A generalized delay-induced SIRS epidemic model with relapse

Shufan Wang; Zhihui Ma; Xiaohua Li; Ting Qi; Shufan Wang; Zhihui Ma; Xiaohua Li; Ting Qi

doi:10.3934/math.2022368

AIMS Mathematics

2022, Volume 7, Issue 4: 6600-6618. doi: 10.3934/math.2022368

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Research article

A generalized delay-induced SIRS epidemic model with relapse

1.
School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu 730000, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received: 17 October 2021 Revised: 23 December 2021 Accepted: 03 January 2022 Published: 21 January 2022
MSC : 37C75, 92B05, 92D25, 93D20

In this paper, a generalized delay-induced $ SIRS $ epidemic model with nonlinear incidence rate, latency and relapse is proposed. Our epidemic model is a generalized one, and the published epidemic models are the special cases of ours under some conditions. By using LaSalle's invariance principle and Lyapunovi's direct method, the dynamical behaviors are investigated and the results show that the disease free-equilibrium $ Q_0 $ is globally asymptotically stable if the basic reproduction number $ R_0 < 1 $ for any time delay. However, if the basic reproduction number $ R_0 > 1 $, there exists a unique endemic equilibrium $ Q_* $ which is locally asymptotically stable under some conditions. Moreover, the effects of latency and relapse on the transmission dynamics of the diseases are analyzed by some numerical experiments which conducted based on $ ODE45 $ in Matlab.
- a delay-induced model,
- generalized nonlinear incidence,
- dynamical behavior,
- numerical experiment
Citation: Shufan Wang, Zhihui Ma, Xiaohua Li, Ting Qi. A generalized delay-induced SIRS epidemic model with relapse[J]. AIMS Mathematics, 2022, 7(4): 6600-6618. doi: 10.3934/math.2022368

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Abstract

In this paper, a generalized delay-induced $ SIRS $ epidemic model with nonlinear incidence rate, latency and relapse is proposed. Our epidemic model is a generalized one, and the published epidemic models are the special cases of ours under some conditions. By using LaSalle's invariance principle and Lyapunovi's direct method, the dynamical behaviors are investigated and the results show that the disease free-equilibrium $ Q_0 $ is globally asymptotically stable if the basic reproduction number $ R_0 < 1 $ for any time delay. However, if the basic reproduction number $ R_0 > 1 $, there exists a unique endemic equilibrium $ Q_* $ which is locally asymptotically stable under some conditions. Moreover, the effects of latency and relapse on the transmission dynamics of the diseases are analyzed by some numerical experiments which conducted based on $ ODE45 $ in Matlab.

References

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