How to model a quasi-exponential function?

Question

This is the graphic of official daily deaths caused by Covid-19 in Brazil.

And this is the same graphic with an exponential model in red. We see the model don't fit the data.

The misadjustment is clearer with a logarithmic Y axis.

Looks like the model that would fit here is closer to a "tilted parabola", like the one I drew in blue with Gimp.

So my question is: what is the equation of a "tilted parabola" in a graph with logarithmic Y axis? Or: how do I model the present data?

Answer 1

You might be looking for functions with sub-exponential growth. From my limited research, this seems to be a topic more commonly studied in algorithm analysis.

Definition

This other answer in stackexchange, provides an intuitive definition of a sub-exponential function $f(x)$:

1. It grows faster than any polynomial

$$\lim_{x\to \infty} \frac{f(x)}{x^a} = \infty\ \forall a$$

2. and it grows more slowly than any exponential

$$\lim_{x\to \infty} \frac{\ln f(x)}{x} = 0$$

Example

The function $f(x) = e^{x^\beta}$, with $0 < \beta < 1$ satisfies both conditions:

1.

$$\lim_{x\to \infty} \frac{e^{x^\beta}}{x^a} = \infty $$

which clearly goes off to infinity.

2.

$$\lim_{x\to \infty} \frac{\ln{e^{x^\beta}}}{x} = \lim_{x\to \infty} x^{b - 1} = 0$$

which equals zero, since $b - 1 < 0$

Caveat with respect to COVID-19 (and epidemiology)

It is possible that your example is better approximated by a polynomial growth rate! At least during the first weeks since the outbreak. This paper suggests that containment policies in China have managed to suppress the infection growth to the point it followed a rate of $t^\mu$ (where $t$ is time).

Take a look at page 2 of that paper. When plotted with a log-log scale (NOT just a single log!), the confirmed cases form a straight line - thus, a polynomial function.

Answer 2

I think the generalized growth model (GGM) is what you're looking for. $$x' = rx^p,$$ where the sub-exponential part happens when $0<p<1$. Sub-exponential growth is actually way more common than exponential growth, especially on a small scale like a district or city. Here is a reference.