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Early Human Development (2007) 83, 141—148

How mathematical models have helped to improveunderstanding the epidemiology of infectionPietro G. Coen ⁎

Infection Control Office, Department of Microbiology, The Windeyer Institute of Medical Sciences,University College London Hospitals NHS Trust, United Kingdom

⁎ Tel.: +44 845 1555 000x8391; fax: +E-mail address: Pietro.coen@uclh.n

0378-3782/$ - see front matter © 200doi:10.1016/j.earlhumdev.2007.01.005

Abstract Mathematical models have proved to be useful tools for addressing questions aboutthe process of infection. This is because the model allows the investigation of the mathematicallink between invisible events (transmission of infection between individuals) and more visibleones (incidence of clinical infection, seroprevalence data etc.). A useful model is often one thatis simple enough to provide answers that are applicable to as general a context as possible, butalso complex enough so that it can address the most relevant questions. Thus, relatively simplemodels of viral (or hospital acquired-) infection were used to establish the existence of criticalthresholds of immunisation (or antibiotic usage) coverage, although more complex models wereneeded for the investigation of the impact of infection control practices within more specificcontexts. Mathematical models have the potential to elucidate key determinants of theepidemiology of infectious disease arising in infants nursed in neonatal intensive care units.© 2007 Elsevier Ireland Ltd. All rights reserved.

KEYWORDSInfection;Control;Model;Dynamics;Deterministic;Stochastic

The act of spreading infection to uninfected ‘susceptible’individuals is a notoriously difficult process to observe directlybecause of the rare and, to the naked eye, invisible nature ofmost infectious agents. This basic fact forces the epidemiol-ogist to make indirect inferences on the infectious processfrom events that are more easily observed (which lie furtherdown the chain of events from the infectious event), such asthe number of cases suffering from clinical infection in apopulation per unit person-years exposure (incidence) or thefraction of the population that has experienced infection atany one time (e.g. as measured via antibody levels in blood —seroprevalence). While most statistical models are descrip-tions of the observable events themselves (e.g. Poisson orlogistic regression for incidence and proportions respectively)‘mechanistic’ models also provide a mechanism as anexplanation for how events came to be in the first place.

44 845 1555 9716.hs.uk.

7 Elsevier Ireland Ltd. All rights re

They may describe the way (unobservable) dynamic interac-tions between infectious and susceptible individuals generate‘observable’ events (e.g. incidence and prevalence data).Once the mechanism is known, such models allow predictionand forecasting of future events, such as the impact of aninfection control intervention. The practice of mathematicalmodelling of infection has changed over the years (from theinterest in control of malaria, to immunisation of viral andbacterial infections, and more recently to infections ofinpatients within hospitals). I overview here a brief history ofinfection modelling, concluding with the lessons learnt frommodelling hospital acquired infections (HAI).

1. Introduction to infection modelling

The relatively simple natural history of viral infections likemeasles, mumps and rubella make their modelling simplerand easier. They are restricted to humans, their mode oftransmission is airborne, and infection generally coincides

served.

142 P.G. Coen

with clinical disease (and reporting). They all share the ‘SIR’pattern whereby the virus infects ‘susceptible’ (S) indivi-duals who become ‘infectious’ (I) and after a few days, they‘recover’ (R) to a category immune to re-infection (assumingthey survive). Imagine an infectious individual moving into apopulation solely made up of susceptible individuals. Theinfection starts to spread, until the whole populationbecomes immune and the infection goes extinct. While thisthought experiment may be a satisfactory ‘model’ forepidemics it cannot explain endemic infections: it requiresthe missing element of the re-introduction of susceptibleindividuals (e.g. via birth or immigration).

For a more quantitative investigation of the SIR modelbehaviour it becomes necessary to build these ideas into amathematical model. This involves drawing a flow diagramwhere compartments are population categories (susceptible,infectious or immune) and arrows represent flow rates ofindividuals between any two compartments (Fig. 1).

A set of differential equations, each representing the rateof change of each compartment, provides the means tocalculate the numbers of individuals within compartments asthey change with time, given a set of starting conditions (e.g.‘all individuals are susceptible except for one’). In themathematical jargon this is known as an ‘initial valueproblem’. The differential equations corresponding to theSIR model are given by the expressions

dSdt

¼ BR−DRðSÞ−IRdIdt

¼ IR−DRðIÞ−RRdRdt

¼ RR−DRðRÞ

The rates BR, DR(.), IR and RR change with time becausethey depend on the ‘numbers’ in compartments S, I and R.Hence, the equations have to be re-written in terms of S, Iand R:

dSdt

¼ μd N−μd S−bd Sd I ð1Þ

dIdt

¼ bd Sd I−μd I−rd I ð2Þ

dRdt

¼ rd I−μd I ð3Þ

The population density is N=S+I+R. Setting the birth rate(BR) equal to the overall mortality rate (DR(S)+DR(I)+DR(R)=μ·N) results in a constant population size, N. In Eqs. (1)–(3) theGreek letters refer to per capita (i.e. per person) rates, whereμ is the mortality rate, σ is the recovery rate and β is the persusceptible-infectious-pair rate of infection. The infection rateterm β·S·I in Eqs. (1) and (2) represents the hypothesis that theprobability of an infectious contact with a given infectiousindividual is randomly distributed among the susceptiblepopulation. This is known as the ‘mass-action’ hypothesis andit has been adopted by all the infection modelling literaturesince Hamer first introduced its concept in 1906 [1,2]. Althoughthe expression does not take into account the possibility of‘wasted’ multiple contacts between any two individuals, thiswas shown to lead to negligible bias when infectious events are

rare [3]. The per-susceptible infection rate, λ=β·I, is known inthe literature as the ‘force of infection’ so that themass actionterm can be re written as λ·S.

The dynamic solution of the SIR model is complex enoughto require a numerical approximation. The simplest type ofapproximation is given by the method

Sðtþ DtÞcSðtÞ þ dSdt

d Dt ð4Þ

for susceptibles, where t is time and Δt is a unit time ‘step’into the future. This method is associated with an unaccep-table approximation error (unless the step size Δt is madevery small). More accurate methods are available that workon the same principle, such as the 4th order Runge–Kuttamethod [4] (see Fig. 2 for the example of a simulation of anendemic infection). For endemic infections, the number ofsusceptible numbers fluctuates about a ‘critical-mass’threshold: when S(t) increases above it, the incidence ofinfection increases, and vice versa. This was first describedby Kermack and McKendrick in the 1920s [5]. Confusingly,most of the literature use the symbols X, Y and Zinterchangeably with S, I and R. Another accepted conventionis the use of low-case letters for proportions (e.g. x, y, z; [6]).

2. Insights from analytic model solutions

Although numerical approximations are useful for makingpredictions of future model behaviour, the ‘exact’ analyticsolutions provide more efficient insights.

2.1. Equilibrium properties

‘Equilibrium’ is said to be reached when the numbers in theinfection compartments stop changing from one timeinterval to the next. At equilibrium the differential Eqs.(1)–(3) are equal to zero and the equilibrium population sizes(S⁎, I⁎ and R⁎) can be calculated as follows

dIdt

¼ bd S⁎d I⁎−Ad I⁎−rd I⁎ ¼ 0

¼ bd S⁎d I⁎−I⁎ðμþ rÞor bd S⁎ ¼ ðμþ rÞand S⁎ ¼ μþ r

b ð5Þ

I⁎ and R⁎ can be derived using the same method:

I⁎ ¼ μμþ r

� �d N−

μþ rb

� �ð6Þ

R⁎ ¼ rμþ r

� �d N−

μþ rb

� �ð7Þ

2.2. The basic case reproduction ratio — R0

This is a statistic that was devised to measure an infection'spotential for spread in a defined population. Anderson andMay define it as the “average number of secondary in-fections produced when one infectious individual is intro-duced into a host population in which every host is

Figure 2 Example of a simulation for the SIR model as definedin Eqs. (1)–(3), starting conditions S(0)=999, I(0)=1, R(0)=0,and μ=0.1, σ=0.5, β=0.0018. The graphs plot the percentageinfectious (grey bars), susceptible (grey diamonds) and immune(empty triangles).

Figure 1 Compartmental representation of the SIR model.BR=birth rate, DR(S), DR(I) and DR(R)=death rate for thesusceptible, infectious and immune respectively, IR=rate ofinfection, RR=recovery rate.

143How mathematical models have helped to improve understanding the epidemiology of infection

susceptible” [6]. This definition means that the infectioncan only spread if R0>1.

The derivation of R0 for the SIR model uses Eq. (2) and therequirements that at time zero S(0)=N, I(0)=1 and dI/dt>0such that

dIdt

¼ bd N−μ−r > 0

bd N > μþ r

bd Nμþ r

> 1

bd Nμþ r

¼ R0

ð8Þ

Biologically speaking this expressionmakes sense because itrepresents the number of new infections (βN) created withinthe ‘lifetime’ of the primary infection (1/(μ+σ)); it also takesinto account the effects of demography (the populationdensity, N, and the mortality rate, μ), the infectious natureof the aetiological agent (β), the host contact behaviour (β), aswell as the immunological response of hosts to infection (σ).Eq. (8) leads to the concept of ‘threshold population density’,NT: when N<NT, R0 is forced below unity and the infectioncannot persist. This argumentwas used to explain the fact that80% vaccination coverage was enough to lead to smallpoxeradication inWest and Central Africa (low population density)and not in Bangladesh (high population density) [7].

Unfortunately, expression (8) is not helpful in estimatingR0 because β cannot be measured directly. More helpful isthe fact that substitution of Eq. (5) in (8) results in theequality between R0 and the reciprocal of the proportionsusceptible at equilibrium, 1/ s⁎=N/S⁎, which is far easier tomeasure. Re-writing Eq. (6) into the form

I⁎ ¼ μdN

μþ r−1b

� �

and writing λ+μ leads to the result

k ¼ μd ðR0−1Þ ð9ÞRearranging we obtain

R0 ¼ 1þ kμ¼ 1þ L

Að10Þ

where L (=1/μ) is the average life expectancy and A (=1/λ)is the average age at first infection, with the proviso thatmortality is age independent. Where the age structure isrectangular (i.e. μ=0 for age <L; μ=∞ for age =L) theexpression changes to R0=L/A [8,9]. Estimates for L and Ahave led to estimates for R0 for several viral infections indifferent populations (e.g. 6 to 7 for Rubella, 11 to 14 formumps, and 16 to 18 for measles in the UK [6]).

2.3. The epidemic pattern and the ‘attack rate’, A

The epidemic pattern of infection is defined as the type ofdynamics whereby the incidence of infection rises to a peakand then declines to zero within a time-span that is muchsmaller than L. An infection with R0>1 can cause an epidemiconly if the mortality rate, μ, is negligible relative to the rateof recovery from infection (μ≪σ and μ≈0). When theepidemic is over, the fraction of the population that hasexperienced infection is known as the ‘attack rate’, A. Thisstatistic can be derived from the SIR model by dividing dS/dt(Eq. (1)) by dR/dt (Eq. (3)):

dSdR

¼ −bd Sr

resulting in the expression

SðtÞ ¼ Sð0Þd expf−½b=r�d RðtÞg

By definition, S(∞)= (1−A)N and R(∞)=AN, and theexpression reveals that

ð1−AÞ ¼ expf−R0d Agwhich, on rearranging, suggests a 1:1 relationship betweenthe attack rate and R0:

R0 ¼ −lnð1−AÞ=A ð11ÞWhile this expression suggests that some individuals may

escape the effects of an epidemic even without priorimmunity, it also suggests that a 50% attack rate may becaused by an infection with the very modest R0 of 1.38(assuming the SIR model form).

144 P.G. Coen

2.4. Effective basic case reproduction ratio andherd immunity

Endemic infections occur in populations made up of amixture of susceptible, infectious and immune individuals.In such conditions, a given infection cannot infect as many asR0 new individuals in its ‘lifetime’ because a proportion of itscontacts (1− s⁎) will be made up of immunes. For this reasonthe ‘effective’ basic case reproduction ratio, R, is equal toR0 · s⁎, which, under endemic conditions, is expected toequal unity. This line of reasoning leads to the expressions⁎=1/R0 already encountered above.

Some vaccines protect against transmission of infection aswell as against disease, with the additional advantage of‘indirect’ protection for the un-immunised. This is known as‘herd immunity’ and it can be modelled within the SIRframework by moving a proportion p of all births to apermanently immune category [10]. This will reduce thenumber susceptible at birth by a fraction (1−p), leading tothe result

R ¼ bd Nð1−pÞμþ r

¼ R0ð1−pÞ

On rearranging, and setting R=1, we obtain the criticalvaccination threshold

pC ¼ 1−1R0

ð12Þ

This result usefully illustrates that it is not necessary toimmunise every child for eradication to be possible (pC≤1).It is sufficient that p be raised above pC and the infection will(eventually) be eliminated. Although this concept wasunderstood as far back as the 19th Century [9], it was Ross[11] who first described the idea as a mathematical modeland Macdonald [12] who used the concept in the 1950s toconvince the World Health Organisation to enter a globalimmunisation campaign against malaria. While Fine [9]hypothesised a non-linear relationship between vaccinecoverage and cumulative incidence, Garnett and Ferguson[13] correctly showed this relationship to be linear,

r⁎ ¼ 1−s⁎−p ð13Þwhere r⁎ is the equilibrium fraction immune as a result ofnatural infection, a measure of cumulative incidence.Substitution of Eq. (12) in (13) shows that when p=pC,r⁎=0. Unfortunately, the other side of the coin to Eq. (12),though, is the fact that a very high immunisation coverage isnecessary to even obtain a limited amount of control, evenfor infections with modest R0s (e.g. R0=4 corresponds withpC=75%).

2.5. Infection ‘control’ is not always beneficial

In situations where immunisation coverage is less than theprotective threshold predicted by R0, the infection will notbe eradicated but it will, in the long run, lead to ‘control’ inthe sense that infection incidence will decline to a levellower than the pre-immunisation equilibrium, I⁎. There arecases where control is not necessarily beneficial to the

population as a whole, most notably where the severity ofinfection increases with age (e.g. rubella and CongenitalRubella Syndrome). This is because reducing the incidence ofinfection leads to an increase in the average age of infection(as A=1/λ). From Eq. (9) we have that

A ¼ LR0−1

and substituting R0 with R0(1− p), this leads to theexpression

A ¼ 1μd

μþ rbð1−pÞN−ðμþ rÞ

� �ð14Þ

This equation shows that the relationship between A, pand R0 is non-linear. McLean used this equation to show thatinadequate vaccination coverage (0.40≤p≤0.60) may becounterproductive in high-income countries (low R0) becausethe average age at infection would rise steeply withincreasing p. In contrast, the same range of coveragesapplied to a low-income country do not lead to a significantchange in A [14–17].

3. Model criticism

The models discussed above have led to useful insights on thegeneral properties of infections. They may, however, sufferfrom a set of unrealistic assumptions which make them lesshelpful when it comes to understanding a specific situation,such as infection in a cohort of individuals, belonging to a setof age groups, such as infants in a particular hospitalintensive care unit. For this sort of modelling, it is necessaryto ‘break’ many assumptions. This leads to increased modelcomplexity, the inevitable increase in the number of var-iables, more complex results and loss of generality in theconclusions. The biology of infection and the nature of theproblem at hand will dictate the degree of model complexitythat is required.

3.1. Contact rate heterogeneity

In neonatal units infants vary in susceptibility to infectionand exposure to risk factors for infection so cannot beconsidered as a homogenous population. The mass-actionmodels described above make the unrealistic assumption of asingle homogeneously mixing population. Real populationsare compartmentalized into sets (geographic, socio-eco-nomic, age, gender etc.) that interlock to varying extents,depending on the biology of transmission of the type ofinfection that is being considered. Although Fox et al. [18]used mathematical models more consistent with epidemicsin closed populations [9], they broke the assumption ofhomogeneous random mixing by modelling a structuredcommunity of 1000 individuals, each assigned to a family, aschool, a social group. Their Monte Carlo simulations suggestthat this type of contact-rate heterogeneity leads to greaterepidemic sizes. Contact rates may also vary with age. Fineand Clarkson [3] showed strong age dependence for measles,probably as a result of aggregation of children in primaryschools from the age of 5 years. Anderson and May [10]

Figure 3 Simulation of the dynamics of an immunisationprogramme with coverage p=0.8 and R0=4.0, aimed against aninfection with SIR infection dynamics (black line) and againstone with SIS (grey line) infection dynamics. The SIS model is theSusceptible–Infectious–Susceptible model, which assumesthat an individual is susceptible for re-acquisition afterrecovery. The expressions for R0 and pC are the same for bothSIR and SIS models.

145How mathematical models have helped to improve understanding the epidemiology of infection

investigated models where the force of infection,λj, is agedependent,

kj ¼Xi¼A

i¼1

bijd Ii ð15Þ

where the transmission coefficients βij are elements of the‘Who Acquires From Whom’ matrix, and they represent thepotential for age group i to contribute differently to theinfection of age group j than other age groups. Their worksuggests that ignoring age-dependence of contact rates canlead to overestimates or underestimates in the ‘true’ mass-action model parameters. In particular, R0 and pC would beunderestimated when forces of infection increase with age,and vice versa.

3.2. Immunisation models

The introduction of age-dependent contact rates leads to thequestion of the effect of age-dependent immunisation rates.Most vaccines cannot (and should not) be given at birthbecause most vaccines only work after maternal antibo-dies start to wane. If the average age of loss of maternalimmunity is M, the ‘effective’ human lifespan shortens by Myears and expression [10] changes to R0=1+(L−M)/(A−M)which shows that ignoring M leads to underestimates of R0and pC. If V is the average age at vaccination then it canreplaceM in this expression, and for developed countries pC=(L−A)/ (L−V), which illustrates the existence of an age‘window’ of immunisation between M and A for which pC≤1.Other assumptions related to an immunisation programmethat may need breaking include the possibility that vaccineefficacy wanes with age [19], and heterogeneity in vaccineuptake, both of which may lead to pC underestimates ifignored.

3.3. Seasonality

Infections like measles and influenza have a strong seasonalincidence pattern. Influenza has winter “peaks” that last 4–6 weeks [20] while measles is more common in the weeksfollowing the annual opening of primary schools [3]. This facthas led to the hypothesis that the contact rate (modelled bythe transmission coefficient, β) varies with the season [21].Yorke et al. used models with seasonal β to suggest thatinfection control is more effective when conducted during aseasonal “low” [22]. Schenzle used an age-structured,compartmental, model for measles in England and Germany,where transmission parameters reflected the seasonalaggregation of successive cohorts of children in schools[23]. The resulting dynamics of this model closely reflectedthe observed impact of the immunisation programme inthese countries. It was also shown by Schenzle that hadseasonality and age structure been ignored, R0 (and pC)would have been overestimated.

3.4. The limitations of R0

The principal limitation of R0 is that it says little about thedynamics of infection. R0 may help predict the minimum

coverage necessary to eradicate infection (pC), but it will nottell us how long it takes to reach eradication or control(where p<pC). Indeed, intervention may lead to stronglynon-linear fluctuations in disease incidence or may increasethe age at first infection. Immunisation against infectionslike measles or chickenpox (best modelled via the SIR modelform), can lead to ‘honeymoon’ periods of low incidencefollowed by a sharp resurgence of incidence [24]. This isthought to be the result of accumulation of susceptibles in(un-immunised) older age groups [25,26]. R0 is openlymisleading where infections with the same R0 responddifferently to intervention because they have differentmodel forms. An example is illustrated in Fig. 3 where twoinfections with R0 =4 are immunised with p=0.8. Asexpected, both lead to eventual eradication, except thatthe ‘SIR’model form takes 3 years while the ‘SIS’model formtakes 60 years to reach it. Hence this statistic can only beused to rank infections that share a similar model form, andmodelling studies must track the dynamics of an infection aswell as the ‘equilibrium’ conditions (summarized by R0). Lastbut not least, R0 suffers from the catch-22 paradigm becausemost infection situations require models that are toocomplex for analytic computation of R0 and where R0 canbe computed, this is probably because the model is toosimplistic to be realistic!

4. Models of hospital acquired infection

Thus far the mathematical models have been deterministic.This means that every model output is completely deter-mined by the set of values assigned to its parameters. Withhospital acquired infections (HAIs) stochastic models arefavoured over deterministic ones because of the smallnumbers of individuals involved (within a model compart-ment or ward location). These models are based on theassumption that a given type of event, i (e.g. a change ofstatus from susceptible to infectious), has a mean rate ri,and the number of such events is then Poisson-distributed

Figure 4 Hypothetical relationships between hand-washingcompliance and its impact on the transmission of infection(measured as the transmission coefficient, β, in the mathema-tical models). The black line is the linear relationship assumed inmost published mathematical modelling studies; the blue linesare alternative relationships. (For interpretation of the refer-ences to colour in this figure legend, the reader is referred to theweb version of this article.)

146 P.G. Coen

over time [27]. Hence, the probability for event not occur-ring by time t has the exponential distribution:

1−p ¼ expð−rd tÞ ð16Þ

Waiting-time stochastic simulations are a series of twostep processes where r represents the sum of the rates overall possible events at a particular moment in time, r ¼ P

ri.In the first step, the time to the next event is first calculatedby substituting a random number uniformly distributedbetween 0 and 1 for p in the related expression

t ¼ −lnðpÞr

ð17Þ

The actual event is then selected via ranking all thepossible events ri and using a randomly generated (uniform)number from 0 to r.

In the past 10 years HAIs have been the subject of stochasticand deterministic modelling studies, each with its idiosyn-cratic set of simplifying assumptions relating to the biology ofHAI spread. All these have assumed an SIS compartmentalstructure (because most are caused by bacterial infectionswith longperiods of asymptomatic carriage, that donot appearto result in long-term immunity, rarely causing morbidity —hence the choice to model carriage, rather than clinicalinfection). Most modelling studies have been concerned withthe spread of a single organism among hospital inpatients. Anotable exception to this rule is represented by studies byAustin et al. [28]who used a deterministicmodel to investigatethe equilibrium conditions for antibiotic resistant strainsinvading a population mostly carrying sensitive strains. Austinet al.'s most significant intuition was to show that, in thecontext of antibiotic usage, the resistant organism will alwaysreplace the sensitive unless there is a fitness cost associatedwith antibiotic resistance (whether via the effects of aplasmid, or indirectly via the effects of infection controlwhen aimed solely at the resistant organism). Under thesecircumstances there is a perverse analogy with herd immunityand vaccination (Eq. (12)) and there exists a critical propor-tion, p′, of antibiotic prescribing above which the resistantorganism coexists with the sensitive one and another one, p″(where p″>p′), above which the resistant organism becomescompletely ‘fixated’. Although Austin et al. did not model anextramural catchment population (all admissions are assumedto be free of carriage), the concept of a fitness cost isconsistent with the observation that some antibiotic resistantHAIs are still relatively rare in the community despite decadesof exposure within hospital inpatients.

More recent modelling focused on the stochastic (prob-abilistic) nature of HAI spread within hospital inpatients anda hospital's catchment population. Furthermore, thesemodels had the realistic touch of allowing for a subset ofthe catchment population composed of individuals with ‘highrisk’ of re-admission, mostly from other hospitals or nursinghomes [29,30]. They convincingly put across the insidiousnature of bacterial HAI infections, characterised by twointerlinked processes: one is the spread of infection withinthe hospital and the other is the process of re-admission ofcarriers who acquired infection in a previous hospitaladmission (most of whom are ‘high risk’ individuals). Thisrealisation has important implications for infection control:efforts that concentrate on controlling infection within the

hospital (improving hand-washing compliance, isolation ofinpatients associated with clinical infection, cohorting) haveto be combined with those that deal with infection brought inby newly admitted individuals (screening ‘high risk’ admis-sions). This has led to the concept of RA, which is the basiccase reproduction number relevant to the single admissionevent, and R0=RA·ξ where ξ is the probability of read-mission. When community-acquired infection is important RAmay be less than 1 and still result in endemic infection [29].Taken to the extreme this becomes a game-theory paradigmwhere some hospital managers may selfishly opt for abolish-ing infection control altogether simply because most of theirinpatients come from local hospitals that do spend money ontheir infection control [31].

5. Questioning the HAI models and suggestionsfor the future

These HAI models have undoubtedly helped reach very usefulinsights about the biology of infection and the relativepotential of different infection control measures. It issalutary, on the other hand, to remind ourselves that youcan only get out of a model what you put in it. Raboud et al.[32], for example, are able to tackle the relative merits ofdifferent cohorting strategies by informing the model withempirical details (e.g. the ratio of nurses to inpatients inday-time and in the night-time, the number of nursing visitsper patient, number of ‘unassigned’ visits, percentage ofvisits with ‘high-risk’ contact etc.).

The question of hand-washing compliance is moreproblematic: all the modelling studies, thus far, assumedthat the only transmission route is hand-to-hand contact andindirectly model its effects via a reduction of the transmis-sion coefficient that is a linear function of hand-washingcompliance [28,33,34]. It is then not surprising that most ofthese studies gratuitously find improvement of hand-washingcompliance as one of the most helpful interventions forinfection control. Unfortunately there is no evidence for alinear relationship between hand-washing compliance andits impact on transmission of infection. Fig. 4 shows how a

147How mathematical models have helped to improve understanding the epidemiology of infection

non-linear relationship may lead to rather different conclu-sions, such as, for example, when other transmission routesexist (e.g. environmental contamination).

An important question is whether it is sensible to modelinfections as though theywere independent of other infections(i.e. caused by other organisms). Where two organisms havebeen modelled, complete competition was assumed forcolonisation of susceptible-space (this was because noallowance was made for a co-carrier compartment) [28,34].Single-species models assume no competition with otherrelevant organisms. A related question is whether the SISmodel makes biological sense: on the one hand it assumes thata carrier's immune system ‘clears’ the organism (in a time scaleusually measured in months), which seems to be incompatiblewith the concept that recovery is then back to the susceptibleclass. More logical would be to assume that recovery involvesimmunity to only one of a number of circulating strains. Mostbacterial species can be classified into genetic strains (or‘Electrophoretic Types’), even themost clonal ones [35]. Couldthese ‘strains’ compete for colonisation space?

Bacteria can certainly harm bacteria of different speciesvia low molecular mass proteins called bacteriocins [36].Much of the evidence in favour of competition in bacterialinfections is anecdotal or at best observational [37], partlybecause few studies are geared for detection of co-carriers[38]. Powerful indirect evidence has come from observa-tional studies carried out during interventions, such as themass immunisation with conjugate-vaccine that targets asubset of Streptococcus pneumoniae (the ‘vaccine sero-types’, VS). Many countries that adopted this programmeexperienced reductions in VS carriage prevalence followedby their replacement with non-vaccine serotypes [39,40].These observations are consistent with competition forcolonisation space, although Lipsitch et al. [41] suggestthis may be explained by acquired immunity betweenserotypes, which is a form of indirect competition. It wasalso observed that Staphylococcus aureus-related otitismedia (OM) significantly increased in incidence after Strep-tococcus pneumoniae OM were allegedly reduced by con-jugate vaccine immunisations [42]. If staphylococci cancompete with pneumococci, would it be unreasonable tosuggest that methicillin-resistant Staphylococcus aureus(MRSA) compete with methicillin-sensitive Staphylococcusaureus (MSSA)? We know from the work of Lipsitch et al. thatinfection control intervention can result in extended effectson some infections through their effects on their (antibioticsensitive) competitors [34].

A mathematical model of infection in a neonatal intensivecare unit would be adapted to reflect the biology of thoseinfections that affect infants such as Staphylococcus aureus,coagulase negative staphylococci, Escherichia coli andGroupBstreptococci [43]. The model would also include expectedrates for diseases specific to infants, such as neonatalmeningitis, pneumonia or necrotizing enterocolitis. Hence anobvious addition to the HAI models described above would bethat of the interventions that put infants in intensive care atincreased risk of infection such as the use of biomedicaldevices. The model would be separated into processes whichinclude factors determining colonisation (asymptomatic/com-mensal carriage), and those determining infection (disease).

To conclude, mathematical models have been extremelyuseful tools for asking questions about the invisible process of

infection and the usefulness of infection control intervention(whether via vaccination of large communities, or via thescreening and isolation of HAIs), especially because they havehelped point out directions for data collection and analysis. Itremains important to avoid making the mistake of believingany individual model: all models are to some extent wrong. Itis the collateral knowledge that comes while doing modellingthat constitutes their most useful heritage. Mathematicalmodels have been little used to study the epidemiology ofinfection in neonatal intensive care units. When there aremany potential strategies to control infection a mathema-tical model can allow predictions to be made about theconsequences of one intervention or another.

Key guidelines

• Understanding mathematical models of infection• Infection control of hospital acquired infection• Hand-washing compliance• Screening admissions• Viral and bacterial infections

Research directions

• Do bacteria compete for colonisation space?• What is the nature of the route of infection (environ-mental vs. airborne vs. hand–hand contact)?

• Is the relationship between hand-washing compliance andinfection spread linear (as assumed by all publishedmathematical modelling studies of HAIs)?

References

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[2] Fine PEM. John Brownlee and the measurement of infectious-ness: an historical study in epidemic theory. J R Stat Soc1979;142:347—62.

[3] Fine PE, Clarkson JA. Measles in England and Wales—I: Ananalysis of factors underlying seasonal patterns. Int J EpidemiolMar 1982;11(1):5–14.

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