Mathematical model of Boltzmann’s sigmoidal equation applicable to the spreading of the coronavirus (Covid-19) waves

• Published in: Environmental science and pollution research international 2021-08, Vol.28 (30), p.40400-40408
• Main Authors: El Aferni, Ahmed, Guettari, Moez, Tajouri, Tahar
• Format: Article
• Language: English
• Subjects: Aquatic Pollution; Atmospheric Protection/Air Quality Control/Air Pollution; Boltzmann transport equation; Coronaviruses; COVID-19; Disease transmission; Earth and Environmental Science; Ecotoxicology; Environment; Environmental Chemistry; Environmental Factors and the Epidemics of COVID-19; Environmental Health; Environmental science; Epidemiology; Forecasting; Humans; Mathematical models; Models, Theoretical; Pandemics; Parameter modification; SARS-CoV-2; Time constant; Viruses; Waste Water Technology; Water Management; Water Pollution Control
• ISSN: 0944-1344; 1614-7499
• DOI: 10.1007/s11356-020-11188-y

Currently, investigations are intensively conducted on modeling, forecasting, and studying the dynamic spread of coronavirus (Covid-19) new pandemic. In the present work, the sigmoidal-Boltzmann mathematical model was applied to study the Covid-19 spread in 15 different countries. The cumulative number of infected persons I has been accurately fitted by the sigmoidal-Boltzmann equation (SBE), giving rise to different epidemiological parameters such as the pandemic peak t p , the maximum number of infected persons I max , and the time of the epidemic stabilization t ∞ . The time constant relative to the sigmoid Δ t (called also the slope factor) was revealed to be the determining parameter which influences all the epidemiological parameters. Empirical laws between the different parameters allowed us to propose a modified sigmoidal-Boltzmann equation describing the spread of the pandemic. The expression of the spread speed V p was further determined as a function of the sigmoid parameters. This made it possible to assess the maximum speed of spread of the virus V p max and to trace the speed profile in each country. In addition, for countries undergoing a second pandemic wave, the cumulative number of infected people I has been successfully adjusted by a double sigmoidal-Boltzmann equation (DSBE) allowing the comparison between the two waves. Finally, the comparison between the maximum virus spread of two waves V p  max 1 and V p  max 2 showed that the intensity of the second wave of Covid-19 is low compared to the first for all the countries studied.