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

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

doi:10.3934/math.2021290

Clara Burgos, Elena López-Navarro + Show 2 more

Open Access

https://doi.org/10.3934/math.2021290

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Journal: AIMS Mathematics	Publication Date: Jan 1, 2021
Citations: 1	License type: cc-by

Affiliation: Universitat Politècnica de València

Abstract

<abstract><p>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.</p></abstract>

Highlights

Interaction between discrete and continuous dynamics gives rise to hybrid models
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, in our analysis we have assumed that all model parameters and the initial condition are random variables having arbitrary distributions

Summary

Introduction

Interaction between discrete and continuous dynamics gives rise to hybrid models. This type of mathematical models mainly appear when describing technological systems, where the continuous dynamics comes from the physical process, while the discrete dynamics appears because of the technological elements of the system [1]. We tackle this interesting scenario by studying a full randomization of an important dynamic model, namely, the linear-quadratic logistic-type differential equation As it will be seen later, for the sake of generality, we will assume that model parameters (the coefficients and the initial condition) involve both discrete and continuous random variables. To the best of our knowledge, in the contributions where the RVT technique has been applied to determine the 1-PDF of the solution stochastic process of ordinary/partial RDEs, all the inputs parameters are assumed to be absolutely continuous random variables. To this end, we use an inverse probabilistic technique, based on the Principle of Maximum Entropy, to assign appropriate probability distributions to model parameters. We point out that in order to facilitate the reading of the paper, a list of all mathematical symbols and abbreviations appearing throughout the paper has been added at the end of the document

Obtaining a PDF of a discrete random variable

Computing the PDF of time until a given population size is reached

Modelling Spotify users with uncertainty in the data

Conclusions

Findings

Conflict of interest