### Organizers

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

- Mon, Jun 18, 2012
- Tue, Jun 19, 2012
- Wed, Jun 20, 2012
- Thu, Jun 21, 2012
- Fri, Jun 22, 2012
- Sat, Jun 23, 2012
- Sun, Jun 24, 2012
- Mon, Jun 25, 2012
- Tue, Jun 26, 2012
- Wed, Jun 27, 2012
- Thu, Jun 28, 2012
- Fri, Jun 29, 2012
- Full Schedule

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 dened. Three well-known stochastic modeling formulations are in- troduced: discrete-time Markov chains, continuous-time Markov chains, and stochastic dierential 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 dierences 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 diï¬€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 diï¬€erential equations. A procedure is described for deriving ItË†o stochastic diï¬€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 diï¬€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 diï¬€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 diï¬€erential equations. A procedure is described for deriving ItË†o stochastic diï¬€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 diï¬€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 diusion. Over large periods of time, the motion is eectively that of a particle with steady drift accompanied a diu- 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 eective behavior is drift toward a desired" location, with enhanced diusivity. In other cases, biological entities are signicantly slowed. Relatively large parti- cles diusing in uids such as mucus, blood, biolms 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 dierential equations, hybrid switching diu- sions and stochastic integro-dierential equations. We will begin with the Langevin model for diusion. 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 eects. 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 diusion. Over large periods of time, the motion is eectively that of a particle with steady drift accompanied a diu- 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 eective behavior is drift toward a desired" location, with enhanced diusivity. In other cases, biological entities are signicantly slowed. Relatively large parti- cles diusing in uids such as mucus, blood, biolms 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 dierential equations, hybrid switching diu- sions and stochastic integro-dierential equations. We will begin with the Langevin model for diusion. 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 eects. 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 - Individualâ€?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 - Individualâ€?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 dierence equations and stochastic dierential 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 eect, 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 PerÃƒÂº |

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 BaÃƒÂ±os |

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 |

useful epidemiological concepts such as the basic reproduction number and the nal size

of an epidemic are dened. Three well-known stochastic modeling formulations are in-

troduced: discrete-time Markov chains, continuous-time Markov chains, and stochastic

dierential 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 dierences 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.

occur in a small time interval. Several examples illustrate the procedure. In particular, stochastic diÃ¯Â¬â‚¬erential equations are derived for predator-prey, competition, and epidemic problems.

are described. Equivalence of SDE systems is explained. Commonly used numerical

procedures are discussed for computationally solving systems of stochastic diÃ¯Â¬â‚¬erential

equations. A procedure is described for deriving ItÃ‹â€ o stochastic diÃ¯Â¬â‚¬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 diÃ¯Â¬â‚¬erential equations are derived for predator-prey, competition, and

epidemic problems.

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 diusion. Over large periods of time,

the motion is eectively that of a particle with steady drift accompanied a diu-

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 eective behavior is drift toward a desired" location, with

enhanced diusivity.

In other cases, biological entities are signicantly slowed. Relatively large parti-

cles diusing in

uids such as mucus, blood, biolms 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 dierential equations, hybrid switching diu-

sions and stochastic integro-dierential equations. We will begin with the Langevin

model for diusion. 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 eects. 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.

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 diusion. Over large periods of time,

the motion is eectively that of a particle with steady drift accompanied a diu-

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 eective behavior is drift toward a desired" location, with

enhanced diusivity.

In other cases, biological entities are signicantly slowed. Relatively large parti-

cles diusing in

uids such as mucus, blood, biolms 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 dierential equations, hybrid switching diu-

sions and stochastic integro-dierential equations. We will begin with the Langevin

model for diusion. 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 eects. 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.

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 dierence equations and stochastic dierential 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 eect, and evolutionary games in stochastic

environments.

**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

**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

**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

**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

**Individualâ€?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

**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

**Individualâ€?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

**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

**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 diï¬€erential equations are introduced and some of their properties

are described. Equivalence of SDE systems is explained. Commonl

**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 diï¬€erential equations are introduced and some of their properties are described. Equivalence of SDE systems is explained. Commonly used

**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 dened. Three well-known stochastic modeling fo