Hyperbolastic type-III diffusion process: Obtaining from the generalized Weibull diffusion process

Antonio Barrera; Patricia Román-Roán; Francisco Torres-Ruiz; Antonio Barrera; Patricia Román-Roán; Francisco Torres-Ruiz

doi:10.3934/mbe.2020043

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 814-833. doi: 10.3934/mbe.2020043

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Hyperbolastic type-III diffusion process: Obtaining from the generalized Weibull diffusion process

1.
Departamento de Análisis Matemático, Estadística e Investigación Operativa y Matemática Aplicada, Facultad de Ciencias, Campus de Teatinos, Universidad de Málaga, Bulevar Louis Pasteur, 31, 29010, Málaga, Spain
2.
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Campus de Fuentenueva, Universidad de Granada, 18071, Granada, Spain
3.
Instituto de Matemáticas de la Universidad de Granada (IEMath-GR), Calle Ventanilla, 11, 18001, Granada, Spain

Received: 29 July 2019 Accepted: 11 October 2019 Published: 04 November 2019

The modeling of growth phenomena has become a matter of great interest in many different fields of application and research. New stochastic models have been developed, and others have been updated to this end. The present paper introduces a diffusion process whose main characteristic is that its mean function belongs to a wide family of curves derived from the classic Weibull curve. The main characteristics of the process are described and, as a particular case, a diffusion process is considered whose mean function is the hyperbolastic curve of type Ⅲ, which has proven useful in the study of cell growth phenomena. By studying its estimation we are able to describe the behavior of such growth patterns. This work considers the problem of the maximum likelihood estimation of the parameters of the process, including strategies to obtain initial solutions for the system of equations that must be solved. Some examples are provided based on simulated sample paths and real data to illustrate the development carried out.
- growth curves,
- generalized Weibull curve,
- Hyperbolastic models,
- inference in diffusion processes
Citation: Antonio Barrera, Patricia Román-Roán, Francisco Torres-Ruiz. Hyperbolastic type-III diffusion process: Obtaining from the generalized Weibull diffusion process[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 814-833. doi: 10.3934/mbe.2020043

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Abstract

The modeling of growth phenomena has become a matter of great interest in many different fields of application and research. New stochastic models have been developed, and others have been updated to this end. The present paper introduces a diffusion process whose main characteristic is that its mean function belongs to a wide family of curves derived from the classic Weibull curve. The main characteristics of the process are described and, as a particular case, a diffusion process is considered whose mean function is the hyperbolastic curve of type Ⅲ, which has proven useful in the study of cell growth phenomena. By studying its estimation we are able to describe the behavior of such growth patterns. This work considers the problem of the maximum likelihood estimation of the parameters of the process, including strategies to obtain initial solutions for the system of equations that must be solved. Some examples are provided based on simulated sample paths and real data to illustrate the development carried out.

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