Asymptotic behaviors and dynamical bifurcation of a stochastic multi-strain epidemic model with jump diffusion

This paper investigated the asymptotic behavior and dynamical bifurcation of a stochastic multi-strain epidemic model with jump diffusion. We defined a threshold parameter $ \lambda $ as the Lyapunov exponent of the total infected population, which incorporates the stationary distribution of strain proportions on the disease-free boundary and a jump-induced correction term arising from the Lévy noise:

$ \lambda = \int_{\Delta} \Bigl[\sum\limits_{i = 1}^n \bigl(\beta_i y_i - (\gamma_i+\mu_i+\eta_i)y_i\bigr) - \frac{1}{2}\bigl(\sum\limits_{i = 1}^n \sigma_{2i} y_i\bigr)^2\Bigr] \mu^*(dy) + \int_{\mathbb{Y}} \Bigl[\ln\bigl(1+\sum\limits_{i = 1}^n y_i f_{2i}(u)\bigr) - \sum\limits_{i = 1}^n y_i f_{2i}(u)\Bigr] \nu(du). $

This threshold provides a necessary and sufficient condition for overall disease persistence ($ \lambda > 0 $) versus extinction ($ \lambda \le 0 $). Moreover, we introduced strain-specific thresholds $ \lambda_i $ and established a competitive exclusion principle: The strain with the largest $ \lambda_i $ dominates, while strains with smaller $ \lambda_i $ go extinct; when two or more strains share the same maximal $ \lambda_i $, they can coexist. Furthermore, $ \lambda $ serves as a dynamical bifurcation point: When $ \lambda \le 0 $, the unique invariant measure is concentrated on the extinction set; when $ \lambda > 0 $, this measure loses stability and a new invariant measure supported on the positive orthant emerges. Numerical simulations confirmed the critical role of $ \lambda $ and illustrated competitive exclusion between strains under different noise intensities.