Global Conference on Mathematics and Applications (GCMA)

We investigate a spatially heterogeneous reaction-diffusion HIV/AIDS model in which recruitment, natural mortality, disease progression, and transmission vary across space. In the reduced SIA framework considered here, the AIDS class is downstream of the infectious class, allowing the invasion mechanism to be reduced to a scalar self-adjoint spectral problem while retaining spatial demographic heterogeneity. We first establish positivity, global existence, uniform boundedness of the susceptible component, and an absorbing total-mass estimate, and characterize the unique strictly positive disease-free susceptible profile as the solution of a heterogeneous elliptic Neumann problem. Linearization about this disease-free state leads to a principal Neumann eigenvalue λ1 that serves as the invasion threshold. We prove that λ1>0 guarantees global extinction of both the infected and AIDS-stage classes together with uniform convergence of the susceptible population to the disease-free profile, whereas λ1<0 implies linear instability of the disease-free equilibrium. For a multiplicative transmission scaling θ, we prove that λ1 is continuous and strictly decreasing in θ, yielding a unique critical value θc separating the extinction and instability regimes; equivalently, the scaled threshold Rsp=θ/θc provides a dimensionless spectral criterion. We further show that λ1 is nondecreasing with respect to the infected diffusion coefficient, identifying a stabilizing diffusion effect within the present self-adjoint structure. To support the analysis, we construct a cell-centered finite-volume discretization consistent with the no-flux boundary condition and perform systematic grid refinement. For the heterogeneous test configuration, the critical value converges from 0.2782379 on 101 cells to 0.2782549 on 801 cells, giving θc≈0.27825. The resulting framework avoids replacing spatial heterogeneity by averaged coefficients and provides a rigorous, transparent, and fully reproducible spectral approach to extinction-threshold analysis in heterogeneous HIV/AIDS dynamics while clearly separating analytical guarantees from illustrative numerical evidence.