Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

Shangzhi Li; Shangjiang Guo

doi:10.3934/dcdsb.2020335

Article Contents

2021, Volume 26, Issue 9: 5101-5134. Doi: 10.3934/dcdsb.2020335

This issue Previous Article Global boundedness of classical solutions to a logistic chemotaxis system with singular sensitivity Next Article Long time localization of modified surface quasi-geostrophic equations

Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps

Shangzhi Li and
Shangjiang Guo^,

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

^* Corresponding author: Shangjiang Guo
^* Corresponding author: Shangjiang Guo

Received: February 29, 2020

Revised: June 30, 2020

Early access: November 2020

Published: September 2021

The first author is supported by China Postdoctoral Science Foundation, and the Post-doctoral Innovative Research Positions in Hubei Province (Grant No. 1232037), the second author is supported by NSFC (Grant Nos. 11671123, 12071446)

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Abstract

This paper is devoted to a stochastic regime-switching susceptible-infected-susceptible epidemic model with nonlinear incidence rate and Lévy jumps. A threshold $ \lambda $ in terms of the invariant measure, different from the usual basic reproduction number, is obtained to completely determine the extinction and prevalence of the disease: if $ \lambda>0 $, the disease is persistent and there is a stationary distribution; if $ \lambda<0 $, the disease goes to extinction and the susceptible population converges weakly to a boundary distribution. Moreover, some numerical simulations are performed to illustrate our theoretical results. It is very interesting to notice that random fluctuations (including the white noise and Lévy noise) acting the infected individuals can prevent the outbreak of disease, that the disease of a regime-switching model may have the opportunity to persist eventually even if it is extinct in one regime, and that the prevalence of the disease can also be controlled by reducing the value of transmission rate of disease.

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Mathematics Subject Classification: Primary: 34K20; Secondary: 92C60.

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Figure 1. Trajectories of solutions to model (2) with parameters (29) in regimes 1 and 2

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Figure 2. Trajectories of susceptible and infected populations of model (2) with parameters (29)

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Figure 3. The joint density distribution of $ (S,I) $ of model (2) with parameters (29). (a) The case without jumps; (b) The case with jumps

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Figure 4. Trajectories of solutions to model (2) with parameters (30) in regimes 1 and 2

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Figure 5. Trajectories of susceptible and infected populations of model (2) with parameters (30)

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Figure 6. The weak convergence of $ S $ to the stationary solution $ \varphi $ of (10) with parameters (30)

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Figure 7. The joint density distribution of $ (S,I) $ of model (2) with parameters (30). (a) The case without jumps; (b) The case with jumps

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Figure 8. Trajectories of solutions to model (2) with parameters (31) in regimes 1 and 2

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Figure 9. Trajectories of susceptible and infected populations of model (2) with parameters (31)

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Figure 10. The weak convergence of $ S $ to the stationary solution $ \varphi $ of (10) with parameters (31)

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Figure 11. The joint density distribution of $ (S,I) $ of model (2) with parameters (31). (a) The case without jumps; (b) The case with jumps

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Figure 12. Trajectories of solutions to model (2) with parameters (32) in regimes 1 and 2

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Figure 13. Trajectories of susceptible and infected populations of model (2) with parameters (32)

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Figure 14. The joint density distribution of $ (S,I) $ of model (2) with parameters (32). (a) The case without jumps; (b) The case with jumps

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Figure 15. Trajectories of solutions to model (2) with parameters (33) in regimes 1 and 2

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Figure 16. Trajectories of susceptible and infected populations of model (2) with parameters (33)

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Figure 17. The joint density distribution of $ (S,I) $ of model (2) with parameters (33). (a) The case without jumps; (b) The case with jumps

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Figure 18. Trajectories of solutions to model (2) with parameters (34) in regimes 1 and 2

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Figure 19. Trajectories of susceptible and infected populations of model (2) with parameters (34) in regimes 1 and 2

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Figure 20. The weak convergence of $ S $ to the stationary solution $ \varphi $ of (10) with parameters (34)

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Figure 21. The joint density distribution of $ (S,I) $ of model (2) with parameters (34). (a) The case without jumps; (b) The case with jumps

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