The Application of Deterministic and Random SEQIR Models in COVID-19 Pandemic and the Study of Threshold Behavior

Up to now, COVID-19 caused by SARS-CoV-2 is still widely spreading. Most patients have a good prognosis, with some critically ill patients dying. This paper considers the SEQIR COVID-19 model with standard incidence. Based on the characteristics of the model, we study the content of threshold behavior in deterministic and stochastic systems. We can first perform dimensionality reduction on the model due to the fact that the reduced model has the same stability as the equilibrium point of the original model. We first express the local stability of boundary equilibrium points for deterministic system after dimension reduction with the method of Lyapunov functions. After considering the perturbation of logarithmic Ornstein-Uhlenbeck processes, we study the existence and uniqueness of positive solutions. Subsequently, the critical value $R^s_0$ related to the basic regeneration number $R_0$ was obtained. And then, the conditions of $R^s_0$ about the persistence and extinction of the disease is in-depth researched, it is a critical condition. When $R^s_0 < 1,$ the disease tends to become extinct, while when $R^s_0 > 1,$ the system exhibits a stationary distribution. And the density function near the positive equilibrium point is described in detail. Finally, our conclusions are well supported through numerical simulation.