# Household epidemic models and McKean-Vlasov Poisson driven stochastic differential equations

## TL;DR

This paper introduces a novel mean field approach to household epidemic modeling using McKean-Vlasov Poisson-driven SDEs, establishing conditions for epidemic persistence or extinction based on a basic reproduction number.

## Contribution

It develops a new mathematical framework for household epidemic models via nonlinear Markov processes and proves a propagation of chaos result, linking epidemic dynamics to McKean-Vlasov equations.

## Key findings

- Derived a basic reproduction number R0 for the model
- Proved existence of a unique ergodic invariant measure when R0>1
- Showed convergence to zero when R0<=1

## Abstract

This paper presents a new view of household epidemic models, where the interaction between the households is of mean field type. We thus obtain in the limit of infinitely many households a nonlinear Markov process solution of a McKean - Vlasov type Poisson driven SDE, and a propagation of chaos result. We also 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, whereas if R0<=1, it converges to 0 as t tends to infinity.