Dynamical analysis and optimal control of an multi-age-structured vector-borne disease model with multiple transmission pathways

Based on the diversity of transmission routes and host heterogeneity of some infectious diseases, a dynamical model with multi-age-structured, asymptomatic infections, as well as horizontal and vectorial transmission, is proposed. First, the existence and uniqueness of the global positive solution of this model is discussed and the exact expression of the basic reproduction number ${\mathcal{R}}_0$ is obtained using the linear approximation method. Further, we deduce that the disease-free steady state ${\mathcal{E}}^0$ is globally asymptotically stable for ${\mathcal{R}}_0<1$, the endemic steady state ${\mathcal{E}}^{\ast }$ exists and the disease is persistent for ${\mathcal{R}}_0>1$. In addition, the locally asymptotically stability of ${\mathcal{E}}^{\ast }$ is also obtained under some certain conditions. Next, our model is extended to a control problem and the existence and uniqueness of the optimal control by using the Gateaux derivative. Finally, numerical simulations are used to explain the main theoretical results and discuss the impact of age-structured parameters and control strategies on the prevention and control of vector-borne infectious diseases.