Optimal control theory and applications in infectious disease modeling

We explore the fundamentals of optimal control theory and the work of Lev Pontryagin's Maximum Principle [39]. The general minimization functional is defined below

[equations]

where x(t) denote the state of the system at time t, and u(t) represent the control, where t spans the interval [t₀, t_f]

In our first application, We use optimal control to analyze the Susceptible-Exposed-Infected-Recovered (SEIR) epidemic model, offering a comparative analysis of minimization and maximization control strategies. This approach aims to minimize the number of infected people and the overall cost of vaccines over a fixed time period T compared to maximizing the overall healthy population N(t).

Our second application focuses on two different compartmental models: the Susceptible-Vaccinated-Infected-Recovered (SVIR) and the Susceptible-Infected-Treated-Recovered (SITR). By comparing and contrasting the outcomes of vaccination and treatment as control strategies, we are able to highlight the effectiveness of each approach within the models. The comparative analysis is extended to different scenarios to simulate different stages of an epidemic.