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:

$$\lambda ={\int }_{\mathrm{\Delta }}[\sum _{i=1}^n({\beta }_i{y}_i-({\gamma }_i+{\mu }_i+{\eta }_i)y_i)-\frac{1}{2}(\sum _{i=1}^n{\sigma }_{2i}{y}_i{)}^2]{\mu }^{\ast }(dy)+{\int }_{\mathbb{Y}}[\ln (1+\sum _{i=1}^n{y}_i{f}_{2i}(u))-\sum _{i=1}^n{y}_i{f}_{2i}(u)]\nu (du).$$

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.