Tuesday, March 10, 2020
CORONA VIRUS EPIDEMIC AND FARR'S LAW
This blog post is based on an excellent article by Santillana et al. from the journal, Infectious Disease Modeling, in 2018. It describes and analyzes the growth of the number of newly infected over time, I(t), in an epidemic, relying on a relationship, Farr's Law, found by an epidemiologist, Dr. William Farr, about a century and a half ago. I will explain part of the research. Here is he link:
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].
The duration of the epidemic can be approximated from t*,
being 2 t* if the curve were symmetric and somewhat
longer due to being skewed with the long tail toward
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.
This link shows the Wuhan incidence data, pretty much a bell-shaped curve with the exception of one day, and it shows a duration of about 40 days:
The Santillana et al. (2018) article has this figure from Farr's analysis of a smallpox epidemic, and we see that the fit is excellent: