[Paper Review] Optimal immunity control by social distancing for the SIR epidemic model

Pierre‐Alexandre Bliman, Michel Duprez+2 more|arXiv (Cornell University)|Jun 10, 2020

COVID-19 epidemiological studies5 references4 citations

TL;DR

This paper formulates and solves an optimal control problem for the SIR epidemic model to minimize infection burden by determining the best social distancing strategy—specifically, the timing and intensity of lockdowns—that maximizes the number of susceptible individuals over an infinite horizon. It proves existence and uniqueness of the optimal control and shows that, under French 2020 lockdown conditions, such a strategy could increase the final susceptible proportion by up to 30% compared to uncontrolled spread.

ABSTRACT

Until a vaccine or therapy is found against the SARS-CoV-2 coronavirus, reaching herd immunity appears to be the only mid-term option. However, if the number of infected individuals decreases and eventually fades only beyond this threshold, a significant proportion of susceptible may still be infected until the epidemic is over. A containment strategy is likely the best policy in the worst case where no vaccine or therapy is found. In order to keep the number of newly infected persons to a minimum, a possible strategy is to apply strict containment measures, so that the number of susceptible individuals remains close to herd immunity. Such an action is unrealistic since containment can only last for a finite amount of time and is never total. In this article, using a classical SIR model, we determine the (partial or total) containment strategy on a given finite time interval that maximizes the number of susceptible individuals over an infinite horizon, or equivalently that minimizes the total infection burden during the curse of the epidemic. The existence and uniqueness of the optimal strategy is proved and the latter is fully characterized. If applicable in practice, such a strategy would lead theoretically to an increase by 30% of the proportion of susceptible on an infinite horizon, for a containment level corresponding to the sanitary measures put in place in France from March to May 2020. We also analyze the minimum intervention time to reach a fixed distance from herd immunity, and show the relationship with the previous problem. Simulations are provided that illustrate and validate the theoretical results.

Motivation & Objective

• To determine the optimal social distancing policy that minimizes total infection burden during an epidemic.
• To maximize the number of susceptible individuals at the end of an infinite-time horizon, equivalent to minimizing cumulative infections.
• To characterize the optimal control strategy (timing and intensity of lockdowns) in the SIR model with a finite intervention window.
• To analyze the minimum time required to reach a desired proximity to herd immunity threshold.
• To validate theoretical results through numerical simulations of the optimal control problem.

Proposed method

• Formulates a modified SIR model with control inputs representing social distancing (u3), vaccination (u1), and isolation (u2).
• Introduces two optimal control problems: (Pα,T) to maximize susceptibles at time T, and (ePα,T) to minimize infection burden over infinite time.
• Uses Pontryagin's Maximum Principle to derive necessary conditions for optimality and characterizes the optimal control as a bang-bang or singular control.
• Employs a bisection-based numerical algorithm to solve the optimal control problem and compute the optimal intervention time T*0.
• Derives an analytical formula for the asymptotic susceptible proportion S∞ using the conservation law S + I - (ν/β)ln S = constant.
• Validates results via numerical simulations over extended time horizons (up to 200 days), comparing optimal and suboptimal trajectories.

Experimental results

Research questions

• RQ1 What is the optimal social distancing strategy that minimizes the total number of infections over the course of an epidemic?
• RQ2 How can the final number of susceptible individuals be maximized by applying a finite-duration lockdown?
• RQ3 What is the minimum duration of intervention required to bring the susceptible population within a fixed distance of the herd immunity threshold?
• RQ4 How does the optimal control strategy depend on the intensity of social distancing (control parameter α)?
• RQ5 Can the theoretical optimal control be numerically approximated and validated through simulation?

Key findings

• The optimal control strategy is a bang-bang or singular control that maintains the susceptible population as close as possible to the herd immunity threshold (Sherd = ν/β) over an infinite horizon.
• For the French 2020 lockdown intensity (α ≈ 0.231), the optimal strategy increases the final susceptible proportion by approximately 30% compared to uncontrolled spread.
• The optimal control problem admits a unique solution, and the optimal intervention time T*0 is characterized by the unimodal behavior of the infection burden function jΦ.
• When the control intensity α is sufficiently low (α ≈ 0.56), the susceptible population can be driven arbitrarily close to the herd immunity threshold (Sherd) over infinite time.
• For sufficiently large T, if α is too high (insufficient lockdown), the optimal control requires no intervention (T*0 = 0), indicating that the epidemic cannot be meaningfully contained.
• Numerical simulations confirm that the optimal trajectories for (Pα,T) and (ePα,T) coincide and match theoretical predictions, validating the analytical framework.