Dynamical behaviors and probability density function of a stochastic multi-strain SEIR model with nonlinear incidence

Abstract In this paper, we propose a stochastic multi-strain SEIR epidemic model with nonlinear incidence rates and study its dynamical behaviors. We first obtain the existence and uniqueness of the global positive solution of the stochastic model with any positive initial value. Then, by constructing some appropriate Lyapunov functions, we derive that there exists a unique ergodic stationary distribution of positive solutions to this stochastic model when the thresholds <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> </m:math> ${R}_{0}^{{s}_{1}}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msubsup> </m:math> ${R}_{0}^{{s}_{2}}$ are greater than one. Furthermore, the sufficient conditions for the extinction of the latent and infectious population are provided. Under the same conditions as the stationary distribution, the exact expression of the log-normal density function near the quasi-infected steady state is also obtained. Finally, the above theoretical results are verified numerically.