## Leonhard Euler

###### Logistics of Covid-19 Army

Eda Toprak

### The Exponential Growth of COVID-19

Figure 1: The Exponential Growth of Covid-19

I believe you have seen this shape before. It is a J-shaped curve, which is seen in the population growth of many natural phenomena, such as pathological organisms. These curves represent an exponential function, which means in terms of an illness, that the number of the infected cases double, triple etc. itself in every specific period of time. In COVID-19 break, this doubling takes place every 2 or 3 days. Although no virus in an epidemic can grow at an exponential rate forever, the first exponential growth is an alarming condition.

Logistic Curves

So we were saying that the epidemic does not continuously grow at the exponential rate, yet it slows down and thus concaves down after the inflection point, which is where the shape changes from concaving up to down. Since the analysis of COVID-19 worldwide leads to logistic growth eventually, let’s further investigate logistic curves. The formula of a logistic curve would be:

where

y(t) → number of cases at any given time

c → limiting value, maximum capacity of y

b > 0.

The number of cases in the beginning (initial value) as t=0:

The maximum growth rate: at t = and y(t) =

The above formula for the logistic curve was derived in the past from a differential formula found by Pierre François Verhulst.

Figure 2: Verhulst's Differential Formula

The green part here represents that this formula is concerning the growth of the population rather than the population size itself as it refers to the derivative and thus the rate of change of population.

y and c in the formula indicates that the growth of population depends on the population size, y, and the maximum capacity, c. When y=c, this will mean that the population is at its maximum size, and the blue part and thus the growth will be equal to zero.

When, on the other hand, y is much smaller than c, the blue part will be close to 1. As a result, the growth is defined by the orange section, which is indeed the formula of the exponential growth.

In general, logistic curves are used to model the population growth. Thus, first we will look at an example of the logistic curve equation used to analyze population with some details about its calculation and then see how this is applied to the spread of Covid-19.

Figure 3: The Logistic Growth of a Population

Figure 4: The Logistic Growth Modelling of Covid-19 in China

To define several variables used in the logistic curve equation, t represents the time. Its units could vary from seconds to years. K represents the carrying capacity. The variable P signifies the population. Because population continuously changes, P will be seen as a function of time, denoted as P(t). The beginning of populations as they exponentially increase until the carrying capacity is reached could be analyzed as an exponential growth as shown in the first example in this paper of Covid-19 graph. Then, an example for the exponential growth function is P(t) = , where is the initial population and the constant r is the growth rate. Those variables will be utilized in the logistic curve equation for the population growth rate. Let’s start if you have grown excited enough.

Solving Logistic Differential Equation

The differential equation of the logistic curve is written in the form (This is the same as colorful expression given before.);

When both sides are multiplied by dt and divided by P(K-P), the equation becomes;

If we multiply both sides with K and integrate, we obtain;

Partial fraction decomposition will be used to integrate the left-hand side of the equation.

If we give a little break here, we can see that :

The equation then becomes:

(check derivative rules for ln; integral is the inverse of differentiation)

(logarithm rule for division is used)

To get rid of the natural logarithm, we will now exponentiate both sides:

We define . Consequently, the equation can be written as:

In order to solve the equation for P(t), we multiply both sides with (K-P) and collect the terms containing P on the left-hand side:

Lastly, we need to determine the value of to solve the equation. To achieve that, let’s substitute t=0 and into P and solve the equation.

Substitute into the function of P(t) that has been found above:

To simplify the expression, expand the fraction by (K - ). Finally, a solution of the differential equation of logistic curve becomes:

So now, when we know the initial population , carrying capacity of an environment K, and the growth rate of the population r, we can find the population at a given time with the logistic curve equation that we have just derived.

Logistic Curve Formula for Covid-19

The logistic formula for Covid-19 pandemic will have the same format as the logistic curve of the population growth with the only difference of variables representing different parameters.

, where Q(t) is the number of cases at a time, a is a constant, b is the incubation rate, K is the capacity value, in other words, the maximum number of Q(t) cases. We do not suggest, but you may now approximately calculate the number of cases at a future date after finding out what the constants in the equation are by modelling the spread of virus in your region.

References:

“Doing the (Coronavirus) Math: Exponentials, Bell Curves and Flattening.” DesignNews, 25 Mar. 2020, www.designnews.com/sitemap/articlepermonth/3/2020?page=2.

Korstanje, Joos. “Modeling Logistic Growth.” Medium, Medium, 29 Mar. 2020, towardsdatascience.com/modeling-logistic-growth-1367dc971de2.

Strang, Gilbert, and Edwin Jed Herman. “8.4: The Logistic Equation.” Mathematics LibreTexts, Libretexts, 29 Aug. 2020, math.libretexts.org/Bookshelves/Calculus/Book:_Calculus_(OpenStax)/08:_Introduction_to_Differential_Equations/8.4:_The_Logistic_Equation.

Wang, Peipei, et al. “Prediction of Epidemic Trends in COVID-19 with Logistic Model and Machine Learning Technics.” NCBI, Elsevier Ltd., 1 July 2020, www.ncbi.nlm.nih.gov/pmc/articles/PMC7328553/.

Figure References:

[1] “Exponential Growth of Covid-19.” DesignNews, Our World in Data, www.designnews.com/sitemap/articlepermonth/3/2020?page=2.

[2] Korstanje, Joos. “Modeling Logistic Growth.” Medium, Medium, 29 Mar. 2020, towardsdatascience.com/modeling-logistic-growth-1367dc971de2.

[3] “An explanation of the logistic growth model.” Lumen Boundless Biology, 2013, https://www.projectrhea.org/rhea/index.php/Logistic_Models.

[4] Wu, K., Darcet, D., Wang, Q. et al. “Mainland China Excluding Hubei.” Nonlinear Dyn, 2020, https://doi.org/10.1007/s11071-020-05862-6.