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William A. Levinson

Health Care

Break the Curve and Keep It Broken

We need to reduce Covid-19’s basic reproduction number to less than 1 to break, rather than flatten, the curve

Published: Tuesday, April 28, 2020 - 11:03

The phrase “flatten the curve” means to slow the transmission of the coronavirus (Covid-19) in order to spread the total number of cases out over a longer period of time. This will avoid overwhelming the healthcare system.1 The model is accurate as presented throughout the internet, but it also overlooks terrible dangers and enormous opportunities.

• Distribution of the case load over a longer period may avoid overwhelming the healthcare system, but given the disease’s high lethality, it could nonetheless kill tens of thousands of Americans by the time it is done. Flattening the curve means only that it would take somewhat longer to kill somewhat fewer people, so it is not enough to just flatten the curve; we have to break it.
• If, however, nonpharmaceutical interventions (NPIs) such as hand hygiene, social distancing, and widespread use of face masks reduce the disease's basic reproduction number (R0) to less than 1, we will lose far fewer lives to Covid-19 than to the 2019–2020 flu season, which has already (as a side bonus) come to a very rapid end.
• Maintenance of some of the countermeasures will ensure that a mutated form of Covid-19 does not make a comeback, and should also prevent or mitigate the 2020–2021 flu season. The latter proposition is not speculation, as the anti-Covid-19 countermeasures have already, in combination with partial vaccination of the population, essentially destroyed the 2019–2020 influenza.

The takeaway from this article is simple. If we do what the Occupational Health and Safety Administration (OSHA), the Centers for Disease Control and Prevention (CDC), Surgeon General Jerome Adams, and physician Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases, tell us to do, this crisis will largely be over by summer, and we can celebrate the defeat of Covid-19 along with Independence Day on July 4. We will be able to go back to work (while maintaining the sensible precautions recommended by OSHA2) to get our economy up and running again. We will be able to go on vacations again, at least in the United States, and we will be able to gather in public subject to common-sense restrictions.

If we disregard their instructions, we could lose, according to current estimates, 100,000 people and suffer enormous economic damage. While nothing in this article constitutes formal engineering or occupational health and safety advice, simple mathematics explains how the recommended countermeasures will defeat Covid-19 if we practice them diligently. The underlying premise is simple: Coronavirus, and also the seasonal flu, must propagate or die. The 2019–2020 seasonal flu seems to have already done the latter, and more than a month before the flu season usually ends.

Viruses live on borrowed time

Covid-19 and also seasonal flu are on borrowed time from the instant they infect somebody. If they don’t find new homes in less than a fortnight (15 days, not the popular game), their host’s immune system will kill or at least inactivate them. “Importantly, the scientists could not grow viruses from throat swabs or sputum specimens after day 8 of illness from people who had mild [coronavirus] infections,” notes Helen Branswell.3 People can, however, display no symptoms at all for several days after they are infected, which is a strong argument for a 15-day quarantine. The CDC adds of seasonal flu, “Most healthy adults may be able to infect others beginning 1 day before symptoms develop and up to 5 to 7 days after becoming sick. Children and some people with weakened immune systems may pass the virus for longer than 7 days.”4 We can infer from this that, if we had a theoretically perfect 15-day quarantine in which each person stayed in total isolation, Covid-19 and flu would be eradicated utterly.

Flatten the basic reproduction number to break the curve

We cannot, of course, have a 100-percent perfect quarantine of all Americans because essential services must still operate, and people must still buy essential items like groceries and medications. We cannot shut our economy down indefinitely, either, as shown by rising unemployment numbers and a stock market that is well into correction territory if not an actual bear market. Here is where the nonpharmaceutical countermeasures differ from occupational health and safety considerations. A partially effective control for a 10-severity failure mode, one that threatens human life or safety, would not be acceptable in a workplace where individuals must be protected from harm. It might, however, be considered “good enough” on a collective basis, where daily life includes acceptance of life-threatening risks such as traffic fatalities (more than 36,000 in 2017) and seasonal flu (more than 34,000 deaths during the 2018–2019 flu season).

The countermeasures therefore do not have to be 100-percent effective from a public health perspective. All they need to do is reduce Covid-19’s basic reproduction number to less than one to break, rather than flatten, the curve.

A disease’s basic reproduction number (R0, or R_nought) is the average number of people whom a contagious person will infect when all individuals in the population are susceptible to infection, and in the absence of any countermeasures such as social distancing. The effective reproduction number Re is “...the number of cases generated in the current state of a population, which does not have to be the uninfected state.”5 It can, in the absence of countermeasures, be calculated simply as:

where S is the number of susceptible individuals out of a population of N5.

The SIR model and the curve

The SIR (susceptible, infected, and recovered) epidemic model is similar to a chemical kinetics problem that involves people rather than chemical reactants.6 Infected people (I) can interact with susceptible people (S) to propagate the disease at a rate expressed by beta (ß), the transmission rate. The pool of infected people is, however, depleted at the recovery rate gamma (Γ), after which they become recovered people (R) who are no longer contagious. Those who die also are removed from the pool of infected people.

The model is therefore very similar to a chemical kinetics problem with two reactants (i.e., susceptible and recovered populations) and two rate constants (transmission and recovery rates), and can be treated mathematically in exactly the same manner. The technical details involve the simultaneous iterative solution of two differential equations by the Runge-Kutta numerical method, and readers who wish to try it themselves can use this online version offered by Hans Nesse.7 I have also provided an Excel spreadsheet and accompanying technical details as to how it works so users can try their own models, and also simulate the effect of partial vaccination on the seasonal flu.

The rate of change in the infected population (I) is:

where dI/dt is mathematical shorthand for the differential (or incremental) change, dI, in the infected population given an incremental change in time, dt. N is the total population, so s = S/N is the susceptible fraction, which would correspond to a concentration in a chemical kinetics problem. The transmission rate beta and recovery rate gamma are meanwhile analogous to rate constants.

Suppose we divide the above equation throughout by N to express the quantities i = I/N as the infected fraction, and r = R/N as the recovered fraction. Now:

Benjamin Ridenhour, Jessica Kowalik, and David Shay add that the basic reproduction number is R0 = ß/Γ, which means ß = R0 × Γ. Then:

Maximization of i (the peak of the “curve”) occurs when the first derivative shown above is zero. This happens when the susceptible fraction s = 1/R0. If, for example, R0 = 3, then the epidemic would peak when s = 1/3 which means two-thirds of the population would have already suffered the disease. In addition, slightly more than 30 percent of the population (or almost 100 million Americans) would be infected at the epidemic’s peak, which would obviously overwhelm our healthcare system. This is why there is so much focus on “flattening the curve,” which may in fact have diverted attention from the prospect of breaking the curve.

Breaking the curve

Where we’re going with this is that, if R0 × s 0 < 1 because 1) the initial susceptible fraction s0 is less than 1 because some people have been vaccinated; and/or 2) nonpharmaceutical interventions (NPIs) suppress R0 sufficiently, there is never a maximum. The infected fraction begins to decline right out of the starting gate, and the would-be epidemic becomes no more than a momentary flash in the pan. Although there is currently no vaccine for Covid-19, there is one for the seasonal flu, which means the NPIs for Covid-19 will be far more effective against the rest of the 2019–2020 flu season and have in fact already ended it as of mid-April. The takeaway here is that if we maintain some of the NPIs to ensure that Covid-19 never makes a comeback in either its current form or a mutated one, and encourage widespread influenza vaccination, we might not have a 2020–2021 flu season.

The rate at which the susceptible population is depleted meanwhile is as follows, and its negative is the rate of increase for the infected population:

The rate of increase for the recovered population, and also the rate of depletion for the infected population, is:

Let’s put some numbers on ß and Γ to explain further, and also bring in the basic reproduction number R0. The recovery rate Γ is the reciprocal of the recovery time, which is believed to be no more than 15 days for Covid-19 from time of infection to the time at which the person is no longer contagious. The reciprocal of 15 days is accordingly 1/(15 days); note that we must include the units of measurement which are, in this case, reciprocal time. If we assume R0 = 3 for a given disease, then ß = R0 × Γ gives us:

This means simply that each infected person will, on average, infect 0.2 other people per day during the 15 days in which he or she is contagious. Now we have transmission and recovery rates to go with the SIR equation, but the interdependent system of differential equations cannot be solved directly (because di/dt is a function of s, and ds/dt is a function of i). The best approach is the Runge-Kutta numerical method, which is described in the downloadable pdf file that goes with this article. This is the source of figure 1, which shows the infected fraction and also recovered fraction (people who have been sick). The latter reflects the area under the infected curve, which exceeds 94 percent in this situation.

Figure 1: Percent infected and recovered vs. time, R0 = 3

Now suppose that limited countermeasures reduce the basic reproduction number to 1.6. The curve gets flatter, to the extent that “only” 8.75 percent of the population would be infected at the peak. About 65 percent of the population, the integral or area under the curve, will, however, get the disease eventually. This could still result in hundreds of thousands of fatalities, so it is not enough to just flatten the curve. We have to break the curve, and figure 3 will show what happens when we do it.

Figure 2: Percent infected and recovered vs. time, R0 = 1.6

The curve breaks when R0 is less than one

Here is the scenario that most “flatten the curve” articles have not considered; the reduction of R0 to less than 1. Donald Wheeler, Al Pfadt, and Kathryn Whyte8 did consider this scenario, which they depict as suppression as opposed to mitigation, in which R0 is reduced but not to below 1. This can be achieved by simply throwing as many obstacles as possible into the virus’s path. None of the obstacles have to be perfect; they just have to cut down the transmission rate by a certain percent.

Suppose for illustration purposes that a virus’s basic reproduction rate is 3; the best current estimate for Covid-19 is about 2.6, although nobody knows for sure. This means that, if we do nothing to impede its spread, each infected person will infect, on average, three others.

Figure 3: R0 = 3; two infected people infect six others

The Health and Safety Executive, the United Kingdom’s counterpart of Occupational Safety and Health Administration (OSHA), tested various forms of respiratory protection and found that even improvised face masks made from bra cups or sanitary towels reduce one’s risk of inhaling infected droplets by a factor of 2, or 50 percent.9 Such masks must, however, be worn properly, put on and taken off properly, and also disinfected between uses.10

Assume for the sake of illustration that social distancing also is 50-percent effective, and hand hygiene only 33 percent effective because the disease is transmitted primarily by droplets in air as opposed to surface contact. These are not known numbers; they are presented only for illustration purposes. The result is a parallel reliability engineering problem in which everything must go wrong (for the humans), or a series reliability problem in which everything must go right (for the virus) for a new infection. This is shown in figure 4.

Figure 4: Nonpharmaceutical interventions in series reduce the effective R0

The result is that the epidemic never peaks; it begins to decline right out of the starting gate, as shown in figure 5, with the assumption that 1 percent (any percentage may be used) of the population is infected at the beginning.

Figure 5: Suppression—percent infected vs. time, R0 = 0.5 due to nonpharmaceutical interventions (NPIs)

Suppression is the scenario that most of the “flatten the curve” depictions have not considered. It is clearly the most desirable scenario because it will not only avoid overwhelming or even burdening the healthcare system, it also will cut down Covid-19 in its tracks before it troubles even a small fraction of our population, and it has apparently ended the 2019–2020 flu season as of early April.

How do we make this happen?

The answer is very simple: Follow the doctors’ orders. Jerome Adams and Anthony Fauci, as well as the CDC and OSHA, have urged people to practice social distancing and diligent hand hygiene. Large-scale gatherings of people should be avoided, and OSHA has issued “Guidance on Preparing Workplaces for Covid-19”11 on what employers should do to protect their workers.

We can innovate even more solutions. Grocery shopping, a vital activity, forces people to come within six feet of one another when they pass one another in aisles. If the stores designate aisles for one-way traffic, people will still have to pass each other while going in the same direction, but not in opposite directions.12 When restaurants are allowed to reopen, partitions between tables will reduce the opportunities for the virus to propagate.13 Best Buy is now offering curbside pickup of orders so customers don’t have to come into the stores and get close to employees or each other.14 Grocery stores are offering curbside pickup and, for an additional fee, home delivery, which is what I am using. Reconsider the desirability of the handshake as a greeting because this requires that people come within six feet of one another, and hand contact can spread the disease. If each nonpharmaceutical countermeasure does its bit to suppress Covid-19’s ability to spread, it should not be a serious menace by summer, although this does not mean we can then drop our guard against it.

Effect of vaccination

This brings us back to the issue of seasonal flu, and it is quite possible that the 2019–2020 flu season will end up causing more fatalities than Covid-19 simply because the flu season is taken for granted, and we tolerate tens of millions of cases per year as a risk of everyday living. The collective mental state is similar to that toward spectacular plane crashes: We take for granted the far greater weekly highway death toll because the fatalities take place in ones or twos, as opposed to dozens or more than a hundred at once. The fact that the countermeasures we used against Covid-19 along with flu vaccination coverage of roughly 45 percent of adults (and a higher fraction of children) have essentially eradicated the 2019–2020 flu season should cause us to reconsider whether we really have to put up with the human and economic costs of seasonal flu at all.

Vaccination, when available, reduces the initial susceptible population R0 as opposed to the basic reproduction number. Suppose the seasonal flu has a basic reproduction of 2.5, which means each infected person will infect 2.5 others while he or she is contagious. This means two sick people would normally infect five others, but if 60 percent of the latter are immune, the former will infect only two others as shown in figure 6.

Figure 6: R0 = 2.5 vs. 60-percent vaccinated fraction

This can be modeled by setting the recovered fraction to 0.6 at the beginning of the calculation, which makes the initial susceptible fraction 0.4. Even if we keep the recovery time of 15 days (it is probably closer to eight for the seasonal flu), the infected fraction starts out flat and then begins to decline.

Figure 7: R0 = 2.5 with initial susceptible fraction of 40 percent

Vaccination plus nonpharmaceutical interventions (NPIs)

Now suppose that only 40 percent of the people are vaccinated, but NPIs such as partitions, one-way supermarket aisles, and better hand hygiene are 50-percent effective against the basic reproduction number. Two infected people would, in the absence of any countermeasures, infect five others before they were no longer contagious. If, however, two of the five are vaccinated, they will not get the seasonal flu, and the NPIs will protect (on average) 1.5 of the three others, as shown in figure 8.

Figure 8: R0 = 2.5 with initial susceptible fraction of 60-percent and 50-percent effective nonpharmaceutical interventions (NPIs)

This situation is modeled by setting R0 to 2.5 × 0.50 = 1.25 and the initial recovered fraction to 0.4. The result is that we probably do not have a flu season as shown in figure 9. This is a strong argument for 1) getting the annual flu vaccine; and 2) maintaining the less-invasive NPIs, including changes in building layouts to reduce opportunities for contagion, along with the better hygiene habits we are acquiring as a result of the coronavirus outbreak.

Figure 9: R0 = 2.5 × 0.50 with initial susceptible fraction of 60 percent

FluView from the CDC reports weekly virologic surveillance results in the United States: “WHO and NREVSS collaborating laboratories, which include both public health and clinical laboratories located in all 50 states, Puerto Rico, and the District of Columbia, report to CDC the total number of respiratory specimens tested for influenza and the number positive for influenza by virus type. In addition, public health laboratories also report the influenza A subtype (H1 or H3) and influenza B lineage information of the viruses they test and the age or age group of the persons from whom the specimens were collected.”15

I obtained and graphed the results (figure 10) for the past four flu seasons and also the current one (total percentages only, no breakdown by types A and B; the axis is the week of the indicated year). The graph shows 1) the 2019–2020 flu season is essentially over as of mid-April even though seasonal flu usually persists into May; and 2) the sharp downturn coincided roughly with the first deployments of social distancing and other countermeasures against Covid-19. This underscores the combined effectiveness of even partial vaccination in combination with NPIs.

Figure 10: Percent of influenza-positive samples

Prevent recurrence

If Covid-19 ever returns, perhaps as a mutated version for which there is no vaccine and even survivors of the current version have no immunity, it is vital to ensure that it is not capable of causing anything that resembles even remotely the kind of human and economic harm it caused during the first several months of 2020. This suggests the desirability of permanently engineered but relatively noninvasive NPIs that will reduce opportunities for contagion without people even having to think about them:
• One-way aisles in grocery stores
• Partitions between restaurant tables
• Partitions between seats in buses and trains, with no standing allowed. This could require more buses and train cars, or adjustment of business starting times to eliminate rush hours.
• Partitions between workstations in offices and factories
• Partitions between cashiers and customers
• Elimination or modification of open offices to reduce opportunities for contagion16
• Transition to distance learning and distance conferencing

These NPIs are also, of course, effective against the seasonal flu and could easily, in conjunction with widespread vaccination, make the 2019–2020 flu season the United States’ last flu season. The same would apply in other countries that initiated these actions.

Administrative controls might include a requirement that people wear disposable masks (provided at the entrances) of grocery stores, shopping malls, and other public places, or at least the capability to implement this countermeasure on notice should another contagious disease ever make its presence known.

Personal protective equipment (PPE) is emphatically a last line of defense rather than an engineering or technical control, but there should never again be a shortage of N95 respirators because U.S. companies have already ramped up production, and the public now realizes the desirability of having them readily available. This means that, should Covid-19 or anything like it ever come back, we will be in a position to ensure that it can never again harm a substantial number of people or shut down our economy.

Disclaimer: This article is based solely on the cited references and mathematical modeling, and does not constitute formal engineering or occupational health and safety advice.

Downloadable resources
SIR_Model.xlsx Excel spreadsheet for the susceptible, infected, recovered (SIR) model. The user can model different basic reproduction numbers to illustrate the effects of non-pharmaceutical interventions, and also initial recovered (immune) fractions to illustrate the effects of vaccination when available.

SIR_Model.pdf Mathematics of the SIR model and directions for using the spreadsheet.

1. Matthews, Dylan. “11 charts that explain the coronavirus pandemic.” Vox, March 17, 2020.
2. Occupational Health and Safety Administration. “Guidance on Preparing Workplaces for Covid-19.” OSHA, March 2020.
3. Branswell, Helen. “People ‘shed’ high levels of coronavirus, study finds, but most are likely not infectious after recovery begins.” STAT, March 9, 2020.
4. Centers for Disease Control and Prevention. “How Flu Spreads.” CDC, last edit August 2018.
5. Wikipedia. “Basic reproduction number.” Wikipedia, last edit April 2020.
6. Ridenhour, Bejamin; Kowalik, Jessica; and Shay, David. “Unraveling R0: Considerations for Public Health Applications.” American Journal of Public Health, e32–e41, February 2014.
7. Nesse, Hans. “Global Health—Susceptible, Infected, Recovered (SIR) Model.” Instructional model, Arizona State University, Spring 2010.
8. Wheeler, Donald; Pfadt, Al; and Whyte, Kathryn. “Tracking Covid-19.” Quality Digest, April 6, 2020.
9. Howard, Harry; and Robinson, Martin. “Coronavirus face masks sell out as prices soar 800%: But do they really work and which one should you buy?” Daily Mail, Feb. 27, 2020.
10. Torres, Monica. “Is It Safe to Reuse a Face Mask to Protect Yourself Against Coronavirus?” Huffington Post, March 24, 2020.
11. Occupational Health and Safety Administration. “Guidance on Preparing Workplaces for Covid-19.” OSHA, March 2020.
12. Mazzoni, Alana. “Calls for Australian supermarket giants Coles, Aldi, and Woolworths to enforce one-way aisles and limit the amount of shoppers in each store.” Daily Mail, March 23, 2020.
13. Tsang, Denise. “Coronavirus: Hong Kong restaurants install physical barriers between diners to allay contagion fears.” South China Morning Post, Feb. 12, 2020.
14. Staff writer, Best Buy. “A Note From Best Buy About Covid-19.” Best Buy website, March 10, 2020.
15. Centers for Disease Control and Prevention. “Weekly U.S. Influenza Surveillance Report.” FluView, April 11, 2020.
16. James, Geoffrey. “Open Plan Offices Spread Diseases More Quickly.” Inc., Feb 2, 2020.


About The Author

William A. Levinson’s picture

William A. Levinson

William A. Levinson, P.E., FASQ, CQE, CMQOE, is the principal of Levinson Productivity Systems P.C. and the author of the book The Expanded and Annotated My Life and Work: Henry Ford’s Universal Code for World-Class Success (Productivity Press, 2013).


COVID-19 Tracking

Excellent Aticle! The readability and overall interest would probably be better if some of the non-technical information (e.g. effects of vaccination, NPIs, Prevention) was more upfront and the tehnical formulas followed. Even as a chemical engineer, I had to rethink some of the terms and symbols used. Some of the nontechnical information would be great for governors to use in daily briefings to explain their decisions. Our governor in Colorado would appreciate it. The addition of downloadable software was also valuable.