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An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations.

ABSTRACT: We investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We formulate the inverse problem as a least squares problem for the probability measure estimation. Using the Prohorov metric framework, we prove existence and consistency of the least squares estimates and outline a discretization scheme for approximating a conditional probability measure. For this scheme, we prove general method stability. The work is motivated by Partial Differential Equation (PDE) models of flocculation for which the shape of the post-fragmentation conditional probability measure greatly impacts the solution dynamics. To illustrate our methodology, we apply the theory to a particular PDE model that arises in the study of population dynamics for flocculating bacterial aggregates in suspension, and provide numerical evidence for the utility of the approach.

SUBMITTER: Mirzaev I

PROVIDER: S-EPMC5352987 | biostudies-literature | 2016

REPOSITORIES: biostudies-literature

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An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations.

Mirzaev Inom I Byrne Erin C EC Bortz David M DM

Inverse problems 20160715 9

We investigate the inverse problem of identifying a conditional probability measure in measure-dependent evolution equations arising in size-structured population modeling. We formulate the inverse problem as a least squares problem for the probability measure estimation. Using the Prohorov metric framework, we prove existence and consistency of the least squares estimates and outline a discretization scheme for approximating a conditional probability measure. For this scheme, we prove general m ...[more]

PMID: 28316360

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Project description:The epidemic increase in the incidence of Human Papilloma Virus (HPV) related Oropharyngeal Squamous Cell Carcinomas (OPSCCs) in several countries worldwide represents a significant public health concern. Although gender neutral HPV vaccination programmes are expected to cause a reduction in the incidence rates of OPSCCs, these effects will not be evident in the foreseeable future. Secondary prevention strategies are currently not feasible due to an incomplete understanding of the natural history of oral HPV infections in OPSCCs. The key parameters that govern natural history models remain largely ill-defined for HPV related OPSCCs and cannot be easily inferred from experimental data. Mathematical models have been used to estimate some of these ill-defined parameters in cervical cancer, another HPV related cancer leading to successful implementation of cancer prevention strategies. We outline a "double-Bayesian" mathematical modelling approach, whereby, a Bayesian machine learning model first estimates the probability of an individual having an oral HPV infection, given OPSCC and other covariate information. The model is then inverted using Bayes' theorem to reverse the probability relationship. We use data from the Surveillance, Epidemiology, and End Results (SEER) cancer registry, SEER Head and Neck with HPV Database and the National Health and Nutrition Examination Surveys (NHANES), representing the adult population in the United States to derive our model. The model contains 8,106 OPSCC patients of which 73.0% had an oral HPV infection. When stratified by age, sex, marital status and race/ethnicity, the model estimated a higher conditional probability for developing OPSCCs given an oral HPV infection in non-Hispanic White males and females compared to other races/ethnicities. The proposed Bayesian model represents a proof-of-concept of a natural history model of HPV driven OPSCCs and outlines a strategy for estimating the conditional probability of an individual's risk of developing OPSCC following an oral HPV infection.

Dataset Information

An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations.

Publications

An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations.

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