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I have recently studied several models which exhibit a common phenomenon, referred to in mathematical epidemiology as a backward bifurcation, which implies that a particular phenomenon (usually an outbreak of an infectious disease) can persist in a population under certain conditions which would not normally (that is, according to its basic reproductive number R0) allow it to do so. Mathematically, this implies that there are multiple transition cycles within the population as a whole, and that although the phenomenon is not "reproducing" well within the population as a whole, it is reproducing sufficiently well within some subgroup of the population to create a "reservoir", if that subgroup begins sufficiently large. Biologically speaking, this means that these processes have a robust structure which makes them more difficult to eradicate. This complicates the control of such diseases (or other phenomena, such as eating disorders, which can be considered to be peer-pressure-driven), but also promises survivability of other phenomena with this type of structure (e.g., cooperative learning environments). In my work with preservice and practicing teachers, I have observed invented computational strategies analogous to those developed by children. These parallels have great potential to shape teachers' classroom practice (in particular, their awareness of student invented strategies). One such recent study of mine involved the division of fractions.

• 1997 - PhD in Mathematics, University of Wisconsin at Madison
• 1991 - MSEE in Electrical Engineering, Georgia Institute of Technology
• 1989 - AB in Mathematics, Duke University
• 1988 - BSEE in Electrical Engineering, Duke University

Office:Pickard Hall Rm 483
kribs@uta.edu
817-272-5513

### Recent Submissions

• #### Graphical analysis of evolutionary trade-off in sylvatic Trypanosoma cruzi transmission modes ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington, 2014-07)
The notion of evolutionary trade-off (one attribute increasing at the expense of another) is central to the evolution of traits, well-studied especially in life-history theory, where a framework first developed by Levins ...
• #### Host switching vs. host sharing in overlapping sylvatic Trypanosoma cruzi transmission cycles ﻿

(Taylor & Francis OpenDepartment of Mathematics, University of Texas at Arlington, 2015-09)
The principle of competitive exclusion is well established for multiple populations competing for the same resource, and simple models for multistrain infection exhibit it as well when cross-immunity precludes coinfections. ...
• #### Agent-based mathematical modeling as a tool for estimating T. cruzi vector-host contact rates ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington, 2015-11)
The parasite Trypanosoma cruzi, spread by triatomine vectors, affects over 100 mammalian species throughout the Americas, including humans, in whom it causes Chagas’ disease. In the U.S., only a few autochthonous cases ...
• #### A metapopulation model for sylvatic T. cruzi transmission with vector migration ﻿

(American Institute of Mathematical ScienceDepartment of Mathematics, University of Texas at Arlington, 2014-06)
This study presents a metapopulation model for the sylvatic transmission of Trypanosoma cruzi, the etiological agent of Chagas' disease, across multiple geographical regions and multiple overlapping host-vector ...
• #### Modeling colony collapse disorder in honeybees as a contagion ﻿

(American Institute of Mathematical SciencesDepartment of Mathematics, University of Texas at Arlington, 2014-12)
Honeybee pollination accounts annually for over $14 billion in United States agriculture alone. Within the past decade there has been a mysterious mass die-off of honeybees, an estimated 10 million beehives and sometimes ... • #### Influence of vectors' risk-spreading strategies and environmental stochasticity on the epidemiology and evolution of vector-borne diseaes: the example of Chagas' disease ﻿ (Public Library of ScienceDepartment of Mathematics, University of Texas at Arlington, 2013) Insects are known to display strategies that spread the risk of encountering unfavorable conditions, thereby decreasing the extinction probability of genetic lineages in unpredictable environments. To what extent these ... • #### The convergence of difference boxes ﻿ (Mathematical Association of AmericaDepartment of Mathematics, University of Texas at Arlington, 2005) 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 R² into itself. We identify the map's unique canonical fixed ... • #### SENDing MORE MONEY in any base ﻿ (Mathematical Association of AmericaDepartment of Mathematics, University of Texas at Arlington, 2006) One of the most significant conceptual jumps involved in algebra is the use of letters (and other symbols) to represent numbers. One way to get students used to this notion is via puzzles. This paper describes a ... • #### Book Review: Differential Equations and Mathematical Biology, D.S. Jones, B.D. Sleeman, in: CRC Mathematical Biology and Medicine Series. Chapman & Hall (2003), Hardback, 408 pages,$79.95, ISBN: 1584882964 ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington; Universidad de Colima, Mexico, 2004)
• #### Modeling nosocomial transmission of rotavirus in pediatric wards ﻿

(Springer-VerlagDepartment of Mathematics and Curriculum & Instruction, University of Texas at Arlington, 2011)
Nosocomial transmission of viral and bacterial infections is a major problem worldwide, affecting millions of patients (and causing hundreds of thousands of deaths) per year. Rotavirus infections affect most children ...
• #### Vector consumption and contact process saturation in sylvatic transmission of T. cruzi ﻿

(Taylor & FrancisDepartment of Mathematics, University of Texas at Arlington; Universidad de Colima, Mexico, 2006)
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 ...
• #### Am I too fat? Bulimia as an epidemic ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington, 2003)
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 ...
• #### A simple vaccination model with multiple endemic states ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington, 2000)
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. ...
• #### Vaccination strategies and backward bifurcation in age-since-infection structured model ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington, 2002)
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 ...
• #### Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity ﻿

(Department of Mathematics, University of Texas at Arlington, 2004)
Genetic studies report the existence of a mutant allele Δ32 of CC R5 chemokine receptor gene at high allele frequencies ( 10%) in Caucasian populations. The presence of the allele is believed to provide partial or full ...
• #### Evaluating treatment of Hepatitis C for hemolytic anemia management ﻿

(ElsevierDepartment of Mathematics, University of Texas at Arlington, 2010)
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 ...
• #### Transmission dynamics and underreporting of Kala-azar in the Indian State of Bihar ﻿

(Department of Mathematics, University of Texas at Arlington, 2010)
"Kala-azar" (or Indian Visceral Leishmaniasis) is a vector-borne infectious dis- ease affecting communities in tropical and subtropical areas of the world. Bihar, a state in India, has one of the highest prevalence and ...
• #### The role of the ratio of vector and host densities in the evolution of transmission modes in vector-borne diseases. The example of sylvatic Trypanosoma cruzi ﻿

(Department of Mathematics, University of Texas at Arlington, 2012)
Pathogens may use different routes of transmission to maximize their spread among host populations. Theoretical and empirical work conducted on directly-transmitted diseases suggest that horizontal (i.e., through host ...
• #### A cross-institutional collaborative model ﻿

(Rapid Intellect, 2011)
In this duoethnography, a narrative framework is used to present the perspectives of members of a cross-institutional collaborative working group of mathematics education researchers. This article provides an example of ...
• #### Sociological phenomena as multiple nonlinearities: MTBI's new metaphor for complex human interactions ﻿

(American Institute of Mathematical SciencesDepartment of Mathematics, University of Texas at Arlington, 2013)
Mathematical models are well-established as metaphors for biological and epidemiological systems. The framework of epidemic modeling has also been applied to sociological phenomena driven by peer pressure, notably in two ...