A SIR epidemic model on a refining spatial grid II-central limit theorem

Abstract

A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein–Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.

Appendix A

Lemma A.1

Let $(h_\varepsilon )_{0<\varepsilon <1}$ be a sequence of $\texttt{H}_{\varepsilon }$. If $(h_\varepsilon )_{0<\varepsilon <1}$ is bounded in $\textrm{H}^{1,\varepsilon }$, then it is relatively compact in $L^2$, and the limit of any convergent subsequence belongs to $\textrm{H}^{1}$.

Proof

By using the fact that the sequence $(h_\varepsilon )$ is bounded in $L^2$ and $\big \Vert \nabla _{\!\!\varepsilon }^+h_\varepsilon \big \Vert _{L^2}\le C\big \Vert h_\varepsilon \big \Vert _{_{\textrm{H}^{1}}}$, then the result of the compactness follows from the compactness theorem of Kolmogorov in $L^2$.

The fact the limit of any convergent subsequence belong to $\textrm{H}^{1}$, follows from the discrete integrating by part

$$\begin{aligned} \int _{\mathbb {T}^1}\nabla _{\!\!\varepsilon }^+h_\varepsilon (x)\varphi (x)dx= -\langle \, \int _0^. h_\varepsilon (y)dy\; , \; \nabla _{\!\!\varepsilon }^+\varphi \; \rangle , \end{aligned}$$

and letting $\varepsilon$ go to zero in this equation. $\square$

Lemma A.2

For all $\displaystyle u_{\varepsilon } \in \texttt{H}_{\varepsilon }$

$\displaystyle \big \Vert \nabla _{\!\!\varepsilon }^- u_{\varepsilon }\big \Vert _{_{\textrm{H}^{-\gamma , \varepsilon }}}^2 = \big \Vert \nabla _{\!\!\varepsilon }^+ u_{\varepsilon }\big \Vert _{_{\textrm{H}^{-\gamma , \varepsilon }}}^2= \sum _m \left( \langle \,u_{\varepsilon } , \varphi _{m}^{\varepsilon }\,\rangle ^2 + \langle \,u_{\varepsilon } , \psi _{m}^{\varepsilon }\,\rangle ^2 \right) \lambda _{m}^{\varepsilon }(1+\lambda _{m}^{\varepsilon })^{-\gamma }.$

Proof

We have

$$\begin{aligned} \nabla ^-_{\!\!\varepsilon } \varphi _m^{\varepsilon } = -b_{m,\varepsilon } \varphi _m^{\varepsilon }- a_{m,\varepsilon }\psi _m^{\varepsilon } \; \; \text { and } \; \; \nabla ^-_{\!\!\varepsilon } \psi _m^{\varepsilon } = a_{m,\varepsilon } \varphi _m^{\varepsilon }-b_{m,\varepsilon }\psi _m^{\varepsilon }, \end{aligned}$$

where $a_{m,\varepsilon }=\varepsilon ^{-1}\sin (\pi m\varepsilon )$ and $b_{m,\varepsilon }=\varepsilon ^{-1} (\cos (\pi m\varepsilon )-1)$.

We have

$$\begin{aligned} a_{m,\varepsilon }^2+b_{m,\varepsilon }^2=\lambda _m^{\varepsilon }. \end{aligned}$$

Let $\displaystyle u_{\varepsilon } \in \texttt{H}_{\varepsilon }$. We have

$$\begin{aligned}{} & {} \big \Vert \nabla _{\!\!\varepsilon }^+ u_{\varepsilon }\big \Vert _{_{\textrm{H}^{-\gamma , \varepsilon }}}^2\\{} & {} \quad =\sum _m\left( \langle \, u_{\varepsilon },\nabla _{\!\!\varepsilon }^- \varphi _m\, \rangle ^2 + \langle \, u_{\varepsilon } , \nabla _{\!\!\varepsilon }^- \psi _m \,\rangle ^2\right) (1+\lambda _m^{\varepsilon })^{-\gamma } \\{} & {} \quad =\sum _m\left( \langle \, u_{\varepsilon }, -b_{m,\varepsilon }\varphi _m^{\varepsilon }-a_{m,\varepsilon } \psi _m^{\varepsilon } \,\rangle ^2+\langle \, u_{\varepsilon }, a_{m,\varepsilon } \varphi _m^{\varepsilon }-b_{m,\varepsilon } \psi _m^{\varepsilon } \,\rangle ^2\right) (1+\lambda _m^{\varepsilon })^{-\gamma } \\{} & {} \quad =\sum _m\left( [-b_{m,\varepsilon } \langle \, u_{\varepsilon },\varphi _m^{\varepsilon }\,\rangle -a_{m,\varepsilon }\langle \, u_{\varepsilon }, \psi _m^{\varepsilon } \,\rangle ]^2+[a_{m,\varepsilon }\langle \, u_{\varepsilon },\varphi _m^{\varepsilon }\,\rangle -b_{m,\varepsilon } \langle \, u_{\varepsilon }, \psi _m^{\varepsilon }\,\rangle ]^2\right) \\{} & {} \qquad \times (1+\lambda _m^{\varepsilon })^{-\gamma } \\{} & {} \quad =\sum _m\left( [a_{m,\varepsilon }^2+b_{m,\varepsilon }^2]\{\langle \, u_{\varepsilon },\varphi _m^{\varepsilon }\,\rangle ^2+\langle \, u_{\varepsilon }, \psi _m^{\varepsilon }\,\rangle ^2\}\right) (1+\lambda _m^{\varepsilon })^{-\gamma } \\{} & {} \quad =\sum _m\left( \langle \, u_{\varepsilon }, \varphi _m^{\varepsilon }\,\rangle ^2+\langle \, u_{\varepsilon }, \psi _m^{\varepsilon }\,\rangle ^2\right) \lambda _m^\varepsilon (1+\lambda _m^{\varepsilon })^{-\gamma } . \end{aligned}$$

The proof of $\displaystyle \big \Vert \nabla _{\!\!\varepsilon }^- u_{\varepsilon }\big \Vert _{_{\textrm{H}^{-\gamma , \varepsilon }}}^2 = \sum _m \left( \langle \,u_{\varepsilon } , \varphi _{m}^{\varepsilon }\,\rangle ^2 + \langle \,u_{\varepsilon } , \psi _{m}^{\varepsilon }\,\rangle ^2 \right) \lambda _{m}^{\varepsilon }(1+\lambda _{m}^{\varepsilon })^{-\gamma }$ is similar by noting that

$$\begin{aligned} \nabla ^+_{\!\!\varepsilon } \varphi _m^{\varepsilon } = b_{m,\varepsilon } \varphi _m^{\varepsilon }- a_{m,\varepsilon }\psi _m^{\varepsilon } \; \; \text { and } \; \; \nabla ^+_{\!\!\varepsilon } \psi _m^{\varepsilon } = a_{m,\varepsilon } \varphi _m^{\varepsilon }+b_{m,\varepsilon }\psi _m^{\varepsilon }. \end{aligned}$$

$\square$