Spatiotemporal modeling and dynamics of a multi-dimensional water-borne disease model with age-space structure and multiple transmission mechanisms

Accounting for spatial heterogeneity, inevitable population movement, and transmission complexity in water-borne diseases, we formulate a novel partially degenerate reaction-diffusion model incorporating multi-pathway transmission and an age-space structure. We rigorously establish the well-posedness of the model, demonstrating the existence, non-negativity, and boundedness of solutions. Utilizing a renewal process approach, we derive the next-generation operator L, enabling the characterization of the basic reproduction number R̃0 as its spectral radius. Significantly, we provide the rigorous derivation of the explicit variational form for R̃0 innovatively. For spatially homogeneous scenarios, threshold dynamics are established, demonstrating that R0 governs disease persistence: the disease-free steady state is globally asymptotically stable when R0<1, whereas a unique endemic steady state is globally asymptotically stable when R0>1. This global stability is proven through comprehensive analysis of the characteristic equation’s root distribution and the construction of Lyapunov functionals. Numerical simulations visually elucidate spatio-temporal-age disease dynamics, validating theoretical findings and quantitatively assessing the critical impact of restricting latent individual mobility and environmental decontamination on disease mitigation.