Dynamic analysis of a stochastic vector-borne model with direct transmission and media coverage

1.   Introduction

Vector-borne disease seriously threatens global health, and it is usually caused by vector-borne parasites, viruses, and bacteria that transmits pathogens between humans or from animals to humans. According to the World Health Organization, the disease accounts for 17% of all infectious and has caused 700,000 deaths annually ^[1]. Despite scientific and technological advances and the growing influence in all regions, vector-borne diseases remain one of the leading causes of global disease. Mathematical modeling has become an essential method for studying epidemics. Since the first attempt to model malaria transmission by Ross ^[2] and subsequent modifications by MacDonald ^[3], a series of vector-borne disease models have been proposed ^[4,5,6,7]. Various disease models based on influencing factors (e.g., time delay, vaccination, age structure, etc.) have been extensively studied ^[8,9,10,11].

It is commonly known that direct and indirect transmissions are two significant ways that various diseases are spread. Although vector-borne diseases are mainly transmitted by vectors, i.e., indirect transmission, vector-borne diseases are often transmitted directly through blood transfusions, organ transplantation, laboratory exposure, or mother-to-baby during pregnancy, childbirth, or breastfeeding. Furthermore Zika can be transmitted through sexual contact^[12]. Thus, direct transmission plays a vital role in the dynamics of vector-borne diseases and has attracted widespread attention ^{[13,14,15,16]}. In the deterministic model proposed by Wei et al. ^[16], the host population is assumed to be divided into three subpopulations, i.e., susceptible, infected, and recovered individuals. The infected individuals will not relapse once recovered, i.e., the recovered individuals will not become susceptible or infected. Let $S(t), I(t)$, and $R(t)$ be the numbers of susceptible, infected, and recovered individuals at time $t$. The vector population is divided into two parts, i.e., susceptible and infected vectors, denoted by $M(t)$ and $V(t)$ as the corresponding numbers at time $t$. The newly recruited vectors are susceptible when vertical transmission is ignored. On the other hand, media coverage is a crucial factor in the control of the spread of epidemics ^[17]. The media helps people to understand the progress of an epidemic and provide beneficial guidance ^[18]. Many scholars have studied the impact of media coverage on disease transmission from the perspective of mathematical modeling ^[19,20].

Based on the above discussion, we introduce media coverage into the epidemic model and investigate the dynamics of vector-borne diseases with direct transmission. Let $\beta_{1}$ be the transmission rate without media intervention, and $\beta_{2} I/(m+I)$ be the effect of media coverage on transmission, where $\beta_{1} > \beta_{2}$ and $m$ measures how quickly people react to media reports ^[21]. During the spread of the vector-borne epidemic, two transmission rates can lead to the susceptible becoming infected : the rate denoted by $\beta_3$ from an infected vector to a susceptible person, and the one denoted by $\beta_4$ from an infected person to a susceptible vector. We propose a vector-borne model with direct transmission and media coverage as follows

where $\Lambda_i$ is a recruitment rate, $d_i \; (i = 1, 2)$, and $\mu$ are the natural, and disease-related death rates of people and vector population, $\alpha_i \; (i = 1, 2, 3)$ denotes the saturated constants during different transmission processes, and $\gamma$ is the recovery rate of infected people. Here, $\beta_{1}{SI}$, $\beta_{3}{SV}$, and $\beta_{4}{MI}$ measure the contagiousness of the vector-borne disease, and $1/(1+\alpha_1I)$, $1/(1+\alpha_2V)$, and $1/(1+\alpha_3I)$ reflect the behavioral change of susceptible individuals. The basic reproduction number is $R_0 = \frac{\beta_{1} \Lambda_1}{d_1\left(d_1+\gamma+\mu\right)}+\frac{\beta_3 \beta_4 \Lambda_1 \Lambda_2}{d_1 d_2^2\left(d_1+\gamma+\mu\right)}$, which determines whether the epidemic occurs. If $R_0 < 1$, system (1.1) has a unique disease-free equilibrium $E_0 = \left(\frac{\Lambda_1}{d_1}, 0, 0, \frac{\Lambda_2}{d_2}, 0\right)$. This represents no infected individuals in either population. If $R_0 > 1$, then model (1.1) has two equilibria: a disease-free equilibrium $E_0$ and an endemic equilibrium $E^* = \left(S^*, I^*, R^*, M^*, V^*\right)$. This means that some individuals of both populations have been infected.

In the real world, random fluctuations are essential to ecosystems ^[22,23,24]. Random factors, such as temperature and humidity, inevitably affect the epidemic's spread. Many stochastic models have been studied in recent years ^[25,26,27]. Considering the complex environmental changes, Liu and Jiang claimed that the random perturbation may depend on the square of the state variables $S$ and $I$ in the system ^[28,29]. Recently, nonlinear perturbations have received much attention ^[30,31,32]. In addition to this, sometimes ecosystems are also affected by violent random perturbations such as typhoons and tsunamis. To reflect reality better, Levy jumps were introduced into the model ^[33,34]. However, this noise differs in detail and often leads to different results. It is worth noting that in the model of vector-borne diseases, Jovanović and Krstić ^[35] proposed that the random perturbation is proportional to the distance. Ran et al. ^[36] studied the dynamics of a stochastic vector-borne model with age structure and saturation incidence, considering the environmental noise on the mosquito bite rate and transmission rate between vector and host. Son and Denu ^[37] provided another stochastic vector-borne model with direct transmission, in which environmental noise affects the mortality of hosts and vectors. We did not want to add complex perturbations to make the model unmanageable; simple perturbations are more likely to reveal the inherent nature of the model. In our work, suppose that the environmental white noise is proportional to the number of subpopulations ^[38,39]. Next, we extend the deterministic model (1.1) to a stochastic model. The recovered class is decoupled from the others in the model and then neglected. Then, we propose the following stochastic model

where $B_i(t) \; (i = 1, 2, 3, 4)$ denotes independent standard Brownian motions, $\sigma_{i} \; (i = 1, 2, 3, 4)$ represents the white noise intensity, and the remaining parameters are the same as in model (1.1).

The rest of this paper is organized as follows. Section 2 reviews some basic concepts and valuable lemmas used later. The uniqueness and positivity of the solution are proved in Section 3. Section 4 provides sufficient conditions for determining whether a disease is extinct. In Section 5, we explore the persistence in the mean. In Section 6, we prove the existence of a unique ergodic stationary distribution under certain conditions. In Section 7, we validate the results of analysis through numerical simulations. A brief conclusion is given in the last section.

2.   Preliminaries

Let $(\Omega, \mathscr{F}, \mathbb{P})$ be a complete probability space with a filtration $\left\{\mathscr{F}_t\right\}_{t \geq 0}$ that satisfies the usual conditions (i.e., it is increasing and right continuous while $\mathscr{F}_0$ contains all $\mathbb{P}$-null sets). Denote $\mathbb{R}_{+}^n = \left\{y \in \mathbb{R}^n: y_i > 0, 1 \leq i \leq n\right\}$. Consider an $n$-dimensional stochastic differential equation of the following form ^[40]

with the initial value $y(0) = y_0 \in \mathbb{R}^n$, where $B(t)$ denotes an $n$-dimensional standard Brownian motion defined on the complete probability space $(\Omega, \mathscr{F}, \left\{\mathscr{F}_t\right\}_{t \geq 0}, \mathbb{P})$. Define a differential operator $\mathscr{L}$ of Eq (2.1) as follows

If $\mathscr{L}$ acts on a nonnegative function $\mathscr{V} \in C^{2, 1}\left(\mathbb{R}^n \times\left[t_0, \infty\right]; \mathbb{R}_{+}\right)$, then

where $\mathscr{V}_t = \frac{\partial \mathscr{V}}{\partial t}, \mathscr{V}_y = (\frac{\partial \mathscr{V}}{\partial y_1}, \ldots, \frac{\partial \mathscr{V}}{\partial y_n}), \mathscr{V}_{y y} = (\frac{\partial^2 \mathscr{V}}{\partial y_i \partial y_j})_{n \times n}$. By Itô's formula, it follows that

Lemma 1. (Strong law of large numbers, ^[41]) Let $M = \left\{M_t\right\}_{t \geqslant 0}$ be a real- valued continuous local martingale vanishing at $t = 0$. Then

Let $Y(t)$ be a regular time-homogeneous Markov process in $\mathbb{R}^n$ in the following form

where the diffusion matrix $\bar{A}(Y) = (b_{ij}(y))$ and $b_{ij}(y) = \sum_{r = 1}^{k}\sigma^i_{r}(y)\sigma^j_{r}(y).$

Lemma 2. ^[42] The Markov process $Y(t)$ has a unique stationary distribution $\pi(\cdot)$ if there is a bounded domain $D \in \mathbb{R}^n$ with a regular boundary and the following holds

(i) There is a positive number $M$ such that $\sum_{i, j = 1}^n b_{i j}(y) \xi_i \xi_j \geq M|\xi|^2, y\in D, \xi \in \mathbb{R}^n$.

(ii) There exists a nonnegative $C^2$ -function $\mathscr{V}$ such that $\mathscr{L}\mathscr{V}$ is negative for any $\mathbb{R}^n\setminus D$; then

3.   Uniqueness and positivity of solution

Theorem 1. For a given initial value $Y(0) = (S(0), I(0), M(0), V(0)) \in \mathbb{R}^4_{+}$, the solution $Y(t) = (S(t), I(t), M(t), V(t))$ of model (1.2) is unique on $t\geq 0$ and will maintain in $\mathbb{R}_{+}^4$ with probability one.

Proof. For a given initial value $(S(0), I(0), M(0), V(0))\in \mathbb{R}^4_{+}$, the coefficient in the model (1.2) satisfies the local Lipschitz continuity condition. Hence, there is a unique local solution when $t\in [0, \tau_e)$, where $\tau_e$ is the explosion time ^[43,44]. To obtain the global property of the solution, we need to prove that $\tau_e = \infty$ almost surely (a.s.). Suppose that $k_0\geq 1$ is sufficiently large such that $S(0), I(0), M(0)$ and $V(0)$ all lie within the interval $[ 1/k_0, k_0]$. For each integer $k\geq k_0$, define a stopping time

where $\emptyset$ is an empty set and $\inf\emptyset = \infty$. It can be seen that $\tau_k$ increases as $k\rightarrow \infty$ and $\tau_{\infty} = \lim _{k\to \infty}\tau_k$ with $0\leq \tau_{\infty}\leq \tau_e$ a.s. In other words, if $\tau_e = \infty$ a.s. does not hold, there must exists constants $T, k_1 > 0$ and $\epsilon \in(0, 1)$ such that $P\left\{\tau_k \leq T\right\} > \epsilon$ for all $k \geq k_1$. Define a $C^2$-function $W :\mathbb{R}^4\rightarrow \mathbb{R}_{+}$ and

Obviously, $W$ is a non-negative function. Applying Itô's formula to Eq (3.2) yields

where $\mathscr{L} W: \mathbb{R}^4_{+}\rightarrow \mathbb{R}_{+}$ can be written in the following form

Hence, we have

Integrate both sides of Eq (3.3) from $0$ to $\tau_k\wedge T$. It is easy to get that

Setting $\Omega = \{\tau_k\leq T\}$ for $k\geq k_1$ and by using Eq (3.1), we get that $P(\Omega_k)\geq \epsilon$. Further, every $\omega$ from $\Omega$ is associated with at least one among $S(\tau_k, \omega), I(\tau_k, \omega), M(\tau_k, \omega),$ and $V(\tau_k, \omega)$ that is equal to $k$ or $1/k$. Hence, $W(S(\tau_k), I(\tau_k), M(\tau_k), V(\tau_k))$ is not less than $k-1-\log k$ or $\frac{1}{k}-1+\log k$. That is to say,

Combining Eqs (3.4) and (3.5), we have

where $1_{\Omega(\omega)}$ denotes an indicator function of set $\Omega$. Letting $k \rightarrow \infty$ leads to the following contradiction

It implies that $\tau_e = \infty$ a.s. The proof is complete.

It is clear that model (1.1) has a disease-free equilibrium $E_0 = (\Lambda_1/d_1 0, \Lambda_2/d_2, 0)$ whereby the disease tends to become extinct within the time limit. However, there is no disease-free equilibrium in the stochastic version of the model, which requires other ways to consider its extinction. Define a threshold value

4.   Disease extinction

Theorem 2. Assume that $d_1 > \frac{\sigma^2_{1}\vee \sigma^2_{2}}{2}$ and $d_2 > \frac{\sigma^3_{1}\vee \sigma^4_{2}}{2}$. Let $(S(t), I(t), M(t), V(t))$ be the solution of system (1.2) with any initial value $(S(0), I(0), M(0), V(0))$. If $\mathscr{R}^S_0 < 1$, then

Moreover,

Proof. According to Reference ^[45], we have

We integrate both sides of the proposed model (1.2) and obtain

It is obvious that

From Eqs (4.1) and (4.2), the limit of Eq (4.3) given by

Similarly, we integrate on both sides of the last two equations of the model (1.2). Hence,

Combining (4.1) and (4.2), we can get the following equation

On the other hand, by Itô's formula, it follows that

The last term here uses the inequality $2IV \leq (I+V)^2$. Integrate on both sides of the equation and divide it by $t$. Thus,

where $\mu_1 = \min\{\mu +d_1+\gamma, d_2\}, \sigma_* = \min\{\sigma_{2}, \sigma_{4}\}$, and $\beta = \max\{\beta_{1}, \beta_{3}\}$. From Eqs (4.4) and (4.5), we can get

According to Lemma 1, it is obtained that

By using Eqs (4.6) and (4.7), we have

It means that $\lim_{t\to \infty}\frac{1}{t}\int_{0}^{t}I(u)du = 0$ and $\lim_{t\to \infty}\frac{1}{t}\int_{0}^{t}V(u)du = 0, a.s.$ Combining Eqs (4.4) and (4.5), it is obvious that

The proof is complete.

5.   Persistence in the mean of the disease

The most interesting aspect in the study of epidemic modeling is the extinction and persistence of the epidemic; in the previous section we studied disease extinction and in this section we will show that diseases are persistent in the mean.

Theorem 3. Assume that $d_1 > \frac{\sigma^2_{1}\wedge \sigma^2_{2}}{2}$ and $d_2 > \frac{\sigma^3_{1}\wedge \sigma^4_{2}}{2}$. If

then for any given initial value $(S(0), I(0), M(0), V(0)) \in R^4_+$, the solution of system (1.2) has the following properties

Proof. (i) From the first equation of system (1.2) integrating the above inequality and dividing both sides by $t$, we get

In view of Theorem 1, for any initial value $(S(0), I(0), M(0), V(0)) \in \mathbb{R}^4_{+}$, there is a unique global solution $(S(t), I(t), M(t), V(t)) \in \mathbb{R}^4_{+}$. Thus,

Through the strong law of large numbers for local martingales, we have

which together with Eq (5.1) yields

This is the required assertion (i).

(ii) From the third equation of system (1.2), integrating the above inequality and dividing both sides by $t$, we get

Then

According to the strong law of large numbers of local martingales, we have

which together with Eq (5.2) yields

This is the required assertion (ii).

(iii) First, define a function $W_2(S, I, V) = -\ln S-\ln I-\ln M-\ln V$. According to the Itô's formula:

where

We integrate the above inequality in the interval $(0, t)$, divide it by $t$, and take the limit to $t$. Thus,

According to the law of large numbers for a martingale,

It follows that Eq (5.3) becomes

The proof is complete.

6.   Stationary distribution

The ergodic property for an epidemic model means that the stochastic model has a unique stationary distribution that forecasts the permanence of the epidemic in the future. That means the disease persists for all time regardless of the initial condition^[46].

In this section, we provide a sufficient condition for the existence of a stationary distribution in the model (1.2). Denote

where $r_1 = \frac{1}{2}\sum_{i = 1}^4\sigma^2_{i} +2d_1+3d_2+\mu+\gamma+\frac{\beta_{1}}{\alpha_1}+\frac{\beta_{3}}{\alpha_2}+\frac{\beta_{4}}{\alpha_3}$, and $r_2 = \frac{1}{2}\sum_{i = 1}^4\sigma^2_{i} +4d_1+2d_2+\mu+\gamma+\frac{\beta_{1}}{\alpha_1}+\frac{\beta_{3}}{\alpha_2}+\frac{\beta_{4}}{\alpha_3}.$

Theorem 4. If $\mathscr{R}^S_2 > 1$, then model (1.2) has a unique stationary distribution $\pi(\cdot)$ with ergodicity.

Proof. The diffusion matrix for model (1.2) is given by

Denote $M = \min _{(S, I, M, V) \in D \subset \mathbb{R}_{+}^4}\left\{\sigma_1^2 S^2, \sigma_2^2 I^2, \sigma_3^2 M^2, \sigma_4^2 V^2\right\}$. It follows that

for $(S, I, M, V) \in D, \xi = \left(\xi_1, \xi_2, \xi_3, \xi_4\right) \in \mathbb{R}^4$, where $D = \left(\frac{1}{k}, k\right) \times\left(\frac{1}{k}, k\right) \times\left(\frac{1}{k}, k\right) \times\left(\frac{1}{k}, k\right)$ and $k$ is a sufficiently large integer. Therefore, the condition (i) in Lemma 2 is satisfied. Next, we prove the condition (ii) in Lemma 2. Let

Denote $\lambda_i = r_i(\mathscr{R}_i-1) \; (i = 1, 2)$, $\sigma^2 = \sigma_1^2\vee\sigma_2^2\vee\sigma_3^2\vee \sigma_4^2$, $b = 2d_1+\mu+\gamma+\frac{\beta_{1}}{\alpha_1}+\frac{\beta_{4}}{\alpha_3}+\frac{\beta_{3}}{\alpha_2}+\frac{\sigma^2_{1}+\sigma^2_{3}}{2}$ and $d = d_1\wedge d_2$. We construct a $C^2$-function $\tilde{\mathscr{V}}: \mathbb{R}_{+}^4 \rightarrow \mathbb{R}$ as follows

where $\varTheta_i \; (i = 1, 2)$ denotes sufficiently large positive constants satisfying $-\varTheta_1 \lambda_1+F_2 \leq-3$, $-\varTheta_2 \lambda_2+F_3 \leq-2$, and

Here, $\theta$ is a positive constant satisfying that $d > (\theta+1)\sigma^2/2.$ It means that

and $\tilde{\mathscr{V}}\left(S, I, M, V\right)$ is a continuous function. Then the function $\tilde{\mathscr{V}}\left(S, I, M, V\right)$ must have a minimum point $\left(\bar{S}, \bar{I}, \bar{M}, \bar{V}\right) \in \mathbb{R}_{+}^4$. Further, we construct a nonnegative $C^2$-function $\mathscr{V}: \mathbb{R}_{+}^4 \rightarrow \mathbb{R}_{+}$ in the following form

Applying Itô's formula to $\mathscr{V}_1$, we get

where

Thus, it follows that

A closed subset is defined as follows:

where $\epsilon > 0$ represents sufficiently small constants satisfying the following conditions

where $H$ is a constant and is determined later. Denote $Y = (S, I, M, V)$. We divide $\mathbb{R}_{+}^4 \setminus D$ into the following eight cases

Now, we will prove that $\mathscr{L}V(S, I, M, V) < -1$ on $\mathbb{R}_{+}^4\setminus D$; this is equivalent to proving that it is valid on the above eight subsets.

Case 1. When $(S, I, M, V)\in D_1$, we can get

where

According to Eq (6.1), we have that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_1$.

Case 2. When $(S, I, M, V)\in D_2$, we have

where

Given Eq (6.2), we get that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_2$.

Case 3. When $(S, I, M, V)\in D_3$, we have that

where $H = d_1+\mu+\gamma+\frac{\beta_{1}}{\alpha_1}+\frac{\beta_{4}}{\alpha_3}+\frac{\beta_{3}}{\alpha_2}+\frac{\sigma^2_{1}+\sigma^2_{3}}{2}+B,$ and, from Eq (6.3), we have that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_3$.

Case 4. When $(S, I, M, V)\in D_4$, then

Again, from Eq (6.3), we find that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_4$.

Case 5. When $(S, I, M, V)\in D_5$, we have

By means of Eq (6.4) we obtain that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_5$.

Case 6. When $(S, I, M, V)\in D_6$, we can get

By Eq (6.5), we have that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_6$.

Case 7. When $(S, I, M, V)\in D_7$, it follows that

Using Eq (6.6), we have that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_7$.

Case 8. When $(S, I, M, V)\in D_8$, then

Again using Eq (6.6), we have that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_8$.

In Cases 1–8, we have chosen sufficiently small values of $\epsilon$ such that $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for any $(S, I, M, V)\in D_i \; (i = 1, 2, \ldots, 8)$. Thus, $\mathscr{L}\mathscr{V}(S, I, M, V) < -1$ for all (S, I, M, V) $\in \mathbb{R}_{+}^4 \backslash D.$ Then, the condition (ii) in Lemma 2 is satisfied. According to Lemma 2, we can obtain that system (1.2) is ergodic and has a unique stationary distribution. This completes the proof.

7.   Numerical simulations

Numerical simulations are presented to support our theoretical findings of the model (1.2) and reveal the impact of media coverage on the spread of disease. Using the Milstein method mentioned by Higham ^[47], we consider the discretized equations as follows:

where the time increment $\Delta t > 0$ and $\zeta_{1i}, \zeta_{2i}, \zeta_{3i}, \zeta_{4i},$ are mutually independent Gaussian random variables which follow the distribution $N(0, 1)$ for $i = 0, 1, 2, ..., n$.

Vector-borne diseases with two transmission routes may be more likely to become endemic than diseases with one transmission route. Therefore, we tend to choose lower transmission rates and recruitment when numerically modeling disease extinction.

Example 1. Let $\Lambda_1 = 100$, $\Lambda_2 = 100$, $\beta_{1} = 0.000012$, $\beta_{2} = 0.0000018$, $\beta_{3} = 0.000039$, $\beta_{4} = 0.000039$, $\alpha_1 = 0.13$, $\alpha_2 = 0.15$, $\alpha_3 = 0.15$, $\mu = 0.13$, $\gamma = 0.13$, $d_1 = 0.1$, $d_2 = 0.1$, $m = 20$, $\sigma_1 = 0.025$, $\sigma_2 = 0.25$, $\sigma_3 = 0.03$, $\sigma_4 = 0.26$, $\mu_1 = \min\{\mu +d_1+\gamma, d_2\}$, $\sigma_* = \min\{\sigma_{2}$, $\sigma_{4}\}$, $\beta = \max\{\beta_{1}, \beta_{3}\}$, and the initial values $(S(0)$, $I(0)$, $M(0)$, $V(0))$ = $(1000$, $15$, $1000$, $50)$. So

According to Theorem 2, the solution of the stochastic model (1.2) will eventually approach zero; this means that the disease will die out almost surely. And, from Figure 1, it is observed that the number of infected individuals tends to zero.

Example 2. We keep the parameters the same as in Example 1, except that $\Lambda_1 = \Lambda_2 = 500$, $\beta_1 = 0.01$ and $\beta_{3} = \beta_{4} = 0.001$. Then

Theorem 3 implies that the disease is persistent in the mean. Interestingly, in Figure 2, it is clear that the number of infected individuals is higher than that of susceptible individuals.

Example 3. Choose the parameters $\Lambda_1 = 35000$, $\Lambda_2 = 30000$, $\beta_{1} = 0.05$, $\beta_{2} = 0.000002$, $\beta_{3} = 0.069$, $\beta_{4} = 0.069$, $\alpha_1 = 0.1$, $\alpha_2 = 0.12$, $\alpha_3 = 0.12$, $\mu = 0.23$, $\gamma = 0.2$, $d_1 = 0.5$, $d_2 = 0.058$, $m = 100$, $\sigma_1 = 0.015$, $\sigma_2 = 0.018$, $\sigma_3 = 0.018$, $\sigma_4 = 0.02$, the initial values $\mu_1 = \min\{\mu +d_1+\gamma, d_2\}$, $\sigma = \min\{\sigma_{2}$, $\sigma_{4}\}$, $\beta = \max\{\beta_{1}$, $\beta_{3}\}$, and the initial values $(S(0), I(0), M(0), V(0)) = (20000, 2000, 20000, 2000)$. Using the parameters

we can obtain that $\mathscr{R}^S_2 = \{\mathscr{R}_3^S, \mathscr{R}^S_4 \}\approx 49.720 > 1$ and the conditions of Theorem 4 are satisfied. Figure 3 shows the histograms of solutions of model (1.2) with white noise. Theoretical conclusions and numerical simulations indicate that the disease will eventually prevail and persist for a long time.

Example 4. Given $\beta_{2} = 0$, different transmission rates $\beta_{1}$, = 0, 0.02, 0.05, 0.08, 0.12. When the transmission rate $\beta_{1} =$ 0.05, we select that $\beta_{2}$ = 0, 0.006, 0.01, 0.012, 0.016, 0.02. The rest parameters are the same as in the Example 1. Figure 4 shows that as $\beta_{1}$ changes from 0, it significantly impacts the system. As $\beta_{2}$ increases, the numbers of infections decreases. This shows that the existence of direct transmission via this transmission route has an significant influence on disease transmission, and that reducing the rate of human-to-human contact through media coverage can reduce the scale of vector-borne infectious diseases.

8.   Conclusions

This paper presented a direct transmission model that is saturated with stochastic vector-borne disease incidence and the associated dynamical behavior. We obtained the positive definiteness and uniqueness of the solution to the stochastic model. Then, we established sufficient conditions for the extinction of the disease in two populations. Furthermore, we have proven the uniqueness and existence of an ergodic stationary distribution of the model when ${\mathscr{R}_2^S > 1}$ by choosing a suitable stochastic Lyapunov function.

On the other hand, from the simulation, we found that the disease under the condition of an increasing transmission rate $\beta_{1}$ showed an increasing transmission scale. It reflected that direct transmission i.e., the transmission route, has a critical influence on the spread of vector-borne diseases. In addition, we observed in the numerical experiments that there is indeed an effect on the number of the infected by increasing the value of the $\beta_{2}$. This also validates the inhibitory impact of media coverage on the spread of the disease.

Finally, reviewing the model we built, we have found that model (1.1) becomes the classical SIR model with media coverage if we set $\beta_{3} = \beta_{4} = 0$. This means that the disease can be endemic in the host population if there is no transmission pathway from the vector to the host. The threshold parameters obtained in this study do not explain this phenomenon. Some algorithms that guarantee the positivity of the solution are more useful when numerical simulations are performed ^[48,49]. We leave these issues for future research.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Natural Science Foundation of China under grant No. 12061070, the Natural Science Foundation of Xinjiang Uygur Autonomous Region under grant No. 2021D01E13, the 2023 Annual Planning Project of the Commerce Statistical Society of China under grant No. 2023STY61, the Innovation Project the of Excellent Doctoral Students of Xinjiang University under grant No. XJU2023BS017, and the Research Innovation Program for Postgraduates of Xinjiang Uygur Autonomous Region under grant Nos. XJ2023G016 AND XJ2023G017.

Conflict of interest

The authors declare that they have no conflict of interest.