## 具有时滞和季节性的炭疽模型的动力学分析

1 长安大学理学院 西安 710064

2 西安工程大学理学院 西安 700048

## Dynamics of an Anthrax Epidemiological Model with Time Delay and Seasonality

Zhang Tailei1, Liu Junli,2, Han Mengjie2

1 School of Science, Chang'an University, Xi'an 710064

2 School of Science, Xi'an Polytechnic University, Xi'an 710048

 基金资助: 国家自然科学基金.  11801431陕西省自然科学基础研究计划项目.  2021JM-445陕西省自然科学基础研究计划项目.  2022JM-023

 Fund supported: the NSFC.  11801431the Natural Science Basic Research Plan in Shaanxi Province.  2021JM-445the Natural Science Basic Research Plan in Shaanxi Province.  2022JM-023

Abstract

In this paper, we developed a time-delayed epidemiological model to describe the anthrax transmission, which incorporates seasonality and the incubation period of the animal population. The basic reproduction number $R_{0}$ can be obtained. It is shown that the threshold dynamics is completely determined by the basic reproduction number. If $R_{0}<1$, the disease-free periodic solution is globally attractive and the disease will die out; if $R_{0} >1$, then there exists at least one positive periodic solution and the disease persists. We further investigate the corresponding autonomous system, the global stability of the disease-free equilibrium and the positive equilibrium is established in terms of $[R_0]$. Numerical simulations are carried out to investigate the sensitivity of $R_0$ about the parameters, the effects of vaccination and carcass disposal on controlling the spread of anthrax is also analyzed.

Keywords： Anthrax model ; Time delay ; Basic reproduction number ; Seasonality ; Threshold dynamics

Zhang Tailei, Liu Junli, Han Mengjie. Dynamics of an Anthrax Epidemiological Model with Time Delay and Seasonality. Acta Mathematica Scientia[J], 2022, 42(3): 851-866 doi:

## 2 模型建立

$$$\left\{\begin{array}{ll} { } \frac{{\rm d}S(t)}{{\rm d}t}= \frac{r(t)N(t)}{1+b(t)N(t)}-\beta(t)S(t)B(t)-\mu(t)S(t), \\ { } \frac{{\rm d}E(t)}{{\rm d}t}= \beta(t)S(t)B(t)-\beta(t-\tau)S(t-\tau)B(t-\tau)e^{-\int^t_{t-\tau}\mu(r){{\rm d}r}}-\mu(t)E(t), \\ { } \frac{{\rm d}I(t)}{{\rm d}t}= \beta(t-\tau)S(t-\tau)B(t-\tau)e^{-\int^t_{t-\tau}\mu(r){{\rm d}r}}-(\mu(t)+\alpha(t))I(t), \\ { } \frac{{\rm d}B(t)}{{\rm d}t}= \eta(t)P(t)-\delta(t)B(t), \\ { } \frac{{\rm d}P(t)}{{\rm d}t}= (\mu(t)+\alpha(t))I(t)-k(t)P(t). \end{array}\right.$$$

($\rm H1$) $\bar{r}>\bar{\mu}, \, \, \forall t\geq 0.$

$$$\left\{\begin{array}{l} { }\frac{{\rm d}x(t)}{{\rm d}t}= \frac{r(t)x(t)}{1+b(t)x(t)}-\mu(t)x(t), \\ x(0)> 0. \end{array}\right.$$$

$C=C([-\tau, 0], {{\Bbb R}} ^5)$, $C^+=C([-\tau, 0], {{\Bbb R}} ^5_{+})$.$(C, C^{+})$是一个序巴拿赫空间.对任意给定的连续函数$x: [-\tau, \sigma)\rightarrow {{\Bbb R}} ^5$, $\sigma>0$, 定义$x_t(\theta)=x(t+\theta)$, $\forall \theta\in[-\tau, 0]$, $\forall t\in[0, \sigma)$.

给定$\phi\in {\cal Z}$, 定义$f(t, \phi)=(f_1(t, \phi), f_2(t, \phi), f_3(t, \phi), f_4(t, \phi), f_5(t, \phi))$如下

$$$\left\{\begin{array}{l} { } \frac{{\rm d}I(t)}{{\rm d}t}=\beta(t-\tau)S^*(t-\tau)e^{-\int^t_{t-\tau}\mu(r){{\rm d}r}}B(t-\tau)-(\mu(t)+\alpha(t))I(t), \\ { } \frac{{\rm d}B(t)}{{\rm d}t}=\eta(t)P(t)-\delta(t)B(t), \\ { } \frac{{\rm d}P(t)}{{\rm d}t}=(\mu(t)+\alpha(t))I(t)-k(t)P(t). \end{array}\right.$$$

$F: {{\Bbb R}} \rightarrow {\cal L}(C, {{\Bbb R}} ^3)$, $V(t) $${{\Bbb R}} 上的一个连续的 3\times3 矩阵函数, 定义如下 \Phi(t, s) 为如下系统的演化矩阵 这里 I 是一个 2\times2 的单位矩阵. C_\omega$$ {{\Bbb R}} $${{\Bbb R}} ^3 上的所有连续的 \omega 周期函数组成的序巴拿赫空间, 其正锥为 定义线性算子 L: C_\omega\rightarrow C_\omega 如下 由文献[14], 系统(2.1)的基本再生数 R_0$$ L$的谱半径, 即$R_0=r(L)$.

$({\cal Y}, {\cal Y}_{+})$是一个序巴拿赫空间.

$P(t)$为系统(2.5)的解映射, 即$P(t)\phi=u_t(\phi)$, 这里$u(t, \phi)$是系统(2.5)唯一的解, $u_0=\phi\in {\cal Y}$.$P:=P(\omega)$为系统(2.5)的庞加莱映射.令$r(P) $$P 的谱半径.由文献[14]中定理2.1, 有如下结论: 引理2.3 R_0-1$$ r(P)-1$有相同的符号.

$g_2(t, y)=(\mu(t)+\alpha(t))y_1(t)-k(t)y$.则有

## 3 阈值动力学行为

$M_1=(0, 0, 0, 0, 0)$, $M_2=(S^*(0), 0, 0, 0, 0)$.因为$S^*(t)$为正周期解, 选取充分小的正数$\varepsilon$使得

$$$\left\{\begin{array}{l} { } \frac{{\rm d}I(t)}{{\rm d}t}\geq \beta(t-\tau)(S^*(t-\tau)-\epsilon)e^{-\int^t_{t-\tau}\mu(r){{\rm d}r}}B(t-\tau)-(\mu(t)+\alpha(t))I(t), \\ { } \frac{{\rm d}B(t)}{{\rm d}t}=\eta(t)P(t)-\delta(t)B(t), \\ { } \frac{{\rm d}P(t)}{{\rm d}t}=(\mu(t)+\alpha(t))I(t)-k(t)P(t). \end{array}\right.$$$

## 4 自治系统

$$$\left\{\begin{array}{l} { }\frac{{\rm d}S(t)}{{\rm d}t}= \frac{r N(t)}{1+b N(t)}-\beta S(t)B(t)-\mu S(t), \\ { }\frac{{\rm d}E(t)}{{\rm d}t}= \beta S(t)B(t)-\beta S(t-\tau)B(t-\tau)e^{-\mu \tau}-\mu E(t), \\ { }\frac{{\rm d}I(t)}{{\rm d}t}= \beta S(t-\tau)B(t-\tau)e^{-\mu \tau}-(\mu+\alpha)I(t), \\ { }\frac{{\rm d}B(t)}{{\rm d}t}= \eta P(t)-\delta B(t), \\ { }\frac{{\rm d}P(t)}{{\rm d}t}= (\mu+\alpha)I(t)-k P(t). \end{array}\right.$$$

($\rm H2$) $r>\mu.$

$$$A_0I^2+A_1I+A_2=0,$$$

$$$(y+\mu)(y+\delta)(y+k)(y+\mu+\alpha)(y+\mu-r)=0.$$$

$$$\upsilon^8+e_1\upsilon^6+e_2\upsilon^4+e_3\upsilon^2+e_4=0,$$$

记$S^\infty=\limsup\limits_{t\rightarrow \infty}S(t), $${ } S_\infty=\liminf\limits_{t\rightarrow \infty} S(t). 类似地定义 E^\infty , E_\infty , I^\infty , I_\infty , B^\infty , B_\infty , P^\infty , P_\infty . 动物的总人口 N(t) 满足下面的微分方程 因为 N(0)=S(0)+E(0)+I(0)>0 , 则 \lim\limits_{t\rightarrow \infty}N(t)=\Theta 全局渐近稳定, 这里 \Theta=\frac{r-\mu}{b\mu} .由于系统(4.1)中的第二个方程不在其他方程中出现, 因此考虑下面的极限方程 $$\left\{\begin{array}{ll} { }\frac{{\rm d}S(t)}{{\rm d}t}= \frac{r-\mu}{b}-\beta S(t)B(t)-\mu S(t), \\ { }\frac{{\rm d}I(t)}{{\rm d}t} = \beta S(t-\tau)B(t-\tau)e^{-\mu\tau}-\mu I(t), \\ { }\frac{{\rm d}B(t)}{{\rm d}t} = \eta P(t)-\delta B(t), \\ { }\frac{{\rm d}P(t)}{{\rm d}t} = \mu I(t)-kP(t). \end{array}\right.$$ h(t)=S(t)+e^{\mu\tau}I(t+\tau) , 得 \lim\limits_{t\rightarrow \infty}h(t)=\Theta 全局渐近稳定.进一步考虑如下的极限方程 $$\left\{\begin{array}{l} { }\frac{{\rm d}\bar{I}(t)}{{\rm d}t} = \beta (\Theta-e^{\mu\tau}\bar{I}(t))\bar{B}(t-\tau)e^{-\mu\tau}-\mu \bar{I}(t), \\ { }\frac{{\rm d}\bar{B}(t)}{{\rm d}t} = \eta \bar{P}(t)-\delta \bar{B}(t), \\ { }\frac{{\rm d}\bar{P}(t)}{{\rm d}t} = \mu \bar{I}(t)-k\bar{P}(t). \end{array}\right.$$ 根据引理2.1的证明可知 是系统(4.12)的正向不变集.与定理3.2的证明类似, 得系统(4.12)是一致持续的, 即存在 \zeta>0 , 对于任意给定的 \psi=(\psi_1, \psi_2, \psi_3)\in {\cal D} , \psi_i(0)>0 , i=1, 2, 3 , 系统(4.12)的解 (\bar{I}(t, \psi), \bar{B}(t, \psi), \bar{P}(t, \psi)) 满足 对任意的 \psi\in {\cal D} , \psi_i(0)>0 , i=1, 2, 3 , 记 (\bar{I}(t), \bar{B}(t), \bar{P}(t))=(\bar{I}(t, \psi), \bar{B}(t, \psi), \bar{P}(t, \psi)) .显然 \Theta e^{-\mu\tau}\geq\bar{I}^\infty\geq \bar{I}_\infty\geq \zeta , \bar{B}^\infty\geq \bar{B}_\infty\geq \zeta , \bar{P}^\infty\geq \bar{P}_\infty\geq\zeta .因此存在序列 t^i_n$$ \sigma^i_n$, $i=1, 2, 3$, 使得

$$$\bar{B}^\infty\geq \frac{\mu \bar{I}^\infty e^{\mu\tau}}{\beta(\Theta-e^{\mu\tau}\bar{I}^\infty)}\geq \frac{\mu \bar{I}_\infty e^{\mu\tau}}{\beta(\Theta-e^{\mu\tau}\bar{I}_\infty)}\geq \bar{B}_\infty.$$$

$$$\frac{\eta}{\delta}\bar{P}^\infty\geq \bar{B}^\infty\geq \bar{B}_\infty\geq\frac{\eta}{\delta}\bar{P}_\infty,$$$

$$$\frac{\mu}{k}\bar{I}^\infty\geq \bar{P}^\infty\geq \bar{P}_\infty\geq\frac{\mu}{k}\bar{I}_\infty.$$$

$$$\frac{\eta}{k\delta}\bar{I}^\infty\geq \frac{\bar{I}^\infty e^{\mu\tau}}{\beta(\Theta-e^{\mu\tau}\bar{I}^\infty)} \geq\frac{\bar{I}_\infty e^{\mu\tau}}{\beta(\Theta-e^{\mu\tau}\bar{I}_\infty)}\geq \frac{\eta}{k\delta}\bar{I}_\infty.$$$

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