Household epidemic models and McKean–Vlasov Poisson driven stochastic differential equations

Abstract

This paper presents a new view of household epidemic models, where we exploit the fact that the interaction between the households is of mean field type. We prove the convergence, as the number of households tends to infinity, of the number of infectious individuals in a uniformly chosen household to a nonlinear Markov process solving a McKean–Vlasov Poisson driven stochastic differential equation, as well as a propagation of chaos result. We also define a basic reproduction number ${\mathit{R}}_{\ast }$ and show that if ${\mathit{R}}_{\ast }>1$, then the nonlinear Markov process has a unique nontrivial ergodic invariant probability measure, whereas if ${\mathit{R}}_{\ast }\le 1$, it converges to 0 as t tends to infinity.