## Qualitative Analysis of a Stochastic SIVS Epidemic Model with Nonlinear Perturbations Under Regime Switching

Zhang Zhonghua,, Zhang Qian,

 基金资助: 国家自然科学基金.  11201277

 Fund supported: the NSFC.  11201277

Abstract

In this paper, we present a stochastic SIVS epidemic model with nonlinear perturbations under regime switching. For the non-autonomous stochastic SIVS epidemic system with white noise, we provide results regarding the stochastic boundedness, stochastic permanence in mean, and we prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii's theory. For the system with Markov conversion, we establish sufficient conditions for existence of ergodic stationary distribution, and the thresholds for persistence in mean and the extinction of infected persons was obtained, respectively. Finally, some numerical simulations are carried out to support the theoretical results.

Keywords： Stochastic SIVS epidemic model ; Nonlinear perturbation ; Markov chain ; Nonlinear incidence

Zhang Zhonghua, Zhang Qian. Qualitative Analysis of a Stochastic SIVS Epidemic Model with Nonlinear Perturbations Under Regime Switching. Acta Mathematica Scientia[J], 2021, 41(4): 1218-1234 doi:

## 2 模型的建立

$$$\left\{ \begin{array}{rl} {\rm d}S = &[(1-q_{r(t)})A_{r(t)}-\beta_{r(t)}f(S(t)g(I(t))-(\mu_{r(t)}+p_{r(t)})S(t)+\gamma_{r(t)}I(t)\\ &+\varepsilon_{r(t)}V(t)]{\rm d}t+\sigma_{1r(t)}S(t){\rm d}B_1(t), \\ {\rm d}I = &[\beta_{r(t)}f(S(t)g(I(t))-(\mu_{r(t)}+\gamma_{r(t)}+\alpha_{r(t)})I(t)]{\rm d}t+\sigma_{2r(t)}I(t){\rm d}B_2(t), \\ {\rm d}V = &[q_{r(t)}A_{r(t)}+p_{r(t)}S(t)-(\mu_{r(t)}+\varepsilon_{r(t)})V(t)]{\rm d}t+\sigma_{3r(t)}V(t){\rm d}B_3(t), \end{array} \right.$$$

ⅰ) $\inf\limits_{\|x\|>l}V(t, x)\rightarrow \infty , $$l\rightarrow \infty, ⅱ) LV(t, x)\leq-1, \|x\|>l, 则系统(2.4)存在 T -周期解, 微分算子 L 由下式定义 其中 为了给出状态转换下微分方程的结果, 定义如下方程 $$\left\{ \begin{array}{rl} {\rm d}X(t) = &f(X(t), r(t)){\rm d}t+g(X(t), r(t)){\rm d}B(t), \\ x(0) = &x_{0}, \quad r(0) = r_{0}, \end{array} \right.$$ 其中 f:\mathbb{R} ^{n}\times\hbar\rightarrow \mathbb{R} ^{n}, g:\mathbb{R} ^{n}\times\hbar\rightarrow \mathbb{R} ^{n\times l}.$$ A(x, k) = g(x, k)g^{T}(x, k) = (A_{ij})_{n\times n}, $$V:\mathbb{R} \times\hbar\rightarrow \mathbb{R} ^{n} 均为二阶连续可微函数. 定义算子 引理2.2[20] 若满足以下条件 ⅰ) 对任意的 i\neq j, \gamma_{ij}>0, ⅱ)对于任意的 k\in\hbar, D(x, k) = (d_{ij}(x, k)) 是对称的, 且对于任意的 x, \xi\in\mathbb{R} ^n 满足 \sigma|\xi|^{2}\leq\langle D(x, k)\xi, \xi\rangle\leq\sigma^{-1}|\xi|^{2}, 其中常数 \sigma\in[0, 1], ⅲ) 存在闭包非空的开集 D, 对于任意的 k\in\hbar, 存在二阶连续可微的非负函数 V(\cdot, k):D^{c}\rightarrow \mathbb{R} ,$$ \alpha>0$时, 对任意的$(x, k)\in D^{c}\times\hbar, $$LV(\cdot, k)\leq-\alpha, 则系统(2.5)是遍历和正常返的, 也就是说有唯一的平稳分布 \mu(\cdot, \cdot), 对任意Borel可测函数 f:\mathbb{R} \times\hbar\rightarrow \mathbb{R} ^{n} 满足 \sum\limits^{N}_{k = 1}\int_{\mathbb{R} ^{n}}|f(x, k)|\mu(dk, k)<+\infty, ## 3 系统(2.3)的随机有界性、持久性以及非平凡正周期解的存在性 定义1[13] 若对于任意的 \delta\in(0, 1) 存在正常数 \varsigma = \varsigma(\delta), 使得对于任意初始值 X(0) = X_0\in\mathbb{R} _+^{3}, 系统(2.3)的解 X(t) 具有以下性质 则称 X(t) 是随机最终有界的. 定义2[21] 若对于任意的 \delta\in(0, 1) 存在一对正常数 \varsigma = \varsigma(\delta), \chi = \chi(\delta), 使得对于任意初始值 X(0) = X_0\in\mathbb{R} _+^{3}, 系统(2.3)的解 X(t) 具有以下性质 则称 X(t) 是随机持久的. 定义参数 定理3.1 若 \lambda_1>0, 且对于任意的初值 (S(0), I(0), V(0))\in \mathbb{R} _+^{3}, 系统(2.3)的解随机最终有界且随机持久. 定义 V = N+\frac{1}{N}, N = S+I+V, 对于 X(t)\in\mathbb{R} _+^{3},$$ |X(t)|\rightarrow \infty$时, 则$V(X(t))\rightarrow \infty.$根据It$\hat{\rm o}$公式可得

根据定理3.1的证明可知, 对于任意的$0<\delta<1,$存在$\Omega_\delta\in\Omega,$满足${\Bbb P}\{\Omega_\delta\}\geq1-\delta,$并且对于所有的$\omega\in\Omega_\delta$都有$N<\frac{H}{\delta} = :\lambda_2,$定义函数

$\omega(t) $$T -周期函数并且满足 \omega'(t) = R_0(t)-\langle R_0\rangle_T, 选取足够小的 n 使 R_0(n, t)>0,$$ n\rightarrow0$时, 可得

## 5 遍历平稳分布

对$i\neq j, \gamma_{ij}>0$成立, 其扰动矩阵为

$0<n<1, 0<\theta<1, M$为正常数. 根据It$\hat{\rm o}$公式得

## 7 数值模拟

### 图 1

(1) 当$r(t) = 1$时, $A = 2, \beta = 3, \mu = 0.65, $$\gamma = 0.2,$$ p = 0.5, q = 0.2, $$\varepsilon = 0.25,$$ \alpha = 0.45, \sigma_{11} = 0.1, \sigma_{12} = 0.01, \sigma_{21} = 0.05, \sigma_{22} = 0.01, \sigma_{31} = 0.05, \sigma_{32} = 0.01.$

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