Fall 2017

Lapin Sergey
Title:  Mathematical Modeling of Human Eye

Open angle glaucoma (OAG) is the second leading cause of blindness in the world and is characterized by changes in the optic nerve head and retinal ganglion cell death, resulting in irreversible vision loss. The lack of understanding of the roles of various risk factors in OAG constitutes a major limitation in the diagnosis, management, and treatment of OAG. One of the known risk factors for glaucoma is elevated intraocular pressure. Mathematical modeling offers an important tool for understanding the relative influence of risk factors by linking the mechanical action of IOP on ocular tissues to the blood flow and oxygen transport within the ocular tissues. The main objective of this work is to study the relationship between intraocular pressure, blood pressure, and blood flow autoregulation in the retinal vasculature and develop mathematical models that can be used to model conditions specific to individual glaucoma patients.

Jordan Culp and Allison Fisher
Title:  Introuduction to Computational Neuroscience

  In this talk, we'll give a mathematical introduction to neuroscience. The presentation will be focused on the basis of the mathematical modeling of neurons and neural networks. For those who are interested in computationl neuroscience, please refer to "Mathmatical Foundations of Neuroscience" by B. Ermentrout and D. Terman.

Xiongzhi Chen
Title: The Functional False Discovery Rate with Applications in Genomics

The false discovery rate (FDR) measures the proportion of false positives among a set of hypothesis tests called significant. This quantity is typically estimated based on p-values or test statistics. In some scenarios, there is additional information available that may be used to more accurately estimate the FDR. We develop the functional FDR (fFDR) framework for formulating and estimating FDRs and q-values when an additional piece of information, which we call an “informative variable”, is available. The FDR is then treated as a function of this informative variable. We consider two applications in genomics. The first is a genetics of gene expression (eQTL) experiment in yeast, where every gene marker and gene expression trait pair are tested for association and the informative variable is the distance between each genetic marker and gene. The second is to detect differentially expressed genes in an RNA-seq experiment carried out in mice, where the informative variable is the per-gene read depth.  In both applications, the fFDR method provides more accurate estimates of q-values and is more powerful than the standard FDR method.

 Elissa Schwartz
Title:  Free-virus and cell-to-cell tranmission in models of equine infectious anemia virus infection

Equine infectious anemia virus (EIAV) is a lentivirus in the retrovirus family that infects horses and ponies. Two strains, referred to as the sensitive strain and the resistant strain, have been isolated from an experimentally-infected pony. The sensitive strain is vulnerable to neutralization by antibodies whereas the resistant strain is neutralization-insensitive. The sensitive strain mutates to the resistant strain. EIAV may infect healthy target cells via free virus or alternatively, directly from an infected target cell through cell-to-cell transfer. The proportion of transmission from free-virus or from cell-to-cell transmission is unknown. A system of ordinary differential equations (ODEs) is formulated for the virus-cell dynamics of EIAV. In addition, a Markov chain model and a branching process approximation near the infection-free equilibrium (IFE) are formulated. The basic reproduction number is defined as the maximum of two reproduction numbers, and one for the sensitive strain and one for the resistant strain. The IFE is shown to be globally asymptotically stable for the ODE model in a special case when the basic reproduction number is less than one. In addition, two endemic equilibria exist, a coexistence equilibrium and a resistant strain equilibrium. It is shown that if the basic reproduction number is greateer than one,  the infection persists with at least one of the two strains. However, for small infectious doses, the sensitive strain and the resistant strain may not persist in the Markov chain model. Parameter values applicable to EIAV are used to illustrate the dynamics of the ODE and the Markov chain models. The examples highlight the importance of the proportion of cell-to-cell versus free-virus transmission that either leads to infection clearance or to infection persistence with either coexistence of both strains or to dominance by the resistant strain.

This is joint work with L.J.S. Allen.

Svetlana Lockwood

Title:  Diffusion of Antimicrobial Resistance

What is common between an urban slum, a chicken farm and atoms in a heated piece of metal? The beauty of mathematics is that it can describe succinctly the disparate processes such as these. In this presentation, I will talk about my current research on population-wide antimicrobial resistance. I will introduce the topic of antimicrobial resistance, application of heat equations to it (graph Laplacian), and promising preliminary results.

Mark Schumaker

Title:  Steady State Analysis of an Immune System Model

This talk describes a steady state analysis of the model of equine infectious anemia proposed by Schwartz et al. (2013).  The model is a system of 5 ODEs with 12 parameters, all assumed to be positive.  There are three sets of biological steady states E0, E1 and E3.  The infection-free equilibria E0 are stable for R0<1, where R0 is the basic reproduction number of virus particles and infected macrophage cells. At R0=1 there is a bifurcation at the intersection of the E0 and E1 sets with an exchange of stability.  The boundary equilibria E1 represent an infection with an immune response involving antibodies only.  They are stable for R0>1 and \hat{R}_1<1, where \hat{R}_1 is the basic reproduction number for cytotoxic T-lymphocytes.  At \hat{R}_1=1 there is a bifurcation at the intersection of the E1 and E3 sets with a second exchange of stability.  The endemic equilibria E3 represent an infection with an immune system response involving both antibodies and cytotoxic T-lymphocytes.  Numerically, when \hat{R}_1>1, we have only observed stable states E3 with complex conjugate leading eigenvalues corresponding to oscillatory decay of antibody and T-cell concentrations to the E3 equilibrium.  The result is qualitatively consistent with experimental observations of antibody and T-cell concentrations in horses with equine infectious anemia.

 Elissa Schwartz
Title:  The Leaky Pipeline for Women in STEM: Global Challenges

More efforts are needed to improve the retention rates of students in STEM disciplines,
especially among historically underrepresented groups. Not only are women and
underrepresented groups important components of the workforce, but STEM fields can
benefit from the wider range of solutions contributed by diversity in its researchers.
Progressing on the path towards STEM careers is not always easy, however, and many
young women disappear along the way. This loss of women is a common phenomenon
in academic systems around the globe and has been described as the “leaky pipeline.”
The leaky pipeline for women in STEM careers starts at early stages, continues
throughout the career trajectory, and gradually results in a small proportion of women
researchers. We initiated a pilot study to begin to remedy the situation, focusing on
Africa. We created a program for professional development, mentoring, and support for
one particular group, female African students in the one-year coursework master’s
program in mathematics at the African Institute of Mathematical Sciences (AIMS) –
South Africa. Exit survey feedback from three initiatives that we implemented will be
presented, focusing on whether these programs increased or decreased the students’
perception of their available resources to succeed, other benefits reported, and what role
these programs may play to increase the likelihood that the women continue in STEM

This is joint work with Dr. Rejoyce Gavhi-Molefe, AIMS-South Africa.

Hongbo Dong
Title: Adaptive Clinical Trial Design, Huge-scale Linear Programs and Non-smooth Optimization

Abstract: TBA

Matthew Mietchen
Title:  Ending the Epidemic: HIV Surveillance Data-to-Care Methods & Other Interesting Tools

Abstract: HIV surveillance has drastically changed within recent years regarding the usage of reported laboratory and medical data. New testing algorithms with earlier detection capabilities has increased acute or early HIV diagnosis. Linkage to HIV care and treatment is currently the primary intervention to reduce new infections by treating infected individuals regardless of stage of infection. Patients with undetectable viral loads are less likely to transmit the virus to sexual partners and has become the primary focus for HIV prevention. Public health department Data-to-Care programs are increasing the usage of HIV surveillance data to reduce community viral loads by ensuring access to HIV antiviral medications and other services. Health Departments are in need to increase the informatics programs to collect, organize, and utilize the increase in data for these programs. Other tools for early detection are also assisting health departments with ending the epidemic in their jurisdictions.    

Adnan Morshed
Title:  Numerical Study of TGFβ-Smad Dependent Activation of Tumor Microenvironment using Interface Resolved Hybrid Technique

Abstract: The TGFβ–SMAD interaction is prevalent in a wide range of tumor microenvironments with autocrine and paracrine mechanisms driving the temporal evolution. However, the dual nature of TGFβ as tumor suppressor and promoter still remains poorly understood. We investigated the dynamic landscapes of TGFβ–SMAD signaling pathway for different levels of extracellular TGFβ. This required a hybrid spatiotemporal description of the extracellular environment with cell membranes being the immersed interfaces. Intracellular reaction network was connected to the extracellular domain through surface reactions. Model tuning with PE25 cells indicates the production of intracellular TGFβ is activated in a switch-like manner depending on the extracellular levels. Additionally, the spatial effect of cellular proximity in a microfluidic cell culture setting is explored where the variation in intercellular distance between different cells show the transformation in intracellular behavior similar to the tumor-mediated transformation of fibroblasts to myofibroblasts. Our model also predicts that TGFβ distribution in the domain to be largely affected by receptor concentrations in different types of cells and diffusivity of the media.

Andreas Vasdekis
Title: A Random Walk through Metabolism, as a Matter of Fat

Abstract: No two individual cells ever “look” the same, even if they share the same genes and grow under identical conditions. This unusual phenomenon has been appreciated since the first ever cell microscopy images; however, its implications in life sciences, medicine and biotechnology have only recently began to be understood. In this talk, I will introduce this phenomenon, its origins, and present our recent findings in the context of lipid metabolism.

Nikolaos Voulgarakis
Title: Mathematical modeling of DNA allostery

Abstract: The role of DNA is not limited only to carrying and protecting the genetic code. DNA may also play a very active role in its own functions. There is increasing experimental evidence that conformational and dynamic changes of the double strand may direct protein aggregations that are responsible for fundamental functions of DNA. Using simple stochastic models, I will show that the coalescence of protein-induced DNA bubbles can mediate allosteric interactions that drive such protein aggregation. We will also discuss how this new type of allostery could regulate (a) the packaging of DNA and (b) the assembly of the transcription machinery.

Anders Omsland
Title:  Nutritional and physicochemical requirements for Chlamydia and Coxiella infectivity and replication

Chlamydia trachomatis and Coxiella burnetii are bacterial obligate intracellular parasites that depend on invasion of a eukaryotic host cell for replication. Pathogen adaptation to intracellular parasitism is believed to drive genome streamlining characterized by loss of biosynthetic capacity and concomitant dependence on scavenging essential nutrients (e.g., amino acids and nucleotides) from the host cell. Both C. trachomatis and C. burnetii undergo developmental transitions between cell forms specialized for either infection / survival, or replication. Because only one cell form is replicative, transition between cell forms is essential for virulence. Nutrient availability as well as specific physicochemical conditions (e.g., pH, and oxygen availability) have emerged as triggers for transitions between cell forms. Multifactorial physicochemical and nutritional requirements for replication shed light in the physiological basis for why the organisms are limited to replicate within specific intracellular niches.

Spring 2017

Kyle Harrington
Title:  Developing Computational Models with Biological Images

Computational biology has been advancing rapidly, and is now a key component of many biomedical research projects. However, the majority of computational models are disconnected from the experiments and data they are designed to explain. I will discuss how computational models are starting to directly incorporate biological images for calibration and validation. Multiscale computational models of vascular biology will be the primary case study discussed in this talk. I will show how we used computational modeling to screen and predict mechanistic models of angiogenesis. The talk will conclude by highlighting ongoing efforts to directly couple image acquisition and model development.

Prashanta Dutta

Title:  Studying Hypoxia in a Microfluidic Environment

Rapid tumor growth can result in localized zones in the tumor microenvironment where cells have far less access to nutrients. The scarcity of nutrients such as oxygen and ascorbate plays a critical role in the fate of tumor microenvironment. For example, hypoxia, the depletion of intracellular oxygen levels below 6%, initiates major changes in cellular dynamics causing tumor cell survival by escaping cellular degradation mechanisms. The intercapillary distance (distance between adjacent blood vessels) across a colony of growing tumor cells and the flow around the colony are believed to be important factors for the initiation of hypoxia. Although cellular dynamics have been studied extensively for a specific hypoxia, all these models consider only the intracellular dynamics and for the most part, treat the species inside as a well-mixed system. However, it is well established that cellular uptake and consumption of nutrients like oxygen, ascorbate and iron from the extracellular environment are continuous processes, which cannot be properly represented only with an intracellular model. In this talk, we will present a hybrid model to study the transport and evolution of different species in both extracellular and intracellular spaces of a hypoxic cellular microenvironment in a microfluidic setting. We will discuss the effect of different key parameters, such as flow strength, the spatial distribution of nutrients, effective diffusion coefficients, and the intercapillary distance on the dynamics of intracellular species inside a colony of tumor cells. We will also discuss how a change in hydroxylation behavior and nutrient supplementation can potentially help us in designing novel therapeutic interventions for cancer.

Todd Coffey

Title: A Comparison of Novel and Common Statistical Methods to Detect Out-of-Trend Stability Results

High-quality stability data are vital to the global medicine supply because they provide rationale for specification limits, product expiration dates, and label storage statements.  Lots placed on stability commonly have out-of-trend (OOT) data points but limited guidance has been given by Health Authorities or the statistical community.  In this talk I will briefly review the four common methods for detecting OOT data and explore two novel approaches based on cumulative sums of differences between the observed and expected stability profiles.

Lynn Schreyer

Title:  Modeling the Movement of Mucus in Lungs

The movement of mucus from lungs is accomplished by a combination of moving hairs called cilia and coughing.   Cilia are hair-like structures that move in unison with the purpose of propelling fluid.  They are found not only in human lungs, but also in gills in molluscs and human female oviducts to carry the ova. Residing in a layer of fluid called periciliary layer (PCL) which keeps the alive, a layer of mucus rests above the PCL and the purpose of the cilia movement is to move the mucus (and all the unwanted debris that is trapped within) out of the pulmonary pathways.   Here we will talk about the status of ongoing research into modeling the movement of mucus.

Nikolaos Voulgarakis

Title:  Mathematical modeling of DNA allostery

The role of DNA is not limited only to carrying and protecting the genetic code. DNA may also play a very active role in its own functions. There is increasing experimental evidence that conformational and dynamic changes of the double strand may direct protein aggregations that are responsible for fundamental functions of DNA. Using simple stochastic models, I will show that the coalescence of protein-induced DNA bubbles can mediate allosteric interactions that drive such protein aggregation. We will also discuss how this new type of allostery could regulate (a) the packaging of DNA and (b) the assembly of the transcription machinery.

Eric Lofgren

Title:  The Patient-Patch: Hospital Epidemiology as an Ecology Problem

Hospital-acquired infections are a pernicious problem in the modern healthcare system, arising from complex interactions between microbes, patients, healthcare providers and the built environment. This talk explores the idea of examining the problem of healthcare associated infections from the perspective of urban disease ecology, using several hospital modeling problems as motivating examples.

Daniel Farber

Title:  The Primary Disease Gradient of Wheat Stripe Rust (Puccinia striiformis f. sp. tritici) Across Spatial Scales

Aerial dispersal of spores or other propagules is a critical element of invasion ecology, including epidemiology. However, it is not well understood, particularly at the disease front where inoculum sources are more likely to be isolated. I inoculated a single wheat leaf with Puccinia striiformis f. sp. tritici (PST), causal agent of wheat stripe rust, and sampled all leaves within two intersecting 0.3 m by 3.0 m transects after a single generation of disease spread. The progeny infections were three-dimensionally mapped, and the primary disease gradient, the proportion of total progeny infections as a function of distance from a source infection, was well fit by the inverse-power, the Modified Pareto, and the Weibull distributions. To examine how spatial scale affects the resulting disease gradient, I combined the above dataset with a field-wide PST dataset (91.4 m) and a region-wide primary disease gradient of wheat stem rust, caused by P. graminis f. sp. tritici (PGT), a close relative of PST. All three datasets were well-fit by a single inverse-power function, of . A mechanistic model was developed to simulate disease spread across multiple spatial scales, comparing compartment sizes of 0.025 m, 1.52 m, and 152.4 m, over several generations, with a range of reproduction rates (R0) and initial infections (P0). Epidemics simulated using differing compartment sizes resulted in very similar disease progress curves. These results suggest it may be possible to extrapolate the results of a dispersal study to a larger scale, given that the study systems remain constant.

Fall 2016

Ilia Karatsoreos
Title:  Circadian Clocks and Disease: Opportunities for Dialogue between Math and Biology

Circadian (daily) rhythms are phylogenetically ancient, and present in nearly all organisms that have a life span greater than 24h. It has been hypothesized that circadian clocks impart adaptive fitness, making them a key component of life. In mammals, the master circadian clock is located in the small hypothalamic suprachiasmatic nucleus (SCN), comprised of nearly 10,000 independent oscillators that couple together to form a cohesive network clock. Disruption of this clock, especially by mistimed light exposure, leads to significant mental and physical health problems. However, the process by which oscillators become disrupted at a cellular level remain elusive. My lab explores how intact and robust circadian timing promotes resilience at various levels of biological organization, and how disrupting these rhythms leads to negative health outcomes. I will present data highlighting what is known about the structure and function of the SCN oscillator, and how disruption of this clock by light can affect health. It is hoped that this presentation can help form a basis for future in-depth discussion and potential collaboration at the interface between math and biology in the context of circadian rhythms and health.
Suggested Readings:
Antle, M. C., Foley, D. K., Foley, N. C. & Silver, R. Gates and oscillators: a network model of the brain clock. J Biol Rhythms 18, 339–350 (2003).
Yan, L. L. et al. Exploring spatiotemporal organization of SCN circuits. Cold Spring Harb Symp Quant Biol 72, 527–541 (2007).
Karatsoreos, I. N. & McEwen, B. S. Psychobiological allostasis: resistance, resilience and vulnerability. Trends Cogn Sci (Regul Ed) – (2011). doi:10.1016/j.tics.2011.10.005
Pauls, S. D., Honma, K.-I., Honma, S. & Silver, R. Deconstructing Circadian Rhythmicity with Models and Manipulations. Trends Neurosci 39, 405–419 (2016).

Mark Schumaker
Title:  Investigations into a model of Virus / Immune System Dynamics


Tim Ginn
Title:  Dose-structured population dynamics

Population response to environmental stimuli such as chemicals, heat, or radiation are so far mostly addressed through individual-based modeling approaches. I will describe an alternative approach where population states are structured on generalized dose, based on a kinematic approach that is a simple generalization of age-structuring. The resulting framework accommodates different memories of exposure as in recovery from toxic ambient conditions, differentiation between exogenous and endogenous sources of variation in population response, and quantification of acute or sub-acute effects on populations arising from life-history exposures to abiotic species. I will summarize three examples involving growth suppression in fish, inactivation of microorganisms with ultraviolet irradiation, and metabolic lag in bacterial growth.

Hong-Ming Yin
Title:  On a Cross-Diffusion System modeling Vegetation Spots and Strips in Arid Landscape

In this paper we study a model which describes the vegetation spots and strips in arid landscape. The mathematical mode consists of a cross-diffusion system with reaction sources. Global existence and uniqueness are proved. Some asymptotic behaviors of the solution will be discussed. Those asymptotic behaviors will demonstrate the vegetation patterns observed in the African drylands.  

Steve Krone
Title:  Does spatial structure mediate bacterial persistence in the presence of phage?

Phage (viruses that infect bacteria) produce a burst of progeny upon lysis of a bacterial cell, and the resulting rapid increase in phage density would seem to doom bacterial populations in some settings. Despite its initial promise, “phage therapy” (as an alternative to antibiotics) has not lived up to expectations. Lab experiments have shown long-term coexistence of bacteria and phage in liquid (chemostat) and spatial (biofilm) cultures. Additionally, surviving cell densities in biofilms are orders of magnitude larger than in liquid cultures. Is there something about spatial structure, per se, that protects cells, or do they derive their protection from mechanisms that only arise spatially? We study spatial agent-based models, as well as ordinary differential equations, to examine the possible effects that spatial structure could have on bacterial survival. Among the mechanisms that we consider are resource concentration, barriers (such as exopolysaccharides (EPS)), cellular debris, and cell signaling. We also propose a kind of “effective burst size” as an indication of the effect of spatial structure. This is joint work with Jim Bull, Kelly Christensen and Carly Scott.

Damilola Olabode
Title:  Optimal Control Applied to a Basic Model of Latent Cell Activation of HIV with a Focus on the Comparison of Objective Functions

The main aim of this paper was to compare three objective functions for an optimal control problem of the HIV treatment. In the administration of the HAART regimen the amount of free viral particles and the amount of T -cells in different time frame of the treatment was observed for each of these objective functions. The basic reproduction number of the model (state equations) were calculated. In this paper, the Pontryangin Maximum Principle is used to achieve the optimal dosage of drug administered according to each of these objective functions based on the basic model of latent cell activation. The forward backward sweep method is used to numerically solve the optimality system, i.e., forward in time for the state equations and backward in time for the adjoint equations using the initial conditions and final conditions(transversality conditions) respectively.

Alex Dimitrov
Title: A vector inhomogeneous Poisson process with stochastic gain as a model of multiunit neural activity

Poisson processes have been used as approximate models of neural spiking activity for a long time. The two main difficulties with this model has been estimating the time-varying rate from neural observations, and observations showing variance inconsistent with the Poisson assumption (typically higher than the mean, vs to equal according to Poisson).

A recent enhancement of the model proposes that the Poisson process has a rate that is modulated by another random variable, reflecting large scale brain state: m(t)=g*f(t) with g ~ p(g) termed a gain random variable [Goris et.al ‘14]. The resultant doubly-stochastic process is essentially a continuous mixture distribution of Poissons weighted by the distribution p(g).

A second enhancement, proposed by us, focuses the estimation on the cumulative count process rather than on the differential point process. That allowed for the cumulative raw rate function F(t) to be approximated well by a low-parameter estimator, which included non-uniform time sampling. It also provided the differential rate function as its derivative,  f(t) = F'(t). It also allows for a natural extension to model the simultaneous activity of multiple neural units, as a vector Poisson process. The main drawback of this approach is that the cumulative Poisson process is heteroschedastic in time, albeit with a known variance function.

This modification demonstrated remarkable explanatory power for a subset of neurons from the ferret primary auditory cortex. I will show some results from applications to neural data, and directions of current and future research. I am particularly interested in hearing about better estimators of the model process parameters.

This is joint work with Stephen David and Zachary Schwartz (OHSU, Portland).

Yuan Wang
Title: Statistical classification for cancer diagnosis

Non-invasive biomedical imaging technologies have been widely used in medical disciplines for interrogation of the pathophysiology and pathogenesis of diseases to access disease features repeatedly over time and often at multiple spatially interdependent units. We are interested in a liver cancer study with the objective of determining the effectiveness of using CT perfusion characteristics to identify and discriminate between regions of liver that contain malignant tissues from normal liver tissue.

This is a classical classification problem where we want to identify which category/class a new observation belongs, on the basis of a training set of data with known classes. However, the CT imaging data provides new challenges because of its increasing size and structure complexity. To reduce model complexity and simplify the resulting inference, possible spatial correlation is often neglected. In this talk, I will introduce several classification approaches that are designed for implementing various correlation structure. The method offers maximal relative improvement in the presence of temporal sparsity wherein measurements are obtainable at only a few time points.

Zhiwu Zhang
Title: Upgrade the Compartments of Mixed Linear Model to Reduce Both False Positives and False Negatives in Gene Mapping

Reporting false discoveries not only damages academic reputation, but also wastes resources by misleading follow-up studies. Population structure and kinship among individuals are two common factors that cause false positives in genome-wide association studies (GWAS). The Mixed Linear Model (MLM) is an effective method to eliminate false positives. MLM fits both testing genetic marker and population structure as fixed effects and incorporates kinship to define the variance and covariance structures of the random individual genetic effect. Unfortunately, the confounding of testing genetic marker with population structure and kinship makes the statistical test on the genetic marker less powerful. Even when the genetic marker is a functional polymorphism, its effect is diluted by the covariate population structure and the kinship. The confounding effect is more problematic on traits that are associated with population structure such as fitness in natural populations or economic traits in cultivars. This presentation describes the principles and practices for developing new MLM methods to simultaneously reduce false positives and retain high statistical power.

Robert Dillon
Title: Lagrangian Mesh Modeling Viscoelasticity

Abstract: We describe an immersed boundary Lagrangian mesh method for modeling complex fluids where the fluid viscoelasticity is represented by a discrete network of Maxwell elements. The rheological properties of the Lagrangian mesh fluid are compared with an Oldroyd-B pde model for complex fluids. We show simulation results from immersed boundary models for peristalsis and sperm motility in Lagrangian mesh and Oldroyd-B fluids.

Xueying Wang
Title: Mathematical models for the Trojan Y-Chromosome eradication strategy of an invasive species

Abstract: The Trojan Y-Chromosome (TYC) strategy, a genetic biocontrol method, has been proposed to eliminate invasive species by introducing sex-reversed trojan females. Because constant introduction of the trojans for all time is not possible in practice, there arises the question: What happens if this injection is stopped after some time? Can the invasive species recover? To answer that question, we study this strategy through deterministic and stochastic models. Our results show that: (1) with the inclusion of an Allee effect, the number of the invasive females is not required to be very low when this injection is stopped, and the remaining trojan population is sufficient to induce extinction of the invasive females; (2) incorporating diffusive spatial spread does not produce a Turing instability, which would have suggested that the TYC eradication strategy might be only partially effective; (3) the probability distribution and expectation of the extinction time of invasive females are heavily shaped by the initial conditions and the model parameters; (4) elevating the constant number of the trojan females being introduced into the population will lead to a decrease in the expected extinction time for wild-type females, as opposed to an increase in the extinction probability within an application time.

Jie Zhao
Title:: Mathematical Modeling and Parameter Estimation of Dynamical Cell Signaling Pathways in Fibroblasts

Abstract: In many types of tumors, stromal fibroblasts become ‘activated’ and express a number of contractile proteins, particularly α-smooth muscle actin (SMA), known as myofibroblasts. The differentiation of fibroblast to myofibroblast may be induced by TGFβ-SMAD and SDF-1-CXCR4 pathways, producing an activated, myofibroblast-rich stromal microenvironment through secretion of various cytokines and growth factors. The myofibroblast-rich stromal microenvironment of different human cancers is associated with an increased risk of invasion and metastasis and a poor clinical prognosis. We developed an ODE model of the differentiation of fibroblast to myofibroblasts. We adapted  a Bayesian inference and MCMC method for ODE systems to estimate model parameters values by using experimental data from a set of cell culture experiments.  We show that the ODE model gives qualitative agreement with experimental results over a wide range of initial TGFβ initial conditions.


Fall 2015

Richard Gomulkiewicz

: An empirically based mathematical model for the potential role of masting by introduced bamboos in North American deer mice population irruptions

Abstract: The ongoing naturalization of frost/shade tolerant Asian bamboos in North America may cause adverse environmental consequences involving introduced bamboos, native rodents and ultimately humans. A specific concern is that the eventual masting by an abundant bamboo within Pacific Northwest coniferous forests could produce a temporary glut of food capable of driving a population irruption of a common native seed predator, the deer mouse (Peromyscus maniculatus), a hantavirus carrier. To address this concern, we conducted feeding trials for deer mice with bamboo and native seeds. Adult deer mice consumed bamboo seeds as readily as they consumed native seeds, and females produced a median litter of 4 pups on a bamboo diet. We used our empirical results to parameterize a modified Rosenzweig-MacArthur consumer-resource mathematical model to project the population-level response of deer mice to a hypothetical pulsed supply of bamboo seeds. The qualitative dynamics of the model, a system of nonlinear ordinary differential equations, predicts rodent population irruptions and declines similar to reported cycles involving Asian and South American rodents but unprecedented in North American deer mice. This is joint work with Melissa Smith and Richard Mack.

Mohammed Kaabar

: Finding the Best Classification Rates for Arabic Sign Language Data using Data Analysis Methods

Abstract: In this talk, we identify several types of methods for finding the classification rates for Arabic sign language data (training and testing data), and these data (feature vectors) are taken and extracted from a published paper: (Recognition of Arabic Sign Language Alphabet using Polynomial Classifiers, Khaled Assaleh and M. AlRousan, EURASIP JASP 2005:13 (2005) 2136-2145. DOI: 10.1155/ASP.2005.2136)). These data represent images that were collected from 30 deaf participants who had to wear colored gloves and then perform their own Arabic sign gestures. We are only using 10 letters (classes) out of the 30 alphabets in Arabic sign language that can be performed in 42 gestures. Firstly, we begin with visualizing the three classes of data in two-dimensional plot using the Principle Component Analysis (PCA). Moreover, we start using linear classifier to generate linear discriminant functions using the pseudo inverse method in order to find the classification rates for both data types: training and testing data. Then, we classify data using neural networks (NN) model, and we implement the k-means algorithm to find the classification rates for our data. Finally, we compare the different types of methods with each other to find the best method for achieving excellent classification rates among the other methods.

Andew Oster

: Laminar Development of the Primary Visual Cortex

Abstract: In this talk, we will introduce the architecture of the visual system in higher order primates and cats. Through activity-dependent plasticity mechanisms, the left and right eye streams segregate in the cortex in a stripe-like manner, resulting in a pattern called an ocular dominance map. We introduce a mathematical model to study how such a neural wiring pattern emerges and extend it to consider the joint development of the ocular dominance map with another feature of the visual system, the cytochrome oxidase (CO) blobs, which appear in the center of the ocular dominance stripes. Since cortex is in fact comprised of layers, we introduce a simple laminar model and perform a stability analysis of the wiring pattern. This intricate biological structure (ocular dominance stripes with 'blobs' periodically distributed in their centers) can be understood as occurring due to two Turing instabilities combined with the first-order dynamics of the system. We show recent numerical simulations showing how monocular deprivation during development can dramatically alter the ocular dominance pattern, while leaving the CO blob distribution nearly unaltered.

Steve Krone

: Directed evolution of phage lysins: using mathematical models to explore feasibility/design of new antibacterial drugs

Abstract: Motivated by a mounting tide of drug resistant bacteria, the search for new antibacterial agents is embracing technologies that lie outside traditional bounds. One promising source of compounds is the lysins encoded by bacteriophages (viruses that infect and kill bacteria). These enzymes degrade the bacterial cell wall from the inside, leading to rupture of the cell and subsequent dispersal of phage progeny. Lysins have evolved to not kill cells from the outside, thus preserving future hosts, but recent lab work has shown that they can be engineered to kill from without. Developing lysins that have desirable properties for therapeutic use is complicated by the fact that the molecular basis of improvement is not yet understood. A possible way forward is provided by directed evolution. Here we propose lab protocols that involve the co-culturing of two bacterial species–one producing a toxin/lysin that kills the other, leading to selective pressure for improved function of the toxin. We use mathematical models and simulations to explore the feasibility of this directed evolution and offer insights into various protocols.

Mark Schumaker

: Improved Models of Equine Infectious Anemia

Abstract: We present an improved model of equine infectious anemia.  The compartmental model introduced by Schwartz et al. (2013) is modified to take into account the saturating effects of viral stimulation of macrophage infection and cytotoxic T-lymphocyte production.  The formula for the basic reproduction number of the infection is not changed by the new terms in the model, but these do expand the domain of one form of infected steady state (the  boundary equilibrium) and change the character of a second form of infected steady state (the endemic equilibrium).  Computation of bifurcation diagrams provides a simple way to look at the effects of the new terms.  The Matcont package for Matlab computes both bifurcation diagrams and eigenvalues, the latter providing important information about the stability of steady states associated with the EIAV infection. This is joint work with Elissa Schwartz.

Robert Dillon

: Mathematical modeling of sperm and cilia motility

Abstract: The motility of sperm flagella and cilia are based on a common physiological structure capable of generating a wide range of dynamical behavior.  We describe a fluid-mechanical model for sperm and cilia coupling the internal force generation of dynein molecular motors through the passive elastic axonemal structure with the external fluid mechanics. As shown in numerical simulations for motile sperm, the model's flagellar waveform depends strongly on viscosity as well as dynein strength.

Haijun Li

: Stochastic Ordering of Epidemics

Abstract: A lesser-known method,  known as stochastic comparison, has been developed for analyzing infectious epidemics. The method is powerful since it removes restrictive  model assumptions commonly used in modeling epidemics processes. The trade-off, however, is that such a comparison approach often leads to extracting qualitative structural features, rather than quantitative information.

In this talk, I will discuss fundamental ideas of stochastic orders (a deep theory in its own right) in the context of analysis of classical epidemic models, for which strong model assumptions are significantly relaxed.

David Wollkind

: The Behavior of a Population Interaction-Diffusion Equation in its Subcritical Regime

Abstract: A model interaction-diffusion equation for population density originally analyzed by Wollkind et al. (1994, SIAM Review 36, pp. 176-214) through terms of third-order in its supercritical parameter range is extended through terms of fifth-order to examine the behavior in its subcritical regime. It is shown that under the proper conditions the two subcritical cases behave in exactly the same manner as the two supercritical ones unlike the outcome for the truncated system. Further there also exists a region of metastability allowing for the possibility of population outbreaks. These results are then used to offer an explanation for the occurrence of isolated vegetative patches and sparse homogeneous distributions in the relevant ecological parameter range where there is subcriticality for a plant-ground water model system as opposed to periodic patterns and dense homogeneous distributions occurring in its supercritical regime. This is joint work with Mitchell G. Davis.

Audrey Fu

: Inferring the cell differentiation trajectory from single-cell gene expression data

Abstract: Cells differentiate at different speeds: even when collected at the same time point, cells may be at different stages of the differentiation process, exhibiting different gene expression profiles.  Additionally, single cells are often a mixture of multiple subtypes, sometimes including previous unknown subtypes and subtypes that were sampled due to imperfection of the experimental procedure.  We aim to infer the cell differentiation trajectory from single-cell gene expression data, while accounting for potentially multiple paths and possibly unknown and unintended subtypes.  We cluster cells by their gene expression profiles into `super cells’ and infer a causal graph among the super cells.  The causal graph corresponds to the differentiation process at a coarse level.  Individual cells are placed along this process depending on their distance from the nearest super cells.  I will illustrate this idea with examples and explain the current development in methods and algorithms for this inference.

Kazuo Yamazaki

: Introduction to differential equations perturbed by random noise

Abstract: This will be a introductory talk on differential equations with a forcing term that is random, in contrast to the classical case. One needs only knowledge from elementary analysis (e.g. Riemann-Stieltjes integral) to understand the talk content. I will discuss the motivation to add the ``noise'' to a differential equation that models natural phenomenon, including the differential equations in biology, how to make sense of the ``noise'' mathematically, basic stochastic calculus, and some of my naive (failed) attempts to prove the well-posedness of differential equations in biology with noise.

Yuan Wang

: Statistical Analysis of Complex Data Objects

Abstract: Non-invasive medical imaging tools such as Magnetic Resonance Imaging (MRI), Computed Tomography (CT), and ultrasound (US) have provided significant assistance to disease diagnosis and treatment monitoring. In the meantime, they also bring exciting challenges in statistical analysis since the data collected by those imaging tools are not only increasing in size, but also in its complexity. In this talk, I will introduce several complex data objects including tree-structured data, functional data, and imaging data.  These complex data objects often lie in non-Euclidean space and traditional statistical tasks such as regression, classification and hypothesis testing become rather difficult. I will talk about the most current techniques for handling these complex objects and the related applications. An important lesson is that analysis in the space of data objects can reveal much deeper scientific insights than the simple analysis of summary statistics.

Elissa Schwartz

: Using compartmental and agent-based modeling to understand the 2009 H1N1 influenza outbreak in Pullman, WA

Abstract: Knowledge of mechanisms of infection in vulnerable populations is needed in order to prepare for future outbreaks. Here, using compartmental and agent-based modeling with a unique dataset collected during the 2009 outbreak of influenza A (H1N1)pdm09 in Pullman, we studied H1N1 infection dynamics in a rural university environment.  Specifically, we evaluated mechanisms of infection, we estimated infection parameters, and we predicted the number of symptomatic individuals that would have resulted given a variety of plausible scenarios.  Our findings are relevant for future influenza epidemics in similar settings.

Svetlana Lockwood

: Application of Topology to Data Analysis

Abstract: In the era of Big Data scientists face the challenging task of the analysis of data. Ideally, they would like to extract some qualitative signal from their data in the hope to better understand the underlying biological processes. In recent years, tools from persistent homology - a subfield of algebraic topology - have been successfully applied in a number of applications. The general idea is that geometric structure found in data may inform us about biological functionality. The key benefits of using topological features to describe data include coordinate-free description of shape, robustness in the presence of noise and invariance under many transformations, as well as highly compressed representations of structures. In this presentation I will review the fundamental tenets of persistent homology and its application to problems from natural sciences as well as available open source computational tools.

Spring 2015

Alex Dimitrov

: Invariant signal processing in auditory biological systems

Abstract: The sense of hearing is an elaborate perceptual process. Sounds reaching our ears vary in multiple features: pitch, intensity, rate. Yet when we parse speech, our comprehension is little affected by the vast variety of ways in which a single phrase can be uttered. This amazing ability to extract relevant information from wildly varying sensory signals is also ubiquitous in other sensory modalities, and is by no means restricted only to human speech. Even though the effect itself is well characterized, we do not understand the approaches used by different neural systems to achieve such performance.
In an ongoing project, we are testing the hypothesis that broadly invariant signal processing is achieved through various combinations of locally invariant elements. The main questions we would like to address are: 1. What are the characteristics of locally-invariant units in auditory pathways? 2. How are biological locally-invariant units combined to form globally invariant processors? 3. What are the appropriate mathematical structures with which to address and model these sensory processes?

The mathematical aspects of the research involve an interesting combination of probability theory (a must in the study of biological sensory systems) and group theory, needed to characterize invariants and symmetries. The combination defines the concepts of a probabilistic symmetry, and expands the scope of probabilities on group structures, originally introduced by Grenander.

Xueying Wang

Mathematical Modeling of Cholera Epidemics

Abstract: Cholera is a severe water-borne infectious disease caused by a bacterium Vibrio cholerae. This disease strikes hardest in underdeveloped regions that lack proper hygiene and sanitation and have limited access to clean water and other resources. Today, cholera remains a major public health threat.  In this talk,  I will present some recent work in mathematical modeling of cholera. First,  a brief introduction to infectious disease modeling will be given. Secondly, we will focus on some recent developments in cholera modeling. Particularly, we will talk about endemic stability, spatial spread, cholera traveling waves and disease threshold dynamics by using ODE and PDE models.

James Kehinde

Analysis of the Transmission of Malaria Parasite

Abstract: In this work, we developed a mathematical model of malaria transmission using ordinary differential equations for the spread of malaria in human and mosquito populations. Particularly, the complex disease transmission pathways are modeled by incorporating not only the recovered humans but also the infected persons return to the susceptible class. Through a rigorous equilibrium analysis, we found that the model can exhibit three equilibria: disease-free equilibrium, mosquito-free equilibrium and endemic equilibrium. The basic reproduction number, R0, is calculated using next generation matrix method. With the derived R0, we analyzed the local dynamics and the disease threshold of malaria infection. Specifically, local stability of the endemic equilibrium is verified using center manifold theory. The result shows that the disease invades the population if R0 is greater than unity and dies out if R0 is less than unity. Additionally, numerical simulation is carried out on the set of values complied for areas of low and high transmission to explore possible behaviors of the model using the baseline parameter values.

Nairanjana Dasgupta

A Look at Multiplicity through Misclassification

Abstract: Multiplicity in large scale studies using, for example, microarray genomic data and functional neuroimaging data (fMRI), has been an intensively researched topic in recent years. One option often used by researchers is a “top r-table”, which involves ranking the hypotheses  in  some  order  (p-values  or  test  statistics)  and  reporting  the  top  r  results.   This has immediate practical applications as what we have is a list of “interesting” results that are worth following up, irrespective of the actual p-value (adjusted or not).  In this manuscript we take another look at multiplicity using top tables.  Our approach is intended to be a compromise between theory and practice.  We look at the relationship between the probability of correct classification, which we call r-power (the units picked in the top-r table do indeed come from the alternative), and the value of r.  We analytically define r-power in terms of order statistics and quantify the probability of correct classification.  We use numerical integration to calculate r-power as a function of effect size, δ; the number of hypotheses tested, N;  the  number  of  hypotheses  coming  from  the  null,  k;  and  r.   We show that r-power is positively related to effect size, and negatively related to k/N.  The relationship to r depends upon whether r < k.  There are two possible uses of our results:  based on a pre-chosen r-power we can calculate r and decide on the number of hypotheses to be followed up or if r is calculated using some other criterion we can use our method to calculate r-power in that context.  We illustrate these ideas using examples from microarrays and functional magnetic resonance imaging data.

Co-authored by:
Nicole A. Lazar Department of Statistics University of Georgia Athens, GA, 30602
Alan Genz Department of Mathematics Washington State University, Pullman, WA, 99163

Nikolay Strigul

New modeling tools for scaling forest dynamics within the hierarchical patch-dynamics framework

Abstract: The forested ecosystem is a complex adaptive system having a complicated hierarchical structure. In this presentation the framework of complex adaptive systems is employed to understand and predict how natural and anthropogenic disturbances occurring at different scales propagate through the forested ecosystems and affect forest structure and dynamics. The original Matreshka modeling framework considers vegetation dynamics as the results of vegetation processes at several hierarchical scales driven by natural and anthropogenic disturbances of different magnitude. The particular processes include growth of individual trees, dynamics of trees within the stand, forest stand mosaic, and changes of the collection of forest stands of different forest types at the landscape level. Recently we have developed models including the Crown Plastic SORTIE, LES, and PPA to address the scaling of vegetation dynamics from the individual to the stand level. All these models employ individual tree plasticity as a crucial factor for forest self-organization. We have also developed a Markov chain model of forest stand dynamics. To parameterize and validate models we have broadly employed data of the USDA Forest Inventory and Analysis Program (FIA data) and of the Quebec forest inventory program. We have also developed an original remote sensing technique to parameterize the spatial component of our individual based models. These new modeling tools will be useful for understanding how climatic changes and different forest management practices can affect forest dynamics and carbon footprint.

Guy Palmer

Diagnostic capacity and vaccine response in space and time

Abstract:  In this presentation, I will talk about diagnostic capacity and vaccine response related to risk by using rabies in Tanzania as an example. Ebola virus disease will be mentioned at the end.

David Wollkind

Vegetative Rhombic Pattern Formation Driven by Root Suction for an Interaction-Diffusion Plant-Ground Water Model System in an Arid Flat Environment

Abstract: A rhombic planform nonlinear cross-diffusive instability analysis is applied to a particular interaction-diffusion plant-ground water model system in an arid flat environment. This model contains a plant root suction effect as a cross-diffusion term in the ground water equation. In addition a threshold-dependent paradigm that differs from the usually employed implicit zero-threshold methodology is introduced to interpret stable rhombic patterns. These patterns are driven by root suction since the plant equation does not yield the required positive feedback necessary for the generation of standard Turing-type self-diffusive instabilities. The results of that analysis can be represented by plots in a root suction coefficient versus rainfall rate dimensionless parameter space. From those plots regions corresponding to bare ground and vegetative patterns consisting of isolated patches, rhombic arrays of pseudo spots or gaps separated by an intermediate rectangular state, and homogeneous distributions from low to high density may be identified in this parameter space. Then, a morphological sequence of stable vegetative states is produced upon traversing an experimentally-determined root suction characteristic curve as a function of rainfall through these regions. Finally, that predicted sequence along a rainfall gradient is compared with observational evidence relevant to the occurrence of leopard bush, pearled bush, or labyrinthine tiger bush vegetative patterns, used to motivate an aridity classification scheme, and placed in the context of some recent biological nonlinear pattern formation studies.

Co-authored by: Inthira Chaiya*,  Richard A. Cangelosi$,   Bonni J. Kealy-Dichone$,   Chontita Rattanakul*
* Department of Mathematics, Faculty of Science, Mahidol University, Rama 6 Road, Bangkok 10400, Thailand. Inthira Chaiya is supported by the Royal Golden Jubilee Ph.D. Program (PHD/0191/2553).
$ Department of Mathematics, Gonzaga University, 502 E. Boone Avenue MSC 2615, Spokane, WA 99258, USA

Krishna Athreya

Introduction to Galton Watson Processes and Coalescence Problems

Abstract: We introduce Galton watson branching processes. We review basic limit theorems. Then we talk about coalesence problems for such processes. We give some basic results and some applications. We outline some interesting open problems.

Marc Evans

  Maximum Likelihood Estimation of Species Richness via Generalized Linear Mixed Models

Abstract: Understanding biological diversity in plant and animal communities is fundamental to understanding community structure and health. Ecologists view species diversity as a function of two components: evenness and richness. Evenness refers to the uniformity of abundance across all species in a region, while richness refers to the total number of unique species present.

Estimators of species richness abound in the statistical literature, including one or two by the likes of Bradley Efron (1975, with R. Thisted) and R. A. Fisher (1943, with A. S. Corbet and C. B. Williams). The more commonly used estimators, with simplifying assumptions, include Good’s (1953) estimator, Darroch’s (1958) estimator where he assumed a zero truncated Poisson model with constant rate parameter, the jackknife estimator of Burnham and Overton (1978) and the nonparametric coverage estimators of Chao (1984) and Chao and Lee (1990). In addition, there are approaches using rarefaction, but these will not be discussed in the talk.

Suppose a sample of n individuals produces Yi individuals from the ith species (i = 1, 2, ..., R), where R represents the total number of species present in the sampled region or sampling frame. It is commonly assumed that counts of this nature follow a Poisson distribution with parameters λi (i = 1, 2, ..., R). However, the ith species appears in the sample if and only if Yi > 0. Thus, the complete set of species is unobservable, and so R is unknown and must be estimated. Under these conditions, the distribution of the observed counts follows a zero truncated Poisson with parameters
λi (i = 1, 2, ..., r), where r (≤ R) represents the total number of observed species. Models for estimating species richness are generally based on simplifying assumptions (e.g., λi=λ for all i as in the case of Darroch’s estimator). Community ecologists have long known that populations are generally composed of a few species with very high abundances as well as a relatively large number of species with very few individuals. This heterogeneity in abundances requires a more complex model structure (e.g., λiλ), but the added complexity has made computation of model parameters difficult.

In this talk I will propose an estimator of species richness based on a generalized linear mixed model (GLMM) framework in which the Yi (i = 1, 2, ..., R) are assumed to follow a Poisson distribution with parameters
λi arising from a distribution with unknown parameters. The estimators based on the use of both continuous and discrete mixing distributions will be discussed. The results of a Monte Carlo simulation comparing the most commonly used estimators of species richness and the GLMM estimators will be presented.

Kazuo Yamazaki

Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model

Abstract: We study the initial boundary value problem of the SIRS-B PDE model for cholera dynamics with advection and diffusion. We obtain the local well-posedness result by relying on the theory of cooperative dynamics system. Via a priori estimates making use of the special structure of the system and continuation of local theory argument, we show that in fact it is globally well-posed. Finally, we show the local asymptotic stability of the solutions considering different values of the basic reproduction number using some spectral analysis. This is a collaboration work with Xueying Wang.

Elissa Schwartz

Identifying the Conditions under which Antibodies Protect Against Infection by Equine Infectious Anemia Virus

Abstract: The ability to predict the conditions under which antibodies protect against viral infection would transform our approach to vaccine development. A more complete understanding is needed of antibody protection against lentivirus infection, as well as the role of mutation in resistance to an antibody vaccine. Recently, an example of antibody-mediated vaccine protection has been shown via passive transfer of neutralizing antibodies before equine infectious anemia virus (EIAV) infection of horses with severe combined immunodeficiency (SCID). Viral dynamic modeling of antibody protection from EIAV infection in SCID horses may lead to insights into the mechanisms of control of infection by antibody vaccination. In this work, such a model is constructed in conjunction with data from EIAV infection of SCID horses to gain insights into multiple strain competition in the presence of antibody control. Conditions are determined under which wild-type infection is eradicated with the antibody vaccine. In addition, a three-strain competition model is considered in which a second mutant strain may coexist with the first mutant strain. The conditions that permit viral escape by the mutant strains are determined, as are the effects of variation in the model parameters. This work extends the current understanding of competition and antibody control in lentiviral infection, which may provide insights into the development of vaccines that stimulate the immune system to control infection effectively.

Hannah Callender

Mathematical Modeling of Integrin Dynamics in Cell Motility: From Stochasticity to Sensitivity

Abstract: A cell’s ability to move to the correct location at the correct time is vital for maintenance of homeostasis; improper movement is often indicative of a pathogenic phenotype. As such, it is critical to understand the molecular phenomena of motility. A key step in the process of cell motility is the development of focal adhesions, which are protein complexes involving cytoskeletal elements, membrane bound proteins, and extracellular matrix components. A fundamental component of a focal adhesion is the transmembrane receptor protein, integrin, that links the actin cytoskeleton to extracellular matrix proteins. Here we develop and analyze a stochastic model of a nascent focal adhesion. The model captures the dynamics of the rate reactions over time between extracellular ligand molecules, intracellular adhesion proteins called talin, and integrins. To better inform our model conclusions, we discuss results from a variety of sensitivity analysis techniques for both deterministic and stochastic models.

Jie Zhao

Mathematical Modeling of the Dynamics of the TGF-beta and SDf-1 Signaling Pathways

Abstract: An important aspect of mathematical systems biology is modeling the dynamics of biochemical networks where molecules are the nodes and the molecular interactions are the edges. It is an extremely complex system if you are trying to include every single related component. Sometimes lots of substrates influence distant enzymes in the network.  This can make the network more complicated.  Including  everything that researchers in biology believe to be important can lead to systems of hundreds or thousands of ODEs with many unknown parameter values. In this talk, I will introduce a reduced mathematical model that incorporates the cross-talk between breast tumor cells and their environments and predicts how the process affects the tumor progress. We focus on two autocrine signaling loops mediated by TGF-beta and SDF-1 and show how the components in these pathways regulate the differentiation of fibroblasts into myofibroblasts and affect cell growth and proliferation.