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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 5: 1083-1106. doi: 10.3934/mbe.2015.12.1083
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Global stability of a multi-group model with vaccination age, distributed delay and random perturbation

  • 1.
    School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049
  • Received: 01 September 2014 Accepted: 29 June 2018 Published: 01 June 2015

  • MSC : 34E10, 37H10, 92D25, 93D20.

  • A multi-group epidemic model withdistributed delay and vaccination age has been formulated and studied.Mathematical analysis shows that the global dynamics of the model is determinedby the basic reproduction number $\mathcal{R}_0$:the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0\leq1$,and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$.Lyapunov functionals are constructed by the non-negative matrix theory and a novel grouping techniqueto establish the global stability.The stochastic perturbation of the model is studied and it is provedthat the endemic equilibrium of the stochastic model is stochastically asymptotically stablein the large under certain conditions.

    Citation: Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation[J]. Mathematical Biosciences and Engineering, 2015, 12(5): 1083-1106. doi: 10.3934/mbe.2015.12.1083

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  • A multi-group epidemic model withdistributed delay and vaccination age has been formulated and studied.Mathematical analysis shows that the global dynamics of the model is determinedby the basic reproduction number $\mathcal{R}_0$:the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0\leq1$,and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$.Lyapunov functionals are constructed by the non-negative matrix theory and a novel grouping techniqueto establish the global stability.The stochastic perturbation of the model is studied and it is provedthat the endemic equilibrium of the stochastic model is stochastically asymptotically stablein the large under certain conditions.


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