Threshold behavior and stability analysis of an age–space structured vector–host epidemic model with multiple ages and relapse

To explore the impact of spatial heterogeneity, population movement inevitability, relapse and vector infection ages prevalence on vector-borne disease transmission, we developed an age–space structured disease model to characterize pathogen spread between vectors and hosts. Initially, the well-posedness of solutions of this model is established by applying the Banach fix point theorem and induction. Subsequently, to establish the threshold condition for disease extinction, we derive the next-generation operator based on renewal process theory and obtain an abstract expression for the basic reproduction number ℛ0. Furthermore, under a spatially homogeneous condition, we derive the exact analytical expression for the ℛ0∗. Based on this, the local and global stability of the disease-free steady state and positive steady state are established through analyzing the distribution of roots of characteristic equations and constructing suitable Lyapunov functionals. Numerical simulations across temporal, age, and spatial dimensions explain the theoretical results. In addition, we also reveal how age-dependent parameters and individual diffusions influence the spatio-temporal transmission of disease. Notably, utilizing reported Malaria cases from Yunnan Province, China, we conduct data comparison and partial rank correlation coefficient to validate the model’s applicability while offering suggestions for disease prevention and control.