Titles and Abstracts for Workshop on Mathematical Models in Biology and Medicine

Invited talks

Linda J. S. Allen, Department of Mathematics and Statistics,Texas Tech University

Deterministic and Stochastic Models for an Emerging Disease: Hantavirus

Hantaviruses are rodent-borne zoonotic agents that cause hantavirus pulmonary syndrome and hemorrhagic fever with renal syndrome in humans. Severe changes in environmental conditions have been linked to outbreaks of hantavirus in rodent and human populations. We formulate hantavirus models for wild rodents based on a male-female SEIR epidemic model. In the models, the environmental carrying capacity has two sources of variation, seasonal and random. Deterministic and stochastic SEIR epidemic models are formulated and numerically simulated. The computational simulations using the models produce realistic dynamics of hantavirus outbreaks in rodent populations.

H. T. Banks, Center for Research in Scientific Computation, North Carolina State U.

Estimation and Prediction with HIV Treatment Interruption Data

We consider longitudinal clinical data for HIV patients undergoing treatment interruptions. We use a nonlinear dynamical mathematical model in attempts to fit individual patient data. A statistically-based censored data method is combined with inverse problem techniques to estimate dynamic parameters. The predictive capabilities of this approach are demonstrated by comparing simulations based on estimation of parameters using only half of the longitudinal observations to the full longitudinal data sets.

Luis M. A. Bettencourt, Los Alamos National Laboratory.

The pace of life in the city: urban population size dependence of the dynamics of disease, crime, wealth and creativity

The historic trend towards concentration of people in urban centers, both in relative terms and in absolute numbers, has just crossed an important threshold, with more than half of the world’s population now living in cities. Yet a quantitative understanding of the dynamics that make cities the central engines of social, economic and intellectual development, as well as the generators of much pollution, crime and disease has remained illusory. Here we give the first integrated and quantitative picture for the general character of urban dynamics, across several nations and time. We shown that wealth creation, intellectual innovation, disease spread and many other important urban properties, are self-similar scaling functions of a city’s size, characterized by exponents greater than unity. The existence of scaling laws for cities points to general principles of social organization common to all urban systems. This perspective leads to the quantification of the analogy between cities and biological organisms. We show that the analogues of metabolic rates and behavioral times in a city scale with its size in a manner that runs opposite to biological organization: cities increase their resource input and informational output per capita as they grow, while behavioral timescales speed up. In turn, these properties have dramatic implications for growth and development, as the increasing returns to scale that bind urban social systems together also create the acceleration of pace that, left unchecked, sows the seeds for their collapse. These general patterns of social dynamics have particular imprints for epidemiology, which we discuss.

Zhilan Feng, Department of Mathematics, Purdue University.
Influence of anti-viral drug treatments on evolution of HIV-1 pathogen
An age-structured model is used to study the possible impact of drug treatment of infections with the human immunodeficiency virus type 1 (HIV-1) on evolution of the pathogen. Different types of drugs (e.g., reverse transcriptase inhibitors, protease inhibitors and entry inhibitors) help to reduce the HIV replication at different stages of the cell infection, and the use of an age structure allows us to more realistically model the effect of these drugs. Inappropriate drug therapy offen leads to the development of drug-resistant mutants of the virus. Previous studies have shown that natural selection within a host favors viruses that maximize their fitness. By demonstrating how drug therapy may influence the within host viral fitness we show that while a higher treatment efficacy reduces the fitness of the drug-sensitive virus, it may provide a stronger selection
force for drug-resistant viruses which allows for a wider range of resistant strains to invade.
 

Mary Ann Horn, Mathematics Department, Vanderbilt University, and NSF.

Mathematical modeling and challenges in the development of drug resistance

Drug resistance has been an emerging problem since the discovery of penicillin. Although penicillin was one of the first antibiotics, resistance was observed in bacteria not long after it was put into public use. Since then, numerous other antibiotics have been developed to combat the effects of resistance, however, bacterial resistance can be seen even in drugs developed quite recently.

Mathematical modelling is a tool that is becoming more important to clinicians and researchers in their study of the effects of the spread of resistance. Systems of differential equations can be used to model many aspects of the issue including the spread of resistant bacteria throughout a closed facility such as a clinic, the evolution and domination of resistant strains of bacteria as a patient is repeatedly treated with antibiotics and the underlying genetic mechanisms that occur during the development of resistance.

This talk will give an overview of some of our recent work on modeling of the development and spread of antibiotic resistance. (Joint work with Erika M. C. D'Agata and Glenn F. Webb.)

Mary Ann Horn, Mathematics Department, Vanderbilt University, and NSF.

Mathematical biology and NSF

Bingtuan Li, Department of Mathematics , University of Louisville.

Spreading speeds and traveling waves in cooperative models

In this talk, we shall discuss spreading speeds and traveling waves in cooperative models that describe the growth, interactions, and migrations of multiple species. The models are formulated in the form of reaction-diffusion equations or integro-difference equations. In a multi-species model, different species may spread at different speeds. We discuss conditions that ensure all the species spread at the same speed, and the speed is that of the system linearized about the leading edge of invasion. We show that the slowest spreading speed can be always characterized as the slowest speed of a suitable class of traveling wave solutions. Applications of the general results to some two-species competition models are presented. (Joint work with Hans F. Weinberger and Mark A. Lewis.)

Irakli Loladze, Department of Mathematics University of Nebraska - Lincoln.

Patterns in molecules and oceans: linking cellular machinery to global N:P ratios

Redfield ratios are one of the largest-scale patterns found in the Biosphere. First identified by Harvard oceanographer Alfred Redfield in 1930s, the pattern refers to the remarkable similarity of carbon(C): Nitrogen (N) : Phosphorus (P) ratios in open ocean and phytoplankton. In particular, Redfield ratios state that for every P atom there are about 16 N atoms in both oceanic water and phytoplankton. Due to the importance of Redfield ratio to biogeochemical cycles, carbon balance and, hence, global climate, numerous attempts have been made to explain it. Here, we derive Redfield ratios by linking molecular processes on cell level with competition among species and global nutrient feedbacks. First, we show that N:P=16 can stem from fundamental molecular constants such as N content in amino acids, and N and P content in nucleotides to manifest itself in biochemically optimal RNA:Protein ratio. Next, we incorporate this biochemical optimum into an ODE model of competition between phytoplankton species to show how the pattern found on molecular scale can propagate itself to global Redfield ratio. The necessary condition for this to happen is nutrient feedbacks that indeed exist in oceans.

Hal Smith and Rosie Renaut, Department of Mathematics and Statistics, ASU

Mathematical and computational biology programs at ASU

The Department of Mathematics and Statistics at ASU now has over 160 graduate students, more than 80% of them are US citizens, 77 of them are in PhD programs, and more than 10% are US citizens from under-represented groups. We have 41 graduate students in mathematical biology and nonlinear dynamics, 25 of them are PhD students. Our mathematical and computational biology research group, consisting of several collaborating subgroups in mathematical biology and dynamical systems, computational mathematics, and statistics and stochastic processes, has expertise in the following related areas:(i) mathematical physiology and medicine including modeling within-host spread of diseases like HIV, cancer and diabetes through nonlinear ordinary and partial differential equations, (ii) neural physiology- modeling of excitatory processes (be they neural dynamics or heart tissue) using networks of nonlinear and possibly chaotic oscillators and nonlinear partial differential equations,(iii) modeling gene regulatory networks via systems of nonlinear differential equations, (iv) evolution and spread of infectious diseases through nonlinear differential equations, transport equations, and agent based simulations, (v) self-organized behavior in biological models for instance in social insects but also in networks of protein machines using nonlinear differential equations, agent based simulations, and discrete event simulations, (vi) medical imaging related to pattern recognition and registration problems-using optimization techniques, statistical methods, numerical analysis and image processing, (vii) biostatistics and stochastic processes with applications to medical imaging and genetic trees. Our research group also has a strong interdisciplinary focus being well connected with local industry, hospitals, and research labs at ASU.

The Department of Mathematics and Statistics at Arizona State University took the lead in the development of an interdisciplinary program in Computational Biosciences. The masters level program, joint with Life Sciences, Computer Science and Business, was implemented in Fall 2002. To date more than 30 students have successfully completed this 42 hour program, and taken jobs in diverse areas, such as Mathworks, Disney, St Jude's Children's Hospital, Arizona Game and Fish, Tgen.. This academic year we admitted the first students to the doctoral Computational Biosciences degree, which is again a collaboration with Life Sciences.  A third of our graduates have continued to doctoral programs. I will provide a brief overview of these programs.

Glenn Webb,  Mathematics Department, Vanderbilt University

Mathematical Models of Prion Proliferation

Prions are infectious proteins that are hypothesized to be the causative agent of diseases such as Creutzfeld-Jacob disease in humams, scrapie in sheep, and bovine spongiform encephalopathies in cows (mad cow disease). This hypothesis is controversial, because prion populations are capable of proliferation even though prions do not contain DNA or RNA. A mathematical model is analyzed to explain prion proliferation. Them model consists of a system of nonlinear ordinary and partial differential equations. An analysis is given of the model, and model simulations are compared to experimental data.

Graduate students and postdoctoral fellows’ talks

Lydia Bilinsky

Not making matters worse: strategies to minimize the evolution of more dangerous cancers in chemotherapy

Traditional chemotherapy drugs work by inflicting damage on the DNA of cancer cells, in the hope that such damage will result in the cells undergoing apoptosis. This does not succeed in killing all cancer cells because (1) some cells will repair the damage, (2) of the cells that don't, a few will escape apoptosis, no treatment being 100% effective. Scenario (2) is dangerous, because the damage may have caused mutations in the cell's DNA which make it a more aggressive cancer. We explored the competition dynamics between two populations of cancer cells that differ in their ability to repair themselves; those with the greater probability of repairing damage were termed “stable” and those with the lesser probability were termed “mutator”. The mutator class is more likely to undergo scenario (2) above, and so the optimal treatment protocol is the one which minimizes both the total number of cancer cells as well as the fraction belonging to the mutator class. Our delay differential equation (DDE) model exhibits the simple dynamics reminiscent of a tradition ODE competition model, but captures the effect of cellular repair time on the replication rate, which an ODE model cannot do. Our model predicts the optimal treatment protocol for a number of cases, among them, those in which apoptosis is largely intact (as when the gene encoding p53 is undamaged) and those in which it is not.

Luis F. Gordillo

Sustained oscillations via coherence resonance in SIR

The deterministic SIR model produce oscillations which damp to the endemic quilibrium. It is well know that the stochastic SIR produces  sustained oscillations. In this work, using multiscale analysis, we analize the oscillatory nature of the stochastic SIR, determine under which conditions the sustained oscillations are expected and how they are modulated by noise.

Mudassar Imran

A model of Antibiotic Resistance in Biofilm

A mathematical model of the effect of antibiotics on bacterial biofilm is considered. According to US National Institute of Health " Biofilms are medically important, accounting for over 80% of microbial infections in the body" such as Otitis media, the most common acute ear infection in children in US. Biofilms are highly resistant to antibiotics. Consequently, very high and/or long-term doses are often required to eradicate biofilm-related infections. We consider a two-compartments, chemostat-based model where one compartment has a very high dilution rate as compared to the other compartment. The high dilution rate compartment represents the fluid environment and the low dilution rate compartment represents the stagnat biofilm environment. A constant supply of nutrient and a periodically fluctuating antibiotic agent is supplied to the high dilution compartment. The model assumes that antibiotic increases the death rate as its concentration is increased. We use persistence as well as global bifurcation results for a mathematical analysis of periodic solutions. The model consists of a system of non-autonomous differential equations which govern the dynamics of the bacteria in biofilm.

Yun Kang

Dynamics of a discrete two dimensional stoichiometric producer-grazer model:

Chaotic extinction and noise effect

The grazer can become extinct while having plenty of producer in a completely deterministic system. An explanation for this lies in the bad nutritional quality of the producer that precludes the grazer from e±ciently converting the consumed food into its own biomass. In experiments, data are collected on discrete time intervals and it is observed that many prey in nature have non-overlapping generations. In our paper, we use a non-overlapping discrete model to see how producer quality can pull the systems out of oscillations and how it halts chaotic dynamics. We discuss the behavior of our discrete dynamic system: in which condition the grazer becomes extinct but the producer still exhibits chaotic behavior and in which condition the grazer becomes extinct but the producer has stable equilibrium. Species diversity in nature is accomplished by coexistence. Utilizing a realistic model that consists of two interacting species producer and grazer, we discovered a stochastic phenomenon where noise can enhance the coexistence and thereby promote species diversity. We use scaling-law and phase plane to explain this interesting phenomenon.

Roxana Lopez-Cruz

Structured SI epidemic models with applications to HIV epidemic

The main objective of this talk is to study the effects of age and social structures on S-I epidemic models, with specific applications to HIV epidemic in Peru and USA. This talk is divided into three main parts, the first two parts deal with effect of age structure on S-I epidemic models. These models are fit with parameters from Peru and USA, HIV epidemic and the outcomes are the same: if nothing changes, both countries will be wiped out by the HIV epidemic in about 200 years. These models exhibit remarkably similar qualitative and quantitative dynamics, indicating the observed dynamic properties are robust. We also present our first attempt on modeling the effects of social, economical and cultural factors on the S-I epidemic diseases. The analysis of these models suggests

that the lack of health care and control of infected pregnant women aided the spread of the disease. Most of findings are supported by systematic mathematical analysis including local and global stability results.

Tufail Malik

A Resource-based Model of Microbial Quiescence

To analyze the ecological features of microbial quiescence, a model is presented that involves "wake-up" rate and "sleep" rate at which the population transitions from a quiescent to an active state and back, respectively. These rates depend continuously on the resources and turn on and off at resource thresholds which may not coincide. The usual dichotomy is observed: the population is washed out under environmental stress and a single "survival" steady state exists otherwise. Proportional nutrient enrichment is used to explore analytically as well as numerically the nature of the steady state which bifurcates from the washout state.

Clinton C. Mason

Improved Modeling of the Glucose-Insulin Dynamical System Leading to a Diabetic State

The dynamics of glucose and insulin involved with the maintenance of normal body metabolism has been modeled for many years. Irregularities of this feedback loop may lead to clinical abnormalities, most notably diabetes. The Beta-I-G model (Topp, 2000) was the first and only dynamical systems model yet proposed to explain the development of this disease state over a long time course. While the bifurcation inherent to the model yields a pathway to hyperglycemia, other modes of attainment are also plausible. Revisions to the model are proposed which produce more realistic long-term glucose profiles in agreement with observed data and recent observational studies.

Karen R. R´ıos-Soto, Carlos Castillo-Chavez, Michael G. Neubert, Edriss S. Titi and Abdul-Aziz Yakubu

Epidemic Spread in Populations at Demographic Equilibrium

We introduce an integrodifference equation model to study the spatial spread of epidemics through populations with overlapping and non-overlapping epidemiological generations. Our focus is on the existence of travelling wave solutions and their minimum asymptotic speed of propagation c_. We contrast the results here with similar work carried out in the context of ecological invasions. We illustrate the theoretical results numerically in the context of SI (susceptible-infected) and SIS (susceptible-infectedsusceptible) epidemic models.

Eunha Shim

A model for rotavirus infection and its vaccination

The impact of rotavirus, the most prevalent diarrheal pathogen in young children worldwide, may be reduced by either a recently approved vaccine or others in development. The dynamics of rotavirus infections are studied using a simple mathematical model that includes the impact of breast feeding, seasonality and the possibility of control via vaccination. Data from Australia are fitted to a model that incorporates the effect of seasonality in the transmission process. The impact of temporary and partially effective vaccines is explored. Finally, the issue of cost is briefly discussed in the context of identifying “optimal” vaccination strategies.

Wolfgang Stefan

Improving the image quality of medical images

State of the art methods in image post processing are presented and demonstrated on medical images like PET or MRI. Image deblurring can significantly improve the image quality however, issues like the extreme ill posedness and the resulting amplification of the naturally occurring noise as well as the estimation of the usually unknown blurring operator have to be taken into consideration. Secondly methods that can highlight or remove certain structures or reconstruction artifacts are applied to medical images. These kind of methods try identify these structures in certain Besov function spaces like B_11. We show that we can improve PET images by partly removing reconstruction artifacts from the filtered back projection by using a wavelet based image decomposition method.

Craig Thalhauser

The Trees for the Forest: A Discrete Cell Model of Tumor Growth, Development, and Evolution

Much of the early work in mathematical modeling of cancer has used a continuous, deterministic framework.  While this approach leads to tractable mathematical analysis and relatively simple simulations, the continuum models lose focus on the fascinating biology of individual cancer cells.  In this talk, I will present a preliminary lattice-free, agent based cellular model for cancer.  I will compare the simulation results of this model with known analyses of existing continuus and discrete cellular automata models and will demonstrate how this approach can inform new research in theoretical biology.

Abdessamad Tridane

Mathematical Analysis of the interaction of cytotoxic T Lymphocyte and epithelial cells in the influenza infection

The recent development of applied mathematics is characterized by ever increasing attempts to apply the modelling approaches to understand  the diseases processes  .The need for a rigorous analysis of the complex system dynamics in immunology has been recognized since more than three decades ago.The aim of this work is to investigate how cytotoxic T lymphocytes  (CTL) interact with epithelial cells infected by the influenza virus. Since the machinery CTL  is not clearly understood, we study different models of this dynamic to determine the conditions for control of an infection by T-cells. This provides a better understanding of the causal relationships between the immune systems response to an infection such as the influenza.

Hao Wang

Top-down Trophic Dynamics produce the 4-year Lemming Cycle and the 10-year Snowshoe Hare Cycle: the maturation delay of predators and the functional response control prey population cycle

The principal purpose of this paper is to answer the following simple question: What ecological factors control population cycles? The answer is the maturation delay of predators and the functional response of predation, especially the maturation delay, which almost determines the period of a population cycle. This result is from sensitivity analysis of each parameter in the lemming-stoat delayed system, which also explains why lemming has a 4-year cycle whereas snowshoe hare has a 10-year cycle. These results are based upon careful model formulation, parameter estimation and empirical data ¯tting. Furthermore, if maturation periods of predators are too short or too long, or the functional response resembles Holling Type I, then population cycles do not appear; intermediate predator maturation periods and other functional responses can generate population cycles for both prey and predators. These results clearly explain why some populations are cyclic whereas others are not.

Xiaohong Wang

On the role of cross-immunity and vaccines on the survival of less fit flu-strains

A pathogen's route to survival involves various mechanisms including their ability to invade (host's susceptibility) and their reproductive success within an invaded host (infectiousness). The immunological history of an individual often plays an important role in reducing host's susceptibility or it helps the host mount a faster immunological response de facto reducing infectiousness. The cross-immunity generated by prior infections to influenza A strains from the same subtype provide a significant example. In this paper, we look at the role of invasion mediated cross-immunity in a population where a precursor related strain (within the same subtype) has already become established. An uncertainty and sensitivity analysis is carried out on the ability of the invading strain to become established for given cross-immunity levels. As it turns out, it is possible for relative low level of cross-immunity to increase the likelihood of strain coexistence even in the case when invading strains are less “fit”, that is, cross-immunity can increase phenotypic diversity. The fears associated with the potential ability of the “bird flu” to propagate among humans may be connected naturally to the low or non-existing levels of herd cross-immunity for such strains. The development of “flu” vaccines that minimally enhance herd cross-immunity levels may, by increasing genotype diversity, may help facilitate the generation and survival of novel virulent strains.