2.1. Boundedness for Fickian diffusion

Throughout this subsection, suppose that $S_0,I_0\in C(\bar {\Omega })$ for (1.5). We first give a classical result that $L^1$ uniform boundedness implies $L^\infty$ boundedness of the solution for the following quasilinear reaction-diffusion system:

(2.1) \begin{equation} \left \{\begin{aligned} &\partial _t u_i-\nabla \cdot (a_i(x,t,u)\nabla u_i)=b_i(x,t,u),\ & x&\in \Omega ,\ t\gt 0,\ i=1,\ldots ,m,\\ &\displaystyle \nabla u_i\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0,\\ &u_i(x,0)=u_i^0(x)\in C(\bar {\Omega }),\ & x&\in \Omega , \end{aligned}\right . \end{equation}

where $u=(u_1,\ldots ,u_m)$ . We assume that for any $(x,t,u)\in \Omega \times \mathbb{R}_+\times \mathbb{R}^m_+$ and $i=1,\ldots ,m$ :

(I) The diffusion operators are uniformly elliptic, i.e., for each $i$ , there exists a positive constant $\tau _0$ such that

\begin{equation*}a_i(x,t,u)\geq \tau _0;\end{equation*}

(II) The reaction functions $b_i$ satisfy the polynomial growth condition, i.e., there exist nonnegative constants $\tau _1$ and $\tau _2$ , and a positive constant $q$ such that

\begin{equation*}|b_i(x,t,u)|\leq \tau _1\sum _{i=1}^m|u_i|^q+\tau _2.\end{equation*}

Under assumptions (I) and (II), we conclude the results in [Reference Dung12, Theorem 1] and [Reference Du and Peng11, Lemma 2.1] in the following lemma.

By virtue of Lemma2.1, we can demonstrate the boundedness of the solution to (1.5).

Theorem 2.1. Assume that (H1)–(H4) hold. There exists a positive constant $N_1$ depending on the initial data such that the nonnegative solution $(S,I)$ of ( 1.5 ) satisfies

\begin{equation*}\|S(\cdot ,t)\|_{L^\infty (\Omega )}+\|I(\cdot ,t)\|_{L^\infty (\Omega )}\leq N_1,\ \forall \ t\in (0,\infty ).\end{equation*}

Moreover, there exists a positive constant $N_2$ independent of the initial data such that

\begin{equation*}\|S(\cdot ,t)\|_{L^\infty (\Omega )}+\|I(\cdot ,t)\|_{L^\infty (\Omega )}\leq N_2,\ \forall \ t\in [T,\infty ),\end{equation*}

for some large time $T\gt 0$ .

2.2. Boundedness for the Fokker–Planck-type diffusion

In this subsection, we assume $S_0\in C(\bar {\Omega })$ and $I_0\in W^{1,\infty }(\Omega )$ for (1.6). Building on the classical results of Amann [Reference Amann2–Reference Amann4] on the local existence of solutions for quasilinear parabolic equations, we present the following preliminary result regarding the local existence of solutions for (1.6).

Lemma 2.2. Assume that (H1)–(H4) hold. There exists $T_{\max }\in (0,\infty ]$ and a uniquely determined pair of nonnegative functions

\begin{equation*}S\in C(\bar {\Omega }\times [0,T_{\max }))\cap C^{2,1}(\bar {\Omega }\times (0,T_{\max })),\end{equation*}

\begin{equation*}I\in C(\bar {\Omega }\times [0,T_{\max }))\cap C^{2,1}(\bar {\Omega }\times (0,T_{\max }))\cap L_{\mathrm{loc}}^\infty ([0,T_{\max });\,W^{1,p}(\Omega )),\end{equation*}

with any $p\gt 1$ such that $(S,I)$ solves ( 1.6 ) classically in $\Omega \times (0,T_{\max })$ . Furthermore, if $T_{\max }\lt \infty$ , then for any $p\gt 1$ ,

\begin{equation*}\|S(\cdot ,t)\|_{L^\infty (\Omega )}+\|I(\cdot ,t)\|_{W^{1,p}(\Omega )}\rightarrow \infty , \mathrm {as}\ t\nearrow T_{\max }.\end{equation*}

The following lemma establishes the $W^{1,\infty }(\Omega )$ boundedness of $I$ for every $t\in (0,T_{\max })$ .

Lemma 2.3. Assume that (H1)–(H4) hold and $\gamma$ is constant. Then, there exist positive constants $M_1$ and $M_2$ depending on $\Omega ,n,\beta ,\gamma ,k,\eta$ and $N$ such that

(2.2) \begin{equation} \|I(\cdot ,t)\|_{L^\infty (\Omega )}\leq M_1,\ \forall \ t\in (0,T_{\max }), \end{equation}

and

(2.3) \begin{equation} \|\nabla I(\cdot ,t)\|_{L^\infty (\Omega )}\leq M_2,\ \forall \ t\in (0,T_{\max }). \end{equation}

Proof. Due to (1.9), we can apply Lemma2.1 with $q=p_0=1$ to the $I$ -equation in (1.6) to yield a positive constant $M_1$ independent of $t$ such that (2.2) holds. According to the variation-of-constants formula, we obtain

(2.4) \begin{equation} I(\cdot ,t)=e^{tg\Delta }I_0+\int _0^t e^{(t-s)g\Delta }\left (\frac {\beta SI}{S+I}-\gamma I\right )(\cdot ,s)ds,\ \forall \ t\in (0,T_{\max }). \end{equation}

We calculate the gradient for the both sides of (2.4) and consider the $L^\infty$ norm to get

\begin{eqnarray*} \|\nabla I(\cdot ,t)\|_{L^\infty (\Omega )}&&\leq \|\nabla e^{tg\Delta }I_0\|_{L^\infty (\Omega )} \\ && \quad +\int _0^t\left \|\nabla e^{(t-s)g\Delta }(\beta +\gamma )I(\cdot ,s)\right \|_{L^\infty (\Omega )}ds,\ \forall \ t\in (0,T_{\max }). \end{eqnarray*}

Let $\lambda _1\,:\!=\,\lambda _1(g)\gt 0$ be the first positive eigenvalue of $-g\Delta$ with homogeneous Neumann boundary condition. Then, by the smoothing properties of the Neumann heat semigroup [Reference Winkler31, Lemma 1.3] and (2.2), we have

\begin{eqnarray*} \|\nabla I(\cdot ,t)\|_{L^\infty (\Omega )}&\leq &C_1\|I_0\|_{W^{1,\infty }(\Omega )}+C_2\int _0^t(1+(t-s)^{-\frac {1}{2}})e^{-\lambda _1g(t-s)}\|I(\cdot ,s)\|_{L^\infty (\Omega )}ds, \\ &\leq &C_3,\ \forall \ t\in (0,T_{\max }), \end{eqnarray*}

where $C_1$ , $C_2$ and $C_3$ are positive constants independent of $T_{\max }$ . This results in (2.3).

Building upon the boundedness of $I$ and $\nabla I$ , we turn to obtain the $L^p$ estimate of $S$ .

Lemma 2.4. Assume that (H1)–(H4) hold and $\gamma$ is constant. For any $p\gt 1$ , we have the following estimate

\begin{equation*} \|S(\cdot ,t)\|_{L^p(\Omega )}\leq C,\ \forall \ t\in (0,T_{\max }). \end{equation*}

where $C$ is a positive constant independent of $t$ .

Proof. We first prove by induction that the lemma holds for $2^k$ for all integers $k\geq 0$ , i.e.,

(2.5) \begin{equation} \|S(\cdot ,t)\|_{L^{2^k}(\Omega )}^{2^k}\leq M_{2^k},\ \forall \ t\in (0,T_{\max }), \end{equation}

where $M_{2^k}$ is a positive constant independent of $t$ . It follows from (1.9) that (2.5) holds with $M_1=N$ for $k=0$ . Let $k\geq 1$ . Suppose that there exists a positive constant $M_{2^{k-1}}$ independent of $t$ such that

(2.6) \begin{equation} \|S(\cdot ,t)\|_{L^{2^{k-1}}(\Omega )}^{2^{k-1}}\leq M_{2^{k-1}},\ \forall \ t\in (0,T_{\max }). \end{equation}

We multiply the first equation in (1.6) by $S^{2^k-1}$ and integrate by parts to produce

\begin{eqnarray*} \frac {1}{2^k}\frac {d}{dt}\int _\Omega S^{2^k}dx&=&\int _\Omega [\nabla \cdot (f\nabla S)+\nabla \cdot (S\nabla f)]S^{2^k-1}dx-\int _\Omega \frac {\beta SI}{S+I}S^{2^k-1}dx+\gamma \int _\Omega IS^{2^k-1}dx\\ &\leq &-\int _\Omega f\nabla S\cdot \nabla S^{2^k-1}dx-\int _\Omega S\nabla f\cdot \nabla S^{2^k-1}dx+\gamma \int _\Omega IS^{2^k-1}dx\\ &=&-\,(2^k-1)\int _\Omega fS^{2^k-2}|\nabla S|^2-(2^k-1)\int _\Omega f_S S^{2^k-1}|\nabla S|^2dx\\ &&-\,(2^k-1)\int _\Omega \big(f_\beta S^{2^k-1}\nabla \beta \cdot \nabla S+f_I S^{2^k-1}\nabla I\cdot \nabla S\big)dx\\ &&+\,\gamma \int _\Omega I S^{2^k-1}dx,\ \forall \ t\in (0,T_{\max }). \end{eqnarray*}

Rearranging the above inequality, one has

(2.7) \begin{eqnarray} &&\frac {1}{2^k}\frac {d}{dt}\int _\Omega S^{2^k}dx+(2^k-1)\int _\Omega f S^{2^k-2}|\nabla S|^2+(2^k-1)\int _\Omega f_S S^{2^k-1}|\nabla S|^2dx\nonumber \\ &\leq &-(2^k-1)\int _\Omega \big(f_\beta S^{2^k-1}\nabla \beta \cdot \nabla S+f_I S^{2^k-1}\nabla I\cdot \nabla S\big)dx+\gamma \int _\Omega I S^{2^k-1}dx\nonumber \\ &\leq &(2^k-1)\int _\Omega S^{2^k-2}[|(f_\beta S\nabla \beta )\cdot \nabla S|+|(f_I S\nabla I)\cdot \nabla S|]dx\nonumber \\ &&+\gamma \int _\Omega I S^{2^k-1}dx,\ \forall \ t\in (0,T_{\max }). \end{eqnarray}

By Young’s inequality, we can obtain

(2.8) \begin{equation} |(f_\beta S\nabla \beta )\cdot \nabla S|\leq \frac {f}{4}|\nabla S|^2+\frac {f_\beta ^2|\nabla \beta |^2 S^2}{f} \end{equation}

and

(2.9) \begin{equation} |(f_I S\nabla I)\cdot \nabla S|\leq \frac {f}{4}|\nabla S|^2+\frac {f_I^2|\nabla I|^2 S^2}{f}. \end{equation}

For any $\epsilon _1\gt 0$ , an application of Young’s inequality again to the last term on the right-hand side of (2.7) gives

(2.10) \begin{equation} \int _\Omega I S^{2^k-1}dx\leq \|I(\cdot ,t)\|_{L^\infty (\Omega )}\left (\epsilon _1\int _\Omega S^{2^k} dx+C_{\epsilon _1}|\Omega |\right ),\ \forall \ t\in (0,T_{\max }), \end{equation}

where $C_{\epsilon _1}$ is a positive constant depending on $\epsilon _1$ . By Lemma2.3, there exists a positive constant $K$ depending on $M_1$ such that $f_\beta ,f_I\leq K$ for all $t\in (0,T_{\max })$ . We substitute (2.8)–(2.10) into (2.7) to yield

(2.11) \begin{align} &\quad \frac {1}{2^k}\frac {d}{dt}\int _\Omega S^{2^k}dx+\frac {2^k-1}{2}\int _\Omega f S^{2^k-2}|\nabla S|^2dx\nonumber \\[3pt] &\leq (2^k-1)\int _\Omega \left ( \frac {f_\beta ^2|\nabla \beta |^2+f_I^2|\nabla I|^2}{f}\right )S^{2^k}dx+\gamma \|I(\cdot ,t)\|_{L^\infty (\Omega )}\left (\epsilon _1\int _\Omega S^{2^k} dx+C_{\epsilon _1}|\Omega |\right )\nonumber \\[3pt] &\leq \left (\frac {(2^k-1)K^2\big(C_4^2+M_2^2\big)}{\eta ^2}+\gamma M_1\epsilon _1\right )\int _\Omega S^{2^k} dx+\gamma M_1C_{\epsilon _1}|\Omega |\nonumber \\[3pt] &\!=\!:\, C_5\int _\Omega S^{2^k} dx+C_6,\ \forall \ t\in (0,T_{\max }), \end{align}

where we assume $\|\nabla \beta \|_{L^\infty (\Omega )}\leq C_4$ for a positive constant $C_4$ in view of the fact that $\beta \in C^1(\bar {\Omega })$ . An application of the interpolation inequality gives that for any $\epsilon _2\gt 0$ ,

(2.12) \begin{equation} \int _\Omega S^{2^k} dx \leq \epsilon _2 \int _\Omega |\nabla S^{2^k-1}|^2 dx +C_{\epsilon _2} \left(\int _\Omega S^{2^{k-1}}dx \right)^2,\ \forall \ t\in (0,T_{\max }). \end{equation}

We now add $\int _\Omega S^{2^k}dx$ to both sides of (2.11) to derive that

(2.13) \begin{eqnarray} & &\frac {1}{2^k}\frac {d}{dt}\int _\Omega S^{2^k}dx+\frac {(2^k-1)\eta }{2^{2k-1}}\int _\Omega |\nabla S^{2^{k-1}}|^2dx+\int _\Omega S^{2^k}dx\nonumber \\ &\leq & (C_5+1)\int _\Omega S^{2^k}dx+C_6,\ \forall \ t\in (0,T_{\max }). \end{eqnarray}

Substituting $\epsilon _2 = \frac {(2^k-1)\eta }{2(C_5+1)2^{2k-1}}$ into (2.12) and applying the result to (2.13), we obtain that

(2.14) \begin{eqnarray} & &\frac {1}{2^k}\frac {d}{dt}\int _\Omega S^{2^k}dx+\frac {(2^k-1)\eta }{2^{2k}}\int _\Omega |\nabla S^{2^{k-1}}|^2dx+\int _\Omega S^{2^k}dx\nonumber \\ &\leq & (C_5+1)C_{\epsilon _2}\left(\int _\Omega S^{2^{k-1}}dx \right)^2+C_6, \ \forall \ t\in (0,T_{\max }). \end{eqnarray}

It follows from (2.6) that

\begin{equation*} \frac {1}{2^k}\frac {d}{dt}\int _\Omega S^{2^k}dx+\frac {(2^k-1)\eta }{4}\int _\Omega |\nabla S^{2^{k-1}}|^2dx+\int _\Omega S^{2^k}dx\leq C_7, \ \forall \ t\in (0,T_{\max }). \end{equation*}

By means of the Gronwall’s inequality, one infers that (2.5) holds.

By virtue of (2.5) and the embedding $L^{k_1}(\Omega )\subset L^{k_2}(\Omega )$ for any $k_1\gt k_2\geq 1$ , we then conclude the lemma holds.

Lemma 2.5. Assume that (H1)–(H4) hold and $\gamma$ is constant. There exists a positive constant $C$ independent of $t$ such that

\begin{equation*} \|S(\cdot ,t)\|_{L^\infty (\Omega )}\leq C,\ \forall \ t\in (0,T_{\max }). \end{equation*}

Combining Lemmas2.2 and 2.5 yields $T_{\max }=\infty$ . We are ready to state the boundedness of the solution to (1.6).

Theorem 2.2. Assume that (H1)–(H4) hold and $\gamma$ is constant. ( 1.6 ) possesses a global classical solution

\begin{equation*}S\in C(\bar {\Omega }\times [0,\infty ))\cap C^{2,1}(\bar {\Omega }\times (0,\infty )),\end{equation*}

\begin{equation*}I\in C(\bar {\Omega }\times [0,\infty ))\cap C^{2,1}(\bar {\Omega }\times (0,\infty ))\cap L_{\mathrm{loc}}^\infty ([0,\infty );\,W^{1,\infty }(\Omega )),\end{equation*}

such that both $S$ and $I$ are nonnegative in $\bar {\Omega }\times (0,\infty )$ . In addition, there exists a positive constant $C$ depending on $N$ such that

(2.15) \begin{equation} \|S(\cdot ,t)\|_{L^\infty (\Omega )}+\|I(\cdot ,t)\|_{W^{1,\infty }(\Omega )}\leq C,\ \forall \ t\in (0,\infty ). \end{equation}

3.1. Spectrum for Fickian diffusion

We linearize the $I$ -equation of (1.5) at the DFE to get that

(3.1) \begin{equation} \left \{\begin{aligned} &\partial _t \tilde {I}-\nabla \cdot (g(\gamma )\nabla \tilde {I})=(\beta -\gamma )\tilde {I},\ & x&\in \Omega ,\ t\gt 0,\\ &\displaystyle \nabla \tilde {I}\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0. \end{aligned}\right . \end{equation}

Letting $\tilde {I}(x,t)=e^{-\nu t}\phi (x)$ in (3.1) leads to

(3.2) \begin{equation} \left \{\begin{aligned} &\nabla \cdot (g(\gamma )\nabla \phi )+(\beta -\gamma )\phi +\nu \phi =0,\ & x&\in \Omega ,\\ &\displaystyle \nabla \phi \cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

It follows from the Krein–Rutman theorem in [Reference Krein and Rutman14] that (3.2) admits a unique eigenvalue $\nu ^*$ with a positive eigenfunction $\phi ^*$ on $\Omega$ . $\nu ^*$ is called the principal eigenvalue, which can be described by the variational characterization:

(3.3) \begin{equation} \nu ^*=\inf _{0\neq \phi \in H^1(\Omega )}\left \{\frac {\int _\Omega g(\gamma )|\nabla \phi |^2dx+\int _\Omega (\gamma -\beta )\phi ^2dx}{\int _\Omega \phi ^2dx}\right \}. \end{equation}

For the case of $g(\gamma )=\eta g_1(\gamma )$ , where $\eta$ is a positive constant and $g_1(\gamma )\gt 0$ is a smooth function, the asymptotic properties of $\nu ^*$ with respect to $\eta$ are given in [Reference Wang, Wang and Kim29, Lemma 2.2].

The exponentially and algebraically dependent diffusion rates are typical examples that satisfy (H5). We consider the asymptotic behaviours of $\nu ^*$ as $k\rightarrow 0$ and $k\rightarrow \infty$ .

Proposition 3.1. Assume that (H1)–(H5) hold. Let $\nu ^*\,:\!=\,\nu ^*(\eta d(k;\,\gamma ),\beta -\gamma )$ and $\phi ^*\,:\!=\,\phi ^*(\eta d(k;\,\gamma ),$ $\beta -\gamma )$ denote the principal eigenvalue and the corresponding eigenfunction of ( 3.2 ), respectively. Then, the following results for $\nu ^*$ hold.

(i) $\nu ^*$ and $\phi ^*$ depend smoothly on parameter $k\in (0,\infty )$ . In addition, $\nu ^*$ is a strictly monotone decreasing function of $k$ if the functions $k\mapsto d(k;\,\gamma )$ and $\beta -\gamma$ are non-constant.

(ii) $\nu ^*\rightarrow \nu ^*(\eta ,\beta -\gamma )$ , as $k\rightarrow 0$ , where $\nu ^*(\eta ,\beta -\gamma )$ is the principal eigenvalue of

(3.4) \begin{equation} \left \{\begin{aligned} &\eta \Delta \phi +(\beta -\gamma )\phi +\nu \phi =0,\ & x&\in \Omega ,\\ &\displaystyle \nabla \phi \cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

(iii) If $d(k;\,\gamma )=e^{-k\gamma }$ or $d(k;\,\gamma )=(1+\gamma )^{-k}$ , then $\nu ^*\rightarrow \min _{x\in \bar {\Omega }}\{\gamma -\beta \}$ as $k\rightarrow \infty$ .

Proof. The smoothness of $\nu ^*$ and $\phi ^*$ on $k$ in assertion (i) is standard in [Reference Cantrell and Cosner6] via implicit function theorem. It suffices to verify the monotonicity of $\nu ^*$ with respect to $k$ . We differentiate both sides of (3.2) with $\nu =\nu ^*$ and $\phi =\phi ^*$ by $k$ to produce

(3.5) \begin{equation} \left \{\begin{aligned} &\eta \nabla \cdot [d^{\prime}(k;\,\gamma )\nabla \phi ^*+d(k;\,\gamma )\nabla (\phi ^*)^{\prime}]+(\beta -\gamma )(\phi ^*)^{\prime}+(\nu ^*)^{\prime}\phi ^*+\nu ^*(\phi ^*)^{\prime}=0,\ & x&\in \Omega ,\\ &\displaystyle \nabla (\phi ^*)^{\prime}\cdot \mathbf{n}=0,\ & x&\in \partial \Omega , \end{aligned}\right . \end{equation}

where the prime notation $^{\prime}$ means differentiation by $k$ . We multiply (3.5) by $\phi ^*$ and (3.2) with $\nu =\nu ^*$ and $\phi =\phi ^*$ by $(\phi ^*)^{\prime}$ , subtract the results and integrate by parts over $\Omega$ to yield

\begin{equation*} (\nu ^*)^{\prime}\int _\Omega (\phi ^*)^2dx=\eta \int _\Omega d^{\prime}(k;\,\gamma )|\nabla \phi ^*|^2dx. \end{equation*}

It follows from $d^{\prime}(k;\,\gamma )\leq 0$ that $(\nu ^*)^{\prime}\leq 0$ . Since $k\mapsto d(k;\,\gamma )$ is non-constant, we find that the equality holds only if $\phi ^*$ is constant on $\Omega$ . Then, by (3.2), we get

\begin{equation*} \beta -\gamma +\nu ^*=0,\ x\in \Omega . \end{equation*}

This leads to a contradiction as $\beta -\gamma$ is non-constant on $\Omega$ . Hence, $\nu ^*\lt 0$ .

It follows from (i) and (H5) that assertion (ii) holds. We turn to prove (iii). According to (3.3), one has

(3.6) \begin{eqnarray} \nu ^*\geq \inf _{0\neq \phi \in H^1(\Omega )}\left \{\frac {\int _{\Omega }(\gamma -\beta )\phi ^2dx}{\int _\Omega \phi ^2dx}\right \} \geq \min _{x\in \bar {\Omega }}\{\gamma -\beta \}. \end{eqnarray}

For any $\epsilon$ , we can choose $\phi _\epsilon \in H^1(\Omega )$ such that

\begin{equation*}\gamma -\beta \leq \min _{x\in \bar {\Omega }}\{\gamma -\beta \}+\epsilon ,\ \forall \ x\in \ \mathrm{supp}(\phi _\epsilon ).\end{equation*}

We find that

\begin{eqnarray*} \lim _{k\rightarrow \infty }\nu ^*&\leq &\lim _{k\rightarrow \infty } \frac {\eta \int _{\mathrm{supp}(\phi _\epsilon )}d(k;\,\gamma )|\nabla \phi _\epsilon |^2dx+\int _{\mathrm{supp}(\phi _\epsilon )} (\gamma -\beta )\phi _\epsilon ^2 dx}{\int _{\mathrm{supp}(\phi _\epsilon )}\phi _\epsilon ^2dx}\\ &\leq &\min _{x\in \bar {\Omega }}\{\gamma -\beta \}+\epsilon . \end{eqnarray*}

This together with (3.6) and the arbitrariness of $\epsilon$ gives (iii).

We then consider the corresponding eigenvalue problem with the weight $\beta$ for (1.5)

(3.7) \begin{equation} \left \{\begin{aligned} &\!-\nabla \cdot (g(\gamma )\nabla \psi )+\gamma \psi =\nu \beta \psi ,\ & x&\in \Omega ,\\ &\displaystyle \nabla \psi \cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

The principal eigenvalue of (3.7) can also be given by the variational characterization

\begin{equation*}\nu _0=\inf _{0\neq \psi \in H^1(\Omega )}\left \{\frac {\int _\Omega g(\gamma )|\nabla \psi |^2dx+\int _\Omega \gamma \psi ^2 dx}{\int _\Omega \beta \psi ^2dx}\right \}\!.\end{equation*}

Then, the basic reproduction for (1.5) $\mathcal{R}_0$ ^(1.5) can be defined as the inverse of $\nu _0$ :

\begin{equation*}\mathcal{R}_0^{(1.5)}(g(\gamma ),\beta ,\gamma )=\frac {1}{\nu _0}=\sup _{0\neq \psi \in H^1(\Omega )}\left \{\frac {\int _\Omega \beta \psi ^2dx}{\int _\Omega g(\gamma )|\nabla \psi |^2dx+\int _\Omega \gamma \psi ^2 dx}\right \}\!.\end{equation*}

By inspection of $\mathcal{R}_0$ ^(1.5), we get the following properties.

Proposition 3.2. Assume that (H1)–(H5) hold. Let $\mathcal{R}_0$ ^(1.5) $:\!=\,\mathcal{R}_0$ ^(1.5) $(\eta d(k;\,\gamma ),\beta ,\gamma )$ denote the basic reproduction number of (1.5). $\mathcal{R}_0$ ^(1.5) satisfies the following properties.

(i) $\mathcal{R}_0$ ^(1.5) depends smoothly on parameter $k\in (0,\infty )$ and is a monotone increasing function of $k$ .

(ii) $\mathcal{R}_0$ ^(1.5) $\rightarrow \mathcal{R}_0$ ^(1.5) $(\eta ,\beta ,\gamma )$ , as $k\rightarrow 0$ , where $\mathcal{R}_0$ ^(1.5) $(\eta ,\beta ,\gamma )$ fulfils the following problem:

(3.8) \begin{equation} \left \{\begin{aligned} &\eta \Delta \psi _0-\gamma \psi _0+\frac {1}{\mathcal{R}_0^{(1.5)}}\beta \psi _0=0,\ & x&\in \Omega ,\\ &\displaystyle \nabla \psi _0\cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

(iii) If $d(k;\,\gamma )=e^{-k\gamma }$ or $d(k;\,\gamma )=(1+\gamma )^{-k}$ , then $\mathcal{R}_0$ ^(1.5) $\rightarrow \max _{x\in \bar {\Omega }}\{\beta /\gamma \}$ as $k\rightarrow \infty$ .

3.2. Spectrum for the Fokker–Planck-type diffusion

Now we explore the eigenvalue problems of (1.6). Linearizing the $I$ -equation of (1.6) at the DFE gives

(3.9) \begin{equation} \left \{\begin{aligned} &\partial _t \tilde {I}-\Delta (g(\gamma )\tilde {I})=(\beta -\gamma )\tilde {I},\ & x&\in \Omega ,\ t\gt 0,\\ &\displaystyle \nabla (g(\gamma )\tilde {I})\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0. \end{aligned}\right . \end{equation}

Letting $\tilde {I}(x,t)=e^{-\mu t}\varphi (x)$ in (3.9) results in

(3.10) \begin{equation} \left \{\begin{aligned} &\Delta (g(\gamma )\varphi )+(\beta -\gamma )\varphi +\mu \varphi =0,\ & x&\in \Omega ,\\ &\displaystyle \nabla (g(\gamma )\varphi )\cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

Setting $\Phi (x)=g(\gamma )\varphi (x)$ , (3.10) becomes

(3.11) \begin{equation} \left \{\begin{aligned} &g(\gamma )\Delta \Phi +(\beta -\gamma )\Phi +\mu \Phi =0,\ & x&\in \Omega ,\\ &\displaystyle \nabla \Phi \cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

In view of the Krein–Rutman theorem [Reference Krein and Rutman14] again, problem (3.11) admits a principal eigenvalue, denoted by $\mu ^*$ , which corresponds to a strictly positive eigenfunction $\Phi ^*$ on $\Omega$ . By the variational characterization, $\mu ^*$ can be represented by

\begin{equation*}\mu ^*=\inf _{0\neq \Phi \in H^1(\Omega )}\left \{\frac {\int _\Omega |\nabla \Phi |^2dx+\int _\Omega \frac {\beta -\gamma }{g(\gamma )}\Phi ^2dx}{\int _\Omega \frac {\Phi ^2}{g(\gamma )}dx}\right \}\!.\end{equation*}

When $g(\gamma )=\eta d(k;\,\gamma )$ , we have the following properties of $\mu ^*$ .

We also consider the eigenvalue problem with the weight for (1.6) as follows:

(3.12) \begin{equation} \left \{\begin{aligned} &\!-\Delta \Psi +\frac {\gamma }{g(\gamma )}\Psi =\mu \frac {\beta }{g(\gamma )}\Psi ,\ & x&\in \Omega ,\\ &\displaystyle \nabla \Psi \cdot \mathbf{n}=0,\ & x&\in \partial \Omega . \end{aligned}\right . \end{equation}

The corresponding principal eigenvalue of (3.12) can also be given via the variational characterization

\begin{equation*}\mu _0=\inf _{0\neq \Psi \in H^1(\Omega )}\left \{\frac {\int _\Omega |\nabla \Psi |^2dx+\int _\Omega \frac {\gamma }{g(\gamma )} \Psi ^2 dx}{\int _\Omega \frac {\beta }{g(\gamma )}\Psi ^2dx}\right \}\!.\end{equation*}

Similarly, the basic reproduction number $\mathcal{R}_0$ ^(1.6) for (1.6) can be characterized as

\begin{equation*}\mathcal{R}_0^{(1.6)}(g(\gamma ),\beta ,\gamma )=\frac {1}{\mu _0}=\sup _{0\neq \Psi \in H^1(\Omega )}\left \{\frac {\int _\Omega \frac {\beta }{g(\gamma )}\Psi ^2dx}{\int _\Omega |\nabla \Psi |^2dx+\int _\Omega \frac {\gamma }{g(\gamma )} \Psi ^2 dx}\right \}\!.\end{equation*}

Proof. We only need to verify the monotonicity of $\mathcal{R}_0$ ^(1.6), as the other properties can be achieved by inspection. We differentiate both sides of (3.12) with $\mu =\frac {1}{\mathcal{R}_0^{(1.6)}}$ and $\Psi =\Psi _0$ by $k$ to get

(3.13) \begin{equation} \left \{\begin{aligned} &\!-\eta d^{\prime}(k;\,\gamma )\Delta \Psi _0-\eta d(k;\,\gamma )\Delta (\Psi _0)^{\prime}+\gamma (\Psi _0)^{\prime}=-\frac {(\mathcal{R}_0^{(1.6)})^{\prime}}{(\mathcal{R}_0^{(1.6)})^2}\beta \Psi _0+\frac {\beta }{\mathcal{R}_0^{(1.6)}}(\Psi _0)^{\prime},\ & x&\in \Omega ,\\ &\displaystyle \nabla (\Psi _0)^{\prime}\cdot \mathbf{n}=0,\ & x&\in \partial \Omega , \end{aligned}\right . \end{equation}

where $\Psi _0$ is the positive eigenfunction corresponding to $\mu _0$ . We multiply (3.13) by $\Psi _0$ and (3.12) with $\mu =\frac {1}{\mathcal{R}_0^{(1.6)}}$ and $\Psi =\Psi _0$ by $(\Psi _0)^{\prime}$ , subtract the results and integrate by parts over $\Omega$ to obtain

\begin{equation*} -\frac {(\mathcal{R}_0^{(1.6)})^{\prime}}{(\mathcal{R}_0^{(1.6)})^2}\int _\Omega \beta (\Psi _0)^2dx=\eta \int _\Omega d^{\prime}(k;\,\gamma )|\nabla \Psi _0|^2dx. \end{equation*}

Since $d^{\prime}(k;\,\gamma )\leq 0$ , one has $(\mathcal{R}_0$ ^(1.6) $)^{\prime}\geq 0$ .

The parameter $k$ plays an important role in the asymptotic profiles of the basic reproduction number. If $k\rightarrow 0$ , the limit of the basic reproduction numbers of (1.5) and (1.6) is $\mathcal{R}_0$ ^(1.1) with $d_I=\eta$ . The corresponding monotonicity and asymptotics of $\mathcal{R}_0$ ^(1.1) on $\eta$ are presented in [Reference Allen, Bolker, Lou and Nevai1, Lemma 2.3]. On the other hand, we find that the basic reproduction numbers of (1.5) and (1.6) have the same limit when $\eta \rightarrow 0$ or $k\rightarrow \infty$ .

It is difficult to compare $\mathcal{R}_0$ ^(1.5) and $\mathcal{R}_0$ ^(1.6). However, they are equal when $\gamma$ is a positive constant. In this case,

\begin{equation*}\mathcal{R}_0^{(1.5)}=\mathcal{R}_0^{(1.6)}=\sup _{0\neq \psi \in H^1(\Omega )}\left \{\frac {\int _\Omega \beta \psi ^2dx}{g(\gamma )\int _\Omega |\nabla \psi |^2dx+\int _\Omega \gamma \psi ^2 dx}\right \}\!,\end{equation*}

as $g(\gamma )$ is also constant.

4.1. Threshold dynamics for Fickian diffusion

We now establish the threshold dynamics in terms of $\mathcal{R}_0$ ^(1.5) for (1.5).

Proposition 4.2. Assume that (H1)–(H4) hold. The following assertions hold.

(i) If $\mathcal{R}_0$ ^(1.5) $\leq 1$ , then the solution $(S,I)$ of ( 1.5 ) satisfies $(S,I)\rightarrow \left (\frac {N}{|\Omega |},0\right )$ as $t\rightarrow \infty$ .

(ii) If $\mathcal{R}_0$ ^(1.5) $\gt 1$ , then ( 1.5 ) is uniformly persistent, i.e., there exists some constant $\rho \gt 0$ such that

\begin{equation*}\liminf _{t\rightarrow \infty }S(x,t)\geq \rho \ \mathrm {and}\ \liminf _{t\rightarrow \infty }I(x,t)\geq \rho \ \mathrm {uniformly for}\ x\in \bar {\Omega }.\end{equation*}

Moreover, there exists at least one EE for ( 1.5 ).

We then show the global stability of the constant EE when $\beta/\gamma$ is a positive constant on $\Omega$ via the Lyapunov functional method.

Proof. Let

\begin{equation*}V(S,I)=\int \frac {S^2-\hat {S}^2}{S^2}dS+\int \frac {I^2-\hat {I}^2}{I^2}dI.\end{equation*}

We construct a Lyapunov functional to verify the global attractiveness of the constant EE as follows:

\begin{equation*} L_1(t)=\int _\Omega V(S(x,t),I(x,t))dx. \end{equation*}

Computing the total differential of $L_1(t)$ leads to

\begin{eqnarray*} \frac {dL_1(t)}{dt}&=&\int _\Omega \left [\frac {S^2-\hat {S}^2}{S^2}\nabla \cdot (f(\beta ,S,I)\nabla S)+\frac {I^2-\hat {I}^2}{I^2}\nabla \cdot (g(\gamma )\nabla I)\right ]dx\\ &&+\int _\Omega \left [\frac {S^2-\hat {S}^2}{S^2}\left ({-}\frac {\beta SI}{S+I}+\frac {\beta I}{\mathcal{R}_0}\right )+\frac {I^2-\hat {I}^2}{I^2}\left (\frac {\beta SI}{S+I}-\frac {\beta I}{\mathcal{R}_0}\right )\right ]dx\\ &=&-\int _\Omega \left [\frac {2f(\beta ,S,I)\hat {S}^2}{S^3}|\nabla S|^2+\frac {2g(\gamma )\hat {I}^2}{I^3}|\nabla I|^2\right ]dx\\ &&+\int _\Omega \beta I\left (\frac {1}{\mathcal{R}_0}-\frac {S}{S+I}\right )\left (\frac {\hat {I}^2}{I^2}-\frac {\hat {S}^2}{S^2}\right )dx,\ \forall \ t\gt 0. \end{eqnarray*}

Since $\frac {\hat {S}}{\hat {S}+\hat {I}}=\frac {1}{\mathcal{R}_0}$ and $f(\beta ,S,I),\ g(\gamma )\gt 0$ , we have

\begin{eqnarray*} \frac {dL_1(t)}{dt}&=&-\int _\Omega \left [\frac {2f(\beta ,S,I)\hat {S}^2}{S^3}|\nabla S|^2+\frac {2g(\gamma )\hat {I}^2}{I^3}|\nabla I|^2\right ]dx\\ &&-\int _\Omega \frac {\beta S\hat {I}^2}{(S+I)(\hat {S}+\hat {I})}\left (\frac {\hat {S}}{S}+\frac {\hat {I}}{I}\right )\left (\frac {\hat {S}}{S}-\frac {\hat {I}}{I}\right )^2dx\leq 0,\ \forall \ t\gt 0. \end{eqnarray*}

The equality holds only at $(\hat {S},\hat {I})$ . Therefore, $L_1(t)$ is a Lyapunov functional for (1.5). It then follows from some standard arguments that

\begin{equation*}(S(\cdot ,t),I(\cdot ,t))\rightarrow (\hat {S},\hat {I})\ \textrm {in}\ (L^2(\Omega ))^2,\ \textrm {as}\ t\rightarrow \infty .\end{equation*}

By Theorem2.1, we have $\|S(\cdot ,t)\|_{L^\infty (\Omega )}$ and $\|I(\cdot ,t)\|_{L^\infty (\Omega )}$ are bounded for all $t\geq 0$ . Thus, we obtain

\begin{equation*}(S(\cdot ,t),I(\cdot ,t))\rightarrow (\hat {S},\hat {I})\ \textrm {in}\ (L^\infty (\Omega ))^2,\ \textrm {as}\ t\rightarrow \infty .\end{equation*}

This indicates the global attractiveness of the constant EE of (1.5). According to the local stability stated in Lemma4.1, we deduce that the constant EE of (1.5) is globally asymptotically stable.

4.2. Threshold dynamics for the Fokker–Planck-type diffusion

Throughout this subsection, we assume that $\beta$ and $\gamma$ are constants on $\Omega$ . We first prove the global stability of the DFE for (1.6) if $\mathcal{R}_0\leq 1$ through energy estimates. Let $U(x,t)=S(x,t)+I(x,t)$ for $x\in \bar {\Omega }$ and $t\geq 0$ . If $\beta$ and $\gamma$ are constant on $\Omega$ , then (1.6) degenerates to

(4.4) \begin{equation} \left \{\begin{aligned} &\partial _t U=\nabla \cdot [(f+f_S S)\nabla U]+\nabla \cdot [(g-f+(f_I-f_S)S)\nabla I],\ &x&\in \Omega ,\ t\gt 0,\\ &\nabla U\cdot \mathbf{n}=0,\ &x&\in \partial \Omega ,\ t\gt 0,\\ &U(x,0)=S_0(x)+I_0(x),\ &x&\in \Omega . \end{aligned}\right . \end{equation}

In view of Theorem2.2, there exists a positive constant $K$ such that $f_S,f_I\leq K$ for all $t\gt 0$ .

Lemma 4.3. Assume that (H1)–(H4) hold and $\beta$ and $\gamma$ are constant on $\Omega$ . The solution $U$ of ( 4.4 ) fulfils that

(4.5) \begin{equation} \int _0^{\infty }\int _\Omega |\nabla U|^2dxdt\lt \infty . \end{equation}

Proof. We multiply the second equation of (1.6) by $I$ and integrate over $\Omega \times (0,T)$ to result in

(4.6) \begin{eqnarray} g(\gamma )\int _0^T\int _\Omega |\nabla I|^2dx&=&\frac {1}{2}\int _\Omega I_0^2dx-\frac {1}{2}\int _\Omega I^2(\cdot ,T)dx+\int _0^T\int _\Omega \left (\frac {\beta S}{S+I}-\gamma\right )I^2dxdt\nonumber \\[4pt] &\leq &\frac {1}{2}\int _\Omega I_0^2dx+\beta \int _0^T\int _\Omega I^2dxdt. \end{eqnarray}

Since $\beta$ and $\gamma$ are constant on $\Omega$ . It follows from [Reference Allen, Bolker, Lou and Nevai1, Lemma 2.3(d)] that $\mu ^*=\nu ^*\gt 0$ as $\mathcal{R}_0\lt 1$ . Since $\partial _tI\leq g(\gamma )\Delta I+(\beta -\gamma )I$ for $x\in \Omega$ and $t\gt 0$ , it can be shown that $Me^{-\nu ^*t}\phi ^*(x)$ is a super-solution to the $I$ -equation for a large enough constant $M\gt 0$ . By comparison principle, one has

(4.7) \begin{equation} I(x,t)\leq Me^{-\nu ^*t}\phi ^*(x),\ x\in \Omega ,\ t\gt 0. \end{equation}

We substitute (4.7) to (4.6) to yield

(4.8) \begin{equation} \int _0^T\int _\Omega |\nabla I|^2dx\leq C\left [\int _\Omega I_0^2dx+\int _0^T\int _\Omega (M e^{-\nu ^*t}\phi ^*(x))^2dxdt\right ]\leq C,\ \forall \ T\gt 0, \end{equation}

for positive constant $C$ independent of $T$ . We then multiply the first equation of (1.6) by $S$ and integrate by parts over $\Omega$ to get

\begin{eqnarray*} \frac {1}{2}\int _\Omega S^2dx&=&-\int _\Omega f|\nabla S|^2dx-\int _\Omega S\nabla f\cdot \nabla Sdx+\int _\Omega \left (\gamma I-\frac {\beta SI}{S+I}\right )Sdx\\[3pt] &\leq &-\int _\Omega f|\nabla S|^2dx-\int _\Omega f_S S|\nabla S|^2dx-\int _\Omega f_IS\nabla S\cdot \nabla Idx+\gamma \int _\Omega SIdx\\[3pt] &\leq &-\eta \int _\Omega |\nabla S|^2dx+K\int _\Omega |S\nabla S\cdot \nabla I|dx+\gamma \int _\Omega SIdx\\[3pt] &\leq &-\frac {\eta }{2}\int _\Omega |\nabla S|^2dx+\frac {K^2}{2\eta }\int _\Omega S^2|\nabla I|^2dx+\gamma \int _\Omega SIdx,\ \forall \ t\gt 0, \end{eqnarray*}

where $K$ is given in the proof of Lemma2.4 and Young’s inequality is used in the last inequality. Since $S$ is bounded by Theorem2.2 for all $t\gt 0$ , one has

(4.9) \begin{eqnarray} \frac {1}{2}\int _\Omega S^2dx&\leq &-\frac {\eta }{2}\int _\Omega |\nabla S|^2dx+\frac {K^2}{2\eta }\|S(\cdot ,t)\|_{L^\infty (\Omega )}^2\int _\Omega |\nabla I|^2dx\nonumber \\ &&+\gamma \|S(\cdot ,t)\|_{L^\infty (\Omega )}\int _\Omega Idx,\ \forall \ t\gt 0. \end{eqnarray}

We integrate both sides of (4.9) from 0 to $T$ to give

(4.10) \begin{eqnarray} \int _0^T\int _\Omega |\nabla S|^2dxdt&\leq & C\left [\int _\Omega S_0^2dx+\int _0^T\int _\Omega |\nabla I|^2dxdt+\int _0^T\int _\Omega M e^{-\mu ^*t}\phi ^*(x)dxdt\right ]\nonumber \\ &\leq & C,\ \forall \ T\gt 0. \end{eqnarray}

Since $C$ is independent of $T$ , consequently, we combine (4.8) and (4.10) to produce (4.5).

We then show that $U_t$ decays at least in the dual space of $H^1(\Omega )$ .

Lemma 4.4. Assume that (H1)–(H4) hold and $\beta$ and $\gamma$ are constant on $\Omega$ . The solution $U$ of ( 4.4 ) satisfies

\begin{equation*} \int _0^\infty \|U_t(\cdot ,t)\|_{(H^1(\Omega ))^*}^2dt\lt \infty , \end{equation*}

where $(H^1(\Omega ))^*$ represents the dual space of $H^1(\Omega )$ .

Proof. By virtue of the forms of $f$ , we can infer that $f\leq K_0$ for some positive constant $K_0\,:\!=\,K_0(\beta ,k,\eta ,\|S(\cdot ,t)\|_{L^\infty (\Omega )})$ if $\|S(\cdot ,t)\|_{L^\infty (\Omega )}$ is bounded for all $t\geq 0$ . We multiply both sides of the first equation of (4.4) by any test function $\theta \in H^1(\Omega )$ and integrate by parts over $\Omega$ to produce

\begin{eqnarray*} \int _\Omega \theta \partial _tUdx&=&-\int _\Omega [(f+f_S S)\nabla U]\cdot \nabla \theta dx-\int _\Omega [(g-f+(f_I-f_S)S)\nabla I]\cdot \nabla \theta dx\\ &\leq &(K_0+K\|S(\cdot ,t)\|_{L^\infty (\Omega )})\int _\Omega |\nabla U\cdot \nabla \theta |dx\\ &&+(g+K_0+2K\|S(\cdot ,t)\|_{L^\infty (\Omega )})\int _\Omega |\nabla I\cdot \nabla \theta |dx\\ &\leq &C\left (\|\nabla U(\cdot ,t)\|_{L^2(\Omega )}\|\nabla \theta \|_{L^2(\Omega )}+\|\nabla I(\cdot ,t)\|_{L^2(\Omega )}\|\nabla \theta \|_{L^2(\Omega )}\right )\\ &\leq &C\left (\|\nabla U(\cdot ,t)\|_{L^2(\Omega )}+\|\nabla I(\cdot ,t)\|_{L^2(\Omega )}\right )\|\nabla \theta \|_{H^1(\Omega )},\ \forall \ t\gt 0, \end{eqnarray*}

for some positive constant $C$ . Hence, $U_t(\cdot ,t)\in (H^1(\Omega ))^*$ and satisfies

\begin{eqnarray*} \|U_t(\cdot ,t)\|_{(H^1(\Omega ))^*}\leq C\left (\|\nabla U(\cdot ,t)\|_{L^2(\Omega )}+\|\nabla I(\cdot ,t)\|_{L^2(\Omega )}\right ),\ \forall \ t\gt 0. \end{eqnarray*}

In light of (4.5) and (4.8), we conclude

\begin{equation*} \int _0^\infty \|U_t(\cdot ,t)\|^2_{(H^1(\Omega ))^*}dt\leq C\int _0^\infty \left (\|\nabla U(\cdot ,t)\|_{L^2(\Omega )}^2+\|\nabla I(\cdot ,t)\|_{L^2(\Omega )}^2\right )dt\lt \infty . \end{equation*}

Now we are able to demonstrate the stability of the DFE for (1.6) with $\mathcal{R}_0\leq 1$ .

Proof. We first consider $\mathcal{R}_0\lt 1$ . By Lemma4.2, we know $I(x,t)\rightarrow 0$ exponentially for $x\in \bar {\Omega }$ as $t\rightarrow \infty$ . We proceed with a similar argument as [Reference Li, Peng and Xiang17, Theorem 3.5] to deduce that

\begin{equation*} \left \|U(\cdot ,t)-\frac {N}{|\Omega |}\right \|_{C(\bar {\Omega })}\rightarrow 0,\ \mathrm {as}\ t\rightarrow \infty . \end{equation*}

This indicates that $S(x,t)\rightarrow \frac {N}{|\Omega |}$ in $C(\bar {\Omega })$ as $t\rightarrow \infty$ .

We then consider the case $\mathcal{R}_0=1$ . This is equivalent to $\beta =\gamma$ . The second equation of (1.6) becomes

(4.11) \begin{equation} \partial _t I=g\Delta I-\frac {\beta I^2}{S+I}\leq g\Delta I-\frac {\beta }{M_5}I^2,\ x\in \bar {\Omega },\ t\gt 0, \end{equation}

where we use the bound of $S+I$  by Theorem2.2. We turn to the following Cauchy problem

\begin{equation*} \left \{\begin{aligned} &\bar {I}_t=-\frac {\beta }{M_5}\bar {I}^2,\ t\gt 0,\\[2pt] &\bar {I}(0)=\|I_0\|_{L^\infty (\Omega )}. \end{aligned}\right . \end{equation*}

It can be verified that $\bar {I}$ is a super-solution of (4.11). It follows from the comparison principle that

(4.12) \begin{equation} I(\cdot ,t)\leq \bar {I}(t)=\left (\frac {1}{\bar {I_0}}+\frac {\beta t}{M_5}\right )^{-1}\rightarrow 0,\ \textrm {as}\ t\rightarrow \infty . \end{equation}

We then multiply the first equation of (1.6) by $S$ and integrate by parts over $\Omega$ to yield

\begin{eqnarray*} \frac {1}{2}\frac {d}{dt}\int _\Omega S^2dx&=&-\int _\Omega f|\nabla S|^2dx-\int _\Omega S\nabla f\cdot \nabla Sdx+\beta \int _\Omega \frac {SI^2}{S+I}dx\\ &\leq &-\frac {\eta }{2}\int _\Omega |\nabla S|^2dx+\frac {K^2}{2\eta }\|S(\cdot ,t)\|_{L^\infty (\Omega )}^2\int _\Omega |\nabla I|^2dx+\beta \int _\Omega I^2dx,\ \forall \ t\gt 0. \end{eqnarray*}

As a result of (4.12) and Theorem2.2, we conclude that

\begin{equation*} \int _0^\infty \int _\Omega |\nabla S|^2dxdt\lt \infty ,\ \forall \ T\gt 0 \end{equation*}

also holds for $\mathcal{R}_0=1$ . We then follow similar arguments as those in $\mathcal{R}_0\lt 1$ . This completes the proof.

We now dedicate to the stability of the constant EE. By the standard Moser-type iteration, we can get the following boundedness for $I$ of (1.6).

Lemma 4.5. Assume that (H1)–(H4) hold. There exists a positive constant $M_6$ independent of $\eta$ and $K$ such that

(4.13) \begin{equation} \|I(\cdot ,t)\|_{L^\infty (\Omega )}\leq M_6,\ \forall \ t\gt 0. \end{equation}

To give the global stability of the constant EE for (1.6), we construct another Lyapunov functional.

Proof. We employ the following Lyapunov functional

\begin{equation*}L_2(t)=\int _\Omega \left [\left (S- \hat {S}-\hat {S}\ln \frac {\hat {S}}{S}\right )+\left (I- \hat {I}-\hat {I}\ln \frac {\hat {I}}{I}\right )\right ]dx.\end{equation*}

We integrate by parts to compute the total derivatives of $L_2(t)$ to give

(4.14) \begin{eqnarray} \frac {dL_2(t)}{dt}&=&\int _\Omega \frac {S-\hat {S}}{S}[\nabla \cdot (f\nabla S)+\nabla \cdot (S\nabla f)]dx+g\int _\Omega \frac {I-\hat {I}}{I}\Delta Idx\nonumber \\ &&+\int _\Omega \frac {S-\hat {S}}{S}\left ({-}\frac {\beta SI}{S+I}+\gamma I\right )dx+\int _\Omega \frac {I-\hat {I}}{I}\left (\frac {\beta SI}{S+I}-\gamma I\right )dx\nonumber \\ &=&\int _\Omega \left [f\nabla S\cdot \nabla \left (\frac {\hat {S}}{S}\right )+S\nabla f\cdot \nabla \left (\frac {\hat {S}}{S}\right )+g\nabla I\cdot \nabla \left (\frac {\hat {I}}{I}\right )\right ]dx\nonumber \\ &&-\int _\Omega \beta I\left (\frac {S}{S+I}-\frac {1}{\mathcal{R}_0}\right )\left (\frac {S-\hat {S}}{S}-\frac {I-\hat {I}}{I}\right )dx\nonumber \\ &=&-\hat {S}\int _\Omega \left (\frac {f|\nabla S|^2}{S^2}+\frac {f_S|\nabla S|^2}{S}+\frac {f_I\nabla S\cdot \nabla I}{S}\right )dx-g\hat {I}\int _\Omega \frac {|\nabla I|^2}{I^2}dx\nonumber \\ &&-\int _\Omega \frac {\beta SI}{(S+I)(\hat {S}+\hat {I})}\left (\frac {\hat {I}}{I}-\frac {\hat {S}}{S}\right )^2dx,\ \forall \ t\gt 0. \end{eqnarray}

Since $f_S\geq 0$ , we drop the second term on the right-hand side of (4.14). By assumption (H2) and $\hat {I}=(\mathcal{R}_0-1)\hat {S}$ , we have

\begin{eqnarray*} \frac {dL_2(t)}{dt}&\leq &-\eta \hat {S}\int _\Omega \frac {|\nabla S|^2}{S^2}dx-g\hat {I}\int _\Omega \frac {|\nabla I|^2}{I^2}dx+K\hat {S}\int _\Omega \frac {|\nabla S||\nabla I|}{S}dx\\[4pt] &&-\frac {\beta \hat {S}}{\mathcal{R}_0}\int _\Omega \frac {[(\mathcal{R}_0-1)S-I]^2}{(S+I)S}dx\\[4pt] &\leq &-\hat {S}\int _\Omega \left (\frac {\sqrt {\eta }|\nabla S|}{S}-\frac {K|\nabla I|}{2\sqrt {\eta }}\right )^2dx-\hat {S}\int _\Omega \left ((\mathcal{R}_0-1)g-\frac {I^2K^2}{4\eta }\right )\frac {|\nabla I|^2}{I^2}dx\\[4pt] &&-\frac {\beta \hat {S}}{\mathcal{R}_0}\int _\Omega \frac {[(\mathcal{R}_0-1)S-I]^2}{(S+I)S}dx,\ \forall \ t\gt 0. \end{eqnarray*}

Since $\mathcal{R}_0\gt 1+\frac {M_6^2 K^2}{4g \eta }$ , by (4.13), one can find a positive constant $C$ such that

\begin{equation*} \frac {dL_2(t)}{dt}\leq -C\int _\Omega \left [|\nabla S|^2+|\nabla I|^2+((\mathcal{R}_0-1)S-I)^2\right ]dx. \end{equation*}

It then follows from the argument in [Reference Li, Peng and Xiang17, Theorem 3.6(i)] that

\begin{equation*}(S(\cdot ,t),I(\cdot ,t))\rightarrow (\hat {S},\hat {I})\ \textrm {in}\ (L^\infty (\Omega ))^2,\ \textrm {as}\ t\rightarrow \infty .\end{equation*}

This completes the proof.

4.3. Numerical examples

We now present numerical examples to validate the theoretical results of the threshold dynamics. We choose $k=\eta =0.1$ and $S_0=I_0=50$ . Simulation results for (1.5) and (1.6) are displayed in Figure 1 in terms of the basic reproduction number $\mathcal{R}_0$ . In Figure 1(a) and (d), the disease is extinct as $\mathcal{R}_0=\frac {1}{3}\lt 1$ . A similar extinction tendency is also observed in Figure 1(b) and (e) as $\mathcal{R}_0=1$ . The disease is persistent in Figure 1(c) and (f) as $\mathcal{R}_0=3\gt 1$ .

Figure 1.

The threshold dynamics are given by (a)–(c) for (1.5) and (d)–(f) for (1.6). (a) and (d): $\beta =0.1$ and $\gamma =0.3$ ; (b) and (e): $\beta =\gamma =0.1$ ; (c) and (f): $\beta =0.3$ and $\gamma =0.1$ .

5.1. Linear dependence

In this subsection, we study the epidemic dynamics in the cases where the dispersal rates are in the linear form, and the infection and recovery rates are taken as Figure 2(a). The simulation for semilinear systems (1.3) and (1.4) is displayed in Figure 3(a) and (d). For Fickian law of diffusion, system (1.5) becomes:

(5.1) \begin{equation} \left \{\begin{aligned} &\partial _t S-\nabla \cdot \left (\left (\frac {k\beta SI}{S+I}+\eta \right )\nabla S \right )=-\frac {\beta SI}{S+I}+\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\partial _t I-\nabla \cdot \left (\left (\frac {k}{\gamma }+\eta \right )\nabla I\right )=\frac {\beta SI}{S+I}-\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\displaystyle \nabla S\cdot \mathbf{n}=\nabla I\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0, \end{aligned}\right . \end{equation}

and the steady state of (5.1) is shown in Figure 3(b). For the Fokker–Planck-type law of diffusion, system (1.6) can be written as:

(5.2) \begin{equation} \left \{\begin{aligned} &\partial _t S-\nabla \cdot \left [\left (\frac {k\beta SI^2}{(S+I)^2}+\frac {k\beta SI}{S+I}+\eta \right )\nabla S +\frac {k\beta S^3}{(S+I)^2}\nabla I\right ]\\[2pt] &=\left (\frac {k S^2I+2kSI^2}{(S+I)^2}\nabla S+\frac {k S^3}{(S+I)^2}\nabla I\right )\cdot \nabla \beta +\frac {k S^2 I}{S+I}\Delta \beta \\[2pt] &-\frac {\beta SI}{S+I}+\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\[2pt] &\partial _t I-\nabla \cdot \left (\left (\frac {k}{\gamma }+\eta \right )\nabla I\right )\\[2pt] &=\frac {2kI|\nabla \gamma |^2}{\gamma ^3}-\frac {kI\Delta \gamma }{\gamma ^2}-\frac {k\nabla \gamma \cdot \nabla I}{\gamma ^2}+\frac {\beta SI}{S+I}-\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\[2pt] &\displaystyle \left [\left (\frac {k\beta SI^2}{(S+I)^2}+\frac {k\beta SI}{S+I}+\eta \right )\nabla S +\frac {k\beta S^3}{(S+I)^2}\nabla I+\frac {kS^2I\nabla \beta }{S+I}\right ]\cdot \mathbf{n}\\[2pt] &=\left [-\frac {kI\nabla \gamma }{\gamma ^2}+\left (\frac {k}{\gamma }+\eta \right )\nabla I\right ]\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0, \end{aligned}\right . \end{equation}

and the steady state of (5.2) is shown in Figure 3(e).

If the diffusion rate of infected individuals is $g(\gamma , I)=k/\gamma I+\eta$ , the $I$ -equation in (5.1) will be replaced by

(5.3) \begin{equation} \partial _t I-\nabla \cdot \left (\left (\frac {k}{\gamma I}+\eta \right )\nabla I\right )=\frac {\beta SI}{S+I}-\gamma I,\ x\in \Omega ,\ t\gt 0 \end{equation}

with the same boundary condition in (5.1). The $I$ -equation in (5.2) will be replaced by

\begin{equation*}\partial _t I-\eta \Delta I\\ =2k\gamma ^{-3}|\nabla I|^2-k\gamma ^{-2}\Delta \gamma +\frac {\beta SI}{S+I}-\gamma I,\ x\in \Omega ,\ t\gt 0\end{equation*}

with the boundary condition

\begin{equation*}({-}k\gamma ^{-2}\nabla \gamma +\eta \nabla I)\cdot \mathbf{n}=0.\end{equation*}

There is the singularity in the diffusion rate of (5.3), since $I$ can be zero at the steady state. The corresponding steady states are shown in Figure 3(c) and (f), respectively.

Figure 3.

The distribution of steady states for $k=1$ , $\eta =0.0001$ , $\beta =6\cos x+6.6$ , $\gamma =\cos x+1.5$ .

In this case, the infection and recovery rate peaks at the same position (see Figure 2(a)). It can be observed from Figure 3(b) and (c) that Fickian law contributes to the spatial homogenization of the distribution of susceptible individuals significantly in the quasilinear systems (5.1) and (5.3), while the Fokker–Planck type never gives rise to this phenomenon. The distribution in Figure 3 for semilinear systems is similar to that for quasilinear systems.

By observing Table 1, we can find that the Fokker–Planck-type law of diffusion helps the spatial segregation of susceptible and infected individuals since the values of $\chi$ are negative for this type. As a consequence, the final infection fraction is relatively small compared with the Fickian law. There is also a segregation phenomenon for the Fickian law if the diffusion rate of infected individuals depends on new recovery cases. Though the segregation occurs, the infection fraction is high, which can be attributed to the high homogenization of susceptible individuals.

Table 1.

Segregation indices and infection fraction of $k=1$ , $\eta =0.0001$ , $\beta =6\cos x+6.6$ , $\gamma =\cos x+1.5$

We now consider the case where the infection rate is the lowest at $x=\pi$ , while the recovery rate is the lowest at $x=\frac {3\pi }{2}$ (see Figure 2(b)). The corresponding steady states are displayed in Figure 4. It can be seen from Table 2 that only the value of $\chi$ for model (5.1) is positive, implying that only this model performs no segregation. For the negative $\chi$ , the values of $\kappa$ for the Fokker–Planck type are bigger than those for the Fickian type, leading to better spatial segregation and a smaller infection fraction for the Fokker–Planck type. The segregation is perfect for the case $g=k/(\gamma I)+\eta$ of the Fokker–Planck type due to $\kappa =0.9626$ being close to one. As a consequence, the infected fraction is close to zero in such a case.

Table 2.

Segregation indices and infection fraction of $k=1$ , $\eta =0.0001$ , $\beta =6\cos x+6.6$ , $\gamma =\sin x+1.5$

Figure 4.

The distribution of steady states for $k=1$ , $\eta =0.0001$ , $\beta =6\cos x+6.6$ , $\gamma =\sin x+1.5$ .

Figure 5.

The distribution of steady states for $k=1$ , $\eta =0.0001$ , $\beta =5e^{-160(x-0.2)^2}+4$ , $\gamma =5e^{-160(x-0.8)^2}+2$ .

The steady-state distributions among different models exhibit significant differences. Susceptible individuals show homogeneous spatial distribution in the Fickian type, especially in quasilinear systems, while displaying heterogeneous distribution in the Fokker–Planck type. In Figure 4(a)(b)(d)(e), the density of infected individuals is not the highest at the position $x=\frac {3\pi }{2}$ where the recovery rate is the lowest. However, in Figure 4(c) and (f), the density of infected individuals is the lowest around this position, suggesting that new recovery cases are more helpful to individual movement. When the diffusion rate of infected individuals is $g=k/(\gamma I)+\eta$ , a disease-free region emerges around $x=\pi$ for both types of diffusion laws. Interestingly, there is a notable aggregation of susceptible individuals in the disease-free region for the Fokker–Planck type. This corresponds to the tendency of susceptible individuals to move towards disease-free regions to avoid infection.

To describe the infection and recovery rates more realistic, we choose

\begin{equation*}\beta =5e^{-160(x-0.2)^2}+4,\ \gamma =5e^{-160(x-0.8)^2}+2,\ x\in (0,1),\end{equation*}

which are shown in Figure 2(c). The infection rate peaks at $x=0.2$ , corresponding to the indoor playground with the high disease infection rate, while the recovery rate reaches its maximum at $x=0.8$ , denoting the hospital with enough medical resources. The distribution of susceptible individuals remains spatially homogeneous, while infected individuals exhibit sensitivity to the infection and recovery rates in Fickian law of diffusion (see Figure 5(a), (b) and (c)). The density of infectives is the highest at $x=0.2$ and the lowest at $x=0.8$ for the Fickian type. However, in the Fokker–Planck type, the density of infectives is the highest at $x=0.8$ (see Figure 5(d) and (e)) when $g$ depends on the recovery rate, while it is the lowest at this position when $g$ depends on new recovery cases (see Figure 5(f)). Consequently, optimal individual movement is observed in Figure 5(f), where a disease-free region occurs around $x=0.8$ .

The spatial separation between the locations with the highest infection and recovery rates leads to the spatial segregation of susceptible and infected individuals in this case, as indicated by the consistently negative values of $\chi$ in Table 3. The infection fraction in the Fokker–Planck type is smaller than that in the Fickian type.

Table 3.

Segregation indices and infection fraction of $k=1$ , $\eta =0.0001$ , $\beta =5e^{-160(x-0.2)^2}+4$ , $\gamma =5e^{-160(x-0.8)^2}+2$

5.2. Exponential and algebraic dependencies

In this subsection, we consider the cases where the diffusion rates are in exponential and algebraic forms. For Fickian law of diffusion with exponential-dependent dispersal rate, model (1.5) becomes

(5.4) \begin{equation} \left \{\begin{aligned} &\partial _t S-\nabla \cdot \left (\eta e^{\frac {k\beta SI}{S+I}}\nabla S \right )=-\frac {\beta SI}{S+I}+\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\partial _t I-\nabla \cdot \left (\eta e^{-k\gamma }\nabla I\right )=\frac {\beta SI}{S+I}-\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\displaystyle \nabla S\cdot \mathbf{n}=\nabla I\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0. \end{aligned}\right . \end{equation}

For the Fokker–Planck type law of diffusion, model (1.6) can be written as

(5.5) \begin{equation} \left \{\begin{aligned} &\partial _t S-\nabla \cdot \left [\eta e^{\frac {k\beta SI}{S+I}}\left (\left (\frac {k\beta SI^2}{(S+I)^2}+1\right )\nabla S+\frac {k\beta S^3}{(S+I)^2}\nabla I\right )\right ]\\[3pt] &=\eta e^{\frac {k\beta SI}{S+I}}\frac {kS^2I}{S+I}\left (\frac {kSI}{S+I}\nabla \beta +\frac {k\beta I^2}{(S+I)^2}\nabla S+\frac {k\beta S^2}{(S+I)^2}\nabla I\right )\cdot \nabla \beta \\[3pt] &+\eta e^{\frac {k\beta SI}{S+I}}\left (\frac {kS^2I+2kSI^2}{(S+I)^2}\nabla S+\frac {kS^3}{(S+I)^2}\nabla I\right )\cdot \nabla \beta \\[3pt] &+\eta e^{\frac {k\beta SI}{S+I}}\frac {kS^2I\Delta \beta }{S+I}-\frac {\beta SI}{S+I}+\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\[3pt] &\partial _t I-\nabla \cdot \left (\eta e^{-k\gamma }\nabla I\right )=\eta k^2e^{-k\gamma } I|\nabla \gamma |^2\\[3pt] &-\eta ke^{-k\gamma }I\Delta \gamma -\eta ke^{-k\gamma }\nabla I\cdot \nabla \gamma +\frac {\beta SI}{S+I}-\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\[3pt] &\displaystyle \left [\eta e^{\frac {k\beta SI}{S+I}}\left (\left (\frac {k\beta SI^2}{(S+I)^2}+1\right )\nabla S+\frac {k\beta S^3}{(S+I)^2}\nabla I+\frac {kS^2I}{S+I}\nabla \beta \right )\right ]\cdot \mathbf{n}\\[3pt] &=[-\eta ke^{-k\gamma }I\nabla \gamma +\eta e^{-k\gamma }\nabla I]\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0. \end{aligned}\right . \end{equation}

If the diffusion rate of infected individuals is $g(\gamma , I)$ , the $I$ -equation in (5.4) will be replaced by

\begin{equation*}\partial _t I-\nabla \cdot \left (\eta e^{-k\gamma I}\nabla I\right )=\frac {\beta SI}{S+I}-\gamma I,\ x\in \Omega ,\ t\gt 0\end{equation*}

with the same boundary condition in (5.4). The $I$ -equation in (5.5) will be replaced by

\begin{eqnarray*} & &\partial _t I-\nabla \cdot \left (\eta e^{-k\gamma I}(1-k\gamma I)\nabla I\right )=\eta e^{-k\gamma I}(k^2I^3\nabla \gamma +k^2\gamma I^2\nabla I)\nabla \gamma -2\eta k e^{-k\gamma I} I\nabla \gamma \cdot \nabla I\\ &&-\eta k e^{-k\gamma I}I^2\Delta \gamma +\frac {\beta SI}{S+I}-\gamma I,\ x\in \Omega ,\ t\gt 0 \end{eqnarray*}

with the boundary condition

\begin{equation*}\left [-\eta ke^{-k\gamma I}I^2\nabla \gamma +\eta e^{-k\gamma I}(1-k\gamma I)\nabla I\right ]\cdot \mathbf{n}=0.\end{equation*}

When we take algebraically dependent functions for the diffusion rates, the steady states for models (1.3) and (1.4) are illustrated in Figure 6(a)(d). For Fickian law of diffusion, system (1.5) becomes

(5.6) \begin{equation} \left \{\begin{aligned} &\partial _t S-\nabla \cdot \left (\eta \left (1+\frac {\beta SI}{S+I}\right )^k\nabla S \right )=-\frac {\beta SI}{S+I}+\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\partial _t I-\nabla \cdot \left (\eta (1+\gamma )^{-k}\nabla I\right )=\frac {\beta SI}{S+I}-\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\displaystyle \nabla S\cdot \mathbf{n}=\nabla I\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0. \end{aligned}\right . \end{equation}

For the Fokker–Planck law of diffusion, system (1.6) is written as

(5.7) \begin{equation} \left \{\begin{aligned} &\partial _t S-\nabla \cdot \bigg [\left (\eta \left (1+\frac {\beta SI}{S+I}\right )^k+\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\frac {\beta SI^2}{(S+I)^2}\right )\nabla S\\ &+\left (\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\frac {\beta S^3}{(S+I)^2}\right )\nabla I\bigg ]\\ &=k(k-1)\left (1+\frac {\beta SI}{S+I}\right )^{k-2}\frac {\eta S^2I}{S+I}\left (\frac {SI\nabla \beta }{S+I}+\frac {\beta I^2\nabla S}{(S+I)^2}+\frac {\beta S^2\nabla I}{(S+I)^2}\right )\cdot \nabla \beta \\ &+\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\left (\frac {(S^2I+2SI^2)\nabla S}{(S+I)^2}+\frac {S^3\nabla I}{(S+I)^2}\right )\cdot \nabla \beta \\ &+\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\frac {S^2I\Delta \beta }{S+I}-\frac {\beta SI}{S+I}+\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\partial _t I-\nabla \cdot \left (\eta (1+\gamma )^{-k}\nabla I\right )=\eta k(k+1)(1+\gamma )^{-k-2}I|\nabla \gamma |^2\\ &-\eta k(1+\gamma )^{-k-1}I\Delta \gamma -\eta k(1+\gamma )^{-k-1}\nabla I\cdot \nabla \gamma +\frac {\beta SI}{S+I}-\gamma I,\ & x&\in \Omega ,\ t\gt 0,\\ &\displaystyle \bigg [\left (\eta \left (1+\frac {\beta SI}{S+I}\right )^k+\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\frac {\beta SI^2}{(S+I)^2}\right )\nabla S\\ &+\left (\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\frac {\beta S^3}{(S+I)^2}\right )\nabla I\\ &+\eta k\left (1+\frac {\beta SI}{S+I}\right )^{k-1}\frac {S^2I\nabla \beta }{S+I}\bigg ]\cdot \mathbf{n}\\ &=[-\eta k(1+\gamma )^{-k-1}I\nabla \gamma +\eta (1+\gamma )^{-k}\nabla I]\cdot \mathbf{n}=0,\ & x&\in \partial \Omega ,\ t\gt 0. \end{aligned}\right . \end{equation}

The steady states of systems (5.6) and (5.7) are shown in Figure 6(b)–(e). If the diffusion rate of infected individuals is $g(\gamma , I)$ , the $I$ -equation in (5.6) will be replaced by

\begin{equation*}\partial _t I-\nabla \cdot \left (\eta (1+\gamma I)^{-k}\nabla I\right )=\frac {\beta SI}{S+I}-\gamma I,\ x\in \Omega ,\ t\gt 0\end{equation*}

with the same boundary condition in (5.6). The $I$ -equation in (5.7) will be replaced by

\begin{eqnarray*} & &\partial _t I-\nabla \cdot \left [\eta \left ((1+\gamma I)^{-k}-k(1+\gamma I)^{-k-1}\gamma I\right )\nabla I\right ]\\ &=&\eta k(k+1)(1+\gamma I)^{-k-2}(\gamma I^2\nabla I+I^3\nabla \gamma )\nabla \gamma -2\eta k(1+\gamma I)^{-k-1}I\nabla \gamma \cdot \nabla I\\ &&-\eta k(1+\gamma I)^{-k-1}I^2\Delta \gamma +\frac {\beta SI}{S+I}-\gamma I,\ x\in \Omega ,\ t\gt 0 \end{eqnarray*}

with the boundary condition

\begin{equation*}\left [-\eta k(1+\gamma I)^{-k-1}I^2\nabla \gamma +\eta ((1+\gamma I)^{-k}-k(1+\gamma I)^{-k-1}\gamma I)\nabla I\right ]\cdot \mathbf{n}=0.\end{equation*}

The corresponding numerical results are given in Figure 6(c) and (f).

The simulation results for exponentially dependent diffusion closely resemble those for algebraically dependent diffusion. Therefore, we only present the numerical results for the algebraic case, as shown in Figure 6. In the Fickian type, the spatial distribution is similar for both semilinear and quasilinear systems. Conversely, in the Fokker–Planck type, susceptible individuals appear to exhibit heterogeneous distribution. From Table 4, there is no segregation in quasilinear systems with Fickian diffusion, whereas segregation occurs in the Fokker–Planck type. The infection fraction is minimized when $g=\eta (1+\gamma I)^{-k}$ for the Fokker–Planck type.

Figure 6.

The distribution of steady states for $k=0.1$ , $\eta =1$ , $\beta =6\cos x+6.6$ , $\gamma =\cos x+1.5$ .

Table 4.

Segregation indices and infection fraction of $k=0.1$ , $\eta =1$ , $\beta =6\cos x+6.6$ , $\gamma =\cos x+1.5$

Upon comparing the numerical simulations (including these in [Reference Wang, Wang and Kim29]) of the semilinear systems (1.3) and (1.4) with the quasilinear systems (1.5) and (1.6), we observe that susceptible individuals display spatial homogeneity in the quasilinear systems with Fickian law of diffusion, and the disease-free region appears in quasilinear systems with the Fokker–Planck-type law of diffusion.