Minimizing the infected peak utilizing a single lockdown: a technical result regarding equal peaks

Abstract

Due to the usage of social distancing as a means to control the spread of the novel coronavirus disease COVID-19, there has been a large amount of research into the dynamics of epidemiological models with time-varying transmission rates. Such studies attempt to capture population responses to differing levels of social distancing, and are used for designing policies which both inhibit disease spread but also allow for limited economic activity. One common criterion utilized for the recent pandemic is the peak of the infected population, a measure of the strain placed upon the health care system; protocols which reduce this peak are commonly said to `flatten the curve." In this work, we consider a very specialized distancing mandate, which consists of one period of fixed length of distancing, and addresses the question of optimal initiation time. We prove rigorously that this time is characterized by an equal peaks phenomenon: the optimal protocol will experience a rebound in the infected peak after distancing is relaxed, which is equal in size to the peak when distancing is commenced. In the case of a non-perfect lockdown (i.e. disease transmission is not completely suppressed), explicit formulas for the initiation time cannot be computed, but implicit relations are provided which can be pre-computed given the current state of the epidemic. Expected extensions to more general distancing policies are also hypothesized, which suggest designs for the optimal timing of non-overlapping lockdowns.

Optimal intervention strategies for minimizing total incidence during an epidemic

“…Morris et al [3] proved that the peak prevalence subject to interventions of bounded duration || u || 0 ≤ c 0 is minimized by an intervention of form and discovered that such interventions induce a second wave having a peak of same height as the first wave. Such a twin peaks phenomenon was also noted in [4]. Miclo, Spiro, and Weibull [5] studied a dual problem of minimizing the intervention cost || u || 1 subject to a bounded peak prevalence || I u || ∞ ≤ c ∞ , and proved that the optimum is of form …”

An explicit formula for minimizing the infected peak in an SIR epidemic model when using a fixed number of complete lockdowns

“…As we discussed, a perfect or near‐perfect lockdown is not practical, but the ideal case helps understand the problem with nonstrict lockdowns, and we presented numerical evidence that the formula is reasonable accurate even in the not totally (but still reasonably) strict case. The fellow‐up paper 36 has recently started the study of the same optimization problem in the nonstrict case, showing that in the optimal case the subsequent peak after release from one lockdown coincides with the infective population at the start of the lockdown. Much further work remains, including the extension of the perfect‐lockdown problem to models such as the ones presented in References 20,25,26.…”