On the Construction of Some Fractional Stochastic Gompertz Models

Abstract

The aim of this paper is the construction of stochastic versions for some fractional Gompertz curves. To do this, we first study a class of linear fractional-integral stochastic equations, proving existence and uniqueness of a Gaussian solution. Such kinds of equations are then used to construct fractional stochastic Gompertz models. Finally, a new fractional Gompertz model, based on the previous two, is introduced and a stochastic version of it is provided.

A generalized Gompertz growth model with applications and related birth-death processes

“…Hence, by noting that equations (28) and (4) have the same form, we get that the mean of the process X(t) is equal to the growth curve proposed in (11), under the assumptions (10) and (29). Some properties of E y (t) and V y (t) are provided in Table 4 of [12].…”

“…Other studies including Gompertz and logistic growth models based on stochastic diffusions can be found in Campillo et al [8], Himadri Ghosh and Prajneshu [21], and Yoshioka et al [36]. Recent advances involving fractional Gompertz growth models in biological contexts have been analyzed in Ascione and Pirozzi [4], Dewanji et al [11], Frunzo et al [14], and in Meoli et al [24]. Nevertheless, our analysis will be restricted to the case of birth-death processes.…”

A generalized Gompertz growth model with applications and related birth-death processes

“…Hence, by noting that equations (29) and (4) have the same form, we get that the mean of the process X (t) is equal to the growth curve proposed in (12), under the assumptions (11) and (30). Some properties of E y (t) and V y (t) are provided in Table 4 of [12].…”

“…Other studies including Gompertz and logistic growth models based on stochastic diffusions can be found in Campillo et al [ 8 ], Himadri Ghosh and Prajneshu [ 22 ], and Yoshioka et al [ 38 ]. Recent advances involving fractional Gompertz growth models in biological contexts have been analyzed in Ascione and Pirozzi [ 4 ], Dewanji et al [ 11 ], Frunzo et al [ 15 ], and in Meoli et al [ 25 ]. Nevertheless, our analysis will be restricted to the case of birth-death processes.…”

Optimal minimum variance‐entropy control of tumour growth processes based on the Fokker–Planck equation

“…One model that can well illustrate tumour growth behaviour is a stochastic Gompertz equation. The stochastic Gompertz model describes the evaluation of morphological variables in low-growth processes [14][15][16]. Many studies that have been done tried to reduce the growth rate of a tumour with a single treatment [5][6][7][8]17].…”