论文标题
空间随机流行模型：大数量和中央限制定理的定律
A Spatial Stochastic Epidemic Model: Law of Large Numbers and Central Limit Theorem
论文作者
论文摘要
我们考虑了连续空间中基于个体的SIR随机流行模型。流行病的演变涉及感染率和个体治愈率。我们假设个体根据独立的布朗尼动作在二维圆环上随机移动。我们定义了经验度量$μ^{s，n} $，$μ^{i，n} $和$μ^{r，n} $，这些内容描述了易感，感染和删除个体的位置的演变。我们证明了序列$（μ^{s，n}，μ^{i，n}）$ to $（μ^{s}，μ^{s}，μ^{i}）$的融合性的融合，为$ n \ rightarrow \ infty $。我们表明序列$（u^{n} = \ sqrt {n}（μ^{s，n}-μ^{s}），v^{n} = \ sqrt {n} {n}（μ^{μ^{i，n} - n} - μ^{i}）$ right a in law a in law as $ n a in a $ nak as $ n as a in a in a a $ rock a a in a in a f in。具有高度奇异高斯驾驶过程的线性PDE系统的解决方案。在个人不动的情况下，我们还定义和研究了序列$（μ^{s，n}，μ^{i，n}）$的大量定律和中心限制定理。
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 $μ^{S,N}$, $μ^{I,N}$ and $μ^{R,N}$ 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 $(μ^{S,N},μ^{I,N})$ towards $(μ^{S},μ^{I})$ solution of a system of parabolic PDEs. We show that the sequence $(U^{N}=\sqrt{N}(μ^{S,N}-μ^{S}),V^{N}=\sqrt{N}(μ^{I,N}-μ^{I}))$ 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 $(μ^{S,N},μ^{I,N})$.
