## Global Stability of an Epidemic Model with Quarantine and Incomplete Treatment

Feng Lixiang,, Wang Defen,

 基金资助: 宁夏自然科学基金.  2021AAC03256

 Fund supported: the NSF of Ningxia Hui Autonomous Region.  2021AAC03256

Abstract

An epidemic model with quarantine and incomplete treatment is constructed. The model allows for the susceptibles to the unconscious and conscious susceptible compartment. We establish that the global dynamics are completely detremined by the basic reproduction number $R_{0}$. If $R_{0}≤1$, then the disease free equilibrium is globally asymptotically stable. If $R_{0}>1$, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to confirm the analytical results.

Keywords： Epidemic model ; Basic reproduction number ; Global stability ; Lyapunov functional

Feng Lixiang, Wang Defen. Global Stability of an Epidemic Model with Quarantine and Incomplete Treatment. Acta Mathematica Scientia[J], 2021, 41(4): 1235-1248 doi:

## 2 模型介绍

 参数 定义 $\Lambda$ 出生率 $\beta_{E}$ 与潜伏者接触的传播概率 $\beta_{I}$ 与感染者接触的传播概率 $\sigma$ 有意识的易感者接触率调节因子 $p$ 易感者由无意识转为有意识的速率 $r_{1}$ 潜伏者转为隔离者的转移率 $r_{2}$ 潜伏者转为感染者的转移率 $\varepsilon$ 感染者被隔离的速率 $\xi$ 接受治疗的速率 $\delta$ 治疗失败的速率 $\mu$ 自然死亡率 $d$ 因病死亡率

### 图 1

$$$\left\{\begin{array}{ll} {S_{1}}' = \Lambda-\beta_{E} S_{1}E-\beta_{I} S_{1}I-(p+\mu)S_{1}, \\ {S_{2}}' = pS_{1}-\beta_{E}\sigma S_{2}E-\beta_{I}\sigma S_{2}I-\mu S_{2}, \\ {E}' = \beta_{E}E( S_{1}+\sigma S_{2})+\beta_{I}I( S_{1}+\sigma S_{2})-(r_{1}+r_{2}+\mu)E, \\ {I}' = r_{2}E-(\mu+d+\varepsilon)I+\delta T, \\ {Q}' = r_{1}E+\varepsilon I-(\mu+d+\xi)Q, \\ {T}' = \xi Q-(\mu+d+\delta)T. \end{array}\right.$$$

## 3 正性和不变集

$$${X}' = G(X),$$$

$$$G(X) = \left( \begin{array}{l} G_1(X) \\ G_2(X) \\ G_3(X) \\ G_4(X) \\ G_5(X) \\ G_6(X) \end{array} \right) = \left( \begin{array}{c} \Lambda-\beta_{E} S_{1}E-\beta_{I} S_{1}I-(p+\mu)S_{1}\\ pS_{1}-\beta_{E}\sigma S_{2}E-\beta_{I}\sigma S_{2}I-\mu S_{2}\\ \beta_{E}E( S_{1}+\sigma S_{2})+\beta_{I}I( S_{1}+\sigma S_{2})-(r_{1}+r_{2}+\mu)E\\ r_{2}E-(\mu+d+\varepsilon)I+\delta T \\ r_{1}E+\varepsilon I-(\mu+d+\xi)Q \\ \xi Q-(\mu+d+\delta)T \end{array} \right).$$$

### 4.1 无病平衡点与基本再生数

$x = (E, \;I, \;Q, \;T, \;S_{1}, \;S_{2})^T$, 系统(2.1) 可改写为

$$$\begin{array}{ll} {\cal F}(x) = \left( \begin{array}{c} \beta_{E}E( S_{1}+\sigma S_{2})+\beta_{I}I( S_{1}+\sigma S_{2}) \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right), \\ {\cal V}(x) = \left( \begin{array}{c} (r_{1}+r_{2}+\mu)E \\ (\mu+d+\varepsilon)I-r_{2}E-\delta T \\ (\mu+d+\xi)Q-r_{1}E-\varepsilon I\\ (\mu+d+\delta)T-\xi Q\\ \beta_{E} S_{1}E+\beta_{I} S_{1}I+(p+\mu)S_{1}-\Lambda\\ \beta_{E}\sigma S_{2}E+\beta_{I}\sigma S_{2}I+\mu S_{2}-pS_{1} \end{array} \right).\end{array}$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Zhou P , Yang X L , Wang X G , et al.

A pneumonia outbreak associated with a new coronavirus of probable bat origin

Nature, 2020, 579, 270- 273

Li Q , Guan X , Wu P , et al.

Early transmission dynamics in Wuhan, China, of Noval Coronavirus-infected Pneumonia

N Engl J Med, 2020, 382, 1199- 1207

Gao Q W , Zhuang J .

Stability analysis and control strategies for worm attack in mobile networks via a VEIQS propagation model

Appl Math Comput, 2020,

Xing W , Gao J F , Yan Q S , et al.

An epidemic model with saturated media/psychological impact

Journal of Northwest University (Natural Science Edition), 2018, 48 (5): 639- 643

Yan Y , Chen Y , Liu K , et al.

Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system (in Chinese)

Sci Sin Math, 2020, 50 (3): 385- 392

Wand X , Tang S Y , Chen Y , et al.

When will be the resumption of work in wuhan and its surrounding areas during COVID-19 epidemic? A data-driven network modeling analysis (in Chinese)

Sci Sin Math, 2020, 50 (7): 969- 978

Xing Y , Zhang L , Wang X .

Modeling and stability of epidemic model with free-living pathogens growing in the environment

J Appl Anal Comput, 2020, 10 (1): 55- 70

Liu S , Zhang L , Xing Y .

Dynamics of a stochastic heroin epidemic model

J Comput Appl Math, 2019, 351, 260- 269

Deng D , Li Y .

Traveling waves in a nonlocal dispersal SIR epidemic model with treatment

Acta Math Sci, 2020, 40A (1): 72- 102

Cao Z W , Wen X D , Feng W , et al.

Dynamics of a nonautonomous SIRI epidemic model with random perturbations

Acta Math Sci, 2020, 40A (1): 221- 233

Huo H F , Feng L X .

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

Applied Mathematical Modelling, 2013, 37 (3): 1480- 1489

Du Z W , Xu X K , Wu Y , et al.

The serial interval of COVID-19 among publicly reported confirmed cases

medRxiv, 2020,

Huo H F , Zou M X .

Stability of a tuberculosis model with vaccination and isolation treatment

Journal of Lanzhou University of Technology, 2016, 42 (3): 150- 154

Zhang J P , Li Y , Jin Z , et al.

Analysis of the relationship between transmission of COVID-19 in Wuhan and soft quarantine intensity in susceptible population

Acta Math Appl, 2020, 43 (2): 162- 173

COVID-19疫情时滞模型构建与确诊病例驱动的追踪隔离措施分析

Li Q , Xiao Y N , Tang S Y , et al.

Modelling COVID-19 epidemic with time delay and analyzing the strategy of confirmed cases-driven contact tracing followed by quarantine

Acta Math Appl, 2020, 43 (2): 238- 250

Yang Y L , Li J Q , Ma Z E , Liu L J .

Global stability of two models with incomplete for tuberculosis

Chaos, Solitons & Fractals, 2010, 43, 79- 85

Huo H F , Feng L X .

Global stability of an epidemic model with incomplete treatment and vaccination

Discrete Dynamics in Nature and Society, 2012,

Yang X , Chen L .

Permanence and positive periodic solution for the single-species nonautonomous delay diffusive model

Comput Math Appl, 1996, 32 (4): 109- 116

van den Driessche P , Watmough J .

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Math Biosci, 2002, 180, 29- 48

LaSalle J P. The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Philadelphia: SIAM, 1976

/

 〈 〉