Title Stochastic models with applications to genetics, cancers, AIDS and other biomedical systems / Tan Wai-Yuan

Published River Edge, N.J. : World Scientific, ©2002

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Description 1 online resource (xv, 441 pages) : illustrations Series Series on concrete and applicable mathematics ; v. 4 Series on concrete and applicable mathematics ; v. 4. Contents 1. Introduction. 1.1. Some basic concepts of stochastic processes and examples. 1.2. Markovian and non-Markovian processes, Markov chains and examples. 1.3. Diffusion processes and examples. 1.4. State space models and hidden Markov models. 1.5. The scope of the book. 1.6. Complements and exercises -- 2. Discrete time Markov chain models in genetics and biomedical systems. 2.1. Examples from genetics and AIDS. 2.2. The transition probabilities and computation. 2.3. The structure and decomposition of Markov chains. 2.4. Classification of states and the dynamic behavior of Markov chains. 2.5. The absorption probabilities of transient states. 2.6. The moments of first absorption times. 2.7. Some illustrative examples. 2.8. Finite Markov chains. 2.9. Stochastic difference equation for Markov chains with discrete time. 2.10. Complements and exercises -- 3. Stationary distributions and MCMC in discrete time Markov chains. 3.1. Introduction. 3.2. The ergodic states and some limiting theorems. 3.3. Stationary distributions and some examples. 3.4. Applications of stationary distributions and some MCMC methods. 3.5. Some illustrative examples. 3.6. Estimation of linkage fraction by Gibbs sampling method. 3.7. Complements and exercises. 3.8. Appendix: A lemma for finite Markov chains -- 4. Continuous-time Markov chain models in genetics, cancers and AIDS. 4.1. Introduction. 4.2. The infinitesimal generators and an embedded Markov chain. 4.3. The transition probabilities and Kolmogorov equations. 4.4. Kolmogorov equations for finite Markov chains with continuous time. 4.5. Complements and exercises -- 5. Absorption probabilities and stationary distributions in continuous-time Markov chain models. 5.1. Absorption probabilities and moments of first absorption times of transient states. 5.2. The stationary distributions and examples. 5.3. Finite Markov chains and the HIV incubation distribution. 5.4. Stochastic differential equations for Markov chains with continuons time. 5.5. Complements and exercises -- 6. Diffusion models in genetics, cancer and AIDS. 6.1. The transition probabilities. 6.2. The Kolmogorov forward equation. 6.3. The Kolmogorov backward equation. 6.4. Diffusion approximation of models from genetics, cancers and AIDS. 6.5. Diffusion approximation of evolutionary processes. 6.6. Diffusion approximation of finite birth-death processes. 6.7. Complements and exercises -- 7. Asymptotic distributions, stationary distributions and absorption probabilities in diffusion models. 7.1. Some approximation procedures and asymptotic distributions in diffusion models. 7.2. Stationary distributions in diffusion processes. 7.3. The absorption probabilities and moments of first absorption times in diffusion processes. 7.4. Complements and exercises -- 8. State space models and some examples from cancer and AIDS. 8.1. Some HIV epidemic models as discrete-time linear state space models. 8.2. Some state space models with continuous-time stochastic system model. 8.3. Some state space models in carcinogenesis. 8.4. Some classical theories of discrete and linear state space models. 8.5. Estimation of HIV prevalence and AIDS cases in the San Francisco homosexual population. 8.6. Complements and exercises -- 9. Some general theories of state space models and applications. 9.1. Some classical theories of linear state space models with continuous-time stochastic system model. 9.2. The extended state space models with continuous-time stochastic system model. 9.3. Estimation of CD4(+) T cell counts and number of HIV in blood in HIV-infected individuals. 9.4. A general Bayesian procedure for estimating the unknown parameters and the state variables by state space models simultaneously. 9.5. Simultaneous estimation in the San Francisco population. 9.6. Simultaneous estimation in the cancer drug-resistant model. 9.7. Complements and exercises Summary This book presents a systematic treatment of Markov chains, diffusion processes and state space models, as well as alternative approaches to Markov chains through stochastic difference equations and stochastic differential equations. It illustrates how these processes and approaches are applied to many problems in genetics, carcinogenesis, AIDS epidemiology and other biomedical systems. One feature of the book is that it describes the basic MCMC (Markov chain and Monte Carlo) procedures and illustrates how to use the Gibbs sampling method and the multilevel Gibbs sampling method to solve many problems in genetics, carcinogenesis, AIDS and other biomedical systems. As another feature, the book develops many state space models for many genetic problems, carcinogenesis, AIDS epidemiology and HIV pathogenesis. It shows in detail how to use the multilevel Gibbs sampling method to estimate (or predict) simultaneously the state variables and the unknown parameters in cancer chemotherapy, carcinogenesis, AIDS epidemiology and HIV pathogenesis. As a matter of fact, this book is the first to develop many state space models for many genetic problems, carcinogenesis and other biomedical problems Bibliography Includes bibliographical references and index Notes Print version record Subject Medicine -- Mathematical models. Stochastic processes. Genetics -- Mathematical models. AIDS (Disease) -- Mathematical models Cancer -- Mathematical models AIDS (Disease) Genetics Neoplasms Acquired Immunodeficiency Syndrome Stochastic Processes MEDICAL -- Family & General Practice. MEDICAL -- Osteopathy. MEDICAL. MEDICAL -- Holistic Medicine. HEALTH & FITNESS -- Reference. MEDICAL -- Essays. MEDICAL -- Alternative Medicine. HEALTH & FITNESS -- Holism. AIDS (Disease) AIDS (Disease) -- Mathematical models Cancer -- Mathematical models Genetics -- Mathematical models Medicine -- Mathematical models Stochastic processes Bio-informatica. Stochastische modellen. Form Electronic book ISBN 9789812777966 9812777962

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