About one indicator applicable for the visual assessment of rapidly growing functions

For many epidemic models — SIR, SEIR, and the like (for details of the mathematical description, see, for example, www.idmod.org/docs/hiv/model-compartments.html ) the following statement is true: at the initial stage of the epidemic, when the number of people infected (I ) is much smaller than the size of the population, the growth rate of the number of cases is proportional to the number of cases:

$$$∂I/∂t=βI$where β is the coefficient characterizing the rate of infection.

The solution to this equation is an exponential function. For exponential function$$$f(t)=a^t$ The following functional equation holds:

$$

$$f(t+log_a⁡2 )=2f(t)$$

T.O. number$$$log_a⁡2$ is the doubling period for a function $$$f(t)=a^t$. By definition, if the doubling period for a smooth non-decreasing function is constant, then the function is exponential.

Like many others at this interesting time, I follow the growth rates of the incidence rate published, for example, on the site .

For quite some time now, the graphs began to resemble something similar to a boomerang or a hockey stick:


Figure 1

The same graphs on a logarithmic scale give a little more information:


Figure 2

It can be seen that the growth rates tend to slow down, since the slope of the logarithm from the corresponding function decreases, but all but there is dissatisfaction from the lack of understanding of how effective the measures taken to contain the epidemic are nevertheless effective.

Real dynamics of the number of infected even in conditions of applicability of the approximation $$$∂I/∂t=βI$differs from exponential, which is primarily due to measures to contain the epidemic, which lead to the fact that $$$β$ceases to be a constant and becomes a decreasing (if effective measures, of course ) function of time.

In connection with the foregoing, it is proposed to use a doubling period as an indicator applicable for the visual assessment of functions similar to the indicative one. In the general case, for a monotonically increasing function$$$f(t)$ doubling period $$$D(t)$can be determined from the following functional equation:

$$

$$f(t+D(t))=2f(t)$$

Difference $$$D(t)$from constant indicates the difference $$$f(t)$from the exhibitor. In relation to the dynamics of incidence rates, growth$$$D(t)$(ideally - to infinity) indicates the effectiveness of measures taken to contain the epidemic.

In the case of functions defined in tabular form on a discrete set, for example, in the form of a table of the dependence of the number of cases on the date, there is an arbitrary$$$D(t)$. As the easiest way to determine$$$D(t)$we can propose the following:

Let t∈ {0; 1; ...; N} be discrete time, I (t) is the number of cases depending on time t. Then it is



also possible to determine the “pessimistic” doubling period



“Pessimism” in this case is due to the fact that the comparison of I (t) is always made with I (o), i.e. with a "low" by definition base. But do we assume that the situation should improve over time? For optimists, there is a definition:



According to the above definitions



The following are examples of the use of the above indicator:


Figure 3


Figure 4

Data on Spain, described in the press as an example of headache


Figure 5

Despite the obvious dizziness at the initial stage, Spain still looks hopeless.

And in conclusion - native penates.


Figure 6 It is

comforting that in Rospotrebnadzor the growth rate of the incidence of COVID-19 in the Russian Federation was considered slow .

The file with the source data, formulas and graphs can be taken here.

Homework:

1. Decide on$$$D(t)$ the equation $$$f(t+D(t))=2f(t)$for the following functions

$$$f(t)=t^t$
$$$f(t)=Γ(t)$where $$$Γ(t)$- gamma function
$$$f(t)=t^n$
$$$f(t)=ln⁡(t)$
Also for $$$f(t)=ln⁡(t)$solve the equation $$$f(t+D(t))=mf(t)$

2. Answer the question: how are the defined period of doubling the function and the logarithmic derivative of the function related?

I ask readers not to post decisions in the comments within a week.