## 2 Empirical Algorithm  
In this section, we present the empirical algorithm that is employed for our estimations. We depart from the approach followed by many studies on the epidemiology of COVID-19, which using standard epidemiological analysis, assume an exponential growth in the number of diseases, based on the idea of a fixed reproduction rate. Our approach relies on kinematic motion models [18]. As in [19], we assume the existence of a theoretical function that represents the epidemic spread over time in a country or region, as follows:  
\[y=f(x) \tag{1}\]  
In Equation (1), \(y\) represents the cumulative number of cases since the pandemic start, and \(x\) is the number of days since the pandemic starting date. The function employed for our research is, essentially, a third-degree polynomial function (cubic function). This choice is empirical, since it is not based on any theoretical model of epidemic development or on any previous hypothesis, but exclusively on the observed capacity to fit satisfactorily, with longer or shorter sections, the epidemic curve that represents the cumulative number of cases as a function of time (1). The adaptability of the cubic function to epidemic curves has been studied in several works, which are discussed below. This approach to epidemic propagation is an analogy to the kinematic description of the rectilinear motion of an object with a variable acceleration, since the equations used are the same:  
Our kinematics-epidemiology analogy is described by the following equations:Space (in meters) \(\equiv\) cumulative number of cases:  
\[y=ax^{3}+bx^{2}+cx+d \tag{2}\]
* Velocity (meters/second) \(\equiv\) daily cases: \[\frac{dy}{dx}=v=3 ax^{2}+2 bx+c\] (3)
* Acceleration (meters/second\({}^{2}\)) \(\equiv\) cases/day:\({}^{2}\) \[\frac{dy}{dx^{2}}=w=6 ax+2b\] (4)  
This analogy imagines a pandemic as a truck with an engine that can reach any speed; the brakes do not work, and the truck is accelerating more and more. The truck will only stop when it runs out of fuel or when there are enough obstacles in its way to stop it. This is also a good description of what happens in a pandemic. If there are no NPIs to stop it, the pandemic will continue to spread until there are no more people susceptible to becoming ill, which is what happens when the truck runs out of gas. The NPIs must make the acceleration zero and then negative (deceleration). The longer it takes to brake the truck, the greater the obstacle must be to stop it, which is equal to the stiffness of the NPIs to be applied when implementation is delayed. Without NPIs, as [PERSON] suggested more than a century ago (20), the daily cases epidemic curve will assume the form of a normal distribution and can be fit quite precisely to a section of the cubic function, except for its asymptotic start and end. As an example of the facility of the cubic function to fit the pandemic curve (y), Figure 1 shows the fit with data from Spain between July 12 (x = 140) and September 19 (\(x=209\)). The seventy values were adjusted to a cubic equation with \(R^{2}=0.9997\), median relative error = 0.13%, and interquartile range (IQR) = 0.92%. The analytical curve in this figure is not a predictor, since it was obtained when all the information was already available on the corresponding dates, but shows a great facility of the cubic function to reproduce the cumulative number of cases on pandemic curve. Figure 1: Second pandemic wave in Spain: fit by linear regression of the observed cumulative number of cases to a third-degree polynomial, \(y=0.08307x^{3}+42.281x^{2}-16,584\)\(x+1,524,253\) (red curve), from 12 July (\(x=140\)) to 19 September (\(x=210\)). The analogy with kinematics equations and the ability to fit the epidemic curve suggest their use for forecasting. Equation (2) describes a rectilinear motion with a variable acceleration that increases or decreases linearly, since the acceleration equation is a line whose slope is \(6a\).