Why exponential?

The change over a given time period of infected person \( \Delta N_{t} \) is proportional to the current number of infected persons \( N_{t} \): $$\Delta N_{t} \propto N_{t}.$$ Taking into account a constant of proportionality that describes the strength of the transmission rate, the following mathematical expression is obtained: $$\Delta N_{t} = c\,N_t.$$ This type of equation, in which the function and also derivatives of the function appear, is called a differential equation. Depending on the type of differential equations there are different ways to solve them. In this simple case the solution is obtained by separating the variables (also known as the Fourier method). The solution to this differential equation is an exponential function: $$N(t) = a\cdot\exp(b\cdot t).$$ The constant of proportionality \(b\) contains, among other factors, information about the probability of infection, which in turn depends on various factors such as the composition of the virus or the average number of infected persons exposed to a person. Using the method of least squares the parameters of the exponential function \((a\) and \(b)\) are determined and the function is fitted to the data.

A doubling time can be derived directly from the estimated parameters. Here the following equation under the question "For which time \( t \) does the initial value of infected person \( a \) double"? is solved: $$a\,\exp(b\cdot t) \stackrel{!}{=} 2\cdot a.$$ The result is: $$t_{\mathrm{double}} = \frac{\ln(2)}{b}.$$

It must be added that the description of infection numbers using an exponential function is only reasonable at the early stage of the infection process. Since the number of persons is naturally finite, the change will certainly decrease. At a later stage, the description of a saturation process by applying a logistic function will doubtlessly be more appropriate.

Data is updated automatically as reported by Johns Hopkins University. Last update: 02/07/23.
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