A Constructive Theory of Banach Lattices and Positive Operators with Applications to Epidemic Modelling in North East Nigeria

This dissertation develops a constructive theory of Banach lattices and positive operator theory, establishing original results on the structure of order-bounded positive operators and their spectral properties, with applications to the mathematical analysis of epidemic spread models in the heterogeneous, conflict-affected populations of North East Nigeria's Lake Chad Basin communities. Banach lattice theory provides the natural setting for the analysis of systems governed by positivity constraints, including population dynamics models and epidemiological models where solution components represent non-negative biological quantities, and constructive approaches are essential for obtaining computationally verifiable criteria for stability and persistence. The dissertation proves original characterisations of AM-spaces and AL-spaces in terms of operator factorisation through C(K) and L1 spaces, extending the Kakutani representation theorem to the case of non-order-complete Banach lattices arising in discretised population models. Novel spectral gap theorems for positive operators on Banach lattices are established, providing sharp quantitative bounds on the rate of convergence of iterates to the spectral radius eigenvector, with constants expressed explicitly in terms of Krivine's function and the lattice order structure. These spectral gap results are applied to derive convergence rate estimates for the power iteration computation of the basic reproduction number in large-scale heterogeneous epidemic models. Applications to a spatially explicit multi-patch lassa fever model for internally displaced person camps in Borno and Adamawa states are developed. Keywords: Banach lattices, positive operators, spectral theory, epidemic models, North East Nigeria