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, hold potential for automation and could offer suitable, adaptable solutions for real-time decision-making. However, despite their potential, the widespread adoption of these techniques has been limited. Several factors may contribute to this challenge, including limited access to specialised software, a perception of high computational costs, or a general 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 (NPIs) 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. By integrating newly available data more quickly, direct methods can enhance the ability to make informed and timely decisions for managing outbreaks effectively.Author summary: This study demonstrates the practical advantages of direct optimisation methods in epidemiological modelling when there is a need to identify effective strategies for disease control while balancing constraints. Through case studies, it examines the effort required to adapt compartmental models for optimisation, the time needed to obtain an optimal solution, and the performance of both open-source and licensed tools. The study begins by contrasting indirect and direct methods using a simple infection model. It then illustrates the application of an accessible mathematical programming framework, JuMP, to optimise control strategies aimed at reducing infections, minimising intervention costs under constraints, and managing multiple interventions. Finally, the study compares the efficiency of different optimisation algorithms. The results show that direct methods, aided by readily available tools like JuMP and IPOPT, enable efficient, flexible, and interpretable modelling with minimal additional implementation effort. This work demonstrates how these techniques can support informed and timely decision-making in the early stages of an epidemic.