Stochastic Models with Applications to Genetics, Cancers, AIDS and Other Biomedical Systems

The following sections are included:

• Dedication
• Preface
• Contents

In the studies of natural systems, the main problem is usually to derive mathematical models for the response or responses in terms of input and/or risk variables as well as time. Almost all of these models are stochastic models because most of the risk variables are subject to random variations and most of the measurements of the responses and input variables are subject to random measurement errors. This is true for medical systems such as AIDS, cancer and genetics as well as for biological sciences, engineering sciences and social and economic systems. To study these systems, therefore, it is important to examine the probability laws governing the behavior of these systems. The purpose of this book is to present a systematic treatment on stochastic models which have been used in genetics, cancer, AIDS and some other medical systems…

In many stochastic systems in natural sciences including genetics as well as in biomedical problems, one may treat both the state space and the parameter space as discrete spaces. In these cases, when the process is Markov, one is then entertaining Markov chains with discrete time. This is especially true in mating types and gene frequencies in natural populations with generation as time unit, see Examples 1.1, 1.2 and 1.11; it is also true in many branching processes involving new mutants and in AIDS epidemics; see Examples 1.9, 1.10 and 1.12. In the past 15 to 20 years, some general theories of Markov chains with discrete time have also been invoked to develop computer algorithms to solve many complicated computational problems in natural sciences. This has been referred to as the MCMC (Markov Chain Monte Carlo) method and has become a very popular method. By using examples from many genetic systems, in this chapter we will develop some general results of discrete time Markov chains and illustrate its applications. In the next chapter, we will develop theories for stationary distributions and illustrate the applications of some MCMC methods.

The following sections are included:

• Introduction
• The Ergodic States and Some Limiting Theorems
• Stationary Distributions and Some Examples
• Applications of Stationary Distributions and Some MCMC Methods
  • The Gibbs sampling method
  • The weighted bootstrap method for generating random samples
  • The Metropolis-Hastings algorithm
• Some Illustrative Examples
• Estimation of Linkage Fraction by Gibbs Sampling Method
• Complements and Exercises
• Appendix: A Lemma for Finite Markov Chains
• References

In the previous two chapters, we have discussed a class of stochastic models which are Markov chains with discrete time. We have developed some basic theories and have demonstrated some applications of this class of stochastic models to some genetic problems and to some biomedical problems. In this chapter, we will extend these results into Markov chains with continuous time. As it turns out, this class of stochastic models is very rich and has a wide range of applications to many biomedical problems as well as ecological systems. It includes stochastic birth-death processes, many genetic processes, filtered Poisson processes, HIV epidemiology models as well as many cancer models.

In Chap. 4, we have discussed some general results of Markov chains with continuous time. For the applications of these chains, we present in this chapter some results on absorption probabilities, first absorption times, stationary distributions as well as some other topics of importance. We will illustrate these results by examples from genetics, cancer and AIDS.

In many biomedical systems, the population size is usually very large so that the state variables are closely approximated by continuous variables. If time is also continuous, then one is entertaining a stochastic process with continuous state space and continuous parameter space. As we shall see, many problems in cancer, genetics and AIDS can be considered as stochastic processes which have both continuous state space and continuous parameter space. This is especially true for evolution processes describing changes of frequencies of certain types or genes in large populations. The latter is justified by the observation that evolution is a slow process which takes place over millions of years and that the population size is usually very large. In this chapter we will thus consider a class of stochastic processes with continuous state space and continuous parameter space involving only the first two moments of the changes. In particular we will consider Markov processes with continuous state space and continuous parameter space which have been referred to as diffusion processes. As we shall see, many processes in biomedical systems can be approximated by diffusion processes; this includes stochastic processes describing population growth, stochastic processes of carcinogenesis, some stochastic process in infectious diseases as well as evolutionary processes involving changes of gene frequencies in natural populations.

In Chap. 6, we have shown that many processes in genetics and biomedical problems can be closely approximated by diffusion processes. Although it is possible to solve the Kolmogorov forward or backward equations to derive the conditional pdf f(x, y; t) in some cases, in most of the cases, the solution is very difficult, if not impossible. If the eigenvalues and eigenfunctions of the equation exist, however, in many cases one may derive asymptotic distributions by approximating these eigenvalues and eigenfunctions. In this chapter, we will thus illustrate how to approximate the eigenvalues and the eigenfunctions whenever exist…

As defined in Definition 1.4, state space models (Kalman filter models) of stochastic systems are stochastic models consisting of two sub-models: The stochastic system model which is the stochastic model of the system and the observation model which is a statistical model based on some data from the system. That is, the state space model adds one more dimension to the stochastic model and to the statistical model by combining both of these models into one model. It takes into account the basic mechanisms of the system and the random variation of the system through its stochastic system model and incorporate all these into the observed data from the system; furthermore, it validates and upgrades the stochastic model through its observation model and the observed data of the system. It is advantageous over both the stochastic model and the statistical model when used alone since it combines information and advantages from both of these models. Given below is a brief summary of the advantages over the statistical model or stochastic model used alone…

In Chap. 8, I have illustrated how to develop state space models in some cancer and AIDS models. As a continuation, in this chapter I proceed to give some general theories and illustrate its application.

In previous chapters, we have developed basic theories of some important stochastic models. To further illustrate applications of these models in medical research, in this chapter we will develop some important biologically supported stochastic models of carcinogenesis and illustrate its applications.

The following sections is included:

• Subject Index