Mortality containment vs. Economics Opening: Optimal policies in a SEIARD model

Abstract

We extend the classic approach (SIR) to a SEAIRD model with policy controls. A social planner's objective reflects the trade-off between mortality reduction and GDP, featuring its perception of the value of statistical life (PVSL). We introduce realistic and drastic limitations to the control available to it. Within this setup, we explore the results of various control policies. We notably describe the joint dynamics of infection and economy in different contexts with unique or multiple confinement episodes. Compared to other approaches, our contributions are: (i) to restrict the class of functions accessible to the social planner, and in particular to impose that they remain constant over some fixed periods; (ii) to impose implementation frictions, e.g. a lag in their implementation; (iii) to prove the existence of optimal strategies within this set of possible controls; iv) to exhibit a sudden change in optimal policy as the statistical value of life is raised, from laissez-faire to a sizeable lockdown level, indicating a possible reason for conflicting policy proposals.

Keywords: Optimal control; Public policy evaluation; SIR; Statistical value of life.

Fig. 1

Mortality and production loss with one single, long lasting intervention. The optimal choice, see Online Material, reduces mortality to 0.34% with a 18.13% GDP loss: the intervention realizes a sharp containment of mortality, but the constraint of protracted measures causes a dramatic GDP loss. This policy has not been followed by any country. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2

An example of two minima. $\mathcal{L}$ as function of the reopening intensity $\overline{c}$ applied from day $150$ to $460$. The social cost of Covid death is fixed at $a\approx 6436.98$. See Section 5.1 for the interpretation of this parameter.

Fig. 3

An example of two minima. Front and side plots of $\mathcal{L}$ as function of the opening level $\underset{\_}{c}$ from Day 85 till Day 150 and reopening intensity $\overline{c}$ applied from day $150$ to $460$. The social cost of Covid death is set at $a\approx 12907.85$.

Fig. 4

Optimization over two opening levels, the first at Day $85$, and the second at Day $150$, planned in advance. The social cost of Covid death is fixed at $a=18000$ and $r=10^{-4}$. Fig. 4.A, Top left: no policy. Fig. 4.B, Top right: Excessive restrictions. Fig. 4.C, Bottom left: optimal policy. Fig. 4.D, Bottom right: Insufficient restrictions.

Fig. 5

Overall optimal control obtained by using BOCOP (Team Commands, 2017, Bonnans et al., 2017) and the piecewise linear optimal control with two level restrictions . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

The numerical results of the experiment where a first lockdown at day 85 is fixed to $\underset{\_}{c}=0.5$ and a reopening level at day 150 has to be determined. Fig. 6.A, Top left: no policy. Fig. 6.B, Top right: high reopening. Fig. 6.C, Bottom left: optimal policy. Fig. 6.D, Bottom right: a limited reopening level.

Fig. 7

An example of the influence of the parameters on the loss functional for the numerical experiment with two restrictions.