## The Use of Re in Forecasting COVID-19 is Misleading

John Simpson kindly drew my attention to the Oxford’s COVID-19 Evidence Service’s method of forecasting the epidemic using their basic reproductive number which in my view is misleading (see www.cebm.net/oxford-covid-19/) and search “when will it be over”).

Their widely cited R parameter is defined as the expected average number of individuals one individual will infect in a susceptible population. If the expected R, Re, is less than 1 they expect the infection to eventually die out; if it is greater than 1, they expect it will continue to spread exponentially in the absence of immunization.

This is somewhat misleading. Instead, the growth of a biological population such as the virus is commonly described with the density-dependent S-shaped logistic function:

which relates the rate of growth of a population such as the virus SARS-CoV-2 which causes the disease COVID-19 at time t (the tangent to the curve relating population size to time) to its existing population size N_{t} and two parameters – r, the intrinsic rate of increase, and K, the ceiling, i.e. the carrying capacity of the environment. Initially, logistic growth approximates exponential growth because the expression in square brackets approximates 1 and hence dN_{t}/dt approximates rN_{t} , the expression for exponential growth. The maximum growth rate (tangent to the curve) is at K/2 and the growth rate declines thereafter symmetrically with its previous increase until it reaches 0 at N = K where population size itself levels off. The reason is obvious. Once half of the carrying capacity of the environment i.e. the susceptible host population is reached, it becomes harder and harder for the virus to find additional hosts; infected individuals become less and less likely to encounter people to infect – sometimes referred to by epidemiologists as herd immunity.

It should be pointed out that Re would not be misleading if it was being assumed that frequent culture-gene coevolution was taking place as described here about five posts ago. If frequent, adaptive mutations were taking place among the viruses, selected for by our culturally spread methods to avoid it (social distancing, mask wearing etc.), then the viral population could potentially continue to grow exponentially even to the ceiling. That extreme at least is very unlikely. Genetic variants of the virus are known, but there is no knowledge that the ones identified have adaptive and certainly that much adaptive significance, as of yet anyway. In any case, they explicitly assume a homogeneous viral population

However, it should also be pointed out that none of this matters for now and for the near future at least. By the best evidence I have been able to find, no country is close to having half of their population infected except possibly farmed Mink in Denmark!

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