My mathematical research interests involve applications of dynamical systems to epidemic modeling. More specifically, my interests include the transmission of Chagas' disease, applications to sociological phenomena driven by peer pressure (such as grassroots political movements, eating disorders, and cooperative learning), and theoretical models of bistability in population biology. Following are brief descriptions of some of these projects.
Dynamical Systems for Biological Modeling: An Introduction -- undergraduate textbook, 2015, Chapman and Hall/CRC Press, ISBN 9781420066418
Abstract. An SIS model for a heterosexually transmitted disease with core and noncore compartments and a generalized recovery function P(t) is analyzed. It exhibits R0 threshold behavior and leads to discussions of stability with respect to choice of P(t), and of the effects of allowing recruitment between core and noncore groups.
Abstract. We consider a set of SIS models for a heterosexually transmitted disease in which there is recruitment between core and noncore subpopulations as a function of prevalence of the disease. Behavior diverges from the traditional R0 threshold behavior and yields an extra pair of endemic equilibria in one case and a limit cycle in the other. Total at-risk population is constant.
Abstract. Epidemic models make use of a function P(t) which describes the rate of recovery; most commonly P(t) is chosen to be an exponential decay or a step function, resulting in models which are ODEs and DDEs, respectively. We provide a way to compare different choices with respect to conduciveness to stability of the corresponding characteristic equations and give some results and examples to indicate to what extent we can try to order the functions in this way.
Abstract. A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign � meant to reduce a disease's reproduction number R(�) below one may fail to control the disease: if the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.
Abstract. Stability analysis of endemic equilibria in epidemic models may be complicated. We apply techniques of center manifolds and normal forms to determine stability through bifurcation analysis. Some examples lead to a discussion of backward bifurcations and multiple endemic equilibria for R0<1. It is observed that backward bifurcations are sufficient but not necessary to cause endemic stationary states to occur when R0<1.
Abstract. We consider models for a disease with acute and chronic infective stages, and variable infectivity and recovery rates, within the context of a vaccination campaign. Models for SIRS and SIS disease cycles exhibit backward bifurcations under certain conditions, which complicate the criteria for success of the vaccination campaign by making it possible to have stable endemic states when R0<1. We also show the extent to which the forms of the infectivity and recovery functions affect the possibility of backward bifurcations. SIR and SI models examined do not exhibit this behavior.
Abstract. For at least the past ten years, eating disorders have had a major impact in the physical and mental health of women, particularly young women. Anorexia and bulimia nervosa are closely linked eating disorders. Anorexia often precedes bulimia. However, there are about 2 million women in college that have been exclusively bulimic. In this article, we focus on the role of college-peer pressure on the dynamics of anorexia-free bulimia. The model looks at bulimia as a progressive disease and explores the impact of intervention (treatment) at two stages of disease progression. The impact of relapse (a common occurrence among bulimics) is taken into account. Analysis indicates that the disorder cannot be wiped out in this population without a shift in cultural pressures; control strategies should include early detection and treatment, as well as preventative education campaigns.
Abstract. Genetic studies report the existence of a mutant allele D32 of CCR5 chemokine receptor gene at high allele frequencies (~10 %) in Caucasian populations. The presence of this allele is believed to provide partial or full resistance to HIV. In this study, we look at the impact of education, temporarily effective vaccines and therapies on the dynamics of HIV in homosexually active populations. In our model, it is assumed that some individuals possess one or two mutant alleles (like D32 of CCR5) that prevent the successful invasion or replication of HIV. Our model therefore differentiates by genetic and epidemiological status and naturally ignores the reproduction process. Furthermore, HIV infected individuals are classified as rapid, normal or slow progressors. In this complex setting, the basic reproductive number R0 is derived in various situations. The separate or combined effects of therapies, education, vaccines, and genetic resistance are analyzed. Our results support the conclusions of Hsu Schmitz that some integrated intervention strategies are far superior to those based on a single approach. However, treatment programs may have effects which counteract each other, as may genetic resistance.
Abstract. Control of the 2002-2003 Severe Acute Respiratory Syndrome (SARS) outbreak was based on two intervention strategies: rapid diagnosis and effective patient isolation. Models of SARS include many more parameters than these two, and it has not been determined what would constitute an optimal control strategy. We use uncertainty and sensitivity analysis of the basic reproductive number (R0) to assess the importance of the roles that all of the parameters from a 10-parameter model play in outbreak control. Our estimate for the reproductive number R0 under perfect isolation of diagnosed cases, and with the other epidemic parameters distributed according to epidemiological data, lies in the interquartile range (IQR) (0.19 -- 1.08) with a median of 0.49 under a reduced effective population size. Even though the median of R0 is less than one, we find that 25% of our R0 distribution lies at R0>1, even when control is achieved. We show that "superspreaders" and isolation effectiveness have the largest fractional effect in determining the magnitude of R0. While effective patient isolation procedures are widely recognized as essential to outbreak control, intervention strategies designed to lower the transmission rate of superspreaders promise an effective mean for lowering R0. We identify the parameter ranges corresponding to R0<1. We briefly discuss approaches that would also consider costs as part of a policy of epidemic control.
Abstract. Many situations in population biology involve a rate --- typically a contact or recruitment rate --- which increases linearly for small populations but reaches a maximum value (saturates) for large populations. Models for populations of variable size need to incorporate both characteristics to predict behavior accurately. This can be done by defining the rate as a continuous, piecewise linear function with a switch point, or via a Verhulst-type (smooth) saturation function. This paper presents several examples of both approaches and draws some conclusions about the differences from a modeling perspective.
Abstract. We consider an elementary mathematical puzzle known as a ``difference box'' in terms of a discrete map from R4 to R4 or, canonically, from a subset of the first quadrant of R2 into itself. We find the map's unique canonical fixed point and answer the general question of how many iterations a given ``difference box'' takes to reach zero.
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Abstract. The AIDS epidemic is having a growing impact on the transport sector of the economy of sub-Saharan Africa, where long-distance truck drivers are at an increased risk of infection due to their frequent contacts with commercial sex workers. The spread of AIDS in the transport industry is especially significant to the economy, as truck drivers are largely responsible for transporting crops and supplies needed for daily subsistence. In this paper we analyze these effects via two models, one employing a switch and the other a Verhulst saturation function, to describe the rate at which new drivers are recruited in terms of the supply and demand for them in the general population. Results provide an estimate of the epidemic's economic impact on the transportation sector through the loss of truck drivers (an estimated 10% per year, with endemic levels near 90%).
Abstract. Recent research in the transmission of the protozoan parasite Trypanosoma cruzi, some strains of which cause Chagas' disease, suggests that consumption of vectors by sylvatic hosts such as raccoons may play a role in maintaining the transmission cycle in the wild. Since both hosts and vectors have been observed to invade new ecological niches, it is important to consider the effect vector consumption may have on vector density. For this reason we consider a per capita contact rate which rises roughly linearly for low vector densities and saturates for high densities. This paper analyzes the effects of these features by superimposing a predator-prey structure on a host-vector infection model (with first one, and then multiple, hosts). Outbreak behavior follows classical threshold behavior via the reproductive number R_0, which allows evaluation of this transmission avenue's relative importance. For sufficiently sharp contact rate saturation, two locally stable vector densities may exist.
Abstract. Understanding the relationship between coinfection with multiple strains of human papillomavirus and cervical cancer may play a key role in vaccination strategies for the virus. In this article we formulate a model with two strains of infection and vaccination for one of the strains (strain 1, oncogenic) in order to investigate how multiple strains of HPV and vaccination may affect the number of cervical cancer cases and deaths due to infections with both types of HPV. We calculate the basic reproductive number $R_i$ for both strains independently as well as the basic reproductive number for the system based on $R_1$ and $R_2$. We also compute the \textit{invasion reproductive number} $\tilde{R_i}$ for strain \textit{i} when strain \textit{j} is at endemic equilibrium ($i\neq j$). We show that the disease-free equilibrium is locally stable when $R_0=max\{R_1,R_2\}<1$ and each single strain endemic equilibrium $E_i$ exists when $R_i>1$. We determine stability of the single strain equilibria using the invasion reproductive numbers. The $R_1,R_2$ parameter space is partitioned into 4 regions by the curves $R_1=1, R_2=1, \tilde{R_1}=1$, and $\tilde{R_2}=1$. In each region a different equilibrium is dominant. The presence of strain 2 can increase strain 1 related cancer deaths by more than 100 percent, but strain 2 prevalence can be reduced by more than 90 percent with 50 percent vaccination coverage. Under certain conditions, we show that vaccination against strain 1 can actually eradicate strain 2.
Abstract. Harvesting and predation occur via contact processes in which the rate at which the managed (prey) population can be found depends upon the population size, usually saturating at high densities. Many models incorporate saturation in this process without considering the effects of the particular function used to describe it. We show that the sharpness with which this saturation occurs has an important effect upon the resulting population dynamics, with bistability (sometimes involving a stable equilibrium and a stable limit cycle) occurring for saturation that is any sharper than the commonly used Michaelis-Menten (Holling type II) functional response. This sharpness threshold occurs across a wide range of model types, from simple harvesting to both density-dependent and ratio-dependent predation.
Abstract. "Kala-azar" (or Indian Visceral Leishmaniasis) is a vector-borne infectious disease affecting communities in tropical and subtropical areas of the world. Bihar, a state in India, has one of the highest prevalence and mortality reported levels of Kala-azar. Yet, the magnitude of the problem is difficult to assess because most cases are handled by private health providers who are not required to and do not report them to the Ministry of Health. The impact of underreporting using district-level reported incidence data from the state of Bihar is the main goal of this manuscript.We derive expressions for, and compute estimates of Kala-azar�s reproduction numbers, an indirect measure of disease prevalence, and levels of underreporting for the 21 most affected districts of Bihar. The average reproduction number (number of secondary cases generated per infective) estimates for Bihar range from 1.3 (2003) to 2.1 (2005) with some districts� estimates with mean values lower than one. Model estimates (using available data and a model-derived expression) show that the proportion of underreported cases declined from an average of 88% in 2003 to 73% in 2005. However, eight districts in 2003 and five districts in 2005 had more than 90% levels of underreporting. Model estimates are used to generate underreporting adjusted incidence rates. The analysis finds that reported data misidentify four of the eight (2003) and three of the nine (2005) districts classified as high-risk. In fact, seven (2003) and five (2005) of the most affected Kalaazar districts had been classified as low-risk when only reported incidence data were used.
Abstract.Although it has been known for nearly a century that strains of Trypanosoma cruzi, the etiological agent for Chagas' disease, are enzootic in the southern U.S., much remains unknown about the dynamics of its transmission in the sylvatic cycles that maintain it, including the relative importance of different transmission routes. Mathematical models can fill in gaps where field and lab data are difficult to collect, but they need as inputs the values of certain key demographic and epidemiological quantities which parametrize the models. In particular, they determine whether saturation occurs in the contact processes that communicate the infection between the two populations.
Concentrating on raccoons, opossums, and woodrats as hosts in Texas and the southeastern U.S., and the vectors Triatoma sanguisuga and Triatoma gerstaeckeri, we use an exhaustive literature review to derive estimates for fundamental parameters, and use simple mathematical models to illustrate a method for estimating infection rates indirectly based on prevalence data. Results are used to draw conclusions about saturation and which population density drives each of the two contact-based infection processes (stercorarian/bloodborne and oral).
Analysis suggests that the vector feeding process associated with stercorarian transmission to hosts and bloodborne transmission to vectors is limited by the population density of vectors when dealing with woodrats but by that of hosts when dealing with raccoons and opossums, while the predation of hosts on vectors which drives oral transmission to hosts is limited by the population density of hosts. The approaches developed here can also be applied to the study of other vector-borne infections.
Abstract.The combination therapy of antiviral peg-interferon and ribavirin has evolved as one of the better treatments for Hepatitis-C. In spite of its success in controlling Hepatitis-C infection, it has also been associated with treatment related adverse side-effects. The most common among them is hemolytic anemia necessitating dose reduction or therapy cessation. The presence of this side-effect leads to trade-off between continuing the treatment and exacerbating the side-effects versus decreasing dosage to relieve severe side-effects while allowing the disease to progress. The drug epoietin is often administered to stimulate the production of RBC in the bone marrow, in order to allow treatment without anemia. This paper uses mathematical models to study the effect of combination therapy in light of anemia. In order to achieve this we introduce red blood cell (RBC) concentration and drug amount as state variables in the usual immunological virus infection model. Analysis of this model provides a quantification of the drug amount that is tolerable by the body without succumbing to hemolytic anemia. Estimation of parameters allow us to calculate the necessary increment in RBC production to be 1.3-2.4 times the patient's original RBC production rate to sustain the entire period of treatment without encountering anemia in a sensitive patient.
Abstract. This paper introduces a simplified framework for the study of the mechanisms behind the growth of cooperative learning in large communities. We begin from the simplifying assumption that individual-based learning focuses on increasing the individual's "fitness" while collaborative learning may result in the increase of the group's fitness. It is not the objective of this paper to decide which form of learning is more effective but rather to identify what types of social communities of learners can be constructed via collaborative learning. The potential value of our simplified framework is inspired by the tension observed between the theories of intellectual development identified with the views of Vygotsky and Piaget. Here they are mediated by concepts and ideas from the fields of epidemiology and evolutionary biology. The community is generated from sequences of successful "contacts" between various types of individuals. Our results allow us to discuss the impact of individual learning on community intellectual development and the resilience of communities constructed via multilevel epidemiological contact processes. Our simple cooperative framework is used to address the generalized belief that sharp community thresholds characterize separate learning cultures. Finally, we provide an example of an application of the model. The example is autobiographical as we are members of the population in this "experiment".
Abstract.The parasite Trypanosoma cruzi, which causes Chagas' disease, is typically transmitted through a cycle in which vectors become infected through bloodmeals on infected hosts and then infect other hosts through defecation at the sites of subsequent feedings. The vectors native to the southeastern United States, however, are inefficient at transmitting T. cruzi in this way, which suggests that alternative transmission modes may be responsible for maintaining the established sylvatic infection cycle. Vertical and oral transmission of sylvatic hosts, as well as differential behavior of infected vectors, have been observed anecdotally. This study develops a model which accounts for these alternative modes of transmission, and applies it to transmission between raccoons and the vector Triatoma sanguisuga. Analysis of the system of nonlinear differential equations focuses on endemic prevalence levels and on the infection's basic reproductive number, whose form may account for how a combination of traditionally secondary infection routes can maintain the transmission cycle when the usual primary route becomes ineffective.
Abstract.A classical epidemiological framework is used to provide a preliminary cost analysis of the effects of quarantine and isolation on the dynamics of infectious diseases for which no treatment or immediate diagnosis tools are available. Within this framework we consider the cost incurred from the implementation of three types of dynamic control strategies. Taking the context of the 2003 SARS outbreak in Hong Kong as an example, we use a simple cost function to compare the total cost of each mixed (quarantine and isolation) control strategy from a public health resource allocation perspective. The goal is to extend existing epi-economics methodology by developing a theoretical framework of dynamic quarantine strategies aimed at emerging diseases, by drawing upon the large body of literature on the dynamics of infectious diseases. We find that the total cost decreases with increases in the quarantine rates past a critical value, regardless of the resource allocation strategy. In the case of a manageable outbreak resources must be used early to achieve the best results whereas in case of an unmanageable outbreak, a constant-effort strategy seems the best among our limited plausible sets.
Abstract.Third political parties are influential in shaping American politics. In this work we study the spread of a third party ideology in a voting population where we assume that party members/activists are more influential in recruiting new third party voters than non-member third party voters. The study uses an epidemiological metaphor to develop a theoretical model with nonlinear ordinary differential equations as applied to a case study, the Green Party. Considering long-term behavior, we identify three threshold parameters in our model that describe the different possible scenarios for the political party and its spread. We also apply the model to the study of the Green Party's growth using voting and registration data in six states and the District of Columbia to identify and explain trends over the past decade. Our system produces a backward bifurcation that helps identify conditions under which a sufficiently dedicated activist core can enable a third party to thrive, under conditions which would not normally allow it to arise. Our results explain the critical role activists play in sustaining grassroots movements under adverse conditions.
My educational research interests focus on the development of mathematics teachers' attitudes, beliefs and teaching practices as they move toward instructional practices based on knowledge about children's learning. Many mathematics teachers, especially in the elementary grades, have had bad experiences with mathematics in their own education, and much research in recent years has gone into determining what (knowledge, experiences, skills) they need in order to be able to help children learn mathematics meaningfully and effectively. My joint appointment allows me to see both the content and the pedagogical sides of this issue. My current interests involve identifying the mathematics (content) particular to teaching, and observing how the changed view some students develop of what it means for themselves to do mathematics interacts with their later education coursework and K-12 classroom experiences to shape their approaches to teaching.
Abstract. A brief case study of prekindergartners puzzling over a discrepancy in informal units of measure serves as a focus for a discussion of how teachers need to observe students' thought processes.
Abstract. A set of exemplary lessons (with teacher's notes) for middle-school mathematics classes.
Abstract. This text gives an explanation, including several thoroughly-discussed sample problems, of how the college mathematics experiences of future mathematics teachers (at all grade levels) can be improved, including clarification of some of the Texas state certification standards.
Abstract. A 30-hour professional development workshop for (primarily middle grades) mathematics teachers, with a focus on problem solving as a way of teaching, rather than an isolated topic.
Abstract. The generalization of a classic cryptarithmetic puzzle provides an opportunity to re-examine the underpinnings of the traditional algorithm for multidigit addition.
Abstract. Este trabajo recoge un estudio sobre las estrategias de c�lculo desarrolladas por estudiantes de sexto a�o de educaci�n primaria y por maestros, el cual revel� el uso de m�todos de dos etapas para problemas de divisi�n de fracciones cuotitiva y partitiva. Estos m�todos a�aden un paso m�s a las estrategias de un solo paso desarrolladas para la divisi�n de n�meros enteros; el paso adicional convierte unidades por medio de multiplicaci�n (la unitizaci�n). Tambien se observa el uso de las unidades de co-medida, y algunos errores comunes.
Abstract. This paper extends a popular high school problem called "Painting the Cube", which relates algebra and geometric patterns in composite cubes, to the problem of a large regular tetrahedron composed of unit tetrahedra and octahedra. Solution of the problem involves measuring area and volume in triangular (and tetrahedral) instead of square (and cubic) units.
Abstract. A study of computational strategies developed by sixth-grade students and prospective and practicing teachers revealed two-step approaches for measurement and partitive division of fractions problems which extend one-step strategies developed for division of whole numbers by adding a step which converts units by multiplying (unitizing), as well as use of co-measure units, and some common errors.
Abstract. This paper applies the counting techniques of "Painting the pyramid" to a composite octahedron and then makes generalizations about the differences in growth patterns between composite simplices (triangles, tetrahedra, etc.) and composite squares and cubes, and how the octahedron shows both types of growth pattern.
Abstract. This article describes how sixth-grade students developed concrete models to solve division of fractions story problems. Students developed separate two-step procedures to solve measurement and partitive problems, drawing on invented procedures for division of whole numbers. Includes 8 story problems and selected student examples. (from NCTM)
Abstract. Originally entitled "Mathematics as a tool for inquiry in the literature classroom," this article examines ways in which mathematics (quantitative literacy) can enhance understanding of children's literature in the middle school classroom, with particular examples involving estimation and probability, the idea of infinity, and different perspectives (number systems as well as cultures).
Abstract. This paper describes observations from a mathematics course for prospective elementary mathematics teachers which suggest that teachers may develop invented computational strategies similar to those invented by children building an understanding of place value and multidigit arithmetic, and discusses the possible implications for the professional development of teachers.
It was during the winter of 1990, while teaching electrical engineering lab courses at Georgia Tech, that I first began to realize that teaching was my calling. Since then, I've taught almost every semester, at either the high school or college level, and even gotten into curriculum design and teacher preparation.
My teaching interests focus on making students active learners in the classroom. It is a real challenge to bring one's students into a dialogue in the classroom, especially in natural sciences such as mathematics, and to get them to understand the motivation and justification for a topic. I have had some success in this area with a variety of different types of class. I am also a firm believer in anchoring mathematical knowledge to real-world applications (especially modeling, which is a big part of my own research). In calculus and linear algebra classes, it is natural to discuss engineering applications, and expose the students to design issues. In classes usually taken by future teachers, the applications to the math they will teach are typically a more integral part of the course. And with regard to both types of classes, I am also interested in bringing appropriate technology into the classroom: it not only puts more mathematical power in students' hands, but helps make it easier to discuss realistic applications and models as well.
Currently I am teaching courses to prepare future elementary school teachers to teach math: a sequence of courses in the math department designed to prepare them mathematically, and one in teacher education designed to prepare them pedagogically. Before pursuing my Ph.D., I taught high school science and math for a year, in a school for principally Native American students, so I bring that experience to both of these classes. My math students work in cooperative small groups to develop their problem-solving skills and intuitive reasoning. These activities give them a firm foundation for the math they will be teaching. I helped develop a sequence of such courses at the University of Wisconsin as a graduate student, and have done so here at UTA as well.
If you have questions about any of this, just drop me a line at kribs@uta.edu
Last modified October 13, 2010