Search Advanced

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1407-1424. doi: 10.3934/mbe.2017073
Previous Article Next Article
 

Dynamics of epidemic models with asymptomatic infection and seasonal succession

  • 1.
    School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, China
  • 2.
    Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
  • Received: 11 March 2017 Accepted: 01 April 2017 Published: 01 October 2017

  • MSC : Primary: 92D25, 34C23; Secondary: 34D23

  • In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0≤ 1$, and as $\mathcal{R}_0 \gt 1$ the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

    Citation: Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1407-1424. doi: 10.3934/mbe.2017073

    Related Papers:

  • In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0≤ 1$, and as $\mathcal{R}_0 \gt 1$ the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.


    加载中
    [1] [ M. E. Alexander,S. M. Moghadas, Bifurcation analysis of SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005): 1794-1816.
    [2] [ R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York, 1991.
    [3] [ J. Arino,F. Brauer,P. van den Driessche,J. Watmoughd,J. Wu, A model for influenza with vaccination and antiviral treatment, J. Theor. Biol., 253 (2008): 118-130.
    [4] [ L. Bourouiba,A. Teslya,J. Wu, Highly pathogenic avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling, J. Theor. Biol., 271 (2011): 181-201.
    [5] [ F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001.
    [6] [ J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math. Biol., 53 (2006): 520-539.
    [7] [ O. Diekmann,J. A. P. Heesterbeek,J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in themodels for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990): 365-382.
    [8] [ K. Dietz, The incidence of infectious diseases under the influence of seasonal fluctuations, Lecture Notes in Biomath, 11 (1976): 1-15, Berlin-Heidelberg-New York: Springer.
    [9] [ D. J. D. Earn,P. Rohani,B. M. Bolker,B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000): 667-670.
    [10] [ Z. Guo,L. Huang,X. Zou, Impact of discontinuous treatments on disease dynamics in an SIR epidemic model, Math. Biosci. Eng., 9 (2012): 97-110.
    [11] [ H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000): 599-653.
    [12] [ Y. Hsieh,J. Liu,Y. Tzeng,J. Wu, Impact of visitors and hospital staff on nosocomial transmission and spread to community, J. Theor. Biol., 356 (2014): 20-29.
    [13] [ W. O. Kermack,A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. R. Soc. London A: Math., 115 (1927): 700-721.
    [14] [ I. M. Longini,M. E. Halloran,A. Nizam,Y. Yang, Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004): 623-633.
    [15] [ R. Olinky,A. Huppert,L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics, J. Math. Biol., 56 (2008): 827-839.
    [16] [ I. B. Schwartz, Small amplitude, long periodic out breaks in seasonally driven epidemics, J. Math. Biol., 30 (1992): 473-491.
    [17] [ H. L. Smith, Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17 (1983): 179-190.
    [18] [ H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Univ. Press, 1995.
    [19] [ L. Stone,R. Olinky,A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007): 533-536.
    [20] [ Y. Tang,D. Huang,S. Ruan,W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008): 621-639.
    [21] [ S. Towers,Z. Feng, Social contact patterns and control strategies for influenza in the elderly, Math. Biosci., 240 (2012): 241-249.
    [22] [ S. Towers,K. Vogt Geisse,Y. Zheng,Z. Feng, Antiviral treatment for pandemic influenza: Assessing potential repercussions using a seasonally forced SIR model, J. Theor. Biol., 289 (2011): 259-268.
    [23] [ P. Van den Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002): 29-48.
    [24] [ W. Wang,S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004): 775-793.
    [25] [ W. Wang,X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008): 699-717.
    [26] [ D. Xiao, Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dynam. Syst. -B, 21 (2016): 699-719.
    [27] [ D. Xiao,S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007): 419-429.
    [28] [ F. Zhang,X. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007): 496-516.
    [29] [ X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(3554) PDF downloads(953) Cited by(13)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog