COVID-19 Math: You, Me, R₀ and Rolling Re-entry

In the war against COVID-19, two math quantities govern our lives: R₀, the basic reproductive number, and H, herd immunity. We all need to know about them and – most importantly – about our roles in determining their values. Welcome to COVID-19 Math, 101!

Let’s first do R₀ (“R naught”). When the pandemic is starting and there is no immunity, R₀ is the average number of new infections caused by a recently infected person. For instance, if R₀ = 2.0, then on average a newly infected person will infect two others before retiring to convalesce. For seasonal flu, we hear R₀ historical values of 1.4 or 1.5. For COVID-19, one hears larger values ranging between 2.5 to 5.0 or so. Why is R₀ important? Because if its value is greater than 1.0, then each generation of the disease is larger than the previous. The number in the next generation is R₀ times the number in the current generation. Suppose we start with one “patient zero.” The next 10 generations will have these numbers of infected people: 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1,024. This is exponential growth. Imagine if R₀ = 3. The 10th generation would have 59,049 infected people. These are truly overwhelming numbers. Let’s not even think about R₀ = 5.0.

Now we get to herd immunity, H, a term often pictured in term of grazing cows. Colloquially, if a sufficient fraction of cows is immune to a certain infectious disease, then the entire herd is immune. How? At herd immunity, if one non-immune cow should become infected (say by a mosquito bite), then when the infecting virus tries to infect other cows, it finds that it can’t grow the numbers of new infections because too many of the cows are immune to the virus. Generation-to-generation infection growth stops. Suppose the infecting virus has an R₀ value of 2.0. When the virus tries to infect two more cows, it will fail to grow new infections if one of the cows it chooses is immune. This happens if 50% of the cows in the herd are immune.

Our finding: For a virus having R₀ of 2.0, herd immunity H equals 50%. R₀ determines H! At a nightmarish extreme, if R₀ = 5.0, then we’d need H = 80% immune, so our newly infected person (or cow) would “try” to infect five others, but four of them (80%) would be immune. Again, no growth in infections. To test your understanding, try to determine the herd immunity value if R₀ = 3.0. Congratulations! That concludes our math lesson! We have found that there is only one parameter that governs the disease spread: R₀.

Past and Future Values of R₀

Here is the key: Past values of R₀ are provided by historical data – depicting human behavior and disease characteristics; future values of R₀ are determined by you (and me and all of us). Imagine that you are infectious and still going about your life, with a “personal R₀” of 4 for a given day. You interact with 20 people each day. You could infect each of those 20, but on average you infect four. The likelihood of infecting any one of them depends on the intensity and type of interaction. A hug or a handshake has high likelihood of transmission. The more careful you are, the less the chance of transmission.

Now suppose you have just decided to interact with 10 of these 20 people via the Internet; for such virtual interactions, there is zero chance of passing on the infection. By reducing the number of face-to-face contacts by 50%, you cut your personal daily R₀ in half, to 2.0! Now, if your remaining 10 face-to-face interactions use social distancing and good hygiene, you could easily achieve another 50% reduction, from 2.0 to 1.0, now a 75% total reduction. While any given disease may be more or less infectious, it is ultimately human behavior that determines the numerical value of R₀. R₀ is not like the constant pi = 3.14159… German Chancellor Angela Merkel believes that Germany achieved an R₀ of about 1.0 as a result of intense social distancing. That would place H at zero, implying that society experiences zero growth in new infections, even if no one is immune.

The extent of infectious spread is determined by our behaviors, individually and collectively. The value of R₀ depends on us. If we were all to stay cooped up in our residences, R₀ could be very close to zero. Yet there would be no real life, no human-to-human interaction, no economy. In the last several weeks, by intense social isolation and excellent hygiene, many parts of the United States have likely reduced R₀ to below 1.0.

Journey Back to More Normal

We need now to emerge from this isolation and begin our journey back to a “more normal” life. In doing this, we must protect the most vulnerable – senior citizens and those with chronic medical conditions. The National Coronavirus Task Force labels our opening strategy “rolling re-entry,” a controlled path back. For example, Boeing and Doosan Bobcat (a farm equipment maker) announced plans to restart some manufacturing with several thousands of their employees, with much attention toward employee safety. Over the coming months, we can expect myriad other examples of carefully crafted rolling re-entries. Merkel recently announced similar gradual openings in Germany.

Doing rolling re-entries well on a national level is a delicate balancing act. We have no vaccine and, to date, only a tiny fraction of our population is recovered and immune. So, immune people circulating will not help us get to H, for just about any positive level of H. Any loosening of social distancing may raise R₀ above 1.0, thus increasing H above 0, and result in more infections. We want the R₀ value to remain small, both to minimize new illnesses and avoid spikes of infection that can swamp our healthcare system. We all have roles to play in making this successful. We are all oarsmen in one huge boat.

Germany is working toward R₀ = 1.1, then perhaps 1.2. The U.S. rolling re-entry has not been described in such scientific terms, but it will likely share the incremental-steps attributes of Germany’s plan. And these steps will vary by state and local conditions. At every step of rolling re-entry, we must maintain strong social distancing and good hygiene, so our R₀ values remain low, perhaps even below 1.0.

We suspect that a significant number of re-openings, if done with meticulous care, will not increase the current value of R₀. Examples could include openings of small shops, restaurants with six-feet separation of tables and traffic-regulated department stores. After each set of re-openings, we stop, evaluate and re-estimate R₀. From this new equilibrium, we may then allow more openings, perhaps resulting in a slightly higher R₀. Keep the lid on, and slowly, ever so slowly, lift it. And then lift it again. Always monitor with testing, then adjust and correct. Eventually we stop, when we will have landed on a managed value of R₀ that we can sustain in a new normal open society – most likely still with masks, social distancing, temperature checks and more. Many months later, with a vaccine, we can party!