Stochastic and Deterministic Analysis of SIS Household Epidemics

We analyse SIS epidemics amongst populations partitioned into households. The analysis considers both the stochastic and deterministic model and unlike previous analysis, we consider general infectious period distributions. For the deterministic model, we prove the existence of an endemic equilibrium for the epidemic if and only if the threshold parameter, $R_\ast >1$. Furthermore, by utilising Markov Chains we show that the total number of infectives converges to the endemic equilibrium as time $t \rightarrow \infty$. For the stochastic model, we prove a law of large numbers result for the convergence of the mean number of infectives per household in the stochastic model to the deterministic limit. This is followed by the derivation of a Gaussian limit process for the fluctuations of the stochastic model