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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 6098-6127. doi: 10.3934/mbe.2020324
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Dynamics of an epidemic model with relapse over a two-patch environment

  • 1.
    School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China
  • 2.
    Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada
  • Received: 23 June 2020 Accepted: 03 September 2020 Published: 14 September 2020

  • In this paper, with the assumption that infectious individuals, once recovered for a period of fixed length, will relapse back to the infectious class, we derive an epidemic model for a population living in a two-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the fixed constant relapse time and a non-local term caused by the mobility of the individuals during the recovered period. We explore the dynamics of the model under two scenarios: (i) assuming irreducibility for three travel rate matrices; (ii) allowing reducibility in some of the three matrices. For (i), we establish the global threshold dynamics in terms of the principal eigenvalue of a 2×2 matrix. For (ii), we consider three special cases so that we can obtain some explicit results, which allow us to explicitly explore the impact of the travel rates. We find that the role that the travel rate of recovered and infectious individuals differs from that of susceptible individuals. There is also an important difference between case (i) and (ii): under (ii), a boundary equilibrium is possible while under (i) it is impossible.

    Citation: Dongxue Yan, Xingfu Zou. Dynamics of an epidemic model with relapse over a two-patch environment[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 6098-6127. doi: 10.3934/mbe.2020324

    Related Papers:

  • In this paper, with the assumption that infectious individuals, once recovered for a period of fixed length, will relapse back to the infectious class, we derive an epidemic model for a population living in a two-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the fixed constant relapse time and a non-local term caused by the mobility of the individuals during the recovered period. We explore the dynamics of the model under two scenarios: (i) assuming irreducibility for three travel rate matrices; (ii) allowing reducibility in some of the three matrices. For (i), we establish the global threshold dynamics in terms of the principal eigenvalue of a 2×2 matrix. For (ii), we consider three special cases so that we can obtain some explicit results, which allow us to explicitly explore the impact of the travel rates. We find that the role that the travel rate of recovered and infectious individuals differs from that of susceptible individuals. There is also an important difference between case (i) and (ii): under (ii), a boundary equilibrium is possible while under (i) it is impossible.


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