Analysis of stochastic SEIR(S) models with random total populations and variable diffusion rates

Abstract

A stochastic SEIR(S) model with random total population, overall saturation constant $K>0$ and general, local Lipschitz-continuous diffusion rates is presented. We prove the existence of unique, Markovian, continuous time solutions w.r.t. filtered, complete probability spaces on certain, bounded 4D prisms. The total population $N(t)$ is governed by kind of stochastic logistic equations, which allows to have an asymptotically stable maximum population constant $K>0$. Under natural conditions on our SEIR(S) model, we establish asymptotic stochastic and moment stability of the disease-free and endemic equilibria. Those conditions naturally depend on the basic reproduction number $\mathcal{R}_0$, the growth parameter $\mu>0$ and environmental noise intensity $\sigma_5^2$ coupled with the maximum threshold $K^2$ of total population $N(t)$. For the mathematical proofs, the technique of appropriate Lyapunov functionals $V(S(t),E(t),I(t),R(t))$ is exploited. Some numerical simulations of the expected Lyapunov functionals $\mathbb{E} [V(S,E,I,R)]$ depending on several parameters and time $t$ support our findings.

Keywords

• stochastic seir(s) model
• stochastic differential equations
• random transition functions
• variable diffusion rates
• lyapunov functionals
• asymptotic stability