convenience, some properties of these processes.

Processes are often modelled as Markov chains because these incorporatein a simple manner dependence from one time point to another. This is truefor phenomena in both continuous time (as in the earlier part of this chap-ter) and discrete time (as in the next chapter). In the case of continuoustime, a further class of Markov chains can arise as embedded processes. Byconcentrating on the jump points or other epochs of a continuous time pro-cess which does not necessarily have a Markovian structure, we may derivea discrete time process which is Markovian (see Section 4.5 for an exam-ple). For a Markov process in continuous time on a countable state space,

3The 'total size' of an epidemic usually includes the initial / infectives, whereasthe 'size' (or, 'ultimate size') of an epidemic usually designates the number of initialsusceptibles that are (ultimately) infected.


the natural process to consider is the one embedded at the jump epochsof the process. Except for certain pathological cases which do not arise inthe models we have considered, the continuous time and embedded jumpprocesses both have the same long-term behaviour, such as the developmentof major and minor epidemics, and the ultimate size of the epidemic. Thissection reviews briefly some results of this kind. The particular usefulnessof embedded jump processes is that they are well-suited to numerical work.

To make these ideas explicit, we use {X(t) : t > 0} in this subsection todenote a Markov process in continuous time on the countable state spaceX = {i,j,...} having time-homogeneous transition probabilities with rates(Qij)- We assume that X is determined by its initial state and the rates q^(jf ^ i) and qi = Yli^j Qij ~ ~Q™ (^J ^ ^0- One dichotomous classificationof states is that any state i is either absorbing or non-absorbing, dependingon whether qi — 0 or qi > 0.

Given a sample path for X for which X(0) = i, the time 7$ = inf{£ > 0 :X(t) ^ i} is well-defined, finite or infinite, with T* < oo a.s. if and only ifqi > 0, i.e. i is non-absorbing. Then Ti is an exponentially distributed r.v.with mean l/q{. Indeed, supposing without loss of generality4 that the sam-ple paths are right-continuous with left-limits, these paths can be describedby a sequence of intervals of lengths T^n\ with epochs tn determined by thepartial sums

(n = 0, l , . . . ) . (3.4.4)

For these,X(t) = in (tn<t< tn + 1, n = 0,1,...), (3.4.5)

for some sequence of states i0, t ' i , . . . , it being understood that if any suchin is an absorbing state the associated interval length T^ is infinite andtr = oo for all r > n + 1. We can now define the embedded jump chain by

{Xn} = {X(tn + 0):tn< oo}, (3.4.6)

with one-step transition probabilities for non-absorbing states i

Pij = Pr{Xn+1 = j | Xn = i} = ^ . (3.4.7)Qi

Also, for all sample paths for which Xn = in, it follows from the strongMarkov property that T^n\ when finite, is exponentially distributed withmean l/qin.

4To be more precise mathematically, suppose that the process is separable, etc.


Markov chains with well-ordered sample paths are easily studied usingthis formalism. Define TT = Pr{X(t) = j for some t | X(0) = i). Thenthe forward Chapman-Kolmogorov equations for the jump chain give therelations

Kik =Pik+ Yl KijPjk' (3.4.8)

In this equation, all quantities on the right-hand side are positive; this is thereason the equation is the basis of a relatively stable numerical procedurefor the computation of the probability

Pik = Pr{ lim X{t) = k | X(0) = i} (3.4.9)t—KX)

of reaching the absorbing state k, starting from the state i.We can also study first passage time r.v.s like

tik = inf{t > 0 : X(t) = fe; X{u) = k for some u > 0 | X(0) = i}. (3.4.10)

The well-ordered property of sample paths, i.e. their strictly evolutionarynature, means that, given a sample path from i to k that passes throughsome intermediate state j , which will necessarily exist if the one-step tran-sition probability pik = 0, then

tik=Uj+tjk. (3.4.11)

Define

Tik = E(tik | X(0) = i, X(u) = k for some u > 0). (3.4.12)

Taking expectations over appropriate sample paths in (3.4.11) and usingthe same forward decomposition as in (3.4.8) gives

] T (7rijTijpjk-\-7rijpjkTjk)= ^ îjPjk{Tij + 1/QJ). (3.4.13)3-P3k>0

More complex relations for higher moments can be developed similarly (seeExercise 3.8).

3.4.3 The total size distribution using the embedded jump chainIt follows from the development of the general stochastic epidemic model inSection 3.3.1 that an embedded Markov chain which is conveniently denoted


Pn

0 . 0 1 5 -

0 . 0 1 -

0.005 -

0 I I I I0 20 40 60 80

PT

0.015 -

0 .01-

0.005 -

00 50

I100

I150

(a) (b)Figure 3.2. Final size distributions of epidemics with p = 150 and(a) (TV, /) = (100,1), (b) (iV, /) = (200,1).

by {(Xn, Yn) : n = 0 ,1, . . .} takes place on the finite region XNj of the two-dimensional integer lattice defined by

For (i, j) egiven by

XNi = {(ij) : i = 0 , 1 , . . . , AT; j = 0 , 1 , . . . , / + N - i}.

, the only non-zero one-step transition probabilities are

f3ij + 7 j13

(3.4.14)

'J ; 0ij + yj i + PConsequently, writing p{iJ) = Pr{(Xn, Yn) = (i,j) for some n \ (X0,YQ) =(-/V,/)}, it follows from (3.4.8) that

P(t+i,i-i)Pt+i H-P(i,i+i)(l -Pi) 0' > 2),

where on the right-hand side we set P(ij) = 0 if (i, j)

Pd o) = Pr{epidemic is of size N — i\ = lim Pio(tv ' ' t-+oo

(3.4.15)

. In particular,

= PN-I (3.4.16)

with Pio(t) and Pn as defined in (3.4.1).Figure 3.2 illustrates most of the distributions of the size of general

stochastic epidemics for which p = 150 and (N,I) = (100,1) for (a) and(200,1) for (b), illustrating sub- and super-critical behaviour respectively.Some probability masses near the origin are off the scale used; in case (a)their values at 0, . . . , 5 are 0.600, 0.145, 0.070, 0.042, 0.029, 0.021, and incase (b) at 0, . . . , 4 they are 0.429, 0.106, 0.052, 0.032, 0.022 respectively.


0 . 0 8 -

0 .06 -

0 .04-

0 .02-

0 -(

1) 20

. . • • * *

140

160

1 °80

TN-n,0

0.08-

0.06-

0.04-

0 . 0 2 -

50I

100I

150

(a) (b)

Figure 3.3. Conditional mean durations of epidemics with p = 150 and(a) (N,I) = (100,1), (b) (N,I) = (200,1). Abscissae give the number ofsusceptibles infected. The means were not computed for large n for whichPn = pjy-n.o < 10~9. Cf. Exercise 3.13 and Daley et al. (1999).

We have also computed the conditional mean durations of these epidemicswith /? = 1 for those final sizes for which the probabilities exceed 10~9. Inthe notation of (3.4.12), we have found ri0 = T(N,i),(i,o) for those i for whichP(i,o) > 10~9. These conditional means satisfy the recurrence relations, for{i, j) € XNI, starting from r^v/ = 0,

U >and

where fiij = l/[j(p + i)] = 1/q^j) for the state (i,j) in the notation of(3.4.13). They are illustrated in Figure 3.3 for the same two epidemics asin the previous figure.

3.4.4 Behaviour of the general stochastic epidemic model: acomposite picture

We now draw together these and other results to describe in general termshow the general stochastic epidemic model behaves, and point out some ofits weaknesses.

First the Threshold Theorem distinguishes between sub- and super-crit-ical conditions. In the case of a sub-critical epidemic, the numbers of infec-tives and removals behave roughly like live and dead individuals in a sub-critical birth-and-death process. This is borne out principally by Whittle's


Threshold Theorem 3.1. Both the duration and final size of the epidemicprocess are roughly the same as for the approximating birth-and-death pro-cess.

The super-critical case is more involved. Here it is possible that only aminor epidemic occurs, with probability approximately (p/N)1. The size ofsuch a minor outbreak is similar to that of a super-critical birth-and-deathprocess conditional on extinction. Prom the work of Waugh (1958), thisis exactly that of a sub-critical birth-and-death process whose linear birth(death) rates are the linear death (birth) rates of the original super-criticalprocess respectively. Otherwise, the early behaviour of the epidemic pro-cess is similar to that of a super-critical birth-and-death process conditionalon non-extinction, until the number of infectives represents an apprecia-ble proportion of the initial number of susceptibles. If the deterministicmodel of the epidemic is any guide, the evolution of the epidemic processthrough a number of generations of infectives is still similar to the birth-and-death process (see Figures 2.7 and 2.8 and Exercise 3.5), until the numberof susceptibles approaches the borderline value p between sub- and super-criticality. At this stage, in terms of numbers of susceptibles and infectives,the process follows approximately the solution of the deterministic epidemicmodel (with E X(t) « x(t) etc., much as for the simple epidemic; cf. Exercise3.1). The 'imprecision' of the approximation, which in model terms incor-porates both the stochastic lag and variability about the mean behaviour(~ deterministic solution), is approximately normally distributed if N andp < N are both sufficiently large. This leads (Barbour, 1972, 1974) to theultimate number of susceptibles being approximately normally distributedabout the mean EX(oo) « NO, where NO is the number of susceptiblessurviving the deterministic epidemic, with a variance that can be computedusing Kendall's 'Principle of Diffusion of Arbitrary Constants' (cf. Daleyand Kendall, 1965, and Section 5.4 below).

If it happens that Ro = N/p is somewhat larger than 1 (e.g. 3 or more),then, conditional on the occurrence of a major outbreak, the number ofsusceptibles surviving the epidemic is approximately Poisson distributed,still with mean NO where 0 is now much closer to 0 (cf. Daniels, 1967).This number can be regarded as representing those individuals having a'small' chance of avoiding infection. If NO is itself not close to 0, then thecloseness of a Poisson distribution to normality ties together this result withthe previous paragraph. Sellke (1983) proved Daniels's result by consideringa sequence of general epidemic models with parameters (Nk, Ik, pk). Theseparameters are such that as k —> oo, all three diverge to oo in such a way that

3.5. The general stochastic epidemic in a stratified population 85

—> M, say, for some finite positive M. Then the number ofsusceptibles Xfc(oo) ultimately surviving the epidemic is shown to convergein distribution to the Poisson with mean M.

This ultimate size distribution is considered in more detail for the differentcases of N and p in Nagaev and Startsev (1970) (see also Nagaev, 1971)using the embedded jump chain. The most general formulation of suchresults may be found in Reinert (1995).

In the limit theorem setting for these results, the duration of the epidemicmay ultimately be exceedingly short; this indicates a need for rescaling timeor else reviewing the conditions under which the model may apply. Whenone considers the observation of measles epidemics in large cities as reviewedin detail by Bartlett (1960), it becomes clear that the time- and generation-dependent evolution of the models need modification if they are to reflectthis aspect more realistically. Fortunately, both the stratification models ofthe next section, and the chain binomial models of the next chapter, offeravenues for overcoming the drawbacks of the overly simple S-I-R model.

3.5 The general stochastic epidemic in a stratified populationThis section starts by describing a continuous time Markov process modelthat extends the ideas of Section 3.3 to a stratified population, consistentwith the deterministic setting of Section 2.4. In that setting we discussedwhat criterion should be used to distinguish a major from a minor out-break; in the stochastic setting two further factors need to be considered,namely the discreteness of the population, and the stochastic modelling ofthe infection and removal processes. These factors lead us to a digressionon the notion of the Basic Reproduction Ratio of an epidemic model. Weconchide the section with an example of a stochastic model for an epidemictaking place in a community of households, a problem to which we returnin Chapter 7.

A stochastic analogue5 of the general deterministic epidemic of Section2.4 assumes that a finite population of susceptibles is stratified into mgroups of sizes N = (TV^,..., ATm)', where as in Section 2.4, vector nota-tion is convenient. At time t — 0 suppose a single infective is added to

5Daley and Kendall (1965) remark that the 'natural' formulation of a populationprocess at the micro-level is discrete and (more than likely) stochastic. From this pointof view, a discrete stochastic model is the prototype of which the deterministic model,with both continuous state space and deterministic (non-stochastic) evolution, is theanalogue or approximation. As Barbour (1994) puts it, a deterministic model 'only hasa meaning as an approximation to some underlying jump process'. Anderson and May(1991) consider deterministic models almost exclusively.


stratum i. At time t > 0 the j t h group consists of Xj(t) susceptibles,Yj(t) infectives and Zj(t) = Nj + 6ij - Xj(t) - Yj(t) removals, so that{Xj.Yj.ZjÔ) = (Nj.Sij.O), (j = l , . . . ,m) , with 6{j the Kronecker delta.We use a continuous time Markov process to model the infection and re-moval mechanisms, analogous to that used earlier in this Chapter. As inSection 2.4, faj denotes the pairwise rate for an infective in i to infect a sus-ceptible in j , and jj the removal rate per infective in j . In vector notation,the property of constant sizes of the several strata is expressible as

X(t) + Y(t) + Z(t) = N + U (all t), (3.5.1)

and the initial condition is that (X, Y, Z)(0) = (N, li , 0), where the m-rowvector li has 1 as its i th component and all other elements zero.

The process {(X,Y, Z)(£) : t > 0} takes place on a finite state space,so it is completely specified as a continuous time Markov process when itsinitial state and infinitesimal transition probabilities are given. We know(X,Y, Z)(0) already, and the constraint (3.5.1) means that it is enough todescribe (X, Y)(t). We assume that for the strata i = 1, . . . , ra, with x andy denoting m-vectors of non-negative integer-valued components,

<5*) = ( x - l i > y + 1,-) | (= * i ( £ £ i A j ! f c ) « + o(6t) - Xj(y'B);«t 4- o(St) {x3 > 1),

Pr{(X,Y)(t + ft) = ( x , y - 1,-) | (X, Y)(t) = (x,y)}= *yjyj6t + o(St) {yj > 1),

Pr{(X,Y)(t + ft) = (x,y) | (X,Y)(t) = (x,

(3.5.2)where B = (Aj)- We can now write down the forward Kolmogorov equa-tions, but they are too unwieldy to give an explicit solution, whether inthe time or transform domain. We have described this model here because(a) it is easily studied via simulation techniques, (b) it is general enough tocapture inhomogeneity, and (c) it provides an adequate setting to discussthe basic threshold phenomenon.

We first consider a method of approximating the behaviour of the process,entirely consistent with Whittle's approach in Section 3.2. This approacheffectively keeps the size of the susceptible population fixed during the earlystage of the epidemic, so that the only changes occurring in the populationare in the numbers of infectives Y say. In the m-strata context we there-fore approximate the process by a multi-type linear birth-and-death process


), in which at time t there are Yj(t) (j = I,..., ra) j-type individuals, forwhom the birth and death rates are respectively YALI 0ijYi(t)Nj anc* lj%\provided no Nj is small these birth rates differ little from YllLi Ajî WÔ Wfor t small enough (cf. (3.5.2)). This process Y(t) is an irreducible multi-type Markovian branching process in continuous time. For such processesit is known (e.g. Athreya and Ney, 1972, §V.7.5) that the vector q whosei th element

qi = Pr{ lim ^ ( t ) = 0 | Y(0) = 1*} (3.5.3)

gives the probability of ultimate extinction, is the least positive solution ofthe equations

YfjLiPtjNjqiqj + 7i = QL'jLiPijNj + li)Qi (i = 1 , . . . , ra). (3.5.4)

Now qi = 1 (all i) is always a solution, but need not be the least positiveone. Also, the process is irreducible if and only if the matrix B is primitive,meaning that for some integer n, which is at most equal to ra, B n has its(z,j)th element positive. Because of irreducibility, either <ft < 1 (all i) orelse qi = 1 (all i), and the former holds if and only if

B diag(N) - diag(7) (3.5.5)

has a real positive eigenvalue; this is equivalent (see Exercise 3.9) to R > 1where R is the largest eigenvalue of the non-negative matrix

Mi = diag(7~1)Bdiag(N). (3.5.6)

In the one-dimensional case equation (3.5.6) gives R = f3N/j = N/p =RQ, and the condition R > 1 is exactly the condition in Whittle's ThresholdTheorem 3.1 for a major outbreak. Thus, we have indeed generalized thatresult to the ra-strata context.

For birth-and-death processes, as for branching processes and randomwalks, conditions for behaviour like extinction or return to the origin, aretypically given both probabilistically and in terms of first moments. Ap-plication of the latter method to epidemics makes use of the Basic Repro-duction Ratio. This is defined for epidemics in homogeneous populationsas 'the average number of persons directly infected by an infectious caseduring its entire infectious period, after entering a totally susceptible pop-ulation' (Giesecke, 1994, p. 111). This definition formalizes the notion thata major epidemic will occur when, in the earlier stages of an outbreak of adisease, each infective on average produces more than one further infective.

3. Stochastic Models in Continuous Time

In particular, in the simplest case (/ = 1) of the general epidemic modelof Section 3.2, the initial infective is placed in a population of N suscep-tibles where it mixes homogeneously during its infectious period T, withT exponentially distributed with mean I/7. Then the number among theN initial susceptibles contacted (and therefore, potentially infected), is abinomial r.v. Bin(iV, 1 — e-/3T). Unconditionally therefore, if susceptiblesare affected only by the initial infective until its removal, the mean numberof new infectives directly contacted by the initial infective equals

* ^ ,3.5.7)P

where i o is as below (3.5.6), and the approximation is valid when p > 1,i.e. (3 <C 7. While this condition is stated in terms of p rather than TV, thecase of super-criticality (i.e. RQ > 1) then implies that N ^$> I. The analogyof the criticality condition, that Ro < or > 1, with that for Galton-Watsonbranching processes, is immediate.

To apply this idea to the m-strata context, we refer to the initial infective,in stratum i say, as a zero generation z-stratum infective. Let m^ denotethe expected numbers of first generation j-stratum infectives (j = 1,. . . , m)to which it gives rise. We ask whether these various first generation in-fectives in different strata, if each is placed in the same environment of Nsusceptibles as the initial infective, would produce on average a larger num-ber of second generation infectives, and so on. Now the expected numberof /c-stratum second generation infectives produced by an initial i-stratuminfective, on the assumption that each first generation infective is placed inthe same initial population of N susceptibles, equals Y^JjLi rriijmjk, i.e. the(i,/c)th component of the matrix M2 where M = (rriij). Continuing thisprocess, the expected numbers of j-stratum infectives after r generations,for r not too large, equals the (i,j)th element of the matrix M r. SinceM is a non-negative matrix (indeed, it is primitive because B is primi-tive) , Perron-Frobenius theory implies that each column of M r is like RQVfor large r, where Ro is the largest eigenvalue of M and v is the right-eigenvector corresponding to Ro. It follows that the expected number ofinfectives after r generations either dies away to 0 when RQ < 1, or elsegrows exponentially for a while when RQ > 1, reflecting the possibility of amajor outbreak.

To identify M, note that an i-stratum infective is infectious for an expo-nentially distributed time Ti with mean 1/7^ If the N susceptibles were nototherwise subject to infection during this period T , the number of j-stratum


infectives produced would be a binomial r.v. Bin(iVj, 1 — e~^T%). Hence,unconditionally, the expected number m^ of first generation j-stratum in-fectives produced by an z-stratum infective is given by

mil - m i - .-*.«) = NS(1 - ^ - ) - M4 , (3.5.8)

the approximation again being valid for 0ij <C 7*. The matrix M is

M = (f^) « ( ^ ) = diag(7-1)Bdiag(N) = Mlt (3.5.9)

with Mi as at (3.5.6). It follows that RQ « R, and we should call RQ theBasic Reproduction Ratio for the epidemic. This terminology is that ofDiekmann, Heesterbeek and Metz (1990) who noted that it is a dimension-less quantity (hence, ratio rather than rate or number), which measures'reproductiveness'. The discussion concerning Examples 3.5.1-2 supportsour contention that this moment-based definition for Ro is more appropriatethan R, defined in more probabilistic terms around (3.5.6). We summarizeour discussion thus far.

Theorem 3.2. A stochastic epidemic in a stratified population such thatno component of N is small, with infection rates faj and removal ratesjj as for (3.5.2), exhibits sub- or super-critical behaviour (i.e. only minoroutbreaks occur, or major outbreaks may occur) according as the BasicReproduction Ratio RQ is < or > 1, where RQ is the eigenvalue of largestmodulus of M = (m^).

The birth-and-death continuous time approximation, leading to Mi, andthe generation-wise approach leading to M, give similar but not exactlyidentical conditions for distinguishing between sub- and super-criticality.Under the first approach the matrix Mi is computed after a single jumpof the process whereas contributions to the matrix M arise from both thissingle jump and possibly several subsequent jumps. Because of this, ap-proximations based on M are better than those based on Mi.

Stochastic analysis of the model defined by (3.5.2), such as the computa-tion of the ultimate size of an epidemic, is complicated by the dimensionalityof the state space needed to provide a Markovian description of the process.Clearly, one possible description is the (2m)-dimensional process (X, Y)(t).If the size of m is appreciable, this may be too large for exact computation.When the parameters fiij and 7 have a simple structure, the analysis canbe correspondingly simplified, as in the following Examples.


Example 3.5.1 (m interacting communities). Consider a population whichconsists of m communities of Ni individuals (i = 1,. . . , ra), all of whom aresusceptible to a disease which, if introduced, spreads by pairwise contact atrates (3u = (3H (all i) and fa = (3C (all (z, j) with i ^ j), where (3H > /3C

(typically, @H ^ Pc)- Removal of infectives occurs at a common rate 7per infective. Suppose that a single infective is introduced into the ithcommunity. Using the approximations at (3.5.9), the Basic ReproductionRatio that gives us a threshold criterion is the largest eigenvalue of them x m matrix Mi with elements

Thus, #0 is the largest positive root of the characteristic equation /(A) = 0where

/(A) = detf}HN2h-\

= det

with Nj =

/(A) =

00

(3HCN2-\0

X 0 0 .. .and /3Hc = PH — Pc- Expanding the determinant,

i =2- A) J$0HcNi - A) + /3CN2(/3HCNI ~ A) HiPncNi - A)

i=3m-1

+ • • • + PcNm(f3HCNi - A) F J (J3HCNi - A)

i - A)) \ P+

This expression is a polynomial of degree m in A; we require the largest Asolving /(A) = 0. By inspection we see that the m-fold product is finite


and non-zero for A > max; PncNi, SO any zero of the second factor in thatregion is a candidate for Ro. Now this second factor is strictly increasing inA for max* 0nc^% < A < OO, ranging from —oo at the left-hand end of thissemi-infinite interval to -f 1 at -foo, so

y ( )

fe i - Xhas exactly one zero there. Therefore Ro equals this zero, i.e.

Ro> {3HCmaxNi. (3.5.13)i

Each term on the right-hand side of (3.5.13) is increasing and convex inso by Jensen's inequality

mfoAT/jr-RfaN/j' K '

where N = Y1T=\ N»/ra; equality holds if and only if either 0H = Pc orNi = Nx (all i). Hence, when 7 > (3H > (3c,

Ro > \PH + (m - $PcWh, (3.5.15)

with equality holding as for (3.5.14).When the fiij and 7 have the structure of this Example but without

the condition 7 » max(f3ci PH), the approximation given in (3.5.8) shouldnot be used. In place of /(A) = det(Mi — A/), we must instead identifyRo with the largest root of det(M — XI) = 0. Exercise 3.10 details thechanges to (3.5.13-15) that then ensue. For example, if /?# = 1 = 7,/3c = 0.001, m = 101 and all JVj = N\, (3.5.15) used as an approximationgives Ro « 1.1 Ni whereas the analysis in Exercise 3.10 gives Ro = 0.60(Wi.

We also emphasize that the discussion and example above are based on theassumption that all components Nj of N are 'large'. To illustrate what mayhappen when this assumption is false, suppose that (3H — 100, 0c — 0.0001,and the other parameters are those just considered. Using Exercise 3.10yields Ro ~ l.OOOliVi, which we interpret as N\ new infectives within thehousehold containing the initial infective, with only a small expected num-ber of infectives in other households. Because the household sizes are small,the disease must be propagated outside the household of the initial infec-tive for a major outbreak to occur. The expected number of such outside


contacts is at most about O.OlJVi per infective per generation, as againstabout iVi infectives produced within the household of any such infective.So in two generations there may be up to about O.OlATf outside infectivesproduced per initial infective, corresponding to O.liVi new infectives pergeneration. Consequently, until N\ is about 10 or more, no major outbreakcan be expected (see also Exercise 3.11), contrary to the interpretation thatfollows from Theorem 3.2 when Ro > 1 which holds for all JV~i > 2. Thiscontradiction stems from assuming that all Nj are large. Fortunately wecan rework the analysis on the basis of distinguishing newly infected in-dividuals, not by the stratum or household to which they belong, but bythe number of susceptible individuals in the same household who remainsusceptible after the initial infective is removed, before the influence of anysecondary infectives.Example 3.5.2 (A community of m households). Assume the model ofthe previous example, where the strata are now households. Suppose agiven infective is initially in a household with i susceptibles, and call it anz-type infective. Assume that the initial infective is infectious for a time T,and that during this infectious period, only the initial infective can makesusceptibles infectious. At the end of this infectious period, identify each ofthe individuals now infected by the number j of susceptibles remaining in itshousehold. Then the probability that j susceptibles remain in the householdof the initial infective at the end of the random time T is PijiPn), where

( 3 5 1 6 )

(see Exercise 3.11). For any other household where there are initially isusceptibles, the probability that j susceptibles remain after the randomtime T is Pij((3c)- Regard the household of the initial infective as beingadditional to the initial population of N susceptibles as described; then theexpected number rrtij of j-type infectives produced by this initial i-typeinfective is given by

(3.5.17)r=l

say. Define M = (m^). Then using the same argument as above (3.5.8),the expected number of second generation fc-type infectives (i.e. infectivesin households with k susceptibles) equals the (i,/c)th element of M2. We


conclude that, with n' = maxi<r<m(7Vr — 1), the Basic Reproduction Ratio.Ro for the process is the largest eigenvalue of the (p! + 1) x (n' + 1) matrix

M =

^Hm0 m\

mii +mi

. . . mn' \

VTL2

^ / o ( ^ i ^ 2

(3.5.18)Explicit computation of this eigenvalue does not appear possible in general.We can compute the limiting value of Ro when /3H > 7 because thenp^ ^ fijo a nd i?o is the largest root of /C(A) = 0, where

/c(A) = det

mo — A1 + 7Tio2 + mo

m i vri2i — A rri2m i m 2 — A

= det

r mo — A m i rri2 - •. rnn>1 + A - A 0 . . . 02 + A 0 - A . . . 0

in' - A

n' 0 0 . . . - A J

say.

Thus, when /?# > 7,

The expressions for A and f? implied by (3.5.19) reduce to

A =

because' + 0c '

A =

(3.5.19)

(3.5.20)

(3.5.21)

where terms in the inner summation are zero for j > ATj, and the innersum equals the expected number of infectives in a given household of iV»


susceptibles produced by a single infective outside the household. IntroduceNi identically distributed indicator r.v.s Ik (k = 1 , . . . , Ni) to denote thatthe kth susceptible in the household is infected. Then EIk is just theprobability that this kth individual is infected before the initial infectivebecomes a removal, so Elk = /?c/(7 +A?)» and the expression for A follows.Similarly,

in which the inner sum is expressible in terms of the same indicator r.v.s as

E

where k ^ k1. Using properties of exponential r.v.s, we see that E(Ik'Ik) =2/?c£c/[(7+2/?c)(7+/3c)]. Collecting terms together then leads to (3.5.21).

To illustrate the relevance of the assumption that (3H ^> 7, suppose that/3H = 1 = 7, /3C = 0.001, m = 100, TV* = Nx = 3 (cf. the example in thetext below (3.5.15) where we should have Ro — 1.8 from Exercise 3.10).Then from (3.5.18), det(M — XI) — 0 is a cubic equation, with largestroot RQ = 0.7864, whereas the approximation at (3.5.20) which depends on0H ^> 7, gives J o ~ 0.9369. The same approximate Ro holds on changing/3H to 3, but the corresponding exact root 0.8992 is closer to it, and closerstill (Ro = 0.9318) for (3H = 9. See Ball, Mollison and Scalia-Tomba (1997)for related work.

3.6 The carrier-borne epidemic

Suppose that a population of X(t) susceptibles of initial size n is beinginfected by W(t) carriers of a disease with W(0) = b, who are themselvesimmune to infection but subject to a pure death process. Any infectedsusceptible is directly removed from the population. Then the stochasticprocess X(t) is subject to the influence of the process W(i) which is inde-pendent of X(t).

Let the death process {W(t) : t > 0} be a homogeneous Markov chain incontinuous time with death rate 7; then we know that its p.g.f. is

(ye'* + 1 - e"^)6 (\v\ < 1), (3.6.1)

3.6. The carrier-borne epidemic 95

so that

Pr{W(t) = fc | W(0) = b} =

andE[W(t) | W(0) = b) = fcT7* - tu(t)

where iu(£) is as in the deterministic carrier model of Section 2.6.To analyse the process {X(t) : t > 0}, assume that {(X, W)(t) : t > 0} is

a bivariate Markov chain with transition probabilities

Pr{(X, W)(* + St) = (i - 1, fc) | (X, W){i) = (i, fc)} = /3ifc6t 4-= (t, fc - 1) | (X, W)(t) = (i, fc)} = jh6t + o(6t),= (i, fc) | (X, W)(«) = (t, fc)} = 1 - (/?* + 7)* «* - o{6t).

The forward Kolmogorov equations for the process are given for 0 < i < n,0 < fc < 6, by

= P(i + l)fcpi+i,fc - ()dc

subject to the initial condition pn6(0) = 1, where

Pik(t) = Pi{(X,W)(t) = (i,k) I (X,W){0) = (n,b)},

and pn+i,fc = Pi,b+i = 0 when either i or fc lies outside its respective range.Use the same transformation tf = (3t as at (3.2.2), and set p = j/(3. Thenfor 0 < i < n, 0 < fc < 6, the forward Kolmogorov equations become

j j ^ i|fc - (t + p)fcpifc + p(fc + lKfc+i (3.6.2)

with the initial condition pnb(fy = 1- For convenience, we use t below forthis new t1.

We discuss two methods for solving this equation: the first is the p.g.f.method discussed earlier, while the second is based on a method of Puri(1975) for a Markov process developing under the influence of another pro-cess. Such a process could be described as a doubly stochastic Markovprocess by analogy with the previously named doubly stochastic Poissonprocess.


Referring to (3.6.2) we derive for the p.g.f.

, v, t) = E ( / W ^ W ) = Yl PikWzW (0<z,v< 1),0<i<n,0<k<b

the p.d.e.

subject to the initial condition (p(z,v,Q) = znv6.To solve this equation, assume that ip{z,v,t) = Z(z)V(v)T(t) and use

the standard method of separation of variables. Then from (3.6.3) we have,writing f = dT/dt, Z' = dZ/dz and V =

t Z1 V V

so that T = e~xt directly. We also see that

Z _ P ( l - 1 ; ) A V _11 Z)~Z~ v W'~~h ( j

where j is a quantity that we discuss shortly. These relations lead to

&___Z3_ V _ A

so that, apart from constants,

V -(-rNow we know that </?(z, v, f) must be a polynomial in z and v, of degree

n in z and 6 in v. Hence j must be an integer in the range 0 < j < n, andso too must 0 < X/(j + p) < b. Corresponding to a particular value of j ,the eigenvalue A will be

It follows that

n bY^'^/C', t / , LJ — 7 7 L-jf^Kj yL — Zl I C — ~ I , I O . D . U )


where we now need to find the values of the coefficients Cjk- We do this bysetting t = 0, when

<p{z,v,0) = znvb =j=0 fe=

Writing £ = 1 — z, 77 = 1; — p/(j + p), this reduces to

j=0 fc=0

whence Cj^, being the coefficient of £J7/fc in the expansion of the left-handside, can be identified as

and

It is easily checked that for z = 1 we recover y?(l, v, t) = [l -f- (v — l)e~pt] ,i.e. the p.g.f. (3.4.1) of the pure death process W(t), following the changein time parameter at (3.2.2).

The joint distribution of (X, W)(t) for any t > 0 can be found by expand-ing (3.6.8). Perhaps more important is the distribution of X(t) whose p.g.f.equals

(3.6.9)

It follows that

= i | X(0) = n} =

C::)[%]


and

For the deterministic model of Section 2.6 the analogous function is x(t),given at (2.5.4). Rewrite it, after changing the time-parameter to make itconsistent with the notation used in this section, as

x(t) = n exp ( —

Thenl l t T I T * i / l TDf* I r ^11x11 JU\V I — ILKZ \

t—• OO

where the inequality, a trivial consequence of ely/p > 1 + 1/p, shows thatthe expected ultimate size of the stochastic epidemic is smaller than theultimate size of the deterministic epidemic. More precisely, we can showthat limôo (E[X(£)] - x(t)]) « 6/2p2, which is o(l) for small to moderateintegers b and somewhat larger integers p. See Exercise 3.12 for more detail.

As in the case of earlier models, we are interested in the number of initialsusceptibles that are ultimately infected. For this, we can readily obtainthe probabilities P^ = \imt-ôo Pn-k,•(£) of the size k of the epidemic from(3.6.10), namely

= (I) B- 1 )^Let T be the duration of the epidemic and F its d.f. Then F(t) =

Pr{no further infection after t} is given by

F(t)=Po.(t)+p.o(t)-Poo(t),

where the terms on the right-hand side can be determined algebraically from(3.6.10), (3.6.1) and (3.6.8), respectively. Hence,

(3.6.13)


In the particular case 6 = 1 this expression simplifies and yields for example

E (r | vr(0) = U = j f [i - "Ml <•« = J - 1 (j) £ f

P Jo P Jo

(Bailey, 1975, p. 196). For integers b > 2, the complete beta integral hereis replaced by one or more incomplete beta integrals that can no longer besimplified.

The following, alternative approach, due to Puri (1975), relies on theobservation that, conditional on W(-)> ^ W 1S itself a pure death processwith the random, time-varying death-rate parameter j3W{-). This implies(e.g. Kendall, 1948) that

{W(u) :0<u<t}) = (ze~W^ + 1 - e -^*>) n , (3.6.14)

where W(t) = /0* W(u)du, the initial condition (X,W)(0) = (n,6) beingunderstood. Expanding (3.6.14) and taking expectations yields the p.g.f.

(3.6.15)

The problem now is to evaluate the expectations on the right-hand side.This can be done^via the joint Laplace transform of the bivariate Markovianprocess (W(t),W(t)),

fl{v, 0, t) = E(v wWe-^W) (M < 1, Re(6) > 0),

for which the corresponding probability terms are

qbk(u, t) = Pr{W(t) - jfe, W(t) 0} is a pure death process: we find


with qkk(u,0) = 1 (all u > 0). Now

subject to n(v, 0,0) = f6. Taking Laplace transforms, equations (3.6.16)become

1 an an _ p an p an0 a t " + v a^ ~~" 0 v & 7 + ^

orf = K« + P )« -P ] f • (3-6.17)

This equation can be solved by using Lagrange's method: write

dt dv d0 dAT ~ {0 + p)v-p ~ T ~ 0 '

whenceIn ((0 + v)p - p) = (0 + p)t -f A, ft = B,

where A and i? are constants. It follows that

ft(t;, e, t) = f{e-(e+^{{9 + p)v - p]), (3.6.18)

subject to fl(v, 9,0) = vb, i.e. vb = f([$ + p]v - p), so

This gives

f ( ^ p{1-;-{e+P)t))b (3.6.19)o + PSetting 0 = j for j = 0 , . . . , n and combining the resulting expressions as in(3.6.15), we recover (3.6.8).

Other examples exploiting this method can be found in Puri (1975).


3.1 The analogue of t3 at (3.1.3) in the deterministic simple epidemic is thesolution t'3 of the equation (cf. (2.1.2)) x{t'j) = j — 1, equivalently, y(t'j) —N + I + l-j. Show that

, 1 . N(N + / + 1 - j)3 /3(N + I)m I(j-l)


and hence deduce that when / > 2,

where JN,I,J is defined below (3.1.4). This quantity is negative, and is some-times called the stochastic lag. An alternative derivation entails the use ofthe approximation ]T^_ i~l ~ / _ i^ u~l û-

3.2 Show that in the special case of a simple stochastic epidemic starting withone infective introduced into a population of N susceptibles, the first twomoments Eii and varti of the duration t\ of the process in Section 3.1 arerelated by

1 A n 1 i»_ 2/?E«x +ig 2 +O(JV- 1 )1 /32(N + 1)2 ^ [ i N + l \

Observe that varti/(E*i)2 = [l/(iVlniV)](l + o(l)) for large AT.

3.3 (a) Check that for real s, the gamma function

/

OO /»OO

e-yy~ts dy = / e1M

J -OO

exp(-a: -

is the characteristic function of the extreme value distribution with p.d.f.exp(—x — e~x). Hence deduce that the asymptotic distribution of W at(3.1.18), having for its Laplace-Stieltjes transform the product of two gammafunctions, is the convolution of two extreme value density functions:

/•oo

(W,W + dw)} = / exp(-v - e'v) exp[-(w - v) - e"^'^] dv dw,J —oo

which reduces to the density at (3.1.21) (Kendall, 1957).(b) To explain why the extreme value distribution occurs in the limit distri-bution for the duration t\ of the simple stochastic epidemic of Section 3.1,take N even, N = 2M say, with 1 = 1 and show that t\ is expressible asthe sum Ui + U2 of i.i.d. r.v.s Uk =d X ^ i Si/[i(2M + l - t ) ] ( l b = 1,2),where Si, ft,., are i.i.d. unit exponential r.v.s. Deduce that each Uk liesbetween U'/M and U'/(2M), where U' is the r.v. U' = Y™ =1 Si/i. Showthat U' =d maxiîf^}, and that (M + \)Uk - lnM and (M + l)U' - lnMboth converge weakly to the same limit r.v. with the extreme value densityof (a). (See Kendall's paper for details.)


3.4 (See also Example 3.3.1.) For the general stochastic epidemic model of Sec-tion 3.3 starting from (N, I) = (1,1), and assuming that 0^% 27, use directd.e. solution methods to solve the equations p n = -(/3 + 7)pn, pio = 7Pn>p02 = pPll - 27/?o2, Poi = 27P02 - 7Poi, Poo = JPoi- For example, show that

7(1 -e^^) p ,F2 /*T

3.5 (Cf. Section 3.3.2 and Figure 2.4.) In a linear birth-and-death process {Y(t) :t > 0} with birth and death rates A and fi per individual, where X > fiand Y(0) — / , suppose that at time t there are Y3(t) jth generation livedescendants of the original / individuals and Zj (t) j th generation descen-dants no longer living (cf. Section 2.5). Show that the expected numbersyj(t) = EYj(t) and Zj(t) = EZj(t) satisfy the system of d.e.s

yj=\yJ-i- HVJ, ZJ=MJ 0 = 0 , 1 , . . . ) ,

provided we set y-i(t) = 0. Deduce the solution

J- o

so that m(t), the 'average' generation number of the {z3(t)}, is given by (cf.Figure 2.4)

u\ Xt X

Waugh (1958) proved that a super-critical linear birth-and-death processconditional on ultimate extinction, behaves like the birth-and-death processwith birth and death rates interchanged, and so is sub-critical. Use this resultto show that

jZ,(u) | HI ) - 0 (I - oo» =

Deduce that Pr{Y(t) —• 0} = (^/A)7, so that for the non-extinction set{Y(t) -+ ex)}, setting r = /i/A,

E[^.jZ3(u)\Y(t) -^ 00 (t 00)}

A [(A - fi)u - l][e ( A~û - r 7 e - ( A -^ ) u ] + 1 - r1

A — \X e(*-»)u _ rI-le-(\-n)u _ I + rI-l


3.6 For the general stochastic epidemic, show that the first passage time distri-bution into any state can be constructed as a weighted sum of convolutions ofdistributions of exponential r.v.s, where the components of the convolutionscorrespond to the sojourn times in the states visited along the paths and theweights are the respective probabilities of those paths.[Remark. The same result holds true for any continuous time Markov pro-cess on countable state space with well-ordered sample paths and boundedtransition rates. This algorithmic principle explains why the p.g.f. methodis applicable to the simple and general stochastic epidemic models.]

3.7 Show that if j is a non-absorbing state in a Markov process with well-orderedsample paths, then -K%3 = q3 /0°° Pr{X(u) = j \ X(0) = i}du (cf. (3.4.8)). Inparticular, for the general stochastic epidemic as in Section 3.4.3,

Pi*,,) = W8 + 7W / Pr{(X,y)(t) = (ij) | (X,Y)(0) = (N,I)}dt.Jo

[Remark. This principle, complementary to the previous exercise, underliesthe simplicity of the jump-chain algorithm for computing both intermediateand final state probabilities of strictly evolutionary Markov chain models.]

3.8 Using the same notation as at (3.4.10), show that the quantities

T™ = E(t2ik | X(0) = i,X(u) = k for some u > 0),

starting from rit = 0, satisfy the recurrence relation

=

3.9 Let the m x m non-negative matrix A be primitive, and let the diagonalmatrix D have all elements positive, much as in Section 2.4. Use Perron-Frobenius theory (e.g. Seneta, 1981, Chapter 1) as at Lemma 2 of Daley andGani (1994) to show that(i) the eigenvalue Amax of A — D that has largest real part, is real;(ii) the eigenvalue /imax of AD'1 that has largest modulus, is real and posi-tive;(iii) Amax <, = or > 0 according as /imax <, = or > 1, respectively.Suppose that the pairwise infectious rates fiio of Sections 2.4 and 3.5 have theproduct form 0i3 = frcty so that B = (0%3) = 0a\ where the vectors 0 anda have components /3t denoting the infectivity rate of infectives in stratum iand relative susceptibility of susceptibles in stratum j . Show that when Miat (3.5.6) is primitive, its largest eigenvalue is R = ^2iNlat/3t/ji. When isit true that R w RQ, the Basic Reproduction Ratio for such an epidemic?


3.10 In the context of Example 3.5.1, with the 0tJ as given there, show that thedominant eigenvalue i f o o f t h e m x m matrix M of (3.5.8), is as at (3.5.12)if we replace Nt by Nx, 0Hc by 0H/(0H + 7) - 0c/(0c + 7) a n d 0c by0c/(0c + 7). Deduce that when N3 = Ni (j = 2, . . . ,ra) (cf. (3.5.13)),

_ {[(m - l)/

Check that this is smaller than the case of equality in (3.5.15), which ap-proximates this expression if 7 ^> max(fc,fe).

3.11 Derive (3.5.16) using the embedded jump technique of Section 3.4.2. Showthat when 0 < 7, PtJ(0) « (0h)%~3(i\/j\)> Use (3.5.20) to conclude that inthe setting of Example 3.5.2, RQ > 1 when N% > l/(ra/fc/7); this conditionyields N\ > 4 when m = 100 and 0C = O.OOI7, and Ni > 10 when m = 100and /3c = O.OOOI7 (cf. the text preceding Example 3.5.2).

3.12 Use the generating function (3.6.8) to show that for the stochastic carrierepidemic,

r 1 - UE[X(t)W(t)} = E[X(t)} E[W(t)]e-* [l

1 + pand deduce that cov(X(t),W(t)) < 0 for 0 < t < oo. Use (3.6.11) toconclude that (d/dt)E[X(t)} = -E[X(t) W(t)] > -E[X(t)]E[W(t)]. Hence,since E|W(£)] = w(t) (see below (3.6.1)), conclude that for the deterministicand stochastic carrier models, the inequality

±(x(t) - E[X(t)]) < (x(t) - E[X(t)])w(t)

is satisfied. Since (X, VF)(0) = (x,ii;)(0), the right-hand side is zero at t = 0and thus is negative for all sufficiently small positive t, hence negative for allfinite positive t, and therefore

0 < E[X(t)] - x(t) T n[(l + p"1)"6 - e~b/p} (0 < t T oo).

[Remark. Alternatively, differentiation shows that when 6 = 1 , E[X(t)]—x(t)is monotonic in t; hence E[JV(£)] > x(t) (all £) for all positive integers 6.The covariance inequality is what we should expect intuitively, namely thatlarger-than-average numbers of carriers should result in smaller-than-averagenumbers of susceptibles.]

3.13 For a deterministic version of a super-critical general epidemic starting from(x,y)(0) = (AT, 1), use Kendall's approach as around equation (2.3.17) todetermine an approximate duration time too by too = inf{£ > 0 : y(t) — 1}.Then find x' = x(\too) and x" — x(\too). Simulate the general stochasticepidemic starting from (X,Y)0 = (AT, 1), and restrict attention to thosesample paths for which Aoo < x". Find empirically the distributions of thephases T\ T" and T'" defined by T = inf{t : Xt < x'}, T' + T" = inf {t :Xt < x"} and Tf + T" + T"' = \nf{t :Yt=0}.

Stochastic Models in Discrete Time

To describe the spread of a disease whose infectious period is relatively shortin comparison with the latent period, especially in smaller populations, it isconvenient to use a discrete-time model with the latent period as the unitof time. There are two such 'classical' models, one tracing back to Reedand Frost in 1928 (Abbey, 1952) and the other to Greenwood (1931). Bothentail sequences of random variables with binomial distributions, hence theterm 'chain binomials'. The fact that they are Markov chains was not fullyappreciated until the work of Gani and Jerwood (1971).

It should be apparent that in this setting it makes little sense to model asimple epidemic as we did at the start of Chapters 2 and 3; here the 'short'infectious period allows us to identify a class of 'former infectives' that arein fact removed. Any individual that becomes an infective, is thus infectiousfor exactly one unit of time. However, there can be mathematical reasonsfor considering such analogues, as for example in Daley and Gani (1999)(see also Section 4.6 below).

It becomes clear in this setting that there are successive generations ofinfectives. When the initial infectives are infectious around the same time,as is necessarily the case with a single initial infective, successive generationswill not overlap in time. The success of the Kermack-McKendrick theory inmodelling the spread of epidemics like measles realistically, can be attributedto the fact that there is a considerable number of smaller scale overlappingepidemics.

4.1 The Greenwood and Reed—Frost models

We use a discrete time unit that may be one or more days; when this unitcan be viewed as the latent period, it models the time between 'generations'of infectives as in the measles data of En'ko (1889) in Exercise 1.3. Let the

105

106 4. Stochastic Models in Discrete Time

numbers of susceptibles and infectives at time t = 0 , 1 , . . . in the populationbe Xt and Yt respectively, with Xo = x0 and Yo = yo > 1. Let p (forsome p € (0,1)) be the probability of contact between an infective and asusceptible, the result of the contact being an infection with probability f3for some (3 € (0,1). Let q = 1 — p be the probability there is no such contact.It is convenient to regard this contact as being instantaneous, so that theinfectious period is concentrated at the contact time £, for some integer t.The probability that there is no infection due to any single infective is

1 - p + p{\ - (3) = 1 - p(3 = a (4.1.1)

say; after contact with the susceptibles, the Yt infectives are identified andso removed. Note that Xt + Yt = Xt-i-

In the simplified Greenwood model (Greenwood, 1931), it is assumedthat the cause of infection is not related to the number of infectives sothat a can be regarded simply as the probability of non-infection. In thiscase the number of infectives yt at time t is determined by x t_i and xt asyt = xt-i - xu and

{X,Y)t = (x,y)t]

Hence, {X* : t = 0,1, . . .} is a Markov chain, with F*+i = Xt — Xt+\.In the Reed-Frost model, an individual susceptible at time t is still sus-

ceptible at time t 4-1 only if contact with all Yt infectives is avoided (morepointedly, if infectious contact is avoided). This event has probabilityay*, independently for each of the Xt individuals susceptible at t. ThusXt+i = B'm(Xt,aYt), where Bin(n,n) denotes a r.v. with the binomialdistribution {(n)?r^(l — /K)n~^j = 0, . . . , n } , and the matrix of one-steptransition probabilities has elements P(x,y)t,(x,y)t+i given by

W ( ) (4.1.3)xt+l/

It follows that a realization (x, y)o,(x,y)i,..., (x, y)r of the epidemic, whereyT = 0 < 2/r_!, will have a cchain binomial' probability of products ofthe form (4.1.3). Here the yt infectives behave independently, resulting ina probability ayt of non-infection and xt = xt+i H- yt+i- The yt+\ newinfectives will now mix with the remaining xt-\.\ susceptibles at time t + l

4.1. The Greenwood and Reed-Frost models 107

xt5 r

4 —

3

2

1Y2 ; = 0

= XA = 0, = 0- • *(f

Figure 4.1. Two sample paths of an epidemic in discrete time,ending at t = 3 ( ) and t = 4 ( ) respectively.

before being removed, to produce the next generation yt+2 of new infectivesat t + 2. Clearly

is a bivariate Markov chain with transition probabilities P(x,y)u(x,y)t+i °fthe binomial form (4.1.3).

The least integer t for which Yt = 0, when Xt = Xt_i or for which Xt = 0,determines the time T = t at which the epidemic terminates. For example,Figure 4.1 (copied from Figure 1.3) shows two possible realizations of achain binomial process; in one of them, Y2 > 0 = Y3 so T = 3 and in theother, F3 > 0 = YA so T = 4.

The probabilities in the Greenwood and Reed-Frost models are usuallydifferent, except for the case of households of two or three individuals witht/o = l (see Table 4.2 for households of size four). Bailey (1975, Chapter14) lists the probabilities for both models for household sizes of up to fiveindividuals (i.e. for XQ + 2/0 < 5) with different initial yo. These individuallyspecified probabilities are the basis of Bailey's (1955) and Becker's (1989)work on inference for epidemic models (cf. also Chapter 6 below).

The fact that the numbers of infectives in successive generations of a chainbinomial epidemic are determined from a binomial distribution, means thatthe expectations of these numbers are in principle readily computable. Inparticular, for a Greenwood model equation (4.1.2) yields E[Jf t+i | Xt] =aXt, so

E[Xt I Xo = x0] = a*xo,E[Yt I Xo = x0] = a'-iQ. - a)x0. { ' ' '

It follows from the first of these equations that EXt —> 0 as t —» oo, so


since Xt is bounded and non-negative, we must ultimately have Xt = 0 forall sufficiently large t. The other equation shows that the expected sizesof successive generations of infectives are geometric. The formulae for theReed-Frost model are not quite so straightforward (see Section 4.2).

4.1.1 P.g.f. methods for the Greenwood modelConsider the Greenwood model with Xo = XQ. Then the matrix P of one-step transition probabilities (4.1.2) is lower triangular of order xo + 1 andequals

11 - a a

2(1 - a)a

)*0-2a2 vô

(4.1.5)

l ( l - a ) x ° xo{l - a)*0"1** (x2°)(l-a)a

We have already noted that when 0 < a < 1, Xt = 0 for sufficiently larget. However, from an epidemiological viewpoint we are interested in the firsttime T when there are no infectives, and in the number W of susceptibleswho have been infected by then. Finding the joint distribution of (W, T)numerically is not difficult. Starting from p® = Pr{Xo = i} = 8Xoi, we have

ptj=Pr{Xt=j,Yt>0} =

where for any integers t > 1 and j = 0, . . . , i,

Then, using (4.1.6) to calculate p^Z^ recursively,

Pr{(W, T) = (A:, n) I Xo = i, YQ > 0} = pn~lpi-ic _fc = pn~1ai~k

We can also use p.g.f. methods to find the distribution of (W, T) as we nowshow. Recall that the epidemic stops for the least integer t for which Yt — 0,or equivalently, for the least integer t for which Xt = Xt-i. Partition thematrix at (4.1.5) as P = P + Q where Q = diag(l, a, . . . , ax°), so that

01 -a 0

p = (I-a)2 2(1-a)a 0

0J

(4.1.7)

4.1. The Greenwood and Reed-Frost models 109

is a lower triangular matrix of transition probabilities for transitions thatdo not allow any repetition, so that they are from some state {1 , . . . ,#o}always to a lower state within the same set. Q is the matrix of probabilitiesfor repetition of the state Xt, i.e. for the transition {Xt-\ = i} *—• {Xt = i}for some state i, and hence Yt = 0. Indeed, we readily see that

pr{T = t} = (t = 1), (4.1.8)

where A = (0,0, . . . , 1) is the vector of initial probabilities correspondingto PT{XO = xo} = 1, and E' = (1 ,1 , . . . , 1) is the unit row vector. Theprocess is allowed to make exactly £ — 1 downward transitions starting fromxo through intermediate states to finish at some state x0 — (t — 1),. . . ,0,without repetition, and then stops at time t when the state {Xt-i = i} isrepeated. The p.g.f. of the distribution is given by

t = i= A1 {I - 6P)-l0QE (4.1.9)

where we have used the fact that P-7 = 0 for j > x0 + 1.Example 4.1.1. Consider a Greenwood model for a family of three sus-ceptibles and one initial infective with the transition probability matrix

1 . .1 — a a

(I-a)2 2(1 -a)a a2

1(1-a)3 3(l-a)2a 3(1 - a)a2 a

The p.g.f. ^ T ( # ) of the duration of the epidemic equals

= (0,0,0,1)(/ 4- OP + 02P2 + 03P3) 0

= (0,0,0,1)

p)

' 11

f 1a

Q 2

3

1 .. 1

1

03a2(l-a) 0

0 .- a ) 3 6a3(l-a)2 0 0

1(4.1.10)


Hence

p r{T = 1} = a3, Pr{T = 2} = (1 - a)3 + 3a2(l - a)2 + 3a4(l - a),Pr{T = 3} = 3a(l + a)(l - a)3 + 6a4(l - a)2,

It is easily checked that these probabilities sum to 1.To find a p.g.f. for the size W of the epidemic, note that each time that

Xt decreases, the number of infectives increases by an equal amount. Thus,in one transition period in which Xt decreases, the matrix P(<p) of p.g.f.sfor the number of new infectives equals

00

2a(l - a)

(4.1.11)where the nth subdiagonal is the set of the equivalent probabilities inP multiplied by <^n, for n = l , . . . , xo- Similarly, P2(ip) has elementsPr{X*+2 — j I Xt — i} multiplied by (pl~i to reflect a total of i — j in-fectives at times t - f1 and t + 2. By using the same argument as underliesthe relation (4.1.9), it now follows that the joint p.g.f. of (W,T), where Wdenotes the total size of the epidemic, is given by

, 0) = A'{I - < 1, |0| < 1). (4.1.12)

Example 4.1.2. For the same Greenwood model for a family of threesusceptibles as Example 4.1.1, the joint p.g.f. ^W,T{V->8) equals

(0,0,0,1)1 . . .. 1 . .. . 1 .. . . 1

+ 0

00

00

(1 - a)V2 2a(l -a)(p 0( l - a ) V 3a(l-a)V 3a2(l-a)ip 0

0 . .2a(l-a)V 0 0 .

a)(l-a)V 6a3(l-a)V 0 0

0 0 . .0 0 0 .

l 6 a 3 ( l - a ) V 3 0 0 0(4.1.13)

4.2. Further properties of the Reed-Frost model 111

All the joint probabilities Pr{W = w, T = t] for w = 0,1,2,3; t = 1,2,3,4can be obtained from 4>W,T(<£S 0) above; for example for T = 3 we obtain

= 3, T = 3} = 3(1 - a)3a(l + a),= 2, T = 3} = 6(1 - a)2a3.

If we require only the distribution of the size W of the epidemic, it is clearthat

= (0,0,0, l)(J-P(</>)) - 1a2

= a3 + y?[3a4(l - a)] + v?2[3a2(l + 2a2)(l - a)2]+ <ps[(l - a)3(l -f 3a + 3a2 + 6a3)]. (4.1.14)

Thus we obtain the probabilities

Pr{W = 0} = a3, Pr{W = 1} = 3a4(l - a),Pr{W = 2} = 3a2(l + 2a2)(l - a)2,pY{W = 3} = (1 - a)3(l + 3a + 3a2 + 6a3).

These probabilities add to 1 as they ought.

4.2 Further properties of the Reed-Frost model

In the Reed-Frost model with Xo — x0 > t/o = Yo, the matrix of transitionprobabilities has elements as at (4.1.3). Whereas in the Greenwood modelit was enough to know Xt for each time-point, we must now keep track ofboth Xt and Yt, or equivalently, since Xt-\ = Xt + Yt, of {{Xt, Xt~\) : t =1,2,...}, which is a Markov chain if we set X-\ = X$ -j- Y$ — XQ -f- t/o- Ineither specification, the model is a bivariate Markov chain. In what followswe use the formulation in terms of (X,Y)t*

We start by presenting a deterministic analogue of the Reed-Frost model.For such an analogue we refer to equation (4.1.3) with the conditional bi-nomial distribution and readily compute

E[(X,Y)t+l ~ a"))- (4.2.1)


(a) (b)Figure 4.2. (a) Susceptibles and (b) infectives in successive generations ofdeterministic analogues of the Reed-Frost or non-overlapping Kermack-McKendrick epidemic, with (JV, /) = (100,1), a = e~1/p = 0.98.

These equations, unlike their Greenwood counterparts at (4.1.4), do notlead to an explicit solution, but their recurrent nature makes them amenableto numerical computation, results of which are shown in Figure 4.2. Thesame figure results from the generation-wise evolution of a deterministicKermack-McKendrick model with non-overlapping generations describedin Exercise 2.7 if we set p = — 1 / log a.

We may write the transition probability matrix for the P(x,y)t,(x,y)t+1 asthe partitioned matrix P , of order (x0 + I)2 x (x0 -f I)2 , given by

P =

PooPio

rx00

0Pn

0

(4.2.2)

Here each submatrix Pij is square, of order x0 + 1> with P{j = (p(k,i),(t,j))where we have from (4.1.3), for k, I = 0 , . . . , x0, and j + £ = k,

(X,Y)t - (M)} =

and P(k,i)t(i,j) = 0 otherwise, i.e. P^ records the transition probabilitiesfrom Yt = i to F£ + 1 = j , with values of Xt and Xt+i indexing the rows andcolumns respectively, ranging from 0 to x$. In particular, Poo = / , POj = 0for j = 1 , . . . , x0, and P^ has zero elements except on the j th subdiagonalwhere one term from each of the binomial distributions for Bin(fc,a*), for

00

- ayo (j-

00

0h l)a*(l -ay •••

00

00

4.2. Further properties of the Reed-Frost model 113

j < k < xo + 1 — j , appears: thus Pij equals

... o

... o

... o

... o: : '•• 0 ••• 00 0 ••• (^âô-rtil-ay ••• 0

Setting A! = (0 • • • 0 10 • • • 0) where the 1 occurs in the place corre-sponding to (x,y)o, it follows as in the previous section that the durationtime T of the epidemic has

Pr{T = t} = A'P^QE, E' = (1 1 • • • 1), (4.2.3)

where P = Q -f- P with Q the square matrix of order (XQ + I)2 comprisingthe transition probabilities into the absorbing states {(#, 0) : x = 0, . . . , XQ},so Q consists of the first x0 + 1 columns of P in (4.2.2) (i.e. of the xo + 1diagonal matrices PJO), and all other elements are zero. Then the non-zeroelements of P are the one-step transition probabilities into transient states(£,j), these being characterized by having the second component j > 0.

As in the previous section, T has p.g.f.

Similarly, the joint p.g.f. of (W, T), where W is the total size of an epidemicand T its duration, is

where P((p) equals the matrix P modified by having the elements in thej t h subdiagonal of its component matrix P^ in the partition at (4.2.2),multiplied by ^ .

It should be clear from this discussion that the principles for the use ofthe Reed-Frost and Greenwood models are identical, though the Reed-Frostmodel requires more computation.Example 4.2.1 (Measles epidemic data from En'ko (1889)). For theepidemics listed in Exercise 1.3 we can extract the following data for thesuccessive numbers of infectives in the several generations of the epidemics.En'ko's paper gives sufficiently detailed data to make this possible.


Table 4.1. Generation sizes in several measles epidemics in St PetersburgEducational Institutions 1865-84

EpidemicGeneration

012345

A

35228

1

B

11325124

C

22814

1

D

14

1410

10

E

13

1012*1

F

14532

G

1120

1

H

1011

*2

I

17045

2

J

499

2121

1

K

1136

* Vacation started

Epidemics A-H occurred at the St Petersburg Alexander Institute, I-Kat the Educational College for the Daughters of the Nobility. EpidemicG occurred in the Autumn after the Spring outbreak in Epidemic E. ForEpidemics A-H excluding G, it is reasonable as a first approximation toassume the same number of initial susceptibles. The numbers and time ofonset of the disease in relation to the vacation period suggest that we shouldexclude H also. The pattern of sizes in successive generations indicate thatEpidemics A, C, I and K may be consistent with a Greenwood-type model,and B, D, E, F and J with a Reed-Frost-type model.

We illustrate a type of fitting procedure on Epidemic B. Let the initialnumber of susceptibles be N; this is unknown, but must be at least as largeas 54, the total number of cases after the initial infection. Given the numberof infectives yt in generation t and the number of new cases zt prior to t (so,z0 — 0 and zt = y\ -\ ht/t-i), Vt has expectation (N-zt)(l— ayt~l). Onemethod of fitting a Reed-Frost model is to minimize the chi-square statistic

T-\ r / AT

t=iT-l

= £t=i

(N - zt)(l - a*-*)

-a*-)

where T == inf{£ : yt = 0} as before, with respect to both N and a, subjectto the constraint that N > J2t=i Vt = Nmin. The minimizing values N anda satisfy the first of the equations

4.3. Chains with infection probability varying between households 115

and the second equation as well if TV > Nmin. We found (N, a) = (54,0.794).For Epidemic D we found (iV,a) = (29.2,0.835), where now iVmin = 29.

4.3 Chains with infection probability varying between households

As noted earlier, the chain binomial models we have been describing were thebasis of data analyses by Abbey (1952), Bailey (1955) and Becker (1989),among others. One of the deficiencies of the simple models is that theassumption of similar susceptibility to infection for all families is invalid.This deficiency may be countered by allowing the non-infection rate a tovary between households; it is usual to suppose that the non-infection rate ais a random variable following a beta distribution, as this is mathematicallyand computationally convenient.

Suppose that within a household of Xo individuals the probability of con-tact in a unit time period is p and the probability of transmission conditionalon a contact is /?. It is arguable that both these probabilities should varybetween the individuals of a household, but such a level of detail may wellmake the model intractable. Consider again, as in (4.1.1), the probabilities

p/3 (of infection) and a = 1 — p/3 (of non-infection),

but assume now that a varies from one household to another: model a asa random variable with a density function f(a). Then for any particularrealization of a chain in a household, denoted (X, V)[O,T] — {(X,Y)t : t =0 , 1 , . . . , T}, with the epidemic ending at time T, the probability of the chaincan be denoted by

P(a) = P((X,Y)[0,n\a). (4.3.1)

Since a varies between households, we need to average P(a) over all possiblevalues of a on (0,1), yielding

/ P(a) / (a )da . (4.3.2)

Below, we assume that the density function /(•) is of the beta function form(1 - a)a~lab-l/B(a, b) for a, b > 0.

We illustrate the procedure for both the Greenwood and Reed-Frost mod-els with transition probabilities as given at (4.1.2) and (4.1.3). In the formercase these probabilities are of binomial form

Pij(a) = Pr{Xt+l =j\Xt=i}= (* ) (1 - af-'of (i > j).


Integrating over all values of a when a has the beta density above, wededuce that for integers i, j and general positive a, 6,

j\B{a + i-j,) B(a,b) ( 4

i\ (a + z - j - l)"-a(b-\-j-1)---6

Thus in a Greenwood chain where we need to record only the valuesXt = xt for t = 0, . . . , T, the probability P(a) of such a realization equals

x\J V XT /(4.3.4)

It follows that, averaging out over all values of a, we obtain

In a Reed-Frost model, if the realization has (Xt,Yt) — {xt,Vt) for t =0,1,. . . ,T, then

i l (^"^(l -a»-1)I'-1"x']aSLi«"-^ (4.3.6)

The last expression reduces to a sum of terms of the typical formJ2j Ajaj; taking its integral over all values of a yields

4.3. Chains with infection probability varying between households 117

Example 4,3.1. Consider both these models for households of four, x0 = 3,starting with a single infective t/o = 1 hi the Reed Frost case. Table 4.2lists the possible realizations and their probabilities under the two modelsin terms of a given value of a. Generic values are also given for use later inSection 6.3.2.

Table 4.2. Epidemics in households of size four with XQ = 3, yo = 1: possiblerealizations and their probabilities under two discrete-time models

t ={Xt} =

Realization033333333

132212210

2

2111000

3 '

1

000

-1]4

0

ModelReed-Frost

3(16(13(16(13(1-

a 3

-a)a4

- a)2a4

- a)2a3

-a ) 3 a 3

- a)3a2

3(1 -<*)3a(l + a)(1 - a ) 3

Greenwooda 3

3(1 - a)a*6(1 - a)2a4

3 ( l - a ) 2 a 2

6(1 - a)3a3

3(1 - a)3a2

3(1 - a)3a(1-a)3

Integration over a of any of these probabilities multiplied by the densityf(a) is readily effected. Only for the case where (Xo • • • X3) = (3 1 0 0) dothey differ between the two models.

In practical applications the parameter values a, b must be estimated fromthe data, as shown e.g. in Bailey (1975, Chapter 14) and Gani and Man-souri (1987). For this purpose, supposing M households of size four with#0 = 3 and y0 = 1 have been found in a survey, the expected numbersof households with the various patterns of infection are equated to the ob-served numbers. For example, the expected number of households with therealization (3 1 0 0) equals

3M ( l - a ) J a v ^ ^ da

3MB(a + 3, b + 1) _ 3M(a + 2)(a + I)a6B(a, b) (a + b + 3)(a + b + 2) (a + b+ l)(a + b)

for the Greenwood model, and for the Reed-Frost model,

- 3M[£(q + 3, b + 1) -f B(a + 3, b + 2)]B(a,b)

3M(a + 2)(q + l)a6 r 6 + 1 i~ (a + 6 + 3)(a + b + 2)(a + b + l)(a + 6) L + a + 6 + 4J'


If similar integrations for the other possible realizations are carried out, thenthe parameter values a, b can be estimated by moment fitting techniques ormaximum likelihood methods using the data. See Bailey (1975) or Ganiand Mansouri (1987) for details.

4.4 Chain binomial models with replacement

There exist situations in which a group that is partially isolated from a largerexternal population, may lose infectives by emigration, and have them re-placed by a mix of infectives and susceptibles immigrating from the externalpopulation. This happens, for example, in groups of intravenous drug users(IVDUs) subject to HIV infection, such as those considered by Gani andYakowitz (1993); here IVDUs who are seropositive leave the group, and arereplaced by recruits from an external population.

We model the process, somewhat crudely, by assuming there is a sequenceof emigration-immigration episodes at times t = 0 , 1 , . . . , of relatively shortduration in comparison with the length of time between them. Let (Xt, Yt)denote the numbers of susceptibles and infectives in the group just beforethe start of the t th such episode; assume that the group is of constant sizeN = Xt + Yt (all £)• In a n emigration-immigration episode, all Yt infectivesare replaced by an equal number of individuals drawn independently andrandomly from a much larger population in which each individual is aninfective with probability p say (0 < p < 1). Denoting the number of suchincoming infectives by F/, it follows from the independence assumption that

Yl =d Bm(Yup) = Bin(N - Xt,p). (4.4.1)

It is of course possible for Y{ to be zero, in which case the group consistsonly of susceptibles, and is assumed not to be subject to further infection.Otherwise, Y( > 0, and in the period until the next emigration- immigrationepisode, we assume that a binomially distributed number Zt of the X[ = N—Y{ susceptibles become infectives, so Xt+\ — X't — Zt- Under a Greenwood-type model, we assume that

For a Reed-Frost-type model, we assume that the Yt' infectives mix withthe X[ susceptibles and produce a further Bin(Xf

;, 1 — aYt) infectives, sothat

;a y0- (4A3)

4.4. Chain binomial models with replacement 119

Since Y( = Bin(N — Xt,p), it follows that under either model, {Xt : t —0,1,...} is a Markov chain for which N is an absorbing state, because assoon as Y( = 0 no further infection can then occur among the X[ = Nsusceptibles.

For simplicity we consider a Greenwood-type model. The transition prob-ability matrix R\ = (r^) say from Xt to X[ is upper triangular with ele-ments

(4.4.4)otherwise,

where 0<p=l-q<l,so thatP

nN-l

*N

ql

If X[ < TV, infection now follows as in the Greenwood model. The transitionprobability matrix R2 = (^ | ) say from X[ to Xt+i = N — Y{ — Zt is lowertriangular and of the form (4.2.1), namely

(4.4.5)r(2) =jk ^ SNk j =. 0 otherwise,

so that, with 7 = 1 — a as the probability of infection,1 . . . . .7 a . . . .

0 0 .. . 0 UIt now follows that transitions from Xt < N to Xt+i < N have distributionsgiven by the elements s^ of the matrix S = Rify, where the N x N matrices.Ri and R2 are derived from Ri and R2 by deleting the last row and column.We obtain

j=max(i,fe)

(N-i)\pN/a^ "-1


where i, k lie in {0,1, . . . , N - 1}. Note also from (4.4.4) and (4.4.5) that

The process {Xt} ultimately ends in the absorbing state AT, but the timeT to absorption (i.e. the duration of the epidemic) need not be small: to getan idea of the expected behaviour of the process we consider E(Xt+\ \ Xt).From (4.4.2) we have

0} + NI{Y[ = 0}YD =

where /{•} denotes the indicator r.v. Prom (4.4.1), Pr{F/ = 0 | Xt} =qN~x* and E(F/ | Xt) = (N - Xt)p. Thus

E(Xt+i | Xt) = [N-(N- Xt)p]a - a)qN~x\

or equivalently,

E(yt+1 - Yt I Yt) = - (1 - pa)Yt

(4.4.7)

(4.4.8)

The expression on the right-hand side is concave in Yu being zero at Yt = 0and negative at F£ = TV because 1 —pa > 1 — a. So when its slope at Yt = 0is positive, i.e. when

I-pa <N(l-a)ln(l/q), (4.4.9)

it has a maximum at a point interior to (0, N). Table 4.3 shows the thresholdvalues NPtOt = (1 — pa)/[(I — a) | ln(l — p)\ ] which are such that when thegroup size N > Np,a there exists a positive value for Yt such that E(Yt+i \Yt) — Yt, i.e. {Yt} remains stationary (in the mean) and endemicity ofinfection is likely.

Table 4.3. Threshold values Np,a of group size for endemicity

pl-cx0.010.020.040.060.080.100.200.300.400.50

= 0.001

99 852.349926.724 963.816 642.912 482.49 986.14993.63 329.42 497.31998.0

0.002

49 851.724 926.412 463.78 309.56 232.34 986.12 493.51662.71 247.3

998.0

0.003

33184.216 592.68 296.85 531.54148.93319.31660.21107.1

830.6664.7

0.005

19851.29926.14 963.63309.42482.31 986.0

993.5662.7497.3398.0

0.010

9851.44926.22 463.61642.71232.3

986.0493.5329.3247.3198.0

0.020

4 851.82426.41213.7

809.5607.3486.1243.5162.7122.398.0

0.030

3 185.61593.3

797.1531.7399.1319.4160.2107.180.664.7

0.040

2 352.71176.8

588.9392.9294.9236-1118.679.459.848.0

0.050

1853.1927.0464.0309.7232.5186.293.662.747.338.0

4.4. Chain binomial models with replacement 121

Let £ be the least positive zero of the right-hand side of equation (4.4.8)rewritten in terms of Xt = N — Yt, i.e. £ is the smallest positive root of

(1 - pa){N - 0 = N(l - a)(l - qN~S); (4.4.10)

some values are shown in Table 4.4 when N — 200. Then £ < TV if and onlyif N > Npja, and the process {Xt} has mean drift away from its absorbingstate when £ < Xt < N. Typically, we should have p small, in which casethe right-hand side of (4.4.9) « Np(l - a) and 7Vp,a « l/[p(l - a)].

Table

P

0.100.200.300.400.50

4.4. Least

= 0.01

200.0200.0200.0200.0198.0

root £

0.02

200.0200.0178.8146.5118.4

of (4.4.10)

0.03

200.0184.9153.6127.5103.9

when N

0.04

200.0171.7144.7121.399.7

= 200

0.05

197.2165.4140.8118.898.0

Another approximation can be obtained by modifying (4.4.2) so as tostudy a process {Xt} satisfying

Xt+i =d Bin(iV- Bin(N-Xt,p),a); (4.4.11)

this is equivalent to changing the last row (0 0 • • • 1) of R2 below (4.4.5) tothe terms of the binomial expansion (7 + a ) ^ , so {Xt} is a Markov chainon {0, . . . , N} with all states recurrent. Then

1 - (nni\t

EXt = [N-(N- EXt-i)p]a = Na(l -p) ^ } + (pa)*EJf0. (4.4.12)1 — poc

Such a process has a stationary version with generic marginal r.v. X forwhich

EX=^-^«ATa, (4.4.13)1 — pa

the approximation holding when p is small. See Exercises 4.1 and 4.2.The process {Xt} satisfies equation (4.4.7), so we might expect EX w £

because Xt has mean drift always towards £, i.e. it has zero mean drift onlywhen Xt = £. Comparison of (4.4.10) and (4.4.13) shows that, for p small,£ « Na only if qN~t « 0, which occurs only if a is not too large. SeeExercise 4.3.


The duration time T until absorption in the state Xt = N is consideredseparately for the two cases determined by the condition (4.4.9) and itscomplement. Crudely speaking, there is drift towards the state N whenN < iVp>Q, whereas when N > NPiCt, Xt will reach N only as a result of arare excursion involving several large positive fluctuations away from EX.

For N < NPjOe, the process {Xt} has a drift to the absorbing state Nwith mean drift E(J*Q+i - Xt \ Xt) lying between N(l ~p)a - N(l - a)qN

and 0. As a rough approximation then, we should have T of the order of(N — XQ)/[N(1 — p)a] = Yo/[N(1 — p)a]; in fact a better approximation forT is given by Y0/[(l - pa)Y0 - N(l - a)(l - qY°)}.

For N > NPta, the process {Xt} no longer has a mean drift towards theabsorbing state so we must use some other approach. Recall that T is thefirst passage time into the state N for the process Xt considered around(4.4.11), so we can approximate the order of T by the mean recurrence timeof Xt for the state N; this equals the reciprocal of Pr{Xt = N} when Xt

has its stationary distribution.In both cases,

n } « Awn (4.4.14)

for sufficiently large n, where A is a positive constant and w is the Perron-Frobenius eigenvalue (i.e. the eigenvalue of largest modulus) of the sub-stochastic matrix S with elements defined at (4.4.6). There are graphsin Gani and Yakowitz (1993) illustrating (4.4.14) when N = 7. We havecomputed ET as

j j j

(4.4.15)where VJ is estimated by Pr{T = n}/Pr{T = n — 1} provided these termsare not excessively small, and n is the number of transitions computed. Ifthe terms are very small, then it is of no consequence, because the last termin (4.4.15) is negligible. Table 4.5 lists some values of ET starting fromXQ = 1 for N = 200 and the same p and a as in Table 4.4.

It is evident from the structure of the state space for the Markov chainXt that the conditional distribution

q\ = Yx{Xt = i | Xt / N} (i = 0,... ,N - 1) (4.4.16)

converges as t —* oo to a probability distribution {^} say, known as thequasi-stationary distribution for Xt (Darroch and Seneta (1965)). Indeed,

4.5. Final size of an epidemic with arbitrary infectious period 123

Table 4.5. Mean duration ET when N = 200, Xo = 1

p =1-a0.100.200.300.400.50

0.01

2.082.352.632.993.44

0.02

2.553.314.476.259.03

0.03

3.014.647.75

13.9126.99

0.04

3.566.6714.3135.20110.64

0.05

4.269.8928.55113.6

1538.0

general theory in Seneta (1981, Chapters 6 and 7) shows that Pi{Xt ^ N}« Avot+l /(I — w) with A and w as in (4.4.14), and the normalized left-eigenvector q for the eigenvalue w has as its elements the terms of the limitdistribution {(&}, i.e. q = [q0 • • • </JV_I]'. It is intuitively clear that whenN > NPyOl, q is approximately the same as the stationary distribution forthe process X considered earlier, and w —• 1 as N —• oo. See Exercise 4.5.

4.5 Final size of an epidemic with arbitrary infectious periodComparison of the mathematical structure of the Reed-Frost chain binomialmodel with the Kermack-McKendrick model of the previous chapter, andthe possibility of viewing of an epidemic as a 'branching process on a finitepopulation', prompt a closer examination of the probabilistic features ofboth models. We show below that these models can be placed in a moregeneral setting, and study their specific features as particular members of abroader class.

Suppose that we regard the population as a set of nodes in a graph,with each node either a susceptible, infective or removal; initially there areN susceptibles and 1 infective. Each node i is the hub of a (small) setof directed links which are 'activated' if i ever becomes infectious. Theactivation of a node's directed links transmits the infection to the nodesat the far ends of those links, turning any susceptible at the latter nodesinto an infective but otherwise having no effect. Consequently, given theset of directed links for each node, we could in principle determine the totalsize of the epidemic: we would trace forwards the spread of infection alongthe links until eventually either the set of nodes that become infectious isexhausted, or there remain no susceptible nodes. In tracing the links froma given node, the change of that node to a removal is noted. Conversely,any node is ultimately infected if and only if there is a path backwards tothe initial infective along some connected set of directed links.

We now impose on this structure some probabilistic assumptions that re-flect the homogeneous mixing and independence properties underlying both


the Kerrnack-McKendrick and Reed-Frost models. To do so, it is helpfulto consider first the Kermack-McKendrick setting. Suppose that at someepoch in time a node V is newly infected, there are X susceptibles, and we'put on hold' the action of all infectives apart from V. If V spreads infectionfor a time 5, then, according to the assumptions of the general stochasticepidemic in Section 3.3, each of the X susceptibles, independently of all oth-ers, becomes infected with probability 1 -e~@s. More widely, / ' is linked toeach node independently with the same probability 1 — e~^5, though a newinfection results only when the target node is a susceptible. The numberof directed links triggered by / " s infection, given 5, is thus a binomial r.v.Bin(Ar, 1 — c~@s). Now S, the period of infectiousness before / " s removal,is itself a random variable, distributed exponentially with mean 1/7. Thus,unconditionally, this r.v. is a mixture of these binomial distributions withmixing parameter 5, while v(X), the number of new infectives, is a mixtureof the binomial r.v.s Bin(X, 1 - e~^s), i.e. for (k = 0 , . . . , X),

Pr{*/(X) = k} = f°° (*\ (1 - e-ê-MX-V-ye-** ds. (4.5.1)

Let t be an index ('time' in the random walk we are about to describe)that counts the number of infectives whose immediate links have been tracedup to the epoch t — 0, at which point there are Xt susceptibles, Yt infectivesand Zt = t removals. Thus, there are Zt infectives whose links have beentraced prior to t — 0. Provided that Yt > 1, we then have

Yt+l = Yt - 1 + u(Xt)9 Xt+x =Xt- i/(Xt), (4.5.2)

where the r.v.s {v(Xt) : t = 0,1, . . .} are independent with the mixed bino-mial distribution (4.5.1), while if Yt = 0, Yt+\ = 0 also.

We can easily evaluate the distribution of Xt. Set

p\ = Pr{Xt = i}, pk(i) = PiMXt) = k I Xt = i}. (4.5.3)

Then when Yt > 1, starting from p® — Sjî,

N-t

fa-iW (* = 0,. . . , AT). (4.5.4)

Modifications for the case where the initial number of infectives is largerthan 1 are easily made (see Exercise 4.6).

4.5. Final size of an epidemic with arbitrary infectious period 125

DefineT = inf{t:Xt + t>N}. (4.5.5)

Inspection of the construction underlying (4.5.2) shows that T is the totalsize of the epidemic (including the initial infectives), i.e. T = AT-f 1 — XT —ZT- It follows from (4.5.4-5) that the distribution of T is given by

Pr{T = k} = Pk = pkN^_k = pk

N-^kPo(N + 1 - k). (4.5.6)

In the argument above, we see that we can replace the exponential distri-bution of S by an arbitrary distribution for the infectious period, with d.f.F(-) say, leading to a different mixed binomial distribution on k = 0 , . . . , X,namely

= k}= f°

The special case in which F(-) is degenerate at 1/7, so that ES = I /7 as forthe exponential distribution, makes v(X) a binomial r.v. Bin(X, 1 — e~^/7).Embedded in the resulting process {(Xt,Yt)} is a Reed-Frost epidemic, aswe now indicate. Define {(Xn, Yn)} recursively, starting from (Xo, Fo> ^0) =(TV, / , 0), in such a way that in 'generation' time n those of the Xn that be-come infectives by direct contact with one of the Yn infectives are identified,totalling Yn+i = Xn — Xn+\ in all. At the same time, the Yn infectives be-come removals, i.e.

Zn (otherwise),

This embedding is effectively the same as that used to define a Galton-Watson branching process in terms of a left-continuous random walk(Spitzer (1964) p. 234). We assert that {(Xn,Yn)} is a Reed-Frost epi-demic process and leave the proof to the reader.

Daley (1990) exploited the representation at (4.5.2) to show that as thedistribution for the infectious period S increases in the sense of stochastic,or decreasing convex, or a transform (Laplace-Stieltjes) ordering, the totalsize distribution increases in the sense of the same ordering (except thatprobability generating functions are the basis of the transform ordering).From a practical viewpoint, this implies that among infectious periods withthe same mean for S, the total size is largest when 5 has the least variance,which is the case in the Reed-Frost model.


Exercise 4.6 indicates how to use the approach of this section to computethe total size distribution {Pk}- See e.g. Ball (1986) for another approach.Exercise 4.7 suggests a relation between the mean total size starting withN + 1 susceptibles to the distribution starting with N,

4.6 A pairs-at-parties model: exchangeable but not homoge-neous mixing

The model described in this section is based on part of Daley and Gani(1999). It attempts to evaluate the importance of the usual homogeneousmixing assumption based on the Law of Mass Action. It also investigatesthe effect of pairwise interaction.

For the simplest construction, we assume a closed population of sizeN = 2M individuals, of whom initially one is an infective and all otherssusceptibles. At discrete epochs in time £ = 1,2,..., these individuals formM pairs, with each individual in contact with exactly one other individualat each time epoch. As a result of this pair formation, we assume that anysusceptible in a susceptible-infective pair becomes an infective. Assumingthat individuals, once infective, remain so, we can ask in the usual fashionhow long it will take for all individuals to become infectives.

Assume that pair formation occurs at random. Denote the number ofsusceptibles after epoch t by Xt, with XQ = N — 1. Then we can useindicator random variables and simple combinatorics to deduce that

To find the distribution of the decrement Xt — Xt+i, we must count thenumber of ways of forming pairs, and in particular infective-susceptiblepairs, i.e. mixed pairs. First, the possible number of distinct sets of Mpairs which can be formed from the 2M distinct individuals equals

J_/ N \ N\ (2M)! = 2MT{M+\)Ml \2 2 • • • 2) M\ (2\)M Ml 2M r(±) " l }

Next, the number of mixed pairs is odd or even according to whether thenumber of infectives Yt = N - Xt is odd or even, so irrespective of Yo, Yt iseven for t = 1,2,... . Define zM{i) = min(i, 2M - i) and set Zt = zM(Xt).Then the number of ways of forming exactly j mixed pairs from the Xt

4.6. A pairs-at-parties model 127

susceptibles and Yt infectives is zero if Zt + j is odd, and

(Zt\ (2M — Zt\ ( the number of ways of forming \V 3 ) \ 3 ) \\(zt~ j) and M - \{Zt + j) non-mixed pairs/

Ztl(2M-Zt)\ (Zt-j)\j\ (Zt - j)\ (2M - Z t - j)\ (\\Zt - j})\

(2M-Zt-j)\ - (4.6.3)

otherwise. It now follows that the one-step transition matrix for the Markovchain {Xt : t = 1,2,...} has elements Pi^-j = rj(i,M) where, with Zt =

above,

rj(i, M) = Pr{Xt+1 =Xt-j\Xt = z, M}(2M-Zt-j)\fZA (2M - Zt\ j\(Zt-j)\

\ j ) \ 3 )(±[Zt-j))\2^-(2M)!M!2M

2J'Zt!(2M-Zt)!M! = 2J'(,- i(z t-j)V^(z t+j))

j \ (\[Zt - j))\ (M - \\Zt + j])! (2M)! (2^)

C7 = Z f , Z t - 2 , . . . , l o r 0 ) ,0 otherwise.

(4.6.4)

We can now use this Markov chain to find numerically the distribution ofthe time T until all 2M individuals are infective.

The assumption that transmission of infection occurs in every infective-susceptible pair is unrealistic. If infection occurs with probability /?' say,then by using indicator variables we should have in place of (4.6.1) therelation

Y\ P'Xt(N - Xt)X) = (4.6.5)

The transition probabilities Vî-j sav> a r e now binomial mixtures of theterms r^(z, M), because from the random number Kt of mixed pairs formedby the Xt susceptibles and N — Xt infectives, the number of new infectivesis a binomially distributed r.v. Bnî^,/?'). We have

min(i,N-i) .

( (4.6.6)=3


11 —

0.8-

0.6-

0.4-

0.2-

o

111<

1

11

1:,1

0 10

//

//

/1

111

i20

/ '/

130

140 12

(a) (b)Figure 4.3. Duration time distributions for the pairs-at-parties model with0' = 1 ( ), /?' = 0.5 ( ), and ff = 0.3 ( ), for N = 20.(a) Actual time Pr{Xt = 0}, (b) rescaled time Pr{Xt/(3f = 0}.

Figure 4.3 plots the distrubution of the duration time of an epidemic in thismodel for three values of /?', with values rescaled in part (b) of the figureindicating convergence as f¥ | 0 to a limit in which the effect of exactly Mpairs being formed in each time unit largely disappears. These distributionscan also be illustrated by plotting the mean numbers of new infectives ina time unit (see Figure 4.4). Rescaling again emphasizes similarities andsuggests convergence (see Exercise 4.8).

We now attempt to understand better the effect of the pair-formationaspect of the model relative to 'classical' homogeneously mixing models,subject of course to appropriate choice of such a classical-style model. For astart, we observe that in the pairs-at-parties model every individual in theclosed population is either susceptible or infective, and that once infected, anindividual remains so forever. In this respect the model resembles the simplestochastic epidemic X(t) of Section 3.1 starting from N — l susceptibles and1 infective, with transmission rate /? which we must determine appropriately.We deduce from Section 3.1 that

d_(4.6.7)

we can immediately see the similarity of functional form with the right-handside of (4.6.5). This derivative best approximates the discrete-time meandecrement at (4.6.5) (which includes the case /?' = 1 in (4.6.1)) by settingP = (3'/(N — 1). The deterministic analogue of this stochastic model givesthe logistic growth curve as in Section 2.1.

4.6. A pairs-at-parties model 129

However, since the pairs-at-parties model is a discrete-time model, withexactly M = \N pairs formed in each unit of time, we should use a discretetime analogue of this simple epidemic model. We consider two possibilities.For the first we start with the discrete time approximating processdefined by X$ = N — 1 and the one step transition probabilities

if j = i,otherwise.

(4.6.8)

We recognize that the term Pi,i-i equals the product of the probabilitiesthat a randomly chosen pair is a mixed pair and that there is infectiontransmission. Since M pairs are formed in each time unit of the pairs-at-parties process {Xt} it is appropriate that it be compared with the process{Xt} =

5 -

4 -

o

2 —

1 -

0 -

/ ,

/ /

/ /it/

10 2

/~v

i4

\

\\\1

16

18

110

_"I12

(a)

2 -

1 . 5 -

0 . 5 -

u(

/

ilif

ij

)

/

/

i5

A\

10I

151

201

25

(b)Figure 4.4. Mean numbers of new infectives in pairs-at-parties model( ), approximating discrete simple epidemic model ( ), andapproximating Reed-Frost-style simple epidemic ( ), for iV = 20and (a) ff = 1, (b) ff = 0.5.

Figure 4.4 compares the two processes {Xt} and {Xt} for N = 20 andthe two cases /?' = 1 and /? = 0.5, plotting for each time unit the meandecrements E(Xt — Xt-\), i.e. the mean numbers of new infectives. Thefigure also shows the corresponding data for the other discrete-time pro-cess {Xt} that is a simple epidemic analogue of the Reed-Frost model ofSections 4.1-2. Conditional on Xti Xt+i is a binomially distributed r.v.


, qN~Xt), where q denotes the probability that a given susceptible isnot infected by a given infective in the time interval (t, t + 1). This processXt has mean decrement

E(Xt - Xt+1 | Xt) = Xt[\ - = Xtp{\ + q^Xt(N-Xt)p. (4.6.9)

By choosing p = (3'/(N — 1) we approximate the decrement (4.6.5), exactlyfor X^ = N — 1 = XQ] otherwise (4.6.9) is a lower bound. Thus, we couldexpect this simple epidemic analogue of a Reed-Frost epidemic to evolvemore slowly than the pairs-at-parties model.

Figure 4.5 provides another comparison of the processes {Xt}, {Xt} and{Xt}, by plotting the increments Pr{Xt = 0} - Pr{Xt_i = 0} of the d.f.shown in Figure 4.3, in the two cases /?' = 1 and 0.5 as for Figure 4.4. Figure4.4 indicates that the spread of infection in the pairs-at-parties model ismore rapid than the Reed-Frost simple epidemic analogue, as noted below(4.6.9). Compared with the simple epidemic analogue, its initial spread isslower; in the middle and later stages, however, its spread is faster so thatthe population becomes totally infected more rapidly (this latter point isunderlined in Figure 4.5).

0.6-1

0 . 4 -

0 . 2 -

0.15-

0 . 1 -

0.05-

o -(

1J1

) 5

/

I1;/ /

I10

\ \

\

115

120

^.1

25

(a) (b)Figure 4.5. Increments Vi{Xt = 0} — ¥x{Xt~\ = 0} in the duration timedistribution of pairs-at-parties model ( ), approximating discrete simpleepidemic model ( ), and approximating Reed-Frost-style simple epi-demic ( ), for N = 20 and (a) ff = 1, (b) ff = 0.5.

Decreasing (3f has the effect of increasing the uncertainty of transmissionof infection. All three Figures 4.3-5 indicate that as /?' decreases, the pairs-at-parties model resembles a classical model more closely.


By way of summary, these properties indicate that, compared with stan-dard homogeneous mixing models, an epidemic spread via the 'regulariza-tion' of pair-formation for all individuals in each discrete time unit, leadsto a more predictable and ultimately faster spread of infection. However,the effect of pair-format ion in each time unit is not great.


4.1 (a) Use a sketch graph to check that when (4.4.9) holds, the root £ of equation(4.4.10) lies between N and £0 = Nct(l - p)/(l - pa).(b) Expand the term qN~* when (N - £0)| ln(l - p)\ is small (i.e. (N - £0)pis small) and deduce that

Compare with the numerical values in Table 4.4.

4.2 Show that for a stationary process {Xt} satisfying (4.4.11),

[1 - (pet)2] var X = a(l - a)(N - pEY) + a2pqEY,

where Y — N — X. What is the asymptotic behaviour of 1 — w for large NlInvestigate conditions for a 'quasi-stable' Y that is O(l).

4.3 Compare the stationary mean EX at (4.4.13) with the least root f of (4.4.10)given in Table 4.4.

4.4 Investigate whether martingale and/or transform methods may assist instudying the process {Xt} of Section 4.4, in particular first passage timevariables and the value of £ (or other quantity) as an approximation.

4.5 Show that the total size distribution {Pk} of an epidemic starting from /infectives and otherwise as in Section 4.5 can be computed from


4.6 (a) Show that the total size distribution {Pk} of the stochastic carrier epi-demic model of Section 3.6 can be computed from Pk = Pn-k,o where, start-ing from Pnb — 1, the quantities PtJ = Pr{for some t there are i susceptiblesand j carriers} are defined recursively by

Pib =P t+ iP<+i ,6 (i = n - l , . . . , 0 ) ,

Pnj =qnPn,j + l (j = 6 - 1, . . . , 0),

Ptj = Pt+iPt+i,j + QiPt,j+i (i = n - 1,. • . , 1; j = b - 1 , . . . , 1),

in which pt = i/(i + p) = 1 — qt.

(b) Justify the alternative computational scheme based on Pk = P^-k w n e r e

p^ = Sm and for j — 6, . . . , 1 and i = n,..., 0,

ti /

[Reinarlc: Either of these computational schemes is numerically more stablethan the explicit formula at (3.6.12).]

4.7 In the Markovian model for a general stochastic epidemic process {(X,Y)t :0 < t < oo} discussed in Section 3.3, starting from (X,Y)0 = (TV, 1), letHt denote the increasing a-fields determined by the sample paths {(X, Y)u :0 < u < t}. Regard a similar model (X,Y)t with (X,Y)o = (N + 1,1) asstarting with the same (N, 1) individuals as determine Ht augmented by adistinguished individual that is initially susceptible and that has no impacton the sample paths (X, Y)t for precisely as long as it remains susceptible.Show that

PN+i(t) = Pr{distinguished individual is susceptible at t}

= E[exp( - / ? / ot y (« )d U ) |H t ] .

Prom this it is plausible that

7 TV + 1

compare with the argument in Ball (1986). Cf. also (2.3.7) and Exercise 5.6.4.8 Investigate appropriately rescaled limits of the pairs-at-parties model and

of the two simple epidemic analogues of Section 4.6, for (a) TV —> oo, and(b) ff - 0.

Rumours: Modelling Spread and its Cessation

This chapter is devoted to further models of spread in a population, mostof them appealing to the Law of Mass Action. Our aim is to exemplify themodelling principles found useful in other problems involving the spread ofan attribute through a population. It is important to note that except formodels in which the only 'mechanism' is that of spreading, as for example inthe case of a simple epidemic, the range of behaviour exhibited by processeswith a removal mechanism is quite varied.

The two key features of previous models involve the spread of a diseaseand the removal of infectious individuals. We have given some emphasis inChapters 2, 3 and 4 to the spread of disease by other than homogeneousmechanisms. In this chapter we look also at a variety of removal mecha-nisms, such as may apply to the spread of a rumour (an 'infection of themind'). We show that different removal mechanisms can lead to somewhatdifferent conclusions from those of earlier chapters, where threshold phe-nomena were of particular interest.

5.1 Rumour modelsThe earliest references in the probability literature to a rumour modelappear in the work of Rapoport and co-workers from 1948 onwards (seeBartholomew (1967) for references), in Feller (1957, Exercises 11.10.21-22)though not in his first (1950) edition, and in Kendall (1957). Here we takeas our basic model that described in Daley and Kendall (1965) (called [DK]below), where the original motivation for its formulation was the possiblesimilarity in the spread of physiological and psychological infections (cf.Daley and Kendall, 1964). The analysis that ensued highlighted a majordifference in modelling the way that spreading ceases for the two types of'infection'.

133

134 5. Rumours: Modelling Spread and its Cessation

As in the simpler models of Chapters 2 and 3, we consider a closed ho-mogeneously mixing population of N -f 1 individuals. At any time they canbe classified as belonging to one of three mutually exclusive and exhaustivecategories consisting at time t > 0 of(a) X(t) individuals who are ignorant of the rumour;(b) Y(t) individuals who are actively spreading the rumour; and(c) Z(t) individuals who know the rumour but have ceased spreading it.

Initially, X(0) = N, Y(0) = 1 and Z(0) = 0, while for all t, X(t) + Y(t) +Z(t) = N 4- 1. We refer to these three types of individuals as ignorants,spreaders and stiflers respectively; they are the equivalents of susceptibles,infectives and removals of the general epidemic model of Chapter 3, but wemodel their behaviour somewhat differently.

The rumour is propagated through the population by contact betweenignorants and spreaders, following the law of mass action just as in Chapter3. Specifically, assume that any spreader involved in any pairwise meetingattempts to 'infect' the other individual involved in the meeting; this 'otherindividual' is either an ignorant, a spreader or a stifler. In the first case,the ignorant becomes a spreader; in the other two cases, either or both ofthose involved in the meeting learn that the rumour is 'known' and so decidenot to tell the rumour any more, thereby joining the stiflers as a result ofthis 'stifling experience'. When the pairwise meeting rate is /?, ignorant-spreader, spreader-spreader and stifler-spreader meetings occur at time t atrelative rates 0X(t)Y(t), \f3Y{t)[Y{t) - 1], and (3Y(t)Z(t) respectively. Itis convenient to write such changes as may occur in a (small) time interval(t, t + h) in the form

Ah(X, Y)(t) = (X, Y)(t + h) - (X, Y)(t), (5.1.1)

so for example an ignorant-spreader meeting yields Ah(X, Y)(t) = (—1,1).We use the meeting rates to set the transition rates for changes in the stateof the population, and so deduce the relations, with the approximationsneglecting terms that are o(h),

Pr{Afc(X, Y)(t) = (-1,1) | (X, Y)(t) = (x, y)} = 0xyh,Pv{Ah(X,Y)(t) = (0,-2) | (X,Y)(t) = (x,y)} = \j3y{y - l)h,Pr{Ah(X, Y)(t) = (0, -1) | (X, Y)(t) = (x,y)} = 0yzh, (5.1.2)

Pr{Ah(X, Y)(t) = (0,0) | (X, Y){t) = (x, y)} = 1 - 0y(N - \[y - l])h,

while all other transitions have probability o(h). It is clear here that (3plays the role of a time constant, so by suitable choice of time unit we mayassume that (3 — 1.

5.1. Rumour models 135

We shall be interested in the proportion / of the ignorant populationX(0) = N which eventually learns the rumour, so that

Km X(t) = N-Nf, limr(t) = 0, lim Z(t) = 1 + Nf. (5.1.3)t—+oo t-+oo t—*oo

We can study the properties of / both in a deterministic version of themodel described by equations (5.1.2) (cf. Chapter 2 and Section 5.2 below),and an embedded jump chain argument (cf. Sections 3.4.2-3 and 5.3). It isalso convenient to use the present context to describe an argument due toD.G. Kendall leading to a quicker computation of asymptotic properties ofthe model for larger population sizes (see Section 5.4).

We now describe two simple variants of the basic [DK] model. For thefirst, which we call the A:-fold stifling model, assume that a spreader doesnot decide to stop propagating the rumour until being involved in A: stiflingexperiences as described earlier. Then by identifying the Y(t) spreaders asbelonging to one of the k mutually exclusive categories of spreaders whohave had j stifling experiences (j = 0, . . . , k — 1), we can still give a Marko-vian description of the process, using the larger, (k + l)-dimensional vector(X, Yi,. . . , Yk)(t) where (Yx + • • • + Yfc)(*) = Y(t) = N+ 1-X(t) - Z(t) asearlier. See Exercise 5.3 for a deterministic version of this model.

For the second variant, which we call the (a,p)-probability variant, ob-serve that the [DK] model is certainly simplistic in supposing that (a) everypairwise meeting involving a spreader results in the spreader attemptingto spread the rumour, and (b) a spreader becomes a stifler as a resultof exactly one stifling experience. Suppose instead that a spreader in-volved in a pairwise meeting attempts to spread the rumour with prob-ability p, and that when such an attempt is made any spreader so in-volved decides with probability a to become a stifler, independently foreach spreader and each meeting. The basic model has p = a = 1; we nowsuppose that 0 < p < 1 and 0 < a < 1. Then when (X,Y)(t) = (x,y),there occurs in (t, t + h) either an (X, Y) meeting resulting in A^(X, Y) =(—1,1) with probability pxyh (neglecting o(h) terms), or a (Y,Y) meet-ing resulting in A/^X, Y) = (0, -1) or (0, -2) with respective probabilitiesp(2 - p)a(l - a)y(y - l)h and p(2 - p)a2\y(y - l)/i, or a (Y", Z) meetingresulting in Ah(X, Y) = (0, -1) with probability payzh, or no change withprobability 1 - p[(l + a)x + (1 - \p)a(2 - a)(y - l)]yh.

Another model which has gained some currency, if only because of itssimplicity and appearance in an undergraduate text, is due to Maki andThompson (1973). As before, the population is classified and counted as(X, Y, Z); a continuous time Markov process version of the model is then


as follows (Watson, 1988). The rumour is spread by directed contact of theY(t) spreaders at time t with others in the population. The outcome ofcontact of a specified spreader with (a) an ignorant is that the ignorant be-comes a spreader, and (b) another spreader or a stifler, is that the initiatingspreader becomes a stifler. No other micro-level transitions occur. Thenthe model has just two elementary transitions, with Ah(X,Y) = (—1,1)or (0, -1) at infinitesimal rates XYh + o(h) and Y(Y - 1 + Z)h + o(h)respectively, i.e. omitting terms o(/i),

Pr{A*(X, Y)(t) = (-1,1) | (X, Y)(t) = (s, y)} = xyh,Pr{A^(X, Y)(t) = (0, -1) | (X, Y)(t) = (*, y)} = y(y + s - l)/i, (5.1.4)

Pr{ A*(X, y)(*) - (0,0) | (X, y)(«) = (*, y)} = 1 - Nyh.

Notice that because there is now directed contact of the initiating spreader,the term \y(y — l) of equations (5.1.2) is replaced by y(y — l) in (5.1.4). Themodel is otherwise like the [DK] model in assuming that it is only throughinteraction between the initiating spreader and others of the population thata spreader becomes a stifler. The resulting deterministic analysis is similar(see the next section and Exercise 5.4), but not so the stochastic analysis(see Section 5.3 and Exercise 5.7).

Both the simple epidemic model and the general epidemic model discussedin Chapter 3 have been suggested as models to describe the spread of a ru-mour (e.g. Goffman (1965), Goffman and Newill (1964, 1967), Bartholomew(1973, Chapters 9 and 10)). Recall that the general epidemic model hastwo basic transitions possible in a small interval of time of length /i, namelyA/j(X, y) = (—1,1) or (0, —1) at rates (3xy and 72/ respectively. Clearly thefirst transition can result from an (X, Y) contact, but the second requiresno interaction between individuals in the population at all. It is enough forthe cessation of rumour-spreading to occur purely as a result of a spreader's'forgetting', irrevocably, to tell the rumour to those of the population withwhom (s)he comes into contact. Now forgetfulness, or disinclination everto tell the rumour, are certainly features we may wish to model as factorscontributing to the decline in the spread of the rumour. But it seems anunrealistic description of human behaviour to make it the sole mechanismcausing the cessation of a rumour's spreading, in the same way as totalreliance on the stifling mechanism may be unrealistic. Just as diseases mayvary in their potency, so rumours may vary in the urgency with which in-dividuals communicate them. On balance we may wish to use both stiflingand forgetfulness mechanisms (for surely, some items of news are eminentlyforgettable!); one possible way of doing so is indicated in Exercise 5.5. It

5.1. Rumour models 137

would then be reasonable to expect the relative rates of pariwise contact(for both spreading and stifling) and of individual forgetfulness to affect theextent of the spread of the rumour.

Threshold phenomena such as occur in branching processes or the spreadof epidemics (see e.g. Chapters 2 and 3) depend on a balance betweenspreading on the one hand and cessation of spread on the other. If, as inthe basic [DK] models, the cessation depends on interaction of the spreaderswith themselves or stiflers in the population, then the effect of the cessa-tion mechanism is negligible relative to that of the spreading mechanismwhen the numbers of spreaders and stiflers are small. On the other hand,when cessation of contagious spread occurs as a result of the evolution ofa spreader independently of the rest of the population, with spreading de-pendent on the state of the population, then it is the state of the rest of thepopulation, relative to the dynamics of spread and its cessation amongst thespreaders themselves, that determines the fate of the spread. In this case,conditions for threshold criteria exist, whether for the spread of a diseaseor a rumour, as in the general epidemic model.

We now describe a distinctly different family of models for the spreadof news or a rumour in a population. For convenience we assume thatthe population is of constant size N + 1, its members being classified asignorants, spreaders and former spreaders (e.g. stiflers or removed cases),much as before; initially there are N ignorants and 1 spreader. Let eachindividual becoming a spreader, including the initial spreader, be identi-fied by a distinct integer-valued index r. Spreader r spreads the rumourto Kr individuals chosen at random (without replacement, for definiteness)from the rest of the population; {Kr : r — 1,2,...} is a family of indepen-dent identically distributed integer-valued random variables with distribu-tion {fk = Pv{Kr = k}, k = 0,1, . . .} and mean m = EKr. Those of theKr individuals who are ignorants become spreaders, while the others arenot affected by the contact. After making these Kr contacts, spreader rplays no further part in spreading the rumour. A convenient identificationof the indices is that matching the order in which the r.v.s {Kr} are takeninto the analysis. Then, writing (Xr,Yr) for the numbers of ignorants andspreaders just before the Kr contacts occur, we have for Yr > 1,

Xr + 1 =Xr- Jr(Kr), Yr+1 =Yr-l + Jr(Kr), (5.1.5)

where Jr(Kr) denotes the number of contacts with ignorants amongst the


Kr contacts of spreader r. These assumptions give

K) -11 * = o - 1 A G) (*:;) /CO • (5L6)K—J

for r such that Yr > 1. Recognizing the hypergeometric distribution prob-abilities as the coefficients of /&, we have E(J r(X r) | Xr = i,Yr > 1) =Sfc fkk{i/N) = mXr/N. The relations (5.1.5) are similar to those at equa-tion (4.5.2) where we considered the analysis of a variant of the generalepidemic model by means of an embedded random walk. The same ap-proach is applicable here.

Such a model is most aptly described as a branching process on a fi-nite population. It can certainly be used to describe the so-called chainletter process. It also includes the general epidemic model when {/&} isthe geometric distribution {pNk~1/(N + p)k : k = 0,1,...}) as far as theultimate extent of the spread of the news or rumour is concerned. Thereexist both deterministic and stochastic threshold theorems for which a ma-jor outbreak can occur only if m > 1. Chapter 10.3 of Bartholomew (1973)includes discussion that refers to related work of Rapoport and co-workers.

5.2 Deterministic analysis of rumour models

Except for the branching process model described at the end of the lastsection, the models we have sketched for the spread of news or a rumourcan be formulated as Markov processes in continuous time. Their analysisby explicit algebra does not proceed easily; in particular their transitionprobabilities are not readily accessible for the range of parameter values ofinterest, most notably the initial number of ignorants N. We resort in thefirst instance to deterministic versions of the models.

For the deterministic version of (5.1.1) we consider the continuous func-tions x(t), y(t) and z(t), differentiate in t, that have the same initial val-ues x(0) = X(0) = N, y(0) = Y(0) = 1, z(0) = Z(0) = 0 and satisfyx(t) 4- y{t) 4- z(t) — N 4-1 for a closed population. The rates of change ofx and y are assumed to coincide with the rates of change of the conditionalmean functions such as l imô E(X(t + h) | (X, Y)(t) = (i, j)) at the latticepoints (£, j) , 'smoothly' interpolated between. These limits are assumed tobe given as below:

x = ^ (change in x) x (rate of change) = (—l)xy, (5.2.1a)simple transitions

5.2. Deterministic analysis of rumour models 139

and similarly,

y = ( + i ) x y + ( - 2 ) I y ( v - l ) + ( - l ) i / « = 2/(x-2/+l-z) = y(2x-N). (5.2.1b)

These two equations yield

^ = - 2 + ^ , (5.2.2)dx x

whose solution consistent with the initial conditions is

2x(t) + y{t) + NIn - ^ = A(a(t), j/(t)) = 2JV + 1 (all t > 0). (5.2.3)x{t)

It follows from (5.2.3) that the limit limôo x(t)/N = ON say, analogous tothe proportion / at (5.1.3) of ignorants who never hear the rumour, satisfies

-1 = 2AT(1 - 0N) + NlnONi (5.2.4)

and that liniAr-*oo ON = 0 satisfies

2(1-0) +In 0 = 0. (5.2.5)

This is the same as the equation for the deterministic model of a generalepidemic considered in Chapter 2.3, having relative removal rate ^N (cf.equations (2.3.6) and (2.3.7)). Unlike the epidemic model, the rumour modelno longer has a family of processes with a parameter determining thresholdbehaviour. Instead, the analogy is with a single process; it is not difficultto check that the dependence on N of the root ON of (5.2.4) is O(N~l) (seeExercise 5.1).

In a similar fashion, the general (a,p)-probability variant of the basic[DK] model leads to the pair of differential equations

x = (~l)pxy,y = (+l)pxy + (-l)[p(2 - p)2a(l - a)\y(y - 1) +payz]

+ (-2)p{2-p)a2\y{y-l)

In this case, the analogue of equation (5.2.2) is

^ = _{1 + a)+a(N-l+P)+a(l-P)ydx x x


1 -

0 . 8 -

0.6-

0 .4 -

0 .2 -

0.001 0.01

0.0001

0.05 0.10 0.15 0.203 0.25

Figure 5.1. Contours of 0(a,p), the proportion of a large population nothearing a rumour with uncertain spreading (p < 1) and stifling (a < 1). Thevalues at the top of each curve are of 0(a,p), the smaller positive root in 0of equation (5.2.8) (see Exercise 5.2 for 0max).

This is still a first order linear differential equation, with the general solution

(1 + a)x NV~~ l - a ( l - p ) 1-p N^pX

for some constant XN,Q,P of integration which is determined by the initialconditions, yielding

(l4-cy)r N11 T ulX IV

The proportion 0N of ignorants who never hear the rumour behaves muchas the root of equation (5.2.4), but with limit 0 = 6(a,p) — linijvôo ONnow satisfying

(2 - - p)0 = 1 - a ( l - p). (5.2.8)

The contour curves in Figure 5.1 depict the non-unit root 0(a,p) as a func-tion of a and p; always, 0 < 0(a,p) < 0max, where 0max « 0.284668 is thesmaller positive root of 0(1 - 2 In 0) = 1 arising from the limit a -> 1 andp —> 0 (see also Exercise 5.2).

The deterministic analysis of Maki and Thompson's model is sketched inExercise 5.4.

5.3. Embedded random walks for rumour models 141

5.3 Embedded random walks for rumour models

We now return to [DKj's stochastic model described by the Markov chainwith infinitesimal transition rates as at (5.1.2). The main vehicle for study-ing the final state of this process (e.g. [DK] and Pittel (1990)) has been theMarkov chain embedded at the jump points occurring at times t\,... ,£n,. . . ,1M (cf. Section 3.4.2 above), where M is the total number of jumps.We note that because the state space is finite and the process is 'strictlyevolutionary' in the sense of having well-ordered sample paths, M must befinite.

Denote the states immediately after these jump epochs by {(X, Y)n : n =0,1,...}, with (X, Y)o = (iV, 1), so that when for some particular n we have

{ (—1, +1) with probability i/fij,(0, -1) with probability (N+l-i- j)/fih

(0, -2) with probability \(j - l ) / / y ,(5.3.1)

where fij = N — ^(j — 1). Note that the first case is excluded if i — 0,when the probability equals zero, or j = 0, while the last case is excluded ifj < 1. This embedded jump process is a random walk on that part of thetwo-dimensional integer lattice for which 0<i<N,0<j < 7V+1 — i, andthe states {(z, 0) : i = 0, . . . , N - 1} are absorbing. Prom equations (5.3.1)we can derive equations for the probabilities P^ on the path of a rumour,defined by

Pit = Pr{(X, Y)(t) = (ij) for some t}= Pr{(X, Y)n = (i,j) for some n}. ( ' ' j

The Markov process {(X,F)(£)} is strictly evolutionary in the sense that,once a state (z, j) is visited and left, it is never visited again. Hence the prob-abilities P^ satisfy a set of equations, derived by a forward decompositionargument, of the general form

jump is A | present state is (ij) - A}, (5.3.3)

where A denotes one of the three jumps (-1,1), (0,-1) and (0,-2) atequation (5.3.1). Specifically, starting from PNi = PN-I,2 = 1, we have for


N-i + D, (5.3.4a)

(J = N- *), (5.3.4b)

J V $ ( j + l ) P f J + a

(2 < j < N - 1 - i), (5.3.4c)

From these relations, in the same way as the distribution of the size ofthe general epidemic process was derived in Section 3.4.3, we can find thedistribution

{Pio} = {Pr{rumour ends with i ignorants} : i = 0 ,1 , . . . , N - l } . (5.3.5)

Figure 5.2(a) illustrates the complementary distribution {PNîy0} of thenumber of ignorants hearing the rumour in this model, for two initial sizesN = 50 and 100. The scales have been changed as indicated in the figurefor N = 100, to facilitate comparison of the two distributions.

In [DK] it was concluded that the proportion £/v = X(oo)/N of ignorantsleft when the rumour stops spreading has

E£N « 0.203188 + 0.273 843/JV = 0 + 0.273 843/iV,JVvax^« 0.310681 +1.232 700/JV.

Observe that the correction term added to 9 — 0.203188, the solution ofequation (5.2.5), shows that ON, the solution of (5.2.4) (see Exercise 5.1),is such that ON ^ E£/v, nor is there any good reason that equality shouldhold. In Section 5.4 we note that

= 0.310681 « « » „ « „ . ,5.3.7,

For the [DK] model the general form of equation (5.3.4) involves threeterms on the right-hand side. For the Maki-Thompson model, however,while the same deterministic equation (5.2.2) applies, the analogous equa-tion has only two terms, corresponding to the simpler possible transitions

5.3. Embedded random walks for rumour models 143

0.15

100

0.05

0.15

0.05

10 20 30 40 50 0 10 20 30 40(a) [DK] model (b) Maki-Thompson model

Figure 5.2. Frequency polygons of the distribution of the numbers ofignorants ultimately hearing a rumour in two rumour models, for N = 50( , scales as shown) and N = 100 ( , double the abscissa scale, halvethe ordinate scale, as indicated at top right-hand corners).

as at (5.1.4). Specifically, starting fromnow have for i = 0, . . . , N — 1,

= P/v-1,2 = 1 as before, we

N1),

j = 0,l).

(5.3.8a)

(5.3.8b)

(5.3.8c)

In principle, generating function methods could now be used to study theseequations and develop a routine for evaluating Pi0. But it is unquestionablymore practical to proceed directly to their numerical evaluation (see Table5.1 for some values of moments), and to contrast their stochastic behaviourfor larger N using the diffusion arguments presented in Section 5.4 (cf.Exercise 5.7). By fitting quadratics in 1/N to the last two columns of Table5.1 for N = 191, 383 and 767, we obtain in place of (5.3.6)

N var 6v0.203188 + 0.117 20/iV + 0.6606/iV2

0.272 735 + 0.529 86/AT + 5.5823/iV2 (5.3.9)

These results differ from the relations at (5.3.6) for the [DK] model exceptfor the commonality of 0 = 0.203188 in E£N. The values for N in the tableare used to facilitate direct comparison with [DKj's Table 1. Note that themoments given are of X(oo) conditioned to be less than N — y/N: condi-tioning is necessary because otherwise the term PN-i,o = I/AT2 contributes


Table 5.1. Maki-Thompson model: mean and varianceof ultimate numbers of ignorants remaining

N95

191383767

EX(oo)19.427438.929577.9398

155.963

varX(oo)26.505152.6515

105.002209.725

0.2044990.2038190.2034980.203342

N var £N

0.2790020.2756620.2741550.273435

significantly to var£/v- The result would then belie the description of thedistribution as approximately normal, and would be reminiscent of the ma-jor/minor humps of the final size distribution of the general epidemic (cf.Figure 3.2). Pittel (1990) illustrates ways around the complications thatresult from this fact when attempting to use a moment method for provingasymptotics of the final size distribution. Watson (1988) remarks that thefact that the asymptotic value of NVSX£N for the Maki-Thompson modelis smaller than for the [DK] model 'accords with the change to smaller morefrequent jumps in the removal process'.

Figure 5.2(b) complements part (a) by presenting the distribution for theMaki-Thompson model. The probability of very few ignorants ever hearingthe rumour, which is approximately ^N~l for the [DK] model and N~2 forthe Maki-Thompson model, shows as a small peak in (a) and not at all in(b), at the scale of the diagram and for the values of N used.

Both the general epidemic and Maki-Thompson models have embeddedrandom walks that are of a simpler structure than that in the [DK] model.Specifically, because the process is strictly evolutionary and the jumps areto 'adjacent' states, much as in a birth-and-death process, the 'time' argu-ment n of the embedded jump process can be used to reduce the dimensionof the state space, from a two-dimensional to a one-dimensional lattice. Tostudy such processes we replace the embedded jump process {{X,Y)n} by{(X',Y')n] which is defined as starting from the same initial state and ashaving one-step transitions to (-1,1) and (0,-1) with the same probabili-ties as for {{X,Y)n}. But now {{X\Y')n} is no longer confined to a firstpassage into the states (0,j) (j = 0 , . . . , N); rather,

(X\ Y')n - (X\ Y

Clearly, (X, Y)n = (X\ Y')n for n = 0, . . . , Af.Now, after n jumps starting from (X',Y')0 = (N, 1), it must be the case,

since these n jumps have increments that are either (-1,1) or (0, -1), that

5.4. A diffusion approximation 145

n = N - X'n + (1 4- N - X'n - Y£), or 2X'n + Y^ + n = 27V + 1. Prom thisrelation we deduce that M = inf{n :n = 2N + 1 — 2X'n).

Also, conditional on the <j-field Tn = O~(XQ, ..., X'n), the expected decre-ment in {X'n} equals X'JN, or equivalent^, E(Xn+1 | ^ n ) = Xf

n{\ -N~l).This means that the functional Vn = [N/(JV - l)]nX^ satisfies

N

x'n) = [ ( T )L V 7 V i y J (5.3.II)

implying that {Vn} is a martingale with respect to the cr-fields {.Fn} o n

which M is a stopping time.Similarly, E([X; + 1 - X^]2 | ^ n ) = X'JN so that

and

and therefore {[AT/(AT - 2)]n(X;2 - X^)} is also a martingale.Sudbury (1985) uses this pair of martingales in conjunction with the

stopping time M to prove that as N —> oo, £# ~~> ^ m probability. Thisis a weaker statement than the convergence in distribution of \/N(^N - 0).Exercise 5.6 sketches a pair of martingales for the analogous embedded jumpprocess {(X;, Y')n} defined on a general epidemic.

5.4 A diffusion approximation

Daley and Kendall (1965) outlined an heuristic method for evaluatingvar X(oo), the variance of the number of ignorants remaining when spread-ing stops. The major step in the procedure is to find the variation in A,the stochastic analogue of the deterministic 'constant of integration' A in(5.2.3). We describe the mechanics of the procedure; mathematical detailsof the martingale and limit arguments involved can be found in Barbour(1972, 1974) and e.g. Watson (1988). Martingale arguments have also beennoted by Pittel (1990) and Lefevre and Picard (1994).


The essence of Kendall's Principle of Diffusion of Arbitrary Constants,as it applies to pure jump Markov population processes such as epidemicsor predator-prey models, embodies two features, one reflecting the meanbehaviour of the process and the other the variability about that meanbehaviour. The mean behaviour is represented by the so-called determin-istic path or deterministic version of the model; ideally, this is representedby an 'invariant of the motion', to borrow a phrase from classical appliedmathematics. Its probabilistic analogue is a martingale. The variability canbe regarded as a diffusion on adjacent paths, each path characterized by adifferent starting point. The local probabilistic analogue is the quadraticvariation of the martingale. The total variation is then the result of accu-mulating this local variability as the process evolves along its deterministicpath.

In the rest of this section we illustrate this approach as it applies to the[DK] rumour model and the general epidemic. Three more applications arestated as exercises.

By analogy with (5.2.3), but recalling that the rumour process (X,Y)(t)is lattice-valued, we start by writing

l (5-4.1)\{X(t),Y(t)) = 2X(t) + Y(t) ^

where A() is clearly a random variable. Recall from (5.2.3) that X(t) =\(x(t),y(t)) = 2x(t)+y(t)-Nln[x(t)/N] = 27V+1 (alH > 0) for (x(t),y(t))denoting the deterministic path. We use this relation later in the form

y(t) -1 = 2[N- x{t)} + JV In . (5.4.2)

Now the process A(t) is a martingale with respect to the cr-fields {î-} ={a({(X, Y)(s) : s < £})}, as is verified by evaluatingE(dA(t) | Tt-)

= [ ( - ! + j^))xWyW + i-iWW) + (-2)iy(t)[r(t) - i]l dt= Y(t) (N -X(t)- Z(t) - Y(t) + 1) dt = 0. (5.4.3)

To find its quadratic variation we find E([dA(£)]2 | Ft-), namely

-Y(t)Z(t) + 2Y(t)[Y(t)-l]\dt

= Y(t)^N-2N+-~+Y(t)~l]dt. (5.4.4)

5.4. A diffusion approximation 147

In order to compute the total quadratic variation over the evolution ofthe spread of the rumour, we replace the stochastic process (X, Y)(t) on theright-hand side by its deterministic version (x(t),y(t)) given by equation(5.4.2) with N > x(t) > NO and x(t) = -x(t)y(t). These substitutions give

fNvar A(oo) « /

Jx=N0N2 x

- I - TV + — + 2(N - x) + ATln ± I dx

Ar f1 l= N /

7dtx

= A r ( i - g ) ( i - 2 g + ag»)> ( 5 4 5 )

using (5.2.5) at the last step. Since \d\/dx\x=Ne = (1 — 20)/0, we arrivefinally at

varA(oc)« ^ L _ i , (5.4.6)

which is the source of the approximation at (5.3.7).Another way of linking (5.4.5) and (5.4.6) is as follows. Recall that in

the evolution of {(X, Y)(t) : t > 0}, the rumour stops spreading at timet = T = M{t > 0 : Y(t) = 0}, when X(T-) = X(T+) = X(oo). Since{A(t),^_} is a bounded martingale, the optional stopping theorem givesEA(TH-) = EA(0) = 2AT + 1. From the definition of A at (5.4,1) it thereforefollows that

E[A(T+) - A(0)] = 0 = 2EX(oo) + ATE[1 + • • • + x(2) + l] -2N-1,

so defining 0N by N0N = EX(oo),

-i=A"(oo) + l

Let X(oo) = EX(oo) + W « N0N + W for some random deviation VT. Then

JVJ-1-7V ^ T «Wf'(EX(oo)),


where f(x) = 2x + Nln(N/x), so f'(x) = 2 - N/x. Hence,

var A(oo) « vax W[f {EX(oo))}2 = (^—^-Yv&vXioo),\ U /

from which (5.4.6) follows.The same relation (5.4.6) follows in a discrete time setting for the jump

process {(X, Y)n} considered in Section 5.3, where we now use

(5.4.7)

and the ^-fields {Tn} = {&({(X,Y)r,r = 0 , . . . ,n})} . It is easy to checkthat E(An+i — An | Fn) = 0, i.e. {A.n,Tn} is a martingale.

We now apply the same argument as from (5.4.1) to the general epidemicmodel, with

for this model (cf. equation (2.3.6)). The jumps of (X, Y)(-) give

at rate pX(t)Y(t)dt,dA(t) \Ft- = { Mt) ' w - ' (5.4.9)

- 1 at rate jY(t)dt.

Prom this the martingale condition E(dA(£) | Tt~) = 0 is readily verified,and its quadratic variation is computed, formally, as

dvarA , „ , . , _ . — = i 8 y W | / ) + - ^ _ | . (5.4.10)dt

Then, using x = —flxy from (2.3.1), and assuming that n = p/N < 1,

NO X NO

.W ( 1 - , ) [ , + fegg]."e-y + «'>, (5.4.U)where we have used p\n8 + N(l — 6) = 0 to deduce the last equation (cf.e.g. equation (2.3.7)). Finally, as an analogue of (5.4.6),

v , , N0(l-0)(0 + K)varX(oo) « V JK

0)2 }- , (5.4.12)

5.5. P.g.f. solutions of rumour models 149

where K = p/N with 9 < n < 1.The epidemic model with K = \ has the same deterministic path as

the basic [DK] rumour model. Then the right-hand side of (5.4.12) equalsN6{\ - 0)(1 + 40)/(l - 20)2, with the same value of 0 « 0.203 as in Sections5.2-3. This asymptotic variance is clearly larger than (5.4.6), as is consistentwith the somewhat different behaviour in the initial stages of a generalepidemic process conditional on a major outbreak and of the [DK] rumourmodel process. In the latter, almost all transitions in the early stage of theprocess are of ignorants becoming spreaders, while in the former the ratioof transitions that increase the number of spreaders to those that decreasetheir number, is about 1 : K.

The Maki-Thompson model is a third model with the same determin-istic version but, as its stochastic origins are again different, so too is itsasymptotic variance (see Exercise 5.9).

The deterministic versions of the A>fold stifling and (a, p)-probability vari-ants of the [DK] model described in Section 5.1 lead to the same propor-tion of surviving ignorants when the rumour stops spreading for the val-ues (a,p) = (k~l, 1). The asymptotic variance 7Vvar£/v is accessible alge-braically in the (a, Invariant (see Exercise 5.7), but only numerically forthe fc-fold variant (see Exercise 5.8).

5.5 P.g.f. solutions of rumour models

In view of our earlier exposition of probability generating function methodsfor studying stochastic models for epidemics, it is proper to note the extentof their applicability to the models described in Section 5.1.

Because all the models are strictly evolutionary and the state space isfinite, it is in principle possible to find Pr{(X, Y)(t) = (ij) \ (X,Y)(0) =(TV, 1)} by enumerating the finite set of paths that connect (N, 1) to (i, j)and have non-zero probability of passage. The extent to which such com-putations can be organized and presented conveniently depends on whetherthe algebraic form of the probabilities and rates involved can be simplifiedsufficiently. These remarks apply equally to the results derived earlier usingp.g.f. methods.

P.g.f. solutions are worthwhile when they afford a simplification of in-formation found by other means (cf. remarks above), or if they provide asolution not available by another route. This, of course, is on condition thatthe solution provides some further understanding of the problem at hand.Certainly (cf. equation (3.3.6)) equations for the p.g.f. provide a useful sum-


mary of the information needed to derive equations for the moments of theprocess concerned.

Consider the [DK] model by way of example. As a continuous timeMarkov chain model it has the infinitesimal transition rates at (5.1.2).Defining Pij(t) = Pt{(X,Y)(t) = (ij) | (X,Y)(0) = (iV,l)} we readilydeduce from the forward Kolmogorov equations that

\{J + 1)0* + 2)Pij+2(t) - j[N - I( j - l)]po-(t), (5.5.1)

where the first term on the right-hand side vanishes for j = 1. Fromthese equations it is easy to find a differential equation for the moments(see Exercise 5.10). If we introduce the generating function F(v,w,t) =E(vx^wY^) (\v\ < 1, |it;| < 1), standard manipulation leads to a partialdifferential equation of first order in t and second order in v and w. Pearce(1998) has shown how this can be solved, by a procedure somewhat moreinvolved than the analogous one for the general epidemic (cf. Section 3.3.1).His method has overtones that recall the recurrence relations of Section 5.3,and for the same reason, namely, the strictly evolutionary nature of theprocess. His setting is more general and includes both the Maki-Thompsonmodel and the (a, p)-variant of the [DK] model.

Introduce

j=0

Then from (5.5.1) we have

Forming the Laplace transforms <&i(w, 6) = f£° e~0t fi(w, i) dt and using theinitial conditions $i(w,0) = SNiw gives

The function $i(w, 0) is a polynomial in w of degree N 4-1 - i, in which thecoefficients are functions of 6 and can be determined by a set of recurrencerelations. The analysis is an algebraic tour de force, describing the ma-nipulations that are possible for strictly evolutionary Markov chains, usingsquare matrices of order up to 3(N -f 1). A formal solution for the $i(w, 6)is obtained. See Pearce (2000) for details.



5.1 Show that the root 0N of equation (5.2.4) satisfies ON = 0 + C/N + O(N~2),where 0 = 0.203 188... and C = 0/(1 - 20) « 0.34.

5.2 Consider the non-unit root 0(a,p) of equation (5.2.8). Rearrange the equa-tion and take limits so as to deduce that 0(a) = limp— i 0{a,p) satisfies theequation (l + a~1)(l — 0) = — ln0 (this equation coincides with (5.2.5) whena = 1). Conclude also that 0(a,p) < 0max on 0 1and p —• 0.

5.3 Formulate a deterministic version of the A;-fold stifling model for which thefunction yi(t) (i = 1, . . . , k) denotes the number of spreaders that have hadcontact with a spreader or stifler on i — 1 occasions. Setting y = yi~\ \-yk,the differential equations are

x = -xy,yi =xy-yi(y-l + z),yi = (jfc-i ~ Vi)(y - 1 + z) (i = 2 , . . . , fc),

Deduce that the quantities x(t), y%(t) (i = 1, . . . , k) satisfy

(k + l)x(t) + kyi(t) + --- + yk(t) + N\n(N/x(t)) = (k + l)N + k (al l t > 0),

and that the analogue $k of 0 at (5.2.5) is the root in (0,1) of

Interpret the equality of 0k and 0{k~l) as in Exercise 5.2 in terms of therespective models defining them. [DK] illustrates the solution curves y andyi (i = 1,2,3) in the case k = 3.

5.4 (Maki-Thompson model). Show that the analogues of equations (5.2.1) thatfollow from the transition probabilities at (5.1.4) are

x = (-l)xy = -xy,y = (+l)xy + (-l)y(N - x) = y(2x - N).

Conclude that dy/dx satisfies equation (5.2.2) so the same solution (5.2.3)and other consequences follow. Investigate the time tu = inft{y(t) < 1}.[Maki and Thompson's original formulation is as a discrete time model inwhich exactly one pairwise contact occurs at each discrete epoch in time.This corresponds to scaling time in our formulation by A (A -f 1).]


5.5 ([DK] model with forget fulness). Modify the [DK] model as follows. Assumethat pairwise meetings occur at rate (3 per unit time, and that each spreadermay also become a stifler by means of 'forgetfulness' or 'loss of interest' at arate 7 per unit time, independently for each individual and of any interactionwith others in the population. To reflect this assumption, replace the lasttwo equations at (5.1.2) by

Pr{Ah(X,Y)(t) = (0,-1) I (X,Y)(t) = (x,y)}Pr{Ah(X,Y)(t) = (0,0) I (X,Y)(t) = (*,»)} = 1 - f3y{N + p - \[y - l))h,

where p = y//3- Deduce that for a deterministic version of the model, inplace of equation (5.2.2) we should have

"j — ^ • •

ax xHence conclude that, unless the individual rate of forget fulness 7 < 0N, therumour spreads to only a small fraction of the population, and that no matterhow small 7 is, about 80% at most of the population would hear the rumour.

5.6 (cf. end of Section 5.3, around equation (5.3.10)). Extend the state spaceXNI to {(ij) : i = 0,...,7V, j = I + N - i,I + n - i - 1,. . . ,0, -1 , . . . } onwhich a random walk {(X1 ,Y')n} is defined so as to coincide on XNI withthe embedded random walk jump process defined on the general epidemicmodel of Chapter 3 (see the start of Section 3.4.3). Show that the processes{Vn} and {Wn} defined by

and'r+dX'r+l-l)

V+X'r+p-2)r = o * + ' •

are martingales with respect to the a-fields {Tn} = {<T(XQ, • • •»^n)}- Find astopping time M such that JV — X'M is the final size of the general epidemic.

5.7 (Variance of final number with uncertain stifling). For the (a, Improbabilityvariant of the [DK] model with deterministic version following the differentialequation (5.2.6), the function \(x,y) = (14- ot)x 4- y — aNmx is invariantalong the path. Use the method of Section 5.4 to find

[2(1 - a) - a ( l + af]0 + a( l + a)202)

where 6 is the non-unit real root of (1 + ct)(l — 0) + a ln0 = 0 (so, 0 = 0(a)in the context of Exercise 5.2).


5.8 (Variance of final number with k-fold stifling). Consider the /c-fold stiflingvariant of the [DK] model and use standardized variables x(t) — X(t)/N,yx(t) = Yt(t)/N. Construct differential equations analogous to those of Ex-ercise 5.3 in conjunction with the procedure of Section 5.4 to conclude thatX(oo), the number of ignorants never hearing the rumour, has approximate

Here, 6k is as in Exercise 5.3, x(0) = 1, y = y{t) = yi -\ 1- yk, V\ (0) « 0(e.g. t/i = 10~6), i/i(0) = 0 (z = 2,.. . ,fc), and for 0 < t < oo, x = -a;j/,2/i = x y - y i ( l - x ) , and t/t = (yi_i - t / r ) ( l - x ) (i = 2 , . . . ,k). [Computationgives var X(oo) « 0.073807V (ife = 2), 0.02199AT (k = 3).]

5.9 (Variance of final number in Maki-Thompson model). Apply the calculus ofSection 5.4 to the model with transition rates at (5.1.4) to deduce that

^ ~°J = 0.272 736.— 2u

5.10 (Differential equations for first moments). Show that the first moments forboth the [DK] and Maki-Thompson models satisfy the same pair of equations(cf. equations (3.3.6) for the general epidemic model), namely

= -EX(t)EY(t) -cov (X(t),Y(t)),

= 2EX(t)EY(t) - EY(t)

6

Fitting Epidemic Data

When epidemic models are used in practice, it is essential to know howwell they fit the available data. This is particularly important if reliablepredictions are to be made, for example, of the number of AIDS cases tobe expected during the next year. Comprehensive accounts of the fitting ofvarious models to sets of epidemic data are given in Bailey (1975), Becker(1989) and Anderson and May (1991), among others. We have alreadyillustrated how certain models were developed to explain observed data.In Chapter 1 we modelled Bernoulli's data in Table 1.3, reviewed Abbey'swork on Aycock's data in Table 1.5, and gave Enko's data in Table 1.6 (seeExercise 1.3) and Wilson and Burke's Providence RI data in Table 1.7 (seeExercise 1.4). Further examples were given in Section 2.8 (Saunders's dataon rabbit populations affected by myxomatosis) and Section 4.2 (Enko'smeasles data again). This chapter provides a more extensive discussion ofepidemic data fitting: we illustrate its principles with five simple examples,two in which the models are deterministic, and three in which they arestochastic. Readers seeking further details are referred to the previouslymentioned books.

In a sense, this chapter serves to introduce the succeeding, final chap-ter which considers the control of epidemics: how can we use an epidemicmodel to evaluate possible strategies for countering a particular epidemicphenomenon? Simple models typically start with an elementary scenario;this may be modified subsequently to bring the model closer to the real-world context in which the phenomenon is occurring. For example, in mostepidemics, those gathering the data may assume a given initial scenario,which is later changed. This implies that subsequent data are no longerdirectly comparable with the initial data. Also, in a very real sense, epi-demic models vary with the historical period which furnishes their setting:one must recognize that many epidemics today typically occur in communi-ties where living conditions have changed much over the past few centuries.

154

6.1. Influenza epidemics: a discrete time deterministic model 155

Such change has been particularly rapid during the twentieth century, withfaster transportation methods and different life styles that affect both in-fection and removal rates.

6.1 Influenza epidemics: a discrete time deterministic model

The use of deterministic models of the Kermack-McKendrick type (see Sec-tion 2.3) for influenza epidemics, with modifications for the transfer of infec-tives between major population centres, dates back to the work of Baroyanet al. (1967, 1971, 1977) in the former USSR. Bailey (1975, Chapter 19)describes their methods. In Britain, Spicer (1979) modified this approachto obtain a discrete time model applicable to influenza epidemics, and todeaths due to influenza and influenzal pneumonia in England and Wales,and Greater London. We referred to this work briefly in Section 2.8; herewe outline his results in greater detail, including some numerical work. Themodel is a discrete time version of the general epidemic model in Section2.3, but with a more general removal rate.

Using the same notation as in Section 2.8, let us consider the discrete timemodel where t — 0 , 1 , . . . , refers to time in days. If in the population thereare, on day t, a total of Yt infectives with influenza and xt susceptibles, thenthe number yt+\ of new infectives produced at time t 4-1 will be

yt+l = j3xtYu (6.1.1)

where /? is the infection parameter. Let ipj be the proportion of infectiveswho are still infectious and mixing in the population j days after initiallycontracting the disease. Baroyan and co-workers found empirical estimatesof these ipj as below (with tpj = 0 for j > 6, i.e. no individual is infectiousfor longer than 6 days):

j 0 1 2 3 4 5 6fy 1.0 0.9 0.55 0.3 0.15 0.05 0.0

We follow Spicer in assuming that these estimates hold universally.The deterministic model starting with XQ susceptibles and Y$ = yo infec-

tives thus leads to the calculations in Table 6.1, where it is clear that

min(5,i)

= (3xtYt = 0xt

= xt-

156 6. Fitting Epidemic Data

Table 6.1. Progress of an influenza epidemic

t Total number of infectives Yt Susceptibles xt New cases yt

0 V>02/0 — YQ XO 01 ^02/1 •+• 12/0 = Y\ x\ — xo — 2/1 2/1 ==: / tooô

3 ^02/3 + "012/2 + </>22/i 4- V>32/o = >3 a?3 = ^2 - 2/3 2/3 =

t il>oyt+ipiyt-i-\ \-ipsyt-s = Yt xt = x t - i - y t yt =/3xt-iYt-i

Suppose that the proportion of individuals dying j days after infection isK<fj, where 'Y^jLoVj — 1* s o that K = Pr{death from influenza | individualis infected by influenza}. Then the number Q of deaths between times t — 1and t is given by

t

£>t~K z2 yt-uVu, (6.1.3)

where the (fu are based on empirical observations, and K is taken as 2 x 10~4,a rough estimate of the influenza death rate. Note that Qt here refers to asubset of all removals whereas z(t) as in (2.3.3) refers to all removals up totime t.

Spicer (1979) reports that, for his analysis, given that notification ofdeaths occurs weekly in England and Wales, the time unit was changedfrom one day to one week. He states that this 'does not seriously affect themodel owing to the short infectious period of influenza, and it introducesa useful smoothing effect'. In order to use the data on the %pj we mustcalculate the evolution of the epidemic via (6.1.2) and (6.1.3) on a dailybasis, and then accumulate them over a week to obtain

7T+6 7T+6

VT = X / y t anc* & = y i Ctt-lT t—lT

(cf. Exercise 6.1 for a continuous time analogue). The fitting procedure wasthe standard one of finding a pair of parameters a = f3x0 (it is impossible toestimate /3 and XQ separately) and yo which minimized the sum of squaresW of the differences between the observed and calculated weekly deathsover a period of V weeks, namely

rW=Y1^T-CT)2, (6.1.4)

T=0

where <^ and ^T denote respectively the observed and fitted numbers ofdeaths in week T. A weighted least squares method was also used.

6.1. Influenza epidemics: a discrete time deterministic model 157

5 0 0 -

4 0 0 -

3 0 0 -

2 0 0 -

100-

00

110

115

Figure 6.1. Weekly deaths from influenza and influenzal pneumoniain England and Wales, 1965-66. (Data from Spicer, 1979.)

Spicer fitted his theoretical results to data1 covering all influenza epi-demics in England and Wales for 1958-73 except those that were bimodal.His paper illustrates his results graphically for nine unimodal data sets ondeaths in this period, and for seven data sets on deaths in Greater Londonfor 1958-72. To the naked eye, the fits appear reasonably good but withseveral of the 16 fitted curves being somewhat too high at the beginningand too low at the end of the epidemic. The graph for 1965-66 for Englandand Wales is reproduced roughly in Figure 6.1; see also the graph of GreaterLondon 1971-72 in Figure 2.12.

One of the two quantities which needs to be estimated is the value a =/?£o, and this is given, for example, by Spicer for the nine sets of data ondeaths in England and Wales mentioned previously, a s 3 . 9 < a < 7 . 7 (seeTable 6.2), and the seven sets of data for Greater London as 4.5 < a < 7.7.The position of the optimum fit was found to be determined more by /3xothan by K and JJQ.

Table 6.2. Estimated values of a = /3x0 for England and Wales, 1958-73

Yeara

1958-595.2

60-615.5

61-626.5

62-634.7

65-66 67-684.5 5.9

69-707.7

71-723.9

72-735.7

Source: Spicer (1979). Note that these data exclude bimodal cases.

If we take the average value a = 5.51, this implies that an infective trans-mits infection to about five susceptibles, which may be an overestimate. The

1Spicer's calculations were 'based on observations reported by Stuart-Harris et al.(1950)'; the data are not otherwise identified; the only data from the 1950 referencematching this description give estimates of Y^=i V5d+j (d = 0,1,2) and *52equal to 0.31, 0.41, 0.26 and 0.02 respectively.


fits are, however, sufficiently good for the model to be used for predictionsof influenza deaths.

6.2 Extrapolation forecasting for AIDS: a continuous time model

One of the most elementary ways of making projections of new AIDS casesin a population is to fit some theoretical curve to the known data empirically,and then extrapolate this curve to predict new cases in the next one or twoyears. This involves no epidemic model of infection as such: the theoreticalcurve is chosen because it seems likely to fit the data; in fact, several curvesmay do so, as we shall see shortly. Practically speaking, the prediction canonly be made for a short period of time ahead, since the conditions whichhave held in the past for the known data may alter over a longer period.

In a paper on extrapolation forecasting, Wilson (1989) uses AIDS diag-noses A by quarter and year in Australia from 1982 to 1988 as the basicdata (see Table 6.3).

Table 6.3. Quarterly AIDS diagnoses in Australia 1982-1988

Quarter 1 2 3 4Year1982198319841985198619871988

01264389101

15365293101

215226083117

1321296895132

Note: Table 6.10 contains extended and updated data.

Wilson (1989) fitted several curves to these quarterly data including thethree empirical curves listed below, namely the log linear (T), log quadratic(Q) and log log (LT) models with Poisson errors, using the statistical pack-age GLIM. These are

T: In A = <* + /?*,Q: In A = a + /ft + 7*2,

LT: In A = a + pint,

where the LT curve was originally proposed by Whyte et al. (1987). Usingt = 1, . . . , 25 to denote the 25 quarters of the data in Table 6.3, Wilson

6.2. Extrapolation forecasting for AIDS: a continuous time model 159

150-

100-

5 0 -

() 15

#

110

«

115

120

•

125

150-

100-

5 0 -

(a)

Figure 6.2. 1982-1988 Australian AIDS data(a) LT ( ) and T (----) , and (b) Q (( ) (text from Wilson (1989).)

estimated values for these three curves as below:

(b)•) with fitted curves:-). (Based on data in

T: & = 1.601,Q: a = -0.446,

LT: a = -1.64,

/? = 0.1386;0 = 0.421,0 = 2.035.

7 = -0.0086;

The fit of T gives 159 for the last quarter (t = 25) and this was consideredunsatisfactory. The fits of both Q and LT curves were considered adequateas indicated in Figure 6.2.

The fit was further refined to take account of two events towards the endof 1987:(a) the extension of the World Health Organization's (WHO's) definition

of AIDS from its 1985 revision; this may have led to an increase inAIDS diagnoses; and

(b) the availability of zidovudine (AZT) at about that time in Australia,with the consequent effect of delaying the onset of AIDS.

Figure 6.3 shows the fitted values taking account of event (b); the disconti-nuity at t = 20 in (6.2.1) reflects (b) also.

Predictions were then made for the next 10 quarters on the basis of these(and other) models, in the cases where the effect of AZT was consideredor neglected, and for the cases where all the 1982-1988 data were used, oronly the data from 1985-1988. Figure 6.4 gives the predictions based on the1982-1988 data, with and without the AZT effect. The two LTjZ models(j = 1,2) considered, in which the effect of AZT was taken into account,


1 5 0 -

(a) (b)

Figure 6.3. 1982-1988 Australian AIDS quarterly data (•) with fittedcurves allowing for possible delayed onset of AIDS: (a) LT,Z ( )and T,Z ( ), and (b) Q,Z ( ). (From Wilson's data, by personalcommunication.)

though they varied only slightly, yielded very different predictions withintwo years.

4 0 0 -

3 0 0 -

2 0 0 -

100 -

400-

300-

2 0 0 -

100-

261

28I

30I

32 34I

361

261

281

301

321

341

36

(a) (b)

Figure 6.4. Predicted numbers of new cases of AIDS diagnosed in Australiafrom the first quarter of 1989, using all the data: (a) T ( — — — ),LT( ), Q (- — );(*>) T,Z( ), LT,Z( ), LT,Z(2)( ), Q,Z ( ). (From Wilson's data, by personal communication.)

The results in Figure 6.5 were obtained using only the last four yearsof data 1985-1988. These were deemed to be more satisfactory than themodels using all the data, and not quite so divergent in their predictions.

6.2. Extrapolation forecasting for AIDS: a continuous time model 161

400-

300-

200-

100-

126

128

130

132

-----

134

136

400-

300-

200-

100-

nU I

26128

i30

132

I34

136

(a) (b)

Figure 6.5. Predicted numbers of new cases of AIDS diagnosed in Australiafrom the first quarter of 1989, using 1985-1988 data only: (a) T ( ),LT( ), Q ( — - ) ; ( b ) T , Z ( - - - ), LT.Z ( ) andQ,Z ( ). (From Wilson's data, by personal communication.)

For yearly predictions up to 1993, the models chosen were

f 1 tTZ(4 years): InDt = \ ".551 + 0.158* (*<20),

1.763+ 0.083* (*>20);LT(4 years): In A = -0.978 + 1.814 In t; (6.2.1)

QZ(4 years): In A "to.764 + 0.272* - 0.004*2 (* < 20),521 + 0.272* - 0.004*2 (* > 20).

These models resulted in the annual predictions of Dt for 1989-1993 shownin Table 6.4.

Table 6.4. Predicted and Actual AIDS diagnoses by year in Australia

Year 1989 1990 1991 1992 1993Model

TZ (4 years)LT (4 years)QZ (4 years)Actual

600600600568

900900700591

12001100700797

17001400600793

23001800500805

While all models predict the same numbers for 1989, they range widelyfor 1993, the last year of the predictions. The actual numbers are nowknown and are shown in the last line of Table 6.4: on this basis the QZmodel came closest. These methods of extrapolation forecasting have nowbeen displaced by back-calculation models, described in Section 6.5.


6.3 Measles epidemics in households: chain binomial models

6.3.1 Final number of cases infectedIn Example 4.3.1 we considered Greenwood and Reed-Frost models involv-ing households of three infected by one infective. For these cases, we ob-tained the probabilities of the final number of individuals infected, so thatif n households in all were observed, their expectations would be as listedin Table 6.5 below.

Table 6.5. Final number of cases in households of threeinfected by one external infective

Final no. Expected numbers of households No. householdsCases Greenwood model Reed-Frost model Observed

0 not3 na 3 a1 3n(l - a)a4 3n(l - a)a4 b2 3 n ( l - a ) 2 a 2 ( l + 2a2) 3n(l - a ) 2 a 3 ( l + 2a) c3 n ( l - a ) 3 ( l + 3a + 3a2 + 6a3) n(l - a)3( l + 3a + 6a2 + 6a3) d

Total no. households n n nNote: Bailey (1975) regards this as a household of four with one member initiallyinfected; the formulations are identical.

For the Greenwood model, the likelihood LQ of the observed numbers ofhouseholds is

from which we can write down lnL^ and maximize it in a to obtain theequation for the maximum likelihood estimator (MLE) d

3a 4- 46 + 2c b 4- 2c + 3d 4ca 3d(l 4- 2d 4- 6d2)d 1 - d + 1 + 2d2 "*" 1 + 3d + 3d2 4-6d3 ~ ' ( ' ' '

Similarly, for the Reed-Frost model, with the different likelihood functionLRF say, the MLE d satisfies

3a 4- 46 4- 2c _ b 4- 2c 4- 3d 2cd 3d(l 4-4d 4-6d2) _d 1 - d + l + 2d"f l4-3d + 6d24-6d3 { }

Wilson et ah (1939) contains data from Providence RI for the total num-bers of infections in 100 households of three infected by one infective (seeTable 6.6).

6.3. Measles epidemics in households: chain binomial models 163

Table 6.6. Fit of the Greenwood and Reed-Frost models to Providence RI data

Fitted valuesTotal no. cases Data Greenwood model Reed-Frost model

0123

Total no. households

439

84100

2.51.5

14.981.1100

4.22.89.0

84.0100

Using the data values (a, 6, c, d) = (4,3,9,84), the values & for the proba-bilities of non-infection in these two models satisfying, respectively, (6.3.1)and (6.3.2) are

6LG = 0.291, aRF = 0.347. (6.3.3)The fitted values given by the second and third columns of Table 6.5 witha = &G, OLRF as at (6.3.3) yield the expected numbers shown in the last twocolumns of Table 6.6. The value for the chi-square goodness-of-fit statisticX% for the Greenwood model, combining the categories with 0 and 1 casesbecause their expectations are both small, equals 3.02/4.0 + 5.92/14.9 4-2.92/81.1 = 4.69; this is significant at the 5% level but not at the 2.5% levelfor x2 on 1 d.f. For the Reed-Frost model, again combining categories with0 and 1 cases, the x2 statistic on 1 d.f. equals zero, reflecting the perfect fit.

6.3.2 Cases infected for different types of chainThe data from Providence RI provide more information than those in Table6.6: the numbers of different types of chain as in Table 4.2 are also given.The values of o, . . . , h for these are listed in Table 6.7.

Table 6.7. Fit of the Greenwood and Reed-Frost models for specific chain types

Type of chainXt= 3 3

3 2 232 113 113 2 1 0 03 2 0 03 1 0 03 0 0

Total no. households

Genericabcdef9hn

Providence RIdata

431843

1067100

Greenwood0.90.40.78.22.76.5

31.049.6100

Fitted valuesmodel Reed-Frost model

1.20.71.02.23.47.3

38.745.5100

For the Greenwood model the log likelihood lnLcr, neglecting a constantterm, equals


so the MLE etc is given by

3a -f 46 + 4c + 2d + 3e + 2 / + .-f 6e + 5/

Similarly, for the Reed-Frost model, In LRF is a linear function of In a,ln(l — a) and ln(l 4- a) , and has derivative

dLRF 3a + 46 + 4c + 3d + 3e 4- 2 / + g gda OLJIF 1 ~f~

6 + 2c + 2a1 + 3e + 3 / + 3g + 3ft tn n r ,(6.3.5)1 -OflF

Setting this function equal to zero yields an equation for a#F-From the data listed in Table 6.7 we evaluate the MLEs as ac = 0.209,

OLRF = 0.231 (cf. equation (6.3.3)). The fitted values are given by sub-stituting these MLE values in the algebraic expressions in Table 4.2, eachmultiplied by the total number of households, n = 100, to give the expectednumbers for the various types of chain. The chi-square statistics, based oncombining chain types 1, 2, 3 and 5 for the Greenwood model, and 1, 2, 3and 4 for the Reed-Frost model, are \h = 26.2, XRF = 57-4 on 3 degrees offreedom; since Pr{X(3) > 11.4} = 0.01, the fit is very poor for both models.

Several reasons may account for the poor fit, not the least that individualsare heterogeneous (see Exercise 6.3), or that infectivity may vary betweenhouseholds, an issue to which we now turn.

6.4 Variable infectivity in chain binomial models

Becker (1981) generalized the concept of chain binomial models by assumingthat, if at time t > 0 there were Xt = x susceptibles and Yt = y infectives,then with ay as the probability of contact between the y infectives and xsusceptibles, Xt+\ would follow the binomial distribution Bin(a:,ay), i.e.

Pr{(X,F)m = (u,x-u) | (X,Y)t = (x,y)} = QaJ(l-ay)*" t t . (6.4.1)

Thus, ay = a for the Greenwood model and ay for the Reed-Frost model.Becker (1981) was concerned with the data collected by Heasman and

Reid (1961) on outbreaks of the common cold in families of 5 with 1 initialinfective (equivalent to households of 4 infected by 1 external infective). Hefitted both the general model, where the ay were estimated directly from the

6.4. Variable infectivity in chain binomial models 165

Table 6.8. Reed-Frost chain binomial probabilities for households of size four

Type of chain Probabilities* Expected values of probabilities*

4 4 a4 zq(3)/z(3)43 3 4a6(l-a) 4zq(b)zp(0)/z(6)43 2 2 12a7(l-a)2 12zq(6)zp(l)/z(S)4 3 2 11 24a7(l - a)3 24zq(6)zp(2)/z{9)4 3 2 10 0 24a6(l - a)4 24zq(5)zp(3)/z(9)43 2 00 12a5(l-a)4 \2zq(4)zp(3)/z(S)4 3 11 12a6(l - a)3 12zq(5)zp(2)/z(S)4 3 10 0 12a4(l - a)3(l - a2) I2zq(3)zp(3)(l + q + 12z)/z(S)4 3 0 0 4a3(l - a)4 4zq(2)zp(3)/z(6)4 2 2 6a6(l - a)2 6zq(b)zp(l)/z{7)4 2 11 12a5(l - a)2(l - a2) I2zq(4)zp(2)(l + q+ 13z)/z(S)4 2 10 0 12a4(l - a)3(l - a2) 12*g(3)zp(3)(l + q + I2z)/z(8)4 2 0 0 6a2(l - a)2(l - a2)2 Qzq(l)zp(3)[76z2 + (17 + 19q)z + (1 + q)2)/z(7)4 1 1 4a4(l - a)3 4zq(3)zp(2)/z(6)4 10 0 4a(l - a)3(l - a3) 4zq{0)zp(3)[3Sz2 + (12 + 9q)z + (1 + q)2]/z(7)4 0 0 (I-a)4 zp(3)/z(3)

*See equation (6.4.5) for definition of p, q, z and zp(-).

data, and the Reed-Frost model where ay = ay, to the data. The Reed-Frost probabilities are shown in column 2 of Table 6.8. Both fits provedadequate but not ideal, with that of the Reed-Frost model the better.

Greenwood (1949) had considered possible variations in a, while Bailey(1975) showed that if a is a random variable with a beta distribution, re-flecting the variable infectivity of households, then chain binomial modelscould provide an improved fit for the Providence RI measles data (see Bailey,1975, Chapter 14, pp. 254-260).

Dietz and Schenzle (1985) reviewed the history of household epidemicstatistics and considered the data of Heasman and Reid (1961) who hadobtained a = 0.886 for the probability of no contact in the Reed-Frostmodel. Becker (1981) found a = 0.884, and Schenzle (1982), after poolingsome of the data (see Table 6.9, column 3), obtained a = 0.893. Schenzle(1982), following Bailey (1975), also considered the case of a Reed-Frostmodel in which a is a beta random variable, but did not provide any details;we reproduce his figures in Table 6.9, without having full knowledge of hismethod of fit.

We now sketch the treatment of Gani and Mansouri (1987), from whichpaper further details can be obtained. Following Bailey (1975) and Section4.3 above, we have for the Reed-Frost model

Piiia) = Pi{(X,Y)t+1 = (j,i-j) I (X,Y)t = (i,y)} = (*Vl - avy-iavi.

(6.4.2)


Suppose that a (0 < a < 1) varies between households and follows the betadistribution with density function

(a,6>0). (6.4.3)B(a,b) ( a ' 6 >

Then, integrating Pij(ot) over all values of a gives

>3

da

(6.4.4)

We can simplify the notation by writing

zr(n) =

a -f- o

+ fcs) (0 < r < 1),

= i - q s = a + b

z(n) = zi(n) = + ks),

(6.4.5)thereby obtaining the formulae in column 3 of Table 6.8.

Table 6.9. Common cold data for households of size four with four model fits

ChainXt= 4 4

4 3 34 3 2 24 3 2 1 14 3 2 1 0 04 3 2 0 04 3 114 3 1 0 04 3 0 04 2 24 2 114 2 1 0 04 2 0 04 114 1 0 04 0 0

TotalX2

Observeddata

423131361442822

241131300

664

Reed-Frostvariable a

420.77133.5240.1710.431.821.096.122.410.53

23.1513.342.413.212.842.180.24

664.232.74

modelsfixed a405.2147.145.310.5

1.40.86.01.70.3

25.612.7

1.72.02.51.10.1

664.09.1

Becker'sgeneral model

403.9147.345.610.7

1.40.86.21.60.3

26.912.2

1.61.83.70.00.1

664.19.5

Schenzle'smodel*

420.9133.440.110.4

1.81.16.12.40.5

23.113.42.43.22.82.00.2

663.85.8

* A Bailey-type variant of the Reed-Frost model; see text for detail.

6.4. Variable infectivity in chain binomial models 167

We now estimate the parameters q and s, and test the goodness-of-fit ofthe model. Prom the third column of Table 6.8 and the observed data incolumn 2 of Table 6.9, we may write the log likelihood function of the chainprobabilities as

In L = C 4- 664 In q + 664 ln(q + s) + 663 \n{q 4- 2s) + 661 ]n(q + 3s)+ 230 ln(q 4- 4s) + 217 \n(q + 5s) + 50 ]n(q + 6s) + 241 lnp+ 110 ln(p + s) 4- 50 ln(p 4- 2s) 4-14 ln(p 4- 3s) + 5 ln(l 4- q 4- 12s)4-11 ln(l 4- <? 4- 13s) + ln(76s2 4- [17 4- 19q]s 4- [14- # )- 6641n(l + s) - 6641n(l 4- 2s) - 6641n(l 4- 3s) - 241 ln(l 4- 4s)- 241 ln(l 4- 5s) - 241 ln(l 4- 6s) - 105 ln(l 4- 7s) - 80ln(l -f 8s)-181n(l4-9s) (6.4.6)

for some constant C. Gani and Mansouri (1987) used the ZXMWD subrou-tine of IMSL to minimize — In L, and found for the MLEs of q and s

q = 0.8887 ± 0.0114, s = 0.0223 ± 0.0222, (6.4.7)

the corresponding information matrix being

T (31747.4 -4727.7 \ ( .1 = (-4727.7 8535.7 J ' ( 6 A 8 )

and variance-covariance matrix

_ / 0.3433 x 10~4 0.1901 x 10"4 \ , .~ \ 0.1901 x 10~4 1.277 x 10"4 ) ' [ }

They obtained as the corresponding estimates of a and b

a = 39.91, 6 = 4.999.

Based on the estimates (6.4.7) of q and s, and the expected probabilitiesin column 3 of Table 6.8, the fitted values for the Reed-Frost model withvariable a were calculated (see Table 6.9, column 3). These were thencompared with columns 4-6 of Table 6.9 carried over from results of Becker(1981) and Schenzle (1982).

To obtain chi-square goodness-of-fit statistics, all chains with fitted valuesless than 5 were combined. For the Reed-Frost model with fixed a (column4 of Table 6.9), Becker obtained \2 = 9.1 with 6 degrees of freedom (P-value = 0.166). For Becker's general model (column 5 of Table 6.9), \2 = 9.5


with 5 d.f. (P-value = 0.091). The Reed-Frost model with variable a givesa closer fit: \2 = 2.74 with 5 d.f. (P-value = 0.728). Schenzle's variant islikewise closer than the fixed a and Becker's model.

These results lend support to the hypothesis, originally made by Green-wood (1949), that there is variation in the susceptibility to infection betweenhouseholds. Bailey (1975) had already shown in his studies of the Provi-dence RI measles data that both Greenwood and Reed-Frost models withvariable a provided improved fits. The present application to common colddata arrives at a similar conclusion.

6.5 Incubation period of AIDS and the back-calculation method

At the end of Section 6.2, we mentioned that simple extrapolation fore-casting had now been replaced by the back-projection or back-calculationmethod as a means of predicting the number of HIV seropositives and AIDSpatients in a population. The models used in this method rely on the rela-tion between HIV incidence in the time interval (s, s + ds) and the subse-quent AIDS incidence in the time interval (t,t + dt) where t > s, followingan incubation period of length x whose distribution f(x) is assumed to beknown. This incubation period is defined as the time from the instant ofinfection by HIV to the onset of AIDS, where the onset is evaluated in termsof external clinical symptoms, or by a patient's T-4 cell count (also calledCD-4 count). When this count falls below a certain threshold, e.g. below200/mm3 blood, say, then the patient is said to have AIDS. The expectednumber a(t) dt of new cases of AIDS in (t, t + dt) is given by

a(t)= / I(s)f(t-s)dsJo

where I(s) ds is the number of HIV cases reported earlier in (s, s + ds). Theintegrated form of this equation expresses the cumulative number A(t) ofAIDS cases to time t in terms of the distribution function F(t) = JQ f(u) duof the AIDS incubation period, namely

A{t)= f F{t-s)I{s)ds. (6.5.1)Jo

The distribution f(u) is usually assumed to be of Weibull or gamma type,with an approximate mean of 10 years. On the basis of past HIV infectioncurves, fitted parametrically or non-parametrically, one can project futurenumbers of AIDS cases.

6.5. Incubation period of AIDS and the back-calculation method 169

Equation (6.5.1) can be used in one of two ways:(a) assuming I(s) and F(t) are known, one can calculate A(t); and(b) assuming A(t) and F(t) are known, one can calculate I(s).

It should be noted that data on A(t) are far more reliable than data on /(s),so that case (b) then allows a reasonable estimate of I(s) to be obtained. Onthe other hand, for predictions of A{t), even an imperfect I(s) can proveuseful. The key in both cases is the distribution F(t) of the incubationperiod, and we begin by considering this.

Several authors have studied the distribution of the incubation period,relying on data from transfusion-associated AIDS patients, for whom thedate of initial HIV infection is known. There are for example early papersby Lui, Lawrence et al. (1986) and Lui, Peterman et al. (1988) in which thedata are fitted by a Weibull distribution, with due allowance for truncationat the reporting date (31 December 1984 for the earlier paper). Denotingthe 3-parameter Weibull distribution by

F(t; A, r, 6) = 1 - (t > 0 > 0), (6.5.2)

these authors estimated the parameters A, r and 6 from the data, andconcluded, on the basis of 83 adult individuals undergoing transfusions from1978 to 1983, that the mean incubation period was 54 months = 4.5 yr, with90% confidence bounds of 2.6 and 14.2 yr. The estimated Weibull densityfunction for this case is shown in Figure 6.6 below. We used the parametersA = 0.000044 5, f = 2.51 and 6 = 2.5 months for this density function, andour time axis is in months. The modal value of the density turns out to belarger than in Lui, Lawrence et al. (1986).

0 .02-

0.015-

0 . 0 1 -

0.005 -

I20

I40

I60 80

I100

I120

Figure 6.6. Estimated Weibull density function for the transfusion-associated AIDS incubation period in months based on 83 adult patients.(Based on Figure 2 of Lui, Lawrence et al. (1986).)


Other authors have used different models for the distribution of the in-cubation period, such as the gamma density function

Among these are Anderson and Medley (1988), who estimated values ofa = 2.7, A = O.lOyr"1, hence a mean of 2.7/0.19 = 14.3 yr, based onfitting a gamma distribution to data from England and Wales up to April1988. They also fitted the 2-parameter Weibull distribution F(t\ p, /?) =1 - e " ( ^ , obtaining {3 = 2.33, p = 0.12 yr"1 and a mean of 7.4yr (cf. theparameters used for Figure 6.6, for the d.f. 1—exp (—[p(t — O')]P), which has/? = f = 2.51,0' = 0.2yr, p = 12(0.000044 5)1/251 = 0.221 yr"1). Lawlessand Sun (1992) considered in addition a log logistic distribution function

( 6 " >

Each of these distributions for the incubation period provides an adequatefit to various transfusion associated data sets. On the other hand, Bac-chetti, Segal and Jewell (1992), who estimated the incubation distributionfrom similar data on haemophiliacs using two different parametric mod-els (including the Weibull distribution as a special case), concluded thattheir analysis showed significant evidence against the Weibull distribution.We would comment simply that fitting any particular distribution withouthaving a model-based rationale for doing so, provides no assurance of thepropriety of the chosen distribution.

An excellent survey of methods used in the estimation of the AIDS incu-bation period, and the back-calculation method, may be found in Section 1of Jewell, Dietz and Farewell's book (1992) on AIDS Epidemiology. Furtherdetails are also available in some of the early papers (Brookmeyer and Gail,1988; Centers for Disease Control, 1990; Brookmeyer, 1991; and Solomonet al., 1991).

We now give some detail of the methods in Lui, Peterman et al. (1988) forpaediatric patients, as an example of the procedures commonly used. Theirdata consisted of the incubation periods U of 32 patients (i = 1,. . . , 32) inthe period 1979-1984, while the AIDS diagnoses were made in the period1982-1985. For patient i let Ri denote the number of months between trans-fusion and 31 December 1985, Li the number of months between transfusionand 1 January 1982 (the month of the first diagnosed case of AIDS). Then


assuming (6.5.2) holds, the likelihood of the observed data can be writtenas

L =

where ^ = 1 if the critical transfusion date for i was before January 1982,Si = 0 otherwise. Thus, when 8i — 1,

F{Ri) - F(Li) = e - A ( L - - ^ r - e~x^-e)r = Pr{Lt <U<

and when 8i — 0,

The maximum likelihood estimates of r and A can now be obtained, assum-ing the value 6 = 3 months which is found to maximize L for the values ofr and A derived from the equations

Writing these out in full detail, we have

a i n L 3 2 **

32

32 32

E ln(^ -«) - E A«< - °r -— OiAylji — uj LllyL/i — uJQ v ' -\- Ayiii — u) in^XL^ — I

The results obtained were f = 1.418, A = 0.00856 with 0 = 3, giving amean incubation period of 29 months or 2.4 yr with an approximate 90%confidence interval of 18 to 86 months. The mean is smaller than the equiv-alent mean for adults (4.5 yr) quoted earlier. The estimated Weibull densityfunction for the 32 paediatric patients is given in Figure 6.7, with the timeaxis in months as before. Note that this differs considerably from Figure6.6, both in its range and its shape.


0.025 -

0 .02 -

0.015 -

0 . 0 1 -

0.005 -

0-0 20 40

I60

I80 100

Figure 6.7. Estimated Weibull density function for the AIDS incubationperiod in months based on 32 paediatric patients. (Based on data in textfrom Lui, Peterman et al. (1988).)

An alternative approach is to use death or birth-and-death processes formodelling the associated T-4 cell count process as in Gani (1991) and Rossi(1991). Whatever model is used, one may assume that the distributionfunction F(t) of the incubation period of AIDS is adequately estimated.We now proceed with the back-calculation method itself.

6.5.1 The distribution of the AIDS incubation periodSolomon et al. (1991) used AIDS data for Australia from 1982 to 1990 toreconstruct HIV prevalence; these data are reproduced in Table 6.10 andthereby bring Table 6.3 up to date to 1990, including revisions of the datagiven previously.

Table 6.10. Quarterly AIDS diagnoses in Australia, 1982-1990

YearQuarter

198219831984198519861987198819891990

01264286112142150

16375196112121131

216296387143153167

13233070102155152143

Using the 2-parameter Weibull distribution in the form

(6.5.7)


with r = 2.55 and A = 0.002078yr~255 = (0.0887)V2.55yr-2.55 for t h eAIDS incubation period distribution, and adopting a parametric approachin which the HIV infection curve is assumed to be either(a) quadratic exponential, I(t) = aoexp(ait — a2t2), or(b) linear logistic, I(t) = (60 + bit)/(I + e62~M),then the AIDS diagnoses in a particular quarter (^,£7+1) are predicted by

A(tj+i) - A(tj) = f J+1 I(s)F(t -s)ds (j = 0,. . . , 33).

Here the increments in A(-) are known from Table 6.10, F(t) is given by(6.5.7) with (r, A) = (2.55,0.002078), and I(t) has one of the two para-metric forms mentioned; it is, of course, possible to reconstruct I(t) non-parametrically.

Given these conditions, assuming the epidemic to have begun in 1981,Solomon et ai. (1991) obtained the following estimates of Jy+ I(s)ds forthe years (y,y + 1) in the period 1981-1990.

Table 6.11. Estimated annual HIV infection incidence in Australia

Model(a) Quadratic

exponential(b) Linear

logistic

Year

(i)(ii)0)(ii)

1981

3434

6262

'82

912912

678678

'83

40464046

40324032

'84

43804380

55225690

'85

11601374

1000

'86

731000

1000

'87

11000

1000

'88

1000

1000

'89

1000

1000

'90

1000

1000

The values at i) for both the quadratic exponential and linear logistic curvesare calculated without adjustment, while at ii) they include various adjust-ments, such as an allocation of 1000 new HIV cases annually from 1986 inA and 1985 in B, or the effects of various treatments, in order to achievegreater realism. From these scenarios, HIV infection curves can be recon-structed under different assumptions.

These HIV infection curves can then be used to make short-term projec-tions of new AIDS diagnoses from (6.5.1). Table 6.12 sets out these for theyears 1991-1995 for Australia as given in Solomon et ai. (1991).

Table 6.12. Estimated annual HIV infection incidence in Australia

Year 1991 '92 '93 '94 '95Quadratic exponential I(t) 765 830 863 863 831Linear logistic I{t) 756 816 844 840 804Note: No actual numbers are available for comparison



6.1 As a setting for a continuous time analogue of (6.1.2) and (6.1.3), let X(t),Y(t), Z(t) denote the number of susceptibles, the (analogue of the) numberof infectives at time t, and the number of deaths by time t. Develop thefollowing relations:

rJo

= -0X(t)Y(t), i.e. ±

v(u)du + J <p(t~u)\dX(u)\\,

where X/J(U) denotes the probability that an infected individual is still in-fectious (and mixing in the population) time u later, and Kip(u)du is theprobability that an infected individual dies in the interval (u,u + du) afterinfection. Observe that, from an algorithmic viewpoint, the discrete timemodel is easily implemented, but not so the continuous time model.

6.2 Table 1.6 presents data of En'ko (1889) recording the daily numbers ofmeasles cases for several years from records at the Smolnyi Institutions. Findestimates of the variance of the dates of identification of cases within eachgeneration, (a) for each year as tabulated, (b) for the pooled data. Is thereany evidence for a model like sÛ) = am + 0mj, for m = a, b accordingas data are from (a) or (b) and j = generation number. How would youinterpret a& = 2aa, 0b = Pa ?

6.3 Consider a chain binomial model in which the probability pt of pairwise in-fection is no longer constant but is monotonically non-increasing in t. Thiscan reflect the propensity of individuals who are more gregarious or moreinfectious or more susceptible, to be infected ahead of others. Investigate theestimation of {pt} and the fit of such models to data like Aycock's measlesepidemic (Table 1.5) or En'ko's data (Tables 1.6 and 4.1).Hem ark: This discrete time model attempts to reproduce the effects of het-erogeneous mixing or infectivity and susceptibility as done for continuoustime models in Cane and McNamee (1982) and Daley, Gani and Yakowitz(2000).

7

The Control of Epidemics

One of the purposes of modelling epidemics is to provide a rational basis forpolicies designed to control the spread of a disease. This aim was alreadyevident in Macdonald's pioneering work on malaria. In his Presidentialaddress to the Royal Society of Tropical Medicine and Hygiene (Macdonald,1965), he referred to the development of a prevention strategy for epidemics,malaria in particular, stating that

there is only one way of doing this—through a working model... madeby assembling all we know, or more aptly, all that we believe to be sig-nificant, of the factors involved in transmission .. . in a form describablein mathematical terms.

A major contribution of Macdonald compared with his predecessors (hementions Farr, Brownlee, Ross, Lotka, Kermack and McKendrick) was topursue this approach to its

logical conclusion—of final interpretation of mathematical reasoninginto non-mathematical explanations of epidemiological happenings suchas could be readily understood by most practising epidemiologists.

In this spirit we consider three models for epidemics to illustrate possibleprevention policies of control by (a) education, (b) immunization, and (c)screening and quarantine. We could add that modelling is of vital impor-tance in evaluating the likely effects of spreading a disease deliberately as ameans of biological control, as with myxomatosis or the calicivirus to reducethe rabbit population in Australia.

Often the data available to decision makers are inadequate, as for examplein the case of HIV/AIDS in Africa or South East Asia. Yet policies needto be formulated, if only on the basis of rough qualitative measures. One

175

176 7. The Control of Epidemics

may, for example, need to know the likely effects of spending funds on twoalternative policies, or the optimal method of immunizing a population.Here, exact models may not be easy to formulate, though one usually tries tomake all modelling as realistic as possible. Accurate data may be impossibleto obtain, but one should always be in a position to minimize the cost ofa policy or to compare the effects of policy A against those of policy B,however approximately. Examination of the control methods discussed inthis chapter shows that they use and extend the simple models discussed inprevious chapters.

When a policy depends only on a single variable, it is relatively easy tominimize the cost. If two policies are to be compared, one can examinetheir respective costs and choose the cheaper policy. Alternatively, if thecriterion is not cost, one can rank the policies with respect to the criterionselected.

The control methods we describe in this chapter can be understood interms of the general epidemic model of Chapters 2 and 3, with pairwisetransmission rate (3 and removal rate 7, or of more complex models thatinclude this as the simplest case. We present strategies aimed at one ormore of the following results:(a) depressing the number of susceptibles in the population, and, where

possible, to below the threshold level p = 7//? described earlier in theKermack-McKendrick criticality theorem (Theorem 2.3.1 above);

(b) accelerating the rate of removal of infectives to reduce their mixingwith the population of susceptibles (i.e. increasing 7, hence p also);

(c) lowering the pairwise rate of infectious contact between infectives andsusceptibles (i.e. decreasing /?, thereby increasing p).

For example, immunizing some or all of the population reduces the initialnumber XQ of susceptibles; operating a screening program or raising publicawareness of higher disease prevalence may raise 7 or lower (5 (or both);discouraging the assembly of large crowds reduces /?.

7.1 Control by educationThe AIDS epidemic has spread rapidly throughout the world. But its ef-fect has been more limited in countries where a campaign for informationand education has been sponsored by the state, or by a foundation, as forexample in Switzerland.

In February 1987 a 'STOP-AIDS' advertising campaign was launched bythe Swiss AIDS Foundation to provide the population with detailed knowl-edge of the AIDS infection and its spread, and to discourage risk-prone

7.1. Control by education 177

Table 7.1. Annual expected increase 32 000(1 — m — pr)nain the number of infectives

na

0.050.100.15

r1

137227444116

Before advertising(m =

= 0.253

4116823212348

= 0.08,

5

68601372920580

p = 0.25)r = 0.50

1 3

1272 38162544 76323816 11448

5

63601272019080

r1

114822963444

After advertising(m =

- = 0.253

3444688810332

0.17,

5

57401148017220

p = 0.45)r = 0.50

1 3

968 29041936 58082904 8712

5

4840968014520

behaviour by recommending (a) the use of condoms in sexual contacts withmultiple or casual partners, (b) mutual faithfulness between sexual partners,and (c) the use of clean needles in drug usage (i.e. no exchanges betweenusers).

It emerged that between January and October 1987, the number of oc-casional sexual contacts of those aged 17-30 years in Switzerland declinedfrom 18 to 14% of the population, and among these the number always us-ing condoms in sexual contacts increased from 8 to 17%, while those usingthem sometimes increased from 25 to 45%. The number of condoms (about9.2 million) placed on the market in this nine-month period increased about60% over the same period in 1986 (see Hausser et al., 1988).

We can calculate roughly the effect of such greater sexual caution onthe development of an AIDS epidemic as follows, using illustrative numbersfor the land of Erehwemos with a population of 6.4 million where beforea campaign like STOP-AIDS begins, we assume that the number of HIVcarriers is 0.5% or 32,000 infectives. If on average each infective has nsexual partners in a year, with each of these partners being infected withprobability a, where 0.05 < a < 0.15, then the number of new infectives tobe expected would be about 32000na. If as a result of a STOP-AIDS typecampaign a proportion m < 1 of these infectives always use condoms, anda further proportion p < 1 — m use condoms for a proportion r of the time,then the number of new infectives expected in a year would be about

32000(1 -m-pr)na. (7.1.1)

This assumes that condoms provide total protection from HIV infection.Table 7.1 lists the number of new infectives expected in a year for different

values of the proportions m of regular and p of occasional condom users,the latter using them a proportion r of the time. The values of m and p inthe table are assumed to be similar to those reported in Switzerland. Theinfection rate is set at 0.05(0.05)0.15, and the number of sexual partners


n == 1, 3 or 5. Depending also on whether r = 0.25 or 0.50, the numberof new infectives computed by the model would decline between 16.3% and23.9%. This indicates the extent to which a STOP-AIDS type campaigncould be expected to modify the course of the epidemic.

One could imagine that a government might wish to minimize the totalcost C of (a) the annual expenditure y on pamphlets, advertisements andother educational material, and (b) the cost of medical treatment of HIVpatients, each of whom requires the sum c per annum. The total expenditureC at the end of the year, starting with N HIV seropositives, would be

C = [N + N(l - m - pr)na]c + y. (7.1.2)

Suppose now that plausible equations for m and p are

m = I ( l - e - f c » ) , p = | ( l - e - * » ) , (7.1.3)

where k > 0 is some constant, and the maximum proportions of regular andoccasional condom users, whatever sum y is spent on education, are | and| respectively. Then the expression for C at (7.1.2) becomes

C = Nc + JVnac[f(l - r) + \{l + 2r)e~ky} + y. (7.1.4)

This function is concave in y, so differentiation yields an equation for thevalue of y giving the least cost C, with solution

y = \ In (\k(l + 2r)Nnac). (7.1.5)

As an example, suppose we take r = 0.25, N = 32000, a = 0.10 andn = 3, while c = 1 is the financial unit. Then y = k~* ln(4800fc). Dependingon the value of h, the value of y rises from 0 to a maximum of 1765.8 beforedecreasing to 0 as k —> oo, as shown in Table 7.2 and Figure 7.1. Inthe optimal scenario, the budget y for educational material will be 1765.8,leading to a minimal cost C = 44451.9 where y/C = 3.97%. Choosing somesmaller y, say y = 61.7 for k = 0.1, yields minimal cost C = 36895.1 sothat y/C = 0.17%.

Table 7.2. Values ofy for different k

0.01 0.1 1.0 10-4.61 -2.30 0 2.30307.12 61.74 8.48 1.08

klnfc

y

0.0002083-8.48

0

0.0003-8.11

1215.48

0.000566309-7.48

1765.82

0.001-6.91

1568.62

7.2. Control by immunization 179

1500-

1000-

500-

0Figure 7.1. Graph of y as a function of In k.

While there is no claim that these are realistic figures, the calculationswe have given indicate the type of decision that policy makers can attemptto make in the allocation of funds. In this example, spending on educationa sum that is approximately 4% of the cost of medical treatment, results ina minimal cost C.

A recent paper of Kaplan (1995) on the merits of needle exchange forintravenous drug users provides an alternative example of HIV control byeducation. O'Neill (1995) has also studied epidemic models in which be-havioural change plays a role.

7.2 Control by immunizationImmunization has long been used as a method for controlling the spreadof an epidemic; in the case of smallpox, a worldwide vaccination campaignsucceeded in eradicating the disease totally (see Fenner et ai., 1988). Thefact that parents are sometimes lax in ensuring that their children are im-munized against preventable diseases like measles and poliomyelitis for ex-ample, has resulted in their random recurrence. Accounts of animal andhuman immunization schemes may be found in Knox (1980), Dietz (1981)and Anderson (1982), among other authors. Readers should, however, notethe cautionary remarks on Knox and Dietz's work in Anderson and May(1991, pp. 152-153).

In considering immunization as a technique for controlling the spread ofa disease, at least two policy questions arise, both subsumed in the pursuitof maximum effect with minimum effort:(a) how widespread can (or, should) the immunization be, and(b) which susceptibles should be immunized for this effort to produce the

best effect (e.g. should individuals be immunized at random, or shouldgroups such as schools or families be targeted)?


These questions involve us in detailed modelling of the population where theimmunization takes place, and in estimating its effect given some descriptionof how the disease spreads. If infection spreads homogeneously through thepopulation then question (b) is void.

Any quasi-realistic description of the spread of contagious infection usu-ally requires recognition that the population in which the process occurs isinhomogeneous. Yet even when the population is subdivided into groups ofindividuals belonging to different strata (however defined), those in a givenstratum are assumed to mix homogeneously amongst themselves, and tobehave similarly towards individuals of other strata. In other words, ho-mogeneity is not entirely dispensed with, for it is this very similarity thatunderlies the statistical approach. As in Section 3.5, we use the neutral term'stratum' to describe such sub-populations within which individuals are re-garded as identical apart from their disease status, noting that it may coverspatial variability, or distinct social behaviour, for example. Anderson andMay (1991, Chapter 12) describe an optimal immunization strategy withina spatially heterogeneous population; Becker and Dietz (1995, 1996) haveconsidered a population consisting of a number of smaller units (households,clubs or schools), and computed the effects of different strategies determinedby the characteristics of these units. The four particular strategies they dis-cussed were(i) random immunization of individuals;

(ii) households chosen at random and all their members immunized;(iii) preferential selection of large households for immunization;(iv) immunization of a fixed fraction of members in every household.

Suppose for the moment that immunization is to be administered uni-formly and at random in a population that is susceptible to a disease whichis characterized by a Basic Reproduction Ratio Ro (see Section 3.5). Recallthat the threshold criterion can be stated in terms of this as

the disease can spread if Ro > 1, whereas it cannot if Ro < 1. (7.2.1)

Then, a result dating back at least to Gordon Smith (1964, Figure 3) statesthat v*, the minimum fraction of the population that should be immunizedso as to prevent a major outbreak, is given by

v* = 1 - l/ifo. (7.2.2)

If immunization is carried out randomly and uniformly in a stratified pop-ulation, then equation (7.2.2), first established for a single homogeneously


mixing population, also holds for stratified populations with the general defi-nition of Ft®. To see this, recall that Ro is defined as the dominant eigenvalueof a linear mapping between strata determined by the mean number raôf j-stratum infectives produced by an z-stratum infective during its entireperiod of infectivity. Now a random uniform immunization programme atlevel v reduces the expected number of susceptibles uniformly by the con-stant factor (1 — v). The m^ are therefore reduced similarly, and thus alleigenvalues are reduced by the same factor. In particular the ReproductionRatio is (1 — v)Ro, and when (7.2.2) holds, this equals the critical level ofunity (see (7.2.1)).

That this relation fails to provide the optimum immunization strategy in agenuinely heterogeneous population, where pairwise infection transmissionrates (3 within groups are typically higher than between groups, is intu-itively understandable (Anderson and May (1991), p. 307). Immunizing thesame fraction in all strata results in relatively too many immunes in smallgroups and too few in large groups; the deficit in the latter is not made upby the excess in the former because the overall rate of infectious contactsis quadratic in the group size. Thus a uniform immunization rate, to beeffective in the population as a whole, must produce more immunes thanwould be needed for uniform immunity within the separate strata. Using adeterministic model, Anderson and May showed that, if the pairwise infec-tion rates (3ij for an z-stratum infective to transmit the disease to a specifiedj-stratum susceptible are fiij = (3H OT /3C depending only on whether i = jor i 7 j respectively, then the optimal policy (i.e. requiring the minimumtotal amount of immunization) is such that the non-immunized suscepti-ble numbers in each stratum are identical. Again, this is understandable,for such an immunization program reduces the susceptible populations to acollection of identical sub-population units.

In more detail and in a stochastic setting, consider a population of mstrata with Ni susceptibles, i = 1,... ,ra, as in Example 3.5.1, in whichthe pairwise infectious contact rates are (3H within each stratum and (3cbetween strata. We saw at (3.5.14) (recall also Exercise 3.10) that when noNi is small,

Ro > P" + (" y1 ) f r l* , (7.2.3)

where N = YîLi Ni/m; equality holds, when (3H ¥" Pc, if and only if all Niare identical. Suppose that as a result of an immunization programme theNi susceptibles are reduced to TV*, say. Then the resulting ReproductionRatio RQ, say, would satisfy (7.2.3) with N replaced by ~N*, and the equality


would now hold if and only if all N* are equal. Consequently, among thoseimmunization programmes that reduce RQ to RQ = 1, the programme re-quiring the smallest number of people to be immunized (i.e. the programmefor which m(N - ~N*) is least), is the one for which each Ni is reduced to

This analysis depends on none of the Ni being small, a condition whichis not generally met when each stratum is a household; in this case analternative setting is needed, such as was discussed in Example 3.5.2. Therewe showed that when (3H > 7, Ro ~ \(A + V'A2 + 4B), where

Here 4, but not 5 , is independent of the detailed household structure.Suppose that under an immunization strategy, the Ni are reduced to AT*,

so that A, B and Ro reduce to A*, B* and RQ respectively, where these aregiven by

B* = l^=* * v l

Y,i=iNi{Ni-l) ' (7.2.6)

Inspection of the expression for B* shows that it is a convex increasingfunction of each N*. Hence a maximal reduction in R$ is achieved for agiven overall reduction AN = £2=1 W""-^*) m t n e number of susceptibles,when max; Ni is reduced maximally subject to this given total reduction.Thus the conclusion immediately above (7.2.4) continues to hold in thischanged setting.

Specifically, given the number AN of susceptibles to be immunized, RQis reduced maximally by finding d for which N* = min(ATi,rf) and AN =Y!j=i(Ni - d)+> w*th 0 < N* < N^ This is a discrete analogue of thecontinuous variable result of Anderson and May (1991, Chapter 12 andAppendix G) referred to above (7.2.3). In fact, unless d is an integer, toattain the optimum exactly we must further adjust the N? for those i forwhich 0 < d — N? < 1 so that all the adjusted N* are integers.

Administratively, this optimal policy may not be as convenient as theuniform fraction (i.e. strategy (i)), or the targeting of specific households(either (ii) or (iii)), and so may prove more costly. But knowing the na-ture of an optimal strategy, provides a rational basis for the assessment of


Table 7.3. Values of the reproduction ratio and immunization fractions fordifferent strategies in the case /3H ^> 7

k™k

R*ôpt

*>unif

Kval l /none

1300

0.49980.53050.7435

0.14990.8592

2400

3200

475

Quasi-optimal strategy1.36260.20190.3007

1.74390.06100.1050

1.89330.01410.0283

All-or-none in households > k0.96400.4836

1.51060.2019

1.80270.0610

520

1.93790.00230.0054

1.91140.0141

65

1.94850.00000.0000

1.94850.0000

other policies. For example, if instead one were to immunize the largesthouseholds completely, how much more immunization would be needed?We discuss such questions in Example 7.2.1 and Exercises 7.1 and 7.2.Example 7.2.1. Consider a population of m = 1000 households consistingof rik households with k susceptibles (k = 1,... ,6) as shown in Table 7.3;let 7 = 1, (5c — 0.0005. Suppose first that /?# » 7 so that all the analysis ofExample 3.5.2 is applicable. From equation (3.5.20) the Basic ReproductionRatio .Ro of the population equals 1.9485 as shown in the last column.The rest of the entries in the table show the effects of three immunizationstrategies.

The effect of the Quasi-optimal strategy, when immunization is carriedout to reduce all households to a maximum of j susceptibles, is shownvia the reproduction ratio RQ that would then be realized. This entailsimmunizing a fraction vopt = Ylk>j(^ ~ J)nk/ Ylk>i ^nk °^ ne population;of course, vopt < fUnif = 1 — RQ/RO, the fraction needing immunizationunder a random strategy to reduce the reproduction ratio by the sameamount.

Consider next the effect of an all-or-none strategy in which all membersin every household of more than k members are immunized (strategy (ill)).Then the reproduction ratio R'o and fraction immunized âii/none a r e a s

shown. The population numbers in the households were chosen so as tomake some of the vaii/none fractions coincide with fractions vopt under thequasi-optimal strategy but for smaller k. Observe that the ratios underthe quasi-optimal strategy are smaller than the ratios under the all-or-nonestrategy when ^li/none = vopt.

We can draw at least two conclusions from this table, for this particulardistribution of households. First, to reduce the value of the reproductionratio to below 1, requires the immunization of about 50% more individ-


Table 7.4. Values of the reproduction ratio and immunization fractions fordifferent strategies in the case /3H = 0.27

knk

R*ôpt

Vunif

R'oâll/none

1300

0.49980.53050.6194

0.14990.8592

2400

3200

475

Quasi-optimal strategy0.96970.20190.2615

1.19330.06100.0912

1.28020.01410.0250

All-or-none in households > k0.65180.4836

1.02350.2019

1.21840.0610

520

1.30650.00230.0050

1.28920.0141

65

1.31310.00000.0000

1.31310.0000

uals under a random strategy than under the optimal one. Second, theall-or-none strategy is marginally more efficient than the random strategy.Strategies (ii) and (iv) are considered in Exercises 7.1 and 7.2.

Suppose now that /3# > 7 does not hold. The analysis we have givenmust then be reworked starting from the matrix M at (3.5.18) and findingits eigenvalue of largest modulus, much as in the concluding paragraph ofSection 3.5. We illustrate this approach in the context of Example 7.2.1 ex-cept that now f3H = O.27 while the other parameter values are unchanged.We computed the dominant eigenvalue by reduction of M to upper Hessen-berg form via Gaussian elimination and use of the QR algorithm as in Presset al. (1987, Chapter 11). This leads to Table 7.4 in place of Table 7.3.

Qualitatively, what is an optimal immunization strategy for an epidemicthat may spread in a community of households, for which the conditionsjustifying the approximations do not hold? Intuitively, since the optimalstrategy described earlier is a function of the graph-theoretic structure ofthe infectious path of the disease, then, assuming the same homogeneitywithin households and the same infectivity relation between households,we would expect the same strategy to remain optimal. In other words, thequantitative details may change, but the qualitative principle would remain.

7.3 Control by screening and quarantine

A third method of controlling the spread of a disease is by screening sus-pected infectives and quarantining those who are thought to pose a risk.This is a procedure often adopted in countries free of malaria (e.g. Aus-tralia) when a returning tourist is suspected of carrying the disease. InCalifornia, prisoners known to be HIV seropositive (i.e. HIV+) are internedin a separate correctional facility at Vacaville, but in general HIV screening

November 1985OctoberOctoberOctoberOctoberOctober

19861987198819891990

November 1992

76612321964313654116985

11565

7.3. Control by screening and quarantine 185

Table 7.5. Numbers of AIDS cases among US correctional inmates, 1985-1992

Survey period Number of cases Per cent increase from preceding survey

61%59%60%72%29%

66% [= 29% p.a.]Source: Thomas and Moerings (1994), p. 139.

is not compulsory. The fact that HIV is more prevalent in prisons thanin the community at large, suggests that HIV may spread more rapidly inconfinement than in the free community (see Exercise 7.2).

HIV is spread in prisons by sexual contacts and by needle sharing amongintravenous drug users (Brewer et al. 1988). The US Department of Justice(1993) reports that there were 11 500 known cases of AIDS in Federal prisonsfor the five months from November 1992 to May 1993, with an averagegrowth of over 50% annually during the past several years (see Table 7.5).This was considerably greater than the rate of increase in the total numberof prisoners (roughly, 75% between 1985 and 1992), and represented anincidence rate of over 1% (cf. less than 0.1% in the wider population then).Thomas and Moerings's book (1994) has documented worldwide concernabout the spread of HIV/AIDS in prisons.

Blumberg and Langston (1991) raised the question of mandatory HIVtesting of prisoners on entry into jail, but such a procedure is not currentlyacceptable. However, voluntary reporting or testing for HIV is possible(Stevens, 1993; see also Hsieh, 1991), and both medical and educationalhelp then becomes available to HIV 4- prisoners; Siegal et al. (1993) statethat HIV-f prisoners who do not report their status do not receive medicalhelp. Hsieh (1991) considered a model of HIV screening that incorporatesthe quarantine of infectives.

We consider in turn models for a single isolated prison, and for a prisoninteracting with the outside world. Then in Section 7.3.3, we discuss ascreening and quarantine procedure for which the overall medical costs overa fixed period of time T, including costs of treating undiagnosed HIV-hprisoners, are minimized.

While only deterministic models are used here, their stochastic equiva-lents can also be analysed (see Yakowitz, Gani and Blount (1996), Gani,Yakowitz and Blount (1997) for details, and Section 4.4 above for a relatedmodel).


7.3.1 The single prison modelSuppose that a prison containing N inmates, allows a simultaneous inflowand outflow of n < N prisoners at time t (t — 0 ,1 , . . .), after which thereare yt prisoners who are HIV-f and N — yt susceptibles. During the interval(t, t -h 1), which we take to be relatively short so that yt does not varygreatly in it, assume that homogeneous mixing occurs with an infection rate/?, so that {3yt(N — yt) new infectives are produced. This is a discrete timeapproximation based on the continuous time simple epidemic; for detailssee Bailey (1975, Chapter 5) or Section 6.1 above (Daley and Gani (1999a)discuss another model analogous to a simple epidemic). Thus at time t +1—there are

yt)) (7.3.1)

infectives. In the outflow of n = rN prisoners at time t -f 1, there are

= ryt[l + 0(N - yt)\ (7.3.2)

infectives, and amongst the inflow of n new prisoners we can expect a pro-portion fit to be infectives, where 0 < nt < 1, so at time t -f 1-f thereare

(1 - r)yt[l + 0{N - yt)] = /(yt,Mt) (7.3.3)

infectives amongst the N prisoners.Are there stable scenarios for this model? We could ask whether a sta-

ble regime is reached starting from t/o infectives at time 0-f under the as-sumption that ^t — M (a^ *)• T° this end we examine the transformationV *-»• fiy,^) f°r anY fixed points lying in [0, AT]. Observe that the functiong(y) = f(y, /i) — y is concave, with #(0) = rtfi > 0 and g(N) = n(/x — 1) < 0under our assumptions on fi. Thus, a unique stable scenario yt = ys (all t)exists, namely

_ (1 + (3N)(1 - r) - 1 + ^[(1 + 0N)(1 - r) - I]2 + 4n/i/?(l - r)Vs ~ 2)9(1 - r)

0(N -n)-r+ y/[P(N - n) - r]2 + 4/xr/?(7V - n)~ 2/3(1 - r) *

(7.3.4)In the limiting case ji = 0, y3 = 0 is also a fixed point and further

argument is needed to decide which of these roots of g(ys) = 0 equalsHindoo yt. The matter is easily decided: equation (7.3.4) always representsthe larger root, and is zero (hence, the only root in [0, AT]) if and only if


Table 7.6. Stable infective sizes in the single prison model for different n, fi, j3N

n25

50

0.010.0050.010.005

0.05.02.55.02.5

0.0539.825.4

9.04.5

0.075160.1154.814.5

7.5

0.1242.3239.631.818.7

0.125293.1291.382.571.2

0.15327.2325.9142.6136.4

/3N0.2

370.2369.3228.3225.3

0.25396.1395.4281.7279.8

0.5448.0447.7390.3389.6

5494.8494.8489.0489.0

(1 -f /3N)(1 — r) < 1. In other words, (7.3.4) always gives the required root:when fji = 0 it simplifies to

0 H/3N <r/(l-r),N — r/[/3(l — r)] otherwise. (7.3.5)

We interpret this equation as indicating that infection is contained to smalllevels within the prison if the former case holds, i.e. if n > f3N2/(l + f3N).

Example 7.3.1. Suppose that N = 500. Table 7.6 indicates the stablevalues ys under a range of values for (3 for n = 25 and 50, and /x = 0.01and 0.005. The time taken to attain these stable values depends on boththe initial infection intensity y0 and the contact rate /3 (see Exercise 7.3).Note that when j3N exceeds the threshold r / ( l — r) the endemic level risesappreciably above the endemic level // of the new prisoners, though thethreshold does depend on interplay of all of /i, f3N and n. For f3N belowthe threshold, ys « nfi/[r - 0N(1 - r)], « fiN for /? -+ 0.

7.3.2 Interaction of a prison with the outside worldConsider a city of fixed population size M with Yt infectives at time t =0 ,1 , . . . , in which there is a single prison with N inmates of whom yt areinfectives. We make the same assumptions as in the previous section con-cerning the prison, so that in (£, t + 1) there are /3yt(N — yt) new prisoninfectives. The same proportion of infectives leaves as before.

For the city population, homogeneous mixing and an infectivity rate /?0per susceptible-infective pair produces /30Yt(M - Yt) new infectives. Sup-pose also that deaths occur at rate 7, and that the number of births, allhealthy, equals the number of deaths, thus keeping the city population sizeconstant. Then at time t + 1—, before exchange with the prison occurs, thecity has

(l~j)Yt+p0Yt(M-Yt)


infectives, so with r = n/JV, R = n/M,

y t + 1 - (l - fl)Yt[i - 7 + A)(M - ^t)] + n/ t[ i + /?(# - yt)] = h{Yt,yt),yt+l = RYt[l - 7 + /?o(M - Yt)} + (1 - r)i/t[l 4- /3(JV - y*)] = /2(*t, Vt).

(7.3.6)Write f{Y,y) for the vector-valued function (/i(Y,i/), f2(Y,y)). For a real-istic model we require that for (V, y) in the rectangle [0, M] x [0, AT], /(Y, y)should also lie within the rectangle. Since / is continuous, and the rectangleis a compact set, there is at least one fixed point (Ys,ys) = f(Ys,ys) whichwe interpret as a vector of stable values for Y and y. Then (Ya,y8) satisfies

(Ys\ _ ((l-fl)[l-7 + A>(M-n)] r[l + (3(N-ys)} ) (Ys\U J " I i ? [ l - 7 + A ) ( M - r s ) ] ( 1 - r ) [ l + / ? ( # - y , ) ] J { y s )

(7.3.7)Addition of these two equations gives

Y.+y.= Ya[l - 7 + A)(M - Y.)\ + ys[l + /J(iV - y.)], (7.3.8)

and hence(3OYS(M - Y.) + Pys{N - ys) = 7 r s . (7.3.9)

Equivalently, the number of new infectives in (t,t + 1) equals the numberof infectives removed in (£, £ + 1). Substitution in (7.3.7) so as to eliminateone or other of the quadratic terms yields

rys = R(l - 7)YS + (1 - R - r)(3ys(N - ys)(l-r)jYs-(l-R-r)(30Ys(M-Ys),

showing that (0,0) and (M, N) are always fixed points, and that they arethe only fixed points when 7 = 0, i.e. when the city population has no birthsand deaths.

Equation (7.3.9) shows that for the disease to be confined to a smallnumber, eM say, of the city population, the various parameters must sat-isfy both (30M(l—e) < 7, i.e. the infection rate /3o must lie below a thresholdlevel determined by the relative removal rate 7/M, and the endemic infec-tion level in prison ys must likewise be contained by a function, namely0ys{N-ys) < (<y-(3oM)eM. Ideally, having (3 < 4(7-/3oM)eM/N2 wouldcontain the incidence in the city population irrespective of the incidencerate in prison. See also Exercise 7.4.

Rewrite (7.3.7) as 0 = gi(Ys,ys) = fi(Ya,y8) - Ya, and 0 = g2(Ys,ys) =f2(Ysyys) — ys. For given z/, <7i(Y, y) is concave in Y, with stationary pointY(y) equal to

v , ^ - A + yM + 4fl>(l - R)ry[\ + (3(N - y)}Y{y)= 2/30(l - R) ' ( 6 '


where A = (1 - R)[l - 7 + PoM] - 1. Similarly, for given Y, 92{Y,y) isconcave in y with stationary point

(Y, - B + y g + 4/3(1 - r)RY[l - 7 + ft(M - Y)}y{Y) = 2/3(1-r) ' ( 7 - 3 ' 1 2 )

where B = (1 — r)/3AT — r. Alternatively, the first expression in (7.3.10)shows that ys and Ya are related by ys = y<i(Ys) where

4)9(1-

with / = R + r and B2 = /?iV(l - f)~r.Example 7.3.2. Suppose we take M = 50000, AT = 1000, fa = 1.5 x 10~7,/? = 1.5 x 10~3. Recall that 7 and n define the equivalents of immigration-emigration rates for the city and prison respectively. For the four pairs ofvalues of 7 and n shown in Table 7.7 the stable values Ys and ys are asshown; note that the infection rate in prison, which is almost 100%, is littleaffected by 7, whereas the infection rate in the city is affected by both 7and n, reflecting respectively the removal rate and rate of introduction ofinfectives 'new' to the city population.

Table 7.7.n

100

50

Stable infective sizes

70.030.0150.030.015

in a prison-and-city modelYs

4096.39404.32132.35446.9

Vs

932.3940.6966.4968.8

7.3.3 A quarantine policy in prisonGani, Yakowitz and Blount (1997) studied a cost effective model for a prisonin which prisoners may be screened regularly, and quarantined if found tobe HIV-K We follow the single prison model of Section 7.3.1 with thedifference that at time t = 0 , 1 , . . . , the prison population of size Af is nowsubdivided into three groups: susceptibles xt, infectives not in quarantineyt, and quarantined prisoners qt, where

xt + yt+qt = N. (7.3.14)

Assuming homogeneous mixing between susceptibles and infectives in (£,t + 1), there are

=Vt+ fixtyt = yt(l + 0xt) (7.3.15)


infectives at time t + 1—. At this epoch we suppose there is an inflow ofn = rN new prisoners of whom a proportion /zt+i are HIV+, so that

wt+i = n/xt+i (7.3.16)

infectives are added. At the same time, n prisoners leave, among them

vt+i = rzt+i and wt+i = rqt (7.3.17)

non-quarantined and quarantined infectives. Thus before a new screen-ing and quarantine of prisoners, the total numbers of non-quarantined andquarantined infectives, at t+ 1—, are, respectively,

(7.3.18a)

andtt-ttft+i = ( l - r ) g t . (7.3.18b)

Let us now screen a proportion a (with 0 < a < 1) of the non-quarantinedprison population, and quarantine those who test HIV-h Then at £ + 1+ thenumbers of non-quarantined and quarantined infectives, and susceptibles,are respectively

yt+1 = (1 - (j)[/i,+1n 4- yt(l + /Ja;t)(l - r)], (7.3.19a)- r) + <7[fit+m -f yt(l + ^ t ) ( l - r)], (7.3.19b)

(7.3.20)

Suppose the costs involved per prisoner in such screening, quarantiningand medical treatment of prisoners are

a for the HIV test,b for quarantining and associated treatment, (7.3.21)c for treating an unquarantined infective.

Then the cost Ct for the time interval (t — 1, t) is

Ct = aa(N - qt) + bqt + cyu (7.3.22)

and the cost over the time interval (0, T) is

Ct. (7.3.23)t=\


8 0 -

6 0 -

4 0 -

2 0 -

0-0

I0.1 0.2

I0.3

T0.4

Figure 7.2. Cost of an epidemic in prison as a function of thescreening rate cr, for finite ( ) and infinite ( ) time horizons.

Example 7.3.3 (cf. Gani, Yakowitz and Blount (1997)). Assume thatb > c > a > 0, so we can expect the optimal screening rate o1 < 1. Weconfirm this expectation for the parameter values TV, /i, n and {3 muchas in Example 7.3.1, under the two scenarios of stability, and of a finitetime horizon T = 50. In general the optimal screening rate differs for theseconditions. It is arguable that the screening rate should be chosen accordingto the worse scenario which, in the case of a low initial incidence rate, isalways the stationary case.

Figure 7.2 shows the average cost J(<r,T)/T and its limit for T —> oo asfunctions of the screening rate a in the case /i = 0.005, n = 25, (3N = 0.15with yo = 0. It shows that the optimal screening rate is about 10% or 15%depending on whether a finite (T = 50) or infinite time horizon is used.Table 7.8 shows the optimal screening rates for the two scenarios for therange of parameter values considered.

Table 7.8. Optimal prison population screening rates for isolated prison model

n25

50

A*0.01

0.005

0.01

0.005

0.050.0

0.0380.0

0.0080.00.00.00.0

0.0750.0270.082

0.00.051

0.00.036

0.00.003

0.10.0650.1190.0320.0860.0260.075

0.00.040

(3N0.1250.1000.1510.0650.1160.0630.1080.0260.073

0.150.1290.1800.0940.1440.0950.1390.0570.102

0.20.1820.2310.1460.1940.1510.1930.1120.154

0.250.2280.2760.1910.2370.2000.2400.1590.199

0.50.3960.4400.3570.3960.3780.4120.3340.367

For each (n,/x), the upper (lower) rows correspond to finite (infinite) time horizon.


In terms of our model with constant prison population size TV, the effectof screening is two-fold: it periodically reduces the number of infectives whomay infect others, and it also reduces the size of the population where in-fectious contact can occur. Another variable that is of concern in managinga screening procedure is its frequency, but this is not investigated here (seeExercise 7.6).


7.1 (Community ofm households with /3H ^> j)- In Example 7.2.1, the assump-tion PH » 7 implies that either none or all of the members of any givenhousehold become infected in the course of an epidemic. Regard such a com-munity of m households as consisting of n' strata, where stratum k consistsof the nfc households with j susceptibles initially and n = max{fc : nk > 0}.The probability that a household with j infectives would infect a specific k thstratum household of susceptibles (assuming no other households have anyeffect in the meantime) equals

T711

VJ^_A 27 + ik/3c \ 7 + fc/fc / 7 Pc '

where pc = j/Pc and the approximation is valid for Pc <C 7. Then ra^.,the mean number of k th stratum households infected by a household of jinfectives, is given by

pc

and the Basic Reproduction Ratio RQ in terms of household units is thelargest eigenvalue of the matrix M H = (m^.). Verify that

- A) «(-xr'-'f ypc

(cf. (3.5.19)), so that

RH ~ V ^ fe2yifc

~ kî PC

In the context of Table 7.3, conclude that in order to reduce RQ below 1,about two-thirds of the households would need to be immunized if Becker andDietz's strategy (ii) were followed; alternatively, since 0 < pc — (n\ 4-4712) <9ri3, it would suffice to immunize all members of households with three ormore susceptibles under a quasi-optimal all-or-none strategy (iii).


7.2 Consider a community of 1000 households as in Example 7.2.1. Supposethat in a randomly chosen proportion p of these households, all membersare immunized, leaving an expected number (1 — p)nk households with ksusceptibles. Assuming /3H > 7, show that Ro is reduced to .R0(l — p) givenby

R'0(l - p) « (1 - p)\(A + V*2 + 45/(1 -p)) > ( 1 - p)Ro,

where A, B are given at (7.2.5). By way of contrast, immunizing the sameproportion of randomly chosen members of the population yields -Ro (1 ~ p)given by

7.3 For the prison model of Section 7.3.1 investigate numerically the rate ofattainment of equilibrium values with N = 500, different /3 and yo.

7.4 In the context of the model of Section 7.3.2, show that when M » AT, thestationary solution (Ys,ys) cannot have Ys/M <C ys/N if (5Q — /3.

7.5 In Example 7.3.2 investigate the rate of convergence to equilibrium (cf. Ex-ercise 7.3).

7.6 In Example 7.3.3, investigate the effect on the optimal screening rate o ofscreening only every d time units.

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Subject Index

Abelian property of Laplace transform,77

AIDS, 154control by education, 176diagnoses in Australia, 158, 172forecastsback-projection method, 168, 173extrapolation method, 158, 161treatment dependent, 159

incidence, vs. HIV incidence, 168incubation period, 168gamma distribution, 170Weibull, 169

transfusion induced infection, 169WHO definition, 159See also HIV

Anopheles mosquito, 46infestation level for endemicity ofmalaria, 47

Aycock's measles dataset, 14model-fitting, 174

back-calculation method, 170Basic Reproduction Ratio RQ, 17, 38,

42, 85, 87, 89characteristic equation root, 90determines vaccination level, 180eigenvalue, 181, 192epidemic in households, 192

Berthollet, C. L., 6Bessel function, 63beta distribution, 116, 166binomial probabilities, 106

Bin(-) notation, 89, 106

birth-and-death process, 42, 73, 102approximates S-I-R epidemic, 74forward Kolmogorov equations, 74

j th-generation descendants, 102limit conditional behaviour, 102

birth process, 57bivariate Markov chain

in chain binomial model, 107branching process on finite graph,

model for epidemic, 123, 137See also Galton-Watson branchingprocess

Breslau, 6Britain, 1

carrier-borne model, deterministic, 45compared with general epidemic, 45control, 45Law of Mass Action, 45modified quasi general epidemic, 55removal process, 45

carrier-borne epidemic, stochastic, 94death process in, 94, 97, 99duration, 98first moment comparison with deter-ministic model, 104

forward Kolmogorov equations, 95,99p.g.f. method, 95

ultimate size distribution, 98, 132chain binomial epidemic, 11, 105, 162

definition, 107fitted to household data, 162for inference in epidemics, 107, 162

205

206 Subject Index

chain binomial epidemic (cont.)non-infection rate a, 106, 115reason for name, 12, 105relation to binomial distribution,

107varying infection probabilities, 115,

164with replacement, 118See also Greenwood model, Reed-Frost model

Chapman-Kolmogorov equations, 81characteristic equation, 90chickenpox data, 14childhood mortality, 3, 6closed population, 9common cold, 20

Heasman's household data, 166fitted by chain binomial, 164

communities, interacting, epidemic in,90

embedded jump technique, 104in m households, 91, 92

asymptotics for large m, 93generation-wise evolution, 92

comparative death risks, 2comparison of epidemic outbreaks, 33competing risks, 3contact, infectious, time of, 106contagious infection, idea of, 8

mechanism, 15control of epidemic, 13, 175

cost minimization, 176, 178education, 176immunization, 13, 179raising public awareness, 176screening and quarantine, 184

costs of epidemic, 13, 178criticality condition, epidemic and

branching process, 88cumulative hazard, 3

death, causes of, 2infant mortality, 3, 6

death process, 57death rate, per capita, 3death risks, comparative, 2

deterministic modelapproximate stochastic model, 11,85

fitting epidemic data, 154from stochastic prototype, 85

d.e.s. for first moments, 68, 104, 153deterministic vs. stochastic methods, 56

modelling, 85stochastic lag, 101

differential equationshypergeometric, 65Lagrange's solution method, 75, 100partial, for carrier-borne p.g.f., 96partial, for S-I p.g.f., 65partial, for S-I-R p.g.f., 68

discrete time general epidemic, 55deterministic model, 48

discrete time stochastic models, 85, 105distribution of infectious contacts, 124doubly stochastic Markov process, 95doubly stochastic Poisson process, 95drug users, HIV incidence among, 177duration time of epidemic

carrier-borne model, 99general epidemic, 83, 104pairs-at-parties model, 127phase decomposition, 63, 104Reed-Frost, 113simple epidemic variant, 130

replacement model, 120simple epidemic, distribution of, 62asymptotic behaviour, 63Laplace-Stieltjes transform, 62mean and variance, 101

embedded jump process, 79Chapman-Kolmogorov equations, 81intermediate states, 81rumour models, 141state probability computation, 103transition probabilities, 80

embedded random walk, 125emigration-immigration model, 118

endemic infection distribution, 121end of epidemic, 12, 20, 21

Subject Index 207

endemic disease, deterministic model,46

emigration-immigration, 120stationary point, 47stable vs. unstable, 47vector-borne disease, 46

epidemic,control of spread, 13end of, 12, 20, 21generation-wise evolution, 38intensity of, 33, 38, 74medical costs, 13overlapping, 105phase of, 63total size, prediction, 13

See also ultimate size of epidemicepidemic curve, 22, 33

fitted by Pearson curve, 8influenza mortality data, 50, 157

epidemic duration, See duration timeof epidemic

epidemic in family, 56chain binomial models, 107

epidemic in prison, 185costs of control policy, 190endemic infection level, 187, 189prison-and-city model, 187quarantine policy, 189single-prison model, 186

epidemic in stratified populationcommunity of households, 92, 192general epidemic, deterministic, 35,

53threshold phenomenon, 37

general epidemic, stochastic, 85immunization effects, 181threshold phenomenon, 89

interacting communities, 90pair wise contact rates, 23, 36, 86simple epidemic, deterministic, 23

epidemic model with replacement, 118See emigration-immigration model

epidemic model, stochastic,See stochastic epidemic model

epidemic model usesevaluate biological control, 51, 175evaluate changing scenario, 154explain data, 154

predictions from, 154working model for prediction, 175

epidemic modellingaims, 13discrete vs. continuous time, 48time-grouped data, 48

epidemic models,Gart, 53See also named model types:carrier-borne epidemic, stochasticchain binomial epidemicemigration-immigration modelepidemic in stratified populationgeneral epidemic, deterministicgeneral epidemic, stochasticpairs-at-partiessimple epidemic, deterministicsimple epidemic, stochastic

epidemic process, 15epidemic with carriers, See carrier-

borne epidemicepidemics in Britain, 1equilibrium level in endemic epidemic,

47Erehwemos, 177Euler's constant, 58Euler's formula, 63exchangeable mixing of population,

126extinction probability

general epidemic, approximation, 73S-I-R epidemic, stratified popula-tion, 87

extrapolation forecastingempirical basis, 158limitation, 158superseded by back-calculation, 161

extreme value distribution, 101, 169

final size, See ultimate size of epidemicfinitencss of population, effect on gen-

eration number distribution, 44first passage time r.v.s, 81fitting data

chain binomial model, 114deterministic model, 154stochastic model, 154

208 Subject Index

forward Kolmogorov equationsbirth-and-death process, 74carrier epidemic, 99general epidemic approximation, 74

Galton-Watson branching process, 44,88, 125

embedded in random walk, 125gamma distribution

AIDS incubation period, 170gamma, Euler's constant, 58gamma function, 63

Laplace transform of duration of S Iepidemic, 101

Gart epidemic model, 53general epidemic, deterministic, 9, 27

branching process on graph, 123comparing several outbreaks, 33d.e.s. approximate first moments, 68discrete time, 55fitting influenza data, 155in stratified population, 35, 53cross-infection mechanism, 38differential equations, 36removal rate vector, 37solution curve (trajectory), 37threshold theorem, 37total size, 37transmission matrix, 37

intensity parameter, 33, 38Kermack-McKendrick thresholdtheorem, 29, 37

latency state, 53final size, 53

realistic for measles, 105total size, 29, 37

general epidemic, intensity of, 33, 38,76

total size, ultimate size, 79general epidemic, stochastic, 66

birth-and-death process approxima-tion, 73

discrete time version, 155optimum fit to data, 157

duration, 84extinction probability, 73final size distribution, 74

See aiso ultimate size of epidemic

in community of households, 85in stratified population, 85forward Kolmogorov equations, 86infection rates of product form, 103inhomogeneity, 86multi-type Markovian branchingprocess, 87birth-and-death process approxi-mation, 87

threshold phenomenon, 86, 90transition matrix, primitive, 87ultimate extinction, 87

infection parameter, 66intensity, 74major outbreak, size, 84asymptotically normal or Poisson,84

variance, 148matrix form of d.e.s. for p.g.f., 69minor/major outbreak dichotomy,

74, 85pairwise infection rates, 86removal rates, 86transition probabilities, 86

non-overlapping generations variant,112

removal parameter, 66relative removal rate, 67

state probabilities, 66transition probabilities, 66

stochastic lag, 84sub- and super-critical behaviour, 82ultimate size distribution, 67, 77, 82by p.g.f. method, 77by embedded jump chain, 81Laplace transform methods, 67limit behaviour, 75geometric, or mixed geometric andnormal or Poisson, 84

variable infectious period, 123Whittle's threshold theorem, 73, 76

generation numberaverage, 42finite population effect on, 44simple/general epidemic compared,

43generation time of disease agent, 17

Subject Index 209

generation-wise evolution, 38general epidemic, 41non-overlapping generations, 54numbers of infectives, 38, 112, 125general epidemic, 41, 125Kermack-McKendrick model, 125Reed-Frost model, 125

simple epidemic, 39, 43Poisson distribution, 39

German measles data, 14graph, structure for epidemic, 123

directed links, 123homogeneous mixing, 123

Graunt's Bills of Mortality, 1Greenwood model, 105, 111, 162

beta density for infection probability,115

fitted to data, 162variable infectivity, 162

p.g.f. methods, 108p.g.f. for total size, 110

transition probability matrix, 108growth (spread) of infection

epidemic, 28logistic curve, 20rumour, 134

Guldberg, 6

haemophiliacstransfusion-induced AIDS, 170

hazard rate, 3cumulative, 3

hierarchical state space, 67See also well-ordered sample path

HIV, 46incidence in prison, 185infection data fitted, 173screening, 185seropositives (HIV+), 178, 184, 190

homogeneous mixing, 7Sec also stratification of population

households, epidemics in, 92, 107, 162,180

Heasman's common cold data, 166optimum vaccination strategy, 182variable infection probability, 115

hypergeometric function, 65

ignorants in rumour model, 134immunization fraction, 182immunization strategies

comparison, 182heterogeneous population, 181household, 180optimum level of, 176random, 180

incubation period, 16infant deaths, 3, 6infection mechanisms, 46infection probability in epidemic model

with replacement, 119infection rate (or infection parameter),

9, 21, 23, 36affected by life style, 155in parasitic disease, 46Kendall's approximation, 31stratified population, 23, 36variable, 31, 115, 164

beta density, 116, 164infectious contact, 106, 126

as directed link on a graph, 124infection transmission, 127via pair-formation, 127

infectious path of disease, 184infectious period, distribution of, 125

effect on total size distribution, 125instantaneous, 105myxomatosis, 50

infectious state, instantaneous, 4infectives, 105

distribution in endemic state, 121generations of, 38, 88

infectivity, variable, 8heterogeneous, 174homo- or heterogeneous spread, 180

inference for epidemics modelschain binomial models, 107, 162, 164Reed Frost, 14, 114

influenza, 49, 156death rate estimate, 156fitting death rates, 156graphical data, 50, 157short infectious period, 156

initial phase of epidemic, 43instantaneous infectious state, 4, 11intensity of epidemic, 33, 38, 74

210 Subject Index

interacting communities, epidemic in,90

intravenous drug users (IVDUs), 179model for epidemic among, 118control via needle exchange, 185

needle sharing, 126, 185

Jensen's inequality, 91

Kermack-McKendrick epidemic modelSee general epidemic

Kermack-McKendrick theorem, 10, 29discrete time deterministic model,

52in control strategy, 176stratified population, 37

stochastic analogue, 89Whittle's stochastic analogue, 76

Kolmogorov equations, forwardcarrier epidemic, 95general epidemic, 66simple epidemic, 59

generating function, 64

Lagrange's expansionfinal size distribution, 76

Lagrange's p.d.e. solution method, 75,100

Laplace and Laplace-Stieltjes trans-forms, 62

Abelian property, 77carrier epidemic, p.g.f. method, 99in p.g.f. solution, 68transform ordering, 125

latent period, 16in S-I-S epidemic, 46myxomatosis, 50relative to infectious period, 54, 105unit of time, 105

Law of Mass Action, 6deterministic carrier model, 45discrete time, 7

deterministic epidemic, 48, 50homogeneous mixing, 126stochastic version, 7stratified population, 36

logistic equation, 21discrete time analogue, 48pairs-at-parties comparison, 128population growth model, 20

major outbreak, 29, 34, 77, 84S-I-R epidemic in stratified popula-tion, 88

malaria, 46, 184threshold condition, 47

Markov chains, 12, 56, 77, 105, 141Markov process

absorbing/non-absorbing states, 80embedded jump chain, 79exponential holding time, 80first passage time r.v.s, 81strong Markov property, 80

martingale methods, final size, 145Mass Action, See Law ofmaximum likelihood, for fitting chain

binomial models, 118measles, 46

datasetsdaily, St Petersburg, 18generation-wise (Aycock), 14generation-wise (En'ko), 113monthly, Providence RI, 19weekly reports, 8

generation number, 174in households, 161large-scale vaccination, 179

method of moments, for fitting chainbinomial models, 118

minor outbreak, 29, 77, 84mixed binomial r.v.s.

number of infectious contacts, 124model-fitting of chain binomial

maximum likelihood method, 118method of moments, 118

mumps, 46myxomatosis, 50, 52

data for rabbits, 52

non-infection probability, 106non-overlapping generations of infec-

tives in general epidemic, 112

Subject Index 211

ordering of r.v.s., 125via transforms, 125

outbreak, major/minor, 29, 34, 77, 84overlapping epidemics, 105

pairs-at-parties epidemic model, 126duration time, 127logistic curve comparison, 128rescaled limits, 132simple epidemic analogue, 128

pairwise contact rate, 36pairwise infection rate, 21, 36

discrete time epidemic, 48in stratified population, 23, 36

parasitic disease spread, 46Pearl-Reed, logistic curve, 21Pearson curves, epidemic curve fit, 8Perron-Probenius eigenvalue, 122

asymptotic behaviour, 131Perron-Probenius theory, 88, 103p.g.f. methods

general epidemic, 67Laplace transforms, 68

Greenwood model, 108number of new infectives, 110

ordering of r.v.s., 125rumour models, 149simple epidemic, 63

phases of epidemic process, 63Poisson distribution, mixed, size of

generations of infectives, 40, 41, 44policy for control, 175poliomyelitis, 45

large-scale vaccination, 179population-size dependence, 51

infection rate, 51population with immigration

stable regime, 186predator-prey model, 146predicting size of epidemic, 13, 158prevention strategy for epidemic, 175primitive matrix, 37, 38Principle of Diffusion of Arbitrary

Constants, 84, 145, 152probability generating function (p.g.f.),

63Providence RI measles data, 19

quarantine of infectives, 184quasi-stationary distribution in epi-

demic with replacement, 122

rabbitscalici virus, 175myxomatosis, 50, 175

Reed-Frost chain binomial model, 11,105, 111, 162

bivariate Markov chain model, 111binomial transition probabilities,111

transient and absorbing states, 113transition probability matrix, 111partitioned matrix, 112

branching process on nodes of agraph interpretation, 123

deterministic analogue, 112duration, 130fitted to data, 114, 162X2 statistic, 114variable infectivity, 115, 164

generations of infectives, 125simple epidemic analogue, 129

Registrar-General's Annual Reports, 7relative infection rate, 47relative recovery rate, 47relative removal rate, 9, 28, 67removal rate parameter, 9

dependent on mode of living, 155stratified population, 36

removals, generations of, 41, 44in carrier model, 45

replacement in Greenwood and Reed-Frost models, 118

reproductiveness, 89See also Basic Reproduction RatioRo

rinderpest in cattle, 7rumour models, 133

chain letter, 137Daley Kendall (or [DK]), 133, 138

(a,p)-variant, 135, 139final size, 142, 146fc-fold stifling variant, 135, 151, 152

epidemic with forgetting, 136, 152history, 133

212 Subject Index

rumour models (cont.)Maki-Thompson, 135, 151, 152directed contact, 135final size, 143, 153

spread, 134rumour model analysis

common deterministic version, 149deterministic analysis, 138embedded jump process, 141embedded random walk, 141generation number, 38martingale methods, 145mechanism interpretation, 136no threshold behaviour, 137, 139p.g.f. analysis, 149

Rushton-Mautner simple epidemic, 23

scarlet fever data, 13serial interval, 16, 48S-I, S-I-R, S-I-S notation, 56S I epidemic, See simple epidemicS-I-R epidemic, See general epidemicsimple epidemic, deterministic, 20

continuous time, 20discrete time approximation, 186

d.e. as first moment analogue, 64discrete time, 130first passage time, 100generations of infectives, 54in stratified population, 23, 53

parametric solution, 24Rushton-Mautner solution, 23two-community example, 27

simple epidemic, stochastic, 57absorbing state, 57termination of epidemic, 61

discrete time, 105duration time, 62variance, 101

pairs-at-parties analogue, 128state probabilities Pi(t), 59, 62

forward Kolmogorov equations, 58generating function, 64Laplace transforms, 59

transition probabilities, 57transition rate matrix eigenvalues, 60

size of epidemic, 123See ultimate size of epidemic

smallpoxEngland epidemic of 1837-39, 7eradication, 1large-scale vaccination, 179population numbers (Bernoulli), 4

spread of infectioncessation of spread, 133, 137rates in different models, 130 spread-ers in rumour model, 134

St Petersburg educational institutions,measles data, 18, 114

state probabilitiescarrier epidemic, 95general epidemic, 66simple epidemic, 58

stationary point, endemic disease, 47stiflers in rumour model, 134stochastic epidemic model, 56

absorbing state, 57continuous time, 56

pairwise infection parameter, 57deterministic model, analogue andapproximation, 11, 12

discrete time, 105prototype of deterministic model, 85

stochastic lag, S-I epidemic, 101S-I-R epidemic, 84

stratification of population, basis for,36

See also epidemic in stratified popu-lation

strictly evolutionary Markov process, 67computation of distributions in, 103first passage time r.v.s., 103sample paths, well-ordered, 67

super-criticality condition, epidemicand branching process, 88

susceptibility, inhomogeneity, 168heterogeneous modelling, 174

susceptibles, surviving, 11Switzerland, 'STOP-AIDS' campaign,

176

threshold immunization leveldetermined by Ro, 180

threshold level for endemicityepidemic with replacement, 120parasitic disease, 47

Subject Index 213

threshold theoremdiscrete time logistic equation, 49divergent/convergent dichotomy, 38general epidemic, deterministic, 29Kermack-McKendrick, 10, 29, 37parasitic disease, 47stratified population, 86Whittle's, for S-I-R epidemic, 73, 76

total size, See ultimate size of epi-demic

transition probability matrixfor chain binomial model, 107

Greenwood model, 108lower triangular, 109Reed Frost model, 111

transmission ratein parasitic disease, 46

tuberculosis, 45typhoid, 45

ultimate size of epidemic, 77, 123vs. total size, 79distribution, 125carrier epidemic, 132general epidemic, 67, 79Greenwood model, p.g.f., 110infectious period dependent, 125with latency state, 53

rumour models, 141

several initial infectives, 131distribution, computation methodembedded jump process, 67, 79embedded random walk, 125Laplace transform, 67, 77martingale for moments, 146p.g.f. methods, 67

unit matrix, 59

vaccination, large-scalemeasles, poliomyelitis, smallpox, 179

variable infectivity, 8variolation, 3

advantages of, 6Verhulst logistic model, 20

Waage, 6Weibull distribution

AIDS incubation period, 169well-ordered sample path, 67, 81

first-passage time r.v.s., 81, 103moments, 81, 103

intermediate state probabilities, 81rumour, 141

Whittle's threshold theorem, 73, 76S-I-R epidemic in stratified popula-tion, 87

Wroclaw, 6

Design

Epidemic modelling an introduction