Math 605: Stochastic Methods for Biology

Stochastic (that is, probabilistic) models have a long history in Biology.  However, their use has greatly increased in recent years as advances in experimental methods in biology, such as Green Fluorescent protein and laser traps, have enabled quantitative measurements at the single cell, and even single molecule, level.  Such experiments show time and time again that the dynamics at this level are intrinsically stochastic, or "noisy."  The (mathematical) implication of this observation is that the standard deterministic models for dynamics (for example, ODE models) need to sometimes be replaced with analogous stochastic models.   
 
In addition to cellular processes, population, epidemic, and birth-death processes (among others) can all be modeled stochastically.  You can also use stochastic models to better understand evolution via models for genetic mutation.
 
This course will focus on  stochastic models of biological phenomena.  We will cover the requisite probability to understand them (although, see the prerequisites below), and will learn different methods of analysis: from the theoretical to the computational.    By the end of this course you will understand how stochastic models arise naturally in biology, and also be able to analyze and apply computational techniques (such as the well known Gillespie or stochastic simulation algorithm) to the models.  
 
Mathematically, we will cover discrete time Markov chains, the basics of point processes, continuous time Markov chains, and diffusion processes (those incorporating Brownian motion).  All topics will be covered with an eye towards understanding the models both analytically and computationally.  Further topics will be covered based on student interest.
 
Intended audience: Advanced undergraduate students and/or graduate students  in

1. mathematics, physics, computer science, engineering, and related disciplines with an interest in biology, and
2. students in biology, biochemistry, and related disciplines, with an interest in quantitative approaches to biology.

 
Prerequisites: It is important that the student has taken Calculus and an introductory probability course (at the level of Math 331/431 or Stat 309/311). Motivated students not meeting all the formal requirements are still welcome, however, as I will quickly review the needed probability at the beginning of the course, and then throughout the course as needed.    A basic knowledge of ordinary differential equations and linear algebra is also important.  Having some experience with Matlab would also be useful, though not strictly necessary as this can be learned during the course.