A New Hybrid Stochastic SIS Co-Infection Model with Two Primary Strains Under Markov Regime Switching and Lévy Jumps

We study a hybrid stochastic SIS co-infection model for two primary strains and a co-infected class with Crowley–Martin incidence, Markovian regime switching, and Lévy jumps. The model is a four-dimensional regime-switching Lévy-driven SDE system with state-dependent diffusion and jump coefficients. Under natural integrability conditions on the jumps and a mild structural assumption on removal rates, we prove uniform high-order moment bounds for the total population, establish pathwise sublinear growth, and derive strong laws of large numbers for all Brownian and Lévy martingales, reducing the long-time analysis to deterministic time averages. Using logarithmic Lyapunov functionals for the infective classes, we introduce four noise-corrected effective growth parameters λ1,…,λ4 and two interaction matrices A,B that encode the combined impact of Crowley–Martin saturation, regime switching, and jump noise. In terms of explicit inequalities involving λk and the entries of A,B, we obtain sharp almost-sure criteria for extinction of all infectives, persistence with competitive exclusion, and coexistence in mean of both primary strains, together with the induced long-term behaviour of the co-infected class. Numerical simulations with regime switching and compensated Poisson jumps illustrate and support these thresholds. This provides, to our knowledge, the first rigorous extinction-exclusion-coexistence theory for a multi-strain SIS co-infection model under the joint influence of Crowley–Martin incidence, Markov switching, and Lévy perturbations.