Sharp Threshold for the Dynamics of a SIRS Epidemic Model With General Awareness- Induced Incidence and Four Independent Brownian Motions

Abstract

In this paper, a stochastic SIRS epidemic model with general awareness-induced and four independent Brownian Motions is established. We verify the global existence of a unique positive solution and find out the noise modified reproduction number R S 0 which is a sharp threshold for the dynamics: If R S 0 < 1, the disease will die out; if R S 0 > 1, the disease persists and there exists a global asymptotically stable stationary distribution under parameter restrictive conditions. Numerical simulations are presented to illustrate the theoretical results.

Artificial Intelligence (AI) and Big Data for Coronavirus (COVID-19) Pandemic: A Survey on the State-of-the-Arts

“…A traditional compartmental model to predict the infectious disease is the SIR (Susceptible-Infected-Removed) model. This SIR model can be expressed by the following non-linear differential equations [64]:…”

“…A traditional compartmental model to predict the infectious disease is the SIR (Susceptible-Infected-Removed) model. This SIR model can be expressed by the following non-linear differential equations [64] : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}\begin{equation*} dS / dt = -\beta S I, \;\; dI / dt = \beta S I - \gamma I, \;\; dR / dt = -\gamma I,\end{equation*} \end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$S $ \end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$I $ \end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$R $ \end{document} denote susceptible, infected, and removed individuals, respectively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\beta $ \end{document} is the transmission rate, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\gamma $ \end{document} is the recovering rate. However, this model is not suitable for the COVID-19 pandemic because of its assumptions, which are 1) the recovered cases will not get infected again and 2) the model simply ignores the time-varying nature of two parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\beta $ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\gamma $ \end{document} .…”

Application of machine learning in the fight against the COVID-19 pandemic: A review

“…SIR (Susceptible-Infected-Removed) is a traditional established model to predict infectious diseases (49). However, this model has some shortcomings including the assumption that recovered patients will not become infected again, ignoring social distancing and quarantine policies and the time-varying nature of some parameters.…”

Impact of Defending Strategy Decision on DDoS Attack

“…e total numbers of computers on the client and server parts are N W and N S , respectively [7]. (H2) Computers on the client part can be classified into three classes: normal clients (W nodes), infected clients (I nodes), and recovered clients (R nodes) [19,20]. Let W(t), I(t), and R(t) represent the proportion of the W, I, and R nodes, respectively, in the total number of computers on the client part at time t. e total number is constantly equal to N W :…”