Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior

Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is addressed and a strategy is established for this purpose. Finally, a simulation study is carried out.This work was supported in part by the Ministerio de Economía, Industria y Competitividad, Spain, under Grant MTM2017-85568-P and by the Consejería de Economía y Conocimiento de la Junta de Andalucía, Spain under Grant A-FQM-456-UGR18

T-Growth Stochastic Model: Simulation and Inference via Metaheuristic Algorithms

The main objective of this work is to introduce a stochastic model associated with the one described by the T-growth curve, which is in turn a modification of the logistic curve. By conveniently reformulating the T curve, it may be obtained as a solution to a linear differential equation. This greatly simplifies the mathematical treatment of the model and allows a diffusion process to be defined, which is derived from the non-homogeneous lognormal diffusion process, whose mean function is a T curve. This allows the phenomenon under study to be viewed in a dynamic way. In these pages, the distribution of the process is obtained, as are its main characteristics. The maximum likelihood estimation procedure is carried out by optimization via metaheuristic algorithms. Thanks to an exhaustive study of the curve, a strategy is obtained to bound the parametric space, which is a requirement for the application of various swarm-based metaheuristic algorithms. A simulation study is presented to show the validity of the bounding procedure and an example based on real data is provided.Ministerio de Economía, Industria y Competitividad, Spain, under Grant MTM2017-85568-PFEDER/Junta de Andalucía-Consejería de Economía y Conocimiento, Spain, Grant A-FQM-456-UGR1

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

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 di usion 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.This work was supported in part by the Ministerio de Economía, Industria y Competitividad, Spain, under Grant MTM2017-85568-P