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

Clara Burgos; Juan Carlos Cortés; Elena López-Navarro; Rafael Jacinto Villanueva; Clara Burgos; Juan Carlos Cortés; Elena López-Navarro; Rafael Jacinto Villanueva

doi:10.3934/math.2021290

AIMS Mathematics

2021, Volume 6, Issue 5: 4938-4957. doi: 10.3934/math.2021290

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

Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, València, Spain

Received: 04 December 2020 Accepted: 24 February 2021 Published: 02 March 2021
MSC : 34A38, 37H10, 34F05, 60H10

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.
Citation: Clara Burgos, Juan Carlos Cortés, Elena López-Navarro, Rafael Jacinto Villanueva. Probabilistic analysis of linear-quadratic logistic-type models with hybrid uncertainties via probability density functions[J]. AIMS Mathematics, 2021, 6(5): 4938-4957. doi: 10.3934/math.2021290

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Abstract

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.

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