﻿ 一类具有非线性传染率的SIS传染病接种模型的研究
 上海理工大学学报  2019, Vol. 41 Issue (4): 339-343 PDF

1. 上海健康医学院 文理教学部，上海 201318;
2. 上海健康医学院 健康信息技术与管理学院，上海 201318

SIS Epidemic Model with Vaccination and Nonlinear Incidence Rate
ZHOU Yanli1, PU Guiping2
1. College of Arts and Science, Shanghai University of Medicine and Health Sciences, Shanghai 201318, China;
2. Faculty of Information Technology and Mangement, Shanghai University of Medicine and Health Sciences, Shanghai 201318, China
Abstract: The global stability of a SIS epidemic model with vaccination and nonlinear incidence rate was investigated. The threshold of epidemic equilibriums-basic reproduction, the number R0 was provided, which determines the extinction or prevalence of the disease. If R0≤1, the disease gradually disappears. If R0>1, the infection pesists in its existence and the disease becomes a local disease. The global asymptotical stabilities of the disease-free equilibrium point and endemic equilibrium point were proved by using the Lyapunov-LaSalle invariant set theory.
Key words: SIS epidemic model     vaccination     nonlinear incidence rate     global stability
1 问题的提出

 $\left\{ \begin{array}{l} \dfrac{{{\rm{d}}S}}{{{\rm{d}}t}} = A(1 - q) - \beta f(I)S - (\mu + p)S + \gamma I +\\ \qquad\; [qA + pS(t - \tau )]{{\rm e}^{ - \mu \tau }}\\ \dfrac{{{\rm{d}}I}}{{{\rm{d}}t}} = \beta f(I)S - (\mu + \gamma )I\\ \dfrac{{{\rm{d}}V}}{{{\rm{d}}t}} = (qA + pS) - [qA + pS(t - \tau )]{{\rm e}^{ - \mu \tau }} - \mu V\\ \end{array} \right.$ (1)
 $S(t) + I(t) + V(t) = N(t)$

(H)　 $f(I)$ $[0,\; + \infty )$ 上连续可导， $\dfrac{{f(I)}}{I}$ $[0,\; + \infty )$ 上单调递减，且 $f(0) = 0,{\kern 1pt} {\kern 1pt} f'(0) > 0$

2 模型研究

 $\begin{split} &\quad (S(\omega ),\; I(\omega ),V(\omega )) =\qquad\qquad\qquad\qquad\qquad\qquad\quad\\ &\qquad ({\varphi _1}(\omega ),\;{\varphi _2}(\omega ),{\varphi _3}(\omega )) \in C([ - \tau ,0],\mathbb{R}_ + ^3), \;\; \\ &\qquad{\varphi _1}(0) \geqslant 0,\;\;{\varphi _2}(0) \geqslant 0,\;\;{\varphi _3}(0) \geqslant 0 \\[-10pt]\end{split}$ (2)

 $D = \left\{ {(S,I,V) \in {\mathbb{R}^3}:S \geqslant 0,I \geqslant 0,V \geqslant 0,S + I + V \leqslant \frac{A}{\mu }} \right\}$

 $m = \mathop {\min }\limits_{0 \leqslant t \leqslant {t_1}}\; \left\{ {\beta \frac{{f(I)S}}{I} - (\mu + \gamma )} \right\}$

 $\frac{{{\rm{d}}I}}{{{\rm{d}}t}} \geqslant mI,\;\;\;\;\;\;\;t \in [0,{t_1}]$

$I({t_1}) \geqslant I(0){{\rm e}^{m{t_1}}} > 0,$ 这与 $I({t_1}) = 0$ 矛盾。因此，对所有的 $\;t > 0$ ，有 $I(t) > 0$

 $\begin{split} &\quad {\left. {\frac{{{\rm{d}}S}}{{{\rm{d}}t}}} \right|_{t = {t_1}}} = A(1 - q) - \beta f(I)S({t_1}) - (\mu + p)S({t_1}) +\qquad\quad \qquad\qquad\qquad\qquad\\ &\qquad\gamma I + [qA + pS(t - \tau )]{{\rm e}^{ - \mu \tau }} > 0\end{split}$

$S(0) = {S_0} > 0$ $S({t_1}) = 0$ ，则一定有 ${\left. {\dfrac{{{\rm{d}}S}}{{{\rm{d}}t}}} \right|_{t = {t_1}}} \leqslant {\rm{0}}$ ，这与上式矛盾，故 $S(t) > 0(t > 0)$

 $V(t) = qA\int_{t - \tau }^t {{{\rm e}^{ - \mu s}}} {\rm{d}}s + p\int_{t - \tau }^t {{{\rm e}^{ - \mu (t - s)}}S(s)} {\rm{d}}s$

 ${R_0} = \frac{{\beta f'(0)[A(1 - q) + qA{{\rm e}^{ - \mu \tau }}]}}{{(\mu + \gamma )[\mu + p(1 - {{\rm e}^{ - \mu \tau }})]}}$

a. 若 ${R_0} \leqslant {\rm{1}}$ ，则有唯一的无病平衡点 ${E_0} = ({S_0},$ ${I_0},\;{V_0})$ ，且全局渐近稳定，

 $\begin{split} &\quad{E_0} = ({S_0},\;{I_0},\;{V_0}) = \qquad\qquad\qquad\qquad\qquad\qquad\qquad\\ &\qquad\quad\left(\frac{{A(1 - q) + qA{{\rm e}^{ - \mu \tau }}}}{{\mu + p(1 - {{\rm e}^{ - \mu \tau }})}},\;0,\frac{{(qA + p{S_0})(1 - {{\rm e}^{ - \mu \tau }})}}{\mu }\right)\end{split}$

b. 若 ${R_0} > {\rm{1}}$ ，则存在唯一的正平衡点（地方病平衡点） $E_\tau ^* = (S_\tau ^*,\;I_\tau ^*,\;V_\tau ^*)$ ，且全局渐近稳定，其中， $S_\tau ^*,\;I_\tau ^*,\;V_\tau ^*$ 由下面的式子决定：

 $\left\{ \begin{split} &S_\tau ^*{\rm{ = }}\frac{{(\mu + \gamma )I_\tau ^*}}{{\beta f(I_\tau ^*)}}\\ &(qA + pS_\tau ^*)(1 - {{\rm e}^{ - \mu \tau }}) = \mu \;V_\tau ^*\\ &A(1 - q) - \beta f(I_\tau ^*)S_\tau ^* - (\mu + p)S_\tau ^* + \gamma I_\tau ^* +\\ &\qquad (qA + pS_\tau ^*){{\rm e}^{ - \mu \tau }} = 0 \end{split}\right.$ (3)

 $\begin{split} &\quad {E_0} = ({S_0},\;{I_0},\;{V_0}) =\\ &\qquad\quad\;\left(\frac{{A(1 - q) + qA{{\rm e}^{ - \mu \tau }}}}{{\mu + p(1 - {{\rm e}^{ - \mu \tau }})}},\;0,\frac{{(qA + p{S_0})(1 - {{\rm e}^{ - \mu \tau }})}}{\mu }\right)\qquad\qquad\qquad\end{split}$

 $\begin{split}&\quad\frac{{{\rm d}W(t)}}{{{\rm d}t}} \leqslant (\mu + \gamma ){I^2}(t)\left(\frac{{\beta f'(0)\left(A(1 - q) + qA{{\rm e}^{ - \mu \tau }}\right)}}{{(\mu + \gamma )\left(\mu + p(1 - {{\rm e}^{ - \mu \tau }})\right)}} - 1\right) =\qquad\qquad\qquad\\ &\qquad (\mu + \gamma ){I^2}(t)({R_0} - 1) \leqslant 0\end{split}$

b. 先证明正平衡点的存在唯一性。由 $\,\beta f(I)S -$ $(\mu + \gamma )I{\rm{ = 0}}$ ，可得 $S{\rm{ = }}\dfrac{{(\mu + \gamma )I}}{{\beta f(I)}}$

 $A(1 - q) - \beta f(I)S - (\mu + p)S + \gamma I + (qA + pS){{\rm e}^{ - \mu \tau }} = 0$

 $A(1 - q) - \mu I - \frac{{(\mu + p)(\mu + \gamma )I}}{{\beta f(I)}} + \left(qA + p\frac{{(\mu + \gamma )I}}{{\beta f(I)}}\right){{\rm e}^{ - \mu \tau }} = 0$

 $\begin{split} &\quad H(I) = A(1 - q) - \mu I - \dfrac{{(\mu + p)(\mu + \gamma )I}}{{\beta f(I)}} +\qquad\qquad\qquad\qquad\qquad\\ &\qquad\left(qA + p\dfrac{{(\mu + \gamma )I}}{{\beta f(I)}}\right){{\rm e}^{ - \mu \tau }} \end{split}$

 $H'(I) = - \mu - [\mu + p(1 - {{\rm e}^{ - \mu \tau }})]\dfrac{{(\mu + \gamma )}}{\beta }\dfrac{{f(I) - If'(I)}}{{{f^{\rm{2}}}(I)}}$

$H'(I) < 0$ 。又

 $\begin{split} &\quad \mathop {\lim }\limits_{I \to \infty }H(I) = \mathop {\lim }\limits_{I \to \infty } \Bigg(A(1 - q) - \mu I -\qquad\qquad\qquad \\ &\qquad \frac{{(\mu + \gamma )}}{\beta }(\mu + p(1 - {{\rm e}^{ - \mu \tau }}))\frac{I}{{f(I)}} + qA{{\rm e}^{ - \mu \tau }}\Bigg) < 0\qquad\qquad \qquad\qquad\qquad\end{split}$

 $f(I) = f(0) + f'(0)I + \frac{{f''(0)}}{{2!}}{I^2} + \cdots \approx f'(0)I$

 $\begin{split} & \quad H(I) =A(1 - q) - \mu I -\qquad\qquad\qquad\qquad\qquad\quad\\ &\qquad \frac{{(\mu + \gamma )(\mu + p(1 - {{\rm e}^{ - \mu \tau }}))}}{\beta }\frac{1}{{f'(0)}} + qA{{\rm e}^{ - \mu \tau }}\qquad\qquad\qquad\qquad\qquad\end{split}$

 $\begin{split} & \quad H(0) = A(1 - q) - \frac{{(\mu + \gamma )(\mu + p(1 - {{\rm e}^{ - \mu \tau }}))}}{{\beta f'(0)}} + qA{{\rm e}^{ - \mu \tau }}= \quad\qquad\qquad\qquad\qquad\qquad \\ & \qquad (A(1 - q) + qA{{\rm e}^{ - \mu \tau }})\left(1 - \frac{{(\mu + \gamma )(\mu + p(1 - {{\rm e}^{ - \mu \tau }}))}}{{\beta f'(0)(A(1 - q) + qA{{\rm e}^{ - \mu \tau }})}}\right)= \\ & \qquad (A(1 - q) + qA{{\rm e}^{ - \mu \tau }})\left(1 - \frac{1}{{{R_0}}}\right) \end{split}$

 $\qquad d \!+\! \gamma \!+\! \alpha \!>\! \beta f'(I_\tau ^*)S_\tau ^*,\quad d \!+\! \gamma \!+\! p \!>\! \beta f'(I_\tau ^*)S_\tau ^*$ (4)

 $\begin{split} & \quad{\lambda ^2} + [2\mu + p + \gamma - \beta f'(I_\tau ^*)S_\tau ^* + \beta f(I_\tau ^*)]\lambda +\qquad\qquad\\ &\qquad [\mu + p + \beta f(I_\tau ^*)][\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*] +\\ & \qquad \beta f(I_\tau ^*)[\beta f'(I_\tau ^*)S_\tau ^* - \gamma ] - [p{{\rm e}^{ - d\tau }}\lambda + \\ &\qquad p{{\rm e}^{ - d\tau }}(\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*)]{{\rm e}^{ - \lambda \tau }} = 0 \end{split}\qquad$

 ${\lambda ^2}{\rm{ + }}{p_1}\lambda + {p_0} + [{q_1}\lambda + {q_0}]{{\rm e}^{ - \lambda \tau }} = 0$ (5)

 $\begin{split}{p_1}&{\rm{ = }}2\mu + p + \gamma - \beta f'(I_\tau ^*)S_\tau ^* + \beta f(I_\tau ^*) \\ {p_0} &= (\mu + p)(\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*) + \beta \mu f(I_\tau ^*)\\ {q_1} &= - p{{\rm e}^{ - \mu \tau }}\\ {q_0}& = - p{e^{ - \mu \tau }}(\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*) \end{split}$

$\tau {\rm{ = 0}}$ 时，由以上表达式可得

 ${p_0}{\rm{ + }}{q_0} = \mu (\mu + \alpha + \gamma - \beta f'(I_\tau ^*)S_\tau ^*) + \beta \alpha f(I_\tau ^*)$
 ${p_1}{\rm{ + }}{q_1} = \mu + \beta f(I_\tau ^*) + \mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*$

 ${p_0}{\rm{ + }}{q_0} > 0,\;\;\;\;{p_1}{\rm{ + }}{q_1} > 0$

$\tau {\rm{ > 0}}$ 时，令 $\lambda = {\rm i}w(w > 0)$ 是方程（5）的解，将 $\lambda = {\rm i}w(w > 0)$ 代入方程（5）并分离实部和虚部，

 ${w^2} - {p_0} = {q_0}\cos\; (w\tau) + {q_1}w\sin\; (w\tau)$
 ${p_1}w = {q_0}\sin\; (w\tau) - {q_1}w\cos\;(w\tau)$

 ${w^{\rm{4}}} + (p_1^2 - 2{p_0} - q_1^2){w^2} + p_0^2 - q_0^2 = 0$ (6)

 ${p_0} - {q_0} = [\mu + p(1 + {{\rm e}^{ - \mu \tau }})](\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*) + \beta \mu f(I_\tau ^*)$
 ${p_0} + {q_0} = [\mu + p(1 - {{\rm e}^{ - \mu \tau }})](\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*) + \beta \mu f(I_\tau ^*)$
 $\begin{split} & \quad p_1^2 - 2{p_0} - q_1^2 = {\mu ^2} + 2\mu p +\\ & \qquad {p^2}(1 - {{\rm e}^{ - 2\mu \tau }}) +{[\mu + \gamma - \beta f'(I_\tau ^*)S_\tau ^*]^2} + \\ &\qquad [\mu + p + \gamma - \beta f'(I_\tau ^*)S_\tau ^*]\beta f(I_\tau ^*) + {\beta ^2}{f^2}(I_\tau ^*)\qquad\qquad\qquad\qquad\qquad\qquad \end{split}$

 $p_0^2 - q_0^2 > 0,\;\;\;\;p_1^2 - 2{p_0} - q_1^2 > 0$

 ${W_1}(t) = \dfrac{1}{2}{(S - S_\tau ^* + I - I_\tau ^{\rm{*}} + V - V_\tau ^{\rm{*}})^2}$

 $\begin{split} & \quad \dfrac{{{\rm d}{W_1}(t)}}{{{\rm d}t}} \!\!=\!\! (S \!\!-\!\! S_\tau ^* \!\!+\!\! I \!\!-\!\! I_\tau ^* \!\!+\!\! V \!\!-\!\! V_\tau ^*)\left(\dfrac{{{\rm d}{S_{}}(t)}}{{{\rm d}t}} \!\!+\!\! \dfrac{{{\rm d}{I_{}}(t)}}{{{\rm d}t}}\!\!+\!\! \dfrac{{{\rm d}{V_{}}(t)}}{{{\rm d}t}}\right) \! =\qquad\qquad\qquad \\ &\qquad (S - S_\tau ^* + I - I_\tau ^* + V - V_\tau ^*)(A - \mu S - \mu I - \mu V) \end{split}$

 $\frac{{{\rm d}{W_1}(t)}}{{{\rm d}t}} = - \mu {(S - S_\tau ^{\rm{*}} + I - I_\tau ^{\rm{*}} + V - V_\tau ^{\rm{*}})^2} \leqslant 0$

3 数值模拟和结论

 图 1 取不同 $\tau$ 值时感染人群随时间的变化曲线 Fig. 1 Variation of infective population with time for different value of delay $\tau$

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