The Path for Economic Revival

We propose an exit strategy from the COVID-19 lockdown, which is based on risk-sensitive levels of social distancing. At the heart of our approach is the realization that the most effective, yet limited in number, resources should protect those at high risk rather than applied uniformly across the population. By generalizing the SEIR model to mixed populations, and based on existing data in Israel, we present an analysis of the maximal load on the health system and the total mortality. We argue that risk-sensitive resource allocation combined with risk-sensitive levels of social distancing enables us to lower the overall mortality toll in parallel to the resumption of economic activity.

Amnon Shashua, Shai Shalev-Shwartz

Introduction

The COVID-19 outbreak toolbox contains available, inexpensive and unlimited measures such as social distancing, hygiene and the use of facial masks. On the other hand, there are “finite resources”, the use of which is limited, such as PCR testing, technological systems for “closing the circuit” for detecting and quarantining infected persons through contact-tracing technologies, and quality epidemiological surveillance for quarantining people who have come into contact with verified infected individuals.

  1. However, it is necessary to determine whether leakage in the containment of the spread among the low-risk population will lead to flooding of the health system.

Guidelines

In a paper [3], the mathematical foundations for bounding the capacity of the health system (i.e., number of respiratory systems) using concentration bounds were developed using statistical confidence intervals and assuming that there is data on the percentage of carriers among the low-risk population by a random sample. In this document, we will use existing data (without the need for a survey) to make a ”back of the envelope” calculation (without confidence intervals) of the maximum number of critical ICU beds (i.e., people on respiratory systems) required to contain the peak of the outbreak from the low-risk population, given an exponential spread of R_0= 1.4 according to the SEIR model (a detailed description of the model deferred to Appendix A). That is, an infected person infects on average 1.4 people. We chose to set the ”basic reproduction number” R_0 to 1.4 as it is in the low range of R_0reported in the literature [2]. The reason for taking the low-range of R_0 is that we would consider releasing the low-risk population from quarantine under social-distancing guidelines, hygiene, facial masks and restrictions of gatherings above certain thresholds. Later on, we will tackle the problem of modeling potential leakage from the low-risk population to the high-risk population.

Figure 1: The percentage of infectious people as a function of time according to the SEIR model (see Appendix A), with the parameters τi = 2.9; τe = 5 and several options for the R₀ parameter.

Extending SEIR Model to Handle Cross-Groups Leakage

So far, we have ignored the high-risk population. In reality, however, there will be some leakage from infectious people in the low-risk population to the high-risk population and leakages among the high-risk population. In Appendix B we generalize the SEIR model so that it will take into account a division to two populations. This is modeled by four values of the basic reproduction number: R_0l,l ;R_0l,h ; R_0h,l ; R_0h,h, representing low-to-low, low-to-high, high-to-low, and high-to-high infections.

Operative Measures

The operative proposal does not change the basket of means and resources but rather their focus. Due to the low sensitivity to leakage in the low risk population, the lockdown upon that population may be released subject to social distancing guidelines, hygiene rules and masks. At the same time, it will be possible to restore the economy and the education system to routine, while the retail, leisure and restaurant sectors may be restored subject to restrictions. The high tolerance to leakage allows for great flexibility in the “dosage” of lockdown release.

Appendix A: The SEIR model

The Susceptible-Exposed-Infected-Recovered (SEIR) model is a simple compartmental mathematical model of infectious disease. In this model, the total population is divided into S susceptible persons (people that did not get the disease yet), E exposed persons (people that has been exposed but are still not ill), I infected persons (people that has been infected and can infect others), and R recovered persons (people that may be recovered/dead/or still be sick, but are not infectious any more). The total population size is the sum N = S + E + I + R. Every day, the relative sizes of the different compartments change. We denote by S[t], E[t], I[t], R[t] the number of persons in each compartment on day t. The update of the sizes is according to the following rules:

Figure 2: The mixed SEIR model with the parameters τi = 2.9, τe = 5, R0l,l = 1.4, R0h,h = 0,7, R0l,h = R0h,l=0.02, N^l = 7,000,000, N^h=2,000,000. The bottom graph is an estimation of the number of severe cases based on pessimistic statistics of 4%; 0:25% probabilities to be severe in the high/low-risk groups respectively. These estimates are based on the same calculations as in Section 2.
Figure 3: SEIR model with the parameters τi = 2.9; τe = 5. Left R0 = 3.0, Right: R0 = 1.5.
Figure 4: The effect of the basic reproduction number.

Appendix B: A SEIR model with a division to High and Low-Risk Populations

We now describe a variant of the SEIR model, in which there are two populations: those of high risk and those of low risk. As in the SEIR model, each sub-population is divided into the four compartments of Susceptible, Exposed, Infectious, and Recovered. We denote the 8 compartments at time t by Sˡ[t], Eˡ[t], Iˡ[t], Rˡ[t], Sʰ[t], Eʰ[t], Iʰ[t], Rʰ[t], where the upper script l or h designates low or high risk groups. We assume mild social distancing within the low-risk group and strict social distancing between the low and high risk groups as well as within the high risk group. This is modeled by four values of the basic reproduction number: R_0l,l ; R_0l,h ; R_0h,l ; R_0h,h, representing low-to-low, low-to-high, high-to-low, and high-to-high infections. The update equations for Iˡ, Iʰ, Rˡ, Rʰ are similar to the vanilla SEIR model. As to the move from suspects to exposed: each person in potentially infects R_0l,l/τi persons in one day if he happens to meet someone from Sˡ (which will happen with probability Sˡ/ where is the size of the low-risk population), and in addition, potentially infects R_0l,h/τi persons in one day if he happens to meet someone from (with probability Sʰ/Nʰ). Similarly for each person in Ih. This gives the following difference equations:

References

[1] T. John. Iceland lab’s testing suggests 50% of coronavirus cases have no symptoms. https://edition.cnn.com/2020/04/01/europe/iceland-testing-coronavirus-intl/index.html.
[2] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K. S. Leung, E. H. Lau, J. Y. Wong, et al. Early transmission dynamics in wuhan, china, of novel coronavirus–infected pneumonia. New England Journal of Medicine, 2020.
[3] A. Shashua and S. Shalev-Shwartz. Can we contain covid-19 without locking-down the economy? CBMM Memo 104, Massachusetts Institute of Technology, 2020.

CEO of Mobileye, SVP at Intel, Co-CEO of OrCam, Chairman of AI21labs & Sachs Prof. of Computer Science at the Hebrew University of Jerusalem

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