A general analytical framework for nonlinear age-structured-spatial SIR epidemic models: threshold and stability results

We develop a unified operator-theoretic framework for SIR-type epidemic models structured simultaneously by age and space. The host population evolves in a bounded domain under a common diffusion mechanism, with age-dependent recovery and mortality, while transmission is encoded by a general nonlocal incidence functional that covers, as special cases, standard, saturated, and kernel-driven contact structures. In an abstract C0-semigroup setting, the infection subsystem is reformulated as a semilinear evolution equation on a convex subset of a Hilbert product space. On this state space we construct a nonlinear order-preserving map whose Fréchet derivative at the disease-free configuration is a positive, compact linear operator. This linearisation furnishes a genuine next-generation operator for nonlinear age-space systems, and its spectral radius defines a basic reproduction index R0. Within this framework we prove global well-posedness, positivity, and invariance of the epidemic dynamics, and we show that R0 yields a sharp existence-nonexistence criterion for endemic equilibria and fully governs the stability of the disease-free state: R0≤1 precludes nontrivial equilibria and ensures global attraction of the disease-free profile, whereas R0>1 guarantees the existence and local asymptotic stability of endemic equilibria under mild structural assumptions. As a proof of concept, we specialise the theory to a one-dimensional age-space SIR model with bilinear incidence, compute numerically the spectral threshold and the spectra of the linearised operators at both disease-free and endemic states, and perform time-dependent simulations in subcritical and supercritical regimes. These experiments provide the first quantitative confirmation, in this setting, of the spectral-radius threshold mechanism for nonlinear age-space epidemic systems.