1. Epidemiologisten tartuntatautimallien perusteita Kari Auranen Rokoteosasto Kansanterveyslaitos Matematiikan ja tilastotieteen laitos Helsingin yliopisto

2. Outline (1) A simple epidemic model to exemplify – dynamics of transmission of infectious disease – epidemic threshold – herd immunity threshold – basic reproduction number R 0 – the effect of vaccination on epidemic cycles – mass action principle

3. 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 0 0

4. 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 Susceptible Case Immune

5. Model equations Dependence of generation t +1 on generation t : C = R x C x S / N S = S - C + B t + 1 0tt t+1t t+1t S = number of susceptibles at time t C = number of cases (infectious individuals) at time t B = number of new susceptibles (by birth) t t t

6. Dynamics of transmission Epidemic threshold : S /N = 1/R 0 e

7. 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 e t+1t t+1t t+1 t+1 t t e0

8. Epidemic threshold C / C = R x S / N = S / S the number of cases increases when S t > S decreases when S t < S the number of cases cycles around B (influx of new susceptibles) t+1t0tte e e t

9. Herd immunity threshold incidence of infection decreases as long as the proportion of immunes exceeds the herd immunity threshold H = 1- S e / 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 e

10. 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., 0 0 0 0 0e e0 0 R x S / N = 1 0 e

11. Basic reproduction number (2) R = 3 0 C C C C C C C C C C C C C

12. Basic reproduction number (3) R = 3 C endemic equilibrium 0 R x S / N = 1 0e C C C

13. Herd immunity threshold and R = 1-1/R H 0 0 (Assumes homogeneous mixing)

14. Effect of vaccination Hamer model under vaccination S = S - C + B (1- VCxVE) Vaccine efficacy (VE) x Vaccine coverage (VC) = 80% t+1 tt+1 Epidemic threshold sustained: S = N / R e 0

15. 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

16. 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 SusceptipleRemovedInfectious

17. The SIR model equations = 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 λ(t) {

18. Endemic equilibrium (SIR) N = 10,000 = 300/10000 (per time unit) = 300/10000 (per time unit) = 10 (per time unit) = 10 (per time unit) = 1 (per time unit) = 1 (per time unit)

19. 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 0 0

20. 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) 0

21. Estimation of R (SIR) Relation between the average age at infection and R (SIR model) = 1/75 (per year) = 1/75 (per year) 0 0

22. A simple alternative formula assume everyone is infected at age A everyone dies at age L (rectangular age distribution) Immunes AL Age (years) Susceptibles 100 % Stationary proportion of susceptibles: S / N = A / L S / N = A / L => R 0 = 1/(S e /N) = L / A => R 0 = 1/(S e /N) = L / A e Proportion

23. Estimation of and R 0 from seroprevalence data 1) Assume equilibrium 2) Parameterise force of infection 3) Estimate 4) Calculate R 0 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 data reasonable fit to the data

24. Estimates of R Anderson and May: Infectious Diseases of Humans, 1991 * * * * 0

25. Indirect effects of vaccination (SIR) vaccinate proportion p of newborns, assume complete protection against infection a new reproduction number R = (1-p) x R 0 if p > H = 1-1/R 0, the infection cannot persist if p < H = 1-1/R 0, in the new endemic equilibrium: S = N/R 0, = (R -1) » proportion of susceptibles remains untouched(!) » force of infection decreases vacc vac c vacc

26. 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 Susceptibles AL p Age 1 S / N = (1-p) A’/L S / N = A/ L => A’ = A/(1-p) i.e., increase in the i.e., increase in the average age of average age of infection infection Proportion ’ e e Immunes with vaccination: without vaccination:

27. 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 Proportion 1 p V L H = 1 - 1/R 0 = 1 - A/L proportion immunised by vaccination p (L-V)/L => p = (L-A)/(L-V) i.e., p bigger than when i.e., p bigger than when vaccination at birth vaccination at birth Immunes Susceptibles A

28. The SIS epidemic model herd immunity threshold still the same: H 0 = 1 - 1/R 0 endemic force of infection: the proportions of susceptibles and immunes different from the SIR model SusceptibleImmune

29. R and the force of infection 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) 0 SIS and SIR

30. 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.

31. 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

32. 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) 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 models

33. 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.