﻿ 高维传染病模型的Lyapunov函数构造

# 高维传染病模型的Lyapunov函数构造Lyapunov Functions for Higher-Dimensional Epidemiological Models

Abstract: Lypunov functions for classical epidemiological models are introduced such as SIR, SIRS, SIS and SEIR. Global stability of some epidemiological models is also established.

1. 引言

2. SIR模型

$\left\{\begin{array}{l}\stackrel{˙}{S}=\gamma N-\beta \frac{SI}{N}-p\gamma I\\ \stackrel{˙}{I}=\beta \frac{SI}{N}-\left(\delta -p\gamma \right)I\end{array}$ (1)

$\beta \frac{{S}^{\ast }{I}^{\ast }}{N}=\gamma N-p\gamma {I}^{\ast }=\left(\delta -p\gamma \right){I}^{\ast }$

$\beta \frac{{S}^{\ast }}{N}=\left(\delta -p\gamma \right)$, ${I}^{\ast }=\frac{\gamma }{\delta }N$.

$V\left(S,I\right)={S}^{\ast }\left(\frac{S}{{S}^{\ast }}-\mathrm{ln}\frac{S}{{S}^{\ast }}\right)+\frac{\delta }{\delta -p\gamma }{I}^{\ast }\left(\frac{I}{{I}^{\ast }}-\mathrm{ln}\frac{I}{{I}^{\ast }}\right)$

$\frac{\partial V}{\partial S}=1-\frac{{S}^{\ast }}{S},\frac{{\partial }^{2}V}{\partial {S}^{2}}=\frac{{S}^{\ast }}{{S}^{2}}$,

$\frac{\partial V}{\partial I}=\frac{\delta }{\delta -p\gamma }\left(1-\frac{{I}^{\ast }}{I}\right),\frac{{\partial }^{2}V}{\partial {I}^{2}}=\frac{\delta }{\delta -p\gamma }\frac{{I}^{\ast }}{{I}^{2}}$

$\frac{{\partial }^{2}V}{\partial S\partial I}=0$

$|\begin{array}{cc}\frac{{\partial }^{2}V}{\partial {S}^{2}}& \frac{{\partial }^{2}V}{\partial S\partial I}\\ \frac{{\partial }^{2}V}{\partial S\partial I}& \frac{{\partial }^{2}V}{\partial {I}^{2}}\end{array}|=\left(\frac{{\partial }^{2}V}{\partial {S}^{2}}\frac{{\partial }^{2}V}{\partial {I}^{2}}-\frac{{\partial }^{2}V}{\partial S\partial I}\right)=\frac{1}{{S}^{2}{I}^{2}}\frac{N\gamma }{\beta }>0$

$\begin{array}{c}\stackrel{˙}{V}\left(S,I\right)=\left(1-\frac{{S}^{\ast }}{S}\right)\left(\gamma N-\beta \frac{SI}{N}-p\gamma I\right)+\left(1-\frac{{I}^{\ast }}{I}\right)\left(\beta \frac{SI}{N}-\left(\delta -p\gamma \right)I\right)\\ =\gamma N-\beta \frac{SI}{N}-p\gamma I-\gamma N\frac{{S}^{\ast }}{S}+\beta \frac{{S}^{\ast }I}{N}+p\gamma I\frac{{S}^{\ast }}{S}+\left(I-{I}^{\ast }\right)\left(\beta \frac{S}{N}-\beta \frac{{S}^{\ast }}{N}\right)\delta \frac{N}{\beta {S}^{\ast }}\\ =\gamma N-\beta \frac{SI}{N}-p\gamma I-\gamma N\frac{{S}^{\ast }}{S}+\beta \frac{{S}^{\ast }I}{N}+p\gamma I\frac{{S}^{\ast }}{S}+\left(I-{I}^{\ast }\right)\left(S-{S}^{\ast }\right)\frac{\delta }{{S}^{\ast }}\\ =\gamma N\left(2-\frac{S}{{S}^{\ast }}-\frac{{S}^{\ast }}{S}\right)-p\gamma I\left(2-\frac{S}{{S}^{\ast }}-\frac{{S}^{\ast }}{S}\right)\\ =\left(2-\frac{S}{{S}^{\ast }}-\frac{{S}^{\ast }}{S}\right)\left(\gamma N-p\gamma I\right)\end{array}$

3. SEIR模型

$\begin{array}{l}{S}^{\prime }=-\lambda {I}^{p}{S}^{q}+b-\mu S\\ {E}^{\prime }=\lambda {I}^{p}{S}^{q}-\left(\epsilon +\mu \right)E\\ {I}^{\prime }=\epsilon E-\left(\gamma +\mu \right)I\\ {R}^{\prime }=\gamma I-\mu R\end{array}$ (2)

$p=1$ 时，参数 $\sigma =\frac{\lambda \epsilon }{\left(\epsilon +\mu \right)\left(\gamma +\mu \right)}$ 表示接触数。

$p>1$ 时，令 $p=2,q=1,\tau =\left(\gamma +\mu \right)t,\alpha =\frac{\mu }{\gamma +\mu },\beta =\frac{\epsilon }{\gamma +\mu },a=\frac{\lambda }{\gamma +\mu }$

$\begin{array}{l}{S}^{\prime }=-a{I}^{2}S+\alpha -\alpha S\\ {E}^{\prime }=a{I}^{2}S-\left(\alpha +\beta \right)E\\ {I}^{\prime }=\beta E-I\end{array}$ (3)

$V\left(S,E,I\right)=\left(S-{S}^{\ast }-{S}^{\ast }\mathrm{ln}\frac{S}{{S}^{\ast }}\right)+\frac{\alpha +\beta }{\beta }I\left(1+\frac{1}{p-1}{\left(\frac{{I}^{\ast }}{I}\right)}^{p}\right)+\left(E-{E}^{\ast }\mathrm{ln}E\right)$

$V\left(S,E,I\right)=\left(S-{S}^{\ast }-{S}^{\ast }\mathrm{ln}\frac{S}{{S}^{\ast }}\right)+\frac{\alpha +\beta }{\beta }I+E$

$\frac{\partial V}{\partial S}=1-\frac{1}{S}$, $\frac{\partial V}{\partial E}=1$, $\frac{\partial V}{\partial I}=\frac{\alpha +\beta }{\beta }$

$\begin{array}{c}{V}^{\prime }=\left(1-\frac{1}{S}\right)\left(-a{I}^{2}S+\alpha -\alpha S\right)+\left[a{I}^{2}S-\left(\alpha +\beta \right)E\right]+\frac{\alpha +\beta }{\beta }\left(\beta E-I\right)\\ =a{I}^{2}-\frac{\alpha +\beta }{\beta }I+\alpha \left(1-S\right)\left(1-\frac{1}{S}\right)\le 0\end{array}$

${R}_{0}=\frac{\lambda \epsilon }{\left(\gamma +\mu \right)\left(\epsilon +\mu \right)}=\frac{a\beta }{\alpha +\beta }$$f\left(I\right)=a{I}^{2}-\frac{\alpha +\beta }{\beta }I$，可得当 $I\in \left(0,\frac{1}{{R}_{0}}\right)$ 时， ${R}_{0}<1$$f\left(I\right)=a{I}^{2}-\frac{\alpha +\beta }{\beta }I<0$

$\begin{array}{l}{X}^{\prime }=-\left(a{I}^{2}+2\alpha +\beta \right)X+2aIS\left(Y+Z\right)\\ {Y}^{\prime }=\beta X-\left(a{I}^{2}+\alpha +1\right)Y\\ {Z}^{\prime }=a{I}^{2}Y-\left(\alpha +\beta +1\right)Z\end{array}$ (4)

$V\left(X,Y,Z;S,E,I\right)=\mathrm{sup}\left\{|X|,\frac{E}{I}\left(|Y|+|Z|\right)\right\}$

$V\left(X,Y,Z;S,E,I\right)\ge c\mathrm{sup}\left\{|X|,|Y|,|Z|\right\}$

$\left(X,Y,Z\right)\in {R}_{+}^{3}$$\left(S,E,I\right)\in \gamma$

$\begin{array}{c}{D}_{+}|X\left(t\right)|\le -\left(a{I}^{2}+2\alpha +\beta \right)|X|+2aIS\left(|Y|+|Z|\right)\\ =-\left(a{I}^{2}+2\alpha +\beta \right)|X|+\frac{2a{I}^{2}S}{E}\left\{\frac{E}{I}\left(|Y|+|Z|\right)\right\}\end{array}$ (5)

${D}_{+}|Y\left(t\right)|\le \beta |X|-\left(a{I}^{2}+\alpha +1\right)|Y|$

${D}_{+}|Z\left(t\right)|\le a{I}^{2}|Y|-\left(\alpha +\beta +1\right)|Z|$

$\begin{array}{c}{D}_{+}\frac{E}{I}\left(|Y|+|Z|\right)=\left(\frac{{E}^{\prime }}{I}-\frac{{I}^{\prime }}{E}\right)\frac{E}{I}\left(|Y|+|Z|\right)+\frac{E}{I}{D}_{+}\left(|Y|+|Z|\right)\\ \le \left(\frac{{E}^{\prime }}{E}-\frac{{I}^{\prime }}{I}\right)\frac{E}{I}\left(|Y|+|Z|\right)+\frac{E}{I}\left(\beta |X|-\left(\alpha +1\right)|Y|+\left(\alpha +\beta +1\right)|Z|\right)\\ \le \frac{\beta E}{I}|X|+\left(\frac{{E}^{\prime }}{E}-\frac{{I}^{\prime }}{I}-\alpha -1\right)\frac{E}{I}\left(|Y|+|Z|\right)\end{array}$ (6)

${D}_{+}V\left(t\right)\le \mathrm{sup}\left\{{g}_{1}\left(t\right),{g}_{2}\left(t\right)\right\}V\left( t \right)$

$\frac{{I}^{\prime }}{I}+1=\frac{\beta E}{I}$, $\frac{{E}^{\prime }}{E}+\alpha +\beta =\frac{a{I}^{2}S}{E}$

${D}_{+}V\left(t\right)\le \mathrm{sup}\left\{\frac{{E}^{\prime }}{E}-\alpha ,\frac{2{E}^{\prime }}{E}+\beta -a{I}^{2}\right\}V\left( t \right)$

${\int }_{0}^{\omega }\mathrm{sup}\left\{{g}_{1}\left(t\right),{g}_{2}\left(t\right)\right\}\text{d}t<0$

4. 结论

Lyapunov函数的构造方法同样也适用于竞争捕食系统 [5] [6]。本文考虑了具有非线性传染率的SIR和SEIR传染病模型，此外，具有非线性传染率的SIRS，SEIRS模型，也可以用同样的构造方法来解决全局稳定性。

[1] Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. Taylor &Francis, London.

[2] Busenberg, S. and Cooke, K. (1993) Vetically Transmitted Diseases in Humans: Models and Dynamics. Springer, Berlin.
https://doi.org/10.1007/978-3-642-75301-5

[3] Anderson, R.M. and May, R.M. (1991) Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford.

[4] Li, M.Y. and Muldowney, J.S. (1995) Global Stability for the SEIR Model in Epidemiology. Mathematical Biosciences, 125, 155-164.
https://doi.org/10.1016/0025-5564(95)92756-5

[5] Goh, B.S. (1980) Management and Analysis of Biological Populations. Elsevier Science, Amsterdam.

[6] Cluskey, M. and Connell, C. (2009) Global Stability for an SEIR Epidemiological Model with Varying Infectivity and Infinite Delay. Mathematical Biosciences and Engineering, 6, 603-610.
https://doi.org/10.3934/mbe.2009.6.603

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