Threshold dynamics and Gaussian stationary distribution of a stochastic SIQR epidemic model with spatial diffusion: a semigroup approach
The spread of infectious diseases is strongly influenced by environmental variability, random disturbances, and population mobility across geographical regions. These factors can significantly affect disease transmission and progression, making purely deterministic models insufficient for capturing realistic epidemic dynamics. Environmental noise may modify transmission rates, recovery processes, and other epidemiological parameters, while the movement of individuals contributes to the spatial propagation of infections. Reaction-diffusion epidemic models provide an effective mathematical framework for describing the spatial spread of diseases, whereas stochastic modeling captures the influence of random fluctuations. Combining stochastic effects with spatial diffusion allows epidemic models to represent more realistic biological and environmental conditions. Despite these advantages, rigorous analytical studies for stochastic reaction-diffusion epidemic models that incorporate quarantine strategies are still relatively limited in the existing literature. In this work, we formulate a stochastic SIQR reaction-diffusion epidemic model to describe the spatio-temporal dynamics of four interacting populations: susceptible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S),$$\end{document} infected \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I),$$\end{document} quarantined \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q),$$\end{document} and recovered \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(R).$$\end{document} Diffusion terms are included to represent the spatial movement of individuals, while stochastic perturbations account for environmental randomness influencing disease transmission. Using stochastic transformations together with linear semigroup theory, we establish the existence and uniqueness of a global positive solution to the nonlinear stochastic system. Two threshold parameters, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{e}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{s},$$\end{document} are introduced to characterize the extinction and strong persistence of the infection. For the special case of a constant transmission rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta,$$\end{document} the endemic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{*}$$\end{document} is derived, and the asymptotic behavior of the system in a neighborhood of this equilibrium is investigated using linear algebra methods and Lyapunov matrix equations. The theoretical analysis shows that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{e} < 1,$$\end{document} the infected population becomes extinct at an exponential rate with probability one. In contrast, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}^{s} > 1,$$\end{document} the disease persists strongly in the population. Furthermore, in the vicinity of the endemic equilibrium \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^{*},$$\end{document} the joint distribution of the system states can be approximated by a multivariate normal probability density function. An explicit representation of the corresponding covariance matrix is obtained under several parameter conditions. Numerical simulations support the analytical results and demonstrate how stochastic perturbations and diffusion rates influence both the stability properties and probabilistic behavior of the epidemic system. The proposed stochastic SIQR reaction-diffusion model provides a rigorous analytical framework for studying the combined influence of spatial diffusion and environmental randomness on epidemic dynamics. The derived extinction and persistence thresholds offer clear criteria for determining long-term disease outcomes. Moreover, the probabilistic characterization near the endemic equilibrium improves understanding of fluctuations around steady states and contributes to the theoretical analysis of stochastic epidemic systems in spatially heterogeneous environments.