Threshold dynamics of a stochastic general SIRS epidemic model with migration

1.   Introduction

Many well-known epidemic models ^{[1,2,3,4,5,6,7]} have been proposed and discussed over the years. For instance, De la Sen et al. ^[8] in their study analyzed an epidemic model that incorporates delayed, distributed disease transmission and a general vaccination policy. Weera et al. conducted a numerical investigation of a nonlinear computer virus epidemic model with time delay effects ^[9]. Li et al. ^[3] examined an SIRS epidemic model with a general incidence rate and constant immigration, which took the following form

where $N = = S +I +R$ and the biological implications are shown in Table 1, and the infectious force $\beta f(N)$ is a continuous and twice differentiable function of total population and $\beta > 0$ is adequate contact rate. Furthermore, $f$ satisfies the following hypotheses

$1)\; f\in C^{2}((0, \infty); (0, \infty))$.

$2) \; f'(N)\leq0$ for any $N > 0$.

$3) \; [f(N)N]'\geq0$ for any $N > 0$.

Their research ^[3] found that

is the basic reproduction number. Furthermore, one gets

● If $b = 0$ and $R_0 < 1$, then system (1.1) has a disease-free equilibrium $E^0 = (S^0, I^0, R^0) = (\frac{A}{\mu}-\frac{cA}{\mu+\delta}, 0, \frac{cA}{\mu+\delta})$, which is globally asymptotically stable (GAS).

● If $R_0 > 1$ and $b = 0$, there exists a unique endemic equilibrium $E^* = (S^*, I^*, R^*)$ which is GAS.

● Otherwise if $b > 0$, there is no disease-free equilibrium in system (1.1) and there exists a unique endemic equilibrium $P^* = (S^*_1, I^*_1, R^*_1)$ which is locally asymptotically stable. In addition, when $\alpha\leq\mu+2\delta$, the endemic equilibrium $P^*$ is GAS.

However, in reality, variations in environmental factors affect the transmission coefficients of infectious diseases. As a result, stochastic modelling is an appropriate way to model epidemics in a variety of situations. For example, stochastic models can account for the randomness of infectious contacts that may occur during potential and infectious periods ^[10]. In comparison to deterministic models, stochastic epidemic models can provide more realism. A growing number of authors have recently focused on stochastic epidemic models ^{[4,5,6,7,11,12,13,14,15,16,17,18,19,20,21,22]}. Cai et al. ^[7] discovered that the global dynamics of a general SIRS epidemic model can determine the existence of either a unique stationary distribution free of disease or a unique stationary distribution with endemic disease. Liu et al. ^[18] found that in a stochastic SIRS epidemic model with standard incidence, in which two threshold parameters $R_0^S$ and $\hat{R_0^S}$ exist.

Inspired by Mao et al. ^[23], this paper posits that fluctuations in the environment primarily manifest as fluctuations in the transmission coefficient,

where $B(t)$ is a standard Brownian motion and $\sigma^{2} > 0$ indicates its intensity. Then we have

Our study is based on the deterministic SIRS epidemic model, which has proven to be an effective tool for investigating the spread of infectious diseases. Our approach incorporates two crucial elements: constant immigration and a general incidence rate, which are essential for understanding the impact of environmental fluctuations on disease dynamics.

One of the main strengths of our study lies in the fact that we have integrated these essential components into a stochastic framework. This has enabled us to analyze the effects of random fluctuations in disease transmission and immigration rates, which are significant factors that can profoundly influence the dynamics of infectious diseases. By examining these effects, we can obtain a more comprehensive understanding of the factors that contribute to the spread and persistence of diseases. Furthermore, our research has established the necessary conditions for the existence of a stationary distribution of positive solutions in the case of disease persistence. This novel contribution to the field has significant implications for the development of effective strategies for managing and controlling infectious diseases. Ultimately, our study provides valuable insights that can inform public health policies and initiatives aimed at reducing the impact of infectious diseases on global health.

The purpose of this paper is to explore the impact of environmental fluctuations on disease dynamics by analyzing the global dynamics of the stochastic SIRS epidemic model (1.2). The paper is structured as follows: In Section 2, we provide some preliminaries. Section 3 outlines the necessary conditions for disease extinction and persistence. We determine sufficient conditions for the existence of stationary distributions for persistent solutions of the model in Section 4. The paper concludes with numerical simulations and conclusions.

2.   Preliminaries

In this paper, unless specified otherwise, let $(\Omega, \mathcal {F}, \{\mathcal {F}_{t}\}_{t\geq0}, \mathbb{P})$ denote a complete probability space with a filtration $\{\mathcal{F}_{t}\}_{t\geq0}$ that satisfies the regular conditions. Let $B(t)$ be defined on this complete probability space. Denote $a\vee b = \max\{a, b\}\; \text{for any}\; a, b\in\mathbb{R}$, and $\mathbb{X} = \{(x_{1}, x_2, x_{3})\in\mathbb{R}^{3}:x_1 > 0, x_2 > 0, x_3 > 0 \}$.

Lemma 1. ^[24] (Strong Law of Large Numbers) Let $M = \{M\}_{ t\geq 0}$ be a real-valued continuous local martingale vanishing at $t = 0$. Then

and

Theorem 1. For any $(S(0), I(0), R(0))\in\mathbb X$, there is a unique solution $(S(t), I(t), R(t))$ of system (1.2) that remain in $\mathbb X$ with probability one.

The proof is standard and hence is omitted here.

Remark 1. From Theorem 2.1, we have

This implies that

is a positively invariant set of system (1.2). Hence throughout this paper we always assume that the initial value $(S(0), I(0), R(0))\in\Gamma$.

3.   Extinction

In contrast to the deterministic system (1.1), the purpose of this section is to study the dynamics of the system (1.2) when $b = 0$ holds. Denote

Theorem 2. Let $b = 0$ and $(S(t), I(t), R(t))$ be a solution of system (1.2). If

or

then

Proof. Making the use of Itô's formula ^[24] to $\ln I$, we have

Integrating the above equality from $0$ to $t$ and then dividing by $t$ on both sides, one obtains

where

Noting that $G(t)$ is a local martingale (since it is a right continuous adapted process defined on $(\Omega, \mathcal {F}, \{\mathcal {F}_{t}\}_{t\geq0}, \mathbb{P})$) whose quadratic variation is

Making the use of Lemma 2.1 leads to $\lim_{t\rightarrow \infty}\frac{G(t)}{t} = 0$ a.s. Combining (3.1), we have

Substituting the above inequality into (3.3) and taking the limit on both sides, we obtain

Consider the case $\sigma^2 < \frac{\beta \mu}{f(\frac{A}{\mu})A}$, we get

Noting that $\frac{A}{\alpha+\mu} < N < \frac{A}{\mu}$, $R > \frac{cA}{\mu+\delta}$ and substituting them into (3.5), we have

From (3.2) and (3.3), we get

Then we have

which means that for arbitrary small $\varepsilon > 0$ there are $t_0$ and $\Omega_\varepsilon$ such that $\mathbb{P}(\Omega_\varepsilon)\geq1-\varepsilon$ and $\alpha I\leq\varepsilon$ for $t\geq t_0$ and $\omega\in\Omega_\varepsilon$. In view of system (1.2), we have

Due to the arbitrariness of $\varepsilon$, one has

Similarly as getting equality (3.8), we have

In view of (3.7)–(3.9), we have

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Remark 2. According to Theorem 3.1, if $R_{0}^s < 1$ and $\sigma$ is not large, the disease will inevitably die out. It is worth noting that the expressions $R_0^s$ and $R_0$ reveal that $R_0^s < R_0$. Furthermore, if $\sigma = 0$, $R_0^s = R_0$. In simpler terms, the conditions for the disease to die out in system (1.2) are considerably easier than those in the corresponding deterministic system (1.1).

4.   Asymptotic stability

In this section, we will prove that if $b = 0$ and $R_0^s > 1$ or $b > 0$, the densities of the distributions of the solutions to system (1.2) can converge in $L^{1}$ to an invariant density.

Theorem 3. The distribution of $(S(t), I(t), R(t))$ has a density $U(t, x, y, z)$ for $t > 0$. If $b = 0$ and $R_0^s > 1$ or $b > 0$, then there is a unique density $U_{*}(x, y, z)$ such that

The following steps constitute the proof of Theorem 4.1 above:

● First, the kernel function of $(S(t), I(t), R(t))$ is absolutely continuous.

● We demonstrate that the kernel function is positive on $\mathbb X$.

● The Markov semigroup is either sweeping with respect to compact sets or asymptotically stable.

● Due to the presence of Khasminski$\check{i}$ function, we exclude sweeping.

For definitions related to Markov semigroups and their asymptotic properties, the reader is referred to the papers ^{[25,26,27,28,29,30,31]}. We will show this by Lemmas 4.1–4.5.

Lemma 2. For $t > 0$ and any initial value $(x_0, y_0, z_0)\in \mathbb{X}$, the transition probability function $P(t, x_0, y_0, z_0, B)$ has a continuous density $k(t, x, y, z; x_0, y_0, z_0)$.

Proof. Similar to the proof method in ^[31], the Lie bracket is given by

Let $a_0(S, I, R) = \begin{pmatrix} aA-\beta f(N)SI-\mu S+\delta R\\ bA+\beta f(N)SI-(\mu +\gamma +\alpha)I\\ cA+\gamma I-(\mu +\delta)R \end{pmatrix}$ and $a_1(S, I, R) = \begin{pmatrix} -\sigma f(N)SI\\ \sigma f(N)SI\\ 0 \end{pmatrix}$. Direct calculation leads to

with

and

where

Therefore, we have

According to H$\ddot{o}$rmander Theorem ^[23], one obtains that $P(t, x_0, y_0, z_0, B)$ has a continuous density $k(t, x, y, z; x_0, y_0, z_0)$.

Next, fixing a point $(x_0, y_0, z_0)\in \mathbb{X}$ and a function $\psi \in L^2(\left [0, T\right]; \mathbb{R})$, we have

where

Let $D_{X_0;\psi}$ be the Fr$\acute{e}$chet derivative. If for some $\psi\in L^2([0, T]; \mathbb{R})$, the rank of $D_{X_0;\psi}$ is 3, then $k(T, x, y, x; x_0, y_0, z_0) > 0$ for $X = X_{\psi}(T)$. Let

where $f'$ and $g'$ are the Jacobians of

For $T\geq t\geq t_0\geq 0$, let $Q(t, t_0)$ be a matrix function such that $Q(t_0, t_0) = Id$, $\frac{\partial Q(t, t_0)}{\partial t} = \Psi (t)Q(t, t_0).$ Then

□

Lemma 3. For each $(x_0, y_0, z_0), (x, y, z)\in \Gamma$, there is $T > 0$ satistying $k(T, x, y, z; x_0, y_0, z_0) > 0$.

Proof. Since we only need to find a continuous control function $\psi$, system (4.1) can be rewritten as follows

First, we verify that the rank of $D_{X_0;\psi}$ is 3. Let $\varepsilon \in (0, T)$ and

where $\chi$ denotes the indicator function of the interval $[T-\varepsilon, T]$. Since

we have

where $v = \begin{pmatrix} -\sigma \\ \sigma \\ 0 \end{pmatrix}$. Direct calculation leads to

where

Thus the rank of $D_{X_0;\psi }$ is 3.

Then let $w_{\psi} = x_{\psi}+y_{\psi}+z_{\psi}$, (4.2) becomes

where

Let

First, we find a positive constant $T$ and a differentiable function

such that $w_\psi(0) = w_0$, $w_\psi(T) = w_1$, $w_\psi'(0) = g_2(x_0, w_0, z_0) = w_0^d$, $w_\psi'(T) = g_2(x_1, w_1, z_1) = w_T^d$ and

We split the construction of the function $w_\psi$ on three intervals $[0, \tau]$, $[\tau, T-\tau]$ and $[T-\tau, T]$, where $0 < \tau < T /2$. Let

If $w_\psi \in \left(\frac{A}{\alpha +\mu }+\theta, \frac{A}{\mu }-\theta\right)$, we have

Then we construct a $C^2$-function $w_\psi$: $[0, \tau] \to \left(\frac{A}{\alpha +\mu }+\theta, \frac{A}{\mu }-\theta\right)$ such that

and for $t\in[0, \tau]$, $w_\psi$ satisfies

Analogously, we can construct a $C^2$-function $w_\psi$: $[T-\tau, T] \to \left(\frac{A}{\alpha +\mu }+\theta, \frac{A}{\mu }-\theta\right)$ such that

and for $t\in[T-\tau, T]$, $w_\psi$ satisfies

Taking $T$ sufficiently large, then we can extend the function $w_\psi$: $[0, \tau]\bigcup [T-\tau, T] \to \left(\frac{A}{\alpha +\mu }+\theta, \frac{A}{\mu }-\theta\right)$ to a $C^2$-function $w_\psi$ defined on the whole interval $[0, T]$ such that

Thus, we can find $C^1$-functions $x_\psi$ and $z_\psi$ that satisfy (4.3). Finally we can determine a continuous function $\psi$. and $T > 0$ such that $x_{\psi}(0) = x_0$, $w_{\psi}(0) = w_0$, $z_{\psi}(0) = z_0$, $x_{\psi}(T) = x$, $w_{\psi}(T) = w$, $z_{\psi}(T) = z$. This completes the proof.□

Lemma 4. If $b = 0$ and $R_0^{s} > 1$ or $b > 0$. For $\left \{ P(t) \right \}_{t\geq 0}$ and every density $g$, one has

Proof. System (1.2) can be rewriten as

From Remark 2.1, we get

For almost every $w\in \Omega$, there is $t_0\in t_0(w)$ such that

As a matter of fact, there exist three possible cases:

1) $N(0, w)\in \left (\frac{A}{\alpha +\mu }, \frac{A}{\mu }\right)$. In this case, our statement is obvious from (4.7).

2) $N(0, w)\in \left (0, \frac{A}{\alpha +\mu }\right)$. Assume that our claim is not satisfied. Then there is $\Omega '\subset \Omega$ with $\mathbb{P}(\Omega') > 0$ such that $N(t, w)\in (0, \frac{A}{\alpha +\mu }), w\in \Omega'$. By (4.7), we obtain that for any $w\in \Omega'$, $N(t, w)$ is strictly increasing on $[0, \infty)$ and

According to system (4.6), we get that $\lim_{t\rightarrow \infty }S(t, w) = \lim_{t\rightarrow \infty }R(t, w) = 0$, $w\in \Omega '$ and thus, $\lim_{t\rightarrow \infty }I(t, w) = \frac{A}{\alpha +\mu}$, $w\in \Omega '$.

Consider the case $b = 0$, making the use of It$\hat{o}$'s formula, we have

Thus

where $G(t): = \frac{1}{t}\int_{0}^{t}\sigma S(\tau)f(N(\tau))dB(\tau)$. Applying Lemma 2.1, we have

Thus, due to $S(t)$, $I(t)$, $f(N(t))$ are continuous,

This contradicts the assumption

Then let us consider the case $b > 0$. Since $\lim_{t\rightarrow \infty }N(t, w) = \frac{A}{\alpha+\mu}$ and $\lim_{t\rightarrow \infty }S(t, w) = \lim_{t\rightarrow \infty }R(t, w) = 0$ for $w\in \Omega'$, which contradicts that $R(t, w) > 0$ for $w\in \Omega'$, $t\in (0, \infty)$ and the claim follows.

3) $N(0, w)\in (\frac{A}{\mu}, \infty)$. We suppose, by contradiction, and analogous arguments to 2), that there is $\Omega'\subset \Omega$ with $\mathbb{P}(\Omega') > 0$ such that

Firstly, consider the case $b = 0$, by the second and third equations of (4.6), for any $w\in \Omega'$, one gets

For all $w\in \Omega'$, one has

Therefore

This contradicts the assumption $\lim_{t\rightarrow \infty }I(t) = 0$ a.s. In other words, for almost all $w\in \Omega$, there is $t_0 = t_0(w)$ such that

When $b > 0$, we get that $I(t, w) > 0$ for $t\in (0, \infty)$ and $w\in \Omega'$. This contradicts the assumption $\lim_{t\rightarrow \infty }N(t, w) = \frac{A}{\mu}$, $w\in \Omega '$ and the claim holds.

Similar to the proof of 2) and 3), one obtains that for almost all $w\in \Omega$, there is $t_1 = t_1(w)$ such that

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Lemma 5. $\left \{ P(t) \right \}_{t\geq 0}$ is asymptotically stable or is sweeping with respect to compact sets.

Proof. In view of Lemma 4.1, $\left \{ P(t) \right \}_{t\geq 0}$ is an integral Markov semigroup with kernel $k(t, x, y, z; x_0, y_0, z_0)$. According to Lemma 4.3, it suffices to consider the restriction of $\left \{ P(t) \right \}_{t\geq 0}$ to the space $L^1(\Gamma)$. By Lemma 4.2, one gets

on $\Gamma$, for every $g\in \mathbb{D}$. Then $\left \{ P(t) \right \}_{t\geq 0}$ is asymptotically stable or is sweeping with respect to compact sets.□

Lemma 6. Assume that $b = 0$ and $R_0^s > 1$ or $b > 0$, then $\{P(t)\}_{t\geq0}$ is asymptotically stable.

Proof. From Lemma 4.4, $\{P(t)\}_{t\geq0}$ satisfies the Foguel alternative. In order to exclude sweeping it is sufficient to construct a nonnegative $C^2$-Khasminskiǐ function $V$ and a closed set $D_\epsilon\in\Sigma$ such that

First of all, we consider the case $b = 0$ and $R_0^s > 1$. Define

where $V_1 = -\ln I-\ell_1N+\ell_2R$, $V_2 = -\ln S$, $V_3 = -\ln(\frac{A}{\mu}-N)$, $V_4 = -\ln(N-\frac{A}{\mu+\alpha})$, $V_5 = -\ln(R-\frac{cA}{\mu+\delta})$, $\ell_1 = \frac{\beta f(\frac{A}{\mu})}{\mu}$, $\ell_2 = \frac{\beta f(\frac{A}{\mu})}{\mu+\delta}$ and $M$ is a positive constant satisfying

It is easy to find that $H$ reaches a minimum at $(S_*, I_*, R_*)$. Then we define

Thus we have

Similarly, we obtain

and

Therefore

Define

where $\epsilon\in(0, 1)$ is sufficiently small satisfying

Denote

Then we prove that $\mathscr{A}^{*}V(S, I, R) < -1$ for any $(S, I, R)\in\Gamma\setminus D_\epsilon = D_1\bigcup D_2\bigcup D_3\bigcup D_4\bigcup D_5$.

Case 1. For any $(S, I, R)\in D_{1}$, from (4.9),

Thus

Case 2. For any $(S, I, R)\in D_{2}$, from (4.8) and (4.10),

Therefore

Case 3. For any $(S, I, R)\in D_{3}$, from (4.11),

Hence

Case 4. For any $(S, I, R)\in D_{4}$, from (4.12),

Then

Case 5. For any $(S, I, R)\in D_{5}$, from (4.12),

Thus

In summary,

Using similar arguments to those in ^[27], we can obtain that $\{P(t)\}_{t\geq0}$ is asymptotically stable.

Next, we consider the case $b > 0$, define

Obviously, $E$ has a minimum point $(S_{1*}, I_{1*}, R_{1*})$ in the interior of $\Gamma.$ Then we define

Then we have

Similarly, define

where $\epsilon_1\in (0, 1)$ is sufficiently small satisfying

For convenience, we divide $\Gamma\setminus U_{\epsilon_1}$ as

The rest of the proof is omitted here due to it is similar to the case of $b = 0$. This completes the proof.

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Remark 4.1. The stationary distribution of the correct solution refers to the long-term behavior of a stochastic system when the probability of the disease persisting is not zero. In other words, if the random threshold $R_0^s$ is greater than 1, the disease may not be eradicated and will persist in the population. In this case, the stable distribution of the correct solution refers to the probability distribution of infected individuals in the population over time once the system has reached a steady state. This distribution is said to be stationary because it does not change over time, while the correct solution refers to the non-zero probability of individuals being infected.

Remark 4.2. According to Theorems 3.1 and 4.1, if $R^s_0 < 1$, the disease will become extinct under mild additional conditions, whereas if $R^s_0 > 1$, the disease will be stochastically persistent. The value of $R^s_0$ can determine the extinction of the disease or not, and thus it can be considered as a threshold for the stochastic system (1.2).

5.   Numerical simulations

In this section, we give several numerical examples to support our results. Employing Milstein's high-order method ^[32], the discretized system is

where the time increment $\Delta t > 0$, $\varrho_k$ for $k = 1, 2, ..., n$ are Gaussian random variables following the standard normal distribution.

5.1.   Threshold dynamics with the standard incidence

In this part, we focus on the dynamical behavior of system (1.2) with standard incidence. Let

Assume

Parameters $b$, $c$ and $\sigma$ will take different values in different examples.

Example 1. (Stationary distribution) Let $b = 0$ and $c = 0.1$, then we obtain $R_0 = 1.3657 > 1$. From ^[3], the disease of the deterministic system (1.1) will persist in a long term (Figure 1).

For system (1.2), let $\sigma = 0.01$ and one obtains

From Theorem 4.1, system (1.2) admits an ergodic stationary distribution (Figure 1).

For the case with $b = 0.1$ and $c = 0$, we choose $\sigma = 0.075$ such that $R_0^s = R_0-\frac{\sigma^2f^2(\frac{A}{\mu})A^2}{2(\mu+\gamma+\alpha)\mu^2} = 0.9983 < 1$. From Theorem 4.1, system (1.2) admits an ergodic stationary distribution (Figure 2).

Example 2. (Extinction) Let $b = 0$, $c = 0.1$, $\sigma = 0.1$, and the other parameters are shown in (5.2) such that

then from Theorem 3.1, the disease of system (1.2) will become extinct, see Figure 3.

Let $b = 0$, $c = 0.1$ and $\sigma = 0.08$ and the other parameters are shown in (5.2) such that

and $\sigma^2-\frac{\beta\mu}{f(\frac{A}{\mu})A} = -0.0036 < 0$. According to Theorem 3.1, the disease of system (1.2) will be extinct (Figure 4).

5.2.   Threshold dynamics with the mass action incidence

In this part, we investigate the threshold dynamics of deterministic system (1.1) and stochastic system (1.2) with mass action incidence. Let

where $\lambda$ is a positive constant. Assume

Parameters $\beta$, $b$, $c$ and $\sigma$ will take different values in different examples.

Example 3. (Stationary distribution) First, consider the persistence of the disease of system (1.2) with $\beta = 0.002$, $b = 0$ and $c = 0.1$. Then we obtain $R_0 = 1.3763 > 1$. From ^[3], the disease of the deterministic system (1.1) will persist in a long term, see Figure 1.

For the stochastic system (1.2), let $\sigma = 0.0002$ and we obtain

From Theorem 4.1, the stochastic system (1.2) admits an ergodic stationary distribution. See Figure 5.

For the case with $b = 0.1$ and $c = 0$, we choose $\beta = 0.001$ and $\sigma = 0.0002$ such that $R_0^s = R_0-\frac{\sigma^2f^2(\frac{A}{\mu})A^2}{2(\mu+\gamma+\alpha)\mu^2} = 0.6944 < 1$. From Theorem 4.1, system (1.2) admits an ergodic stationary distribution. See Figure 6.

Example 4. (Extinction) Let $b = 0$, $c = 0.1$, $\beta = 0.0015$ and $\sigma = 0.002$, and the other parameters are shown in (5.3) such that

thus from Theorem 3.1, the disease of system (1.2) will be extinct exponentially in a long term (Figure 7).

Let $b = 0$, $c = 0.1$, $\beta = 0.0014$ and $\sigma = 0.0015$ and the other parameters are shown in (5.3) such that

and $\sigma^2-\frac{\beta\mu}{f(\frac{A}{\mu})A} = -5.5\times 10^{-7} < 0$. According to Theorem 3.1, the disease of system (1.2) will be extinct exponentially in a long term (Figure 8).

6.   Conclusions

In this study, we present a stochastic SIRS epidemic model with constant immigration and general incidence rate. Our results show that the threshold parameter

for this model is lower than its deterministic counterpart ($R_0^s < 1 < R_0$). In this scenario, the deterministic system may have an endemic state, while the stochastic system leads to disease extinction with probability one (Theorem 3.1). On the other hand, if $R_0^s > 1$, the distribution of solution converge in $L^{1}$ to an invariant density (Theorem 4.1), indicating that environmental fluctuations can positively impact the control of infectious diseases. Moreover, if there is a constant influx of infected population, i.e. $b > 0$, the stationary distribution will always exist and the disease will persist. We contend that conducting a comprehensive analysis of the influence of migration on the dynamics of our model will yield valuable insights into the intricate interplay between migration and disease transmission.

Acknowledgments

This work was supported by Department of Science and Technology of Jilin Province (No. 20210509040RQ), National Natural Science Foundation of China (No. 12271201), Innovation and Entrepreneurship Talent Funds of Jilin Province (No. 2022ZY22) and the Research Funds of Jilin University of Finance and Economics (No. 2022YB025).

Conflict of interest

The authors declare there is no conflict of interest.