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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 2: 445-461. doi: 10.3934/mbe.2013.10.445
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Dynamics of an infectious diseases with media/psychology induced non-smooth incidence

  • 1.
    Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049
  • 2.
    Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, 710049
  • 3.
    College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062
  • Received: 01 May 2012 Accepted: 29 June 2018 Published: 01 January 2013

  • MSC : 92C50, 92C60, 54B20.

  • This paper proposes and analyzes a mathematical model on aninfectious disease system with a piecewise smooth incidence rateconcerning media/psychological effect. The proposed models extendthe classic models with media coverage by including a piecewisesmooth incidence rate to represent that the reduction factor becauseof media coverage depends on both the number of cases and the rateof changes in case number. On the basis of properties of Lambert Wfunction the implicitly defined model has been converted into apiecewise smooth system with explicit definition, and the globaldynamic behavior is theoretically examined. The disease-free isglobally asymptotically stable when a certain threshold is less thanunity, while the endemic equilibrium is globally asymptoticallystable for otherwise. The media/psychological impact although doesnot affect the epidemic threshold, delays the epidemic peak andresults in a lower size of outbreak (or equilibrium level ofinfected individuals).

    Citation: Yanni Xiao, Tingting Zhao, Sanyi Tang. Dynamics of an infectious diseases with media/psychology induced non-smooth incidence[J]. Mathematical Biosciences and Engineering, 2013, 10(2): 445-461. doi: 10.3934/mbe.2013.10.445

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  • This paper proposes and analyzes a mathematical model on aninfectious disease system with a piecewise smooth incidence rateconcerning media/psychological effect. The proposed models extendthe classic models with media coverage by including a piecewisesmooth incidence rate to represent that the reduction factor becauseof media coverage depends on both the number of cases and the rateof changes in case number. On the basis of properties of Lambert Wfunction the implicitly defined model has been converted into apiecewise smooth system with explicit definition, and the globaldynamic behavior is theoretically examined. The disease-free isglobally asymptotically stable when a certain threshold is less thanunity, while the endemic equilibrium is globally asymptoticallystable for otherwise. The media/psychological impact although doesnot affect the epidemic threshold, delays the epidemic peak andresults in a lower size of outbreak (or equilibrium level ofinfected individuals).


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