Dynamical Behavior and Numerical Simulation of an Influenza A Epidemic Model with Log-Normal Ornstein–Uhlenbeck Process

Abstract

Influenza remains one of the most widespread epidemics, characterized by serious pathogenicity and high lethality, posing a significant threat to public health. This paper focuses on an influenza A infection model that includes vaccination and asymptomatic patients. The deterministic model examines the existence and local asymptotic stability of equilibria. In light of the influence of environmental disruption on the spread of disease, we develop a stochastic model in which the transmission rate follows a log-normal Ornstein–Uhlenbeck process. To demonstrate the dynamic behavior of the stochastic model, we verify the existence and uniqueness of the global positive solution. The establishment of suitable Lyapunov functions allows for the determination of sufficient conditions for the stationary distribution and extinction of the disease. Furthermore, the expression of the local density function around the quasi-endemic equilibrium is represented. Eventually, numerical simulations are conducted to support theoretical results and explore the effect of environmental noise. Our findings indicate that high noise intensity can expedite the extinction of the disease, while low noise intensity can facilitate the disease reaching a stationary distribution. This information may be valuable in developing strategies for disease prevention and control.

Proof of Theorem 3.1

Proof

As the coefficients of model (3) are locally Lipschitz continuous, there is a unique solution on $t \in \left[ 0, \tau _{e}\right)$ for model (3), where $\tau _{e}$ is defined as an explosion time. We can get the conclusion that the solution is global as long as the explosion time $\tau _{e}=+\infty$, a.s..

Let $n_{0}$ be a positive integer which ensure that $(S(0), A(0), I(0), \beta (0)) \in R_{+}^{4}$, model (3) stays in the interval $[\frac{1}{n_{0}},n_{0}]$.For each $n>n_{0}$,

$$\begin{aligned} \tau _{n}{} & {} = inf \{ 0<t<\tau _{e}|\min \{S(t),A(t),I(t),\beta (t) \}\\{} & {} <\frac{1}{n} \ or \ \max \{S(t), A(t), I(t), \beta (t)\}>n \}. \end{aligned}$$

Obviously, $\tau _{n}$ is a monotone increasing function about variable n. For convenience, we define $\inf \{\varnothing \}=+\infty$ and $\tau _{\infty }=\lim _{n \rightarrow +\infty } \tau _{n}$. As $\tau _{\infty } \le \tau _{e}$ a.s., $\tau _{e} = \infty$ as long as $\tau _{\infty } = \infty$. Next, we need to verify $\tau _{\infty }=\infty$.

Assuming $\tau _{\infty }<\infty$ a.s., then there exist three constants $(\zeta , T, n_{1})$, satisfying $\zeta \in (0,1)$, $T \in R_{+}$, $n_{1} \in N^{*}$ such that

$$\begin{aligned} \begin{aligned} P\left( \tau _{n} \le T\right) \ge \zeta , \ \forall n \ge n_{1}. \end{aligned} \end{aligned}$$

(A.1)

Then we can construct a non-negative $C^2$-function $W(S,A,I,\beta ): R_{+}^{4}\rightarrow R_{+}$ as follows

$$\begin{aligned} W(S, A, I, \beta )=S-1-\log S+A-1-\log A+I-1-\log I+\beta -1-\log \beta . \end{aligned}$$

By It$\hat{o}$’s formula, one derives that

$$\begin{aligned} {\mathcal {L}}(-\log S)={} & {} -\frac{(1-p+\frac{\gamma _{3}}{\mu })\Lambda }{S}+\mu +\varphi +\beta (I+\delta A)-\frac{\gamma _{1} A+\gamma _{2} I}{S}+\frac{\gamma _{3}(S+A+I)}{S}, \nonumber \\ {\mathcal {L}}(-\log A)={} & {} -\delta \beta A+\mu +\nu +\gamma _{1}, {\mathcal {L}}(-\log I)=-\beta S-\nu \frac{A}{I}+\mu +\gamma _{2}. \end{aligned}$$

(A.2)

$$\begin{aligned} d\beta =\beta \left[ \alpha (\log \bar{\beta }-\log \beta )+\frac{1}{2}\sigma ^{2}\right] dt+\sigma \beta dB(t). \end{aligned}$$

(A.3)

Thus, referring (A.2) and (A.3), applying It$\hat{o}$’s to $W(S,A,I,\beta )$ yields

$$\begin{aligned}{} & {} {\mathcal {L}} W(S, A, I, \beta )=(1-p+\frac{\gamma _3}{\mu })\Lambda -(\mu +\gamma _3)(S+A+I)-\varphi S+\mu +\varphi +\beta (I+\delta A) \nonumber \\{} & {} \qquad -\frac{\gamma _1 A+\gamma _2 I}{S}-\frac{1}{S}[(1-p+\frac{\gamma _3}{\mu })\Lambda -\gamma _{3}(S+A+I)]-\delta \beta S+(\mu +\nu +\gamma _1)-\beta S\nonumber \\{} & {} \qquad -\nu \frac{A}{I}+(\mu +\gamma _2)+\beta [\alpha (\log \bar{\beta }-\log \beta )+\frac{1}{2}\sigma ^{2}]-\alpha (\log \bar{\beta }-\log \beta )\nonumber \\{} & {} \quad =(\mu +\gamma _{3})[\bar{\Lambda }-(S+A+I)]-\varphi S-\frac{1}{S}[(1-p+\frac{\gamma _3}{\mu })\Lambda -\gamma _3(S+A+I)]+\mu +\varphi \nonumber \\{} & {} \qquad +\beta (I+\delta A)-\frac{\gamma _1 A+\gamma _2 I}{S}-\delta \beta S+(\mu +\nu +\gamma _1)+(\mu +\gamma _2)-\beta S-\nu \frac{A}{I}\nonumber \\{} & {} \qquad +\beta [\alpha (\log \bar{\beta }-\log \beta )+\frac{1}{2}\sigma ^{2}]-\alpha (\log \bar{\beta }-\log \beta )\nonumber \\{} & {} \quad \le 3\mu +\varphi +\nu +\gamma _1+\gamma _2+(\mu +\gamma _{3})\bar{\Lambda }+\beta (I+\delta A)-\frac{(1-p+\frac{\gamma _{3}}{\mu })\Lambda -\gamma _{3}(S+A+I)}{S}\nonumber \\{} & {} \qquad +\beta [\alpha (\log \bar{\beta }-\log \beta )+\frac{1}{2}\sigma ^{2}]-\alpha (\log \bar{\beta }-\log \beta ).\nonumber \\ \end{aligned}$$

(A.4)

Summing the first three equations in model (3) obtains the following differential equation

$$\begin{aligned} d(S+A+I)= & {} \left[ \left( 1-p+\frac{\gamma _{3}}{\mu }\right) \Lambda -(\mu +\gamma _{3})(S+A+I)-\varphi S\right] dt \\\le & {} (\mu +\gamma _{3})[\bar{\Lambda }-(S+A+I)], \end{aligned}$$

which means

$$\begin{aligned} S(t)+A(t)+I(t) \le {\left\{ \begin{array}{ll} S(0)+A(0)+I(0), &{} if\ S(0)+A(0)+I(0) \ge \bar{\Lambda },\\ \bar{\Lambda }, &{} if\ S(0)+A(0)+I(0) < \bar{\Lambda }. \end{array}\right. } \end{aligned}$$

Denote $K = \max \{S(0)+A(0)+I(0), \bar{\Lambda }\}$, thus the following inequation holds

$$\begin{aligned} S(t)+A(t)+I(t) \le K. \end{aligned}$$

(A.5)

Combining inequation (A.5), (A.4) can be transformed into

$$\begin{aligned} {\mathcal {L}}W\le & {} 3\mu +\varphi +\nu +\gamma _1+\gamma _2+(\mu +\gamma _{3})\bar{\Lambda }+\beta (I+\delta A) \nonumber \\{} & {} -\frac{(1-p+\frac{\gamma _{3}}{\mu })\Lambda -\gamma _{3}(S+A+I)}{S} \nonumber \\{} & {} +\beta [\alpha (\log \bar{\beta }-\log \beta )+\frac{1}{2}\sigma ^{2}]-\alpha (\log \bar{\beta }-\log \beta ), \nonumber \\\le & {} 3\mu +\varphi +\nu +\gamma _1+\gamma _2+(\mu +\gamma _{3})\bar{\Lambda }+\beta (1+\delta )K \nonumber \\{} & {} +\beta [\alpha (\log \bar{\beta }-\log \beta )+\frac{1}{2}\sigma ^{2}]-\alpha (\log \bar{\beta }-\log \beta ),\nonumber \\ {}:= & {} H(\beta ). \end{aligned}$$

(A.6)

Especially, if $S(0)+A(0)+I(0)<\bar{\Lambda }$, then $K=\bar{\Lambda }$, which indicates the solution (S(t), A(t), I(t)) has an invariant set $\Gamma ^{*}=\{(S(t),A(t),I(t))\in R_{+}^{4}|S(t)+A(t)+I(t)<\bar{\Lambda }\}$. It is obvious that $\lim \nolimits _{\beta \rightarrow 0} H(\beta )=-\infty$ and $\lim \nolimits _{\beta \rightarrow +\infty } H(\beta )=-\infty$. Then there is a positive constant $H _{0}$ which satisfies

$$\begin{aligned} \sup \limits _{\beta \in R_{+}} H(\beta ) < H_{0}. \end{aligned}$$

The rest proof is similar to paper [48]. Then we finish the proof. $\square$