The dynamics and application of a stochastic delayed SIS epidemic model with vaccination

Abstract

In this article, based on the existing literature, we extend a deterministic delayed SIS epidemic model with vaccination to the stochastic version. First, we show that the solution of system is globally unique, positive and stochastically bounded. Then, constructing suitable Lyapunov function, we study the stochastic stability of system at the disease-free equilibrium. Moreover, the threshold ${\widehat{R}}_0$ which determines the extinction or persistence in time mean of the disease is proposed. When ${\widehat{R}}_0$ is less than 1, the disease will disappear; while ${\widehat{R}}_0$ is greater than 1, the disease will persist. Moreover, we study the existence of the ergodicity and the stationary distribution of the model, which can help us better understand the dynamic behavior and statistical characteristics of stochastic delayed biological models. Finally, we apply the results to hepatitis B in Chinese mainland. Using the nonlinear Least-Square method, we estimate the parameter of the model and obtain the threshold ${\widehat{R}}_0=0.0556.$ We forecast that hepatitis B infections in Chinese mainland will drop to about 20 million in 2100 from 90 million in 2015, and give the probability densities of Hepatitis B infections each year.

Keywords:

• The threshold
• stochastic perturbations
• vaccination
• time delay
• extinction
• stationary distribution

Acknowledgments

The authors would like to thank the anonymous referees and the editor for their very helpful comments and suggestions.

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Funding

This work is partially supported by the Nature Science Foundation of China (Grant No. 11661050, 11861044), the NSF of Gansu Province of China (061707), the Development Program for HongLiu Outstanding Young Teachers in Lanzhou University of Technology (Q201308), the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.