Epidemiologisten tartuntatautimallien perusteita

Kari AuranenRokoteosastoKansanterveyslaitos

Matematiikan ja tilastotieteen laitosHelsingin yliopisto

Outline (1)

• A simple epidemic model to exemplify – dynamics of transmission of infectious disease– epidemic threshold– herd immunity threshold

– basic reproduction number R0

– the effect of vaccination on epidemic cycles – mass action principle

Outline (2)

• The Susceptible - Infected - Removed (SIR) model– endemic equilibrium– force of infection– estimation of the basic reproduction number R– effect of vaccination

• The SIS epidemic model– R and the choice of the model type– age-specific proportions of susceptibles/infectives

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A simple epidemic model (Hamer, 1906)

• Consider an infection that– involves three states/compartments of infection:

– proceeds in discrete generations (of infection)– is transmitted in a homogeneously mixing population of

size N

SSusceptibleusceptible CCasease ImmuneImmune

Model equations

• Dependence of generation t+1 on generation t:

C = R x C x S / N

S = S - C + B

t + 1t + 1 00 tt tt

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

SS = number of susceptibles at time t = number of susceptibles at time tCC = number of cases (infectious individuals) at time t = number of cases (infectious individuals) at time tBB = number of new susceptibles (by birth) = number of new susceptibles (by birth)

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Dynamics of transmission

Dynamics (Ro = 10; N = 10,000; B = 300)

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Epidemic threshold : S /N = 1/REpidemic threshold : S /N = 1/R00ee

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

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t+1t+1

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Epidemic threshold

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

• the number of cases increases when St > S

decreases when St < S

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

t+1t+1 tt 00 tt tt ee

ee

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Herd immunity threshold

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

H = 1- Se / N

• a complementary concept to the epidemic threshold

• implies a critical vaccination coverage: what proportion of the population needs to be effectively vaccinated to eliminate infection

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.,

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

Basic reproduction number (2)

R = 3R = 300

CC

CC

CC

CC

CC

CC

CC

CC

CC

CC

CC

CC

CC

Basic reproduction number (3)

R = 3 R = 3 CC

endemic equilibriumendemic equilibrium00

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

CC

CCCC

Herd immunity threshold and R

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herd immunity

threshold H

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Ro

= 1-1/RHH 00

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(Assumes homogeneous mixing)(Assumes homogeneous mixing)

Effect of vaccination

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

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Hamer model under vaccinationHamer model under vaccination

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

Vaccine efficacy (VE) Vaccine efficacy (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 the analogy to modelling the rate of chemical reactions – is responsible for indirect effects of vaccination– assumes homogenous mixing

• in the whole population or• in appropriate subpopulations/strata

The SIR epidemic model

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

three epidemiological “compartments” • formally defined through 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

λλ(t)(t)

{{

Endemic equilibrium (SIR)

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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)

0R

The basic reproduction number (SIR)

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

R = = rate of infectious contacts x

“mean duration” of infection

• R (usually) not directly observable• need to derive relations to observable quantities

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Force of infection (SIR)

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

λ(t) = x I(t) / N

• incidence rate of infection: (t) x S(t)

• endemic force of infection (SIR): = x (R - 1)00

Estimation of R (SIR)

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average age at infection A

bas

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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

10

/ R /1

0R

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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 %

Stationary proportion of susceptibles:Stationary proportion of susceptibles: S / N = A / LS / N = A / L

=> R=> R00 = 1/(S = 1/(See/N) = L / A/N) = L / A

ee

ProportionProportion

Estimation of and R0 from seroprevalence data

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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 R4) Calculate R00

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*

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Indirect effects of vaccination (SIR)

• vaccinate proportion p of newborns, assume complete protection against infection

• a new reproduction number R = (1-p) x R0

• if p > H = 1-1/R0 , the infection cannot persist

• if p < H = 1-1/R0 , in the new endemic equilibrium:

S = N/R0 , = (R -1)

» proportion of susceptibles remains untouched(!)» force of infection decreases

vaccvacc

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

AgeAge

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

with vaccination:with vaccination:

without vaccination:without vaccination:

Vaccination at age V > 0 (SIR)

• assume proportion p vaccinated at age V (instead of at birth)• every susceptible infected at age A• what proportion p should be vaccinated to obtain herd immunity threshold H?

Age Age

ProportionProportion

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pp

VV LL

H = 1 - 1/RH = 1 - 1/R00 = 1 - A/L = 1 - A/L

proportion immunised proportion immunised by vaccination p (L-V)/L by vaccination 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

The SIS epidemic model

• herd immunity threshold still the same: H0 = 1 - 1/R0

• endemic force of infection: • the proportions of susceptibles and immunes different

from the SIR model

SusceptibleSusceptible ImmuneImmune

)10 R

R and the force of infection

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force of infection (per year)

Ro

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force of infection (per year)

Ro

birth rate = 1/75 (per year)rate of clearing infection = 2.0 (per year)

birth rate = 1/75(per year)

no immunity to infection (SIS) lifelong immunity to infection (SIR)

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SIS and SIRSIS and SIR

Extensions of simple models (1)

• so far all models assumed– homogeneous mixing– constant force of infection across age (classes)

• more realistic models incorporate– heterogeneous mixing

• age-dependent contact/transmission rates• social structures: families, day care groups, schools,

neighbourhoods etc.

Extensions of simple models (2)

– seasonal patterns in risks of infection– latency, maternal immunity etc.– different vaccination strategies– different models for the vaccine effect

• stochastic models to– model chance phenomena– time to eradication– apply statistical techniques/inference

Contact structures (WAIFW)

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

a a c d ea b c d ec c c d ed d d d ee 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.

• WAIFW matrix non-identifiable from age-specific incidence !

Example: structured modelsExample: structured models

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 Haber M., "Estimation of the direct and indirect effects of vaccination", Statistics in Medicine, 18, 2101-2109, 1999

4 Halloran M.E., Cochi S., Lieu T.A., Wharton M., Fehrs L., "Theoretical epidemologic and mordibity effects of routine varicella immunization of preschool children in the U.S.", AJE, 140, 81-104, 1994

5 Levy-Bruhl D., lecture notes in the EPIET course, Helsinki, 1998.

6 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

7 Lipsitch M., "Vaccination against colonizing bacteria with multiple serotypes", Population Biology, 94, 6571-6576, 1997

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