Discrete-Time Two-Strain Epidemic Dynamics on Complex Networks

We investigate a discrete-time two-strain symbiotic epidemic model on complex networks with both random and long-range interactions. Our analysis examines how the co-infection recovery rate ($\mu$), the long-range decay exponent ($\alpha$), and the scale-free connectivity exponent ($\gamma$) shape epidemic persistence under cooperative dynamics. Comparison with a two-strain competition model shows how these parameters control strain dominance, coexistence, or extinction. The results demonstrate that contagion dynamics are strongly affected by environmental randomness and long-range couplings. In facultative symbiosis, the co-infection recovery rate undergoes a clear phase transition, separating persistence from extinction. In the competitive setting, regimes with $\alpha < 2$ and $\gamma < 3$ markedly lower the epidemic threshold, allowing persistence even at small contagion rates ($\sigma$). Statistical analysis further reveals that $\gamma$ and $\alpha$ exert pronounced, nonlinear, and time-dependent effects on strain survival.