Confinement Strategies in a Simple SIR Model

Abstract

We propose a simple deterministic, differential equation-based, SIR model in order to investigate the impact of various confinement strategies on a most virulent epidemic. Our approach is motivated by the current COVID-19 pandemic. The main hypothesis is the existence of two populations of susceptible persons, one which obeys confinement and for which the infection rate does not exceed 1, and a population which, being non confined for various imperatives, can be substantially more infective. The model, initially formulated as a differential system, is discretised following a specific procedure, the discrete system serving as an integrator for the differential one. Our model is calibrated so as to correspond to what is observed in the COVID-19 epidemic, for the period from February 19 to April 16. Several conclusions can be reached, despite the very simple structure of our model. First, it is not possible to pinpoint the genesis of the epidemic by just analysing data from when the epidemic is in full swing. It may well turn out that the epidemic has reached a sizeable part of the world months before it became noticeable. Concerning the confinement scenarios, a universal feature of all our simulations is that relaxing the lockdown constraints leads to a rekindling of the epidemic. Thus, we sought the conditions for the second epidemic peak to be lower than the first one. This is possible in all the scenarios considered (abrupt or gradualexit, the latter having linear and stepwise profiles), but typically a gradual exit can start earlier than an abrupt one. However, by the time the gradual exit is complete, the overall confinement times are not too different. From our results, the most promising strategy is that of a stepwise exit. Its implementation could be quite feasible, with the major part of the population (perhaps, minus the fragile groups) exiting simultaneously, but obeying rigorous distancing constraints.

Recruitment Effects on the Evolution of Epidemics in a Simple SIR Model

“…This is even more complicated by the fact that the cases detected by the performed tests may represent a serious undercounting of the existing infections. This was stressed in the publication by Russel and collaborators in [9] and was also our conclusion [4]. Pulano and collaborators [10] point out that the under-detection of cases (by a factor they estimate at around 10 for France) might compromise the epidemic control effort, allowing the virus to propagate silently.…”

“…While the two aforementioned works were based on stochastic approaches, Ianni and Rossi [3] used a simple, time-dependent, variant of the standard SIR model in order to describe the situation prevailing during lockdown and social distancing. In this sense their model is quite similar to the one we proposed in [4], which focused on the possible strategies for a satisfactory exit from lockdown. Exit strategies have been the object of a workshop organised in the Newton Institute.…”

“…Another assumption, not totally unrelated to the first, is that the populations are homogeneous. While this can be overcome by introducing various subgroups (just as we did in [4]), the basic SIR model is still built upon the homogeneity assumption.…”

“…Incorporating confinement and seasonality effects into SIR is straightforward. In [4], where we investigated the various strategies for a successful confinement exit, we had introduced two subpopulations of susceptible: one strictly confined and one which, due to employment imperatives, was unconfined. The two groups had, of course, different reproduction numbers.…”

“…They acknowledged the fact that uncertainties in the model parameters and the initial conditions can result in different outcomes of the epidemic, leading, or not, to a second wave of infections. In [4] we addressed the question of a possible existence of a second wave, our conclusion being that it depended crucially on the aforementioned strategies.…”