Joint 2012 MBI-NIMBioS-CAMBAM Summer Graduate Workshop: Stochastics Applied to Biological Systems

(June 18,2012 - June 29,2012 )

Organizers


Linda Allen
Department of Mathematics and Statistics, Texas Tech University
Laura Kubatko
Statistics/EEOB, The Ohio State University
Suzanne Lenhart
Mathematics, University of Tennessee
Libby Marschall
Evolution, Ecology and Organismal Biology, The Ohio State University
Lea Popovic
Dept of Mathematics and Statistics, Concordia University

The Workshop will be held at the Mathematical Biosciences Institute and will have instructors from across North America whose research expertise is stochastic modeling in biological systems. Some of the topics to be covered include Markov chains, birth and death processes, branching processes, Brownian motion and diffusion processes, stochastic differential equations, and agent-based models. Applications of stochastic processes will come from epidemiology, ecology, phylogenetics, microbiology, evolutionary biology, and genetics. The workshop will consist of lectures on mathematical and statistical methods for stochastic processes in biological systems and daily computer and analysis activities. In addition, each student will work on a research project over the duration of the program with a team of four or five participants. Applications received by January 13, 2012 will receive full consideration. Members of the organizing committee are: Linda Allen (Texas Tech), Laura Kubatko (Ohio State University), Suzanne Lenhart (University of Tennessee, Knoxville); Libby Marschall (Ohio State University), and Lea Popovic (Concordia University).

Accepted Speakers

Edward Allen
Mathematics and Statistics, Texas Tech University
David Anderson
JJ Chai
NIMBioS, University of Tennessee
Steve Krone
Mathematics, University of Idaho
Nicolas Lanchier
Mathematics and Statistics, Arizona State University
Scott McKinley
Mathematics, University of Florida
Michael Reed
Mathematics, Duke University
Gregory Roth
Evolution and Ecology, University of California, Davis
Pavitra Roychoudhury
Mathematics, University of Idaho
Sebastian Schreiber
Center of Population Biology, University of California, Davis
Monday, June 18, 2012
Time Session
12:00 AM
11:00 AM
- An introduction to thinking like a probabilist about biology
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.
Tuesday, June 19, 2012
Time Session
09:00 AM
12:00 PM
Linda Allen - An Introduction to Stochastic Epidemic Models
A brief introduction is presented to modeling in stochastic epidemiology. Several
useful epidemiological concepts such as the basic reproduction number and the nal size
of an epidemic are de ned. Three well-known stochastic modeling formulations are in-
troduced: discrete-time Markov chains, continuous-time Markov chains, and stochastic
di erential equations. Methods for derivation, analysis and numerical simulation of the
three types of stochastic epidemic models are presented. Emphasis is placed on some of
the di erences between the three stochastic modeling formulations as illustrated in the
classic SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered)
epidemic models. In addition, some of the unique properties of stochastic epidemic
models, such as the probability of an outbreak, nal size distribution, critical commu-
nity size, and expected duration of an epidemic are demonstrated in various models of
diseases impacting humans and wildlife.
02:00 PM
05:30 PM
Edward Allen - A Practical Introduction to Stochastic Differential Equations in Mathematical Biology
Properties of the Wiener process are reviewed and stochastic integration is explained. Stochastic di´Č€erential equations are introduced and some of their properties are described. Equivalence of SDE systems is explained. Commonly used numerical procedures are discussed for computationally solving systems of stochastic di´Č€erential equations. A procedure is described for deriving It╦†o stochastic di´Č€erential equations from associated discrete stochastic models for randomly-varying problems in biology. The SDEs are derived from basic principles, i.e., from the changes in the system which
occur in a small time interval. Several examples illustrate the procedure. In particular, stochastic di´Č€erential equations are derived for predator-prey, competition, and epidemic problems.
Wednesday, June 20, 2012
Time Session
12:00 AM
11:00 AM
Edward Allen - A Practical Introduction to Stochastic Differential Equations in Mathematical Biology
Properties of the Wiener process are reviewed and stochastic integration is explained. Stochastic di´Č€erential equations are introduced and some of their properties are described. Equivalence of SDE systems is explained. Commonly used numerical procedures are discussed for computationally solving systems of stochastic di´Č€erential equations. A procedure is described for deriving It╦†o stochastic di´Č€erential equations from associated discrete stochastic models for randomly-varying problems in biology. The SDEs are derived from basic principles, i.e., from the changes in the system which
occur in a small time interval. Several examples illustrate the procedure. In particular, stochastic di´Č€erential equations are derived for predator-prey, competition, and epidemic problems.
Thursday, June 21, 2012
Time Session
12:00 AM
11:00 AM
Scott McKinley - Anomalous diffusion in biological fluids
Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargo-laden
vesicles in axons undergo stark periods of forward and backward motion, inter-
rupted by sudden pauses and periods of free di usion. Over large periods of time,
the motion is e ectively that of a particle with steady drift accompanied a di u-
sive spread greater than what can be explained by thermal
uctuations alone. As
another example, E. coli and other bacteria are known to respond to the local con-
centration of nutrients in such a way that they can climb gradients toward optimal
locations. Again, the e ective behavior is drift toward a desired" location, with
enhanced di usivity.
In other cases, biological entities are signi cantly slowed. Relatively large parti-
cles di using in
uids such as mucus, blood, bio lms or the cytoplasm of cells all
experience hinderances due to interactions with the polymer networks that consti-
tute small-scale biological environments. Researches repeatedly observe sublinear
growth of the mean-squared displacement of particle paths. This signals to theo-
reticians that the particles are not experiencing traditional Brownian motion. In-
terestingly, many viruses are actually small enough to avoid this type of hinderance
when moving through human mucus. However, the body's immune response in-
cludes teams of still smaller antibodies that can immobilize virions by serving as an
intermediary creating binding events between virions and the local mucin network.
Underlying the mathematical description of all these phenomena is a modeling
framework that employs stochastic di erential equations, hybrid switching di u-
sions and stochastic integro-di erential equations. We will begin with the Langevin
model for di usion. This is the physicist's view of Brownian motion, derived from
Newton's Second Law. We will see how the traditional mathematical view of Brow-
nian motion arises by taking a certain limit. The force-balance view permits a
variety of generalizations that include particle-particle interactions, the in
uence of
external energy potentials, and viscoelastic force-memory e ects. We will use sto-
chastic calculus to derive important statistics for the paths of such particles, develop
simulation techniques, and encounter a number of unsolved theoretical problems.
09:00 AM
12:00 AM
- Continuous time Markov chain models used in biology: models, approximations, and simulation
Continuous time Markov chain models used in biology: models, approximations, and simulation
Friday, June 22, 2012
Time Session
12:00 AM
11:00 AM
Scott McKinley - Anomalous diffusion in biological fluids
Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargo-laden
vesicles in axons undergo stark periods of forward and backward motion, inter-
rupted by sudden pauses and periods of free di usion. Over large periods of time,
the motion is e ectively that of a particle with steady drift accompanied a di u-
sive spread greater than what can be explained by thermal
uctuations alone. As
another example, E. coli and other bacteria are known to respond to the local con-
centration of nutrients in such a way that they can climb gradients toward optimal
locations. Again, the e ective behavior is drift toward a desired" location, with
enhanced di usivity.
In other cases, biological entities are signi cantly slowed. Relatively large parti-
cles di using in
uids such as mucus, blood, bio lms or the cytoplasm of cells all
experience hinderances due to interactions with the polymer networks that consti-
tute small-scale biological environments. Researches repeatedly observe sublinear
growth of the mean-squared displacement of particle paths. This signals to theo-
reticians that the particles are not experiencing traditional Brownian motion. In-
terestingly, many viruses are actually small enough to avoid this type of hinderance
when moving through human mucus. However, the body's immune response in-
cludes teams of still smaller antibodies that can immobilize virions by serving as an
intermediary creating binding events between virions and the local mucin network.
Underlying the mathematical description of all these phenomena is a modeling
framework that employs stochastic di erential equations, hybrid switching di u-
sions and stochastic integro-di erential equations. We will begin with the Langevin
model for di usion. This is the physicist's view of Brownian motion, derived from
Newton's Second Law. We will see how the traditional mathematical view of Brow-
nian motion arises by taking a certain limit. The force-balance view permits a
variety of generalizations that include particle-particle interactions, the in
uence of
external energy potentials, and viscoelastic force-memory e ects. We will use sto-
chastic calculus to derive important statistics for the paths of such particles, develop
simulation techniques, and encounter a number of unsolved theoretical problems.
09:00 AM
12:00 PM
- Continuous time Markov chain models used in biology: models, approximations, and simulation
Continuous time Markov chain models used in biology: models, approximations, and simulation
Saturday, June 23, 2012
Time Session
Sunday, June 24, 2012
Time Session
09:00 AM
12:00 PM
Steve Krone - Individual�based stochastic spatial models
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.
02:00 PM
05:30 PM
- Flux and fixation for the voter model and the Axelrod model
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.
Monday, June 25, 2012
Time Session
09:00 AM
12:00 PM
Steve Krone - Individual�based stochastic spatial models
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.
02:00 PM
05:30 PM
- Flux and fixation for the voter model and the Axelrod model
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.
Tuesday, June 26, 2012
Time Session
09:00 AM
12:00 PM
- Markov Models of Molecular Evolution, Bayesian Phylogenetics, and the MCMC approach
Markov Models of Molecular Evolution, Bayesian Phylogenetics, and the MCMC approach
02:00 PM
05:30 PM
Sebastian Schreiber - Persistence, coexistence and spatial spread in a fluctuating environment
All populations experience stochastic
uctuations in abiotic factors such as temperature, nutrient avail-
ability, precipitation. This environmental stochasticity in conjunction with biotic interactions can facilitate
or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic
forces is the construction and analysis of stochastic di erence equations and stochastic di erential equations.
Many theoretical biologists are interested in whether the models are stochastically bounded and persis-
tent. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact
sets. In contrast, stochastic persistence requires that the population process tends to be epelled" by some
"extinction set". Here, I will review recent results on both of these proprieties are reviewed for models
of multi-species interactions and spatially-structured populations. Basic results about random products of
matrices, Lyapunov exponents, stationary distributions, and small-noise approximations will be discussed.
Applications include bet-hedging, coexistence via the storage e ect, and evolutionary games in stochastic
environments.
Wednesday, June 27, 2012
Time Session
09:00 AM
12:00 PM
- Markov Models of Molecular Evolution, Bayesian Phylogenetics, and the MCMC approach
Markov Models of Molecular Evolution, Bayesian Phylogenetics, and the MCMC approach
Thursday, June 28, 2012
Time Session
Friday, June 29, 2012
Time Session
Name Affiliation
Allen, Linda linda.j.allen@ttu.edu Department of Mathematics and Statistics, Texas Tech University
Allen, Edward edward.allen@ttu.edu Mathematics and Statistics, Texas Tech University
Aminzare, Zahra aminzare@math.rutgers.edu Mathematics, Rutgers University
Anderson, David dendersn@gmail.com
Antwi, Shadrack saantwi@email.wm.edu Applied Science, College of William and Mary
Arat, Seda sedag@vbi.vt.edu Virginia Bioinformatics Institute, Virginia Tech
Auerbach, Jeremy jauerbac@utk.edu Mathematics, University of Tennessee
Batty, Christopher cbatty89@comcast.net Mathematics, The Ohio State University
Bowden, Sarah sarah@drakeresearchlab.com Odum School of Ecology, University of Georgia
Buckalew, Richard rb301008@ohio.edu Mathematics, Ohio University
Chai, JJ chai@nimbios.org NIMBioS, University of Tennessee
Charteris, Nicholas opnickc@gmail.com Physics, Oakland University
cheong, sami cheongs@uwm.edu Mathematical Sciences, University of Wisconsin
Childs, Parker parker@math.utah.edu Mathematics, University of Utah
Durney, Clinton durney.1@math.osu.edu Mathematics, The Ohio State University
Eager, Eric s-eeager1@math.unl.edu Mathematics, University of Nebraska
Emanuel, George george.emanuel@colorado.edu JILA, University of Colorado
Enyeart, Jack jenyeart@math.duke.edu Mathematics, Duke University
Ford, Mauntell mauntell@gmail.com Mathematics, The Ohio State University
Gasior, Kelsey kelsgasior@gmail.com Biomathematics, North Carolina State University
Gautam, Raju rgautam@cvm.tamu.edu Veterinary Integrative Biosciences, Texas A & M University
Greene, James jmgreene@math.umd.edu Mathematics, University of Maryland
Hellmann, Jennifer hellmann.13@osu.edu Evolution, Ecology, and Organismal Biology, The Ohio State University
Herrera Reyes, Alejandra donajialej@gmail.com Mathematics, University of British Columbia
Kinderknecht, Kelsy kinderknecht.1@math.osu.edu Mathematics, The Ohio State University
Konrad, Bernhard konradbe@math.ubc.ca Mathematics, University of British Columbia
Krone, Steve krone@uidaho.edu Mathematics, University of Idaho
Lahodny, Glenn glenn.lahodny@ttu.edu Mathematics & Statistics, Texas Tech University
Lanchier, Nicolas lanchier@math.asu.edu Mathematics and Statistics, Arizona State University
Lapid, Carlo cmlapid@wustl.edu Department of Biology, Washington University in St. Louis
Li, Chunlei cli2@nd.edu Applied and Computational Math & Statistics, University of Notre Dame
Linder, Daniel dlinder@georgiahealth.edu Biostatistics, Georgia Health Sciences University
Lipshutz, David dlipshut@math.ucsd.edu Mathematics, University of California, San Diego
long, yunhan ylong@email.wm.edu applied science, College of William and Mary
Marschall, Libby marschall.2@osu.edu Evolution, Ecology and Organismal Biology, The Ohio State University
MARTINEZ, LUIS luis27@ciencias.unam.mx Applied Mathematics, University of Mexico (UNAM)
Massaro, Tyler tyler.massaro@gmail.com Mathematics, University of Tennessee
Matamba Messi, Leopold lmatamba@gmail.com Department of Mathematics, University of Georgia
McKinley, Scott scott.mckinley@ufl.edu Mathematics, University of Florida
Najera Chesler, Aisha aisha.najera@cgu.edu Mathematics, Claremont Graduate University
Narayan, Monisha narayan.28@math.osu.edu Mathmatics, The Ohio State University
Norton, Jacob jfnorton@ncsu.edu Biomathematics Graduate Program, North Carolina State University
Oliveros-Ramos, David ricardo.oliveros@gmail.com Modeling Research Center, Instituto del Mar del Perú
Popovic, Lea lpopovic@mathstat.concordia.ca Dept of Mathematics and Statistics, Concordia University
Powell, Kristin kipowell@wustl.edu Biology, Washington University
Rabajante, Jomar jfr_jomar@yahoo.com Mathematics Division, University of the Philippines Los Baños
Reed, Michael reed@math.duke.edu Mathematics, Duke University
Roth, Gregory greg.roth51283@gmail.com Evolution and Ecology, University of California, Davis
Roychoudhury, Pavitra padm3003@vandals.uidaho.edu Mathematics, University of Idaho
Ryu, Hwayeon hwayeon@math.duke.edu Mathematics, Duke University
Schreiber, Sebastian sschreiber@ucdavis.edu Center of Population Biology, University of California, Davis
Shoemaker, Lauren lauren.shoemaker@colorado.edu Ecology and Evolutionary Biology, University of Colorado
Smith, Heather blythe.32@math.osu.edu Mathematics, The Ohio State University
Stefan, Thorsten thorsten.stefan@glasgow.ac.uk Institute of Biodiversity, Animal Health & Comparative Medicine, University of Glasgow
Stepien, Tracy tls52@pitt.edu Mathematics, University of Pittsburgh
Wang, Xiao xwang06@email.wm.edu Applied Science, College of William and Mary
Wei, Feng weifeng@umich.edu Mathematics, University of Michigan
Weigang, Helene helene.weigang@gmail.com DTU Aqua / Department of Plant Biology, Technical University of Denmark (DTU Aqua) / Michigan State University (Kellogg Biological Station)
Weinberg, Daniel daerwe@math.umd.edu Mathematics, University of Maryland
Weiss-Lehman, Christopher christopher.weisslehman@colorado.edu Ecology and Evolutionary Biology, University of Colorado
Yates, Andrew yates.115@osu.edu Computer Science and Engineering, The Ohio State University
An Introduction to Stochastic Epidemic Models
A brief introduction is presented to modeling in stochastic epidemiology. Several
useful epidemiological concepts such as the basic reproduction number and the nal size
of an epidemic are de ned. Three well-known stochastic modeling formulations are in-
troduced: discrete-time Markov chains, continuous-time Markov chains, and stochastic
di erential equations. Methods for derivation, analysis and numerical simulation of the
three types of stochastic epidemic models are presented. Emphasis is placed on some of
the di erences between the three stochastic modeling formulations as illustrated in the
classic SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered)
epidemic models. In addition, some of the unique properties of stochastic epidemic
models, such as the probability of an outbreak, nal size distribution, critical commu-
nity size, and expected duration of an epidemic are demonstrated in various models of
diseases impacting humans and wildlife.
A Practical Introduction to Stochastic Differential Equations in Mathematical Biology
Properties of the Wiener process are reviewed and stochastic integration is explained. Stochastic di├»┬ČÔéČerential equations are introduced and some of their properties are described. Equivalence of SDE systems is explained. Commonly used numerical procedures are discussed for computationally solving systems of stochastic di├»┬ČÔéČerential equations. A procedure is described for deriving It├őÔÇáo stochastic di├»┬ČÔéČerential equations from associated discrete stochastic models for randomly-varying problems in biology. The SDEs are derived from basic principles, i.e., from the changes in the system which
occur in a small time interval. Several examples illustrate the procedure. In particular, stochastic di├»┬ČÔéČerential equations are derived for predator-prey, competition, and epidemic problems.
A Practical Introduction to Stochastic Differential Equations in Mathematical Biology
Properties of the Wiener process are reviewed and stochastic integration is explained. Stochastic di├»┬ČÔéČerential equations are introduced and some of their properties
are described. Equivalence of SDE systems is explained. Commonly used numerical
procedures are discussed for computationally solving systems of stochastic di├»┬ČÔéČerential
equations. A procedure is described for deriving It├őÔÇáo stochastic di├»┬ČÔéČerential equations
from associated discrete stochastic models for randomly-varying problems in biology.
The SDEs are derived from basic principles, i.e., from the changes in the system which
occur in a small time interval. Several examples illustrate the procedure. In particular, stochastic di├»┬ČÔéČerential equations are derived for predator-prey, competition, and
epidemic problems.
Individual�based stochastic spatial models
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.
Individual�based stochastic spatial models
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.
Anomalous diffusion in biological fluids
Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargo-laden
vesicles in axons undergo stark periods of forward and backward motion, inter-
rupted by sudden pauses and periods of free di usion. Over large periods of time,
the motion is e ectively that of a particle with steady drift accompanied a di u-
sive spread greater than what can be explained by thermal
uctuations alone. As
another example, E. coli and other bacteria are known to respond to the local con-
centration of nutrients in such a way that they can climb gradients toward optimal
locations. Again, the e ective behavior is drift toward a desired" location, with
enhanced di usivity.
In other cases, biological entities are signi cantly slowed. Relatively large parti-
cles di using in
uids such as mucus, blood, bio lms or the cytoplasm of cells all
experience hinderances due to interactions with the polymer networks that consti-
tute small-scale biological environments. Researches repeatedly observe sublinear
growth of the mean-squared displacement of particle paths. This signals to theo-
reticians that the particles are not experiencing traditional Brownian motion. In-
terestingly, many viruses are actually small enough to avoid this type of hinderance
when moving through human mucus. However, the body's immune response in-
cludes teams of still smaller antibodies that can immobilize virions by serving as an
intermediary creating binding events between virions and the local mucin network.
Underlying the mathematical description of all these phenomena is a modeling
framework that employs stochastic di erential equations, hybrid switching di u-
sions and stochastic integro-di erential equations. We will begin with the Langevin
model for di usion. This is the physicist's view of Brownian motion, derived from
Newton's Second Law. We will see how the traditional mathematical view of Brow-
nian motion arises by taking a certain limit. The force-balance view permits a
variety of generalizations that include particle-particle interactions, the in
uence of
external energy potentials, and viscoelastic force-memory e ects. We will use sto-
chastic calculus to derive important statistics for the paths of such particles, develop
simulation techniques, and encounter a number of unsolved theoretical problems.
Anomalous diffusion in biological fluids
Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargo-laden
vesicles in axons undergo stark periods of forward and backward motion, inter-
rupted by sudden pauses and periods of free di usion. Over large periods of time,
the motion is e ectively that of a particle with steady drift accompanied a di u-
sive spread greater than what can be explained by thermal
uctuations alone. As
another example, E. coli and other bacteria are known to respond to the local con-
centration of nutrients in such a way that they can climb gradients toward optimal
locations. Again, the e ective behavior is drift toward a desired" location, with
enhanced di usivity.
In other cases, biological entities are signi cantly slowed. Relatively large parti-
cles di using in
uids such as mucus, blood, bio lms or the cytoplasm of cells all
experience hinderances due to interactions with the polymer networks that consti-
tute small-scale biological environments. Researches repeatedly observe sublinear
growth of the mean-squared displacement of particle paths. This signals to theo-
reticians that the particles are not experiencing traditional Brownian motion. In-
terestingly, many viruses are actually small enough to avoid this type of hinderance
when moving through human mucus. However, the body's immune response in-
cludes teams of still smaller antibodies that can immobilize virions by serving as an
intermediary creating binding events between virions and the local mucin network.
Underlying the mathematical description of all these phenomena is a modeling
framework that employs stochastic di erential equations, hybrid switching di u-
sions and stochastic integro-di erential equations. We will begin with the Langevin
model for di usion. This is the physicist's view of Brownian motion, derived from
Newton's Second Law. We will see how the traditional mathematical view of Brow-
nian motion arises by taking a certain limit. The force-balance view permits a
variety of generalizations that include particle-particle interactions, the in
uence of
external energy potentials, and viscoelastic force-memory e ects. We will use sto-
chastic calculus to derive important statistics for the paths of such particles, develop
simulation techniques, and encounter a number of unsolved theoretical problems.
Persistence, coexistence and spatial spread in a fluctuating environment
All populations experience stochastic
uctuations in abiotic factors such as temperature, nutrient avail-
ability, precipitation. This environmental stochasticity in conjunction with biotic interactions can facilitate
or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic
forces is the construction and analysis of stochastic di erence equations and stochastic di erential equations.
Many theoretical biologists are interested in whether the models are stochastically bounded and persis-
tent. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact
sets. In contrast, stochastic persistence requires that the population process tends to be epelled" by some
"extinction set". Here, I will review recent results on both of these proprieties are reviewed for models
of multi-species interactions and spatially-structured populations. Basic results about random products of
matrices, Lyapunov exponents, stationary distributions, and small-noise approximations will be discussed.
Applications include bet-hedging, coexistence via the storage e ect, and evolutionary games in stochastic
environments.
video image

Persistence, coexistence and spatial spread in a fluctuating environment
Sebastian Schreiber All populations experience stochastic
uctuations in abiotic factors such as temperature, nutrient avail-
ability, precipitation. This environmental stochasticity in conjunction with biotic interactions can facilitate
or disrupt

video image

Anomalous diffusion in biological fluids
Scott McKinley Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. C

video image

Anomalous diffusion in biological fluids
Scott McKinley Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. C

video image

Flux and fixation for the one-dimensional Axelrod model
Nicolas Lanchier 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 - t

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Individual�based stochastic spatial models
Steve Krone 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 ingredient

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Flux and fixation for the voter model and the Axelrod model
Nicolas Lanchier 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 - t

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Individual�based stochastic spatial models
Steve Krone 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 ingredient

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An introduction to thinking like a probabilist about biology
Louis Gross 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, M

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A Practical Introduction to Stochastic Differential Equations in Mathematical Biology
Edward Allen Properties of the Wiener process are reviewed and stochastic integration is explained. Stochastic di´Č€erential equations are introduced and some of their properties
are described. Equivalence of SDE systems is explained. Commonl

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A Practical Introduction to Stochastic Differential Equations in Mathematical Biology
Edward Allen Properties of the Wiener process are reviewed and stochastic integration is explained. Stochastic di´Č€erential equations are introduced and some of their properties are described. Equivalence of SDE systems is explained. Commonly used

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An Introduction to Stochastic Epidemic Models
Linda Allen A brief introduction is presented to modeling in stochastic epidemiology. Several
useful epidemiological concepts such as the basic reproduction number and the nal size
of an epidemic are de ned. Three well-known stochastic modeling fo