Introducing a novel mean-reverting Ornstein–Uhlenbeck process based stochastic epidemic model

Abstract

The major objective of this paper is to examine a novel mean-reverting Ornstein–Uhlenbeck process-based stochastic SIRD model for transmission the epidemic disease that is a great crisis in numerous societies. For this purpose, the deterministic model is further converted into the stochastic form by allowing the infection rate satisfies the mean-reverting Ornstein–Uhlenbeck process to account the uncertainties involved in epidemic spread. At first using Lyapunov functions, the solution’s uniqueness and positivity will be demonstrated. Subsequently, the stochastic epidemic threshold ℜ0S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re_{0}^{S}$$\\end{document} that controls the disease’s extinction and persistence in the mean is identified analytically. It has been established that when ℜ0S<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re_{0}^{S} < 1$$\\end{document} the disease will extinguish, whereas if ℜ0S>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re_{0}^{S} > 1$$\\end{document} the disease is persistent. At last, several numerical simulations are presented to demonstrate the findings of the hypothetical investigation results. These simulations served to vividly illustrate and validate the implications derived from the hypothetical analysis.