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Mathematical Biosciences and Engineering

2014, Volume 11, Issue 3: 449-469. doi: 10.3934/mbe.2014.11.449
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The global stability of an SIRS model with infection age

  • 1.
    Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi
  • Received: 01 November 2012 Accepted: 29 June 2018 Published: 01 January 2014

  • MSC : Primary: 34D23; Secondary: 34G20, 35B35, 92D30.

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    Infection age is an important factor affecting the transmission ofinfectious diseases. In this paper, we consider an SIRS modelwith infection age, which is described by a mixed system ofordinary differential equations and partial differentialequations. The expression of the basic reproduction number$\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then themodel only has the disease-free equilibrium, while if$\mathscr{R}_0>1$ then besides the disease-free equilibrium themodel also has an endemic equilibrium. Moreover, if$\mathscr{R}_0<1 then="" the="" disease-free="" equilibrium="" is="" globally="" asymptotically="" stable="" otherwise="" it="" is="" unstable="" if="" mathscr="" r="" _0="">1$ then the endemicequilibrium is globally asymptotically stable under additional conditions. The local stabilityis established through linearization. The global stability of thedisease-free equilibrium is shown by applying the fluctuationlemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.

    Citation: Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age[J]. Mathematical Biosciences and Engineering, 2014, 11(3): 449-469. doi: 10.3934/mbe.2014.11.449

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  • Infection age is an important factor affecting the transmission ofinfectious diseases. In this paper, we consider an SIRS modelwith infection age, which is described by a mixed system ofordinary differential equations and partial differentialequations. The expression of the basic reproduction number$\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then themodel only has the disease-free equilibrium, while if$\mathscr{R}_0>1$ then besides the disease-free equilibrium themodel also has an endemic equilibrium. Moreover, if$\mathscr{R}_0<1 then="" the="" disease-free="" equilibrium="" is="" globally="" asymptotically="" stable="" otherwise="" it="" is="" unstable="" if="" mathscr="" r="" _0="">1$ then the endemicequilibrium is globally asymptotically stable under additional conditions. The local stabilityis established through linearization. The global stability of thedisease-free equilibrium is shown by applying the fluctuationlemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.


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