Date of Award

5-27-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Science

First Advisor

William Barker

Abstract

This dissertation investigates traveling wave solutions for two classes of delayed nonlocal dispersal susceptible--infected--recovered (SIR) epidemic models incorporating biologically realistic mechanisms such as delayed infectivity, delayed dispersal, nonlocal transmission, demographic turnover, and nonlinear incidence effects. These models extend classical spatial epidemic frameworks by allowing long-range population movement through nonlocal dispersal operators and incorporating temporal memory into both diffusion and transmission processes. The primary objective is to establish the existence of traveling wave solutions connecting disease-free equilibria to endemic states and to characterize threshold conditions governing epidemic propagation. The simultaneous presence of nonlocal dispersal, multiple delays, and non-monotone nonlinear incidence terms creates substantial analytical challenges, including loss of compactness, failure of regularizing effects, and the breakdown of standard monotone semiflow techniques. To overcome these difficulties, this dissertation develops an analytical framework combining upper and lower solution constructions, monotone iteration methods adapted to delayed nonlocal operators, and fixed point arguments in weighted Banach spaces. For each epidemic model considered, sufficient conditions are established for the existence of traveling wave solutions above an associated critical wave speed. In addition, asymptotic properties of the constructed wave profiles are derived, and biologically relevant threshold criteria involving the basic reproduction number are identified. The results obtained herein extend existing traveling wave theory for epidemic systems by rigorously incorporating delay directly into nonlocal dispersal mechanisms and by treating delayed nonlocal transmission within spatially heterogeneous SIR frameworks. More broadly, this work contributes to the mathematical theory of delayed functional integro-differential equations and provides new analytical tools for studying wave propagation in nonlocal systems with memory effects.

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