Stochastic SIRC epidemic model with time-delay for COVID-19

F A Rihan; H J Alsakaji; C Rajivganthi

doi:10.1186/s13662-020-02964-8

Stochastic SIRC epidemic model with time-delay for COVID-19

Adv Differ Equ. 2020;2020(1):502. doi: 10.1186/s13662-020-02964-8. Epub 2020 Sep 18.

Authors

F A Rihan¹, H J Alsakaji¹, C Rajivganthi²

Affiliations

¹ Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain, 15551 UAE.
² School of Applied Mathematics, Getulio Vargas Foundation, Rio de Janeiro, RJ 22250-900 Brazil.

Abstract

Environmental factors, such as humidity, precipitation, and temperature, have significant impacts on the spread of the new strain coronavirus COVID-19 to humans. In this paper, we use a stochastic epidemic SIRC model, with cross-immune class and time-delay in transmission terms, for the spread of COVID-19. We analyze the model and prove the existence and uniqueness of positive global solution. We deduce the basic reproduction number $R_{0}^{s}$ for the stochastic model which is smaller than $R_{0}$ of the corresponding deterministic model. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov function, and conditions for the extinction of the disease are obtained. Our findings show that white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can occur due to the existence of feedback time-delay (or memory) in the transmission terms.

Keywords: Brownian motion; COVID-19; Cross-immunity; Extinction; Stationary distribution; Stochastic SIRC model.