6:46 a.m. - 2020-08-05
VIRUSES AND EXPONENTIAL GROWTH
The spread of viruses can be expressed mathematically, but it can be difficult to make predictions because of biological differences between viruses and different behaviors in a population. Any mathematical model of a viral contagion contains built-in assumptions. Some viruses (measles, for example) are inherently more contagious than others, and factors outside the virus can have a big impact as well.

The equation for exponential growth is

(N+1) = (N)(E)(p)

where N is the number of cases on a given day
(N+1) is the number of cases on the following day
E is the average number of exposures
P is the probability of infection for one exposure

The rate of contagion is the product of E and p. It's the number by which we multiply today's cases to predict tomorrow's cases. In the absence of a vaccine, measles has a rate of contagion of about 15, so every infected person, on average, infects 15 others. Covid has a contagion rate of around 2. This seems low at first glance, but it means that the number of cases can double every day. An old puzzle gives the reader the choice between receiving one million dollars or starting with one penny and doubling that every day for a month. The second choice results in a net gain of $536,870,912: a terrific deal if you're making an investment, but not if you're following new infections in a population.

Consequently, societies take steps to lower the rate of contagion. Ideally, we would administer a vaccine, reducing P, the probability of exposure, to zero (assuming a 100% effective vaccine and no anti-vaxxers). In the absence of a vaccine, the value of E, the average number of exposures, can be reduced through social distancing, sheltering at home, self-quarantine, sterilizing surfaces, and wearing masks. The value of p, the probability of infection for one exposure, can be reduced by hand washing and not touching one's face. The value of p also goes down as more people within a population develop immunity to the virus, either through vaccination or having been exposed and developing antibodies naturally. This latter phenomenon is referred to as "herd immunity." The lower the product of E and p, the lower the contagion rate of the virus.

It should be apparent why the contagion rate is usually expressed as a range: Behaviors such as wearing masks, social distancing, and hand washing will vary from person to person and, on average, from population to population, and so will the product of E and p. Behaviors that increase E and p, such as large social gatherings in small areas with poor ventilation and talking loudly while not wearing a mask, will also increase the contagion rate. A drop in contagion rate doesn't mean the virus has gone away; it only means that the behaviors are working. Reversing the behaviors can also reverse the trend in contagion rate.

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