SOME STOCHASTIC MODELS IN MATHEMATICAL BIOLOGY

Abstract

Abstract Many recent scientific works have addressed the need for a better understanding of the underlying theory of modeling in biology. However, much more attention has been paid to the area of deterministic modeling in biology than to stochastic modeling, although it is more realistic to consider biological processes as stochastic rather than deterministic. The goal in writing this thesis is to ~ introduce a contribution to fill the arising gap, ~ provide some of deterministic and stochastic biological models, and to 0 compare between such models. To achieve this, we present the thesis in the following chapters: Chapter One: Measure Theorv & Basic Modern Probability The underlying mathematical theory of stochastic modeling is stochastic processes, and the theory of stochastic processes is based on probability theory. The axiomatic development of probability theory was initiated by Kolmogorov in the early l930’s. Modem probability theory is technically a branch of measure theory, but it has developed characteristics and methods of its own, so any systematic exposition of the subject must begin with some basic measure-theoretic facts. The fundamental concept in this approach to probability theory is the probability space. In this Chapter, topics from measure theory and probability theory are reviewed which are particularly relevant to stochastic processes. Chapter Two: Stochastic Process In this chapter some basic concepts from the theory of stochastic processes are presented, in particular those are needed for an exploration of stochastic differential equations. ln the first section, after defining the stochastic process, the concept of Brownian motion is introduced as one ofthe most important examples of such process. The Markov process is a stochastic process exhibits specific property, called Markov dependence, is presented in the second section. Another related concepts such as: Random Walks, Martingales, Ito Integration and Ito Formula are introduced in the remain sections. Chapter Three: Stochastic Differential Equations One ofthe main problems in the stochastic modeling is how to solve the analog stochastic differential equation explicitly or at least numerically, so this chapter is devoted to many related concepts to the solutions of stochastic differential equations: the first part of this chapter is devoted to Analytical Solutions of SDEs, many related concepts are introduced such as: Interpretation of Stochastic Differential Equations, an Existence and Uniqueness Solutions, Strong and Weak Solutions, Martingale Problem,.... However, except in simple cases, it is generally not possible to obtain explicit solutions to SDEs, so the second part is Numerical Methods for Solving SDEs, to reach the last part, which is the main stone in the remain chapters, that is the Diffusion Processes and SDEs. Chapter Four: Deterministic Models in Biology Numerous d6I8tTl1lItlSIlC models from biology for single and two interacting species are presented, such as: Exponential Growth Model for Single population, Logistic Growth Model, Competition Model, Predator-Prey Model and Harvesting Problem. For these models, the behavior of deterministic model is discussed, and will be developed to the corresponding stochastic model, in the next chapter. Chapter Five: Stochastic Models in Biology Both deterministic and stochastic models have important roles to play and should therefore be considered together. We start with an introduction to show the importance of considering stochastic models, and then introduce a stochastic analogue to each one of the previous deterministic models. The explicit solutions of some few models are presented. For the majority, however, it is impossible to get such solutions. Ito SDEs for Interacting Populations offer much help to solve many population models numerically, therefore, we devote a section to discuss these equations, and then use them to solve many models in the remain sections. In biology we are often asked to infer the nature of population development from a single data set, yet different realizations of the same process can vary enormously. Since even stochastic solutions are only of limited help here, we shall construct simple computer simulation procedures which provide much needed insight into the underlying generating mechanisms. Indeed, such model-based simulations can highlight hitherto unforeseen features of a process and thereby suggest further profitable lines of biological investigation. All MAT HEMATICA-4.1 and MATLAB-6.2 programs used to generate the graphs are provided in the Appendix for easy referencing and developing to further studies. Chapter six: Conclusion and General Remarks Although full conditions for agreement between deterministic and expected stochastic solutions are at present unknown, it could be, however, compared them. In this chapter, a comparison between the Cl6l6I'I1'lll'llSliC & stochastic models is given to show the feature and the better use of each one.