A Spatial Stochastic Epidemic Model: Law of Large Numbers and Central Limit Theorem

Abstract

We consider an individual-based SIR stochastic epidemic model in continuous space. The evolution of the epidemic involves the rates of infection and cure of individuals. We assume that individuals move randomly on the two-dimensional torus according to independent Brownian motions. We define the empirical measures $\mu^{S,N}$, $\mu^{I,N}$ and $\mu^{R,N}$ which describe the evolution of the position of the susceptible, infected and removed individuals. We prove the convergence in propbability, as $N\rightarrow \infty$, of the sequence $(\mu^{S,N},\mu^{I,N})$ towards $(\mu^{S},\mu^{I})$ solution of a system of parabolic PDEs. We show that the sequence $(U^{N}=\sqrt{N}(\mu^{S,N}-\mu^{S}),V^{N}=\sqrt{N}(\mu^{I,N}-\mu^{I}))$ converges in law, as $N\rightarrow\infty$, towards a Gaussian distribution valued process, solution of a system of linear PDEs with highly singular Gaussian driving processes. In the case where the individuals do not move we also define and study the law of large numbers and central limit theorem of the sequence $(\mu^{S,N},\mu^{I,N})$.