Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation

In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data.This research has been funded by “Programa Operativo FEDER de Andalucía 2014-2020 A-FQM228-UGR18

Statistical Inference for Partially Observed Markov Processes via the R Package pomp

Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, iterated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory matching. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP models, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp.Comment: In press at the Journal of Statistical Software. A version of this paper is provided at the pomp package website: http://kingaa.github.io/pom

A Note on Estimation of Multi-Sigmoidal Gompertz Functions with Random Noise

The authors would like to thank the three anonymous reviewers for their suggestions that have improved the content of the paper.The behaviour of many dynamic real phenomena shows different phases, with each one following a sigmoidal type pattern. This requires studying sigmoidal curves with more than one inflection point. In this work, a diffusion process is introduced whose mean function is a curve of this type, concretely a transformation of the well-known Gompertz model after introducing in its expression a polynomial term. The maximum likelihood estimation of the parameters of the model is studied, and various criteria are provided for the selection of the degree of the polynomial when real situations are addressed. Finally, some simulated examples are presented.Ministerio de Economía, Industria y Competitividad, Spain, under Grant MTM2017-85568-P

Modeling Tumor Clonal Evolution for Drug Combinations Design

Cancer is a clonal evolutionary process. This presents challenges for effective therapeutic intervention, given the constant selective pressure toward drug resistance. Mathematical modeling from population genetics, evolutionary dynamics, and engineering perspectives are being increasingly employed to study tumor progression, intratumoral heterogeneity, drug resistance, and rational drug scheduling and combinations design. In this review we discuss the promising opportunities that these interdisciplinary approaches hold for advances in cancer biology and treatment. We propose that quantitative modeling perspectives can complement emerging experimental technologies to facilitate enhanced understanding of disease progression and improved capabilities for therapeutic drug regimen designs.David H. Koch Cancer Research Fund (Grant P30-CA14051)National Cancer Institute (U.S.). Integrative Cancer Biology Program (Grant U54-CA112967)National Institute of General Medical Sciences (U.S.). Interdepartmental Biotechnology Training Program (5T32GM008334

A Bayesian Model of COVID-19 Cases Based on the Gompertz Curve

The COVID-19 pandemic has highlighted the need for finding mathematical models to forecast the evolution of the contagious disease and evaluate the success of particular policies in reducing infections. In this work, we perform Bayesian inference for a non-homogeneous Poisson process with an intensity function based on the Gompertz curve. We discuss the prior distribution of the parameter and we generate samples from the posterior distribution by using Markov Chain Monte Carlo (MCMC) methods. Finally, we illustrate our method analyzing real data associated with COVID-19 in a specific region located at the south of Spain.Ministerio de Economía y Competitividad. Gobierno de España; 2014-2020 ERDF Operational Programme; Consejería de Economía, Conocimiento, Empresas y Universidad.Junta de Andalucí

Penalised maximum likelihood estimation for multi-state models

Multi-state models can be used to analyse processes where change of status over time is of interest. In medical research, processes are commonly defined by a set of living states and a dead state. Transition times between living states are often interval censored. In this case, models are usually formulated in a Markov processes framework. The likelihood function is then constructed using transition probabilities. Models are specified using proportional hazards for the effect of covariates on transition intensities. Time-dependency is usually defined by parametric models, which can represent a strong model assumption. Semiparametric hazards specification with splines is a more flexible method for modelling time-dependency in multi-state models. Penalised maximum likelihood is used to estimate these models. Selecting the optimal amount of smoothing is challenging as the problem involves multiple penalties. This thesis aims to develop methods to estimate multi-state models with splines for interval-censored data. We propose a penalised likelihood method to estimate multi-state models that allow for parametric and semiparametric hazards specifications. The estimation is based on a scoring algorithm, and a grid search method to estimate the smoothing parameters. This method is shown using an application to ageing research. Furthermore, we extend the proposed method by developing a computationally more efficient method to estimate multi-state models with splines. For this extension, the estimation is based on a scoring algorithm, and an automatic smoothing parameters selection. The extended method is illustrated with two data analyses and a simulation study

Stochastic Processes with Applications

Stochastic processes have wide relevance in mathematics both for theoretical aspects and for their numerous real-world applications in various domains. They represent a very active research field which is attracting the growing interest of scientists from a range of disciplines.This Special Issue aims to present a collection of current contributions concerning various topics related to stochastic processes and their applications. In particular, the focus here is on applications of stochastic processes as models of dynamic phenomena in research areas certain to be of interest, such as economics, statistical physics, queuing theory, biology, theoretical neurobiology, and reliability theory. Various contributions dealing with theoretical issues on stochastic processes are also included

Flexible multistate models for interval‐censored data: Specification, estimation, and an application to ageing research

Continuous‐time multistate survival models can be used to describe health‐related processes over time. In the presence of interval‐censored times for transitions between the living states, the likelihood is constructed using transition probabilities. Models can be specified using parametric or semiparametric shapes for the hazards. Semiparametric hazards can be fitted using P‐splines and penalised maximum likelihood estimation. This paper presents a method to estimate flexible multistate models that allow for parametric and semiparametric hazard specifications. The estimation is based on a scoring algorithm. The method is illustrated with data from the English Longitudinal Study of Ageing

Glosarium Matematika

273 p.; 24 cm