Global dynamics and optimal control of an environmentally-driven epidemic model on heterogeneous networks

Considering environmental transmission and the heterogeneity of contacts between people, an infectious disease model with environmental transmission on a heterogeneous network is proposed. First, the non-negativity and boundedness of solutions for this model are verified. Subsequently, the basic reproduction number $R_0$ is derived, serving as the critical threshold governing the model’s dynamics. Specifically, the disease-free equilibrium is globally asymptotically stable when $R_0<1$, disease is uniformly persistent for $R_0>1$. Furthermore, the endemic equilibrium is proved to be globally asymptotically stable when $R_0>1$ and under some additional conditions. To evaluate intervention efficacy, we extend the model to incorporate control strategies, where the effects of uniform immunization, target immunization and acquaintance immunization on disease control are compared. The results demonstrate that targeted immunization achieves superior disease suppression under equivalent immunization intensities. Additionally, applying Pontryagin’s maximum principle, we prove that simultaneous implementation of multiple controls maximizes intervention efficacy while minimizing operational costs. Finally, the theoretical results are explained by numerical simulations and the effects of immunization and comprehensive control are compared under different degree distributions. The results show that comprehensive control is the best choice to prevent and control the spread of the disease under different degree distributions.