Stationary distribution and extinction of a stochastic influenza virus model with disease resistance

1.   Introduction

Influenza, usually known as the flu, is a virus-borne illness that mostly affects the throat, nose and lungs. It can be transferred from person to person by sneezes, contaminated air, or coughing, as well as through direct contact with infected people. Many authors have considered the mathematical models of the influenza virus ^[1,2,3,4,5]. In real life, there is an exposed time following infection transfer from susceptible to possibly infective person. However, these potentially infective people without any acquired symptoms are capable of spreading the flu. In 2016, Khanh ^[6] proposed a novel human virus transmission SEIR model with disease resistance. Briefly, the model, reports the recovery of an infected group without any prior therapy. Depending on the treatment, the other groups can be returned to the susceptible group. It describes the susceptible $(S)$, the exposed $(E)$, the infected $(I)$ and the recovered $(R)$ as four unique epidemiological subgroups of persons in the overall population. A set of ordinary differential equations provides the model as

where $\Lambda$ is the susceptible recruitment rate; $\alpha$ is the susceptibility constant rate for recovered individuals; the contact transmission rates of the virus is denoted by $\beta$; the susceptibility rate of individual exposed and infected people is expressed as $c$ and $b$; $\gamma$ is the persistent pace of recovery; $\epsilon$ is the continuous infectious proportion of the exposed population and $\mu$ is the population's natural mortality rate. Positive values are expected for all parameters. In System (1.1), $R_0 = \frac{\beta(\gamma+b+\mu+\epsilon)}{(\gamma+b+\mu)(c+\epsilon+\mu)}$ represents the basic reproduction number. Importantly, infection persists in the population when $R_0 > 1$ but finally dies off when $R_0\leq 1$.

Epidemic models are powerful tools to understand the dynamics of the tramission of an infectious disease. There is a large number of researchers who pay attention to the study of epidemic models. For example, Allegretti et al. ^[7] derived an SIR model to describe the evolution in time of the infectious disease by Sars-Cov-2; Kumar and Erturk ^[8] considered a cholera epidemic model from the perspective of generalized Liouville-Caputo fractional derivatives; Özk$\ddot{\rm{s}}$e and Yavuz ^[9] established a fractional-order pandemic model to discuss interactions between COVID-19 and diabetes by using real data from Türkiye; Naik et al. ^[10] proposed and analyzed a fractional-order epidemic model with a classic Caputo operator and the Atangana-Balenu-Caputo operator for the transmisson during the COVID-19 epidemic; Yavuz and Özdemir ^[11] used an SIR model to simulate the transmission dynamics of diseases where individuals acquire permanent immunity. Besides, there are also several works devoted to population systems, such as ^[12,13,14] and their references.

The population system for the actual world is unavoidably influenced by ambient white noise. The population size may be changed greatly in a short time due to fuctuations in the environment, such as earthquakes and tsunamis. Hence, the parameters of the system may not be absolute constant, and they may fluctuate around some constants. Thus, incorporating environmental noise into the epidemic models seems to be a good way to describe these phenomena. Additionally, in population dynamics, stochastic differential equation model is one of the significant types models because they offer a more realistic description than deterministic models. Biological and epidemiological stochastic models have been examined by many researchers ^{[15,16,17,18,19]}. Therefore, in the current research, Imhof and Walcher's technique is followed under the assumption that ambient white noise is a function of $S(t)$, $R(t)$, $E(t)$ and $I(t)$. For sufficiently small $\Delta t$, one can model $X = (S, E, I, R)^T$ as a Markov process with the following specifications

and

Following is the outline of the model,

where $\sigma_i^2 > 0 (i = 1, 2, 3, 4)$ signifies the intensity of the white noise. Brownian movements $B_i(t) (i = 1, 2, 3, 4)$ are defined on a complete probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq 0}, \mathbb {P})$ with a filtration $\{\mathcal{F}_t\}_{t\geq 0}$ that meets the standard criteria of mutually independent i.e., while $\mathcal{F}_0$ includes all P-null sets, it is increasing and right continuous. Comparatively, the parameters of Systems (1.1) and (1.2) imply the same meaning. And $\mathbb{R}_+^4 = \{x = (x_1, x_2, x_3, x_4)\in \mathbb{R}^4;x_i > 0, i = 1, 2, 3, 4\}$ in this paper. We also need to assume that these variables $\Lambda, \beta, c, b, \alpha, \mu, \epsilon$ and $\gamma$ should be positive constants.

We should ponit out that Ikram et al. ^[20] considered a stochastic epidemic model consisting of four human classes, i.e., susceptible, infected, vaccinated and removed. They established the global dynamcis of the model in terms of the stochastic basic reproduction number $R_0^s$ and used a stochastic Runge-Kutta method to implement the numerical simulations. In real life, there is an exposed period after the transmission of infection from susceptible to potentially infective members but before these potential infectives develop symptoms and can transmit infection. Due to this reason, the method in ^[20] cannot be applied to System (1.2). Thus, we need to construct a new auxiliary function to consider this system. On the other hand, a stochastic model may not have the positive equilibrium state. Therefore, we hope to study the existence and stability of "stochastic positive equilibrium" and the existence of a stationary distribution, for this stochastic influenza virus model. As we all know, there are no results about this system. Since the model described by System (1.2) consists of four equations, it is very difficult to construct the Lyapunov function if we consider the existence of an ergodic distribution. Thus, we need to construct a new Lyapunov function and a rectangular set. We also should note that the conditions for extinction of the influenza virus have a close relationship with the basic reproduction ratio $R_0$ in System (1.1).

The structure of the paper is outlined below. In Section 2, we show that there exists a unique global positive solution of System $(1.2)$ with initial value $(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4$. The existence of ergodic stationary distribution of System $(1.2)$ is established in Section 3. These conclusions can be generated by constructing a suitable stochastic Lyapunov function and a rectangular set. Furthermore, the sufficient condition for the extinction of disease can be found in Section 4. In Section 5, numerical simulations involving Milstein's order method are introduced to illustrate our theoretical results. To wrap up the article, certain conclusions are offered.

2.   Existence and uniqueness of the positive solution

Investigating an epidemic model's dynamic behavior begins with determining whether or not the solution is both positive and global. In this section, inspired by the methods presented in ^[21], we demonstrate that there exsits a solution for System $(1.2)$ that is a unique, global positive one.

Focus on a stochastic differential equation with an initial value of $z(0) = z_0\in\mathbb{R}^l$ in an $l$-dimensional, i.e.,

where $f\in L^1(\mathbb R^l\times \mathbb R_+, \mathbb R^l)$ and $g\in L^1(\mathbb R^l\times \mathbb R_+, \mathbb R^{l\times m})$. Equation ($2.1$) is connected with a differential operator $L$, which is defined as follows

Theorem 2.1. There exists a unique solution $(I(t), E(t), R(t), S(t))$ to System (1.2) for each given initial value $(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4$, and this solution will persist with a probablity of $1$, i.e., the solution will remain in $\mathbb{R}^4_+$ almost surely (a.s.).

Proof. Because the local Lipschitz condition is met by the coefficients of System $(1.2)$, there exists a unique local solution $(S(t), E(t), I(t), R(t))$ on $t\in [0, \tau_e)$ for every initial value $(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4$, while $\tau _e$ is the explosion time. To show that this solution is global, we only need to verify $\tau_e = \infty$ a.s.. To begin, we demonstrate that $S(t), \ E(t), \ I(t)$ and $R(t)$ do not explode to infinity in a limited period. Assume that $S(0), \ R(0), \ E(0)$ and $I(0)$ all fall inside the interval $[\frac{1}{k_0}, k_0]$ when $k\geq k_0$. For each integer value $k\geq k_0$, we determine the stopping time to be

We have specified $\inf \emptyset = \infty$ throughout this work (as is customary, $\emptyset$ signifies an empty set). Clearly, as a result, when $k\to \infty$, $\tau_k$ is increasing. Denote $\tau _{\infty} = \lim_{k\to \infty}\tau_k$ a.s. There exist two constants $T > 0$ and $\epsilon \in (0, 1)$ such that if this assumption is false, $\mathbb {P}\{\tau_{\infty}\geq T\} > \epsilon$. Hence, there exists an integer $k_1\geq k_0$ satisfying

Let a $C^2$-function $V:\mathbb{R}_+^4\to \mathbb{R}_+$ be defined by

For any $u > 0$, this function's nonnegativity may be determined by using $u-1-\ln u\geq 0$. It does not matter what number is in $k\geq k_0$ and $T > 0$. It$\hat{o}$'s formula may be applied to $V(S, E, I, R)$; then

where $LV:\mathbb{R}_+^4\to \mathbb{R}$ and

Then

where $K$ is a positive constant in this equation. As a result, we can get

Integrating Eq $(2.3)$ from $0$ to $\tau_k \land T$, we apply the mathematical expectation

Thus,

Setting $\Omega_k = \{\tau_k\leq T\}$ for $k\geq k_1$ and by virtue of Eq $(2.2)$, we get $\mathbb{P}(\Omega_k)\geq \epsilon$. It is worth noting that for each $\omega \in \Omega_k$, there exists $S(\tau_k, \omega)$, $I(\tau_k, \omega)$, $E(\tau_k, \omega)$ or $R(\tau_k, \omega)$ that is equal to either $\frac{1}{k}$ or $k$. As a result, either $k-1-\ln {k}$ or $\frac{1}{k}-1-\ln {\frac{1}{k}} = \frac{1}{k}-1+\ln k$ is no more than $V(S(\tau_k, \omega), E(\tau_k, \omega), I(\tau_k, \omega), R(\tau_k, \omega))$. As a result, we may acquire

It then follows from Eq $(2.4)$ that

where the indicator function of $\Omega_k$, i.e., $I_{\Omega_k}$, is inconsistent with what we have allowed. Letting $k\to\infty$,

Thus, we have a contradiction. Finally, we have $\tau_{\infty} = \infty$, and $S(t), I(t), R(t)$ and $E(t)$ do not seem to be exploding in a limited period.

3.   Stationary distribution

Eventually, it becomes important to know when an epidemic dynamical system will continue and propagate through a population. Using deterministic models, this issue may be resolved by considering evidence indicating that the model's global attractor or globally asymptotically stable endemic equilibrium exists. On the other hand, System (1.2) may not have an endemic equilibrium. Considering the hypothesis of Khasminskii ^[22], we demonstrate that the disease is likely to persist in the mean that there is a stationary distribution. We shall go over a theory introduced in ^[22] on stationary distributions. The following stochastic differential equation in D-dimensional Euclidean space defines $X(t)$ as a Markov homogeneous process in D-dimensional Euclidean space $E_d$ ($E_d$ denotes d-dimensional Euclidean space)

where $b\in L^1(\mathbb R^d, \mathbb R^d)$ and $g_r\in L^2(\mathbb R^d, \mathbb R^{d})$, $r = 1, 2, \cdots, k$. And, following is the definition of the diffusion matrix,

Lemma 3.1. [22,Chapter 4.3] The Markov process $X(t)$ has a unique ergodic stationary distribution $\mu$ if there is a bound open set $D\subset E^d$ that has a regular boundary $\Gamma$ and the following conditions:

$(1)$ there exists a positive number $M$ s.t. $\sum_{i, j = 1}^da_{i, j}(x)\xi_i\xi_j\geq M\|\xi\|^2, x\in D, \xi\in R^d.$

$(2)$ for any $E_d\backslash D$, there is a nonnegative $C^2$-function $V$ s.t. $LV$ is negative.

Let $f$ be an integrable function that is integrable with respect to the measure $\mu(\cdot)$; then,

We will give a definition of the parameter $\hat{R}_0^s$,

Theorem 3.2. Consider that $\hat{R}_0^s > 1$ and $(S_0, E_0, I_0, R_0)\in R_+^4,$ there exists a unique stationary distribution $\mu(\cdot)$ for System (1.2) that has the ergodic property.

Proof. It then follows from Theorem 2.1 that there exists a unique global solution $(S(t), E(t), I(t), R(t))\in \mathbb{R}_+^4$ with the initial value of $(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4$. We simply need to confirm Conditions $(1)$ and $(2)$ in Lemma 3.1 to prove this theorem. Now, we shall verify the situation given by Condition (1). The diffusion matrix of System (1.2) is provided by

Selecting $M = \min\limits_{(S, E, I, R)\in D_k\subset \mathbb{R}_+^4}\{\sigma_1^2S^2, \sigma_2^2E^2, \sigma_3^2I^2, \sigma_4^2R^2\},$ we get

where $D_k = [k, \frac{1}{k}]\times[k, \frac{1}{k}]\times[k, \frac{1}{k}]\times[k, \frac{1}{k}]$. Then, Condition (1) of Lemma $3.1$ is satisfied.

Create a $C^2$-function $Q:\mathbb{R}_+^4\to \mathbb{R}$ as follows:

where $0 < \theta < \frac{2\mu}{\sigma_1^2\lor\sigma_2^2\lor\sigma_3^2\lor\sigma_4^2}$,

Furthermore, selecting $M > 0$ fulfills the following requirement

where

It is simple to verify this

where $U_k = (\frac{1}{k}, k)\times(\frac{1}{k}, k)\times(\frac{1}{k}, k)\times(\frac{1}{k}, k)$. Additionally, $Q(S, E, I, R)$ is a continuous function. This means that just a minimum point $(S_0, E_0, I_0, R_0)$ which is in the interior of $\mathbb{R}_+^4$ is essential for $Q(S, E, I, R)$. Thus a nonnegative $C^2$-function $V :\mathbb{ R}_+^4 \to \mathbb{R}_+^4$ is defined by

If $V_1(S, E, I, R) =$ $-c_1\ln S-c_2\ln E-c_3\ln I+c_4(S+E+I+R)$-$c_5\ln S+c_6(S+E+I+R)$, it holds

If $V_2(S, E, I, R) = -\ln S-\ln E-\ln I-\ln R$, it holds that

If $V_3(S, E, I, R) = S+E+I+R$, it holds that

If $V_4(S, E, I, R) = \frac{1}{\theta+1}(S+E+I+R)^{\theta+1}$, it holds that

where

Hence, we have

Now, we will prove the Condition $(2)$ holds. Set

Let

To make it easier, we've divided $\mathbb{R}_+^4 \backslash D$ into eight domains

where $\epsilon_i(i = 1, 2, 3, 4)$ are sufficiently small positive constant factors that fulfill the following conditions:

Then, we show that $\forall (S, E, I, R)\in \mathbb{R}_+^4\backslash D$, $LV(S, E, I, R)\leq -1$, which is comparable to demonstrating it on the above eight domains.

Case 1. If $(S, E, I, R)\in D_1,$ then one has

From Eq (3.3), $\forall (S, E, I, R)\in D_1 and LV(S, E, I, R)\leq -1.$

Case 2. If $(S, E, I, R)\in D_2,$ then one has

Based on Eqs (3.4) and (3.5), we have

Case 3. If $(S, E, I, R)\in D_3,$ then one has

From Eq (3.6), $LV(S, E, I, R)\leq-2\Big(\frac{\mu\beta}{\epsilon_1}\Big)^{\frac{1}{2}}+F < -1, \forall (S, E, I, R)\in D_3.$

Case 4. If $(S, E, I, R)\in D_4,$ then one has

It then follows from Eq (3.7) that $LV(S, E, I, R)\leq-\frac{\gamma}{\epsilon_1}+F < -1, \forall (S, E, I, R)\in D_4.$

Case 5. If $(S, E, I, R)\in D_5,$ then one has

It then follows from Eq (3.8) that $LV(S, E, I, R) < -1, \forall (S, E, I, R)\in D_5.$

Case 6. If $(S, E, I, R)\in D_6,$ then one has

According to Eq (3.9), $LV(S, E, I, R) < -1, \forall (S, E, I, R)\in D_6.$

Case 7. If $(S, E, I, R)\in D_7,$ then one has

From Eq (3.11), $LV(S, E, I, R) < -1, \forall (S, E, I, R)\in D_7.$

Case 8. If $(S, E, I, R)\in D_8,$ then one has

From Eq (3.10), $LV(S, E, I, R) < -1, \forall(S, E, I, R)\in D_8.$

As a result, for sufficiently small values $\epsilon_i, i = 1, 2, 3, 4$, we can demonstrate that

And, as a consequence, the Condition $(2)$ of Lemma $3.1$ is satisfied. It follows from Lemma $3.1$ that System $(1.2)$ has a unique ergodic stationary distribution.

Remark 3.3. Theorem 3.2 states that if $\hat{R}_0^s = \frac{\beta\mu(\gamma+\mu+b+\epsilon+\frac{\sigma_3^2}{2})}{(\mu+\frac{\sigma_1^2}{2})(\gamma+\mu+b+\frac{\sigma_3^2}{2})(c+\epsilon+\mu+\frac{\sigma_2^2}{2})} > 1$, then System (1.2) has a unique ergodic stationary distribution $\mu(\cdot)$. If we ignore the white noise, the expression $\hat R_0^s$ is equal to the threshold for the deterministic autonomous system System (1.1). This demonstrates that the deterministic system's results may be generalized.

4.   Extinction of the disease

We shall focus on the disease extinction in this section. Defining the following notations for ease of use and simplicity in the following analysis, we have

and

where $f$ is an integrable function on $[0, \infty)$.

Lemma 4.1. If $(S(t), E(t), I(t), R(t))$ is the solution of System (1.2) satisfying any initial value$(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4,$ then

The proof is omitted since it is comparable to [24,Lemma 3.1].

Lemma 4.2. [21,Theorem 1.3.4] Denote $M = \{M_t\}_{t\leq 0}$ as a real-valued continuous local martingale vanishing at $t = 0$. It holds that

and

Theorem 4.3. Assuming that System $(1.2)$ has the solution $(S(t), E(t), I(t), R(t))$ with any initial value $(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4.$ If $\hat{R}_0 < 1,$the solution $(S(t), E(t), I(t), R(t))$ of the System (1.2) satisfies

Proof. By System (1.2), we have

where $Q(t) = S(t)+E(t)+I(t)+R(t)$. Integrating both sides of Eq $(4.1)$ from $0$ to $t$ and then dividing by $t$, it follows that

It then follows from Lemma 4.1 that

Let $C(t) = \frac{\gamma+b+\mu+\epsilon}{c+\epsilon+\mu}E(t)+I(t)$. Note that

Using It$\hat{o}$'s formula, it holds that

Integrating both sides of Eq $(4.4)$ from $0$ to $t$ and then dividing by $t$, we have

Let $M(t) = \int_0^t\frac{\frac{\gamma+b+\mu+\epsilon}{c+\epsilon+\mu}E(s)}{C(s)}dB_2(s)$. It is clear that,

According to Lemma 4.2, we have

Similarly, we get

Based on Eqs (4.5)–(4.7), it holds that

Since $\widehat{R}_0 < 1$, we can get

based on Eq $(4.8)$.

From System $(1.2)$, we conclude that $\lim_{t\to\infty}R(t) = 0$ a.s. when $\lim_{t\to\infty}I(t) = 0\ a.s.$ According to Eq (4.2), $\lim_{t\to\infty}\langle S\rangle_t = \frac{A}{\mu}\ a.s.$ This completes the proof.

Remark 4.4. If $R_0 = \frac{\beta(\gamma+b+\mu+\epsilon)}{(c+\epsilon+\mu)(\gamma+b+\mu)} > 1$ in the determined model, the infection will continue to exist. However, we can come to this conclusion based on the expression of $\hat R_0$ at the beginning of this section which indicates that if the density of the noise is sufficiently large, the disease may die out.

5.   Numerical simulations

Several examples are presented in this section to exemplify the theoretical findings. Our results are presented as a result of using Milstein's higher method suggested in ^[23]. Consider the following discretization equation

where the intensities of white noise are represented by the time increment $\Delta t > 0, \sigma_i^2 > 0 (i = 1, 2, 3, 4)$ and Gaussian random variables $\epsilon_{i, j}\sim N(0, 1) (i = 1, 2, 3, 4)$. We subdivide the time interval into $1000$ equidistant time steps. Moreover, $(S_j, E_j, I_j, R_j)^T$ is the value of the $i$th iteration of the discretized equation.

For the parameters and initial values, refer to the data in ^[6]. We chose $\beta = 0.34, c = 0.25, b = 0.2, \epsilon = 0.15, \alpha = 0.2,$ $\gamma = 0.1, \Lambda = 0.015, \mu = 0.01,$ $\sigma_1^2 = 0.002, \ \sigma_2^2 = 0.02, \ \sigma_3^2 = 0.04\ $ and $\ \sigma_4^2 = 0.005$. In this example, numerical simulations were used to show the trajectory images and distribution function graphs of $E(t)$ and $I(t)$, as shown in Figure 1. In the figure, one can find that the number of people making up the infectious population fluctuates around a nonzero value, which means that both will persist in the community. According to Theorem $3.2$, $\hat R^s_0 = 1.05785$, and there exists a unique stationary distribution of System $(1.2)$ in this circumstance. The results are supported by the results of numerical simulations.

Now, we will examine the influences of the contact transmission rate, susceptibility rate of exposed individuals and infected people.

Let $\beta$ vary in $[0, 1]$ and keep the other parameters as shown in Figure 1. Figure 2(a) shows that $R_0, \hat R_0^s$ and $\hat R_0$ are increasing functions. If we lower the viral transmission contact rate $\beta$ to $0.278$ and keep the other parameters the same as in Figure 1, we get the following results. In this situation, the simulations using the initial value $(S_0, E_0, I_0, R_0) = (0.5, 0.1, 0.1, 0.1)$ show that $E(t)$ and $I(t)$ of the solutions of System (1.2) tend to $0$ as $t\to+\infty$. This indicates that the disease is no longer present (see Figure 3). In this case, we may compute $\hat{R}_0 = 0.9751143 < 1$. The disease will die out with a probability of $1$ based on Theorem $4.3$. The results are supported by numerical simulations. We may deduce that the disease will be eradicated if the rate of infection is reduced.

Following that, let $b$ vary in $[0, 1]$ and keep the other parameters as shown in Figure 1. Figure 2(b) shows that $R_0, \hat R_0^s$ and $\hat R_0$ are decreasing functions. We raise the amount of $b$ to $0.55$, as well as the other parameters as shown in Figure 1. The simulations with the initial value $(S_0, E_0, I_0, R_0) = (0.5, 0.1, 0.1, 0.1)$ show that when $t$ approaches $+\infty$, $E(t)$ and $I(t)$ of solutions of System (1.2) tend to $0$. This indicates that the disease is no longer present (see Figure 4). In this case, $\hat{R}_0 = 0.96 < 1$. Hence the disease will die out with a probability of $1$ based on Theorem $4.3$. The results are supported by the numerical simulations.

Let $c$ vary in $[0, 1]$ and keep the other parameters as shown in Figure 1. Figure 2(c) shows that $R_0, \hat R_0^s$ and $\hat R_0$ are decreasing functions. Then, we increase the value of $c$ to $0.3$ and keep the other parameters the same as in Figure 1. In this situation, the simulations using the initial value $(S_0, E_0, I_0, R_0) = (0.5, 0.1, 0.1, 0.1)$ show that when $t$ approaches $+\infty$, $E(t)$ and $I(t)$ of the solutions of System (1.2) tend $0$ as $t\to+\infty$. This indicates that the disease is no longer present (see Figure 5). In this case, $\hat{R}_0 = 0.67 < 1$. According to Theorem $4.3$, the disease will die out with a probability of $1$. There is a good correlation between the numerical simulations and the results. In conclusion, the disease will die out by enhancing disease resistance.

From Figures 3–5, one can find that the components $E(t)$ and $I(t)$ of the stochastic epidemic model will approach $0$ as $t\to+\infty$. We should also point out that the components $E(t)$ and $I(t)$ of the deterministic epidemic model will tend toward positive values as $t\to+\infty$. This fact shows that white noise is beneficial for disease extinction. On the other hand, Figure 2 shows that, no matter how we change the values of $\beta, b$ and $c$, we have $R_0 > \hat R_0 > \hat R_0^s$. This result also shows that white noise is beneficial for disease extinction.

Consequently, we chose $\sigma_2^2 = 0.2, \sigma_3^2 = 0.3$, and the values shown in Figure 1 for the other variables. In this case, we observed that the $(S(t), E(t), I(t), R(t))$ of System (1.1) tended to $(1.2190, 0.1693, 0.0793, 0.0378)$ as $t\to +\infty$, whereas the $I(t)$ of System (1.2) gradually decreased to $0$ as t approached $\infty$ (see Figure 6). Ultimately, it could be concluded that white noise is beneficial for disease extinction.

6.   Conclusions

Briefly, the current study yielded a stochastic influenza virus model with disease resistance. According to mathematical approaches, in reality, the circumstances required for the disease to die out are rough to some extent. In this study, we focused on a stochastic influenza virus model with disease resistance. First, we obtained the existence of one unique global positive solution satisfying the initial value of $(S_0, E_0, I_0, R_0)\in \mathbb{R}_+^4$. Second, we get the existence of a stationary distribution for the positive solutions by using a stochastic Lyapunov function method. Third, we established the sufficient conditions for extinction of the disease. Finally, we conducted some numerical simulations. The results of the numerical simulations indicate that enhancing disease resistance is beneficial for the control of the disease. In addition, a simple example showed that the white noise is beneficial for the extinction of the disease.

Furthermore, there is a parametric region between the extinction conditions and a stationary distribution for positive solution values. As a result, it is only logical to wonder about the dynamics in this region. We shall address these incidents in future works.

Acknowledgments

This work was supported by the Qian Duansheng Distinguished Scholar Support Program of the China University of Political Science and Law (DSJCXZ180403).

Conflict of interest

The authors declare that there is no conflict of interest.