key: cord-0144430-sf78e5ub
authors: Nikolaou, Michael
title: Simple Formulas for a Two-Tier Strategy to Flatten the Curve
date: 2020-04-08
journal: nan
DOI: nan
sha: c52ee7529d67210fca78d3ed8ed8d12a067aa564
doc_id: 144430
cord_uid: sf78e5ub

Some basic facts that enable or constrain efforts to"flatten the curve"by non-pharmaceutical interventions (NPI) are summarized and placed in the current context of the COVID-19 epidemic. Analytical formulas are presented for a simple two-tier NPI strategy that places different social distancing targets for high- and low-risk groups. The results aim to facilitate rapid what-if analysis rather than replace detailed simulation, which remains indispensable for informed decision making. The appearance of the oft neglected Lambert function in the resulting formulas is noted.

In the absence of effective vaccines or antiviral drugs against COVID-19, non-pharmaceutical interventions (NPI)measures aiming at reducing virus transmission in the population through reduction of human contacts -remain the only viable response to the epidemic. 3 The inexorable realities faced by NPI are best accentuated by the simple plot in Figure  1 resulting from analysis of the standard SIR epidemiology model: With the basic reproductive ratio, 0 , for COVID-19 being between 2 and 3, 2 the total fraction of a population infected by the end of the epidemic -whenever that might come -would be around 90% (for 0 ≈ 2.5 in Figure 1 ) in the absence of NPI. At a fraction of 1% mortality for infected individuals, fatalities for a population the size of the US would be in the millions. This bleak estimate is in agreement with results from more elaborate calculations earlier presented by the US Centers for Disease Control and Prevention (CDC). 4 NPI measures should bring 0 down from 2.5 to about 1.0 for the infection rate to eventually reverse sign to negative, thus eventually resulting in low single digits for total percentage of infected individuals in a population ( Figure 1 ); or to below 1, for the epidemic to immediately start contracting. For example, it is reported that travel restrictions in Wuhan, China quickly brought 0 down from 2.35 to 1.05. 2 Similarly, recent projections by the US federal government for 200,000 fatalities correspond to 0 ≈ 1.03. 5 As a practical matter, current data on exponential growth of infections in various countries (widely available on the internet 6 ) indicate that extraordinary measures are needed for 0 to be drastically reduced. With a sizable asymptomatic period for infected individuals, universal restrictions on social contacts quickly emerged as a preferred NPI measure. 1 Figure 1 . Total fraction of a population infected and removed by the end of an epidemic as a function of the basic reproductive ratio 0 -the average number of secondary infections arising from an average primary infection in an entirely susceptible population, eqn. (2) -according to eqn. (7) . For 0 < 1 the virus does not spread and the epidemic is contained. Reports on data from Wuhan, China, estimate 0 dropping from 3 to 1 to 0.3 with each successive round of NPI measures implemented, 1 or from 2.35 to 1.05. 2 The sensitivity of the total infected fraction (and resulting critical care needs and fatalities) to small uncertainty in 0 makes accurate predictions difficult when 0 ≈ 1.

In addition to the role of 0 already discussed, it turns out that reducing 0 helps mitigate the overwhelming of hospitalization facilities by infected individuals in need of treatment, and thus prevent fatalities from inadequately treated infections. This is because the fraction of infectious individuals reaches its peak value of 1 − 1/ 0 − ln( 0 )/ 0 at a time approximately proportional to 1/( 0 − 1) (Appendix B). In this context, lowering that peak by reducing 0 achieves "Flattening the Curve," 6-8 as discussed in more detail in section 3.

These realizations have culminated into the policy "Stay Home", as a practical if universal NPI measure to effect the necessary reduction in human contacts and thus lower 0 . However, in addition to flattening the curve, this reduction also has adverse effects felt by many, not least in terms of drastically reduced economic activity. Facing the inexorable conflict summarized as "saving lives vs. saving the economy" appears inevitable. The purpose of this publication is to help understand that conflict and contribute towards its mitigation by exploiting a simple and widely observed fact: That hospitalization and fatality rates are higher by at least an order of magnitude among the elderly and the vulnerable who are roughly 1/5 to 1/6 of the population ( Figure 2 ). 9, 10 Therefore, by shielding human contact for this high-risk group more than the low-risk group, the effective combined 0 could be improved, with commensurate reduction in hospitalization and fatality rates, without increasing adverse effects on the economy, as the most economically active group would face no additional restrictions on distancing. While numerical simulations on sophisticated models have included many cases of this kind of enhanced social distancing for the elderly, 9 the contribution of this paper is a simple way to quickly assess the effects of design parameters, such as cut-off age or distancing targets for high-and low-risk groups. The standard SIR model with two-subpopulations is used. Exact or approximate analytical solutions are obtained that can easily asses the effects of decision variables on outcomes such as hospitalization and critical care needs or fatalities. It is emphasized that the intent of the paper is not to advocate any particular policy. Rather, it purports to provide shortcut tools that both scientists and public officials can easily use in rapid what-if analysis of related ideas.

The standard SIR model [12] [13] [14] [15] comprises the basic equations

for the fractions, ( ) and ( ), referring to susceptible to a spreading infection and infectious, respectively, in a population of constant size. The third fraction,

refers to individuals removed from the infectious group, by either recovery or death. Clearly, then, =

where (0) = 0, (0) = , (0) = 1 − ; is the infection spread rate; is the removal rate from the infectious group; and 0 ≝ / is the basic reproductive ratio. NPI aims at designing interventions, such as closing schools, churches, and social venues, workplace distancing, or quarantine, in order that the resulting universal 0 -as uncertain as 0 may be -best achieve desired outcomes. 1, 16 Despite its utter simplicity, the SIR model provides valuable insights for the course of an epidemic, 15 which are relevant to the two-tier strategy to be presented in the next section. In summary, eqn. (2) implies that an epidemic does not occur iff

For 0 > 1, eqns. (1)-(4) imply that (a) the infection spreads and an epidemic occurs (b) the infectious fraction, , initially rises exponentially at rate − > 0, (c) peaks at time * with

and (d) asymptotically goes to 0, achieving herd immunity by leaving the remaining uninfected fraction of the population, (∞), at Figure 2 . Distribution of symptomatic infection cases requiring critical care and resulting in fatalities over US age groups. 3, 11 arxiv.org Michael Nikolaou 3 ( Figure 1) where ( ) is the Lambert function. 17,18 * It is eqn. (7) that underlies the stark predictions for anticipated fatalities from COVID-19 and the challenges faced by NPI in the absence of a vaccine, as already discussed in the Introduction.

Responses to the epidemic clearly can involve sophisticated policies that adjust 0 over time, over geographical areas, or over different population groups. What follows is not an attempt to replace these indispensable tools. Rather, it is an attempt to present shortcut methods for rapid what-if analysis.

As already discussed, "Flattening the Curve" by effecting a low 0 (well below 2.5, near or below 1) is a universal approach to mitigating the total number of fatalities throughout the epidemic (Figure 1 ) as well as the number of infected patients in need of hospitalization or critical care at any time. It can be shown (Appendix B) that the appropriate 0 , given a maximum infectious fraction, * , is given analytically by

where −1 is the lower branch of the Lambert function 18 with values less than −1.

Reducing the universal 0 value, however, inevitably prolongs the time, * , to the lowered peak of the infectious fraction, ( * ), hence the duration of the epidemic as well. This can be roughly captured by the time, , needed for to reach an inflection, according to the equation Eqn. (8) also suggests that while the time to peak decreases as (0) = increases, the peak value of ( * ) remains approximately constant over a wide range of values of , given a fixed value of 0 as illustrated in Figure 4 . This has obvious implications for designing strategies to flatten the curve. 

In a population consisting of high-risk and low-risk groups a number of possibilities exist for adjusting binary contacts between groups. Assuming different rates of infection in contacts between the groups, the SRI model becomes

with initial conditions (0) = − , (0) = − , (0) = , (0) = . The positive parameters ≤ 1 and ≤ 1 capture the additional measures taken by the high-risk group, to avoid contacts.

Eqns. (12) and (13) imply that ( ) declines after * if 0 ( * ) + 0 ( * ) < 1/

where 0 ≝ , 0 ≝ (Appendix A). If = = 1, the herd immunity inequality, eqn. (6), is recovered. In the rest of the paper we assume = and = .

Dividing eqn. (10) by eqn. (11) and integrating implies that

For = 1 eqn. (17) simply confirms that both groups are infected at the same rate. In the extreme, if unrealistic case of = 0 (quarantine of the high-risk group) it would follow from eqn. (17) that ( ) = (0), suggesting that only the low-risk group would be infected, with much lower needs for hospitalization and fatality rates. For practically more interesting intermediate values of , realistically closer to 1, it turns out (as explained next) that both fatalities and critical care needs are reduced, as shown in Figure 5 . 19)). Age is used as a simple, if rough proxy to separate between high-and low-risk groups ( Figure 2 ).

To generate the results in Figure 5 , it is first assumed that the population fraction of low-risk infected by the end of the epidemic,

, is set at an upper bound, to cap fatalities, given a level of restrictions 0 on that group; whereas the high-risk group can manage additional restrictions 0 , to experience a corresponding population fraction of infected (∞) = (0) − (∞) ≤ (0). Then, the fatality fraction, , for the entire population by the end of the epidemic will be

It can be shown (Appendix C) that the ratio of the total fatality fraction for < 1 and for = 1 is approximately independent of the selected value of (∞) for reasonably small values of (∞), e.g. below 0.1 − 0.2, as ( )

A similar expression can be derived in the same way for hospitalization and critical care needs. In both cases, derivation relies in the simple approximation of eqn. (17)

Note that the results in this section have been derived with minimal assumptions for the underlying model used.

Similar to eqn. (7) , the total fraction of infected through the epidemic for each of the high-and low-risk groups can be shown (Appendix D) to be (∞)

Note that for = 1, eqns. 

Simple analytical formulas for a two-tier NPI strategy to "flatten the curve" were collected. The formulas are based on a simple two-tier model of the SIR variety, involving highand low-risk groups, and are meant to be used for rapid answers to what-if questions, rather than to provide accurate numbers. For the latter, numerical simulation remains the tool of choice. Taming the COVID-19 epidemic is a multifaceted problem that will require diverse resources and approaches. It is hoped that this text will contribute to that effort.

All computations were done in Mathematica, available at the University of Houston. Sharing of teaching material about the SIR model on Github by Jeff Kantor of Notre Dame is also gratefully acknowledged.

Evolving Epidemiology and Impact of Non-pharmaceutical Interventions on the Outbreak of Coronavirus Disease

Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The Lancet Infectious Diseases

Report 9 -Impact of nonpharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand

White House Projects Grim Toll From Coronavirus

Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilator-days and deaths by US state in the next 4 months. medRxiv

An interactive web-based dashboard to track COVID-19 in real time

The Global Impact of COVID-19 and Strategies for Mitigation and Suppression

The Novel Coronavirus Pneumonia Emergency Response Epidemiology Team. The Epidemiological Characteristics of an Outbreak of 2019 Novel Coronavirus Diseases (COVID-19) -China

Age and Sex Composition in the United States

A Contribution to the Mathematical Theory of Epidemics

Mathematical Biology: I. An Introduction

Infectious diseases of humans: dynamics and control

Modeling Infectious Diseases in Humans and Animals

Interventions to mitigate early spread of SARS-CoV-2 in Singapore: a modelling study

On the LambertW function

The Lambert function belongs in the Function Hall of Fame. Chemical Engineering Education In preparation

Appendix A. Proof of eqn. (16) Eqns. (12) and (13) implyThe eigenvalues of the matrix are in the left half-plane iff det( ) > 0 (24) which yields eqn. (16) , andwhich is guaranteed by eqn. (16) if ≈ .Appendix B. Proof of eqns. (8) and (9) Standard analysis of eqns. (1) and (2) proceeds as follows:If is maximized at * , then 

Given a reasonable range of values foris accurate, as indicated by the following figure. Therefore, eqns. (18) and (17)