Modelling as a tool for planning vaccination policies

Kari Auranen

Department of VaccinesNational Public Health Institute, Finland

Department of Mathematics and StatisticsUniversity of Helsinki, Finland

Outline

• Basic concepts and models – dynamics of transmission– herd immunity threshold– basic reproduction number– herd immunity and critical coverage of vaccination – mass action principle

Outline continues

• Heterogeneously mixing populations– more complex models and new survey data to learn about routes of

transmission

• Use of models in decision making– example: varicella vaccinations in Finland

• Ude of models in planning contaiment strategies– example: a simulation tool for pandemic influenza

Basic concepts and modelsBasic concepts and models

A simple epidemic model (Hamer, 1906)

• Consider an infection that– involves three “compartments” of infection:

– proceeds in discrete generations (of infection)– is transmitted in a homogeneously mixing (“everyone meets

everyone”) population of size N

SSusceptibleusceptible CCasease ImmuneImmune

Dynamics of transmission

• Numbers of cases and susceptibles at generation t+1

C = R * C * S / N

S = S - C + B

t + 1t + 1 00 tt tt

t+1t+1 tt t+1t+1 tt

S = number of susceptibles at time t (i.e. generation t)S = number of susceptibles at time t (i.e. generation t)C = number of cases (infectious individuals) at time t C = number of cases (infectious individuals) at time t B = number of new susceptibles (by birth)B = number of new susceptibles (by birth)

tt

tt

tt

Dynamics of transmission

R0 = 10; N = 10,000; B = 300

0

200

400

600

800

1000

1200

1400

time period

nu

mb

ers

of

ind

ivid

ual

s

susceptibles

cases

epidemicthreshold

Epidemic threshold : S = N/REpidemic threshold : S = N/Ree 00

Epidemic threshold S

S - S = - C + B

• the number of susceptibles increases when C < B decreases when C > B• the number of susceptibles cycles around the

epidemic threshold S = N / R• this pattern is sustained as long as transmission is

possible

ee

t+1t+1 tt t+1t+1 tt

t+1t+1

t+1t+1

tt

tt

ee 00

Epidemic threshold

C / C = R x S / N = S / S

• the number of cases increases when S > S decreases when S < S

• the number of cases cycles around B (influx of new susceptibles)

t+1t+1 tt 00 tt tt ee

ee

ee

tt

Herd immunity threshold

• incidence of infection decreases as long as the proportion of immunes exceeds the herd immunity threshold

H = 1- S / N

• a complementary concept to the epidemic threshold

• implies a critical vaccination coverage

ee

Basic reproduction number (R )

• the average number of secondary cases that an infected individual produces in a totally susceptible population during his/her infectious period

• in the Hamer model : R = R x 1 x N / N = R• herd immunity threshold H = 1 - 1 / R

• in the endemic equilibrium: S = N / R , i.e.,

00

0000

00

00ee

ee 00

00

R x S / N = 1R x S / N = 100 ee

Basic reproduction number (2)

R = 3R = 300

Basic reproduction number (3)

R = 3R = 3endemic equilibriumendemic equilibrium

00

R x S / N = 1R x S / N = 100 ee

Herd immunity threshold and R

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

herd immunity

threshold H

0 1 2 3 4 5

Ro

= 1-1/RHH 00

00

(Assumes homogeneous mixing)(Assumes homogeneous mixing)

Effect of vaccination

Ro = 10; N = 10,000; B = 300

0

500

1000

1500

2000

time period

nu

mb

ers

of

ind

ivid

ual

s

susc.

cases

epidemicthreshold

Hamer model under vaccinationHamer model under vaccination

S = S - C + B (1- VCxVE)S = S - C + B (1- VCxVE)

Vaccine effectiveness (VE)Vaccine effectiveness (VE) xxVaccine coverage (VC) = 80%Vaccine coverage (VC) = 80%

t+1t+1 tt t+1t+1

Epidemic threshold sustained: S = N / R Epidemic threshold sustained: S = N / R ee 00

Mass action principle

• all epidemic/transmission models are variations of the use of the mass action principle which– captures the effect of contacts between individuals– uses an analogy to modelling the rate of chemical reactions – is responsible for indirect effects of vaccination– assumes homogenous mixing

• in the whole population• in appropriate subpopulations (defined by usually by age

categories)

The SIR epidemic model

• a continuous time model: overlapping generations• permanent immunity after infection• the system descibes the flow of individuals between the

epidemiological compartments • uses a set of differential equations

SusceptipleSusceptiple RemovedRemovedInfectiousInfectious

The SIR model equations

dtdI )()( tS

NtI )(tI )(tI

dtdS N )()( tS

NtI )(tS

dtdR )(tI )(tR

)()()( tRtItSN

= birth rate= birth rate

= rate of clearing infection= rate of clearing infection

= rate of infectious contacts= rate of infectious contacts

by one individual by one individual

= force of infection= force of infection

Endemic equilibrium (SIR)

0

200

400

600

800

1000

1200

1400

03.

07,

0012

,00

19,0

028

,00 46

time

nu

mb

ers

of

ind

ivid

ual

s

susceptibles

infectives

epidemicthreshold

N = 10,000N = 10,000

= 300/10000 = 300/10000 (per time unit)(per time unit)

= 10 = 10 (per time unit)(per time unit)

= 1 = 1 (per time unit)(per time unit)

R00

The basic reproduction number

• Under the SIR model, Ro given by the ratio of two rates:

R = = rate of infectious contacts x

mean duration of infection

• R not directly observable• need to derive relations to observable quantities

00

00

)

Force of infection

• the number of infective contacts in the population per susceptible per time unit:

(t) = x I(t) / N

• incidence rate of infection: (t) x S(t) • endemic force of infection = x (R - 1)

00

Estimation of R

0

10

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90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

average age at infection A

bas

ic r

epro

du

ctio

n n

um

ber

Relation between the average age at infection and R (SIR model)Relation between the average age at infection and R (SIR model)

= 1/75 = 1/75 (per year)(per year)

)10 R

ALR /10

1/ R /1R

00

00

75/1 L

A simple alternative formula

• Assume everyone is infected at age A everyone dies at age L (rectangular age distribution)

ImmunesImmunes

AA LL

Age (years)Age (years)

SusceptiblesSusceptibles100 %100 %

Proportion of susceptibles:Proportion of susceptibles: S / N = A / LS / N = A / L

R = N / S = L / AR = N / S = L / A

ee

00 ee

ProportionProportion

Estimation of and Ro from seroprevalence data

0

10

20

30

40

5060

70

80

90

100

1 5 10 15 20 25 30

age a (years)

proportion with rubella antibodies

observed [8]

model prediction

1) Assume equilibrium 1) Assume equilibrium

2) Parameterise force of infection 2) Parameterise force of infection

3) Estimate3) Estimate

4) Calculate Ro4) Calculate Ro

Ex. constant Ex. constant Proportion not yet infected: Proportion not yet infected: 1 - exp(- a) ,1 - exp(- a) , estimate = 0.1 per year gives estimate = 0.1 per year gives reasonable fit to the datareasonable fit to the data

Estimates of R

Infection Location R0

Measles England and Wales (1950-68) 16-18

Rubella England and Wales (1960-70) 6-7

Poliomyelitis USA (1955) 5-6Hib Finland, 70’s and 80’s 1.05

Anderson and May: Infectious Diseases of Humans, 1991*

*

*

**

00

Critical vaccination coverage to obtain herd immunity

• Immunise a proportion p of newborns with a vaccine that offers complete protection against infection

• R = (1-p) x R• If the proportion of vaccinated exceeds the herd

immunity threshold, i.e., if p > H = 1-1/R , infection cannot persist in the population (herd

immunity)

vaccvacc 00

00

Critical vaccination coverage as a function of R0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Basic reproduction number

Crit

ical

vac

cina

tion

cove

rage

p = 1 – 1/Rp = 1 – 1/R00

Indirect effects of vaccination

• If p < H = 1-1/R , in the new endemic equilibrium: S = N/R , = (R -1)

» proportion of susceptibles remains untouched» force of infection decreases

00ee

ee

00

vaccvacc

vaccvacc

Effect of vaccination on average age A’ at infection (SIR)

• Life length L; proportion p vaccinated at birth, complete protection • every susceptible infected at age A

SusceptiblesSusceptibles

AA LL

pp

Age (years)Age (years)

11

S / N = (1-p) A’/L S / N = (1-p) A’/L

S / N = A/ LS / N = A/ L

=> A’ = A/(1-p) => A’ = A/(1-p)

i.e., increase in thei.e., increase in the average age of average age of infectioninfection

ProportionProportion

’’

ee

ee

ImmunesImmunes

Vaccination at age V > 0 (SIR)

• Assume proportion p vaccinated at age V• Every susceptible infected at age A• How big should p be to obtain herd immunity threshold H

Age (years)Age (years)

ProportionProportion

11

pp

VV LL

H = 1 - 1/R = 1 - A/L H = 1 - 1/R = 1 - A/L

H = p (L-V)/L H = p (L-V)/L

=> p = (L-A)/(L-V) => p = (L-A)/(L-V)

i.e., p bigger than when i.e., p bigger than when vaccination at birth vaccination at birth

ImmunesImmunes

SusceptiblesSusceptibles

AA

Modelling transmission in a Modelling transmission in a heterogeneously mixing heterogeneously mixing populationpopulation

More complex mixing patterns

• So far we have assumes (so called) homogeneous mixing– “everyone meets everyone”

• More realistic models incorporate some form of heterogeneity in mixing (“who meets whom”)– e.g. individuals of the same age meet more often each other

than individual from other age classes (assortative mixing)

Example: WAIFW matrix

• structure of the Who Acquires Infection From Whom matrix for varicella , five age groups (0-4, 5-9, 10-14, 15-19, 20-75 years)

a a c d e a b c d e c c c d e d d d d e e e e e e

table entry = rate of transmission between an infective and a susceptible of respective age groups

e.g., force of infection in age group 0-4: a*I1 + a*I2 + c*I3 + d*I4 + e*I5

I1 = equilibrium number of infectives in age group 0-4, etc.

POLYMOD contact survey

• Records the number of daily conversations in study participants in 7 European countries

• Use the number of contacts between individuals from different age categories as a proxy for chances of transmission

• Is currently being used to aid in modelling the impact of varicella vaccination in Finland

POLYMOD contact survey: the mean number of daily contacts

Country Number of daily contacts

Relative (95% CI)

DE 7.95 1FI 11.06 1.34 (1.26-1.42)IT 19.77 2.33 (2.19-2.48)LU 17.46 2.02 (1.90-2.14)NL 13.85 1.78 (1.63 -1.95)PL 16.31 1.90 (1.79 – 2.01GB 11.74 1.40 (1.31 – 1.48)

POLYMOD contact survey: numbers of daily contacts

00

-04

05

-09

10

-14

15

-19

20

-24

25

-29

30

-34

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-39

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-44

45

-49

50

-54

55

-59

60

-64

65

-69

70

+

00-04

05-09

10-14

15-19

20-24

25-29

30-34

35-39

40-44

45-49

50-54

55-59

60-64

65-69

70+

Age of participant

Ag

e o

f c

on

tac

t

Finland

0.00-0.31 0.31-0.63 0.63-0.94 0.94-1.25 1.25-1.56

1.56-1.88 1.88-2.19 2.19-2.50

POLYMOD contact survey:where and for how long

Duration

0%

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100%

<5 min 5-15 min 15-60 min 1-4 h 4+ h

non-physical

physical

0%

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100%

daily weekly monthly less often first time

non-physical

physical

0%

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100%

daily weekly monthly less often first time

4+ h

1-4 h

15-60 min

5-15 min

<5 min

0%

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100%

home school leisure work transport other multiple

non-physical

physical

Use of models in policy making

• Large-scale vaccinations usually bring along indirect effects– the mean age at disease increases– population immunity changes

• Population-level experiments are impossibleNeed for mathematicl modelling

– to predict indirect effects of vaccination– to summarise the epidemiology of the infection– to identify missing data or knowledge about the natural history of

the infection

References

1 Fine P.E.M, "Herd immunity: History, Theory, Practice", Epidemiologic Reviews, 15, 265-302,1993

2 Fine P.E.M., "The contribution of modelling to vaccination policy, Vaccination and World Health, Eds. F.T. Cutts and P.G. Smith, Wiley and Sons, 1994.

3 Nokes D.J., Anderson R.M., "The use of mathematical models in the epidemiological study of infectious diseases and in the desing of mass immunization programmes", Epidemiology and Infection, 101, 1-20, 1988

4 Anderson R.M. and May R.M., ”Infectious Diseases of Humans”; Oxford University Press, 1992.

5 Mossong J et al, Social contacts and mixing patterns relevant to the spread of infectious diseases: a multi-country population-based survey, Plos Medicine, in press

6 Duerr et al, Influenza pandemic intervention planning using InfluSim, BMC Infect Dis, 2007