Optimal control frameworks for a class of epidemiological and oncological models

Abstract

In this thesis, we employ optimal control frameworks in two distinct contexts: Human immunodeficiency virus (HIV) and esophageal cancer. For HIV, we introduce a comprehensive data-driven nonlinear optimization framework designed for personalized therapies. This framework utilizes a deterministic in-host nonlinear ordinary differential equation (ODE) model and formulates two optimization problems using individual patient data. The first problem focuses on estimating patient-specific parameters through constrained optimization, while the second problem determines optimal combination therapies to reduce viral load to undetectable levels. Several numerical experiments suggest that our framework can provide robust and effective optimal dosages with lower toxicity levels to control HIV.

In esophageal cancer, we present an innovative approach to model and control aberrant signaling pathways. This involves leveraging an It\^o stochastic process to capture signaling pathway dynamics governed by a degenerate Fokker-Planck partial differential equation. Our study proposes a refined treatment strategy targeting aberrant signaling pathways, specifically focusing on epidermal growth factor (EGF) pathways. This strategy includes developing a pharmacokinetic model considering pathway heterogeneities, preceded by constrained optimization to obtain model parameters from patient data. Subsequently, we propose a personalized optimal treatment strategy targeting aberrant EGF using a non-smooth open-loop control for a stochastic process modeled by the Fokker-Planck equation. The solution to these problems is characterized within the framework of Pontryagin's minimum principle (PMP), and the optimization problem is solved using a sequential quadratic Hamiltonian (SQH) method. Experimental results are presented to validate the proposed framework successfully.