Dynamics and optimal control of a two-strain epidemic model with imperfect vaccine efficacy and saturated incidence on complex networks

High prevalence and frequent outbreaks of infectious diseases are often accompanied by the coexistence and competition of multiple strains, which are significantly influenced by vaccination coverage, vaccine efficacy, and individual behavioral heterogeneity. Therefore, this study constructs a two-strain epidemic model on complex networks incorporating imperfect vaccination and a saturated incidence function. Through mathematical analysis, we proved the existence of a global positive solution, thereby verifying its well-posedness. We then derived the invasion reproduction numbers characterizing the persistence or extinction of the two strains. Through further theoretical analysis, when both basic reproduction numbers are below unity, the disease-free equilibrium is globally asymptotically stable, indicating disease eradication.Furthermore, threshold analysis determines the conditions for the existence and global stability of both the dominant-strain and coexistence equilibria. To explore the impact of personal protective strategies and screening-and-isolation measures on disease dynamics, the model is extended into an optimal control system, and the existence of an optimal solution is established using Pontryagin's maximum principle. Finally, numerical simulations revealed that network topology can reverse competitive outcomes. Higher connectivity in Poisson networks favors strain 1, whereas strong heterogeneity in scale-free networks suppresses it. Moreover, a degree-based dynamic control strategy is particularly effective in scale-free networks, providing new insights into optimizing interventions by leveraging network heterogeneity.