A simple asymmetry

Suppose the number of people infected grows exponentially with time as $latex ce^{rt} &s=0$. Consider two cases: in the first, the true rate is less than r by δ percent; in the second, it is greater than r by δ percent. The absolute value of the error in the estimated number of infections is $latex ce^{rt}-ce^{r(1-\delta)t}$ in the former and $latex ce^{r(1+\delta)t}-ce^{rt}$ in the latter case. The ratio of the latter to the former is simply

$latex \displaystyle R_{\delta}(t)=\frac{ce^{r(1+\delta) t} – ce^{rt}}{ce^{rt}-ce^{r(1-\delta) t}} = \frac {e^{r\delta t} – 1}{1-e^{-r\delta t}}. &s=1$

As t gets large enough, $latex R_\delta (t)$ increases, in essence, exponentially. Of course one could argue that the real-word infection curve would soon reach its inflection point and get concave. If 100% get infected, it would eventually look, normalized by population size, like a distribution function. For the exponential growth phase, however, $latex R_\delta (t)$ is a strictly increasing function and is greater than one for any positive t.

This ratio would still be (strictly) above one for any (strictly) convex function, as follows from Jensen’s inequality:

$latex \displaystyle R_a(t)=\frac{f(x+a)-f(x)}{f(x)-f(x-a)} > 1. &s=1$

Things get more complicated with functions that look like the logistic curve, but it’s still worth bearing in mind this asymmetry, striking for fast-growing convex functions. This is just my two cents on the Taleb–Ioannidis debate, or should I call it Taleb’s criticism of Ioannidis and others: underestimating the spread rate by a certain amount or percentage can lead to a much greater error in the estimate of the number of the sick than overestimating it by the same amount or percentage.

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