In this talk I will present an epidemic model where susceptible and infected individuals are distributed in discrete households of finite size. Infected individuals can either infect other individuals in the same household or individuals chosen uniformly in the whole population. When the number of households tends to infinity, we obtain a limit for the epidemic process given in terms of a nonlinear Markov process which solves a McKean-Vlasov Poisson driven SDE. We also prove a propagation of chaos result. Finally, we define a basic reproduction number R0, and show that if R0>1, then the nonlinear Markov process has a unique non trivial ergodic invariant probability measure (i.e. the epidemic spreads to a large proportion of the population with positive probability), whereas if R0<=1, the non-linear Markov process converges to 0 as t tends to infinity (the epidemic quickly dies out). To conclude I will mention some results on the fluctuations of the epidemic process.
This is joint work with Étienne Pardoux.