Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity
Abstract Infected individuals often obtain or lose immunity after recovery in medical studies. To solve the problem, this paper proposes a stochastic SIRS epidemic model with a general incidence rate and partial immunity. Through an appropriate Lyapunov function, we obtain the existence and uniquene...
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Format: | Article |
Language: | English |
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SpringerOpen
2022-11-01
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Series: | Boundary Value Problems |
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Online Access: | https://doi.org/10.1186/s13661-022-01668-0 |
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author | Tao Chen Zhiming Li |
author_facet | Tao Chen Zhiming Li |
author_sort | Tao Chen |
collection | DOAJ |
description | Abstract Infected individuals often obtain or lose immunity after recovery in medical studies. To solve the problem, this paper proposes a stochastic SIRS epidemic model with a general incidence rate and partial immunity. Through an appropriate Lyapunov function, we obtain the existence and uniqueness of a unique globally positive solution. The disease will be extinct under the threshold criterion. We analyze the asymptotic behavior around the disease-free equilibrium of a deterministic SIRS model. By using the Khasminskii method, we prove the existence of a unique stationary distribution. Further, solutions of the stochastic model fluctuate around endemic equilibrium under certain conditions. Some numerical examples illustrate the theoretical results. |
first_indexed | 2024-04-11T08:03:39Z |
format | Article |
id | doaj.art-68a6e63da054483eae3b76767df0d59d |
institution | Directory Open Access Journal |
issn | 1687-2770 |
language | English |
last_indexed | 2024-04-11T08:03:39Z |
publishDate | 2022-11-01 |
publisher | SpringerOpen |
record_format | Article |
series | Boundary Value Problems |
spelling | doaj.art-68a6e63da054483eae3b76767df0d59d2022-12-22T04:35:37ZengSpringerOpenBoundary Value Problems1687-27702022-11-012022112210.1186/s13661-022-01668-0Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunityTao Chen0Zhiming Li1College of Mathematics and System Science, Xinjiang UniversityCollege of Mathematics and System Science, Xinjiang UniversityAbstract Infected individuals often obtain or lose immunity after recovery in medical studies. To solve the problem, this paper proposes a stochastic SIRS epidemic model with a general incidence rate and partial immunity. Through an appropriate Lyapunov function, we obtain the existence and uniqueness of a unique globally positive solution. The disease will be extinct under the threshold criterion. We analyze the asymptotic behavior around the disease-free equilibrium of a deterministic SIRS model. By using the Khasminskii method, we prove the existence of a unique stationary distribution. Further, solutions of the stochastic model fluctuate around endemic equilibrium under certain conditions. Some numerical examples illustrate the theoretical results.https://doi.org/10.1186/s13661-022-01668-0SIRS epidemic modelGeneral incidenceStationary distributionPartial immunity |
spellingShingle | Tao Chen Zhiming Li Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity Boundary Value Problems SIRS epidemic model General incidence Stationary distribution Partial immunity |
title | Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity |
title_full | Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity |
title_fullStr | Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity |
title_full_unstemmed | Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity |
title_short | Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity |
title_sort | dynamics and stationary distribution of a stochastic sirs epidemic model with a general incidence and immunity |
topic | SIRS epidemic model General incidence Stationary distribution Partial immunity |
url | https://doi.org/10.1186/s13661-022-01668-0 |
work_keys_str_mv | AT taochen dynamicsandstationarydistributionofastochasticsirsepidemicmodelwithageneralincidenceandimmunity AT zhimingli dynamicsandstationarydistributionofastochasticsirsepidemicmodelwithageneralincidenceandimmunity |