Abstracts for the BioMathDays 2008


Plenary Speakers

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Chris Bauch     

TITLE
Wealth as a source of density dependence in human population growth: evidence from the demographic transition

ABSTRACT
The phenomenon of density dependence, whereby higher population density results in a reduced population growth rate due to resource limitations, is a topic of intensive research in natural populations.  In modern industrialized nations, human birth rates have been declining persistently for decades, and have now fallen below the replacement threshold in many countries.  However, unlike in natural populations, lower birth rates in modern industrialized countries appear to be positively correlated with resource availability, e.g. gross domestic product (GDP) per capita.  Here, we show that declining birth rates in human populations   are actually a manifestation of density-regulated population growth brought on by socioeconomic development, as reflected by GDP per capita.  This is demonstrated by combining empirical power law relations between population size,  GDP per capita, and fertility in a simple theoretical model describing population dynamics in developed countries.  For a closed population, the model exhibits growth to a globally-stable equilibrium population size, for both  national and city populations.  The growth dynamics are highly sensitive to demographic and economic power law exponents.  An extended model for a country that is open to the flow of labor, technology and capital form other countries exhibits a good fit to long-term time series data on population size, GDP per capita, and births rates for the United States, France, and Japan, and provides future projections as well.   An implication of this work for sustainability is that the earth's human population may limit itself before environmental constraints cause widespread premature death, as predicted by Malthusean models.


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Gerda de Vries

TITLE
Introduction to Models in Radiation Biology:
From Cell Population Models to Tumour Control Probability Curves

ABSTRACT
In this talk, I will review concepts in radiation biology that allow us to model the effectiveness of radiation therapy in the treatment of cancer. In particular, I will focus on the Tumour Control Probability (TCP), which is the probability that no cancerous cells survive the treatment.  Early TCP formulae are based on simple binomial and Poissonian statistics.  They are of limited value, since they do not take cell proliferation during the treatment period into account.  Recent TCP formulae are based on dynamic models of a cell population, taking cell proliferation as well as the cell cycle into account.  I will conclude with a discussion of how and when the TCP formulae are related to each other, and how they can be used to compare the efficacy of different treatment schedules.

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Karl Peter Hadeler

TITLE
Quiescent Phases in Dynamical Systems and in Biological Modeling

ABSTRACT
In biology there are many situations where all of the dynamics or certain components go to rest for some time and then resume activity again and again. Examples are formation of spores, hibernation, taking refuge from predators, etc. Such quiescent phases can affect the whole system like in a seasonal or diurnal cycle, or individual units (molecules, cells, animals) in a stochastic fashion. In the first case modeling quiescence leads to  time-dependent differential equations. In the second case we end up with what we call dynamical systems with quiescent phases. In mathematical terms these are systems where a given dynamics  is coupled diffusely to the trivial dynamics (which changes nothing).

Our  problem is to determine how quiescent  phases change a given dynamics and to understand these changes in terms of a given biological situation. We shall present some general results on reduction of growth rates, stabilization against oscillations, and apply them to classical examples like prey predator dynamics.


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Pauline van den Driessche

TITLE
Models for Influenza

ABSTRACT
The spread of an infectious disease, such as influenza, can be modeled by a dynamical system that includes specific features of the disease. The basic reproduction number R0 is an important threshold parameter that depends on the model formulation and parameter values related to the disease as estimated from data.  In ordinary differential equation systems, R0 can be determined as the spectral radius of the next generation matrix.  This is illustrated for a simple influenza model, and for a metapopulation model with individuals who travel between discrete spatial patches.  Using matrix inequalities, useful bounds on R0 are derived for this metapopulation model. A control reproduction number is calculated for an extended influenza model that contains vaccination and antiviral treatment.  This gives insight into quantifying strategies to control an influenza epidemic.


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Contributed Talks

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Majid Bani-Yaghoub

TITLE
Pattern Formation of Reaction-Diffusion Systems:  Destabilizing Effect of Interaction Between Signaling Pathways

ABSTRACT
Signaling pathways are known as a series of biological reactions that affect major cellular functions (such as migration and differentiation) by targeting specific genes. While signaling pathways regulate cellular functions, interaction among signaling pathways (i.e. cross-talk through endocrine, paracrine and autocrine mechanisms) may impact the effects of signaling pathways. In the present work, interaction of signaling pathways is considered as a perturbation to a general reaction-diffusion (RD) system representing the chemical concentrations in cells in a bounded domain. It is shown that, under some conditions, weak interactions between signaling pathways may have negligible effects on the spatial patterns generated by RD systems. This corresponds to the topological equivalency between the perturbed and unperturbed RD systems that may result in generation of qualitatively the same spatial patterns. On the contrary, high levels of interaction may result in loss of stability of Turing-type or Target patterns and formation of new spatial patterns. Destabilizing effect of interaction between Signaling Pathways is demonstrated by considering an activator-inhibitor model of axon-forming potential, where the stability of the Turing-type and Target patterns is numerically examined.


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Aquino L. Espindola

TITLE
Probabilistic Cellular Automata to TB spread and drug resistence emergence

ABSTRACT
The spread and evolution of bacterial resistance within human communities is addresses in this presentation. Both, bacterial growth dynamics within individuals and the interaction among individuals are taken into account. We have developed a probabilistic cellular automata model to analyse tuberculosis transition dynamics and emergence of resistant bacteria among individuals subjected to varying levels of antibiotic treatment. Different interaction levels among individuals can be tested adjusting local and global contact rates. These variables are key to determine the severity of the epidemic prevalence. We assume that antibiotic treatment is independent of bacterial colonization. Individuals are randomly chosen to receive treatment with antibiotics for a given time interval. Moreover, this baseline model will allow us to test some treatment protocols, including vaccination.

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Aquino L. Espindola

TITLE
Continuous and discrete growth models in terms of generalized exponential functions

ABSTRACT
Here, we consider one-parameter generalizations of the logarithmic and exponential functions, which are obtained fom the integration of non-symmetric hyperbolas. These generalizations coincide with the ones obtained in the context of non-extensive thermostatistics. We show that these functions are suitable fo describe and unify a great majority of continuous growth models. Furthermore, we also show that such a generalization of the exponential function is also suitable to unify most of the popular scramble and contest one-species, discrete population dynamics models into a simple formula.

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Daihai He
TITLE
Time series analysis via mechanistic models

ABSTRACT
The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae.


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Olga Krylova

TITLE
Smallpox: history, virology, data analysis and more

ABSTRACT
Smallpox is one of the most devastating of human diseases. In its 12,000-year history, it has probably killed more people than any other illness, including the plague. Interest in the study and modeling of smallpox spread has increased in the last few years. While this infectious disease was eradicated by the World Health Organization in 1979, it still presents a potential threat as a bio weapon. In fact, smallpox is considered one of the two most dangerous BW agents (the other being anthrax) because of its high case-fatality rate (>30%), person-to-person transmission, lack of population immunity and lack of treatment. We have analyzed weekly mortality data from the 17th, 18th and 19th centuries in London, UK, where frequent smallpox epidemics occurred. The data that we have cover almost 200 years and include many smallpox epidemics. We examined the patterns in these historical data using time series analysis. Interesting changes in smallpox dynamics are evident over the centuries. I will discuss the history of the disease, its virology and the results of our data analysis.


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Frithjof Lutscher     

TITLE
The effect of landscape heterogeneity on spread and persistence in integrodifference equations

ABSTRACT
The spread of non-indigenous species and diseases poses a major risk to ecosystems and human health worldwide. The key challenges to management and control of such invasions are to understand the conditions of spread and the different factors influencing the speed of spread. Of particular interest is the effect of landscape heterogeneity on the spread of organisms. We formulate a discrete-time model for growth and dispersal, where both of these processes vary in space. We then present approximation formulas for the spread rate in such a heterogeneous landscape and demonstrate their validity by comparison with numerical simulation. We also give rules of thumb for the conditions under which a species is able to spread in a heterogeneous landscape. We separately consider the two cases with and without Allee effect in the population growth function. Our results provide simple recipes for calculation of spread rates in complex landscapes together with their limits of validity.

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Jeff Musgrave

TITLE
An integrodifference equation model for the spread of the Emerald Ash Borer

ABSTRACT
A biological invasion is the introduction and spread of exotic organisms outside their native range.  In this talk, the Emerald Ash Borer, (EAB) (Agrilus planipennis), an invasive species first discovered in Michigan in 2002, is studied. 

An important prediction of biological invasions is the spread rate at which a population front is moving.  In this talk,  I present a two-dimensional probabilistic model to predict the spread of EAB.  The model predicts the natural spread rate of EAB and the spread rate when containment strategies are used.  Results of the model are compared with a one-dimensional integrodifference equation model.


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Joe Tien

TITLE
Parameter estimation for bursting neural models

ABSTRACT
Parameter estimation for differential equation models is a challenging problem. This talk will consider parameter estimation for an ODE model of respiratory neurons that "burst".  I'll discuss an approach which uses geometric features of the differential equation to aid the estimation, and consider possible ways that the neuronal output may be modulated.


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Eunha Shim

TITLE
Antiviral intervention during pandemix influenza: prophylaxis and treatment coverage levels driven by individual and societal interest

ABSTRACT
Antiviral agents will play a critical role in mitigating the next influenza pandemic. The administration of drugs has epidemiological and evolutionary repercussions that affect both the individual patient and the community. An individual benefits from reduced probability and/or severity of infection but potentially suffers adverse effects related to antiviral drugs. From the community perspective, the positive externality of antiviral intervention is reduced transmission, while the negative externality is selection for drug resistance.  We develop an epidemiological game-theoretic model of pandemic influenza in order to evaluate how the balance among these factors determines the discrepancy between coverage levels optimized from the perspectives of individual and societal interest.  We parameterize the model with survey data on actual perceptions regarding infection risk, the level of resistance, the efficacy and adverse effects of antivirals, and the willingness to pay for antivirals during pandemic influenza.  We find that the demand for antivirals driven by self-interest during pandemic influenza would likely be far lower than those that would maximize overall utility for the population. At current drug pricing, this discrepancy is larger when antivirals are used for treatment than for prophylaxis.  We find that public education about the correct information regarding the infection risk and the level of resistance associated with antiviral drug use may be a beneficial measure to achieve the optimal levels of antiviral drug use for population.


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Robert Smith?

TITLE
Explicitly accounting for antiretroviral drug uptake in theoretical HIV models.

ABSTRACT
Mathematical models of HIV therapy have traditionally amalgamated the action of antiretroviral drugs, trading the complexity of the situation in favour of simpler - and hence mathematically tractable - models. However, the effects of ignoring such dynamics remain underexamined. In this paper, the traditional method of dosing (where the dose is modelled implicitly as a proportional inhibition of viral infection and production) is compared to a model that accounts for drug dynamics via explicit compartments. Four limiting cases are examined: frequent dosing of both major classes of drugs, absence of either drug, frequent dosing of one drug alone, or frequent dosing of the other drug alone. When both drugs are absent, both models predict that the disease-free equilibrium will be unstable and the infection equilibrium will be stable. When reverse transcriptase inhibitors are given frequently, both models predict that the disease-free equilibrium will be stable and the infection equilibrium will be unstable; this is true regardless of whether the reverse transcriptase inhibitors act alone or in conjunction with protease inhibitors. However, if protease inhibitors alone are given frequently, then the implicit model predicts that the disease-free equilibrium will be stable and the infection equilibrium will be unstable, whereas the (more realistic) explicit model predicts that the reverse situation may occur. It follows that the impact of drug regimens consisting only of protease inhibitors must be urgently re-examined.



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Naveen Vaidya

TITLE
Modeling and analysis of HIV and tuberculosis co-infection dynamics

ABSTRACT
We develop a co-epidemic model of HIV and Tuberculosis infection dynamics. In addition to calculating the basic reproduction number, the equilibria-analysis of the model is carried out. We present an illustrative numerical analysis of the HIV-TB co-epidemics in India, which we use to explore the effects of prevention and treatment scinario. We also discuss the importance of including the effects of each disease on the transmission and progression of the other disease.