Analysis of a heterogeneous SEIRS patch model with asymmetric mobility kernel

1.   Introduction

There is mounting evidence in epidemiology that spatial-temporal heterogeneity and human mobility have a substantial impact on the spread of infectious diseases ^[1,2,3,4]. Different population, social contact rates, individual response, control measures, and medical resources (sickbeds and physicians per thousand people) at different regions all contribute to spatial heterogeneity. Many reaction diffusion or patch models have been presented in recent years to examine the role of diffusion and spatial heterogeneity in disease transmission ^{[3,5,6,7,8,9,10,11,12,13,14,15]}. Among these works, Allen et al. ^[5] developed a susceptible-infected-susceptible (SIS) patch system that collaborated with spatial heterogeneity in the following way:

where $\Omega = \{1, 2, 3, \cdots, n\}$ with $n\geq2$ denoting the patch numbers, $\overline{S}_{j}(t)$ and $\overline{I}_{j}(t)$ represent the number of susceptible and infected individuals, respectively, in patch $j$ at any given time $t$. The transmission rate of the disease in patch $j$ is denoted by $\beta_j$, while the recovery rate is denoted by $\gamma_j$. Furthermore, $d_{S}$ and $d_{I}$ are the diffusion rates of the susceptible and infected populations, respectively. Matrix $(L_{jk})_{n\times n}$ is symmetric with $L_{jk}$ representing the degree of movement from patch $k$ into patch $j$.

In ^[5], a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady states. More works related to system (1.1) can be seen in ^[6] (e.g., asymptotic behaviors of the endemic equilibrium ($EE$) as the diffusion rate of the infected individuals ($d_{I}$) approaches to zero) and ^[16,17] (e.g., asymmetric matrix $(L_{jk})_{n\times n}$).

However, these models neglected the mobility of exposed persons and did not incorporate the class of exposed individuals. Exposed individuals, without exhibiting symptoms, can be seen in various epidemic diseases, such as malaria, HIV/AIDS, and SARS-COV-2. The disease can spread spatial if exposed individuals with no symptoms travel, making it more difficult to control ^[18,19]. As a result, it appears critical to incorporate the exposed individuals subclass and examine the effects of exposed individuals' mobility on infection disease spread, which is mathematically linked to the basic reproduction number's dependence on the diffusion rates of exposed individuals. Earlier studies have investigated discrete-space multi-patch models concerning this issue ^[20,21]. The exposed and recovery classes will be extended to the SIS patch model, and the associated SEIRS patch model will be analyzed in this study.

1.1.   SEIRS patch model

We consider a system of differential equations:

where $S$, $E$, $I$, and $R$ represent the number of individuals in the susceptible, exposed, infectious, and recovered compartments, respectively. The parameters $\beta$, $\sigma$, $\gamma$, and $\alpha$ denote the rates of transmission, latent period, recovery, and loss of immunity, respectively.

Here $\Omega = \{1, 2, 3, \cdots, n\}$ with $n\geq2$ denoting the patch numbers, $S_{i}(t)$, $E_{i}(t)$, $I_{i}(t)$ and $R_{i}(t)$ denote the population of susceptible, exposed, infected and recovered individuals at time $t$ and patch $i$, respectively. Furthermore, $d_{S}$, $d_{E}$, $d_{I}$ and $d_{R}$ represent the diffusion coefficients associated with susceptible, exposed, infected and recovered individuals, respectively. We assume that $\beta: = (\beta_{i})_{n \times 1}$ and $\gamma: = (\gamma_{i})_{n \times 1}$ depends on the environment, could be spatial heterogeneous and are assumed to be positive throughout this paper. Given that the latent period $1/\sigma$ which represents the time takes for an infection to become detectable after exposure and the rate of loss of immunity $\alpha$ which describes the proportion of individuals who lose their immunity to a disease over time, are inherent properties of the individual and not influenced by the external environment, it is reasonable to consider $\sigma$ and $\alpha$ as constants in this study.

$\mathbf{L} = (\ell_{ij})$ is an $n \times n$ cooperative, irreducible matrix with $\ell_{ii} = -\sum_{j\neq i}\ell_{ij}$ referred as Laplacian matrix, where $\ell_{ij}$, denoting the degree of movement from patch $k$ into patch $j$, is independent of $t$. Here an $n\times n$ matrix is said to be cooperative if all of its off-diagonal entries are non-negative, and to be irreducible if its index set $\{1, 2, \ldots, n\}$ cannot be split into two complementary sets (without common indices) $\{p_{1}, p_{2}, \ldots, p_{r}\}$ and $\{q_{1}, q_{2}, \ldots, q_{s}\}$ $(r + s = n)$ such that $a_{p_{i}q_{j}} = 0$ for all $1\leq i\leq r, 1\leq j\leq s$. An $n\times n$ matrix is called quasi-positive (also called Metzler matrix) if $l_{ij}\geq 0$ for all $i\neq j$. Here we mention that $\mathbf{L}$ can be asymmetric. Moreover, to match the real human mobility patterns and transportation phenomena such as gravity ^[22,23] or radiation mobility model ^[24], we assume that $\mathbf{L}$ can be asymmetric. We also assume that the following initial conditions are satisfied throughout the study.

(A1) $S_{i}(0), E_{i}(0), I_{i}(0), R_{i}(0)\geq 0$ for $i \in \Omega$ and $\sum_{i = 1}^{n}I_{i}(0) > 0$.

It is easy to verify that $\frac{\beta_{i}S_{i}I_{i}}{S_{i}+I_{i}+E_{i}+R_{i}}$ is a Lipschitz continuous function of $S_{i}$ and $I_{i}$, therefore we define the quantity to be zero in cases where $S_{i} = 0$ or $I_{i} = 0$. According to [34, Chapter I], it can be stated that the system (1.2) has a unique classical solution $S, E, I, R \in C^{1}((0, \infty); \mathbb{R}^{n})$, and $S_{i}(t), E_{i}(t), I_{i}(t)$, $R_{i}(t) > 0$ for $i\in\Omega$ and $t > 0$.

The total population at time $t$ is defined as

and let the total number of individuals is a fixed positive constant at the initial time $t = 0$, denoted by $N_{0}$. By the system (1.2), we conclude that the total population size is constant, i.e.,

1.2.   Equilibrium problems

Let $u, v\in \mathbb{R}^{n}$. We write $u\geq v$ if $u_{i}\geq v_{i}$ for any $i \in \Omega$; $u > v$ if $u_{i}\geq v_{i}$ for any $i \in \Omega$, and there exists $j$ such that $u_{j} > v_{j}$; $u\gg v$ if $u_{i} > v_{i}$ for any $i\in\Omega$. We say $u$ is non-negative, positive and strongly positive if $u\geq 0, u > 0, u\gg 0$, respectively.

Our primary focus will be on equilibrium solutions of the system (1.2), i.e., non-negative solutions of the following system:

where the population of susceptible, exposed, infected and recovered individuals at equilibrium are represented by the variables $\widetilde{S}$, $\widetilde{E}$, $\widetilde{I}$, and $\widetilde{R}$, respectively. A solution of (1.4) in which $\widetilde{I}_{i} = 0$ for every $i \in \Omega$ is a disease-free equilibrium (DFE), while a solution in which $\widetilde{I}_{i} > 0$ for some $i \in \Omega$ is referred to as an endemic equilibrium (EE). It is straightforward to observe that DFE is unique, given by $E_{0} = (\widetilde{N}, 0, 0, 0)$, where $\widetilde{N}$ is the unique positive eigenvector corresponds to the eigenvalue of $0$ in the following eigenvalue problem:

Moreover, for any EE, $\widetilde{S}, \widetilde{E}, \widetilde{I}, \widetilde{R}\gg 0$.

1.3.   Main results

The purpose of this research is to examine how changes in human mobility and environment heterogeneity affect the persistence or eradication of infectious diseases. Our research will investigate the threshold-type dynamics of system (1.2), the characteristics of the basic production ratio, and when the diffusion rate of the susceptible individuals approaches zero, the asymptotic behaviors of the endemic equilibria of (1.4).

The basic reproduction number is a fundamental concept in the study of infectious disease transmission. It is defined as the expected number of secondary infections generated by a single infected individual in a completely susceptible population ^[25,26]. More importantly, it is frequently used to determine the behaviour of various epidemic models' thresholds. Specifically, when the basic reproduction number is less than unity, the disease is expected to fade out, while a basic reproduction number greater than unity indicates the potential for the disease to establish itself within the population. The next-generation operator method has been widely utilized for the computation of fundamental reproduction number, see ^[25,27,28] and for related investigations, see ^[3,28,29].

By ^[28,29], the basic reproduction number, denoted as $\mathcal{R}_{0}$, can be expressed as $\mathcal{R}_{0} = \frac{1}{\mu_{0}}$ (see Lemma 2). Here, $\mu_{0}$ represents the unique positive eigenvalue associated with a positive eigenvector of the following problem:

In terms of $\mathcal{R}_{0}$, we state the dynamics for system (1.2) as follows:

Theorem 1.1. (i) If $\mathcal{R}_{0}\leq 1$, then DFE is globally asymptotically stable, that is $E(t), I(t), R(t) \rightarrow 0$ as $t\rightarrow \infty,$ and $S \rightarrow \widetilde{N}$ as $t\rightarrow \infty;$

(ii) If $\mathcal{R}_{0} > 1$, then there exists a positive constant $\epsilon_{0}$ such that any positive solution of (1.2) satisfies

Furthermore, system described by (1.2) has at least one endemic equilibrium.

It is not hard to see that the value of the basic reproduction number gives an indication on the persistence or extinction of infectious diseases. Therefore, we focus on the dependence of $\mathcal{R}_{0}$ on $d_{E}, d_{I}$, $d_{S}$ and $d_{R}$. It is easy to know that $\mathcal{R}_{0}$ is independent of $d_{S}$ and $d_{R}$. The dependence of $\mathcal{R}_{0}$ on $d_{E}, d_{I}$ can have important implications for disease control because it explores how changes in human mobility affect the persistence or extinction of infectious diseases. ^[16] demonstrates that $\mathcal{R}_{0}$ for model (1.1) is decreasing in $d_{I}$. However, for the $SEIRS$ system (1.2), the monotonicity of $\mathcal{R}_{0}$ with respect to $d_{I}$ is more subtle due to the presence of mobility of exposed individuals.

To start with, we give the asymptotic properties of $\mathcal{R}_{0}$ when $d_{E}, d_{I}$ tend to $0$ or infinity in Theorem 1.2. Define $\beta = (\beta_{i})_{n\times1}, \gamma = (\gamma_{i})_{n\times1}$ and set

Moreover, $\xi$ is defined as the positive eigenvector of $\mathbf{L}$ that is unique, satisfying $\sum_{i\in\Omega} \xi_{i} = 1$.

Theorem 1.2. (i) Fix $d_{I} > 0$. Then $\mathcal{R}_{0}\rightarrow \rho((-d_{I}\mathbf{L}+Y)^{-1}B)$ as $d_{E} \rightarrow 0$, and $\mathcal{R}_{0}\rightarrow \frac{\|B(-d_{I}L+Y)^{-1}\xi\|_{1}}{\|\xi\|_{1}}$ as $d_{E} \rightarrow \infty$.

$\rm(ii)$ Fix $d_{E} > 0$. Then $\mathcal{R}_{0}\rightarrow \rho(\sigma(-d_{E}\mathbf{L}+\sigma I_{n\times n})^{-1}Z)$ as $d_{I} \rightarrow 0$ and $\mathcal{R}_{0}\rightarrow \frac{\sum_{i = 1}^{n}\xi_{i}\beta_{i}}{\sum_{i = 1}^{n}\xi_{i}\gamma_{i}}$ as $d_{I} \rightarrow \infty$.

$\rm(iii)$ As $d_{E}, d_{I} \rightarrow 0$, then $\mathcal{R}_{0}\rightarrow \max\left\{\frac{\beta_{i}}{\gamma_{i}}, i\in\Omega\right\}$;

$\rm(iv)$ As $d_{E} \rightarrow \infty$ and $d_{I} \rightarrow 0$, then $\mathcal{R}_{0}\rightarrow \sum_{i = 1}^{n}\frac{\xi_{i}\beta_{i}}{\gamma_{i}}$.

The following result is concerned with the monotonicity of $\mathcal{R}_{0}$ with respect to $d_{E}, d_{I}$:

Theorem 1.3. If $\mathbf{L}$ is symmetric and either $\beta$ or $\gamma$ is a multiple of vector $\mathbf{1}$, then $\mathcal{R}_{0}$ exhibits a monotone decreasing behaviour with respect to the parameters $d_{E}$ and $d_{I}$. Furthermore, strict monotonicity is observed if and only if one of them is not multiple of vector $\mathbf{1}$.

If both $\beta$ and $\gamma$ are not multiples of vector $\mathbf{1}$, Theorem 1.3 may be not valid. Refer to Theorem 3.3.

Theorem 1.4. Let $\ell_{ij} = 1, i\neq j, \ell_{ii} = -(n-1)$ and

holds for any $i, j \in \Omega$. Then $\mathcal{R}_{0}$ is nonincreasing in $d_{E}, d_{I}$.

Under the conditions in Theorems 1.3 and 1.4, the nonincreasing behaviour of $\mathcal{R}_{0}$ with respect to $d_{E}$ and $d_{I}$ seems counter-intuitive. If no traffic control or transportation restrictions policies are adopted during the disease spread, large diffusion rate also means accessing more health resources ($\beta$ is a multiple of vector $\mathbf{1}$; condition (1.8)) or escaping from high risk areas ($\gamma$ is a multiple of vector $\mathbf{1}$; condition (1.8)).

Finally, to better understand the influence of the suspected individuals movement on the spread of disease, we demonstrate that the asymptotic behavior of the endemic equilibrium as the diffusion rate of susceptible individuals approaches zero. By Theorem 1.1, when $\mathcal{R}_{0} > 1$, there exist at least one endemic equilibrium. To this end, let's examine this linear eigenvalue problem

And here represent the minimum eigenvalue of (1.9) as $\Lambda_{1}\left(-d_{R}\mathbf{L}+\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right)$ ([33, Theorems 1 and 2]).

Theorem 1.5. Assume $\mathcal{R}_{0} > 1$ and $\Lambda_{1}\left(-d_{R}\mathbf{L}+\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right) < 0$. Then

${\rm(i)}$ There exist positive constants $C_{1}, C_{2},$ independent of $d_{S}$, such that for sufficiently small $d_{S}$,

${\rm(ii)}$ As $d_{S}$ approaches zero, with respect to a given sequence,

where $M^{*} \in \mathbb{R}^{n}$ satisfying $0\leq M^{*} \leq \xi$ and $|\{i\in\Omega: M^{*} _{i} = \xi_{i}\}|\in (0, n).$

In particular, Theorem 1.5 demonstrates that as $d_S$ approaches zero, the variables $\widetilde{E}$, $\widetilde{I}$, and $\widetilde{R}$ converge uniformly to zero in $\Omega$. Additionally, $\widetilde{S}^{*}$ is positive and there exists a non-empty subset $\{i\in\Omega: \widetilde{S}^{*}_{i} = 0\}$. From a biological standpoint, limiting the diffusion rate of susceptible individuals can effectively reduce the number of exposed and infected individuals in a population. We point out that if $\Lambda_{1}\left(d_{R}\mathbf{L}-\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right) > 0$, Theorem 1.5 may be not valid.

Most of the results of SEIRS patch model (1.2) match the contents of SEIRS reaction diffusion model in ^[19], which deals with the continuous-space case to the discrete case. Compared with ^[19], the main improvement of this paper is as follows: investigation of threshold dynamics and asymptotic behaviours of $\mathcal{R}_{0}$ with respect to $d_{E}$ and $d_{I}$ are extended to asymmetric mobility kernel such as gravity ^[22,23] or radiation mobility model ^[24]. In [19, Theorem 1.2], under the condition that $\gamma$ is a constant function, the proof of the monotonicity of $\mathcal{R}_{0}$ with respect to $d_{E}, d_{I}$ is complicated and difficult for readers to access the core ideas. For SEIRS patch model in this paper, the proof is shortened (see Theorem 3.2) and can give more intuitiveness to understand the proof of [19, Theorem 1.2]. Moreover, though the results in Theorem 1.4 is similar in spirit to [19, Theorem 1.3], the proof for patch model is novel and totally different from continuous-space case.

This paper is organized as follows. In Section 2, we establish the well-posedness of the system (1.2), introduce the basic reproduction number $\mathcal{R}_{0}$, and analyze the system's dynamics in terms of $\mathcal{R}_{0}$. Section 3 is dedicated to exploring the asymptotic stability and directional monotonicity of $\mathcal{R}_{0}$ in relation to $d_{E}$ and $d_{I}$. Finally, Section 4 focuses on the behavior of the endemic equilibrium as $d_{S}$ approaches zero.

2.   The stability of disease free equilibrium

To start with, we have the following uniform bound result, which can be directly derived from (1.3).

Lemma 2.1. For some positive constants $C_1$, which are independent of initial values, and $T > 0$, the solution $(S, E, I, R) \in C^1((0, \infty); \mathbb{R}^{4n})$ of the system (1.2) satisfies

We now make use of the theory developed in ^[28] to derive $\mathcal{R}_{0}$ of system (1.2).

Lemma 2.2. (i). $\mathcal{R}_{0} = \rho(FV^{-1}) = \rho(\sigma B (-d_{E}\mathbf{L}+\sigma I_{n\times n})^{-1}(-d_{I}\mathbf{L}+Y)^{-1})$.

(ii). The eigenvalue problem (1.6) has a positive eigenvalue that is unique, denoted by $\mu_{0}$, along with a corresponding positive eigenvector. Additionally, the basic reproduction number of system (1.2), denoted by $\mathcal{R}_{0}$, satisfies

Proof. (i) The linearization of system (1.2) at $E_{0}$ is given by

Note that $E$ and $I$ represent the infected population in (1.2). Furthermore, here $F$ and $V$ in ^[28] is denoted as

where $L = \mathrm{diag}\left\{{-d_{E}\mathbf{L}, -d_{I}\mathbf{L}} \right\}$. Part (i) follow immediately from ^[28]. We only need to prove part (ii). By Perron-Frobenius theorem [33, Theorems 1 and 2], it is possible to establish the existence and uniqueness of a positive eigenvalue $\mu_{0}$ with respect to a positive eigenvector $\varphi$ within the system described in Eq (1.6), i.e.,

Note that there exists a positive eigenvector $\phi$ such that

which implies

By the uniqueness of $\mu_{0}$, we obtain $\mathcal{R}_{0} = \frac{1}{\mu_{0}}$. This completes the proof.

We now demonstrate that the stability of $E_0$ can be determined by the basic reproduction number $\mathcal{R}_{0}$.

Lemma 2.3. The disease-free equilibrium $E_{0}$ is locally asymptotically stable if $\mathcal{R}_{0} < 1$, unstable if $\mathcal{R}_{0} > 1$.

Proof. The proof of Lemma 2.3 can follow from ^[28]. For readers' convenience, we also give some details here. To prove the locally asymptotically stability if $\mathcal{R}_{0} < 1$, we only need to prove $\inf\{\mathrm{Re} \lambda, \ \lambda \in \Lambda\} > 0$, where $\Lambda$ is the spectrum of the following eigenvalue problem

Note that $\Lambda = \Lambda_{\{(\phi_{E}, \phi_{I})\neq0\}}\cup\Lambda_{\{(\phi_{E}, \phi_{I}) = 0\}}.$ Obviously, $\inf\{\mathrm{Re} \lambda, \lambda \in \Lambda_{\{(\phi_{E}, \phi_{I}) = 0\}}\} > 0.$ Moreover, it can be shown that $\Lambda_{{(\phi_{E}, \phi_{I})\neq0}}$ is a subset of the set of $\sigma(L-F+V)$, where $L, F, V$ are defined in (2.4) and $\sigma(L-F+V)$ denotes the eigenvalues of the matrix $L-F+V$. The infimum of the real parts of the eigenvalues of $L-F+V$ is positive, here denoted by $\lambda_{1}$, i.e., $\inf\{\mathrm{Re} \lambda, \lambda \in \sigma(L-F+V)\} = \lambda_{1} > 0$. Moreover, $\lambda_{1}$ represents the principal eigenvalue of

Consider the left eigenvector corresponding to principal eigenvalue of (2.6), i.e.,

Multiply (1.6)'s first equation by $(\phi_{E}^{*})^{T}$ and (2.7)'s first equation by $\varphi_{E}^{T}$. Subtracting the resulting equations yields

Moreover, multiply (1.6)'s second equation by $(\phi_{I}^{*})^{T}$ and (2.7)'s second equation by $\varphi_{I}^{T}$. Subtracting the resulting equations yields

Adding two Eqs (2.8) and (2.9) yields

Then we have $\mathrm{sign}(1-\mathcal{R}_{0}) = \mathrm{sign}(\lambda_{1})$ for the positive of $\varphi_{E}, \phi_{E}^{*}, \varphi_{I}, \phi_{I}^{*}, \beta$. We obtain $\inf\{\mathrm{Re} \lambda, \lambda \in \Lambda_{\{(\phi_{E}, \phi_{I})\neq 0\}}\} > 0.$ Therefore, if $\mathcal{R}_{0} < 1$, $E_{0}$ is locally asymptotically stable.

To establish the linear instability of $E_{0}$ when $\mathcal{R}_{0} > 1$, it is necessary to demonstrate the existence of a non-trivial solution to (2.5) with $\mathrm{Re} \lambda < 0$. To this end, let $\lambda = \lambda_{1} < 0$, where $\lambda_{1}$ denotes the principal eigenvalue of (2.6) and select $(\phi_{E}, \phi_{I})$ as the eigenvector of (2.6) associated with $\lambda_{1}$ and solving for $\phi_{S}$ and $\phi_{R}$ in (2.5). Therefore, $E_0$ is unstable if $\mathcal{R}_{0} > 1$.

We now demonstrate that the disease will be extinct if $\mathcal{R}_{0} < 1$, i.e., the $E_0$ is globally asymptotically stable, and we show that when $\mathcal{R}_{0} > 1$ at least one EE exists and that both $\widetilde{S}$ and $\widetilde{E}$ are positive.

Proof of Theorem 1.1. (i). In order to analyze the behavior of the system (1.2), we make use of LaSalle's invariance principle (Theorem 1 in ^[30]) and construct Lyapunov function. We consider the ordered space $X = \mathbb{R}^{4n}$ equipped with the supremum norm, and observe that $X$ has a nonempty interior, which we denote by $int(P)$, where the cone $P$ is composed of all functions in $X$ that are nonnegative. Set

and $U = P\cap X_{0}.$ It can be shown that system (1.2) and (1.3) defines a dynamic system on $U$. Moreover, for any initial condition $(s_{0}, e_{0}, i_{0}, r_{0}) \in U$, the unique solution of the system is denoted by $\Phi_{t}(s_{0}, e_{0}, i_{0}, r_{0}) = (S, E, I, R)$ for any $t > 0$. It's worth noting that $\Phi_{t}$ is compact, and for each $u_{0} \in U$, the orbit of $u_{0}$ under the dynamical system generated by (1.2) has compact closure in $U$.

Define

for $u\in U,$ where $(\phi_{E}^{*}, \phi_{I}^{*})$ is the eigenvector corresponding to the principal eigenvalue $\lambda_{1}$ of (2.7), i.e., left eigenvector of (2.6). Next, we show that $L(u)$ serves as a Lyapunov function for (1.2). Considering an arbitrary solution of the system (1.2) coupled with (1.3), we obtain:

Following the same process as in the proof of Lemma 2.3, $\mathcal{R}_{0}\leq 1$ yields that $\lambda_{1} \geq 0$. Furthermore, $S, E, I, R \geq 0$, and $\beta, \phi_{E}^{*}, \phi_{I}^{*} > 0$. Consequently, we can assert that $\frac{d}{dt}{L(u(t))}\leq 0$, which indicates that $L(u)$ functions as a Lyapunov function for the system (1.2).

Next define

where $u$ is the unique solution of (1.2) with $u_{0} = (s_{0}, e_{0}, i_{0}, r_{0})\in U$ as initial condition. By (2.10), we have $M = \{u_{0} = (s_{0}, e_{0}, i_{0}, r_{0})\in U| i_{0} = 0\}$ if $\lambda_{1} = 0$, and $M = \{u_{0} = (s_{0}, e_{0}, i_{0}, r_{0})\in U| e_{0} = i_{0} = 0\}$ if $\lambda_{1} > 0$. Using (1.2), we can infer that when $\lambda_{1}\geq0$, the maximal invariant set in $M$ can be characterized by:

Consequently, we can utilize the LaSalle invariant principle (stated in Theorem 1 in ^[30]) to obtain:

By combining the above equation with (1.2), we can infer that $R(t) \rightarrow 0$ as $t\rightarrow \infty.$ Consequently, utilizing (1.3), we obtain $\sum_{i = 1}^{n}S(t) \rightarrow \widetilde{N}$ as $t\rightarrow \infty$, where $\widetilde{N}$ is defined in (1.5).

We omit the proof of Theorem 1.1 (ii) because it is standard.

3.   The properties of $\mathcal{R}_{0}$

In terms of the basic reproduction number $\mathcal{R}_{0}$, we established the threshold dynamics of system (1.2) in the previous section. In this part, we will examine the asymptotic characteristics and monotonicity of the basic reproduction number in relation to the parameters $d_{E}$ and $d_{I}$. The aim is to explore the influence of population movement on the long-term behavior of infectious diseases, particularly persistence or eradication. In this section, we will establish the proofs of Theorems 1.2 and 1.4. Theorem 1.3 is a consequence of Theorems 3.1 and 3.2.

3.1.   Preliminary results

For further conveniences, we give some preliminary results here. To start with, we give some definitions. Let $u, v\in \mathbb{R}^{n}$. We define the relation between $u$ and $v$ as follows: $u\geq v$ if $u_{i}\geq v_{i}$ for $\forall i \in \Omega$; $u > v$ if $u_{i}\geq v_{i}$ for $\forall i \in \Omega$, and $\exists j$ such that $u_{j} > v_{j}$; $u\gg v$ if $u_{i} > v_{i}$ for $\forall i\in\Omega$. We say that $u$ is non-negative, positive and strongly positive if $u\geq 0, u > 0, u\gg 0$, respectively. Moreover, we define the vector norm

Consider an $n \times n$ matrix, denoted as $A$, with the set of eigenvalues $\sigma(A)$. Let $r(A)$ to be the spectral radius of $A$, i.e.,

Denote $s(A)$ be the spectral bound of $A$, i.e.,

Recall that a principal eigenvalue of the matrix $A$ means a real and simple eigenvalue with positive eigenvectors.

Now we have the following preliminary results.

Lemma 3.1. $-\mathbf{L}$ admits a unique principal eigenvalue $0$ with a unique positive eigenvector $\xi$ satisfying $\sum_{i\in\Omega} \xi_{i} = 1$. Moreover, for any vector $\varphi$, we have $-\varphi^{T} \mathbf{L} \varphi\geq 0.$

Throughout this paper, we define $\xi$ to be the positive eigenvector that is unique satisfying $\sum_{i\in\Omega} \xi_{i} = 1$.

Lemma 3.2 (^[17]). Consider an $n\times n$ quasi-positive irreducible matrix, denoted as $A = (a_{ij})_{n\times n}$. Let $Q$ be a diagonal matrix, denoted as $Q = \mathrm{diag}\left\{{q_i} \right\}$. Then, we have the subsequent results:

(i). If $s(A) < 0$, then $s(\mu A + Q)$ is strictly decreasing in $\mu\in\mathbb{R}^{+}$. Furthermore,

and

(ii). If $s(A) = 0$, then $s(\mu A + Q)$ is strictly decreasing provided that $Q$ is not a multiple of $I_{n\times n}$. Furthermore,

and

where $\xi$ refers to the positive eigenvector of $s(A)$ that is unique and satisfies the condition $\sum_{i\in\Omega} \xi_{i} = 1$. (It is worth noting that if every row of matrix $A$ has a sum of zero, then $\xi$ denotes the left positive eigenvector of $A$.)

3.2.   Asymptotic properties of $\mathcal{R}_{0}$ associated with $d_{E}, d_{I}$

Lemma 3.3. For any $d_{E} > 0, d_{I} > 0$,

Proof. According to Lemma 2.2, the reciprocal of the basic reproduction number, denoted as $\frac{1}{\mathcal{R}_{0}}$, represents principal eigenvalue that is unique of the eigenvalue problem expressed in Eq (1.6). Therefore, it follows that:

It follows from the addition of two equations of (3.2) that

which yields

Since $\gamma_{i}$ and $\varphi_{I}$ are positive, we obtain (3.1).

Lemma 3.3 showed an estimate of $\mathcal{R}_{0}$ and implied that if $\frac{\beta_{i}}{\gamma_{i}}\equiv \mathrm{Constant}$, then $\mathcal{R}_{0}$ is constant, which is independent of $d_{E}, d_{I}$.

Proof of Theorem 1.2. We only need to prove (i) here because (ii) can be shown through similar arguments. The case $d_{E} \rightarrow 0$ can be derived by the continuity of eigenvalue with respect to $d_{E}$ [32, Proposition 2.1]. Now we explore the case $d_{E} \rightarrow \infty$. By Lemma 3.3, if required, we can pass to a sequence such that $\mathcal{R}_{0} \rightarrow \tilde{R}_{0}$ as $d_{E}\rightarrow \infty$. Without loss of generality, we may assume $\|\varphi_{E}\|+\|\varphi_{I}\| = 1$. Passing to a sequence if necessary, $\varphi_{E}\rightarrow \tilde{\varphi}_{E}, \varphi_{I}\rightarrow \tilde{\varphi}_{I}$ in $\mathbb{R}_{+}^{n}$ as $d_{E} \rightarrow \infty$. As shown in Lemma 3.2, $\tilde{\varphi}_{E}$ is a multiple of $\xi$ and $\tilde{\varphi}_{I}$ is a solution of a certain equation

Summarizing the first equation of (3.2) yields that $\mathcal{R}_{0}\rightarrow \frac{\|B(-d_{I}L+Y)^{-1}\xi\|_{1}}{\|\xi\|_{1}}$. This ends the proof of (i).

(iii) The statement can be derived by the continuity of eigenvalue with respect to $d_{E}$ [32, Proposition 2.1].

(iv) The statement can be directly deduced from Lemma 3.2 and statement (ii).

3.3.   Monotonicity of $\mathcal{R}_{0}$ with respect to $d_{E}, d_{I}$

Now, we show some cases that $\mathcal{R}_{0}$ has monotonicity with respect to $d_{E}, d_{I}$ results as follows.

Theorem 3.1. If $\beta$ is a multiple of vector $\mathbf{1}$ and $\mathbf{L}$ is symmetric, then $\mathcal{R}_{0}$ is monotone decreasing in $d_{E}, d_{I}$. Moreover, the strict monotonicity holds if $\gamma_{i}$ is not a multiple of vector $\mathbf{1}$.

Proof. Firstly, we demonstrate that $\mathcal{R}_{0}$ is monotone decreasing with respect to $d_{E}$. By [32, Proposition 2.1], we know that $\mathcal{R}_{0}$ and the corresponding eigenvectors $(\varphi_{E}, \varphi_{I}), (\varphi_{E}^{*}, \varphi_{I}^{*})$ are both differentiable functions of $d_{E}, d_{I}$. By direct differentiating both sides of the equations in (3.2) by $d_{E}, d_{I}$, we obtain

and

respectively. Since no confusion occurs in subsequent proofs, here the prime notation is utilized to signify differentiation in relation to either $E$ or $I$, for the purpose of convenience. The initial equation of (3.2) is multiplied by $\varphi_{E}'$ and the initial equation of (3.4) is multiplied by $\varphi_{E}$. The resulting equations are then subtracted to obtain the desired outcome,

Analogously, the second equation of (3.2) is multiplied by $\varphi_{I}'$ and the second equation of (3.4) is multiplied by $\varphi_{I}$. The resulting equations are then subtracted to obtain the desired outcome, (It should be noted that, since $\sigma$ is constant, it can be treated as a constant throughout the procedure)

If $\beta$ is a multiple of vector $\mathbf{1}$, we can see that

By Lemma 3.1, $\varphi_{E}^{T}\mathbf{L}\varphi_{E}\leq 0$. We obtain $\mathcal{R}_{0}'\leq 0$ as $\varphi_{I}, \varphi_{E}$ are positive. In addition, the equality is only possible if $\varphi_{E}$ is a multiple of vector $\mathbf{1}$. This fact together with the first equation of (3.2) leads to the conclusion that $\varphi_{I}$ must be a multiple of vector $\mathbf{1}$. The obtained expression, along with the second equation of (3.2), implies that the parameter $\gamma$ must be a multiple of the vector $\mathbf{1}$. Consequently, it follows that the quantity $\mathcal{R}_{0}$ exhibits a monotone decreasing behavior with respect to the variable $d_{E}$, with strict monotonicity being guaranteed only if the parameter $\gamma$ is not a multiple of vector $\mathbf{1}$.

Next we show the monotonicity of $\mathcal{R}_{0}$ with respect to $d_{I}$. The initial equation of (3.2) is multiplied by $\varphi_{E}'$, and initial equation of (3.5) is multiplied by $\varphi_{E}$. The resulting equations are subtracted to obtain the desired outcome,

Analogously, the second equation of (3.2) is multiplied by $\varphi_{I}'$, and the second equation of (3.5) is multiplied by $\varphi_{I}$. The resulting equations are subtracted to obtain the desired outcome,

If $\beta$ is a multiple of vector $\mathbf{1}$. We obtain

Similar to the previously mentioned arguments, it can be stated that the quantity $\mathcal{R}_{0}$ is exactly monotonically decreasing in behaviour with respect to the variable $d_{I}$ if and only if the parameter $\gamma$ is not a multiple of vector $\mathbf{1}$.

Theorem 3.2. If $\gamma$ is a multiple of vector $\mathbf{1}$ and $\mathbf{L}$ is symmetric, then $\mathcal{R}_{0}$ is monotone decreasing function of $d_{E}, d_{I}$ Furthermore, it is only when the parameter $\beta$ is not a multiple of vector $\mathbf{1}$ that the monotonicity of $\mathcal{R}_{0}$ becomes strictly decreasing.

Proof. Note from Lemma 2.2 that $\mathcal{R}_{0} = \rho(\sigma B (-d_{E}\mathbf{L}+\sigma I_{n\times n})^{-1}(-d_{I}\mathbf{L}+Y)^{-1})$. Let $\gamma = y \mathbf{1}$. Then $\mathcal{R}_{0}$ satisfies

where $\varphi$ is a strongly positive eigenvector.

We only prove that $\mathcal{R}_{0}$ is monotone decreasing function of $d_{E}$ and the proof of monotonocity with respect to $d_{I}$ is similar. Differentiating (3.6) by $d_{E}$ yields that

Multiplying (3.6) by $\varphi^{T}$ and (3.7) by $\varphi^{', T}$, and subtracting the resulting equations yield that

Therefore, $\mathcal{R}_{0}$ is monotone decreasing function of $d_{E}$. Applying analogous arguments to that utilized in the proof of Theorem 3.1, it can be asserted that the quantity $\mathcal{R}_{0}$ is exactly monotonically decreasing behaviour with respect to $d_{E}$ if and only if the parameter $\beta$ is not a multiple of vector $\mathbf{1}$.

Now we prove Theorem 1.4.

Proof of Theorem 1.4. Note that the sign conditions mean that for any $i, j\in\Omega$, $\beta_{i}\geq\beta_{j} (\gamma_{i}\leq\gamma_{j})$ or $\beta_{i}\leq\beta_{j} (\gamma_{i}\geq\gamma_{j})$ holds true. We divide the proof of Theorem 1.4 into two steps.

Step 1. First we show that in (1.6),

holds for any $i, j \in \Omega$. Fix $i, j$, set $\rho_{E} = \varphi_{E, i}-\varphi_{E, j}, \rho_{I} = \varphi_{I, i}-\varphi_{I, j}$, and without loss generality, let $\beta_{j}\geq\beta_{i}, \gamma_{j}\leq\gamma_{i}$. Note from (1.6) that

Subtracting the last and first two equations yields that

Moreover, it follows from (1.6) that

Denote $h_{0} = [(\gamma_{i}-\gamma_{j})\varphi_{I, j}, \mu(\beta_{j}-\beta_{i})\varphi_{I, j}]^{T}, h_{1} = [d_{I}\sum_{j\in \Omega}\varphi_{I, j}, d_{E}\sum_{j\in \Omega}\varphi_{E, j}]^{T}$, $h_{2} = [\varphi_{E, i}, \varphi_{I, i}]^{T}$ and

Thus, (3.10) can be rewritten as $Mh_{2} = h_{1}\gg 0$. By [31, Fact 6.11.13 (xii)], $M$ is M-matrix and $M^{-1}$ is a positive matrix. Here a matrix $A = (a_{ij})_{n\times n}$ is called an M-matrix if $a_{ij} \leq 0$ for all $i\neq j$ and $A = sI-B$ with B having all off-diagonal elements negative and $s \geq r(B)$. Therefore, $[\rho_{E}, \rho_{I}]^{T} = M^{-1}h_{0}\geq 0$, which implies (3.8).

By similar arguments, we can also obtain that

Step 2. Moreover, the reciprocal of the basic reproduction number, denoted as $\frac{1}{\mathcal{R}_{0}}$, represents the principal eigenvalue that is unique of the eigenvalue problem expressed in Eq (1.6). Using the same notations $F, V$ of (2.4), (1.6) can be written as

where $\varphi = (\varphi_{E}, \varphi_{I})$. Thus the adjoint problem of (1.6) can be written as

i.e.,

where $\varphi^{*} = (\varphi_{E}^{*}, \varphi_{I}^{*})$ corresponds to a positive eigenvector. We can derive the following equation by multiplying the first equation of (3.4) by the transpose of $\varphi_{E}^{*}$, denoted by $\varphi_{E}^{*, T}$, and multiplying the first equation of (3.12) by the transpose of the derivative of $\varphi_{E}$, denoted by $(\varphi_{E}')^{T}$, then subtracting the resulting equations:

The second equation of (3.4) is multiplied by $\varphi_{I}^{*}$, and the second equation of (3.12) is multiplied by $\varphi_{I}'$, which yields

As a consequence of (3.13) and (3.14), we have

This combined with (3.15) suggests that $\mathcal{R}_{0}' \leq 0$. By similar arguments as the proof of Theorem 3.1, if and only if both the parameters $\beta$ and $\gamma$ are constant vectors the equality holds.

We then prove that $R_{0}$ decreases monotonically with respect to $d_{I}$. Using similar arguments as before, we demonstrate that,

and the remaining arguments are analogous to the previous ones.

3.4.   Non-monotonicity of $\mathcal{R}_{0}$ with respect to $d_{E}, d_{I}$

In previous subsections, we show that $\mathcal{R}_{0}$ is monotone decreasing in $d_{E}, d_{I}$ in some cases. In this part, we will prove that $\mathcal{R}_{0}$ isn't always monotone decreasing with respect with $d_{E}, d_{I}$.

Theorem 3.3. There exist $d_{E}^{0}$ and $d_{I}^{1} < d_{I}^{2}$ such that $\mathcal{R}_{0}(d_{E}^{0}, d_{I}^{1}) < \mathcal{R}_{0}(d_{E}^{0}, d_{I}^{2}),$ if

Proof. Based on Theorem 1.2, it can be shown that for any fixed value of $d_{E} > 0$, $\mathcal{R}_{0}\rightarrow \frac{\sum_{i = 1}^{n}\xi_{i}\beta_{i}}{\sum_{i = 1}^{n}\xi_{i}\gamma_{i}}$ as $d_{I} \rightarrow \infty$, and $\mathcal{R}_{0}\rightarrow \sum_{i = 1}^{n}\frac{\xi_{i}\beta_{i}}{\gamma_{i}}$ as $d_{I} \rightarrow 0, d_{E} \rightarrow \infty$. For any arbitrarily small positive value of $\epsilon$, there exists a sufficiently large constant $C_{1}(\epsilon)$ such that for any $\frac{1}{d_{I}^{1}}, d_{E}\geq C_{1}(\epsilon)$, we have

Furthermore, there exists $C_{2}(\epsilon, d_{E})$ such that for any $d_{I}^{2}\geq C_{2}(\epsilon, d_{E})$,

Since

we can choose $\epsilon_{0}$ small enough such that

and let $d_{E}^{0} = C_{1}(\epsilon_{0}), d_{I}^{1} = \frac{1}{C_{1}(\epsilon_{0})}, d_{I}^{2} = C_{2}(\epsilon_{0}, d_{E}^{0})$, we have $\mathcal{R}_{0}(d_{E}^{0}, d_{I}^{1}) < \mathcal{R}_{0}(d_{E}^{0}, d_{I}^{2})$.

Theorem 3.4. Let $\nu_{0} = \frac{\sum_{i = 1}^{n}\xi_{i}\gamma_{i}}{\sum_{i = 1}^{n}\xi_{i}\beta_{i}}$ and $\varphi_{1}, \phi_{1}$ be the unique solutions of

and

respectively. If

there exist $d_{I}^{0}$ and $d_{E}^{1} < d_{E}^{2}$ such that $\mathcal{R}_{0}(d_{E}^{1}, d_{I}^{0}) < \mathcal{R}_{0}(d_{E}^{2}, d_{I}^{0})$.

Proof. Consider the principal eigenvalues of the following two eigenvalue problems, which referred to as $\mu$ and $\nu$, respectively,

and

with $\sum_{i = 1}^{n}\varphi_{i} = \sum_{i = 1}^{n}\phi_{i} = 1.$ Note from Theorem 1.2 that for any fixed $d_{I} > 0$, $\mathcal{R}_{0}\rightarrow \frac{1}{\mu}$ as $d_{E} \rightarrow 0$ and $\mathcal{R}_{0}\rightarrow \frac{1}{\nu}$ as $d_{E} \rightarrow \infty$. Now we take $\epsilon = \frac{1}{d_{I}}$ and perform regular expansions on $(\phi, \nu)$ and $(\varphi, \mu)$ to obtain the following expressions.

Our objective is to establish that the inequality $\mu > \nu$ holds true for small values of $\epsilon$ subject to the condition (3.16). Upon performing direct computation, we obtain the following expressions, $\varphi_{0} = \phi_{0} = \xi,$ $\mu_{0} = \nu_{0} = \frac{\sum_{i = 1}^{n}\xi_{i}\gamma_{i}}{\sum_{i = 1}^{n}\xi_{i}\beta_{i}}$ and $\varphi_{1}, \phi_{1}$ satisfy

and

respectively. Furthermore, we have

and

Therefore, by condition (3.16), (3.20), (3.21) and $\mu_{0} = \nu_{0}$, we obtain

Thus $\mu > \nu$ for large $d_{I}$. Therefore, we can find $d_{I}^{0}$ large, and $d_{E}^{2}$ large, $d_{E}^{1}$ small such that $\mathcal{R}_{0}(d_{E}^{1}, d_{I}^{0}) < \mathcal{R}_{0}(d_{E}^{2}, d_{I}^{0}).$

Remark: We present an example in which condition (3.16) holds. Let $\Omega = \{1, 2\}$, $\beta_{1} = 1.01, \beta_{2} = 1.424$ and $\gamma_{1} = 1.01, \gamma_{2} = 2.01$, by direct computation, we can see

4.   Asymptotic behaviours of the endemic equilibrium

Theorem 1.5 is a consequence of Lemma 4.1, Theorems 4.1 and 4.2. To analyze the behavior of endemic equilibria as $d_{S}$ tends towards zero, we shall examine the asymptotic properties of the system. Now we consider the alternative statements of the endemic equilibrium problem (1.4).

Lemma 4.1. $(\widetilde{S}, \widetilde{E}, \widetilde{I}, \widetilde{R})$ satisfies ${\rm(1.4)}$ if and only if $(S, E, I, R)$ satisfies the following system,

Moreover, $\widetilde{S} = \kappa S, \widetilde{E} = \kappa E, \widetilde{I} = \kappa I, \widetilde{R} = \kappa R$ and

Proof. Note that

which implies that

where $\xi$ is the unique positive eigenvector of the unique principal eigenvalue $0$ of $\mathbf{L}$ satisfying $\sum_{i\in\Omega} \xi_{i} = 1$. Set $S = \widetilde{S}/\kappa, E = \widetilde{E}/\kappa, I = \widetilde{I}/\kappa, R = \widetilde{R}/\kappa$. ${\rm(4.1)}$ can be obtained from (1.4).

Now we investigate the asymptotic behavior of the endemic equilibria when $d_{S}\rightarrow 0$. Recall that $\Lambda_{1}\left(-d_{R}\mathbf{L}+\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right)$ is the smallest eigenvalue of (1.9).

Theorem 4.1. Suppose that $\mathcal{R}_{0} > 1$. Under this assumption, we have

(i) As $d_{S} \rightarrow 0$, $E\rightarrow {E}^{*}, I\rightarrow {I}^{*}, {R}\rightarrow {R}^{*}$

(ii) The set $J^{+}: = \{i|M^{*}_{i} = \xi_{i}, i\in\Omega\}$ is nonempty, where $M^{*}_{i}: = d_{E}E_{i}^{*}+d_{I}I_{i}^{*}+d_{R}R_{i}^{*}$;

(iii) If further assume $\Lambda_{1}\left(-d_{R}\mathbf{L}+\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right) < 0$, then the set $J^{-}: = \{i|M^{*}_{i} < \xi_{i}, i\in\Omega\}$ is not empty.

Proof. (i). Note that $E_{i}, I_{i}, R_{i} > 0$ for any $i\in \Omega, d_{S} > 0.$ $d_{S}S+d_{E}E+d_{I}I+d_{R}R = \xi$ yields that $\frac{\beta_{i}S_{i}I_{i}}{S_{i}+E_{i}+I_{i}+R_{i}}$ exhibit uniform boundedness for any $d_{S} > 0.$ After extracting a subsequence if necessary, $E\rightarrow E^{*}$ as $d_{S} \rightarrow 0$ where $E^{*} \geq 0$. Using analogous analysis, it can be shown that $I\rightarrow I^{*}, R\rightarrow R^{*}$ as $d_{S} \rightarrow 0$ where $I^{*}, R^{*}\geq 0$, which satisfy

Now we show that $E^{*} \neq 0$, i.e., $E^{*} > 0$. It can be proved by contradiction. Assuming $E^{} = 0$, it follows from Eq (4.2) that $I^{} = R^{*} = 0$. Consequently, $S\rightarrow \infty$ almost everywhere as $d_{S}\rightarrow 0$. Thus $\frac{\beta_{i}S_{i}}{S_{i}+E_{i}+I_{i}+R_{i}} \rightarrow \beta_{i} \ \ $ as $d_{S}\rightarrow 0$. Define

Observe that $\hat{E}, \hat{I}$ and $\hat{R}$ are strictly positive, and $\| \hat{E}\|_{1}+\| \hat{I}\|_{1}+\| \hat{R}\|_{1} = 1.$ After extracting a subsequence if necessary, we obtain $\hat{E}, \hat{I}, \hat{R}$ approaches $\hat{E}^{*}, \hat{I}^{*}, \hat{R}^{*}$ respectively as $d_{S} \rightarrow 0,$ where $\hat{E}^{*}_{i}, \hat{I}^{*}_{i}, \hat{R}^{*}_{i}\geq 0$ for $i \in \Omega$ and

It follows from $\frac{\beta_{i}S_{i}}{S_{i}+E_{i}+I_{i}+R_{i}} \rightarrow \beta_{i}\ \ $ as $d_{S}\rightarrow 0$ that $\hat{E}^{*}$ is a solution of

which gives

It can be deduced from Eq (4.3) that the values of $\hat{E}^{*}, \hat{I}^{*}, \hat{R}^{*}$ are significantly greater than zero, which indicates that $\mathcal{R}_{0} = 1$. This lead to a contradiction, implying $E^{*} > 0$. Therefore, we obtain $I^{*}, R^{*} \gg0$.

To establish the claim that $|J^{+}| > 0$, we prove by contradiction. Suppose that $|J^{+}| = 0$, then as $d_{S}\rightarrow 0$, it follows that $S\rightarrow \infty$, and thus $\frac{\beta_{i}S_{i}}{S_{i}+E_{i}+I_{i}+R_{i}} \rightarrow \beta_{i} \ \ $ as $d_{S}\rightarrow 0$. Therefore, $E^{*}$ is a solution of

which yields

Based on Eq (4.5) in conjunction with $I^{} \gg0$ and $R^{} \gg0$, it can be inferred that $E^{*}\gg0$. Consequently, we have $\mathcal{R}_{0} = 1$, which leads to a contradiction. Therefore, $|J^{+}| > 0$.

Part (iii) can be shown through contradiction. Assume now that $|J^{-}| = 0$. Denote $h_{i}: = \frac{\beta_{i}S_{i}I_{i}}{S_{i}+E_{i}+I_{i}+R_{i}}-\alpha R_{i}$ and choose $\varphi \in \mathbb{R}^{n}$ subject to the condition that $\varphi \geq 0$. By multiplying the first three equations of (4.1) by $\varphi^{T}$ and adding the results together, we obtain

As $|J^{-}| = 0$, $M^{*} = 0$. Thus, we obtain

$\varphi \in \mathbb{R}^{n}$ such that $\varphi \geq 0$.

Let $\phi_{0}$ be a positive left eigenvector of $\Lambda_{1}\left(-d_{R}\mathbf{L}+\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right)$, i.e.

Since $S, E, I, R\gg0$ on $\Omega$ and $d_{R}\mathbf{L} R+YI-\alpha R = 0$, we obtain

Multiplying (4.9) by $\phi_{0}$ and applying (4.8), we obtain

Let $d_{S}\rightarrow 0$, it follows from (4.7) that $\Lambda_{1}\phi_{0}^{T}R \geq 0$. Given that $\phi_{0}, R^{*} > 0$ on $\Omega$, it follows that $\Lambda_{1} > 0$. This contradiction finishes the proof of (iii).

Theorem 4.2. Assume $\mathcal{R}_{0} > 1$ and $\Lambda_{1}\left(-d_{R}\mathbf{L}+\alpha\mathrm{diag}\left\{{1-\frac{\gamma_{i}}{\beta_{i}}} \right\}\right) < 0$. Then we have

(i) As $d_{S}\rightarrow 0$,

(ii) There exist positive constants $C_{1}, C_{2},$ independent of $d_{S}$ such that for sufficiently small $d_{S}$,

Proof. (i). For further purposes, denote $M: = d_{E}E+d_{I}I+d_{R}R.$ By (4.1), we have

Given that $S, E, I, R \gg0$ and $d_{S}S+d_{E}E+d_{I}I+d_{R}R = \xi$, we can conclude that $E, I$ and $R$ exhibit uniform boundednes with respect to $d_{S}$. Therefore,

By Theorem 4.1(i), (ii),

Therefore,

Furthermore, (4.1) implies $\widetilde{S} = \frac{\kappa}{d_{S}}(\xi-M).$ Based on (4.10) and Theorem 4.1(i), we obtain

as $d_{S}\rightarrow 0$.

Now we proceed to the proof of (ii). By $d_{S}S+d_{E}E+d_{I}I+d_{R}R = \xi$ and $\widetilde{E} = \frac{\kappa}{d_{S}} d_{S} E, \ \ \widetilde{I} = \frac{\kappa}{d_{S}} d_{S} I, \ \ \widetilde{R} = \frac{\kappa}{d_{S}} d_{S} R$, we get

Hence, (i) implies

Now we prove

by contradiction. Assume that $\min\{ \widetilde{E}_{i}, \widetilde{I}_{i}, \widetilde{R}_{i}, i\in\Omega\}/d_{S} = o(d_{S}).$ By (1.4) and direct calculation, $\sum_{i = 1}^{n} \widetilde{E}dx, \sum_{i = 1}^{n} \widetilde{I}_{i}, \sum_{i = 1}^{n} \widetilde{R}_{i} = o(d_{S}),$ which implies

Note that

Let $d_{S}\rightarrow 0$, it follows from (i), (4.11) and (4.13) that

As a result, we have $|J^{-}| = 0$. This contradiction finishes the proof of (ii).

Acknowledgments

The authors of this work receive financial support from various sources. S.Y. is partially funded by the National Natural Science Foundation of China (NSFC) grant number 12101487 and the China Scholarship Council under grant number 202006280442. J.W. is supported by the Canada Research Chair Program with reference number 230720. P.S. receives support from the National Natural Science Foundation of China (NSFC) grants 12101487 (PS) and 12220101001 (PS), the China Postdoctoral Science Foundation under grant number 2020M683445, and the Postdoctoral Fellowship of York University, Toronto, Canada. The authors also express their gratitude to the referees for their valuable assistance in reviewing the manuscript.

Conflict of interest

The authors declare there is no conflict of interest.