Date of Defense
6-4-2026 5:00 PM
Location
F3-033
Document Type
Dissertation Defense
Degree Name
Doctor of Philosophy in Mathematics
College
COS
Department
Mathematical Sciences
First Advisor
Prof. Abdessamad Tridane
Keywords
Epidemiological models; Local diffusion; Nonlocal diffusion; Social behavior models; Awareness campaigns; Basic reproduction number; Digraph analysis; Global asymptotic stability; Uniform persistence; Backward Bifurcation; Bistability.
Abstract
This dissertation explores how epidemiological and behavioral processes interact across spatial and temporal scales to shape the dynamics of infectious diseases and addiction. Bringing together mathematical rigor and biological realism, it develops and analyzes a series of nonlinear models that capture the effects of spatial heterogeneity, mobility patterns, behavioral adaptation, and intervention strategies. Using tools from semigroup theory, functional analysis, stability analysis, and numerical simulation, the study provides a unified framework for understanding how awareness programs, fear-driven protection, vaccination, and social reinforcement influence transmission thresholds, persistence, and long-term population outcomes.
awareness-based interventions, where a basic reproduction number is derived and Lyapunov methods establish the global stability of both drug-free and drug-use equilibria, yielding practical insight into how awareness can reduce addiction prevalence. It then turns to spatial epidemic models with heterogeneous mobility, including an SIR framework in which infected individuals undergo restricted local diffusion while susceptible and recovered populations disperse nonlocally. This analysis establishes well-posedness, threshold dynamics, persistence, and asymptotic endemic behavior, showing that limiting infected movement can concentrate transmission in favorable locations. A related delayed SPIR model captures fear-driven protection under mixed local and nonlocal diffusion, providing a variational characterization of the reproduction number and clear conditions for either disease extinction or sustained spread, while also identifying the protection level needed to suppress transmission relative to the classical SIR case. The dissertation further studies a frequency-dependent behavioral system linking social contact and addiction, revealing bistability and bifurcation, and concludes with a vaccination model demonstrating how mobility differences and vaccine coverage jointly shape spatial transmission patterns and inform public health strategy.
Included in
LOCAL-NONLOCAL DISPERSAL, BEHAVIORAL RESPONSES, AND INTERVENTION STRATEGIES IN INFECTIOUS DISEASE AND ADDICTION DYNAMICS
F3-033
This dissertation explores how epidemiological and behavioral processes interact across spatial and temporal scales to shape the dynamics of infectious diseases and addiction. Bringing together mathematical rigor and biological realism, it develops and analyzes a series of nonlinear models that capture the effects of spatial heterogeneity, mobility patterns, behavioral adaptation, and intervention strategies. Using tools from semigroup theory, functional analysis, stability analysis, and numerical simulation, the study provides a unified framework for understanding how awareness programs, fear-driven protection, vaccination, and social reinforcement influence transmission thresholds, persistence, and long-term population outcomes.
awareness-based interventions, where a basic reproduction number is derived and Lyapunov methods establish the global stability of both drug-free and drug-use equilibria, yielding practical insight into how awareness can reduce addiction prevalence. It then turns to spatial epidemic models with heterogeneous mobility, including an SIR framework in which infected individuals undergo restricted local diffusion while susceptible and recovered populations disperse nonlocally. This analysis establishes well-posedness, threshold dynamics, persistence, and asymptotic endemic behavior, showing that limiting infected movement can concentrate transmission in favorable locations. A related delayed SPIR model captures fear-driven protection under mixed local and nonlocal diffusion, providing a variational characterization of the reproduction number and clear conditions for either disease extinction or sustained spread, while also identifying the protection level needed to suppress transmission relative to the classical SIR case. The dissertation further studies a frequency-dependent behavioral system linking social contact and addiction, revealing bistability and bifurcation, and concludes with a vaccination model demonstrating how mobility differences and vaccine coverage jointly shape spatial transmission patterns and inform public health strategy.