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

Authors
• 1 United Arab Emirates University, Al-Ain, 15551, UAE , Al-Ain (United Arab Emirates)
• 2 Getulio Vargas Foundation, Rio de Janeiro, RJ, 22250-900, Brazil , Rio de Janeiro (Brazil)
Type
Published Article
Journal
Publisher
Springer International Publishing
Publication Date
Sep 18, 2020
Volume
2020
Issue
1
Identifiers
DOI: 10.1186/s13662-020-02964-8
Source
Springer Nature
Keywords
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 R0s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{R}}_{0}^{s}$\end{document} for the stochastic model which is smaller than R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{R}}_{0}$\end{document} 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.