CORONA VIRUS AND FARR'S LAW

This post is based largely on an excellent article by Santillana et al. (2018) from the journal, Infectious Disease Modeling, in 2018. It describes and analyzes the growth of the number of infected over time, I(t), in an epidemic, relying on a relationship, Farr's Law, found by an epidemiologist, Dr. William Farr, almost two centuries ago. I will explain part of the research, which would seem to relate to our corona virus epidemic. Here is the Santillana et al. link:

https://www.sciencedirect.com/science/article/pii/S2468042718300101

Farr defined a series of time increments, t=0, 1, 2, 3...etc., for which we have the number of persons newly infected, I(t). He found as time goes on, the tendency to infect an individual slowly decreases, by a factor 1/(1+d), where d>0 and depends on the time interval chosen. So, I(t+1)/I(t)= 1/(1+d). 

Farr used ratios of numbers infected at the time intervals t to express his "law":

[I(t+3)/I(t+2)] / [I(t+1)/I(t)] = K = [1/(1+d)]^4.

The symbols ^4 mean "to the 4th power."

A century ago John Brownlee showed that Farr's law gives an exponential function:

I(t) = exp[-At^2 + Bt + C] = exp(C) exp[-At^2 + Bt]

this is a constant term times an exponential term of growth (Bt) and one of decrease (-At^2).

From the beginning, t near zero, to the end, t >> 1 or -At^2+Bt << -1, I(t) looks somewhat like the bell-shaped curve of the Normal, Gaussian, statistical distribution, exp(-a(t-b)^2). 

Setting -At^2+Bt << -1 lets one predict the effective end of the epidemic, given A and B from fitting the curve function to the data. 

The maximum of the infection curve is when the derivative of the argument of the exponential 

(d/dt) [-At^2 + Bt]= 0.

2 At = B, at time

t = B/2A

Thus, I(t, max) = exp(C) exp[B^2/4A]. 

B can be estimated from the initial slope of I(t).

A might be obtained from curve-fitting, perhaps 

made more convenient by taking the logarithm of I(t).

The logarithm of I(t) is -At^2 + Bt + C and this

quadratic can be solved readily, given the value for log I(t)

at the particular t.  

These mathematical approaches should enable epidemiologists to give reliable predictions about the course of our corona virus epidemic.

Douglas Winslow Cooper, Ph.D.