Analysis of a new stochastic Gompertz diffusion model for untreated human glioblastomas

Author:

Phan Tuan Anh1,Wang Shuxun2,Tian Jianjun Paul1ORCID

Affiliation:

1. Department of Mathematical Sciences, New Mexico State University, 1290 Frenger Mall, MSC 3MB/Science Hall 236, Las Cruces, New Mexico 88003-8001, USA

2. School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong, Shaanxi, 723000, P. R. China

Abstract

In this paper, we analyze a new Ito stochastic differential equation model for untreated human glioblastomas. The model was the best fit of the average growth and variance of 94 pairs of a data set. We show the existence and uniqueness of solutions in the positive spatial domain. When the model is restricted in the finite domain [Formula: see text], we show that the boundary point 0 is unattainable while the point [Formula: see text] is reflecting attainable. We prove there is a unique ergodic stationary distribution for any non-zero noise intensity, and obtain the explicit probability density function for the stationary distribution. By using Brownian bridge, we give a representation of the probability density function of the first passage time when the diffusion process defined by a solution passes the point [Formula: see text] firstly. We carry out numerical studies to illustrate our analysis. Our mathematical and numerical analysis confirms the soundness of our randomization of the deterministic model in that the stochastic model will set down to the deterministic model when the noise intensity approaches zero. We also give physical interpretation of our stochastic model and analysis.

Funder

National Foundation for Science and Technology Development

National Institutes of Health

Publisher

World Scientific Pub Co Pte Ltd

Subject

Modeling and Simulation

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