# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2015 , Volume 12 , Issue 4

Special issue dedicated to the 70th birthday of Glenn F. Webb

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2015, 12(4): v-xvi doi: 10.3934/mbe.2015.12.4v +[Abstract](939) +[PDF](390.8KB)
Abstract:
It is my distinct pleasure to introduce this volume honoring the 70th birthday of Professor Glenn F. Webb. The existence of this compiled volume is in itself a testimony of Glenn's dedication to, his pursuit of, and his achievement of scientific excellence. As we honor Glenn, we honor what is excellent in our profession. Aristotle clearly articulated his concept of excellence. We are what we repeatedly do. Excellence, then, is not an act, but a habit." As we look over the course of his career we observe ample evidence of Glenn Webb's habitual practice of excellence. Beginning with Glenn's first paper [1], one observes a constant stream of productivity and high impact work. Glenn has authored or co-authored over 160 papers, written one research monograph, and co-edited six volumes. He has delivered plenary lectures, colloquia, and seminars across the globe, and he serves on the editorial boards of 11 archival journals. He is a Fellow of the American Mathematical Society. Glenn's scientific career chronicles an evolution of scientific work that began with his interest in nonlinear semigroup theory and leads up to his current activity in biomedical mathematics. At each stage we see seminal contributions in the areas of nonlinear semigroups, functional differential equations, infinite dimensional dynamical systems, mathematical population dynamics, mathematical biology and biomedical mathematics. Glenn's work is distinguished by a clarity and accessibility of exposition, a precise identification and description of the problem or model under consideration, and thorough referencing. He uses elementary methods whenever possible but couples this with an ability to employ power abstract methods when necessitated by the problem.

2015, 12(4): xvii-xxii doi: 10.3934/mbe.2015.12.4xvii +[Abstract](653) +[PDF](336.7KB)
Abstract:
I began working with Glenn F. Webb in 1997. At that time, I was on the faculty of Vanderbilt University, in the School of Medicine, in the Department of Medicine, in its Division of Infectious Diseases. As with mathematics, modern medicine has its different disciplines (e.g. Surgery and Internal Medicine), and then further subdivisions (e.g. Cardiology and Infectious Diseases). Within Internal Medicine, most of the divisions are based on the treatment of conditions that relate to a single organ or group of organs -- the heart, the lungs, the kidneys, the digestive system. But the discipline of Infectious Diseases was based on a different concept: the war between humans and microbes.

2015, 12(4): i-iv doi: 10.3934/mbe.2015.12.4i +[Abstract](646) +[PDF](115.3KB)
Abstract:
This special issue is dedicated to the 70th birthday of Glenn F. Webb. The topics of the 12 articles appearing in this special issue include evolutionary dynamics of population growth, spatio-temporal dynamics in reaction-diffusion biological models, transmission dynamics of infectious diseases, modeling of antibiotic-resistant bacteria in hospitals, analysis of Prion models, age-structured models in ecology and epidemiology, modeling of immune response to infections, modeling of cancer growth, etc. These topics partially represent the broad areas of Glenn's research interest.

2015, 12(4): 643-660 doi: 10.3934/mbe.2015.12.643 +[Abstract](936) +[PDF](464.5KB)
Abstract:
An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).
2015, 12(4): 661-686 doi: 10.3934/mbe.2015.12.661 +[Abstract](982) +[PDF](474.1KB)
Abstract:
A model of ordinary differential equations is formulated for populations which are structured by many stages. The model is motivated by ticks which are vectors of infectious diseases, but is general enough to apply to many other species. Our analysis identifies a basic reproduction number that acts as a threshold between population extinction and persistence. We establish conditions for the existence and uniqueness of nonzero equilibria and show that their local stability cannot be expected in general. Boundedness of solutions remains an open problem though we give some sufficient conditions.
2015, 12(4): 687-697 doi: 10.3934/mbe.2015.12.687 +[Abstract](619) +[PDF](421.0KB)
Abstract:
In the current study, probit and logistic models were employed to fit experimental mortality data of the Khapra beetle, Trogoderma granarium (Everts) (Coleoptera: Dermestidae), when fumigated with three plant oils of the gens Achillea. A generalized inverse matrix technique was used to estimate the mortality model parameters instead of the usual statistical iterative maximum likelihood estimation. As this technique needs to perturb the observed mortality proportions if the proportions include 0 or 1, the optimal perturbation in terms of minimum least squares ($L_2$) error was also determined. According to our results, it was better to log-transform concentration and time as explanatory variables in modeling mortality of the test insect. Estimated data using the probit model were more accurate in terms of $L_2$ errors, than the logistic one. Results of the predicted mortality revealed also that extending the fumigation period could be an effective control strategy, even, at lower concentrations. Results could help in using a relatively safe and effective strategy for the control of this serious pest using alternative control strategy to reduce the health and environmental drawbacks resulted from the excessive reliance on the broadly toxic chemical pesticides and in order to contribute safeguard world-wide grain supplies.
2015, 12(4): 699-715 doi: 10.3934/mbe.2015.12.699 +[Abstract](829) +[PDF](2913.5KB)
Abstract:
In this paper, we study a diffusive plant-herbivore system with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability of spatially homogeneous steady states is established. We also derive conditions ensuring the occurrence of Hopf bifurcation and steady state bifurcation. Interesting transient spatio-temporal behaviors including oscillations in one or both of space and time are observed through numerical simulations.
2015, 12(4): 717-737 doi: 10.3934/mbe.2015.12.717 +[Abstract](880) +[PDF](551.1KB)
Abstract:
This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.
2015, 12(4): 739-760 doi: 10.3934/mbe.2015.12.739 +[Abstract](997) +[PDF](773.3KB)
Abstract:
The fundamental problem of collecting data in the best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical $D$-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.
2015, 12(4): 761-787 doi: 10.3934/mbe.2015.12.761 +[Abstract](675) +[PDF](2782.2KB)
Abstract:
A model of epidemic bacterial infections in hospitals is developed. The model incorporates the infection of patients and the contamination of healthcare workers due to environmental causes. The model is analyzed with respect to the asymptotic behavior of solutions. The model is interpreted to provide insight for controlling these nosocomial epidemics.
2015, 12(4): 789-801 doi: 10.3934/mbe.2015.12.789 +[Abstract](756) +[PDF](377.6KB)
Abstract:
We consider the so-called prion equation with the general incidence term introduced in [14], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [11]. The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted $L^1$ spaces and the analysis of a nonlinear system of three ordinary differential equations.
2015, 12(4): 803-840 doi: 10.3934/mbe.2015.12.803 +[Abstract](1048) +[PDF](1952.9KB)
Abstract:
Evidence suggests a strong correlation between the prevalence of HSV-2 (genital herpes) and the perseverance of the HIV epidemic. HSV-2 is an incurable viral infection, characterized by periodic reactivation. We construct a model of the co-infection dynamics between the two diseases by incorporating a time-since-infection variable to track the alternating periods of infectiousness of HSV-2. The model considers only heterosexual relationships and distinguishes three population groups: males, general population females, and female sex workers. We calculate the basic reproduction numbers for each disease that provide threshold conditions, which determine whether a disease dies out or becomes endemic in the absence of the other disease. We also derive the invasion reproduction numbers that determine whether or not a disease can invade into a population in which the other disease is endemic. The calculations of the invasion reproduction numbers suggest a new aspect in their interpretation - the class from which the initial disease carrier arises is important for understanding the invasion dynamics and biological interpretation of the expressions of the reproduction numbers. Sensitivity analysis is conducted to examine the role of model parameters in influencing the model outcomes. The results are discussed in the last section.
2015, 12(4): 841-858 doi: 10.3934/mbe.2015.12.841 +[Abstract](1024) +[PDF](423.8KB)
Abstract:
The past century's description of oncolytic virotherapy as a cancer treatment involving specially-engineered viruses that exploit immune deficiencies to selectively lyse cancer cells is no longer adequate. Some of the most promising therapeutic candidates are now being engineered to produce immunostimulatory factors, such as cytokines and co-stimulatory molecules, which, in addition to viral oncolysis, initiate a cytotoxic immune attack against the tumor.
This study addresses the combined effects of viral oncolysis and T-cell-mediated oncolysis. We employ a mathematical model of virotherapy that induces release of cytokine IL-12 and co-stimulatory molecule 4-1BB ligand. We found that the model closely matches previously published data, and while viral oncolysis is fundamental in reducing tumor burden, increased stimulation of cytotoxic T cells leads to a short-term reduction in tumor size, but a faster relapse.
In addition, we found that combinations of specialist viruses that express either IL-12 or 4-1BBL might initially act more potently against tumors than a generalist virus that simultaneously expresses both, but the advantage is likely not large enough to replace treatment using the generalist virus. Finally, according to our model and its current assumptions, virotherapy appears to be optimizable through targeted design and treatment combinations to substantially improve therapeutic outcomes.
2015, 12(4): 859-877 doi: 10.3934/mbe.2015.12.859 +[Abstract](865) +[PDF](474.9KB)
Abstract:
In this paper, we study an age-structured virus dynamics model with Beddington-DeAngelis infection function. An explicit formula for the basic reproductive number $\mathcal{R}_{0}$ of the model is obtained. We investigate the global behavior of the model in terms of $\mathcal{R}_{0}$: if $\mathcal{R}_{0}\leq1$, then the infection-free equilibrium is globally asymptotically stable, whereas if $\mathcal{R}_{0}>1$, then the infection equilibrium is globally asymptotically stable. Finally, some special cases, which reduce to some known HIV infection models studied by other researchers, are considered.
2015, 12(4): 879-905 doi: 10.3934/mbe.2015.12.879 +[Abstract](869) +[PDF](6066.4KB)
Abstract:
Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumor's aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.

2017  Impact Factor: 1.23