Abstract
In this paper, we propose and analyze a stochastic SIVS model with saturated incidence and Lévy jumps. We first prove the existence of a global positive solution of the model. Then, with the help of semimartingale convergence theorem, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. At last, we further study the threshold dynamics of a stochastic SIRS model with saturated or bilinear incidence by a similar method and carry out some numerical simulations to demonstrate our theoretical results. Comparing with the method given by Zhou and Zhang (Physica A 446:204–216, 2016), we find that the method used in this paper is simple and effective.
Highlights
In recent years a large number of mathematical models based on the mechanism of disease transmission have been formulated to help us better understand how the disease spreads in the future because explicit elements of biology and behavior are included in the models [2,3,4]
Epidemic models are inevitably affected by random environmental fluctuations, which play an important role in the study of transmission dynamics of infectious diseases [5,6,7]
Fan et al [15] established a class of SIR epidemic models with generalized nonlinear incidence rate and gave some sufficient conditions guaranteeing the extinction and persistence of the epidemic disease
Summary
In recent years a large number of mathematical models based on the mechanism of disease transmission have been formulated to help us better understand how the disease spreads in the future because explicit elements of biology and behavior are included in the models [2,3,4]. Liu et al [14] considered a stochastic SIRS epidemic model with standard incidence and established sufficient conditions for the existence of ergodic stationary distribution of the model. Fan et al [15] established a class of SIR epidemic models with generalized nonlinear incidence rate and gave some sufficient conditions guaranteeing the extinction and persistence of the epidemic disease. Rifhat et al [16] studied the dynamics of a class of periodic stochastic SIS epidemic models with general nonlinear incidence and gave sufficient conditions for the existence of a stochastic nontrivial periodic solution. Cai et al [17] proposed a stochastic SIRS epidemic model with nonlinear incidence rate and presented a stochastic threshold that determines the outcome of the disease
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
