Joint 2012 MBI-NIMBioS-CAMBAM Summer Graduate Workshop
on Stochastics Applied to Biological Systems
(June 18-29, 2012)
Abstract: PDF
Project: Description, Project, Exercises
Reading Material:
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Abstract: PDF
Project: PDF
Reading Material: PDF
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Abstract and Project: PDF
Notes:
- GILL_LV_timetrack.m
- NRM_LV_timetrack.m
- CFD_main.m
- CFD_path.m
- CMC_main.m
- CMC_path.m
- CRN_main.m
- CRN_path.m
- Likelihood_main.m
- Likelihood_path.m
Reading Material:
- David F. Anderson and Thomas G. Kurtz. Continuous time Markov chain models for chemical reaction networks.
- David F. Anderson. An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time Markov Chains.
- Sergey Plyasunov and Adam P. Arkin. Efficient stochastic sensitivity analysis of discrete event systems.
- Muruhan Rathinam, Patrick W. Sheppard, and Mustafa Khammash, Efficient computation of parameter sensitivities of discrete stochastic chemical reaction networks.
Anderson's Group Presentation: Parameter Sensitivities and Continuous Markov Chain
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This set of lectures and discussions will provide a quick one-day conceptual overview of stochastic issues in biology. Time permitting, I will point out the major conceptual approaches to stochasticity as typically applied in biology (random walks, Markov chains, birth and death processes, branching processes, agent-based models, stochastic DEs, diffusion processes, statistical modeling, Bayesian methods) and make the connection between these and deterministic analogs.
The learning objectives for this day are:
- Assist attendees in developing some intuition concerning how to think about biology from the perspective of probability distributions;
- Encourage attendees to realize that there are diverse methods and models that can be applied across many fields of biology that have similar mathematical underpinnings, and these may be related to simpler deterministic models; and
- Provide attendees with some hands-on experience with analysis of a stochastic process using simple computer tools.
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Presentation: PDF
Abstract: We will work through some of the basic ideas involved in modeling various types of interactions in spatial population biology using interacting particle systems (sometimes referred to as stochastic cellular automata). Some of the essential ingredients and behaviors come from simple models like the contact process and the voter model. These components can be combined and tweaked to obtain models with more biological detail, including epidemic behavior for host-pathogen systems, the spread of antibiotic resistance genes, etc. These models can be informative since real biological populations exhibit a high degree of spatial structure and this structure affects the interactions between individuals and species in ways that can dramatically alter dynamics compared to well-mixed systems. The computer exercises will allow students to alter some existing MATLAB code to simulate various processes. A preview of these models can be found in the WinSSS software that can be downloaded from Steve Krone's webpage.
Reading Material: In the Spatial Models paper, the part on "particle flip dynamics" is the most relevant for this workshop. There is also Durrett's out-of-print book "Lecture Notes on Particle Systems and Percolation." Other papers of mine, as well as the WinSSS software, are available on my webpage: http://www.webpages.uidaho.edu/~krone/
- Spatial Models: Stochastic and Deterministic. S.M. Krone.
- Spatial invasion by a mutant pathogen. W. Wei and S.M. Krone.
Possible group project: Consider the effects of spatial heterogeneity on spatial population dynamics. For example, a basic SIR-type epidemic model using interacting particle systems would have a single host species and one would track whether the hosts are uninfected, infected, or dead. What happens when host quality (from the point of view of the pathogen) can vary? In practice, these differences in host quality can be related to host age or host species. They can be considered to be static as the pathogen spreads, or they can be changing dynamically.
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Abstract: As a warming up, we will start with a brief overview of the main results about the voter model: clustering versus coexistence, cluster size and occupation time. The voter model is an example of interacting particle system - individual-based model - that models social influence, the tendency of individuals to become more similar when they interact. Each vertex of the lattice is characterized by one of two possible competing opinions and updates its state at rate one by mimicking one of its neighbors chosen uniformly at random. We will conclude with recent results about the one-dimensional Axelrod model which, like the voter model includes social influence, but unlike the voter model also accounts for homophily, the tendency of individuals to interact more frequently with individuals who are more similar. In the Axelrod model, each vertex of the lattice is now characterized by a culture, a vector of F cultural features that can each assumes q different states. Pairs of neighbors interact at a rate proportional to the number of cultural features they have in common, which results in the interacting pair having one more cultural feature in common.
Reading Material:
- The Dissemination of Culture: A Model with Local Convergence and Global Polarization
- The Axelrod Model for the Dissemination of Culture Revisited
- Consensus in the two-state Axelrod model
- Fixation in the one-dimensional Axelrod model
Project: Understanding the Voter Model
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Abstract: PDF
Project: PDF
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Lectures & Abstract: PDF
Project: PDF
Presentation materials: MolecularEvolutionLab.zip
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Overview: PDF
Project: PDF
R-files: Schreiber-1A.R, Schreiber-1B.R, Schreiber-2.R
