Policy Analysis
COVID-19 Infection Dynamics
How to Know When It Is Safe


Garrett A. Hughes
Principal Architect


Featured Article #2

Summary of Learnings

In Featured Article #2 we learned how to quantify the conditions under which the SARS-CoV-2 virus will result in a self-sustaining infection rate in a given population. We learned how to calculate the fraction of that population that needed to be immune to the virus before that population as a whole could be said to have acquired herd immunity. This turned out to be approximately 55 percent of that total population. This percentage is called the herd protection threshold or HPT. That population was characterized as having a maximum infection multiplier, MAX_infMul, of 0.375, which corresponds to a reproduction number, R-zero, of 2.25.

We simulated exposing a population with herd immunity to an attack by the virus, and observed that the virus was able to infect only a small number of individuals before dying out of its own accord. This was a truly amazing result, given that a population without herd immunity, and of the same size, was devastated by the virus. The total population in both simulations consisted of over one million individuals.

We also simulated an attack by the coronavirus on a population with a significantly smaller percentage of immune individuals than that required to confer herd immunity. The result was a self-sustaining infection rate that decimated that population in a manner similar to the population that had no immune members.

Finally, we pointed out that herd immunity will only be conferred on populations that are uniformly mixed, and where contacts between individuals occur at random. Populations that do not meet these criteria will require a different percentage of immune individuals to acquire herd immunity, and in some cases such as nursing homes, nothing short of complete immunity would be required to protect the entire nursing home from infection.

Population Characteristics Determine When It Is Safe to "Open Up"

We can characterize a population by the nature of the encounters taking place between its members. The variables of interest are the spacing between individuals, and the duration of the encounters. On one end of the spectrum we have encounters that take place with distant spacing for a short duration of time, and at the other end we have close encounters of lengthy duration. The former might occur while walking through a mall, while the latter may take place when sitting next to a stranger in a theater or at a sporting event. Populations as large as a city will have encounters taking place over a range of values for both spacing and duration.

To determine when it is safe to "open up", it is useful to examine the impact of the novel coronavirus on a population with a uniform population density throughout, where contacts between individuals occur completely at random, and where no members of the population are currently infected. Assuming uniform mixing of population members, we can define an infection multiplier known as the reproduction number. This number, also known as as R-zero, and written Ro, represents the average number of susceptible individuals that can be infected by one infectious person while they remain infectious. Ro devolves into the effective reproduction number, R, as the susceptible population dwindles. As the susceptible population dwindles, chance encounters that produce infections will at first rise due to the increase in the number of infectious individuals, but will then fall as the susceptible members of the population dwindle. The scarcity of susceptible members can be represented by the ratio of the remaining susceptible members to the total population.

At some point this ratio will be such that the infectious members of the population can only find one replacement for themselves during the entire period that they are infectious. The effective reproduction number, R, at that point will equal one, and the infection rate will cease to grow. The complement of the susceptible to total population ratio at that point will equal the herd protection threshold. For our population this characteristic turned out to be approximately 55 percent.

Note: Close observation of the simulation will show that the more rapid the infection rate, the less likely that the infection rate will peak at the herd protection threshold, especially if the infectious period is distributed over a number of days among the infected population. This fact does not impact the results of what we expect with regard to a quiescent population at the HPT where infected individuals are inserted into that population.

The herd protection threshold is considered by many to be the "safe point" at which to open up a population of susceptible individuals. Note that the HPT depends upon the characteristic value of Ro for a given population. In a population at its HPT susceptible individuals will neither have to isolate themselves from encounters (suppression) , nor take steps to protect themselves by physical distancing or the wearing of masks when encountering other individuals (mitigation).

Caveat: it would be a "safe point" if all populations were of uniform density and mixed randomly with a fixed mode of encounter. This is hardly the case for real populations. The difference in population characteristics from a densely populated city center to that of a small town on the outskirts of a city can be large, and both much different than the characteristics of a nursing home population.

When we speak of the maximum infection multiplier, MAX_infMul, or the reproduction number, Ro, we are speaking of averages derived empirically (or mathematically) for given populations. The larger the population, numerically, the less likely that Ro can be applied to any given neighborhood within that population. Therefore, "opening up" the entire population based on an average value, creates the same problem as assuming it safe to wade across a river of average depth of two feet.

Nevertheless, Ro and HPT are currently the best indicators of population characteristics that epidemiologists have to offer the general public at this time. Note, however, that respiratory droplets From COVID-19 environments are calculated to be capable of producing values of Ro from 1.4 to 5.7 (Wikipedia: basic reproduction number). And we have nothing better to offer here except our words of caution in using these numbers as point estimates for population characteristics. At some time in the near future, however, we will present an entirely different approach to modeling the tranmissibility of an infectious agent based on the stochastic nature of the encounters themselves.

Some Proposals On When To "Open Up" Susceptible Populations

First, let's be clear on what we mean by "open up".

By "opening up" we mean a return to all activities, public or private, that existed prior to entry of the SARS-CoV-2 pandemic into the United States, with or without efforts to mitigate the future spread of the infection to the susceptible population.

Opening up without mitigation
We have simulated the results of opening up a susceptible population to a few infected individuals and the results were disastrous. But that is exactly what some people want to do. Their rational seems to be that the symptoms aren't that bad, and that they themselves will most likely return to normal health in a couple of weeks. They have given absolutely no thought to the 20 percent who will become more seriously ill, the 10 percent that will require hospitalization, the 5 percent ending up in the ICU , and the one percent or more who will succumb.

What these people fail to understand is that when opening up without mitigation almost everyone will be ill at the same time. The amount of virus exchanged during an encounter will be large, and that the larger the initial dose, the more difficult it will be for the newly infected individual to create a timely antibody response. They also don't realize that with all hospitals filled to capacity, that if they themselves require hospitalization, and are unable to obtain it, they will likely die of their infection.

One reason the mortality rate hasn't gone higher in many areas of the country is due to the heroic efforts of the health care providers. That, despite the impediments that the Trump administration has placed in their paths. This includes impediments to obtaining the necessary equipment to assist a patient's breathing, and securing the personal protective equipment required while treating patients.

The resulting chaos of such an opening would very likely destroy the tattered fabric of this society, and result in a state of lawlessness and panic that would destroy this country.

Opening up with mitigation
We have shown in Featured Article #1 that holding the infection rate at a very low value using mitigation efforts will eventually result in an epidemic of severe proportions in a population without herd immunity. We have also suggested that this will result in a search for scapegoats in the same way that Trump is now searching for a scapegoat to exonerate himself from any culpability in the rapid spread of the infection in the United States. He had plenty of time to react. He did nothing because of his fear of mitigation efforts affecting the economy. Nevertheless, his worst fears were realized despite his recalcitrance. That is the way with microbes: they just don't have any political affiliation.

Opening up with mitigation and testing
In order to be safe, with the virus still extant, people whose job it is to meet with other people on a regular basis, or care for them, or handle goods that will be delivered to them including food, would need to be tested on a daily basis to determine whether they were infectious. Inexpensive, reliable test kits that can be used by anyone are currently not available. It is also possible that even if someone in this position tests positive, they may ignore the results, if not working threatens their livelihood.

People working in environments that offer protective measures between themselves and those with whom they are working or serving are not much better off. At some point, due simply to the frequency of encounters, the protective measures will fail, and multiple infections will result.

Even if an individual tests themselves favorably in the morning, how do you keep a person who has become infected at work, school, or play on that same day, from returning home and infecting everyone in their household. Now you are mandating at least two tests per day per individual.

Our Only Option

We do not have any viable choices to open up. That is, if we don't want to cause another rise in infection rates more serious than the ones that we have just experienced. We must continue acting to both suppress the spread of the virus by sheltering a large segment of the susceptible population, and mitigate against the spread of the virus by taking protective measures for all other activities. We should be especially cognizant of the needs of those working in public to provide us with essential services.

Yes, there will be hardships if we continue to shelter. Especially for those who cannot continue to work and earn a living. But many who are working from home or in essential services are in a position to help those who are unable to work. A special tax on income and dividends should be placed on those working to help support those who have lost their jobs. It is the best way of sharing the burden until a vaccine becomes available. A vaccine will, in a matter of a year to 18 months be made available. In the meantime we had better learn to take care of each other. You could be the next person who needs help

And if you still remain skeptical, and want to chance it out with the virus, we suggest you read the feature article in a recent issue of the journal Science: "A Rampage Through the Body: The lungs are ground zero, but COVID-19 also tears through organ systems from brain to blood vessels", by Meredith Wadman et al. 24 April 2020

A Future Direction That Will Better Help Us Understand How To Solve Problems Like This One

The website on which you are reading this article was created to help people learn how to solve problems that arise in the context of complex systems. COVID-19 is one of those problems. Solutions to such problems will require a paradigm shift in the way we educate people. So will problems that arise from any viruses that attack the human biosystem, or problems that arise in human induced climate dynamics, or even more difficult problems that we cannot yet foresee. We need to prepare ourselves to be able to solve these problems before their consequences destroy us. For a methodology being designed to do just that we suggest reading over the contents of this website —