Titles & Abstracts
Department of Mathematics and Statistics
Texas Tech University
Thresholds for Population or Pathogen Survival in Stochastic Models Population or pathogen survival depends on the magnitude of the basic reproduction number R0. If R0>1, then the populations or pathogens may survive and become
established. In stochastic theory, population or pathogen survival probabilities also depend on the initial population size or the initial number of infectious individuals. For example, in the SIS Markov chain epidemic model, if the basic
reproduction number R0>1 and if I(0) =a is small, then an approximation to the probability of an outbreak is 1-(1/R0)a. This classic result can be easily derived from branching process theory. We apply results from multitype branching process theory to generalize approximations for probability of population or pathogen survival to models with multiple species and multiple pathogens.
Metapopulation Research Group
University of Helsinki
Deriving simple models from complex models
The complexity of mathematical models employed in ecology varies greatly, and it is often difficult to judge in advance which level of complexity would strike the optimal balance between the conceptual clarity of simple models and the apparent realism and potentially greater predictive power of complex models. Greater complexity provides flexibility but complex models are often criticized because they involve a large number of parameters that can rarely be rigorously estimated. In my talk I simplify complex models with the aim of achieving more tractable models but not oversimplifying the underlying ecology. I do model simplification either by considering scaling limits, integrating over intermediate states, or finding effective parameter combinations. As the first example, I unify a broad family of animal movement models by re-parameterizing them with two parameters only: the characteristic spatial and temporal scales of movement. As the second example, I simplify models of animal movement in two-dimensional heterogeneous space by taking a scaling limit at which elongated habitats (such as edges, corridors and barriers) become one-dimensional. As the third example, I simplify a complex individual-based model of a butterfly metapopulation into a parameter sparse patch occupancy model, and compare how the complex and simple models perform in terms of their fit to data.
Sarah (Sally) Otto
Department of Zoology
University of British Columbia
Inferring the past for traits that alter
speciation and extinction
I will describe BiSSE, a likelihood-based approach to infer how speciation and extinction rates depend on the state of a particular character. The phylogenetic tree of a group of species contains information about character transitions and about diversification: higher speciation rates, for example, give rise to shorter branch lengths. The likelihood method that we have developed uses the information contained in a phylogeny and integrates over all possible evolutionary histories to infer the speciation and extinction rates for species with different character states. Our method can be used to provide more detailed information than previous methods, allowing us to disentangle whether a particular character state is rare because species in that state are prone to extinction, are unlikely to speciate, or tend to move out of that state faster than they move in.
Department of Ecology and Evolutionary Biology
University of Toronto
How is, and how should ecological theory be structured?
Ecological theory is challenged by the need to represent many diverse systems consisting of unique components using a manageable set of mathematical models. A complete theory should ideally give guidelines for when specific simple models yield reasonable or unreasonable predictions of the dynamics of different types of complex systems. Such a theory should also provide guidance in how to select component functions when building a model of a complex system. These have not been major goals of actual theoretical research over the past several decades, and historical precedent has played too large a role in the choice of both models and their functional components. At the same time, empirical work has not addressed our ignorance of these functional forms, but has instead focused on documenting relationships between poorly defined aggregate variables. Illustrations of some of these claims will be presented, primarily using models of competing species and consumer-resource interactions.
Department of Psychiatry and Biobehavioral Sciences
University of California Los Angeles
Recent insights from modeling the potential spread of HIV drug resistance in San Francisco & Botswana
I will discuss our recent research on modeling the evolutionary dynamics of complex networks of HIV drug-resistant strains in the community of me who have sex with men (MSM) in San Francisco. Over the past two decades, HIV resistance to antiretrovirals (ARVs) has risen to high levels in the wealthier countries of the world able to afford widespread treatment. We have gained insights into the evolution and the transmission dynamics of ARV resistance by designing a biologically complex multi-strain network model. Using this model, we have traced the evolutionary history of ARV resistance in San Francisco and predicted the future dynamics. Using classification and regression trees, we have identified the key immunologic, virologic, and treatment factors that increase ARV resistance. Our modeling shows that 60% of the currently circulating ARV-resistant strains in San Francisco are capable of causing self-sustaining epidemics, as each individual infected with one of these strains can cause on average more than one new resistant infection. Our results suggest that a new wave of ARV-resistant strains that pose a significant threat to global public health is emerging.
I will also discuss our recent research on modeling the introduction of pre-exposure prophylaxis (PrEP) into the MSM community in San Francisco. The administration of antiretrovirals before HIV exposure to prevent infection (i.e., PrEP) is currently under evaluation in clinical trials. Because PrEP is based on antiretrovirals, there is considerable concern that it could substantially increase transmitted resistance, particularly in resource-rich countries. We have used mathematical modeling to predict the effect of PrEP interventions on the HIV epidemic in San Francisco. The model is calibrated using Monte Carlo filtering and analyzed by constructing nonlinear response hypersurfaces. We predict PrEP interventions could substantially reduce transmission, but significantly increase the proportion of new infections caused by resistant strains. Two mechanisms can cause this increase. If risk compensation occurs, the proportion increases due to increasing transmission of resistant strains and decreasing transmission of wild-type strains. If risk behavior remains stable, the increase occurs because of reduced transmission of resistant strains coupled with an even greater reduction in transmission of wild-type strains. We define this as the paradox of PrEP (i.e., resistance appears to be increasing, but is actually decreasing). We determine this paradox is likely to occur if the efficacy of PrEP regimens against wild-type strains is greater than 30% and the relative efficacy against resistant strains is greater than 0.2 but less than the efficacy against wild-type. Our modeling shows, if risk behavior increases, that it is a valid concern that PrEP could significantly increase transmitted resistance. However, if risk behavior remains stable, we find the concern is unfounded and PrEP interventions are likely to decrease transmitted resistance.
Finally, I will discuss our current research on modeling the dynamic interactions between PrEP interventions and treatment programs in the heterosexual population in Botswana. The objective of this work is to predict the effect on HIV transmission and resistance. Two recent clinical trials have demonstrated the effectiveness of PrEP in preventing HIV infection. Consequently, PrEP may soon be rolled out as an epidemic control strategy. We are using modeling to predict the impact of PrEP interventions in resource-constrained countries. Our results show the "quality" of treatment programs will determine the success of PrEP interventions and, conversely, the "quality" of PrEP interventions will determine the success of treatment programs. We predict the number of treatment-naïve individuals needing second-line therapies (SLT) will increase, but the number of treatment-experienced individuals needing SLT will decrease. Our results indicate the optimal strategy for rolling out PrEP in resource-constrained countries is to begin around the "worst" treatment programs. If the rollout of PrEP is carefully planned it could decrease resistance and increase sustainability of treatment programs. If it is not, resistance could increase and the sustainability of treatment programs in resource-constrained countries could be compromised.
A Mathematical Model for Hyperparasitic Regulation of an Infectious Pathogen
College of Saint Rose, Albany NY
Collaborators: Thomas Caraco, Kurt A. McKean, Heather Wilson (University of Albany) and Christopher Dottino (College of Saint Rose, Albany, NY)
Infections within individual hosts often encompass multiple interactions, involving competing pathogen strains, symbiotic mutualists, hyperparasites, and (for human and agricultural hosts) antibiotics. Here, we present a mathematical model investigating interactions among bacterial pathogen growth, host immune responses, and hyperparasitic pathogen regulation by a bacteria-consuming virus. The model links within-host dynamics to both time-dependent host survival and between-host disease transmission. We develop predictions for both acute and chronic infection. The former are being tested in a three-species experimental system.
A Spatiotemporal Model of B/CYDV Transmission Dynamics with Seasonality, Age Structure, and Plant Competition
We model the transmission of a generalist pathogen within a patch framework that incorporates the movement of vectors between discrete host patches to investigate the effects of local host community composition and vector movement rates on disease dynamics. We use barley and cereal yellow dwarf viruses (B/CYDV), a suite of generalist, aphid-vectored pathogens of grasses, and their interactions with a range of host species as our case study. We examine whether B/CYDV can persist locally or in a patch framework across a range of host community configurations. We then determine how pathogen-mediated interactions between perennial and annual competitors are altered at the local and regional scale when the host populations are spatially structured. We find that the spatial configuration of the patch system, host composition within patches, and patch connectivity affect not only the ability of the pathogen to invade a fragmented system, but also determine whether the pathogen facilitates the invasion of a non-native host species. Further, our results suggest that connectivity can interact with arrival time and host infection tolerance to determine the success or failure of establishment for newly arriving species.
A Stochastic Model for Guiding Field Cage Experiments for Testing Transgenic Mosquitoes
North Carolina State University
We demonstrate the utility of mathematical model as an aid in the design and assessment of experiments aimed at measuring the effects of proposed vector population control strategies. We develop a stochastic, age-structured model and use numerical simulations to explore field cage experiments that test the ability of a female-killing (FK) strain of the mosquito Aedes aegypti (L.) to suppress a wild-type population. Model output predicts that choices of release ratio and population size can impact mean extinction time and variability in extinction time among experiments. We find that unless fitness costs are greater than 60% they will not be detectable in experiments with high release ratios. At lower release ratios the predicted length of the experiment increases significantly for fitness costs greater than 20%. We explore field cage designs that specifically aim to study the impact of density dependence; in some cases, predictions indicate that population eradication may not be obtainable in an operationally realistic time frame. We propose a method to predict the extinction time of a population based on the rate of population reduction with the goal of shortening the duration of the experiment. We demonstrate the utility of this model in guiding and assessing experiments by fitting the model to data collected in recent field cage trials in Tapachula, Mexico.
Possible Impact of Dengue Fever Modeling: Challenges to Public Health Officials
Arizona State University
Dengue fever has been a burden to public health officials for decades. Despite strong efforts to 'control' the spread of the disease it keeps spreading at an alarming rate, mostly in tropical countries where healthcare is a privilege. Two distinct mathematical models will be discussed to highlight public health challenges and their role in the spread of the disease. Possible 'intervention/control' strategies will be discussed.
A Mathematical Model for within-host Toxoplasma gondii Invasion Dynamics
University of Tennessee
Toxoplasma gondii is a protozoan parasite that infects a wide range of intermediate hosts, including all mammals and birds. Up to 20% of the human population in the US and 30% in the world are chronically infected. This paper presents a mathematical model to describe intra-host dynamics of T. gondii infection. The model considers the invasion process, egress kinetics, interconversion between fast-replicating tachyzoite stage and slowly replicating bradyzoite stage, as well as the host's immune response. Analytical and numerical studies of the model can help to understand the influences of various parameters to the transient and steady-state dynamics of the disease infection.
Statistical Analysis of the Growth Rate of the Green Treefrog (Hyla cinerea)
University of Louisiana, Lafayette
Hyla cinerea (Green Treefrog) is a common wetlands species in the southeastern US. To better understand the growth rate, we followed a relatively isolated population of Green Treefrogs starting in the springtime to early autumn from 2005 until 2011 at a federal office complex in Lafayette, LA. Weekly, Green Treefrogs were caught, measured, marked with VIE tags, and released. The data were used to estimate the growth rate of the frogs. To model the growth rate, we used MATLAB to obtain an 8th degree polynomial regression. After excluding frogs with only two recaptures, we found that the growth rate from 30mm to 46mm, where most of our data points lie, is similar in each year. The growth rate for each curve rises dramatically until reaching a peak between 33 and 34 mm, then decreases at varying rates as the body length increases until it nears zero. The fluctuations at the edges of each curve are because of a lack of data points for frogs outside of the given size range.
A theoretical approach to a semi-discrete model for intraguild predation
Arizona State University
Food web dynamics have long-since been a topic of interest in the ecological world. Various food web configurations can be used to depict the interactions between species of different trophic levels. However, it is important to note that interactions do exist between species of the same trophic level, whether it is a predatorprey relationship or a competitive relationship. Competition and predation have repeatedly been recognized as important factors in community ecology, but it was not until recently that they were considered together. Ecologists began to acknowledge an interaction between the two where potentially competing species are also involved in a predator-prey relationship. Thus, intraguild predation (IGP) has commonly been defined as this mixture of competition and predation. Here, we look at the intraguild relationship between three species: gypsy moth, white-footed mouse, and oak. Gypsy moths are a primary defoliator of oak trees, while whitefooted mice are significant predators of the gypsy moth. Acorns, produced by oak trees, are also a primary resource for the white-footed mouse. Populations typically remain stable, but gypsy moths continue to experience intermittent outbreaks. Previous studies have recognized acorn abundance as the main cause for these outbreaks. We develop a semi-discrete model based on theoretical analyses and specific assumptions in order to isolate principal parameters leading to gypsy moth outbreaks.
Evolution of Host Resistance to Parasite Infection in the Snail-Schistosome-Human System
The University of Tennessee
The evolutionary strategies that emerge within populations can be dictated by numerous factors, including interactions with other species. We investigate the consequences of such a scenario using a host-parasite system of human concern. Mathematical models are used to investigate evolutionary outcomes resulting from interactions between S. mansoni and its snail and human hosts. The model includes two types of snail hosts representing resident and mutant types. Using this approach, we focus on establishing evolutionary stable strategies (ESS) under conditions where snail hosts express different life-histories and when drug treatment is applied to an age-structure population of human hosts. Results from this work demonstrate that the evolutionary trajectories of host-parasite interactions can be varied, and at times, counter-intuitive, based on parasite virulence, host resistance, and drug treatments.
A deterministic model for influenza with multiple stains and drift
The influenza virus continuously mutates via antigenic drift, resulting in continuous creation of new variants able to re-infect hosts that have become immune to earlier types. In large part because of this continuous drift property, influenza presents a significant morbidity and mortality burden. On first world countries of the temperate zones, control is largely achieved through mass-production and dissemination of vaccines, however this requires prediction of the dominant circulating strain well in advance of the up-coming season. The dynamics of influenza patterns for the tropics are less understood and few mathematical models have try to address this topic. To aid the control of the disease, models are needed that can accurately simulate the spread of influenza over the course of several years, with strainspecific dynamics, including the process of drift and pre-existing immunity to newly circulating strains. Ideally the model should be flexible enough to reproduce the dynamics of both temperate zones and the tropics. Here we describe a novel SIR deterministic model for the spread of influenza within a population. The model incorporates appearance of new strains due to antigenic drift, and partial immunity within the population to circulating strains due to prior infection with related strains. Because seasonality is important to consider in temperate regions, the model includes optional seasonal forcing of the transmission rate. Our model successfully reproduces several key features in influenza data observed in tropical and temperate regions.
Numerical study on freezing-thawing phase change heat transfer in biological tissue as a porous media
St. Margaret Engineering College, Neemrana
Abstract-Biological tissue can be treated as porous media as it is composed of dispersed cell separated by connective voids which allow flow of nutrients, minerals etc. to reach all cell within tissue. In the present study, a mathematical model has been developed to study the phase change heat transfer phenomena during freezing and thawing process in biological tissues considered as porous media. Effective heat capacity formulation is used for phase change problem. Numerical simulation is used to study the effect of porosity, on the motion of freezing and thawing front and transient temperature distribution in biological tissue. It is observed that porosity has significant effect on transient temperature profile and phase change interfaces, further decrease in freezing rate has been found with increased value of porosity.
The effect of using e-education in academic achievement For students of the Faculty of Education Razi Article Statistics
Current research aims to know the impact of substance use education as an electronic Statistics in academic achievement for students of fourth phase of the science sections Computers and Life Sciences College of Education, Al-Razi.
To achieve this goal, the researchers put the following null hypothesis:
No statistically significant difference in average levels of academic achievement between students of the experimental group who use e-learning education and the control group students who use the standard way. After the completion of the search experience on the results obtained researcher as follows: The average degrees of students of the experimental group (71.1) degrees, and the standard deviation of (9.003), while the average degree students the control group (61.66) degrees, and the standard deviation of (6.358) were (v) calculated equal to (.), while (v) spreadsheet equal to (1.67) at 0.001, this means that there are significant differences at the level of 0.001 for the experimental group used a computer in the degrees of achievement, and it appears from the results of academic achievement that the use of educational electronic therapeutic techniques to master the learning has contributed in raising student achievement. as The researchers recommended that the university is working on programming courses for the teachers to facilitate training and use. and to work with the university to provide the processing of all colleges with modern computer, and the training of the students in order to develop their attitudes towards education and electronic uses.
Basic and type- reproduction numbers for a compartmental model of an infectious disease with free-living pathogen
Texas A&M University
The basic reproduction number R0 is a well-known threshold quantity used for control and prevention of infectious diseases. This study highlights the issue of calculating R0 for compartmental models of infectious diseases with a free-living pathogen (FLP) capable of survival and growth in the environment. We consider a susceptible-infectious-recovered-susceptible model with FLP in the environment. In addition to the stability analysis of the equilibria, different R0 expressions are derived based on the hypothesized role of the environment in the ecology of the infectious disease. Specifically, the environment could serve as an extended stage of host infectiousness, as a reservoir of FLP or a combination of both. Each R0 expression leads to a different estimate of the effort required to control an infectious disease. The issue of non-unique estimates can be resolved under certain conditions. In particular, when the environment is unable to serve as a reservoir of FLP and the host population is targeted, we compute a unique threshold quantity known as type-reproduction number T. The main outcomes of this study are numerically verified via examples of salmonellosis and cholera infections in animal and human populations, respectively.
Periodic Multidrug Therapy in a Within-Host Virus Model
University of Florida
Floquet theory and perturbation techniques are used to analyze a classical within-host virus model with periodic drug treatment. Both single and multidrug treatment strategies are investigated. Specifically, the effects of both RT-inhibitors and Pinhibitors on the stability of the infection-free steady state are studied. It is found that when both classes of drugs have periodic drug efficacy functions, then shifting the phase of these functions can critically affect the stability of the infection-free steady state. A numerical study is conducted to illustrate the theoretical results and provide additional insights.
The implications of social contact structure for the economics of disease control
Disease control strategies, such as vaccination, aim to reduce disease incidence and therewith the disease costs incurred by a population. However, control strategies also have the indirect effect of changing the age distribution of infected cases. Because the severity and cost of infection are heavily dependent on age for many diseases, control strategies may therefore also impact the economic costs of a disease in this way. Here, we evaluate whether and to what extent these effects are governed by social contact patterns. We use a simple bio-economic framework to analyze the implications of different social contact structures on the value of vaccination. We base our analysis on an age structured SIR model with varying degrees of assortative mixing by age and a given function for how the cost of an infection depends on age.
Comparing Malaria Control Interventions in the Presence of Insecticide Resistance with a Mathematical Model
Swiss Tropical and Public Health Institute
Malaria is an infectious disease, spread through mosquito bites, that is responsible for substantial morbidity and mortality around the world. Although significant progress has been made in recent years in reducing transmission and disease burden, most control strategies depend on chemicals that lead to the development of insecticide resistance in mosquito populations. To help in better planning malaria control, we have developed a mathematical model of malaria transmission and mosquito population dynamics to determine the effectiveness of interventions in reducing transmission and their effect on the spread of resistance. We present a periodically-forced difference equation model for the population dynamics of mosquitoes with density-dependent death in the larval stage. We connect this model with a deterministic model for malaria dynamics in mosquitoes and a stochastic individual-based model for malaria in humans. We extend the model of mosquito population dynamics to include the spread of a resistant gene in response to insecticidal treatment and the corresponding effect on malaria transmission and disease.
A Discrete Dispersal Model with Constant and Periodic Environments
University of Louisiana, Lafayette
We study a discrete juvenile-adult model which describes the dynamics of a population that reproduces and disperses between two patches constantly or seasonally. When breeding and dispersal rates are constant, the model has a unique interior equilibrium that is globally attractive, provided the net reproductive number is greater than one. If net reproductive number is less than one, then the extinction equilibrium is globally asymptotically stable. When breeding and dispersal rates are periodic of period two, the extinction equilibrium is globally asymptotically stable if the net reproductive number is less than one. If the net reproductive number is greater than one, then there exists a unique globally attractive periodic solution. We then use bifurcation analysis to compare constant and seasonal breeding strategies, to explore the effects of different birth and dispersal periodicities and to understand the influence of strong nonlinearities on the dynamics of the model.
Evolutionary distributions as alternative to evolutionary games theory
University of Minnesota
Evolutionary distributions (ED) mimic evolution-by-natural-selection via reaction-diffusion PDE. They incorporate random mutations, inheritance and natural selection that follow the dynamics of the distributions of the populations of phenotypes.
(1) A stable ED encapsulates the idea of evolutionary stable strategies (ESS);
(2) ED captures the games among phenotypes with integral PDE;
(3) A game of competition where like compete most with like results in high density of phenotypes at the boundaries of the phenotypic-trait (strategy) space and low density of phenotypes in the interior of the space;
(4) patterns may emerge in predator-prey coupled distributions;
(5) because ED are based on the mechanistic principles of evolution, they explain why certain patterns survive while others do not.
Evolutionary game theoretic, discrete-time matrix models for semelparous population dynamics
University of Arizona
Evolutionary game theoretic versions of discrete time matrix models will be used to study the dynamic outcome of a semelparous life history strategy for R0 near 1. Bifurcation theoretic methods will be used to investigate the evolutionary consequences presented by dynamic dichotomy between equilibration with overlapping generations and periodic oscillations with non-overlapping generations that is created when R0 increases through 1.
Optimal Control Applied to Native-Invasive Species Competition via a PDE Model
Tennessee State University
Co-Authors: Volodymyr Hrynkiv (University of Houston), Xiaoyu Mu (University of Tennessee)
We consider an optimal control problem of a system of parabolic partial differential equations modeling the competition between an invasive and a native species. The motivating example is cottonwood-salt cedar competition, where the effect of disturbance in the system (such as flooding) is taken to be a control variable. Flooding being detrimental at low and high levels, and advantageous at medium levels led us to consider the quadratic growth function of the control. The objective is to maximize the native species and minimize the invasive species while minimizing the cost of implementing the control. A new existence result for an optimal control with these quadratic growth functions is given. Numerical examples are given to illustrate the results. Our findings will provide suggestions to the natural resource managers for controlling the invasive species.
General Allee Effect in Planar Dynamical Systems
Saber Elaydi and George Livadiotis*
Trinity University and Southwest Research Institute
Abstract: We present a general framework for the notion of Allee effect in planar dynamical systems. Here the basic assumption is that the extension equilibrium (0,0) is locally attracting. The boundary of the basin of attraction of (0,0) will be called “the Allee curve”, which corresponds to the Allee-point in one-dimensional dynamics. We show how a “phase space core” of only three or four equilibrium points is sufficient to describe the essential dynamics that characterize the notion of the Allee effect. The traditional three types of equilibrium point stability (Attractor/Repeller/Saddle) allow the existence of only one case of a 3-point core and three cases of a 4-point core. A richer dynamics occurs if we add to those three stability types the notion of semistability. This phenomenon may be present only if one of the eigenvalues of the Jacobian of the map is unity. We provide the sufficient conditions for the existence of a semistable equilibrium, using the center manifold theory. Then we show that the existence of semi-stable equilibrium points increases dramatically the number of the possible cases of 3- or 4-point cores. Several examples will be provided to illustrate our theory.
Wentzell Boundary Conditions in the Context of Structured Populations and Mathematical Epidemiology
University of Louisiana at Lafayette
We introduce and analyze a nonlinear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment which carries mass, hence the model is equipped with Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for example Wolbachia in mosquitoe populations, hence the only nonlinearity arises in the recruitment term. Well posedeness of the model and the Principle of Linearised Stability follow from standard semilinear theory. In our main result we establish existence of non-trivial steady states to the model. Our method utilizes an operator theoretic framework with a fixed point approach.
Dynamics of a Prey-Dependent Digestive Model with State-Dependent Impulsive Control
University of Texas – Pan American
In this talk, we study the dynamical behavior of a prey-dependent digestive model with a state-dependent impulsive effect. Using the Poincare map and the Lambert
W-function, we find the analytical expression of discrete mapping. Sufficient conditions are established for the transcritical bifurcation and the period-doubling bifurcation through an analytical method. Exact locations of these bifurcations are explored. Numerical simulations of an example are illustrated which agree well with our theoretical results.
Cattle exhibit different dynamics of fecal shedding depending on the infecting strain of Escherichia coli O157:H7
Texas A&M University
Escherichia coli O157:H7 is an important human pathogen with a natural reservoir in cattle. An experimental inoculation study was conducted using three E. coli O157: H7 strains to investigate the strain effect on the pattern and dynamics of E. coli O157:H7 fecal shedding in cattle. Twelve steers were randomly assigned to strain FRIK47, isolated from a human patient (human isolate), while also known to be prevalent in cattle, and 6 steers each were randomly assigned to strains FRIK1641 and FRIK2533, both of which were originally isolated from cattle farms (farm strains). Fecal samples were collected the day after inoculation and every two days thereafter until 30 days post inoculation to monitor shedding of E. coli O157:H7 in feces. Shedding of one of the farm strain (FRIK2533) was markedly different from the other two strains; this strain was shed by only two steers and only on the second day following inoculation. The other two strains were shed repeatedly by most inoculated animals. To compare the dynamics and pattern of fecal shedding between steers inoculated with these two strains, the human isolate (FRIK47) and farm (FRIK1641) strain, a multistate Markov chain model was developed. The model included 3 transient (representing latency, shedding, and non-shedding) states and one absorbing state representing recovery. The risk of progression from latency and shedding to recovery for the human isolate was marginally significantly lower compared to the farm strain FRIK1641 (hazard ratio=0.47, 80% CI= 0.22, 0.99), suggesting a longer overall duration of host infection with the human isolate. Likewise, the average total time spent in the shedding state during infection was considerably higher for the human isolate (15 days) compared to the farmFRIK1641 strain (4 days). Considering the crucial role that pathogen shedding has on transmission, the findings of this study indicate that genetic composition of E. coli O157:H7 could strongly affect its transmissibility in the host population. Furthermore, the existing genetic diversity of E. coli O157:H7 in cattle herds could be the reason for variability in the observed prevalence in fecal shedding of E. coli O157:H7 among cattle populations.
Modeling and Analysis of Controlled Forest Dynamics under Environmental Changes
Prarie View A&M University
Co-Authors: Yuri Yatsenko (Houston Baptist University) and Renan-Ulrich Goetz (University of Girona, Campus Montilivi)
A mathematical model is suggested to depict the dynamics of a size-structured population of trees with intra-species competition and the environmental impact on the forest population. The dynamics of forest and carbon accumulation in the wood and soil are described by a system of integral and partial-differential equations. The objective function includeds the revenue from timber production, operational expenses, and profit from carbon sequestration. The possible climate change scenarios are taken from the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. The proposed environmental-economic model has been approbated on real data on Spanish forest industry. The goal of a qualitative analysis is to understand how environmental changes impact the biological processes of forest growth and carbon sequestration, and find the optimal regime of forest management.
Stability analysis of the spatially distributed dynamical model of the wound healing
Texas Tech University
There is an intensive research interest in the area of wound healing modeling. In this work we are modeling wound healing as an inflammatory response of the immune system. The basic frame work is based on the Keller–Segel chemotaxis model. This paper introduces a kinetics-based model for analyzing reactions of various cells/proteins and biochemical processes as well as their transient behavior during wound healing in 3-D dimensional bounded domain. Inflammatory response initiation is studied through a linear, asymptotic stability analysis of non-trivial equilibrium state. In the main theorem we provide sufficient condition on the set of the parameters which guarantee conditional stability of the system in the appropriate Sobolev space. Constraints of the stability have clear biological interpretation and supported by set of counterexamples.
Public Avoidance and Epidemics: Insights from an Economic Model
Wake Forest University
We present a mathematical model of infectious disease transmission in which people can engage in public avoidance behavior to minimize the likelihood of acquiring an infection. The framework employs the economist's theory of utility maximization to model people's decision regarding their level of public avoidance. We derive the reproductive number of a disease which determines whether an endemic equilibrium exists or not. We show that when the contact function exhibits saturation, an endemic equilibrium must be unique. Otherwise, multiple endemic equilibria that differ in disease prevalence can coexist, and which one the population gets to depends on initial conditions. Even when a unique endemic equilibrium exists, people's preferences and the initial conditions may determine whether the disease will eventually die out or become endemic.
Interactions Between Allee Effects and Competition in Discrete Time Two Species Models
Arizona State University
Many species can experience both Allee effects and competitions with consequences that their populations do not grow optimally at low densities and individuals compete with one another at high densities. Understanding how interactions between Allee effects and competition affecting population dynamics can advance our understanding of the extinction and establishment of species in ecology communities, with implications for conservation programs. In this lecture, we first introduce basic ecological concepts of Allee effects and competition for population models. Then we compare the population dynamics of single species models subject to Allee effects and two-species competition models without Allee effects to the dynamics of two-species competition models subject to Allee effects. The comparison indicates that:
1. Weak Allee effect induced by Predator saturated can promote permanence of two competing species.
2.Inter-specific competition can save endangered species subject to strong Allee effects from extinction at high densities.
Biological implications of these two interesting phenomenon are given.
Mixed Strategies, Evolution and the Tragedy of the Commons in Ecological Niche Construction
Arizona State University
A number of theoretical models that are focused on ecological interactions of individuals and their environment assume population homogeneity. In this paper we argue that ignoring heterogeneity results in ignoring a large component of natural selection, since intraspecies competition may impose just as much of a selective pressure on each individual as do interactions with the environment and other species. The proposed mathematical model of ecological niche construction describes the interactions of a parametrically heterogeneous population with a common renewable resource and illustrates the importance of individual variation within the population using analytical methods. We classify the possible dynamic regimes of the system and demonstrate that if one wants to predict where a system will evolve, just knowing the rules that govern its dynamics is not enough to make an accurate prediction. One will also need to know the composition of the population that is playing by these rules.
Natural Selection on Seed Size in Dithyrea Californica
University of Arizona
Introduction: Fitness consequences in seedling survival and reproductive success in relation to seed size and plant density were investigated for four consecutive years on a population of a desert annual plant, Dithyrea californica. Seed size in this system can be readily measured in its natural setting even after the seed has germinated, making our system ideal for the direct evaluation of the fitness consequences of seed size in a natural setting and hence natural selection on seed size in the wild. Seedling survival was measured in a year where the cohort of plants had initial conditions for germination but not subsequent rains to fuel growth. Remaining mericarps were recovered in mortality censuses to relate survivorship with the size of their parent seeds.
Reproductive success was measured as the number of seeds produced by individuals in the population and then relating it to the size of their parent seeds and the crowdedness around them as competition.
Results/Conclusions: Survivorship analysis demonstrated that seedlings from bigger seeds survive better in the face of drought (X2=6, d.f.=1, p=0.015). Bigger seeds are better provisioned with maternal resources that help them survive better.
In adulthood, seed size was positively correlated with fecundity. However, plant density exerted a negative effect on seed size selection in such a way that very dense patches make seed size irrelevant (F2,86 = 25.63, p= 0.001, R2= 0.37). These results suggest that competition favors the production of smaller seeds, less competitive but with more colonization abilities. This counterintuitive pattern might be the consequence of a loss in variation in seed sizes due to selection for larger seeds in seedling survival and establishment.
Discrete-Time Models for the Transmission of Mosquito-Borne Disease and Their Dynamics
University of Alabama, Huntsville
We formulate simple discrete-time SEIR (susceptible-exposed-infective-recovered) epidemic models for the transmission of mosquito-borne diseases, based on different time-steps and different selections of model dynamics in the absence of infection. Fundamental investigations for the dynamics of these models, such as the derivation of a formula for the reproductive number and the determination of existence of an endemic equilibrium, and numerical simulations to demonstrate the model dynamics are presented. The modeling and further research are also discussed.
Modeling the Dynamics of Woody Plant-Herbivore Interactions with Age-Dependent Toxicity
University of Wyoming
We study the effects that woody plant chemical defenses may have on interactions between boreal hares that in winter feed almost entirely on twigs. We focus particularly on the fact that toxin concentration often varies with the age of twig segments. The model incorporates the fact that early in the growth of twigs, segments are often highly defended by toxins and are, therefore, highly unpalatable to hares. But that after a year or two, the toxin concentration of older twig segments is sufficiently reduced for the their biomass to be more palatable. This age-dependent toxicity of twig segments is modeled using age-structured model equations which are reduced to a system of delay differential equations involving multiple delays in the woody plant - hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems.
St. Mary’s University
A gradostat model of resource competition and allelopathy
Resource competition and persistence of algal populations in a reservoir can be dependent on the rate of flow. At a shoreline or flow impediment, reduced flow may result in storage of both algal population and allelopathic toxins. A gradostat model of resource competition, incorporating both allelopathy and resource recycling, was developed and analyzed. The study includes stability conditions and validation for parameters taken from Prymnesium parvum.
A Refuge-mediated Apparent Competition Model
The influence of a resource subsidy on predator-prey interactions
University of Central Florida
We study the influence of a donor-controlled resource subsidy on predator-prey interactions. The prey increases logistically, the subsidy appears arithmetically, and the predator experiences satiation. In one model, the prey and subsidy are found together, and in a second they are spatially separated. Criteria for feasibility and stability of the different equilibrium states are discussed. Implications for a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy) are considered.
Theoretical Investigation of the Effects of Large Scale Iron Fertilization of the Oceans
S.C. Oukouomi Noutchie
North-West University (South Africa)
The Iron Hypothesis, formulated by oceanographer John Martin, and first tested in 1993, suggested that fertilizing the oceans with iron would draw down atmospheric CO2. Unfortunately, the overwhelming majority of the scientific evidence available to date indicates that fertilization results in a short term increase in phytoplankton biomass that is then consumed and respired by heterotrophs, returning the CO2 immediately to the atmosphere. In this paper we use mathematical tools to investigate the validity of the Iron Hypothesis. The model which we study recognizes the role played by the phyotoplankton aggregates which are viewed as individual building blocks labeled by their size. The evolution of the system is captured by nonlinear integro-differential equation containing terms responsible for the coalescence, fragmentation, growth and removal of the aggregates. Based on the outcomes of the analysis, we reformulate the Iron Hypothesis and demonstrate that the level of atmospheric CO2 could be dramatically reduced. In particular, the prescribed iron fertilization method happens to accelerate fish production.
Marine bacteriophage infection with constant latency period - a model based study
University of Kalyani
We consider and analyze a mathematical model for the marine bacteriophage infection with latency period (L). In recent time many researchers work on marine virus and they show that the marine bacteriophage infection induces bacterial lysis by which viruses release into the marine environment on average rate. In addition with this we assume that within the constant latency period L, the released virus newly infect susceptible bacteria at the rate ψ for which the growth of virus is at the rate The dynamical behavior of the modified system is studied by showing bounds of solution, stability analysis, hopf bifurcation and persistence criterion. A threshold b* exists and the disease free equilibrium Ef is locally and globally stable for 1 < b < b* and for b > b* the positive equilibrium E+ bifurcates from Ef . A concluding discussion using numerical simulation is then presented.
The evolution of alternative transmission modes in sylvatic Trypanosoma cruzi
University of Texas at Arlington
Pathogens may use different routes of transmission to maximize their spread among host populations. Horizontal (i.e., through host contacts) and vertical (i.e., from mother to offspring) transmission modes are generally associated with different parasite strategies. Highly virulent pathogens are more efficiently transmitted horizontally because they usually produce larger parasite loads. On the other hand, because the efficiency of vertical transmission is dependent on host fitness, less virulent pathogens causing less harm to their hosts seem be associated with this transmission route. Among directly-transmitted diseases, more virulent pathogens also tend to be favored in large host populations in which impairing host fitness does not compromise disease transmission. In vector-borne diseases, pathogen transmission rate not only depends on the total population size but on the relative proportion of vectors and hosts. How this could affect the evolution of different transmission modes has been largely overlooked. The parasite Trypanosoma cruzi which is transmitted by Triatomine bugs to several Vertebrate hosts is responsible of Chagas disease in Latin America. T. cruzi is also widespread in the Southeastern U.S. where it infects sylvatic hosts such as raccoons, opossums or rodents. Besides classical transmission involving vectors feeding on hosts, other alternative transmission routes such as vertical and oral transmission (via the consumption of vectors by hosts) have been reported in the sylvatic T. cruzi cycles. The two major T. cruzi strains occurring in the U.S. show different efficiencies at vertical and horizontal transmission, which suggests that a trade-off occurs between these two transmission modes. We were able to show mathematically with an epidemiological two-strain model that the outcome of the competition between the two parasites is affected by the ratio of the number of vectors per host. Our model does not involve parasite virulence (typically, T. cruzi does not harm sylvatic hosts) and the result holds providing that oral transmission also occurs and that oral and horizontal transmission rates saturate for different vector / host ratios. This result opens new perspectives for the understanding of the competition and specialization of the T. cruzi strains occurring in the U.S. but also provides new insights into the evolution of transmission modes in vector-borne diseases.
Modeling Cell Migration in the Presence of Spatially Varying Chemoatractans
Alicia Prieto Langarica
University of Texas at Arlington
Cell migration plays a vital role throughout the life span of organisms. In many cases, this movement is governed by the presense of certain chemicals which attract or repell the cells. We use a previously developed cellular automata model that describes the movement of cells which are attracted to a chemicalt in the environment. We assume that the concentration of the chemical varies in space and the cells move toward greater chemical concentrations with greater probability.
In addition, we allow cell to remain in their current place with certain probability depending on the concentration of the chemical in the current site. Next, we upscale the dicrete mathematical model to a continuous partial differential equation model using transition probabilities dependent on space and time. Other works have done the upscaling of discrete models were cell are constantly moving and the direction of the movemnt is governed by a constant concentration of chemoattractants, while in our CA, we allow cells to stay in their current site and the chemoatractant varies in space and time.
Species abundance distributions in a stochastic competition model
Pacific Ecoinformatics and Computational Ecology Lab
Species abundance distributions may reflect the dynamic processes that influence fitness in a community, but the question of how they do so remains open in ecology. Recent studies focus on the relative importance of two types of mechanisms: niche and neutral dynamics. Neutral dynamics are based on demographic stochasticity and immigration and niche dynamics are generated by trait differences that affect the fitnesses of competing species. One recent study showed that abundance patterns produced under niche and neutral dynamics are very similar, especially when diversity is relatively high compared with the number of niches. In this study, species in separate niches are essentially non-interacting. To investigate these issues, we compare the patterns resulting from a stochastic Lotka-Volterra competition model in which all species are interacting with one another. This model generates neutral communities when competition does not depend on traits, and communities with niches when competition declines with trait distance. We find more substantial differences in species abundance distributions of these two types of communities than have been previously shown, suggesting that mechanisms that produce niches also influence abundance patterns. In particular we show that interactions between niches increases mean species abundance, and that fitness differences within a niche largely influence the differences we see species abundance distributions between the two types of communities.
Non-Iterative Numerical Integration Method for Singular Perturbation Problems Exhibiting Internal and Twin Boundary Layers
National institute of Technology, Warangal
In this paper, a non-iterative numerical integration method is developed on a uniform mesh for a class of singularly perturbed two-point boundary value problems exhibiting internal and twin boundary layers.
This method is non-iterative on a small deviating argument which converts the original second order boundary value problem to the first order differential equation with the deviating argument. By applying numerical integration method on first order differential equation, tridiagonal scheme is obtained and is solved efficiently. This method is noniterative and very easy to implement. Relative errors with L2-norm are presented to illustrate the proposed method.
A Model for Antibiotic-Resistant Infections with Application of Optimal Control Theory
Joaquin Rivera Cruz
Healthcare associated infections caused by medical-facility-borne antibiotic-resistant bacteria have become a costly problem that compromises medical care in hospitals worldwide. We developed a compartmental model that focuses on the evolution of two bacterial strains (drug-resistant and non-drug resistant) residing within the patient population and healthcare workers in a hospital. Reformulating the model as an optimal control problem, the model predicts that the non-resistant bacteria will decrease and eventually reach a very low level after the control is applied. Moreover, in contrast to the differential equation model, the drug-resistant bacteria will persist, possibly because of mutation, but it will take about five times longer to reach the maximum levels.
Modeling the spread of waterborne disease: Incorporating heterogeneity in multiple transmission pathways
Ohio State University
Heterogeneity is a fundamental issue in mathematical epidemiology. We expect many factors influencing disease transmission to vary across populations and different spatial scales. Many results exist for the effect of heterogeneity on the spread of disease for SIR type models, where transmission occurs as a result of direct contact with infected individuals. Waterborne disease, such as cholera, may be spread through contact with a contaminated water source as well as through direct person-person transmission. We investigate the effect of heterogeneity in both transmission pathways on the value of the basic reproductive number R0 in multi-patch SIWR models, specifically a system of N patches sharing a common water source.
Texas Tech University
We consider a model of three differential equations for a food resource and the competition of two plant species while the competition is apparent: an indirect interaction where the invading plants provide a refuge for a shared consumer, subsequently increasing consumer pressure on the resident plant species. We assume that the resident species is the superior competitor. The equilibria and the local stability of all equilibria are shown. A Lyapunov function is found to show the global stability of the equilibrium in which only the resident species survives.
Within-host dynamics of influenza virus infection
Influenza virus infection remains a public health problem worldwide. The biological mechanisms underlying viral control during infection are not fully understood. Here, we developed a new mathematical model including both innate and adaptive immune responses to study the within-host dynamics of equine influenza virus infection in horses. By comparing modeling predictions with both interferon and viral kinetic data, we examined the relative roles of target cell availability, innate and adaptive immune responses in controlling the virus. This study provides a detailed and quantitative understanding of the biological factors that can explain the virus and interferon kinetics during a typical influenza virus infection.
Some New Results in Periodic Difference Equations of Mathematical Biology: The Underestimated Role of Semigroups
University of Southern California
We present some generalizations of recent results found in the literature. A general theorem developed during a Summer NSF Session on Research Experience for Undergraduates will be presented and applied to obtain more general theorems with much shorter proofs.
Uniform Persistence and Competitive Exclusion of Pathogens in a Nonautonomous SIR
Epidemic Model with Variable Population Size
University of Louisiana at Lafayette
In this paper we consider a nonautonomous version of the SIR epidemic model in , for competition of n infection strains in a host population. We give sufficient conditions for the robust uniform persistence of the total population, as well as of the susceptible and infected subpopulations. The first two forms of persistence depend entirely on the rates at which the disease is vertically transmitted to offspring. We also discuss the competitive exclusion among the n infection strains, namely when a single infection strain survives and all the others go extinct.
Physiologically-based pharmacokinetic (PBPK) modeling of metabolic pathways of bromochloromethane
North Carolina State University
Co-Authors: William Cuello (University of California Berkeley), Tyler James (University of Colorado Boulder), Jill Jessee (Simpson College), Melissa Venecek (The College of Wooster), Marina Evans and Chris Eklund (U.S. Environmental Agency)
Bromochloromethane (BCM) is a volatile compound that is transformed into a toxic compound via metabolism in the liver and kidne. Using a physiologically-based pharmacokinetic model, we explore two hypotheses describing the metabolic pathways of BCM in rats: a two-pathwaymodel using both the enzyme CYP2E1 and the enzyme glutathione transferase for metabolism, and a two-binding site model where metabolism can occur at a hypothesized second binding site on CYP2E1. This project will help determine the difference in metabolic mechanisms, which is important in determining the mode of action for this chemical and similarly structured compounds. After optimizing for the metabolic pathway models generate curves that fit our data well; however, the two-binding site model more accurately fits the data obtained at higher concentrations of BCM. In addition, we explore the sensitivity of different parameters for each model using our obtained optimized values as well as for regions around these values. (This abstract does not represent EPA policy).
Controlling of Chaos in a Food Chain Model
Indian Institute of Technology Roorkee
The control of chaotic behavior occurred in a food chain model is attempted through various techniques. Feedback and Non-feedback control techniques are proposed to suppress chaos to unstable equilibrium or periodic solutions. The stabilization of unstable fixed point of the chaos is achieved by bounded feedback method. Stability of controlled system is analyzed by Routh-Hurwitz Criterion. Non-feedback method is based on delayed feed back control are used to control chaos to periodic orbits. Numerical results substantiate the analytical findings.
Delay Model of the Adaptive Immune response in HBV infection and optimal antiviral therapy
Arizona State University at Polytechnic Campus
The characterization of the functional defects of adaptive immunity in patients with HBV infection is still largely incomplete. The aim of this talk is to introduce mathematical analysis of two models that describe the adaptive immune response to Hepatitis B virus (HBV) infection and the antiviral therapy. The contribution of the B- cells is very important in the HBV control although these components of the adaptive immunity have less attracted scientific attention in comparison to T cells . The findings of the first model suggest a possible explanation of the failure of the immune response to clear the HBV infection. This situation is described by the over domination of humoral immune response.
Moreover, the immune response by CD8+ is not immediate and the presence of the time delay between infection and this immune response was observed in HBV infection, particularly for patients with co-infection and immunocompromise. The second model we will study this situation and will also present an optimal control strategy of the antiviral therapy.
Mathematical Population Dynamics and Resource Competition
St. Mary's University
A new model for two-species chemostat with allelopathy will be presented. The model incorporates an assumption that toxin-production carries a 'metabolic load,' inflicting a poison-production penalty on the growth of the environment-poisoning species. Examining the local asymptotic behavior the parameter ranges were found for the four stable states: extinction of both species, survival of poison-producer, survival of poison-susceptible, coexistence. Simulations will be presented with parameters coming from P. Parvum.
An Optimal Control Study for Cholera
Old Dominion University
Cholera is a severe water-borne infectious disease which remains an important global cause of morbidity and mortality. Building on a recently developed mathematical model, we perform an optimal control study for cholera epidemics. We apply optimal control theory to seek a cost-effective balance of different intervention methods, including vaccination, antibiotic treatment, and water sanitation. We also present numerical simulation results to quantify the analysis.
Statistical Models for Health Care Utilization of Hispanic Elder Diabetics
University of Texas-Pan American
Diabetes is a costly disease, whose complications can be significantly delayed or relieved by regular health care and self-management. Access to healthcare is important for managing diabetes; however, little is known about predictors of healthcare utilization among minorities. This study focuses on determining personal and social correlates to health care utilization among border Hispanic seniors with diabetes. A community assessment survey is conducted. We use statistics models to study personal and social correlates to health care utilization among border Hispanic seniors with diabetes.
**Updated September 23, 2011