Influence of stochastic perturbation on an SIRI epidemic model with relapse

ABSTRACT

In this paper, we investigate a stochastic susceptible-infective-removed-infective (SIRI) epidemic model with relapse. We show that the densities of the distributions of the solutions can converge in $L^1$ to an invariant density or can converge weakly to a singular measure under certain condition. We also find the support of the invariant density. Moreover, we establish sharp sufficient criteria for the extinction of the disease in two cases. The results show that the smaller white noise can assure the existence of a stationary distribution which implies the persistence of the disease while the larger white noise can lead to the extinction of the disease.

COMMUNICATED BY:

• Lukasz Stettner

KEYWORDS:

• Stochastic SIRI epidemic model
• Markov semigroups
• stationary distribution
• asymptotic stability
• extinction

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34E10
• 60H10
• 92B05
• 92D30

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ahmed Alsaedi http://orcid.org/0000-0003-3133-7119

Additional information

Funding

This work was supported by the National Natural Science Foundation of P.R. China [No. 11371085], Natural Science Foundation of Guangxi Province [No. 2016GXNSFBA380006].