2. Preliminary and formulation

Let $\mathcal{O}$ be a bounded domain in $\mathbb{R}^l$ (with $l\geq 1$ ) having a $C^2$ boundary, and let $H\,:\!=L^2(\mathcal{O};\,\mathbb{R})$ be the separable Hilbert space, endowed with the scalar product

\begin{equation*}\langle u, v \rangle_H\,:\!=\int_{\mathcal{O}} u(x) v(x) \, {\rm d} x \end{equation*}

and the corresponding norm $\left\vert{u}\right\vert_H = \sqrt {\langle u, u \rangle_H}$ . We will say $ u\geq 0$ if $ u(x)\geq 0$ almost everywhere in $\mathcal{O}$ . Moreover, we denote by $L^2(\mathcal{O},\mathbb{R}^2)$ the space of all functions $u(x)=(u_1(x),u_2(x))$ where $ u_1,u_2\in L^2(\mathcal{O},\mathbb{R})$ , on which the inner product is defined as

\begin{align*}\langle u,v \rangle_{L^2(\mathcal{O},\mathbb{R}^2)}&\,:\!=\int_\mathcal{O} \big\langle u(x),v(x) \big\rangle_{\mathbb{R}^2} \, {\rm d} x=\int_{\mathcal{O}}(u_1(x)v_1(x)+u_2(x)v_2(x)) \, {\rm d} x\\&\,=\langle u_1,v_1 \rangle_{L^2(\mathcal{O},\mathbb{R})}+\langle u_2,v_2 \rangle_{L^2(\mathcal{O},\mathbb{R})},\end{align*}

for all $u,v\in L^2(\mathcal{O},\mathbb{R}^2)$ . Note that $L^2(\mathcal{O},\mathbb{R}^2)$ is a separable Hilbert space. In what follows, we use u to denote a function that is either real valued or $\mathbb{R}^2$ -valued (as will be clear from the context). Denote by E the Banach space $C(\mathcal{ \overline{O}};\,\mathbb{R})$ endowed with the sup-norm

\begin{equation*}\left\vert{u}\right\vert_E\,:\!=\sup_{x\in \mathcal{\overline{O}}}\left\vert{u(x)}\right\vert.\end{equation*}

Let $(\Omega, \mathcal{F},\{\mathcal{F}_t\}_{t\geq 0},\mathbb{P})$ be a complete probability space, and $L^{\;p}(\Omega;\,C([0,t],C(\mathcal{\overline{O}},\mathbb{R}^2)))$ be the space of all predictable $C(\mathcal{\overline{O}},\mathbb{R}^2)$ -valued processes u in $C([0,t],C(\mathcal{\overline{O}},\mathbb{R}^2))$ , P almost surely (a.s.), with the norm $L_{t,p} $ as follows:

\begin{equation*}\left\vert{u}\right\vert^p_{L_{t,p}}\,:\!=\mathbb{E} \sup_{s\in [0,t]}\left\vert{u(s)}\right\vert^p_{C(\mathcal{\overline{O}},\mathbb{R}^2)},\end{equation*}

where

\begin{equation*}\left\vert{u}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}=\Bigg(\sum_{i=1}^2\sup_{x\in \mathcal{\overline{O}}}\left\vert{u_i(x)}\right\vert^2\Bigg)^\frac 12\qquad\text{if}\ u=(u_1,u_2)\in C(\mathcal{\overline{O}},\mathbb{R}^2).\end{equation*}

For $\varepsilon>0,p\geq 1$ , denote by $W^{\varepsilon,p}(\mathcal{O},\mathbb{R}^2)$ the Sobolev–Slobodeckij space (the Sobolev space with non-integer exponent) endowed with the norm

\begin{equation*}\left\vert{u}\right\vert_{\varepsilon,p}\,:\!=\left\vert{u}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)}+\sum_{i=1}^2\int_{\mathcal{O}\times\mathcal{O}}\dfrac{\left\vert{u_i(x)-u_i(y)}\right\vert^p}{\left\vert{x-y}\right\vert^{\varepsilon p+l}} \, {\rm d} x \, {\rm d} y.\end{equation*}

Assume that $B_{k,1}(t)$ and $B_{k,2}(t)$ with $k=1,2,\dots$ are independent $\{\mathcal{F}_t\}_{t\geq 0}$ -adapted one-dimensional Wiener processes. Now, fix an orthonormal basis $\{e_k\}_{k=1}^{\infty}$ in H and assume that this sequence is uniformly bounded in $L^{\infty}(\mathcal{O},\mathbb{R})$ , i.e.

\begin{equation*}C_0\,:\!=\sup_{k\in\mathbb{N}}\left\vert{e_k}\right\vert_{L^{\infty}(\mathcal{O},\mathbb{R})}=\sup_{k\in\mathbb{N}}\text{ess}\,\sup\limits_{\!\!\!\!\!\!\!x\in\mathcal{O}} e_k(x)<\infty.\end{equation*}

We define the infinite-dimensional Wiener processes $W_i(t)$ , which are driving noise in (1.1), as follows:

\begin{equation*}\displaystyle W_i(t)=\sum_{k=1}^{\infty}\sqrt {a_{k,i}}B_{k,i}(t)e_k,\qquad i=1,2,\end{equation*}

where $\{a_{k,i}\}_{k=1}^{\infty}$ are sequences of non-negative real numbers satisfying

(2.1) \begin{equation}a_i\,:\!=\sum_{k=1}^{\infty}a_{k,i}<\infty,\qquad i=1,2.\end{equation}

Let $A_1$ and $A_2$ be Neumann realizations of $k_1\Delta$ and $k_2\Delta$ in H, respectively, i.e.

\begin{equation*}D(A_i)=\big\{u\in H \mid \Delta u\in H\;\text{and}\;\partial_\nu u=0\;\text{on}\;\partial \mathcal{O}\big\},\end{equation*}

\begin{equation*}A_i u=k_i\Delta u,\qquad u\in D(A_i),\end{equation*}

where the Laplace operator in the above definition is understood in the distribution sense. Then, $A_1$ and $A_2$ are infinitesimal generators of analytic semigroups ${\rm e}^{tA_1}$ and ${\rm e}^{tA_2}$ with corresponding Neumann heat kernels denoted by $p_\mathcal{O}^{N,1}(t,x,y)$ , $p_\mathcal{O}^{N,2}(t,x,y)$ , i.e.

\begin{equation*}({\rm e}^{tA_i}u)(x)=\int_\mathcal{O} p_\mathcal{O}^{N,i}(t,x,y)u(y) \, {\rm d} y, \qquad i=1,2,\end{equation*}

respectively. In addition, if we denote by $A\,:\!=(A_1,A_2)$ the operator defined in $L^2(\mathcal{O},\mathbb{R}^2)$ by $Au\,:\!=(A_1u_1,A_2u_2)$ for $u=(u_1,u_2)\in L^2(\mathcal{O},\mathbb{R}^2)$ , then it generates an analytic semigroup ${\rm e}^{tA}$ with ${\rm e}^{tA}u=({\rm e}^{tA_1}u_1,{\rm e}^{tA_2}u_2)$ . In [Reference Davies13, Theorem 1.4.1], it is proved that the space $L^1(\mathcal{O},\mathbb{R}^2 )\cap L^\infty(\mathcal{O},\mathbb{R}^2)$ is invariant under ${\rm e}^{tA}$ , so that ${\rm e}^{tA}$ may be extended to a non-negative one-parameter semigroup ${\rm e}^{tA(p)}$ on $L^{\;p}(\mathcal{O};\,R^2 )$ , for all $1\leq p\leq\infty$ . All these semigroups are strongly continuous and consistent in the sense that ${\rm e}^{tA(p)}u={\rm e}^{tA(q)}u$ for any $u\in L^{\;p}(\mathcal{O},\mathbb{R}^2)\cap L^q(\mathcal{O},\mathbb{R}^2)$ (see [Reference Cerrai8]). So, we will suppress the superscript p and denote them by ${\rm e}^{tA}$ whenever there is no confusion. Moreover, if we consider the part $A_i^E$ of $A_i$ in the space of continuous functions E, it generates an analytic semigroup (see [Reference Arendt3, Chapter 2]), which has no dense domain in general. However, since we have assumed that $\mathcal{O}$ has a $C^2$ boundary in our boundary condition, $A_i^E$ has dense domain in E (see [Reference Da Prato and Zabczyk12, Appendix A.5.2]), and hence this analytic semigroup is strongly continuous. Finally, we recall some well-known properties of the operators $A_i$ and analytic semigroups ${\rm e}^{tA_i}$ for $i=1,2$ as follows (for further details, we refer the reader to [Reference Arendt3, Reference Davies13, Reference Ouhabaz31] and the references therein):

1. • For all $u\in H$ , $\int_ 0^t {\rm e}^{sA_i}u \, {\rm d} s\in D(A_i)$ and $A_i(\int_0^t {\rm e}^{sA_i}u \, {\rm d} s)={\rm e}^{tA_i}u-u$ .

2. • By Green’s identity, it can be proved that $A_i$ is symmetric, that $A_i$ is self-adjoint in H, and that, $\text{for all}\ u\in D(A_i)$ , $\int_\mathcal{O} (A_iu)(x) \, {\rm d} x=0$ .

3. • For any $t>0$ , $x,y\in\mathcal{O}$ ,

   \begin{equation*}0\leq p_\mathcal{O}^{N,i}(t,x,y)\leq c_1 (t\wedge 1)^{-\frac l2}\exp\bigg\{-c_2\frac{\left\vert{x-y}\right\vert^2}{t}\bigg\}\end{equation*}
   for some constants $c_1$ and $c_2$ which depend on $\mathcal{O}$ but are independent of u and t.

4. • The semigroup ${\rm e}^{tA}$ satisfies

   (2.2) \begin{equation}\begin{aligned}\left\vert{{\rm e}^{tA}u}\right\vert_{L^\infty(\mathcal{O},\mathbb{R}^2)}\leq c\left\vert{u}\right\vert_{L^\infty(\mathcal{O},\mathbb{R}^2)}\quad\text{and}\quad\left\vert{{\rm e}^{tA}u}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}\leq c\left\vert{u}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}\end{aligned}\end{equation}
   for some constant c which depends on $\mathcal{O}$ but is independent of u and t.

5. • For any $t,\varepsilon>0$ , $p\geq 1$ , the semigroup ${\rm e}^{tA}$ maps $L^{\;p}(\mathcal{O},\mathbb{R}^2)$ into $W^{\varepsilon,p}(\mathcal{O},\mathbb{R}^2)$ and, $\text{for all}\ u\in L^{\;p}(\mathcal{O},\mathbb{R}^2)$ ,

   (2.3) \begin{equation}\left\vert{{\rm e}^{tA}u}\right\vert_{\varepsilon,p}\leq c(t\wedge 1)^{-\varepsilon/2}\left\vert{u}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)} \end{equation}
   for some constant c independent of u and t.

Now, we rewrite (1.1) as a stochastic differential equation in an infinite-dimension space:

(2.4) \begin{equation}\begin{cases}{\rm d} S(t)=\bigg[A_1 S(t) +\Lambda-\mu_1S(t)-\dfrac{\alpha S(t)I(t)}{S(t)+I(t)}\bigg] \, {\rm d} t + S(t) \, {\rm d} W_1(t),\\[0.3cm]{\rm d} I(t)=\bigg[A_2 I(t)-\mu_2I(t) + \dfrac{\alpha S(t)I(t)}{S(t)+I(t)}\bigg] \, {\rm d} t+I(t) \, {\rm d} W_2(t),\\S(0)=S_0,I(0)=I_0.\end{cases}\end{equation}

As usual, we say that $(S(t),I(t))$ is a mild solution to (2.4) if

(2.5) \begin{equation}\begin{cases}\displaystyle S(t)={\rm e}^{tA_1}S_0+\int_0^t {\rm e}^{(t-s)A_1}\bigg(\Lambda-\mu_1S(s)-\dfrac{\alpha S(s)I(s)}{S(s)+I(s)}\bigg) \, {\rm d} s+W_S(t),\\[0.3cm]\displaystyle I(t)={\rm e}^{tA_2}I_0+\int_0^t {\rm e}^{(t-s)A_2}\bigg(-\mu_2 I(s)+\dfrac{\alpha S(s)I(s)}{S(s)+I(s)}\bigg) \, {\rm d} s+W_I(t),\end{cases}\end{equation}

where

\begin{equation*}W_S(t)=\int_0^t {\rm e}^{(t-s)A_1}S(s) \, {\rm d} W_1(s)\quad\text{and}\quad W_I(t)=\int_0^t {\rm e}^{(t-s)A_2}I(s) \, {\rm d} W_2(s),\end{equation*}

or, in vector form,

(2.6) \begin{equation}\displaystyle Z(t)={\rm e}^{tA}Z_0+\int_0^t {\rm e}^{(t-s)A}F(Z(s)) \, {\rm d} s+\int_0^t {\rm e}^{(t-s)A}Z(s) \, {\rm d} W(s),\end{equation}

where $ Z=(S,I)$ , $F(Z)=(F_1(Z),F_2(Z))\,:\!=\bigg(\Lambda-\mu_1S-\dfrac{\alpha SI}{S+I},-\mu_2I+\dfrac{\alpha SI}{S+I}\bigg)$ , and

\begin{equation*}{\rm e}^{(t-s)A}Z(s) \, {\rm d} W(s)\,:\!=({\rm e}^{(t-s)A_1}S(s) \, {\rm d} W_1(s)\;,\; {\rm e}^{(t-s)A_2}I(s) \, {\rm d} W_2(s)).\end{equation*}

Because we are modeling SIR epidemic systems, we are only interested in the positive ( $\geq 0$ ) solutions. Therefore, we define a positive mild solution of (2.4) as a mild solution S(t, x), I(t, x) such that $S(t,x),I(t,x)\geq 0$ , almost everywhere $x\in\mathcal{O}$ , for all $t\geq 0$ . Moreover, to have the term $\dfrac{si}{s+i}$ well defined, we assume that it is equal 0 whenever either $s=0$ or $i=0$ .

Remark 2.1. The integrals on the right-hand side of (2.5) are understood as Bochner integrals (in the Banach space H), while $W_S(t)$ and $W_I(t)$ are stochastic integrals (stochastic convolutions). The S(s) (resp. I(s)) in the stochastic integrals is understood as a multiplication operator, i.e.

\begin{equation*}S(s)(u)=S(s)u \qquad \text{for all}\ u\in H.\end{equation*}

The stochastic integral $\int_0^t {\rm e}^{(t-s)A_i}U(s) \, {\rm d} W_i(s)$ (see [Reference Da Prato and Zabczyk12, Chapter 4] for more details on stochastic integrals) is well defined if the process U(s) satisfies

\begin{equation*}\int_0^t \sum_{k=1}^{\infty}a_{k,i}\big\vert{{\rm e}^{(t-s)A_i} U(s) e_k}\big\vert^2_H \, {\rm d} s<\infty.\end{equation*}

Finally, in the vector form, to simplify the notation we do not write the vectors in column form. However, the calculations involving vectors are understood as in the usual sense.

To investigate epidemic models, an important question is whether the infected individuals will die out in the long term. That is, the consideration of extinction or permanence. Since the mild solution is used, let us introduce the definitions in the weak sense as follows.

Definition 2.1. A population with density u(t, x) is said to be extinct in the mean if

\begin{equation*}\limsup_{t\to\infty}\dfrac 1t\int_0^t\mathbb{E}\int_\mathcal{O} u(s,x) \, {\rm d} x \, {\rm d} s=0,\end{equation*}

and is said to be permanent in the mean if there exists a positive number $R_I$ , independent of the initial conditions of the population, such that

\begin{equation*}\liminf_{t\to\infty}\dfrac 1t\int_0^t\bigg(\mathbb{E}\int_\mathcal{O} (u^2(s,x)\wedge 1) \, {\rm d} x\bigg)^{\frac 12} \, {\rm d} s\geq R_I.\end{equation*}

3. Existence and uniqueness of a positive mild solution

In this section we shall prove the existence and uniqueness of a positive mild solution of the system as well as its continuous dependence on the initial conditions. In what follows, without loss of the generality we can assume $\left\vert{\mathcal{O}}\right\vert=1$ , where $\left\vert{\mathcal{O}}\right\vert$ is the volume of the bounded domain $\mathcal{O}$ in $\mathbb{R}^l$ , and the initial values are non-random for simplicity.

3.1. Proof of Theorem 3.1

In this proof, the letter c denotes a positive constant whose value may change in different occurrences. We will write the dependence of the constant on parameters explicitly if it is essential. First, we rewrite the coefficients by defining f and $f^*$ as follows:

\begin{equation*}f(x,s,i)=\bigg(\Lambda(x)-\mu_1(x)s-\dfrac{\alpha(x) si}{s+i},-\mu_2(x)i+\dfrac{\alpha(x) si}{s+i}\bigg),\qquad x\in\mathcal{O},\ (s,i)\in \mathbb{R}^2,\end{equation*}

and

\begin{equation*}f^*(x,s,i)=f(x,s\vee 0,i\vee 0).\end{equation*}

Writing $z=(s,i)$ , by noting that we are assuming the term $\dfrac {si}{s+i}$ will be equal to 0 whenever either $s=0$ or $i=0$ , it is easy to see that $f^*(x,\cdot,\cdot):\,\mathbb{R}^2 \mapsto\mathbb{R}^2$ is Lipschitz continuous, uniformly in $x\in\mathcal{O}$ , so that the composition operator $F^*(z)$ associated with $f^*$ , i.e.

\begin{equation*}F^*(z)(x)=(F^*_1(z)(x),F_{2}^*(z)(x))\,:\!=f^*(x,z(x)),\qquad x\in \mathcal{O},\end{equation*}

is Lipschitz continuous in both $L^2(\mathcal{O}, \mathbb{R}^2)$ and $C(\mathcal{\overline{O}},\mathbb{R}^2)$ . Now, we consider the following problem:

(3.1) \begin{equation}{\rm d} Z^*(t)=\big[AZ^*(t)+F^*(Z^*(t))\big] \, {\rm d} t+(Z^*(t)\vee 0) \, {\rm d} W(t), \quad Z^*(0)=Z_0=(S_0,I_0),\end{equation}

where $Z^*(t)=(S^*(t),I^*(t))$ and $Z^*(t)\vee 0$ is defined by

\begin{equation*}(Z^*(t)\vee 0)(x)=(S^*(t,x)\vee 0,I^*(t,x)\vee 0).\end{equation*}

For any

\begin{equation*}u(t,x)=(u_1(t,x),u_2(t,x))\in L^{\;p}(\Omega;\, C([0,T],C(\mathcal{\overline{O}},\mathbb{R}^2))),\end{equation*}

consider the mapping

\begin{equation*}\gamma(u)(t)\,:\!={\rm e}^{tA}Z_0+\int_0^t {\rm e}^{(t-s)A}F^*(u(s)) \, {\rm d} s+\varphi(u)(t),\end{equation*}

where

\begin{align*}\varphi(u)(t)&\,:\!=\int_0^t {\rm e}^{(t-s)A}(u(s)\vee 0) \, {\rm d} W(s)\\&\,:\!=\bigg(\int_0^t {\rm e}^{(t-s)A_1}(u_1(s)\vee 0) \, {\rm d} W_1(s),\int_0^t {\rm e}^{(t-s)A_2}(u_2(s)\vee 0) \, {\rm d} W_2(s)\bigg).\end{align*}\vskip-\lastskip\pagebreak

We will prove that $\gamma$ is a contraction mapping in $L^{\;p}(\Omega;\, C([0,T_0],C(\mathcal{\overline{O}},\mathbb{R}^2)))$ for some $T_0>0$ and any $p\geq p_0$ for some $p_0$ .

Lemma 3.1. There exists $p_0$ such that, for any $p\geq p_0$ ,

1. • the mapping $\varphi$ maps $L^{\;p}(\Omega;\, C([0,t],C(\mathcal{\overline{O}},\mathbb{R}^2)))$ into itself,

2. • for any $u=(u_1,u_2),v=(v_1,v_2)\in L^{\;p}(\Omega;\, C([0,t],C(\mathcal{\overline{O}},\mathbb{R}^2))),$

   (3.2) \begin{equation}\left\vert{\varphi (u)-\varphi (v)}\right\vert_{L_{t,p}}\leq c_p(t)\left\vert{u-v}\right\vert_{L_{t,p}}\!,\end{equation}
   where $c_p(t)$ is some constant satisfying $c_p(t)\downarrow 0$ as $t\downarrow 0$ .

Proof. Let $p_0$ be sufficiently large to ensure that, for any $p\geq p_0$ , we can choose simultaneously $\beta,\varepsilon>0$ such that

\begin{equation*}\frac 1p<\beta<\frac 12\quad\text{and}\quad \frac lp<\varepsilon<2\bigg(\beta-\frac 1p\bigg).\end{equation*}

Now, for any fixed $p\geq p_0$ , let $\beta,\varepsilon$ be chosen as above. By using a factorization argument (see, e.g., [Reference Da Prato and Zabczyk12, Theorem 8.3]), we have

\begin{equation*}\varphi(u)(t)-\varphi(v)(t)=\dfrac{\sin \pi \beta}{\pi}\int_0^t (t-s)^{\beta-1}{\rm e}^{(t-s)A}Y_\beta(u,v)(s) \, {\rm d} s,\end{equation*}

where

\begin{equation*}Y_\beta(u,v)(s)=\int_0^s (s-r)^{-\beta}{\rm e}^{(s-r)A}(u(r)\vee 0-v(r)\vee 0) \, {\rm d} W(r).\end{equation*}

If

\begin{equation*}\int_0^t \big\vert{Y_\beta(u,v)(s)}\big\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)}^p{\rm d} s<\infty \hbox{ a.s.},\end{equation*}

then it is easily seen from the properties in (2.3) of the semigroup ${\rm e}^{tA}$ and Hölder’s inequality that

(3.3) \begin{equation}\begin{aligned}&\left\vert{\varphi(u)(t)-\varphi(v)(t)}\right\vert_{\varepsilon,p}\\&\quad \leq c_\beta\int_0^t (t-s)^{\beta-1}((t-s)\wedge 1)^{-\varepsilon/2}\left\vert{Y_\beta(u,v)}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)} \, {\rm d} s\\&\quad \leq c_{\beta,p}(t)\bigg(\int_0^t ((t-s)\wedge 1)^{\frac p{p-1}(\beta-\varepsilon/2-1)} \, {\rm d} s\bigg)^{\frac{p-1}p}\bigg(\int_0^t \left\vert{Y_\beta(u,v)(s)}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)}^p \, {\rm d} s\bigg)^{\frac 1p}\\&\quad\leq c_{\beta,p}(t)\bigg(\int_0^t \left\vert{Y_\beta(u,v)(s)}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)}^pds\bigg)^{\frac 1p} \quad \text{a.s.} ,\end{aligned}\end{equation}

where $c_{\beta,p}(t)$ is some positive constant satisfying $c_{\beta,p}(t)\downarrow 0$ as $t\downarrow 0$ . Rewriting $Y_\beta(u,v)(s)=(Y_{1\beta}(u,v)(s),Y_{2\beta}(u,v)(s))$ , where

\begin{equation*}Y_{i\beta}(u,v)(s)\,:\!=\int_0^s (s-r)^{-\beta}{\rm e}^{(s-r)A_i}(u_i(r)\vee 0-v_i(r)\vee 0) \, {\rm d} W_i(r),\qquad i=1,2.\end{equation*}\vskip-\lastskip\pagebreak

Therefore, applying the Burkholder inequality, we obtain that for all $s\in [0,t]$ and almost every $x\in\mathcal{O}$ ,

\begin{equation*}\begin{aligned}&\!\!\!\mathbb{E} \left\vert{Y_{i\beta}(u,v)(s,x)}\right\vert^p\leq c_p\mathbb{E}\bigg[\int_0^s(s-r)^{-2\beta}\sum_{k=1}^{\infty}a_{k,i}\left\vert{M_i(s,r,k,x)}\right\vert^2 \, {\rm d} r\bigg]^{\frac p2} , \end{aligned}\end{equation*}

where

\begin{equation*}M_i(s,r,k)={\rm e}^{(s-r)A_i}(u_i(r)\vee 0-v_i(r)\vee 0)e_k.\end{equation*}

In the above, we used the notations

\begin{equation*}Y_{i\beta}(u,v)(s,x)\,:\!=Y_{i\beta}(u,v)(s)(x),\quad M_i(s,r,k,x)\,:\!=M_i(s,r,k)(x),\qquad i=1,2.\end{equation*}

As a consequence,

(3.4) \begin{equation}\begin{aligned}\mathbb{E} \int_0^t &\left\vert{Y_\beta(u,v)(s)}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)}^p \, {\rm d} s\\&\leq c_p(t)\mathbb{E}\int_0^t\int_\mathcal{O} \Big(\left\vert{Y_{1\beta}(u,v)(s,x)}\right\vert^p+\left\vert{Y_{2\beta}(u,v)(s,x)}\right\vert^p\Big) \, {\rm d} x \, {\rm d} s\\&\leq c_p(t) \int_0^t\mathbb{E}\bigg(\int_0^s (s-r)^{-2\beta}(a_1+a_2)\sup_{k\in\mathbb{N}}\left\vert{M(s,r,k)}\right\vert_{L^{\infty}(\mathcal{O},\mathbb{R}^2)}^2 \, {\rm d} r\bigg)^\frac p2 \, {\rm d} s,\end{aligned}\end{equation}

where $M(s,r,k)\,:\!=(M_1(s,r,k),M_2(s,r,k))$ and $a_1,a_2$ are defined in (2.1). Moreover, from the uniformly boundedness property of $\{e_k\}_{k=1}^\infty$ and (2.2), we have

(3.5) \begin{equation}\sup_{k\in\mathbb{N}}\left\vert{M(s,r,k)}\right\vert_{L^\infty(\mathcal{O},\mathbb{R}^2)}\leq c\left\vert{u(r)-v(r)}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)} \end{equation}

for some constant c independent of s, r, u, and v. Combining (3.4) and (3.5) implies that

(3.6) \begin{equation}\begin{aligned}\mathbb{E} \int_0^t &\left\vert{Y_\beta(u,v)(s)}\right\vert_{L^{\;p}(\mathcal{O},\mathbb{R}^2)}^p \, {\rm d} s\\&\leq c_p(t)\int_0^t\mathbb{E}\sup_{r\in [0,s]}\left\vert{u(r)-v(r)}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p\bigg(\int_0^s (s-r)^{-2\beta} \, {\rm d} r\bigg)^\frac p2 \, {\rm d} s\\&\leq c_{\beta,p}(t)\int_0^t\mathbb{E}\sup_{r\in [0,s]}\left\vert{u(r)-v(r)}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p \, {\rm d} s\leq c_{\beta,p}(t)\left\vert{u-v}\right\vert^p_{L_{t,p}}<\infty,\end{aligned}\end{equation}

where $c_{\beta,p}(t)$ is some positive constant satisfying $c_{\beta,p}(t)\downarrow 0$ as $t\downarrow 0$ . Therefore, the inequality (3.3) holds and, as a consequence, $\varphi(u)(t)-\varphi(v)(t)\in W^{\varepsilon,p}(\mathcal{O},\mathbb{R}^2)$ . Since $\varepsilon >l/p$ , the Sobolev embedding theorem implies that $\varphi(u)(t)-\varphi(v)(t)\in C(\mathcal{\overline{O}},\mathbb{R}^2).$ Finally, (3.3) and (3.6) imply that

\begin{equation*}\left\vert{\varphi(u)-\varphi(v)}\right\vert_{L_{t,p}}\leq c_p(t)\left\vert{u-v}\right\vert_{L_{t,p}}\end{equation*}

for some constant $c_p(t)$ satisfying $c_p(t)\downarrow 0$ as $t\downarrow 0.$ The lemma is proved.

Therefore, for $p\geq p_0$ , with sufficiently large $p_0$ , $\gamma$ maps $L^{\;p}(\Omega;\, C([0,t],C(\mathcal{\overline{O}},\mathbb{R}^2)))$ into itself. Moreover, by using (2.2) and the Lipschitz continuity of $F^*$ , we have

(3.7) \begin{equation}\begin{aligned}\int_0^t&\left\vert{{\rm e}^{(t-s)A}\big[F^*(u(s))-F^*(v(s))\big]}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p \, {\rm d} s\leq c\int_0^t \left\vert{(u(s)-v(s))}\right\vert^p_{C(\mathcal{\overline{O}},\mathbb{R}^2)} \, {\rm d} s\\&\leq c\int_0^t \sup_{r\in[0,s]}\left\vert{(u(r)-v(r))}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p \, {\rm d} s\leq ct\sup_{s\in [0,t]}\left\vert{u(s)-v(s)}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p.\end{aligned}\end{equation}

Hence, (3.2) and (3.7) imply that

\begin{equation*}\left\vert{\gamma (u)-\gamma (v)}\right\vert_{L_{t,p}}\leq c_p(t)\left\vert{u-v}\right\vert_{L_{t,p}}\!,\end{equation*}

where $c_p(t)$ is some constant depending on p and t satisfying $c_p(t)\downarrow 0$ as $t\downarrow 0$ . Therefore, for some $T_0$ sufficient small, $\gamma$ is a contraction mapping in $L^{\;p}(\Omega;\, C([0,T_0]$ , $C(\mathcal{\overline{O}},\mathbb{R}^2))).$ By a fixed point argument we can conclude that (3.1) admits a unique mild solution in $L^{\;p}(\Omega;\, C([0,T_0],C(\mathcal{\overline{O}},\mathbb{R}^2))).$ Thus, by repeating the above argument in each finite time interval $[kT_0,(k+1)T_0]$ , for any $T>0$ and $p\geq p_0$ (3.1) admits a unique mild solution $Z^*(t)=(S^*(t),I^*(t))$ in $L^{\;p}(\Omega;\, C([0,T],C(\mathcal{\overline{O}},\mathbb{R}^2))).$ We proceed to prove the positivity of $S^*(t),I^*(t)$ .

Lemma 3.2. Let $(S^*(t),I^*(t))$ be the unique mild solution of (3.1). Then, for all $t\in [0,T]$ , $S^*(t),I^*(t)\geq 0$ a.s.

Proof. Equivalently, $(S^*(t),I^*(t))$ is the mild solution of the equation

(3.8) \begin{equation}\begin{cases}{\rm d} S^*(t)=\big[A_1 S^*(t) + F_{1}(S^*(t)\vee 0,I^*(t)\vee 0)\big]{\rm d} t + (S^*(t)\vee 0){\rm d} W_1(t),\\{\rm d} I^*(t)=\big[A_2 I^*(t) + F_{2}(S^*(t)\vee 0,I^*(t)\vee 0)\big]{\rm d} t+(I^*(t)\vee 0){\rm d} W_2(t),\\S^*(0)=S_0,I^*(0)=I_0.\end{cases}\end{equation}

For $i=1,2$ , let $\lambda_i\in \rho(A_i)$ be the resolvent set of $A_i$ and $R_i(\lambda_i)\,:\!=\lambda_i R_i(\lambda_i,A_i)$ , with $R_i(\lambda_i,A_i)$ being the resolvent of $A_i$ . For each small $\varepsilon>0$ , $\lambda=(\lambda_1,\lambda_2)\in \rho(A_1)\times\rho(A_2)$ , by [Reference Liu27, Proposition 1.3.6], there exists a unique strong solution $S_{\lambda,\varepsilon}(t,x),I_{\lambda,\varepsilon}(t,x)$ of the equation

\begin{equation*} \begin{cases}\displaystyle{\rm d} S_{\lambda,\varepsilon}(t)=\big[A_1S_{\lambda,\varepsilon}(t)+R_1(\lambda_1)F_{1}(\varepsilon\Phi(\varepsilon^{-1} S_{\lambda,\varepsilon}(t)),\varepsilon\Phi(\varepsilon^{-1} I_{\lambda,\varepsilon}(t)))\big]{\rm d} t\\\quad\quad\quad\quad\quad\quad+R_1(\lambda_1)\varepsilon\Phi(\varepsilon^{-1} S_{\lambda,\varepsilon}(t)){\rm d} W_1(t),\\\displaystyle dI_{\lambda,\varepsilon}(t)=\big[A_2 I_{\lambda,\varepsilon}(t)+ R_2(\lambda_2)F_{2}(\varepsilon\Phi(\varepsilon^{-1} S_{\lambda,\varepsilon}(t)),\varepsilon\Phi(\varepsilon^{-1} I_{\lambda,\varepsilon}(t)))\big]{\rm d} t\\ \quad\quad\quad\quad\quad\quad+R_2(\lambda_2)\varepsilon\Phi(\varepsilon^{-1} I_{\lambda,\varepsilon}(t)){\rm d} W_2(t),\\S_{\lambda,\varepsilon}(0)=R_1(\lambda_1)S_0,\quad I_{\lambda,\varepsilon}(0)=R_2(\lambda_2)I_0,\end{cases}\end{equation*}

where

\begin{equation*}\Phi(\xi)=\begin{cases}0 \qquad\text{if}\;\xi\leq 0,\\3\xi^5-8\xi^4+6\xi^3 \qquad\text{if}\;0<\xi < 1,\\\xi \qquad\text{if}\;\xi\geq1,\end{cases}\end{equation*}

satisfying

\begin{equation*}\begin{cases}\Phi\in C^2(\mathbb{R}),\\\varepsilon\Phi(\varepsilon^{-1}\xi)\to \xi\vee 0\quad \text{as}\;\varepsilon\to 0.\end{cases}\end{equation*}

Combined with the convergence property in [Reference Liu27, Proposition 1.3.6], we obtain that $(S_{\lambda(k),\varepsilon}(t)$ , $I_{\lambda(k),\varepsilon}(t)) \to (S^{*}(t),I^{*}(t))$ in $L^{\;p}(\Omega;\, C([0,T],L^2(\mathcal{O},\mathbb{R}^2)))$ for some sequence $\{\lambda(k)\}_{k=1}^{\infty}\subset \rho(A_1)\times\rho(A_2)$ and $\varepsilon\to 0$ .

Now, let

\begin{equation*}g(\xi)=\begin{cases}\xi^2-\dfrac 16\quad\quad\;\;\;\ \text{if}\;\xi\leq -1,\\-\dfrac {\xi^4}2-\dfrac{4\xi^3}{3}\quad\text{if}\;-1<\xi < 0,\\0 \quad\quad\quad\quad\quad\;\ \text{if}\;\xi\geq 0.\end{cases}\end{equation*}\vskip-\lastskip\pagebreak

Then $g'(\xi)\leq 0\;\text{for all}\ \xi$ and $g''(\xi)\geq 0\;\text{for all}\ \xi$ . Hence, we are to compute

\begin{equation*}d_t\bigg(\int_\mathcal{O} g(I_{\lambda,\varepsilon}(t,x){\rm d} x)\bigg).\end{equation*}

Since $g'(\xi)\Phi(\xi)=g''(\xi)\Phi(\xi)=0\;\text{for all}\ \xi$ , by Itô’s Lemma [Reference Curtain and Falez10, Theorem 3.8], we get

\begin{equation*}\begin{aligned}\int_\mathcal{O} g(I_{\lambda,\varepsilon}(t,x))\,{\rm d} x&=k_2\int_0^t\int_\mathcal{O} g'(I_{\lambda,\varepsilon}(s,x))\Delta I_{\lambda,\varepsilon}(s,x)\,{\rm d} x\,{\rm d} s\\&=-k_2\int_0^t \int_\mathcal{O} g''(I_{\lambda,\varepsilon}(s,x))\left\vert{\nabla I_{\lambda,\varepsilon}(s,x)}\right\vert^2\,{\rm d} x\,{\rm d} s\\&\leq 0.\end{aligned}\end{equation*}

Since $g(\xi)>0$ for all $\xi<0$ , we conclude that, for all $\lambda \in \rho(A_1)\times \rho(A_2)$ , $\varepsilon> 0$ , $ I_{\lambda,\varepsilon}(t,x)\geq 0$ for all $t\in [0,T]$ , almost everywhere in $\mathcal{O}$ . Similarly, we have

\begin{equation*}\begin{aligned}\int_\mathcal{O} g(S_{\lambda,\varepsilon}(t,x))\,{\rm d} x&=\int_0^t\int_\mathcal{O} g'(S_{\lambda,\varepsilon}(s,x))(k_1\Delta S_{\lambda,\varepsilon}(s,x)+(R_1(\lambda_1)\Lambda)(x))\,{\rm d} x\,{\rm d} s\\&=-k_1\int_0^t \int_\mathcal{O} g''(S_{\lambda,\varepsilon}(s,x))\left\vert{\nabla S_{\lambda,\varepsilon}(s,x)}\right\vert^2\,{\rm d} x\,{\rm d} s\\&\quad+\int_0^t\int_\mathcal{O} g'(S_{\lambda,\varepsilon}(s,x))(R_1(\lambda_1)\Lambda)(x)\,{\rm d} x\,{\rm d} s\\&\leq 0,\end{aligned}\end{equation*}

where the last inequality above follows from the fact that

\begin{equation*}R_1(\lambda_1,A_1)=\int_0^\infty{\rm e}^{-\lambda_1 t}{\rm e}^{tA_1}\,{\rm d} t\end{equation*}

preserves positivity. Again, since $g(\xi)>0$ for all $\xi<0$ , we obtain the positivity of $S_{\lambda,\varepsilon}(t,x)$ . Hence, $S^{*}(t,x),I^{*}(t,x)\geq 0$ almost everywhere in $\mathcal{O}$ for all $t\in [0,T]$ , a.s.

Returning to the proof of the theorem, since $(S^*(t),I^*(t))$ is a unique mild solution of (3.8) and is positive, it is a mild solution of (2.4). Therefore, (2.4) admits a unique positive mild solution $(S(t),I(t))$ .

Now we prove the second part. For convenience, we use subscripts to indicate the dependence of the solution on the initial value. Let $Z_{z_0}(t),Z_{z^{\prime}_0}(t)$ be the positive mild solutions of (2.6) with the initial conditions $Z(0)=z_0$ and $Z(0)=z^{\prime}_0$ , respectively. That means,

\begin{equation*}Z_{z_0}(t)={\rm e}^{tA}z_0+\int_0^t {\rm e}^{(t-s)A}F^*(Z_{z_0}(s))\,{\rm d} s+\int_0^t {\rm e}^{(t-s)A} Z_{z_0}(s)\,{\rm d} W(s)\end{equation*}

and

\begin{equation*}Z_{z^{\prime}_0}(t)={\rm e}^{tA}z^{\prime}_0+\int_0^t {\rm e}^{(t-s)A}F^*(Z_{z^{\prime}_0}(s))\,{\rm d} s+\int_0^t {\rm e}^{(t-s)A}Z_{z^{\prime}_0}(s){\rm d} W(s).\end{equation*}

This implies that

\begin{equation*}\begin{aligned}Z_{z_0}(t)-Z_{z^{\prime}_0}(t)&={\rm e}^{tA}z_0-e^{tA}z^{\prime}_0+\int_0^t {\rm e}^{(t-s)A}(F^*(Z_{z_0}(s))-F^*(Z_{z^{\prime}_0}(s)))\,{\rm d} s\\&\quad+\int_0^t {\rm e}^{(t-s)A}(Z_{z_0}(s)-Z_{z^{\prime}_0}(s))\,{\rm d} W(s).\end{aligned}\end{equation*}

Since (3.3) and (3.6) hold, we can obtain that

(3.9) \begin{equation}\begin{aligned}\mathbb{E} \sup_{s\in[0,t]}&\left\vert{\int_0^s {\rm e}^{(s-r)A}(Z_{z_0}(r)-Z_{z^{\prime}_0}(r))\,{\rm d} W(r)}\right\vert^p_{C(\mathcal{\overline{O}},\mathbb{R}^2)}\\&\leq c_p(t)\int_0^t \mathbb{E}\sup_{r\in [0,s]}\left\vert{Z_{z_0}(r)-Z_{z^{\prime}_0}(r)}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p{\rm d} s\\&\leq c_p(t)\int_0^t \left\vert{Z_{z_0}-Z_{z^{\prime}_0}}\right\vert_{L_{s,p}}^p{\rm d} s .\end{aligned}\end{equation}

Therefore, by virtue of (3.7) and (3.9), it is possible to get

\begin{equation*}\left\vert{Z_{z_0}-Z_{z^{\prime}_0}}\right\vert_{L_{t,p}}^p\leq c_p\left\vert{z_0-z^{\prime}_0}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p+c_p(t)\int_0^t \left\vert{Z_{z_0}-Z_{z^{\prime}_0}}\right\vert_{L_{s,p}}^p{\rm d} s.\end{equation*}

Hence, it is easy to obtain, from Gronwall’s inequality, that

\begin{equation*}\left\vert{Z_{z_0}-Z_{z^{\prime}_0}}\right\vert_{L_{T,p}}^p\leq c_p(T)\left\vert{z_0-z^{\prime}_0}\right\vert_{C(\mathcal{\overline{O}},\mathbb{R}^2)}^p.\end{equation*}

Therefore, the continuous dependence of the solution on the initial values is proved.

4. Long-time behavior

This section investigates the properties of the positive mild solution $(S(t),I(t))$ of system (2.4) when $t\to\infty$ . In particular, we provide the sufficient conditions for extinction and permanence. For each function $u\in E$ , denote

\begin{equation*}u_*=\inf_{x\in\mathcal{\overline{O}}}u(x).\end{equation*}

Define the number

\begin{equation*}\widehat R=\int_\mathcal{O}\alpha(x)\,{\rm d} x-\int_\mathcal{O}\mu_2(x)\,{\rm d} x-\dfrac {a_2}2.\end{equation*}

Theorem 4.1. If $\Lambda_*>0$ and $\widehat R>0$ , then the infected class is permanent in the sense that, for any initial values $0\leq S_0,I_0\in E$ satisfying

\begin{equation*}\int_\mathcal{O} -\ln I_0(x)\,{\rm d} x<\infty,\end{equation*}

we have

\begin{equation*}\liminf_{t\to\infty} \dfrac1t\int_0^t\left(\mathbb{E}\int_\mathcal{O} (I^2(s,x)\wedge 1) \,{\rm d} x\right)^{\frac12}{\rm d} s\geq R_I \end{equation*}

for some $R_I>0$ independent of initial values.

Proof. To obtain the long-time properties of $(S(t),I(t))$ , one of tools we use is Itô’s formula. Unfortunately, in general Itô’s formula is not valid for mild solutions. Hence, our idea is to approximate the solution by a sequence of strong solutions when the noise is finite dimensional. First, we assume that $S_0,I_0\in D(A_i^E)$ , where $D(A_i^E)$ is the domain of $A_i^E$ , the part of $A_i$ in E. For each fixed $n\in \mathbb{N}$ , let $\overline S_n(t,x),\overline I_{n}(t,x)$ be the strong solution (see [Reference Da Prato and Zabczyk12] for more details about strong, weak, and mild solutions) of the following equations:

(4.1) \begin{equation}\begin{cases}{\rm d} \overline S_n(t,x)=\bigg[A_1\overline S_n(t,x)+\Lambda(x)-\mu_1(x)\overline S_n(t,x)-\dfrac{\alpha(x) \overline S_n(t,x)\overline I_n(t,x)}{ \overline S_n(t,x)+\overline I_n(t,x)}\bigg]{\rm d} t \\ \qquad\qquad\quad + \displaystyle \sum_{k=1}^n \sqrt{a_{k,1}}e_k(x)\overline S_n(t,x){\rm d} B_{k,1}(t),\\{\rm d} \overline I_n(t,x)=\bigg[A_2 \overline I_n(t,x)-\mu_2(x) \overline I_n(t,x) + \dfrac{\alpha(x) \overline S_n(t,x)\overline I_n(t,x)}{ \overline S_n(t,x)+\overline I_n(t,x)}\bigg]{\rm d} t\\\qquad\qquad\quad +\displaystyle \sum_{k=1}^n \sqrt{a_{k,2}}e_k(x)\overline I_n(t,x){\rm d} B_{k,2}(t),\\\overline S_n(x,0)=S_0(x),\quad \overline I_n(x,0)=I_0(x).\end{cases}\end{equation}

The existence and uniqueness of a strong solution of (4.1) follow the results in [Reference Da Prato and Tubaro11] or [12, Section 7.4]. To see that the conditions in these references are satisfied, we note that the semigroups ${\rm e}^{tA_1}$ and ${\rm e}^{tA_2}$ (as well as their restrictions to E) are analytic (see [Reference Arendt3, Chapter 2]) and strongly continuous (see [Reference Da Prato and Zabczyk12, Appendix A.5.2]). Moreover, from the characterizations of fractional power of elliptic operators in [Reference Yagi38, Chapter 16] or [Reference Da Prato and Zabczyk12, Appendix A], it is easy to confirm that the coefficients in (4.1) satisfy condition (e) in Hypothesis 2 of [Reference Da Prato and Tubaro11]. Moreover, a detailed argument can be also found in [Reference Nguyen and Yin29, Reference Nguyen and Yin30].

In addition, from the continuous dependence on parameter $\xi$ of the fixed points of the family of uniform contraction mappings $T(\xi)$ , by a similar ‘parameter-dependent contraction mapping’ argument, it is easy to obtain (see [Reference Da Prato and Zabczyk12] or [Reference Nguyen and Yin30, Proposition 4.2]) that, for any fixed t,

\begin{equation*}\lim_{n\to\infty}\mathbb{E}\left\vert{S(t)-\overline S_n(t)}\right\vert_H^2\to 0\end{equation*}

and

\begin{equation*}\lim_{n\to\infty}\mathbb{E}\left\vert{I(t)-\overline I_n(t)}\right\vert_H^2\to 0.\end{equation*}

To proceed, we state and prove the following auxiliary lemmas.

Lemma 4.1. Let $\mu_*\,:\!=\inf_{x\in\mathcal{\overline{O}}}\min\{\mu_1(x),\mu_2(x)\}$ . If $\mu_*>0$ then

\begin{equation*}\mathbb{E}\int_\mathcal{O}(S(t,x)+ I(t,x))\,{\rm d} x\leq {\rm e}^{-\mu_* t} \int_\mathcal{O}(S_0(x)+I_0(x))\,{\rm d} x+ \dfrac{ |\Lambda|_E}{\mu_*}.\end{equation*}

Proof. In view of Itô’s formula ([Reference Curtain and Falez10, Theorem 3.8]), we can obtain

\begin{equation*}\begin{aligned}\mathbb{E}\, {\rm e}^{\mu_*t}\int_\mathcal{O}(\overline S_n(t,x)+ \overline I_n(t,x))\,{\rm d} x&\leq \int_\mathcal{O}(S_0(x)+I_0(x))\,{\rm d} x+\mathbb{E}\int_0^t {\rm e}^{\mu_*s}\int_\mathcal{O}\Lambda(x)\,{\rm d} x \,{\rm d} s\\&\leq \int_\mathcal{O}(S_0(x)+I_0(x))\,{\rm d} x+\dfrac{ |\Lambda|_E}{\mu_*}{\rm e}^{\mu_* t}.\end{aligned}\end{equation*}

Letting $n\to\infty$ , we obtain the desired result.

Now we are in a position to estimate $\mathbb{E}( \int_\mathcal{O} \frac 1{\overline S_n^p(t ,x)}\,{\rm d} x)$ by the following lemma.

Lemma 4.2. For any $p>0$ , if $\int_\mathcal{O}\frac 1{S_0^p(x)}\,{\rm d} x<\infty$ , there exists $\widetilde K_p>0$ , which is independent of n and the initial conditions, such that

\begin{equation*}\begin{aligned}\mathbb{E}&\int_\mathcal{O} \dfrac 1{\overline S_n^p(t ,x)}\,{\rm d} x\leq {\rm e}^{-t}\int_\mathcal{O}\frac 1{S_0^p(x)}\,{\rm d} x+\widetilde K_p.\end{aligned}\end{equation*}

Proof. For any $0<\varepsilon<\frac{p\Lambda_*}{2}$ , using Itô’s Lemma ([Reference Curtain and Falez10, Theorem 3.8]) and by direct calculations, we have

(4.2) \begin{equation}\begin{split}&\hspace*{-6pt} {\rm e}^{t }\!\!\int_\mathcal{O} \dfrac 1{(\overline S_n(t ,x)+\varepsilon)^p}\,{\rm d} x \\[6pt] &\hspace*{-6pt} \ =\int_\mathcal{O}\dfrac 1{(S_0(x)+\varepsilon)^p}\,{\rm d} x+\int_0^{t } {\rm e}^s\int_\mathcal{O} \dfrac 1{(\overline S_n(s,x)+\varepsilon)^p}\,{\rm d} x\,{\rm d} s +\int_0^{t } {\rm e}^s\int_\mathcal{O}\dfrac {-p}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\\[6pt]&\hspace*{-6pt}\qquad \times \bigg(k_1\Delta \overline S_n(s,x)+\Lambda(x)-\mu_1(x)\overline S_n(s,x)-\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}\bigg)\,{\rm d} x\,{\rm d} s\\[6pt]&\hspace*{-6pt}\qquad+\dfrac 12 \int_0^{t } {\rm e}^s\sum_{k=1}^n\int_\mathcal{O} \dfrac{p(p+1)a_{k,1}e^2_k(x)\overline S_n^2(s,x)}{(\overline S_n(s,x)+\varepsilon)^{p+2}}\,{\rm d} x\,{\rm d} s\\[6pt]&\hspace*{-6pt}\qquad +\sum_{k=1}^n\int_0^{t } {\rm e}^s\bigg[\sqrt{a_{k,1}}\int_\mathcal{O} \dfrac{-pe_k(x)\overline S_n(s,x)}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\,{\rm d} x\bigg]\,{\rm d} B_{k,1}(s)\\[6pt]&\hspace*{-6pt}\leq \int_\mathcal{O}\dfrac 1{(S_0(x)+\varepsilon)^p}\,{\rm d} x+\int_0^{t } {\rm e}^s\int_\mathcal{O}\dfrac{-pk_1\Delta\overline S_n(s,x)}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\,{\rm d} x\,{\rm d} s\\[6pt]&\hspace*{-6pt}\qquad+\int_0^{t } {\rm e}^s\int_\mathcal{O} \dfrac{p}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\bigg(-\Lambda (x)+\frac{\varepsilon}p+\bigg(\left\vert{\mu_1}\right\vert_E+\left\vert{\alpha}\right\vert_E+\dfrac 1p+\dfrac{p+1}{2}a_1C_0^2\bigg)\\[6pt]&\hspace*{-6pt}\qquad \times \overline S_n(s,x)\bigg)\,{\rm d} x\,{\rm d} s+\sum_{k=1}^n\int_0^{t } {\rm e}^s\bigg[\sqrt{a_{k,1}}\int_\mathcal{O} \dfrac{-pe_k(x)\overline S_n(s,x)}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\,{\rm d} x\bigg]\,{\rm d} B_{k,1}(s)\\[6pt]&\hspace*{-6pt}\leq \int_\mathcal{O}\dfrac 1{(S_0(x)+\varepsilon)^p}\,{\rm d} x+\int_0^{t } \dfrac{pK_p^{p+1}2^p}{\Lambda_*^p}{\rm e}^s\,{\rm d} s\\[6pt]&\hspace*{-6pt}\qquad +\sum_{k=1}^n\int_0^{t } {\rm e}^s\bigg[\sqrt{a_{k,1}}\int_\mathcal{O} \dfrac{-pe_k(x)\overline S_n(s,x)}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\,{\rm d} x\bigg]\,{\rm d} B_{k,1}(s),\end{split}\end{equation}

where $K_p=\left\vert{\mu_1}\right\vert_E+\left\vert{\alpha}\right\vert_E+\frac 1p+\frac{p+1}{2}a_1C_0^2$ . In the above, we used the facts that

\begin{equation*}\int_\mathcal{O}\dfrac{-pk_1\Delta\overline S_n(s,x)}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\,{\rm d} x=-p(p+1)k_1\int_\mathcal{O}\dfrac{\left\vert{\nabla \overline S_n(s,x)}\right\vert^2}{(\overline S_n(s,x)+\varepsilon)^{p+2}}\,{\rm d} x\leq 0 \ \hbox{ a.s.}\end{equation*}

and

\begin{equation*}\begin{aligned}\int_\mathcal{O} &\dfrac{p}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\bigg(-\Lambda (x)+\dfrac{\varepsilon}p+\bigg(\left\vert{\mu_1}\right\vert_E+\left\vert{\alpha}\right\vert_E+\dfrac 1p+\dfrac{p+1}{2}a_1C_0^2\bigg)\overline S_n(s,x)\bigg)\,{\rm d} x\\&\leq\int_\mathcal{O}\dfrac{p}{(\overline S_n(s,x)+\varepsilon)^{p+1}}\bigg(-\dfrac{\Lambda_*}2+K_p\overline S_n(s,x)\bigg)\boldsymbol{1}_{\big\{\overline S_n(s,x)\geq \frac{\Lambda_*}{2K_p}\big\}}\,{\rm d} x\\&\leq \dfrac{pK_p^{p+1}2^p}{\Lambda_*^p}\ \text{a.s.\;}\end{aligned}\end{equation*}\vskip-\lastskip\pagebreak

Hence, (4.2) implies that, for all $t\geq 0$ and $n\in \mathbb{N}$ ,

(4.3) \begin{equation}\begin{aligned}\mathbb{E} \int_\mathcal{O} \dfrac 1{(\overline S_n(t ,x)+\varepsilon)^p}\,{\rm d} x\leq {\rm e}^{-t}\int_\mathcal{O}\dfrac 1{(S_0(x)+\varepsilon)^p}\,{\rm d} x+ {\rm e}^{-t }\int_0^{t }\dfrac{pK_p^{p+1}2^p}{\Lambda_*^p}{\rm e}^s\,{\rm d} s.\end{aligned}\end{equation}

Letting $\varepsilon\to 0$ , we have, from the monotone convergence theorem,

\begin{equation*} \begin{aligned}\mathbb{E} \int_\mathcal{O} \dfrac 1{\overline S_n^p(t ,x)}\,{\rm d} x\leq {\rm e}^{-t}\int_\mathcal{O}\frac 1{S_0^p(x)}\,{\rm d} x+ {\rm e}^{-t }\int_0^{t }\dfrac{pK_p^{p+1}2^p}{\Lambda_*^p}{\rm e}^s\,{\rm d} s.\end{aligned}\end{equation*}

The proof of the lemma is completed.

Noting that our initial conditions are not assumed to satisfy $\int_\mathcal{O} \frac 1{S^2_0(x)}\,{\rm d} x<\infty$ , we will prove in the following lemma that, after some finite time, the solutions have inverse functions that belong to $L^2(\mathcal{O},\mathbb{R})$ .

Lemma 4.3. For any $n\in\mathbb{N}$ ,

\begin{equation*}\begin{aligned}\mathbb{E} &\int_\mathcal{O} \dfrac 1{\overline S_n^2(4 ,x)}\,{\rm d} x\leq \ell_0,\end{aligned}\end{equation*}

where $\ell_0$ depends only on the initial conditions (independent of n).

Proof. Using

\begin{equation*}\begin{aligned}\mathbb{E}\int_\mathcal{O} \overline S_n(t,x)\,{\rm d} x&=\int_\mathcal{O} S_0(x)\,{\rm d} x +\int_0^t \mathbb{E}\int_\mathcal{O} \bigg(k_1\Delta \overline S_n(s,x)+\Lambda(x)\\&\quad\quad\quad-\mu_1(x)\overline S_n(s,x)-\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{\overline S_n(s,x)\overline I_n(s,x)}\bigg)\,{\rm d} x\,{\rm d} s\\&\leq \int_\mathcal{O} S_0(x)\,{\rm d} x +t\left\vert{\Lambda}\right\vert_E \end{aligned}\end{equation*}

and $s^q\leq s+1$ for all $s\in\mathbb{R}>0, q \in [0,1]$ , it is easy to show that there exists $\ell_{1}>0$ such that

(4.4) \begin{equation}\mathbb{E} \int_\mathcal{O} \overline S_n^q(t ,x)\,{\rm d} x\leq \ell_{1} \qquad \text{for any } t\in[0,1], q\in[0,1],\end{equation}

where $\ell_1$ is independent of n. For any $\varepsilon>0$ , using Itô’s Lemma ([Reference Curtain and Falez10, Theorem 3.8]) again, we have

(4.5) \begin{equation}\begin{aligned}&\!\!\!\mathbb{E} \int_\mathcal{O} (\overline S_n(1 ,x)+\varepsilon)^{\frac12}\,{\rm d} x \\ & \ =\int_\mathcal{O}(S_0(x)+\varepsilon)^{\frac12}{\rm d} x+\int_0^{1}\mathbb{E} \int_\mathcal{O}\dfrac {1}{2(\overline S_n(s,x)+\varepsilon)^{\frac12}}\bigg(k_1\Delta \overline S_n(s,x)\\\\&\qquad+\Lambda(x)-\mu_1(x)\overline S_n(s,x)-\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}\bigg)\,{\rm d} x\,{\rm d} s\\&\qquad-\dfrac 18 \int_0^{1} \mathbb{E}\sum_{k=1}^n\int_\mathcal{O} \dfrac{a_{k,1}e^2_k(x)\overline S_n^2(s,x)}{(\overline S_n(s,x)+\varepsilon)^{\frac32}}\,{\rm d} x\,{\rm d} s \\&\geq \frac12\int_0^1\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(s,x)+\varepsilon)^{\frac12}}\,{\rm d} x\,{\rm d} s-N_1 \int_0^1\left(\mathbb{E}\int_\mathcal{O} \overline S_n^{\frac12}(s ,x)\,{\rm d} x\right){\rm d} s,\end{aligned}\end{equation}

where

\begin{equation*}N_1=\dfrac{\left\vert{\mu_1}\right\vert_E+\left\vert{\alpha}\right\vert_E+\frac {a_1C_0^2}4}{2}.\end{equation*}

In view of (4.4) and (4.5), we have

\begin{equation*}\frac12\int_0^1\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(s,x)+\varepsilon)^{\frac12}}\,{\rm d} x\,{\rm d} s\leq (1+N_1)\ell_{1}+\sqrt{\varepsilon} \qquad\text{for all}\ \varepsilon>0,\end{equation*}

which implies that

\begin{equation*}\int_0^1\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(s ,x))^{\frac12}}\,{\rm d} x\,{\rm d} s\leq 2(1+N_1)\ell_{1} ,\end{equation*}

or there exists $t_1=t_1(n)\in[0,1]$ such that

\begin{equation*}\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(t_1,x))^{\frac12}}\,{\rm d} x\leq \dfrac{2(1+N_1)\ell_{1}}{\Lambda_*}.\end{equation*}

Applying Lemma 4.2 and the Markov property of $(S_n, I_n)$ , we have

\begin{equation*}\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(t ,x)+\varepsilon)^{\frac12}}\,{\rm d} x\leq\ell_{2} \qquad \text{for all}\ t\in[1,2],\end{equation*}

for some $\ell_{2}$ independent of n. We again have

(4.6) \begin{equation}\begin{aligned}&\!\!\!\mathbb{E} \int_\mathcal{O} (\overline S_n(2 ,x)+\varepsilon)^{-\frac12}{\rm d} x \\[6pt] & \ =\mathbb{E}\int_\mathcal{O}(\overline S_n(1,x)+\varepsilon)^{-\frac12}{\rm d} x -\int_1^{2}\mathbb{E} \int_\mathcal{O}\dfrac {1}{2(\overline S_n(s,x)+\varepsilon)^{\frac32}}\bigg(k_1\Delta \overline S_n(s,x)\\[6pt]&\qquad+\Lambda(x)-\mu_1(x)\overline S_n(s,x)-\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}\bigg)\,{\rm d} x\,{\rm d} s\\[6pt]&\qquad+\dfrac 38 \int_1^{2 } \mathbb{E}\sum_{k=1}^n\int_\mathcal{O} \dfrac{a_{k,1}e^2_k(x)\overline S_n^2(s,x)}{(\overline S_n(s,x)+\varepsilon)^{\frac32}}\,{\rm d} x\,{\rm d} s\\[6pt]&\leq -\frac12\int_1^2\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(s ,x)+\varepsilon)^{\frac32}}\,{\rm d} x\,{\rm d} s+\mathbb{E}\int_\mathcal{O}(\overline S_n(1,x)+\varepsilon)^{-\frac12}{\rm d} x\\[6pt]&\qquad +N_2 \int_1^2\left(\mathbb{E}\int_\mathcal{O} \overline S_n^{-\frac12}(s ,x)\,{\rm d} x\right){\rm d} s,\end{aligned}\end{equation}

where

\begin{equation*}N_2=\dfrac{\left\vert{\mu_1}\right\vert_E+\left\vert{\alpha}\right\vert_E+\frac {3a_1C_0^2}4}{2}.\end{equation*}\vskip-\lastskip\pagebreak

Thus,

\begin{equation*}\int_1^2\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(s ,x)+\varepsilon)^{\frac32}}\,{\rm d} x\,{\rm d} s\leq \ell_{3} \end{equation*}

for some $\ell_{3}$ depending only on the initial conditions. Letting $\varepsilon\to0$ , we obtain that, for some $t_2=t_2(n)\in[1,2]$ ,

\begin{equation*}\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{\overline S_n^{\frac32}(t_2,x)}\,{\rm d} x\leq \ell_{3},\end{equation*}

which together with Lemma 4.2 implies that

\begin{equation*}\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(t ,x)+\varepsilon)^{\frac32}}\,{\rm d} x\leq \ell_{4},\qquad \text{for all}\ t\in[2,3],\end{equation*}

where $\ell_4$ is some constant independent of n. Using the same process we can obtain that there exist $t_3=t_3(n)\in [0,4]$ and $\ell_5$ such that

\begin{equation*}\mathbb{E} \int_\mathcal{O} \dfrac{\Lambda(x)}{(\overline S_n(t_3 ,x))^{\frac52}}\,{\rm d} x\leq \ell_{5}.\end{equation*}

Therefore, it is possible to obtain the existence of two constants, $t_4=t_4(n)\in [0,4]$ and $\ell_6$ , satisfying

\begin{equation*}\mathbb{E} \int_\mathcal{O} \dfrac 1{(\overline S_n(t_4,x))^2}\,{\rm d} x<\ell_6.\end{equation*}

The lemma is proved by applying Lemma 4.2.

In view of Lemmas 4.2 and 4.3, we have

(4.7) \begin{equation}\begin{aligned}\mathbb{E} \int_\mathcal{O} \dfrac 1{\overline S_n^2(t ,x)}\,{\rm d} x\leq {\rm e}^{-t}\ell_0+ \widetilde K_2\qquad \text{for all}\ n\in\mathbb{N}, t\geq 4.\end{aligned}\end{equation}

Note that both $\ell_0$ and $\widetilde K_2$ are independent of n, and $\ell_0$ may depend on the initial point but $\widetilde K_2$ is independent. By Itô’s Lemma ([Reference Curtain and Falez10, Theorem 3.8]) again, and with calculations similar to obtaining (4.3), we have

\begin{equation*}\begin{aligned}\mathbb{E}\! \int_\mathcal{O}\overline I_n(t,x)\,{\rm d} x& \geq\mathbb{E} \int_\mathcal{O}\ln (\overline I_n(t,x)+\varepsilon)\,{\rm d} x \\&=\int_\mathcal{O}\ln (I_0(x)+\varepsilon)\,{\rm d} x+\int_0^t \mathbb{E}\int_\mathcal{O}\dfrac 1{\overline I_n(s,x)+\varepsilon}\bigg(k_2\Delta \overline I_n(s,x)\\&\quad\quad\quad-\mu_2(x)\overline I_n(s,x)+\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}\,{\rm d} x\,{\rm d} s\bigg)\\&\quad\quad\quad-\dfrac 12\int_0^t\mathbb{E}\sum_{k=1}^n \int_\mathcal{O}\dfrac{a_{k,2}\overline I_n^2(s,x)e_k^2(x)}{(\overline I_n(s,x)+\varepsilon)^2}\,{\rm d} x\,{\rm d} s\\&\geq \int_\mathcal{O}\!\!\ln (I_0(x)+\varepsilon){\rm d} x-\bigg(\dfrac {a_2}2+\left\vert{\mu_2}\right\vert_E\bigg)t \ \text{for all}\ n\in\mathbb{N}, t>0, 0<\varepsilon<1.\end{aligned}\end{equation*}

As a consequence,

\begin{equation*} \mathbb{E}\! \int_\mathcal{O}\!\overline I_n(t,x)\,{\rm d} x\geq\mathbb{E}\!\int_\mathcal{O}\! \ln \overline I_n(t,x)\,{\rm d} x\geq\!\int_\mathcal{O}\!\ln I_0(x)\,{\rm d} x-\bigg(\dfrac{a_2}2+\left\vert{\mu_2}\right\vert_E\bigg)t>-\infty\ \text{for all}\ t>0.\end{equation*}

That means,

(4.8) \begin{equation}\mathbb{P}\big\{\overline I_n(t,x)>0\;\text{almost everywhere in}\;\mathcal{O}\big\}=1 \qquad\text{for all}\ n\in\mathbb{N}, t>0.\end{equation}

On the other hand, combining Itô’s Lemma and basic calculations implies that

\begin{equation*}\begin{aligned}0&\geq \mathbb{E}\int_\mathcal{O}\ln \dfrac{\overline I_n(t,x)+\varepsilon}{1+\overline I_n(t,x)}\,{\rm d} x\geq \int_\mathcal{O} \ln\dfrac{I_0(x)+\varepsilon}{1+I_0(x)}\,{\rm d} x+\widehat Rt\\[4pt]&\quad\quad-\int_0^{t}\mathbb{E}\int_\mathcal{O} \bigg(\dfrac{\alpha(x) \overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}+\dfrac{\alpha(x) \overline S_n(s,x)\overline I_n(s,x)}{(\overline S_n(s,x)+\overline I_n(s,x))(\overline I_n(s,x)+1)}\bigg)\,{\rm d} x\,{\rm d} s\\[4pt]&\quad\quad-\int_0^t\mathbb{E}\int_\mathcal{O} \dfrac{\alpha(x)\varepsilon}{\overline I_n(s,x)+\varepsilon}\,{\rm d} x\,{\rm d} s \qquad \text{for all}\ t>0,n\in \mathbb{N}, 0<\varepsilon<1.\end{aligned}\end{equation*}

Thus, for all $t> 0$ , $n\in\mathbb{N}$ , $0<\varepsilon<1$ ,

(4.9) \begin{equation}\begin{aligned}\int_0^t \mathbb{E}\int_\mathcal{O} &\bigg(\dfrac{\alpha(x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}+\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{(\overline S_n(s,x)+\overline I_n(s,x))(\overline I_n(s,x)+1)}\bigg)\,{\rm d} x\,{\rm d} s\\[4pt]& \geq \mathbb{E}\int_\mathcal{O} \ln\dfrac{I_0(x)+\varepsilon}{1+I_0(x)}\,{\rm d} x+\widehat Rt-\left\vert{\alpha}\right\vert_E\int_0^t \mathbb{E}\int_\mathcal{O} \dfrac{\varepsilon}{\overline I_n(s,x)+\varepsilon}\,{\rm d} x\,{\rm d} s.\end{aligned}\end{equation}

Let $\varepsilon\to 0$ ; using (4.8) and (4.9), we have

(4.10) \begin{equation}\begin{aligned}\int_0^t \mathbb{E}\int_\mathcal{O} &\bigg(\dfrac{\alpha(x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}+\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{(\overline S_n(s,x)+\overline I_n(s,x))(\overline I_n(s,x)+1)}\bigg)\,{\rm d} x\,{\rm d} s\\[4pt]\geq& \int_\mathcal{O}\ln \dfrac{I_0(x)}{1+I_0(x)}\,{\rm d} x+\widehat Rt\qquad\text{for all}\ t>0,n\in\mathbb{N}.\end{aligned}\end{equation}

We have the following estimates:

\begin{equation*}\begin{aligned}\left\vert{\alpha}\right\vert_E\left(\mathbb{E}\int_\mathcal{O} \dfrac{\overline I_n^2(s,x)}{(1+\overline I_n(s,x))^2}\,{\rm d} x\right)^{\frac12}&\geq \mathbb{E}\int_\mathcal{O}\dfrac{\alpha(x)\overline I_n(s,x)}{1+\overline I_n(s,x)}\,{\rm d} x \\[4pt]&\geq\mathbb{E} \int_\mathcal{O} \dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{(\overline S_n(s,x)+\overline I_n(s,x))(\overline I_n(s,x)+1)}\,{\rm d} x \end{aligned}\end{equation*}

and

\begin{equation*}\begin{aligned}\left\vert{\alpha}\right\vert_E&\left(\mathbb{E}\int_\mathcal{O} \dfrac{\overline I_n^2(s,x)}{(1+\overline I_n(s,x))^2}\,{\rm d} x\right)^{\frac12}\left(\mathbb{E} \int_\mathcal{O}\left(\dfrac 1{\overline S_n(s,x)}+1\right)^2{\rm d} x\right)^{\frac12}\\[4pt]&\geq \mathbb{E}\int_\mathcal{O}\dfrac{\alpha(x)\overline I_n(s,x)}{1+\overline I_n(s,x)}\left(\dfrac1{\overline S_n(s,x)}+1\right){\rm d} x \geq \mathbb{E}\int_\mathcal{O}\dfrac{\alpha(x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}\,{\rm d} x,\end{aligned}\end{equation*}

since

\begin{equation*}\frac{1+I}{S+I}=\frac{1}{S+I}+\frac{I}{S+I}\leq \frac1S+1.\end{equation*}

Therefore, after some basic estimates, we can get from (4.10) that

\begin{equation*}\begin{aligned}\int_4^t \left\vert{\alpha}\right\vert_E&\left(\mathbb{E}\int_0 \dfrac{\overline I_n^2(s,x)}{(1+\overline I_n(s,x))^2}\,{\rm d} x\right)^{\frac12}\left(1+\left(\mathbb{E} \int_\mathcal{O}\left(\dfrac 1{\overline S_n(s,x)}+1\right)^2{\rm d} x\right)^{\frac12}\right){\rm d} s\\[3pt]\geq&\int_4^t \mathbb{E}\int_\mathcal{O} \bigg(\dfrac{\alpha(x)\overline I_n(s,x)}{\overline S_n(s,x)+\overline I_n(s,x)}+\dfrac{\alpha(x)\overline S_n(s,x)\overline I_n(s,x)}{(\overline S_n(s,x)+\overline I_n(s,x))(\overline I_n(s,x)+1)}\bigg)\,{\rm d} x\,{\rm d} s \\[3pt]\geq&\int_\mathcal{O}\ln \dfrac{I_0(x)}{1+I_0(x)}\,{\rm d} x+\widehat R t- 8\left\vert{\alpha}\right\vert_E,\end{aligned}\end{equation*}

which together with (4.7) leads to

\begin{equation*}\begin{aligned}\int_4^t \left\vert{\alpha}\right\vert_E&\left(\mathbb{E}\int_0 \dfrac{\overline I_n^2(s,x)\,{\rm d} x}{(1+\overline I_n(s,x))^2}\right)^{\frac12}\left(2\sqrt{{\rm e}^{-s}\ell_0}+2\widetilde K_2^{\frac 12}+3\right) {\rm d} s \\\geq& \int_\mathcal{O}\ln \dfrac{I_0(x)}{1+I_0(x)}\,{\rm d} x-8\left\vert{\alpha}\right\vert_E+\widehat R t.\end{aligned}\end{equation*}

Letting $n\to\infty$ yields

(4.11) \begin{equation}\begin{aligned}\int_4^t \left\vert{\alpha}\right\vert_E&\left(\mathbb{E}\int_\mathcal{O} \dfrac{I^2(s,x)}{(1+I(s,x))^2}\,{\rm d} x\right)^{\frac12}\left(2\sqrt{{\rm e}^{-s}\ell_0}+2\widetilde K_2^{\frac 12}+3\right) {\rm d} s \\\geq& \int_\mathcal{O}\ln \dfrac{I_0(x)}{1+I_0(x)}\,{\rm d} x-8\left\vert{\alpha}\right\vert_E+\widehat R t,\end{aligned}\end{equation}

which is easily followed by

\begin{equation*}\begin{aligned}\liminf_{t\to\infty}\dfrac1t\int_0^t \left(\mathbb{E}\int_\mathcal{O} \dfrac{I^2(s,x)}{(1+I(s,x))^2}\,{\rm d} x\right)^{\frac12}{\rm d} s\geq \dfrac{\widehat R}{\left\vert{\alpha}\right\vert_E(2\widetilde K_2^{\frac 12}+3)}.\end{aligned}\end{equation*}

As a consequence,

\begin{equation*}\liminf_{t\to\infty}\dfrac1t\int_0^t\left(\mathbb{E}\int_\mathcal{O} (I^2(s,x)\wedge 1) \,{\rm d} x\right)^{\frac12}{\rm d} s\geq R_I>0,\end{equation*}

where $R_I$ is independent of the initial points. The proof of the theorem is completed by using the dense property of $D(A_i^E)$ in E and the continuous dependence on initial data of the solution. In more detail, since the constants $\widetilde K_2$ and $\widehat R$ are independent of the initial points, the estimates (4.7) and (4.11) still hold for the solution starting from arbitrary initial points $S_0,I_0\in E$ with $\int_\mathcal{O} -\ln I_0(x)\,{\rm d} x<\infty$ .

Theorem 4.2. For any non-negative initial data $S_0,I_0\in E$ , if

\begin{equation*} (\mu_2-\alpha)_*= \inf_{x\in\mathcal{\overline{O}}}(\mu_2(x)-\alpha(x))>0,\end{equation*}

then the infected class will be extinct with exponential rate.

Proof. First, we define the linear operator $J\,:\,H \mapsto\mathbb{R}$ as, for all $u\in H$ ,

\begin{equation*}Ju\,:\!=\int_\mathcal{O} u(x)\,{\rm d} x.\end{equation*}

By the properties of ${\rm e}^{tA_i}$ , for all $u\in H$ , $J({\rm e}^{tA_i}u-u)=0$ or $Ju=J{\rm e}^{tA_i}u$ , for all $i=1,2$ .

Now, as in the definition of the mild solution, we have

\begin{equation*}I(t)={\rm e}^{tA_2}I_0+\int_0^t{\rm e}^{(t-s)A_2}\bigg(-\mu_2I(s)+\dfrac{\alpha S(s)I(s)}{S(s)+I(s)}\bigg)\,{\rm d} s+\int_0^t {\rm e}^{(t-s)A_2}I(s)\,{\rm d} W_2(s).\end{equation*}

Hence, applying the operator J to both sides, using the properties of operator J and stochastic convolution (see [12, Proposition 4.15]), we obtain

\begin{equation*}\begin{aligned}\int_\mathcal{O}I(t,x)\,{\rm d} x=&\int_\mathcal{O}I_0(x)\,{\rm d} x+\int_0^t \int_\mathcal{O}\bigg(-\mu_2(x) I(s,x)+\dfrac{\alpha(x) S(s,x)I(s,x)}{S(s,x)+I(s,x)}\bigg)\,{\rm d} x\,{\rm d} s\\&\quad+\int_0^t J({\rm e}^{(t-s)A_2}I(s))\,{\rm d} W_2(s),\end{aligned}\end{equation*}

where $J({\rm e}^{(t-s)A_2}I(s))$ in the stochastic integral is understood as the process taking values in spaces of the linear operator from H to $\mathbb{R}$ that is defined by

\begin{equation*}J({\rm e}^{(t-s)A_2}I(s))u\,:\!=\int_\mathcal{O} ({\rm e}^{(t-s)A_2}I(s)u)(x)\,{\rm d} x\qquad\text{for all}\ u\in H.\end{equation*}

From (2.1), it is easy to see that these integrals are well defined. By taking the expectation on both sides and using the properties of stochastic integral [10, Proposition 2.9],

\begin{equation*}\begin{aligned}\mathbb{E} \int_\mathcal{O}I(t,x)\,{\rm d} x&=\int_\mathcal{O}I_0(x)\,{\rm d} x+\mathbb{E}\int_0^t\int_\mathcal{O}\bigg(-\mu_2(x) I(s,x)+\dfrac{\alpha(x) S(s,x)I(s,x)}{S(s,x)+I(s,x)}\bigg)\,{\rm d} x\,{\rm d} s\end{aligned}\end{equation*}

As a consequence,

\begin{equation*}\begin{aligned}\mathbb{E}\! \int_\mathcal{O}I(t,x)\,{\rm d} x-\mathbb{E}\! \int_\mathcal{O} I(s,x)\,{\rm d} x&=\int_s^t\!\mathbb{E}\!\int_\mathcal{O}\bigg(-\mu_2(x) I(r,x)+\dfrac{\alpha(x) S(r,x)I(s,x)}{S(r,x)+I(r,x)}\bigg)\,{\rm d} x\,{\rm d} r\\&\leq -(\mu_2-\alpha)_*\int_s^t\mathbb{E}\int_\mathcal{O} I(r,x)\,{\rm d} x\,{\rm d} r .\end{aligned}\end{equation*}

Hence, we can obtain the following estimate for the upper Dini derivative:

\begin{equation*}\dfrac{{\rm d}}{{\rm d} t^+}\mathbb{E}\int_\mathcal{O}I(t,x)\,{\rm d} x\leq -(\mu_2-\alpha)_* \mathbb{E}\int_\mathcal{O}I(t,x)\,{\rm d} x\qquad\text{for all}\ t\geq 0.\end{equation*}

Since $(\mu_2-\alpha)_*>0$ , we can get that $\mathbb{E}\int_\mathcal{O}I(t,x)\,{\rm d} x$ converges to 0 with exponential rate as $t\to \infty$ . Hence, it easy to claim that the infected class goes extinct.

Theorem 4.3. Suppose that $W_2(t)$ is a space-independent Brownian motion with covariance $a_2 t$ . For any non-negative initial data $S_0,I_0\in E$ , if

\begin{equation*} (\mu_2-\alpha)_*+\frac{a_2}2\,:\!=\inf_{x\in\mathcal{\overline{O}}}(\mu_2(x)-\alpha(x))+\frac{a_2}2>0,\end{equation*}

then when $p>0$ is sufficiently small that

\begin{equation*}R_p\,:\!=(\mu_2-\alpha)_*+\frac{(1-p)a_2}2>0,\end{equation*}

we have

\begin{equation*}\limsup_{t\to\infty}\dfrac{\ln \mathbb{E}\left(\int_\mathcal{O} I(t,x)\,{\rm d} x\right)^p}t\leq -pR_p<0.\end{equation*}

Proof. Since $W_2(t)$ is a space-independent Brownian motion, as in the arguments in the proof of Theorem 4.1, the mild solution I(t) is also the solution in the strong sense if $I_0\in D(A_i^E)$ . Hence, with an initial value in $D(A_i^E)$ , we have

\begin{equation*}\int_\mathcal{O}\! I(t,x)\,{\rm d} x = \int_0^t\!\int_\mathcal{O}\bigg(-\mu_2(x) I(s,x)+\dfrac{\alpha(x) S(s,x)I(s,x)}{S(s,x)+I(s,x)}\bigg)\,{\rm d} x\,{\rm d} s + \int_0^t\!\int_\mathcal{O}\! I(s,x)\,{\rm d} W_2(s) .\end{equation*}

By Itô’s formula, we obtain that

\begin{equation*}\begin{aligned}\bigg(&\int_\mathcal{O} I(t,x)\,{\rm d} x\bigg)^p\\&= \int_s^t\bigg[ p\bigg(\int_\mathcal{O} I(r,x)\,{\rm d} x\bigg)^{p-1}\int_\mathcal{O}\bigg(-\mu_2(x) I(r,x)+\dfrac{\alpha(x) S(r,x)I(r,x)}{S(r,x)+I(r,x)}\bigg)\,{\rm d} x\bigg]\,{\rm d} r\\& \quad + \int_s^t p(1-p)\frac{a_2}2 \left(\int_\mathcal{O}I(r,x)\,{\rm d} x\right)^p{\rm d} r + \int_s^t\left(\int_\mathcal{O}I(r,x)\,{\rm d} x\right)^p{\rm d} W_2(r)\\&\leq -pR_p \int_s^t \left(\int_\mathcal{O}I(r,x)\,{\rm d} x\right)^p{\rm d} r + \int_s^t\left(\int_\mathcal{O}I(r,x)\,{\rm d} x\right)^p{\rm d} W_2(r).\end{aligned}\end{equation*}

Since $\mathbb{E} \left(\int_\mathcal{O}I(t,x)\,{\rm d} x\right)^p<\infty$ , we have

\begin{equation*}\mathbb{E} \left(\int_\mathcal{O} I(t,x)\,{\rm d} x\right)^p= \mathbb{E}\left(\int_\mathcal{O} I(s,x)\,{\rm d} x\right)^p-pR_p \int_s^t \mathbb{E}\left(\int_\mathcal{O}I(r,x)\,{\rm d} x\right)^p{\rm d} r ,\end{equation*}

which easily leads to

\begin{equation*}\dfrac{{\rm d}}{{\rm d} t^+} \mathbb{E} \left(\int_\mathcal{O} I(t,x)\,{\rm d} x\right)^p\leq -pR_p \mathbb{E} \left(\int_\mathcal{O} I(t,x)\,{\rm d} x\right)^p.\end{equation*}

An application of the differential inequality shows that

(4.12) \begin{equation}\mathbb{E} \left(\int_\mathcal{O} I(t,x)\,{\rm d} x\right)^p \leq {\rm e}^{-pR_pt} \left(\int_\mathcal{O} I(0,x)\,{\rm d} x\right)^p \end{equation}

for any $t\geq 0$ and initial values in $D(A_i^E)$ . Since $D(A_i^E)$ is dense in E, (4.12) holds for each fixed t and any initial values in E. Then the desired result can be obtained.

5. An example

In this section, to demonstrate our results we consider an example where the noise processes in (1.1) are standard Brownian motions and the recruitment, death, infection, and recovery rates are independent of the space variable. Precisely, we consider

(5.1) \begin{equation}\begin{cases}{\rm d} S(t,x)=\bigg[k_1\Delta S(t,x)+\Lambda-\mu_1S(t,x)- \dfrac{\alpha S(t,x)I(t,x)}{S(t,x)+ I(t,x)}\bigg]{\rm d} t \\ \qquad\qquad +\, \sigma_1S(t,x){\rm d} B_1(t)\quad \text{in } \mathbb{R}^+\times\mathcal{O}, \\[1ex]{\rm d} I(t,x)=\bigg[k_2\Delta I(t,x)-\mu_2I(t,x) +\dfrac{\alpha S(t,x)I(t,x)}{S(t,x)+ I(t,x)}\bigg]{\rm d} t \\\qquad\qquad +\,\sigma_2 I(t,x){\rm d} B_2(t)\quad\text{in } \mathbb{R}^+\times\mathcal{O},\\[1ex]\partial_{\nu}S(t,x)=\partial_{\nu}I(t,x)=0\quad\quad\quad\quad\quad\quad\quad\text{in} \;\;\;\;\mathbb{R}^+\times\partial\mathcal{O},\\S(x,0)=S_0(x),I(x,0)=I_0(x)\quad\quad\quad\quad\;\ \text{in} \;\;\;\;\mathcal{\overline{O}},\end{cases}\end{equation}

where $\Lambda$ , $\mu_1$ , $\mu_2$ , and $\alpha$ are positive constants, and $B_1(t)$ , $B_2(t)$ are independent standard Brownian motions. As we obtained above, for any initial values $0\leq S_0,I_0\in E$ , (5.1) has a unique positive mild solution $S(t,x),I(t,x)\geq 0$ . Moreover, the long-time behavior of the system is shown in the following theorem.

6. Concluding remarks

Being possibly one of the first papers on spatially inhomogeneous stochastic partial differential equation epidemic models, we hope that our effort will provide some insights for subsequent study and investigation. For possible future study, we mention the following topics.

• • First, there is growing interest in using the so-called regime-switching stochastic models in various applications; see [Reference Yin and Zhu39] for the treatment of switching diffusion models, in which both continuous dynamics and discrete events coexist. Such switching diffusion models have gained popularity with applications ranging from networked control systems to financial engineering. For instance, in a financial market model, one may use a random switching process to model the mode of the market (bull or bear). Such a random switching process can be built into the stochastic partial differential equation models considered here. The switching is used to reflect different random environments that are not covered by the stochastic partial differential equation part of the model.

• • Second, instead of systems driven by Brownian motions, we may consider systems driven by Lévy process; some recent work can be seen in [Reference Bao, Yin and Yuan5]. One could work with stochastic partial differential equation models driven by Lévy processes. Recent work on switching jump diffusions [Reference Chen, Chen, Tran and Yin9] may also be adopted in stochastic partial differential equation models.

• • Finally, in terms of mathematical development, various estimates of the long-time properties were given in the average norm, although the solution is in the better space E. Future work will be to obtain stochastic regularity of the solution by using the methods in [Reference Brzeźniak7, Reference Veraar and Weis34, Reference van Neeren, Veraar and Weis35] so that it is possible to provide estimates in the sup-norm ( $\left\vert{\cdot}\right\vert_E$ ). Nevertheless, some mathematical details need to be carefully worked out. The result, in turn, will be of interest to people working on real data. Some other properties such as the strict positivity of the solutions and sharper conditions for extinction and permanence are worthy of consideration.