Global dynamics and optimal control of a vector-borne disease model with class-age structured incubation and discrete incubation, and multiple transmission pathways

Addressing the significant differences in the incubation period dynamics of pathogens between hosts and vectors, as well as their biological heterogeneity, we innovatively construct a hybrid dynamical model that integrates ordinary differential equations (ODEs), partial differential equation (PDE), and delay differential equation (DDE). The core of this model highlights the coupled effects of delays and class-age structure on the spread of infectious diseases, thereby addressing the limitation of traditional models in capturing the combined influence of incubation period delays and structural heterogeneity. First, we prove the existence and uniqueness of global positive solutions for the model and derive the exact analytical expression of the basic reproduction number $\mathcal{R}_{0}$. Second, by constructing some Lyapunov functional adapted to this hybrid structure, we establish the global asymptotic stability of the disease-free and endemic steady states, which are entirely determined by $\mathcal{R}_{0}$, thereby enriching the stability theory system for infectious disease models with delays and class-age structure. Furthermore, based on public health prevention and control practices, we integrate personal protection awareness and vector control measures into the model, and systematically derive the optimal control problem for this model, rigorously prove the existence and uniqueness of the optimal solution, and resolve the key challenge of deriving optimal control strategies for models coupling delays and structural features. Finally, numerical simulations are conducted to verify the reliability of the theoretical conclusions, quantify the sensitivity effects of key parameters (including delay parameters and class-age structural parameters) on $\mathcal{R}_{0}$, and evaluate the regulatory efficacy of different control strategies on the transmission dynamics of infectious diseases. The research outcomes provide core theoretical support and quantitative basis for formulating precise prevention and control strategies for infectious diseases with complex delay structures.