Complete characterization of dynamical behavior of stochastic epidemic model motivated by black-Karasinski process: COVID-19 infection as a case

To capture the underlying dynamics of the COVID-19 pandemic, we develop a stochastic SEIABR compartmental model, where the concentration of coronaviruses in the environment is considered. This paper is the first attempt to introduce the Black-Karasinski process as the random fluctuations in the modeling of epidemic transmission, and it is shown that Black-Karasinski process is a both mathematically and biologically reasonable assumption compared with existing stochastic modeling methods. We first obtain two critical values R-0(s) and R-0(s) related to the basic reproduction number R-0 of deterministic system. It is theoretically proved that (i) if R-0(s)>1, the stochastic model has a stationary distribution & ell;(<middle dot>), which implies the long-term persistence of COVID-19; (ii) the disease will go extinct exponentially when R-0(s)<1; (iii) R-0(s)=R-0(s)=R-0 if there is no environmental noise in COVID-19 transmission. Then, we study the local stability of the endemic equilibrium P* of deterministic system under R-0>1. By developing an important lemma for solving the relevant Fokker-Planck equation, an approximate expression of probability density function of the distribution & ell;(<middle dot>) around P* is further derived. Finally, several numerical examples are performed to substantiate our theoretical results. It should be mentioned that the techniques and methods of analysis in this paper can be applied to other complex high-dimensional stochastic epidemic systems.