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Mathematical Biosciences and Engineering

2016, Volume 13, Issue 1: 227-247. doi: 10.3934/mbe.2016.13.227
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A note on dynamics of an age-of-infection cholera model

  • 1.
    School of Mathematical Science, Heilongjiang University, Harbin 150080
  • 2.
    Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501
  • Received: 01 March 2014 Accepted: 29 June 2018 Published: 01 October 2015

  • MSC : Primary: 92D30, 92B05; Secondary: 35B35.

  • A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite ifi(a,t)/i∗(a)=0 on some age interval.In this note, we give a supplement to above paper with necessary mathematical arguments.

    Citation: Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an age-of-infection cholera model[J]. Mathematical Biosciences and Engineering, 2016, 13(1): 227-247. doi: 10.3934/mbe.2016.13.227

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  • A recent paper [F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite ifi(a,t)/i∗(a)=0 on some age interval.In this note, we give a supplement to above paper with necessary mathematical arguments.


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