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Dynamics of an epidemic model with relapse over a two-patch environment

  • 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

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  • 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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    [1] S. W. Martin., Livestock Disease Eradication: Evaluation of the Cooperative State-Federal Bovine Tuberculosis Eradication Program, National Academy Press, Washington, 1994.
    [2] J. Chin., Control of Communicable Diseases Manual, American Public Health Association, Washington, 1999.
    [3] K. E. VanLandingham, H. B. Marsteller, G. W. Ross, F. G. Hayden, Relapse of herpes simplex encephalitis after conventional acyclovir therapy, JAMA, 259 (1988), 1051-1053.
    [4] P. van den Driessche, X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.
    [5] J. Arino, P. van den Driessche, Disease spread in metapopulations, Fields Institute Communications, 48 (2006), 1-12.
    [6] T. Dhirasakdanon, H. R. Thieme, P. van Den Driessche, A sharp threshold for disease persistence in host metapopulations, J. Biol. Dyn., 1 (2007), 363-378.
    [7] M. Salmani, P. van den Driessche, A model for disease transmission in a patchy environment, Discret. Cont. Dyn. Syst. Ser B, 6 (2006), C185-C202.
    [8] W. Wang, X. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006), 1454-1472.
    [9] Y. Xiao, X. Zou, Transmission dynamics for vector-borne diseases in a patchy environment, J. Math. Biol., 69 (2014), 113-146.
    [10] R. M. Almarashi, C. C. McCluskey, The effect of immigration of infectives on disease-free equilibria, J. Math. Biol., 79 (2020), 1015-1028.
    [11] S. Chen, J. Shi, Z. Shuai, Y. Wu, Asymptotic profiles of the steady states for an SIS epidemic patch model with asymmetric connectivity matrix, J. Math. Biol., 80 (2020), 2327-2361.
    [12] J. Li, X. Zou, Generalization of the Kermack-McKendrick SIR model to patch environment for a disease with latency, Math. Model. Natl. Phenom., 4 (2009), 92-118.
    [13] J. Li, X. Zou, Dynamics of an epidemic model with non-local infections for diseases with latency over a patch environment, J. Math. Biol., 60 (2010), 645-686.
    [14] J. W. H. So, J. Wu, X. Zou, Structured population on two patches: modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51.
    [15] J. Yang, H. R. Thieme, An endemic model with variable re-infection rate and application to influenza, Math. Biosci., 180 (2002), 207-235.
    [16] M. V. Barbarossa, G. Röst, Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting, J. Math. Biol., 71 (2015), 1737-1770.
    [17] A. Berman, R. J. Plemmons, Non-negative matrices in the mathematical sciences, Academic Press, London 1979.
    [18] J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Spring, New York, 1993.
    [19] H. L. Smith, Monotone dynamical systems, An introduction to the theory of competitive and cooperative systems, Amer. Math. Soc., Providence, 1995.
    [20] H. L. Smith, P. Waltman, The theory of the chemostat, Cambridge University Press, Cambridge, 1995.
    [21] C. Castillo-Chaves, H. R. Thieme, Asymptotically autonomous epidemic models, In: Arino O et al (eds) Mathematical population dynamics: analysis of heterogeneity, I. Theory of epidemics. Wuerz, Winnipeg, 1995, 33-50.
    [22] K. Mischaikow, H. Smith, H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
    [23] W. M. Hirsch, H. Hanisch, J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Commu. Pure Appl. Math., 38 (1985), 733-753.
    [24] J. A. J. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer-Verlag, New York, 1986.
    [25] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
    [26] P. van den Driessche, L. Wang, X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.
    [27] X. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Can Appl Math Q, 3 (1995), 473-495.
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