Dynamic analysis of the recurrent epidemic model

Hui Cao; Dongxue Yan; Ao Li; Hui Cao; Dongxue Yan; Ao Li

doi:10.3934/mbe.2019299

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5972-5990. doi: 10.3934/mbe.2019299

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Dynamic analysis of the recurrent epidemic model

Hui Cao ¹,
Dongxue Yan ^{2
,
,},
Ao Li ³

1.
Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, 710021, P.R. China
2.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, P.R. China
3.
Department of Applied Mathematics, The University of Western Ontario, London, N6A 5B7, Canada

Received: 28 February 2019 Accepted: 11 June 2019 Published: 27 June 2019

In this work, an SIRS model with age structure is proposed for recurrent infectious disease by incorporating temporary immunity and delay. We formulate the model as an abstract non-densely defined Cauchy problem and derive the conditions for the global stability of disease free equilibrium, the local stability of endemic equilibrium, and the existence of Hopf bifurcation. Both non-periodic and periodic behaviors are possible when the disease persists in population, where time delay plays an important role. Numerical examples are provided to illustrate our theoretical results.
- age-structured model,
- recurrent epidemic,
- C₀-semigroup,
- asymptotical stability,
- Hopf bifurcation
Citation: Hui Cao, Dongxue Yan, Ao Li. Dynamic analysis of the recurrent epidemic model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5972-5990. doi: 10.3934/mbe.2019299

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Abstract

In this work, an SIRS model with age structure is proposed for recurrent infectious disease by incorporating temporary immunity and delay. We formulate the model as an abstract non-densely defined Cauchy problem and derive the conditions for the global stability of disease free equilibrium, the local stability of endemic equilibrium, and the existence of Hopf bifurcation. Both non-periodic and periodic behaviors are possible when the disease persists in population, where time delay plays an important role. Numerical examples are provided to illustrate our theoretical results.

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