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Non-parametric estimates of transmission functions

Investigating theoretical aspects of epidemic spread, sources of individual variation, problems and possibilities of inference, suggesting inferential methods with non-parametric assumptions.

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Non-parametric estimates of transmission functions

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  1. Non-parametric estimates of transmission functions Epidemic models, generation times and Inference Åke Svensson Stockholm University DIMACS-oct 2008

  2. General aspects of epidemic spread • Strength (R0) How many becomes infected • Speed (Generation times) How fast does the epidemic grow DIMACS-oct 2008

  3. 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 Generation times demography as a model DIMACS-oct 2008

  4. Models for epidemic spread in a population • Individual variation contact intensity • Natural history of infection latency time infectious time and/or time-varying infectivity DIMACS-oct 2008

  5. Sources of variation • Within individual variation here contacts according to a Poisson process (intensity k). Alternative more clustered (superspreading events) or self-avoiding contacts. • Natural history variation here infectivity varies according to a random measure (t(s)), different for different infected • Environmental variation here immunity (only) DIMACS-oct 2008

  6. 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 DIMACS-oct 2008

  7. 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 kt(s) Possible infectious contacts with immune individuals are wasted DIMACS-oct 2008

  8. Population transmission function DIMACS-oct 2008

  9. Epidemic curve in a homogeneous mixing closed population(n=10000, R0=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) DIMACS-oct 2008

  10. 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. DIMACS-oct 2008

  11. Estimate based on entire epidemic curve(n population size) DIMACS-oct 2008

  12. Estimate based on entire curve(with individual variation)derivative at 0 is R0x(mean generation time) DIMACS-oct 2008

  13. Estimates based on epidemic treeNow disregarding individual variation DIMACS-oct 2008

  14. Estimates based on epidemic tree(before immunity is important) DIMACS-oct 2008

  15. Estimates based on epidemic tree(before immunity is important) DIMACS-oct 2008

  16. 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 (Wi(t)) isknown at each time The number of infected is a process with intensity Wi(t)kt(t-si) DIMACS-oct 2008

  17. 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 DIMACS-oct 2008

  18. 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) DIMACS-oct 2008

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