Efficient Function Integration and a Case Study with Gompertz Functions for Covid-19 Waves

Abstract

Numerical algorithms are widely used in different applications, therefore, the execution time of the functions involved in numerical algorithms is important, and, in some cases, decisive, for example, in machine learning algorithms. Given a finite set of independent functions A(x), B(x), …, Z(x) with domains defined by disjoint, consecutive, and not necessarily adjacent intervals, the main goal is to integrate into a single function F(x) = k1×A(x) + k2×B(x) + … + kn×Z(x), where each activation coefficient k, is one if x is in the interval of the respective domain and zero otherwise. The novelty of this work is the presentation and formal demonstration of two general forms of integration of functions in a single function: The first is the mathematical version and the second is the computational version (with the AND function at the bit level), which is characterized by its efficiency. The result is applied in a case study (Peru), where two regression functions were obtained that integrate all the waves of Covid-19, that is, the epidemic curve of the variable global number of deaths/infected per day, the adjustment provided a highly statistically significant measure of correlation, a Pearson's product-moment correlation of 0.96 and 0.98 respectively. Finally, the size of the epidemic was projected for the next 30 days