# American Institute of Mathematical Sciences

October 2017, 14(5&6): 1407-1424. doi: 10.3934/mbe.2017073

## 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

* Corresponding author

Received  March 11, 2017 Accepted  April 2017 Published  May 2017

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 $\mathcal{R}_0>1$ 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. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073
##### References:
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Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [12] Y. Hsieh, J. Liu, Y. Tzeng and J. Wu, Impact of visitors and hospital staff on nosocomial transmission and spread to community, J. Theor. Biol., 356 (2014), 20-29. doi: 10.1016/j.jtbi.2014.04.003. [13] W. O. Kermack and 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 and Y. Yang, Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092. [15] R. Olinky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics, J. Math. Biol., 56 (2008), 827-839. doi: 10.1007/s00285-007-0140-4. [16] I. B. Schwartz, Small amplitude, long periodic out breaks in seasonally driven epidemics, J. Math. Biol., 30 (1992), 473-491. doi: 10.1007/BF00160532. [17] H. L. Smith, Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17 (1983), 179-190. doi: 10.1007/BF00305758. [18] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511530043. [19] L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. [20] Y. Tang, D. Huang, S. Ruan and 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. doi: 10.1137/070700966. [21] S. Towers and Z. Feng, Social contact patterns and control strategies for influenza in the elderly, Math. Biosci., 240 (2012), 241-249. doi: 10.1016/j.mbs.2012.07.007. [22] S. Towers, K. Vogt Geisse, Y. Zheng and Z. Feng, Antiviral treatment for pandemic influenza: Assessing potential repercussions using a seasonally forced SIR model, J. Theor. Biol., 289 (2011), 259-268. doi: 10.1016/j.jtbi.2011.08.011. [23] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [24] W. Wang and S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793. doi: 10.1016/j.jmaa.2003.11.043. [25] W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [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. doi: 10.3934/dcdsb.2016.21.699. [27] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [28] F. Zhang and X. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085. [29] X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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##### References:
 [1] M. E. Alexander and S. M. Moghadas, Bifurcation analysis of SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816. doi: 10.1137/040604947. [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 and J. Wu, A model for influenza with vaccination and antiviral treatment, J. Theor. Biol., 253 (2008), 118-130. doi: 10.1016/j.jtbi.2008.02.026. [4] L. Bourouiba, A. Teslya and J. Wu, Highly pathogenic avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling, J. Theor. Biol., 271 (2011), 181-201. doi: 10.1016/j.jtbi.2010.11.013. [5] F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [6] J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math. Biol., 53 (2006), 520-539. doi: 10.1007/s00285-006-0382-6. [7] O. Diekmann, J. A. P. Heesterbeek and 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. doi: 10.1007/BF00178324. [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. doi: 10.1007/978-3-642-93048-5_1. [9] D. J. D. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670. doi: 10.1126/science.287.5453.667. [10] Z. Guo, L. Huang and X. Zou, Impact of discontinuous treatments on disease dynamics in an SIR epidemic model, Math. Biosci. Eng., 9 (2012), 97-110. doi: 10.3934/mbe.2012.9.97. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [12] Y. Hsieh, J. Liu, Y. Tzeng and J. Wu, Impact of visitors and hospital staff on nosocomial transmission and spread to community, J. Theor. Biol., 356 (2014), 20-29. doi: 10.1016/j.jtbi.2014.04.003. [13] W. O. Kermack and 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 and Y. Yang, Containing pandemic influenza with antiviral agents, Am. J. Epidemiol., 159 (2004), 623-633. doi: 10.1093/aje/kwh092. [15] R. Olinky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics, J. Math. Biol., 56 (2008), 827-839. doi: 10.1007/s00285-007-0140-4. [16] I. B. Schwartz, Small amplitude, long periodic out breaks in seasonally driven epidemics, J. Math. Biol., 30 (1992), 473-491. doi: 10.1007/BF00160532. [17] H. L. Smith, Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17 (1983), 179-190. doi: 10.1007/BF00305758. [18] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Univ. Press, 1995. doi: 10.1017/CBO9780511530043. [19] L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. [20] Y. Tang, D. Huang, S. Ruan and 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. doi: 10.1137/070700966. [21] S. Towers and Z. Feng, Social contact patterns and control strategies for influenza in the elderly, Math. Biosci., 240 (2012), 241-249. doi: 10.1016/j.mbs.2012.07.007. [22] S. Towers, K. Vogt Geisse, Y. Zheng and Z. Feng, Antiviral treatment for pandemic influenza: Assessing potential repercussions using a seasonally forced SIR model, J. Theor. Biol., 289 (2011), 259-268. doi: 10.1016/j.jtbi.2011.08.011. [23] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [24] W. Wang and S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793. doi: 10.1016/j.jmaa.2003.11.043. [25] W. Wang and X. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [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. doi: 10.3934/dcdsb.2016.21.699. [27] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [28] F. Zhang and X. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085. [29] X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.
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