Exploring epidemic control policies using nonlinear programming and mathematical models

Optimal control theory in epidemiology has been used to establish the most effective intervention strategies for managing and mitigating the spread of infectious diseases while considering constraints and costs. Using Pontryagin's Maximum Principle, indirect methods provide necessary optimality conditions by transforming the control problem into a two-point boundary value problem. However, these approaches are often sensitive to initial guesses and can be computationally challenging, especially when dealing with complex constraints. In contrast, direct methods, which discretise the optimal control problem into a nonlinear programming (NLP) formulation, could offer robust, adaptable solutions for real-time decision-making. Despite their potential, the widespread adoption of these techniques has been limited, which may be due to restricted access to specialised software, perceived high costs, or unfamiliarity with these methods. This study investigates the feasibility, robustness, and potential of direct optimal control methods using nonlinear programming solvers on compartmental models described by ordinary differential equations to determine the best application of various interventions, including non-pharmaceutical interventions and vaccination strategies. Through case studies, we demonstrate the use of NLP solvers to determine the optimal application of interventions based on single objectives, such as minimising total infections, "flattening the curve", or reducing peak infection levels, as well as multi-objective optimisation to achieve the best combination of interventions. While indirect methods provide useful theoretical insights, direct approaches may be a better fit for the fast-evolving challenges of real-world epidemiology.