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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Main Authors: Tao Chen, Zhiming Li
Format: Article
Language:English
Published: SpringerOpen 2022-11-01
Series:Boundary Value Problems
Subjects:
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.
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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