Global Dynamics and Bifurcation Theory of High-Dimensional Epidemic Models with Heterogeneous Mixing and Environmental Reservoirs

This dissertation advances the global dynamics and bifurcation theory of high-dimensional epidemic compartmental models that incorporate heterogeneous population mixing structures and environmental transmission reservoirs, motivated by and applied to the epidemiology of cholera, lassa fever, and meningitis in Nigeria's diverse ecological and demographic settings. Standard epidemic models typically assume homogeneous mixing and ignore environmental pathogen persistence, assumptions that generate systematically misleading predictions for diseases whose transmission is structured by age, geography, and environmental contamination levels. The dissertation formulates a general class of n-group heterogeneous mixing epidemic models coupled with partial differential equation-governed pathogen concentration dynamics in environmental reservoirs, establishing conditions for the well-posedness of the resulting infinite-dimensional dynamical system in appropriately defined function spaces. The global basic reproduction number is derived via an operator-theoretic generalisation of the next-generation matrix method applicable to systems with continuous heterogeneity. Novel sufficient conditions for the global asymptotic stability of the disease-free equilibrium and for the uniform persistence of the disease are proved using abstract persistence theory and Lyapunov functionals constructed via graph-theoretic methods applied to the network structure of inter-group mixing. Backward bifurcation and transcritical bifurcation are characterised through the application of Castillo-Chavez and Song bifurcation theorems. Applications to Nigeria's cholera epidemiology in the Lake Chad basin and to lassa fever in Edo State are developed with parameter estimation from published surveillance data. Keywords: epidemic models, global stability, bifurcation theory, heterogeneous mixing, environmental reservoir