In rplzzz/CovMitigation: COVID-19 infection model for Virginia

library(dplyr)
library(ggplot2)
library(tibble)

td <- 8
b <- 60

Given an exponentially growing population (as, for example, in the early stages of an epidemic, when $S/N \approx 1$), and given a targeted testing regime, how does the growth rate of confirmed cases compare to the growth rate in the general population? It turns out to be a lot lower.

Consider an example, let the growth rate in the general population be $r = r signif(log(2)/td, 2)$, for a doubling time of about $\tau_d = r td$. Define the test targeting factor $b$ as a multiplier on the odds ratio of infected to not infected. That is, if the fraction infected is $f$, then the odds ratio is $\frac{f}{(1-f)}$. The odds ratio for positive to negative tests would then by assumption be $b \frac{f}{1-f}$. We will take $b = r signif(b,2)$. Both this and the value of $\tau_d$ are approximately the values recovered in our MAP parameter set.

r <- log(2)/td
f1 <- function(x) {0.01*exp(r*x)}
f2 <- function(x) {b*f1(x)/(1+(b-1)*f1(x))}

x <- seq(0,15, length.out=100)
pd <- bind_rows(
  tibble(t=x, y=f1(x)/f1(0), type='population'),
  tibble(t=x, y=f2(x)/f2(0), type='confirmed')
)
ggplot(pd, aes(x=t, y=y, color=type)) + geom_line(size=1.2) + 
  scale_y_log10() + 
  theme_bw() + 
  scale_color_brewer(type='qual') + ylab('y/y(0)')

This plot shows the growth of confirmed cases and population infection, relative to their initial values, over time. The y-axis is a log scale, so a straight line indicates exponential growth. By construction the population growth line crosses $y/y_0 = 2$ at $t = r td$. The confirmed cases show a lower initial growth rate, and the growth is sub-exponential. Thus, in this case trying to infer the growth rate in the population by looking at the growth in confirmed cases would result in a significant underestimate of the true growth rate.

rplzzz/CovMitigation documentation built on June 7, 2021, 8:48 a.m.