Stochastic Asymptotic Stability of SIR Model with Variable Diffusion Rates

Abstract

We introduce random fluctuations on contact and recovery rates in deterministic SIR model with disease deaths in nonparametric manner and obtain stochastic counterparts with general diffusion coefficients (functional contact and recovery rates). $$\begin{aligned} \displaystyle dS&= \,\Big (-\beta S I +\mu (K-S)\Big )~dt - S I~ F_1\big (S,I,R\big ) ~dW_1\nonumber \\ dI&= \,\Big (\beta S I-\big (\alpha +\gamma +\mu \big )I\Big )~ dt + S I~ F_1\big (S,I,R\big ) ~dW_1-I~F_2\big (S,I,R\big ) ~dW_2 \\ dR&= \,\Big (\alpha I - \mu R\Big )~dt+ I~F_2\big (S,I,R\big ) ~dW_2. \nonumber \end{aligned}$$ (1) The introduced stochastic model has functional diffusion coefficients which contains arbitrary local Lipschitz-continuous functions \(F_i\)’s defined on $$\begin{aligned} {\mathbb {D}}=\{(S,I,R) \in {\mathbb {R}}^3: ~S\ge 0, ~I \ge 0, ~R \ge 0, ~S+I+R\le K\}. \end{aligned}$$ In this paper we prove the global existence of a unique strong solution and discuss stochastic asymptotic stability of disease free and endemic equilibria of the model and visualize our results with some simulations to confirm them.