Theoretical Properties of an Adaptive Spatial Hierarchical Bayesian SEIR Model for HIV Transmission Dynamics

This study establishes theoretical foundations of an adaptive spatial hierarchical Bayesian SEIR model for HIV transmission dynamics. We prove existence and uniqueness of solutions, derive equilibrium conditions, analyze stability properties, and establish convergence guarantees. The enhanced SEIR model incorporates spatial heterogeneity through time-varying connectivity weights and employs hierarchical Bayesian methods for robust parameter estimation. Key results include global existence of nonnegative solutions, stability conditions for disease-free equilibrium, derivation of spatial basic reproduction number R₀ = ρ(FV⁻¹), and geometric ergodicity of MCMC estimation. Stability analysis demonstrates global asymptotic stability when R₀ ≤ 1. These theoretical properties provide rigorous mathematical foundations for HIV control strategies in resource-limited settings.