A simple asymmetry

Suppose the number of people infected grows exponentially with time as ce^{rt} . 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 ce^{rt}-ce^{r(1-\delta)t} in the former and ce^{r(1+\delta)t}-ce^{rt} in the latter case. The ratio of the latter to the former is simply

\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}}.

As t gets large enough, 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, 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:

\displaystyle R_a(t)=\frac{f(x+a)-f(x)}{f(x)-f(x-a)} > 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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