Two Multi-Sigmoidal Diffusion Models for the Study of the Evolution of the COVID-19 Pandemic
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AuthorBarrera, Antonio; Román Román, Patricia; Serrano Pérez, Juan José; Torres Ruiz, Francisco De Asís
COVID-19Diffusion processesMulti-sigmoidal curvesInference on diffusion processesFirst-passage times
Barrera, A.; Román-Román, P.; Serrano-Pérez, J.J.; Torres-Ruiz, F. Two Multi-Sigmoidal Diffusion Models for the Study of the Evolution of the COVID-19 Pandemic. Mathematics 2021, 9, 2409. https://doi.org/ 10.3390/math9192409
SponsorshipMinisterio de Economía, Industria y Competitividad, Spain, under Grant MTM2017-85568-P; FEDER, Consejería de Economía y Conocimiento de la Junta de Andalucía, Spain under Grant A-FQM-456-UGR18
A proposal is made to employ stochastic models, based on diffusion processes, to represent the evolution of the SARS-CoV-2 virus pandemic. Specifically, two diffusion processes are proposed whose mean functions obey multi-sigmoidal Gompertz and Weibull-type patterns. Both are constructed by introducing polynomial functions in the ordinary differential equations that originate the classical Gompertz and Weibull curves. The estimation of the parameters is approached by maximum likelihood. Various associated problems are analyzed, such as the determination of initial solutions for the necessary numerical methods in practical cases, as well as Bayesian methods to determine the degree of the polynomial. Additionally, strategies are suggested to determine the best model to fit specific data. A practical case is developed from data originating from several Spanish regions during the first two waves of the COVID-19 pandemic. The determination of the inflection time instants, which correspond to the peaks of infection and deaths, is given special attention. To deal with this particular issue, point estimation as well as first-passage times have been considered.