Optimal control of an SIR epidemic through finite-time non-pharmaceutical intervention

Abstract

We consider the problem of controlling an SIR-model epidemic by temporarily reducing the rate of contact within a population. The control takes the form of a multiplicative reduction in the contact rate of infectious individuals. The control is allowed to be applied only over a finite time interval, while the objective is to minimize the total number of individuals infected in the long-time limit, subject to some cost function for the control. We first consider the no-cost scenario and analytically determine the optimal control and solution. We then study solutions when a cost of intervention is included, as well as a cost associated with overwhelming the available medical resources. Examples are studied through the numerical solution of the associated Hamilton-Jacobi-Bellman equation. Finally, we provide some examples related directly to the current pandemic.

Beyond just “flattening the curve”: Optimal control of epidemics with purely non-pharmaceutical interventions

“…Moreover, we show that our numerically computed optimal control obeys two fundamental stability criteria, which impose an upper limit on the transmission rate and its rate of change on the way out of the initial lockdown. The precise structure of the optimal control (i.e., three phases of the intervention) obtained in this paper differs from the results described in similar works [42][43][44]. After the initial submission of this paper, the preprint by Charpentier et al [48] appeared, who studied a similar optimization problem on the basis of an extended SIR-type model with parameters adjusted to the COVID-19 pandemic in France.…”

“…We point out that the optimal transmission control described above differs from the results obtained for similar optimization problems considered in Refs. [43,44,47], which do not exhibit the distinct structural features of the intervention (initial lockdown, critical period, final phase intervention) presented here. A comparable result was described in Ref.…”

“…The paper by Kruse and Strack [42] is devoted to the analysis of the optimal timing of social distancing measures under the constraint that the overall (temporal) budget for NPIs is limited. Ketcheson [43] presented a detailed analysis for optimal transmission control in a SIR (susceptible-infected-recovered) epidemic model with the aim of achieving a stable equilibrium ("herd immunity") within a fixed finite time interval while simultaneously avoiding hospital overflow. A similar problem (including a simple state-dependent mortality rate) was studied by Alvarez et al [44], who focussed on minimizing the lockdown costs and included further economic aspects such as the assumed value of statistical life.…”

“…Next to adaptive feedback strategies for iterative loosening of the social distancing policies after an initial lockdown, the authors also examined the possibility of eradicating the virus. All of these papers on optimal control deal with deterministic epidemiological models, in particular the basic SIR model [25,[42][43][44]47] or various extended SEIR-type models [40,41,46]. We remark that this survey on optimal control of COVID-19 is not exhaustive.…”

COVID-19 in Kerala: analysis of measures and impacts

“…For comparison with other countries, a snapshot at the end of May (Table 1b) shows a high level of success in Kerala, in keeping the number of infections down. But it is evident from the inferred ℛ 0 that relaxation came too early, and that longer-term control strategies are needed to prevent further escalation through community transmission 18 .…”