Probabilistic analysis of linear-quadratic logistic-type models with hybrid uncertainties via probability density functions

We provide a full stochastic description, via the first probability density function, of the solution of linear-quadratic logistic-type differential equation whose parameters involve both continuous and discrete random variables with arbitrary distributions. For the sake of generality, the initial c...

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Bibliographic Details
Published inAIMS mathematics Vol. 6; no. 5; pp. 4938 - 4957
Main Authors Burgos, Clara, Cortés, Juan Carlos, López-Navarro, Elena, Villanueva, Rafael Jacinto
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2021
Subjects
first probability density function
hybrid uncertainty
principle maximum entropy
random linear-quadratic logistic differential equation
random variable transformation method
uncertainty quantification
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Summary:We provide a full stochastic description, via the first probability density function, of the solution of linear-quadratic logistic-type differential equation whose parameters involve both continuous and discrete random variables with arbitrary distributions. For the sake of generality, the initial condition is assumed to be a random variable too. We use the Dirac delta function to unify the treatment of hybrid (discrete-continuous) uncertainty. Under general hypotheses, we also compute the density of time until a certain value (usually representing the population) of the linear-quadratic logistic model is reached. The theoretical results are illustrated by means of several examples, including an application to modelling the number of users of Spotify using real data. We apply the Principle Maximum Entropy to assign plausible distributions to model parameters.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2021290