Infection with the Human Papilloma Virus (HPV) is a prerequisite for the development of cervical cancer, the second most common cancer in women in the developing world. While about 80% of women get infected with HPV during their lifetime, most clear the virus within 2 years. However, if the infection persists, further cellular events can lead to high-grade lesions and eventually invasive carcinoma. To date, various aspects of the carcinogenesis remain poorly understood at the cellular level. In this talk, we develop and discuss a stochastic model of the cervical epithelium, coupling the dynamics of HPV infection to a multi-stage model of carcinogenesis.

We use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $\ZZ^d$, $d\ge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $\sigma^2I$. The games we consider have matrices of the form ${\bf 1} + wG$ where ${\bf 1}$ is matrix of 1's and $w$ is small and positive. We prove that the effect of space is equivalent to (i) changing the entries of the game matrix and (ii) replacing the replicator ODE by a related PDE. The first idea is due to Ohtsuki and Nowak (for the pair approximation) while the second is well known in the theory of stochastic spatial processes. A remarkable aspect of our result is that the limiting PDE depends on the interaction kernel only through the values of two simple noncoalescence probabilities. Due to results of Aronson and Weinberger, and Fife and McLeod, we can analyze any 2x2 game. However, when there are three strategies the limiting object is a system of reaction diffusion equations, so we only have results for special cases.

Biology is a tremendously diverse field covering systems operating at vastly different scales, with differing levels of interaction between these. Much of the effort in mathematical biology has been driven by the desire to develop the general principles by which biological systems are organized and operate. Yet at the same time there are demands for answers to quite specific questions to better manage natural systems, to enhance human health, and to plan for the future impacts of human actions. I will give a variety of examples of projects in mathematical ecology that lie at the interface between theory and practice, providing some indication of the utility of quantitative methods to elucidate general patterns of natural system response to management actions. These will include applications of optimal control methods to problems in wildlife management and disease ecology as well as a discussion of individual-based models. An objective of these approaches is to develop hypothetical "best" methods to manage a system, and use this as a template to compare and contrast management scenarios arising from the differing view points of diverse stakeholders in a relative assessment framework.

The interface between the mathematical sciences and the biosciences is two-way and may be summarized as "math -> bio" and "bio -> math." This talk will have two parts. First, I will describe previous work on gaits of four-legged animals (based on distinguishing gaits, such as walk, trot, and pace, by their spatio-temporal symmetries). Second, I will discuss how the application to gaits has led to results about phase-shift synchrony in periodic solutions of coupled systems of differential equations.

******* CANCELED ****************** In recent years it has become increasingly clear that stochasticity plays an important role in many biological processes. Examples include bistable genetic switches, noise enhanced robustness of oscillations, and fluctuation enhanced sensitivity or “stochastic focusing".. Numerous cellular systems rely on spatial stochastic noise for robust performance. We examine the need for stochastic models, report on the state of the art of algorithms and software for modeling and simulation of stochastic biochemical systems, and identify some computational challenges.

Glioblastomas (GBM) are the most common and malignant primary tumors of the brain and are commonly treated with radiation therapy. Despite modest advances in chemotherapy and radiation, survival has changed very little over the last 50 years. Radiation therapy is one of the pillars of adjuvant therapy for GBM but despite treatment, recurrence inevitably occurs. Here we develop a mathematical model for the tumor response to radiation that takes into account the plasticity of the hierarchical structure of the tumor population. Based on this mathematical model we develop an optimized radiation delivery schedule. This strategy was validated to be superior in mice and nearly doubled the efficacy of each Gray of radiation administered. This is based on joint work with Ken Pitter, Eric Holland, and Franziska Michor.

Given the difficulty in finding a cure for HIV/AIDS, a promising prevention
strategy to reduce HIV transmission is to directly block infection at the
portal of entry. The recent Thai RV144 vaccine trial offered the first
evidence that a vaccine may provide location protection and block HIV
transmission in the vagina. Unfortunately, the underlying mechanisms for
protection remain unclear. In this talk, we examine theoretically a
hypothesis that builds on Sam Lai's recent laboratory observation that
virus-specific antibodies (Ab) may be capable of trapping individual
virions in genital mucus secretions. Ab are known to have a weak –
previously considered inconsequential – binding affinity with the mucin
fibers that constitute cervicovaginal mucus (CVM). However, several Ab
may be bound to a single virion at the same time, multiplying the Ab-mucin
binding effect, thereby creating an indirect virion-mucin affinity. Our
model takes into account biologically relevant length and time scales,
while incorporating known HIV-Ab affinity and the respective diffusivities
of viruses and Ab in semen and CVM. The model predicts that HIV-specific
Ab in CVM can effectively immobilize HIV in a shock-like front near the
semen-CVM interface, far from the vaginal epithelium. The robustness of
the result implies that even weak Ab-mucin affinity can markedly reduce the
flux of virions reaching target cells. Beyond this specific application,
the model developed here is adaptable to other pathogens, mucosal barriers,
geometries, kinetic and diffusional effects, providing a tool for hypothesis
testing and producing quantitative insights into dynamics of immune-
mediated protection.

This talk will discuss previous work on quadrupedal gaits and recent work on a generalized model for binocular rivalry proposed by Hugh Wilson. Both applications show how rigid phase-shift synchrony in periodic solutions of coupled systems of differential equations can help understand high level collective behavior in the nervous system.

Most cells possess the ability to change morphology or migrate in response
to environmental cues. To understand the molecular mechanisms that drive
cell movement requires a systems-level approach that combines computational
approaches, including mathematical modeling and image analysis tools, with
high-resolution microscopy of living cells. Here we present several
examples for how such an integrated research strategy has been
successfully applied. First, we combine stochastic modeling with novel
biosensors for monitoring the spatiotemporal dynamics of Rho GTPase
activity to investigate the role of RhoG in cell polarization and
migration. Next, mathematical modeling and quantitative image analysis
methods are used to establish the role of cerebral cavernous malformation
(CCM) proteins in vascular tube formation. Finally, we present a novel
computational method for tracking and quantifying changes in cell shape.

Clinical oncology generates patient data spanning from the molecular scale to the whole-body scale, which tend to be used in isolation when planning patient care. There is no current technique to quantitatively combine these with novel in vitro experimental data into comprehensive models that can illuminate complex, systems-level emergent phenomena and improve therapeutic and surgical planning. In this talk, we will discuss efforts by my lab, the USC Physical Sciences Oncology Center, and the Consortium for Integrative Computational Oncology to solve these issues. With a focus on patient pathology-calibrated breast cancer modeling and multidisciplinary modeling of liver metastases, we will explore agent-based and continuum model calibration to individual patient data, integration with novel experimental measurements, and emergent predictions of macroscopic and systems-level behavior. We will discuss the implications for making and quantitatively testing biological hypotheses, and the role of computational modeling in facilitating a deeper understanding of biology, pathology, and radiology. More information can be found at MathCancer.org.

Sleep and wake states are regulated by the interactions among a number of
brainstem and hypothalamic neuronal populations and the expression of
their neurotransmitters. Based on experimental studies, several different
structures have been proposed for this sleep-wake regulatory network with
particular debate over components involved in rapid-eye movement (REM)
sleep regulation. We have developed a mathematical modeling framework
that is uniquely suited for investigating the structure and dynamics of
proposed sleep-wake regulatory networks. Using this framework, we are
analyzing the competing proposed network structures for the regulation of
REM sleep to determine how the structure of the sleep-wake regulatory
network determines sleep-wake behavior and the dynamics of behavioral
state transitions.

"I will talk about tumor growth modelling in xenograft experiments. It's a statistical approach (see e.g. the ref. below), but I'd like to incorporate more mathematical modelling." Roy Choudhury, K., Kasman, I., Plowman, G., 2010, Analysis of multi-arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models, Statistics in Medicine, 29, 2399-2409

Living systems are characterized by variability; they are subject to constant evolution through the three processes of population growth, selection and mutations, a principle established by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the'fittest trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variations of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait? An area of population biology that aims at describing mathematically these processes is born in the 1980's under the name of 'adaptive dynamics' and, compared to population genetics, considers usually asexual reproduction, a continuous phenotypical trait and population growth. We will give a self-contained mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations. The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed. Several other modeling methods have been proposed, stochastic individual based models, evolutionary game theory, dynamical systems. Connections will be presented too. This talk is based on collaborations with G. Barles, J. Carrillo, S. Cuadrado, O. Diekmann, M. Gauduchon, S. Genieys, P.-E. Jabin, S. Mirahimmi, S. Mischler and P. E. Souganidis.

Social contacts provide the backbone over which infectious diseases are transmitted. The dynamic networks that describe the contact patterns of social systems over time make predicting disease outbreaks difficult. In this talk, I'll discuss some computational experiments that show how disease patterns on static networks are observably different from those on dynamic networks. I'll then provide some intuition about how to prove boundary conditions about transmission on networks that explain why and under what circumstances we are likely to see those differences.

A mechanistic understanding of biome distributions is a critical issue in
modern ecology, especially in the context of predictive models of past and
future climate change. While we can explain the current distribution of
many biomes accurately, our predictions are less successful in dynamic
systems where vegetation-environment feedbacks are significant. The
challenge is to integrate feedbacks more fundamentally into a coherent
theory of ecological process that determines biome distributions
currently, and that will shape them into the future. This is among the
most urgent questions to be addressed in ecology today. Savannas and
grasslands cover ~40% of the land surface and forests cover another ~30%.
Understanding the dynamics among these biomes will help explain biosphere
dynamics, past, present and into the future. I will combine empirical and
theoretical work for insights into the mechanisms that give rise to the
emergent stability of savanna, despite variability in vegetation structure
within the biome.