CONTROL PROBLEMS IN INFECTIOUS DISEASE MANAGEMENT

Open Access
Author:
Zhang, Dongmei
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 22, 2014
Committee Members:
  • Alberto Bressan, Dissertation Advisor
  • Timothy Reluga, Dissertation Advisor
  • Alberto Bressan, Committee Chair
  • Timothy Reluga, Committee Chair
  • Jinchao Xu, Committee Member
  • Lyle Norman Long, Special Member
  • Yuxi Zheng, Special Member
Keywords:
  • control theory
  • PDE
  • infectious disease management
  • economic epidemiology
  • optimization
Abstract:
This dissertation presents applications of mathematical control theory to better under- stand and prevent infectious disease transmission in epidemiology. Control of structured epidemic models with age heterogeneity and spatial heterogeneity are formulated and an- alyzed separately. Age structured model is crucial for disease transmission such as HIV. We introduce continuous age structure in both course of infection of individuals and the transmission rate in the population. Not only we analyzed the disease transmission under such detailed age-structured model, we further applied game theory to identify the ratio- nal behavior accordingly. Notice that disease often spread into population form a disease reservoir nearby, we also researched a simplified spatial model in one dimension regarding how to control disease from spreading into nearby community. In both age and spatially structured model, we provide mathematical analysis of the existence and derived necessary conditions of the possible optimal strategy. Numerical simulations are also included to show ways to calculate the optimal strategy. In order to fundamentally prevent the disease spread from disease reservoir to the community nearby, we further investigate the controllability problems of the disease reservoir in two dimension, including the confinment and steering problems. The asymptotic shape of the reachable set are also analyzed.