
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 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 randomlyvarying 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 predatorprey, competition, and
epidemic problems.

Scott McKinley
Rapid recent progress in advanced microscopy has revealed that nanoparticles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargoladen
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 smallscale biological environments. Researches repeatedly observe sublinear
growth of the meansquared 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 integrodierential 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 forcebalance view permits a
variety of generalizations that include particleparticle interactions, the in
uence of
external energy potentials, and viscoelastic forcememory 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.

Scott McKinley
Rapid recent progress in advanced microscopy has revealed that nanoparticles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargoladen
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 smallscale biological environments. Researches repeatedly observe sublinear
growth of the meansquared 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 integrodierential 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 forcebalance view permits a
variety of generalizations that include particleparticle interactions, the in
uence of
external energy potentials, and viscoelastic forcememory 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.

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 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 hostpathogen 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 wellmixed 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.

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  individualbased 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 onedimensional 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.

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 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 hostpathogen 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 wellmixed 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.

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  individualbased 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 onedimensional 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.

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 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 multispecies interactions and spatiallystructured populations. Basic results about random products of
matrices, Lyapunov exponents, stationary distributions, and smallnoise approximations will be discussed.
Applications include bethedging, coexistence via the storage eect, and evolutionary games in stochastic
environments.

Louis Gross
This set of lectures and discussions will provide a quick oneday 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, agentbased 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 handson experience with analysis of a stochastic process using simple computer tools.

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 wellknown stochastic modeling formulations are in
troduced: discretetime Markov chains, continuoustime 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 (susceptibleinfectedsusceptible) and SIR (susceptibleinfectedrecovered)
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.

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 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 randomlyvarying 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 predatorprey, competition, and epidemic problems.