1. DIMACS-oct 20081 Non-parametric estimates of transmission functions Epidemic models, generation times and Inference Åke Svensson Stockholm University

2. DIMACS-oct 20082 General aspects of epidemic spread Strength (R 0 ) How many becomes infected Speed (Generation times) How fast does the epidemic grow

3. DIMACS-oct 20083 Generation times demography as a model Demography: Official statistics Generation times (when do mother get daughters) Maternal net function Mass action Epidemics: Not observable events Immunity Interaction in a population Early phase

4. DIMACS-oct 20084 Models for epidemic spread in a population Individual variation contact intensity Natural history of infection latency time infectious time and/or time-varying infectivity

5. DIMACS-oct 20085 Sources of variation Within individual variation here contacts according to a Poisson process (intensity  ). Alternative more clustered (superspreading events) or self-avoiding contacts. Natural history variation here infectivity varies according to a random measure (  (s)), different for different infected Environmental variation here immunity (only)

6. DIMACS-oct 20086 Purpose of this talk A purely theoretical investigation (with a purpose) based on non-parametric assumptions The importance of different sources of individual variation The problems of inference The possibilities of inference Suggestions of inferential methods

7. DIMACS-oct 20087 Basic elements of the epidemic model (individual transmission function) Transmission function (combined effect of natural history of the infection and contact patterns) defines a Cox- process (double stochastic Poisson process) of possible infectious contacts after infection with random intensity  (s) Possible infectious contacts with immune individuals are wasted

8. DIMACS-oct 20088 Population transmission function

9. DIMACS-oct 20089 Epidemic curve in a homogeneous mixing closed population (n=10000, R 0 =1.7, latent period = exp(1.2), infectious period = exp(1.85)) | | Individual variation Population mean population mean important without immunity and immunity Wexp(rt)

10. DIMACS-oct 200810 Counting infected in a homogeneous mixing closed population The number of infected up till time t, N(t), is a counting process with intensity, Epidemic started by one infected, population size n.

11. DIMACS-oct 200811 Estimate based on entire epidemic curve (n population size)

12. DIMACS-oct 200812 Estimate based on entire curve (with individual variation) derivative at 0 is R 0 x(mean generation time)

13. DIMACS-oct 200813 Estimates based on epidemic tree Now disregarding individual variation

14. DIMACS-oct 200814 Estimates based on epidemic tree (before immunity is important)

15. DIMACS-oct 200815 Estimates based on epidemic tree (before immunity is important)

16. DIMACS-oct 200816 Network-models (individual trees) Same method of estimate as above if the number of contacts of each infected is known. Late in the epidemic the number of non-immune contacts (W i (t)) is known at each time The number of infected is a process with intensity W i (t)  (t-s i )

17. DIMACS-oct 200817 Household models (disregarding several infections from outside) Observing the tree Same procedure as before But the model for how contacts occur in time may be unrealistic (the contact processes of the members of the houshold may be dependent) Note that the contact variation has to be separated from natural history variation

18. DIMACS-oct 200818 Household models (disregarding several infections from outside) Observing the curve the epidemic may reach all members of the household In that case no information on the tail of the transmission function is obtained Possible solutions: Only regard the time till first secondary infection. Construct a possible epidemic tree (cf Wallinga-Teunis) and use an iterative procedure Calculate time under risk for different sections of the transmission time (in analoge with Nelson-Aalen estimators in survival analysis)