On optimal control at the onset of a new viral outbreak

Abstract

We propose a versatile model with a flexible choice of control for an early-pandemic outbreak prevention when vaccine/drug is not yet available. At that stage, control is often limited to non-medical interventions like social distancing and other behavioral changes. For the SIR optimal control problem, we show that the running cost of control satisfying mild, practically justified conditions generates an optimal strategy, u(t), t ∈ [0, T], that is sustainable up until some moment τ ∈ [0, T). However, for any t ∈ [τ, T], the function u(t) will decline as t approaches T, which may cause the number of newly infected people to increase. So, the window from 0 to τ is the time for public health officials to prepare alternative mitigation measures, such as vaccines, testing, antiviral medications, and others. In addition to theoretical study, we develop a fast and stable computational method for solving the proposed optimal control problem. The efficiency of the new method is illustrated with numerical examples of optimal control trajectories for various cost functions and weights. Simulation results provide a comprehensive demonstration of the effects of control on the epidemic spread and mitigation expenses, which can serve as invaluable references for public health officials.

Keywords: Compartmental model; Epidemiology; Optimal control; Transmission dynamic.

Fig. 1

Comparison of different cost functions c.

Fig. 2

Weight λ = 0.1.

Fig. 3

Weight λ = 0.05.

Fig. 4

Weight λ = 0.01.

Fig. 5

Weight λ = 10⁻⁷.