# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2008 , Volume 5 , Issue 2

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2008, 5(2): 219-238 doi: 10.3934/mbe.2008.5.219 +[Abstract](633) +[PDF](396.7KB)
Abstract:
We consider an SIR metapopulation model for the spread of rabies in raccoons. This system of ordinary differential equations considers subpop- ulations connected by movement. Vaccine for raccoons is distributed through food baits. We apply optimal control theory to find the best timing for dis- tribution of vaccine in each of the linked subpopulations across the landscape. This strategy is chosen to limit the disease optimally by making the number of infections as small as possible while accounting for the cost of vaccination.
2008, 5(2): 239-260 doi: 10.3934/mbe.2008.5.239 +[Abstract](535) +[PDF](434.4KB)
Abstract:
The FitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron FIring and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, be- sides giving rise to the typical fast traveling wave solution exhibited in the original ''diffusion'' FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where ''normal'' impulse propagation is possible.
2008, 5(2): 261-276 doi: 10.3934/mbe.2008.5.261 +[Abstract](389) +[PDF](1107.3KB)
Abstract:
We propose BioS (Bio-potential Study) as a new virtual data anal- ysis and management environment.It was devised to cope with the physiological signals, in order to manage different data using advanced methods of analy- sis and to find a simple way to decode and interpret data. BioS has been structured as a flexible, modular, and portable environment. It includes sev- eral modules as data importing and loading, data visualization (1D, 2D, 3D), pre-processing (frequency and saturation filtering, statistical analysis), spatio- temporal processing such as power spectrum, independent component analysis (ICA) in spatial and time domain, and nonlinear analysis for the extraction of the maximum Lyapunov exponent and d (d-inifnite) using optimized al- gorithms. The environment provides a user-friendly Graphic User Interface that allows inexperienced users to perform complex analyses and to speed up experimental data processing.
2008, 5(2): 277-298 doi: 10.3934/mbe.2008.5.277 +[Abstract](637) +[PDF](557.0KB)
Abstract:
In this paper, we consider a mathematical model for the forma- tion of spatial morphogen territories of two key morphogens: Wingless (Wg) and Decapentaplegic (DPP), involved in leg development of Drosophila. We define a gene regulatory network (GRN) that utilizes autoactivation and cross- inhibition (modeled by Hill equations) to establish and maintain stable bound- aries of gene expression. By computational analysis we find that in the presence of a general activator, neither autoactivation, nor cross-inhibition alone are suf- ficient to maintain stable sharp boundaries of morphogen production in the leg disc. The minimal requirements for a self-organizing system are a coupled system of two morphogens in which the autoactivation and cross-inhibition have Hill coefficients strictly greater than one. In addition, the GRN modeled here describes the regenerative responses to genetic manipulations of positional identity in the leg disc.
2008, 5(2): 299-313 doi: 10.3934/mbe.2008.5.299 +[Abstract](529) +[PDF](391.3KB)
Abstract:
The mathematical modeling of tumor growth allows us to describe the most important regularities of these systems. A stochastic model, based on the most important processes that take place at the level of individual cells, is proposed to predict the dynamical behavior of the expected radius of the tumor and its fractal dimension. It was found that the tumor has a characteristic fractal dimension, which contains the necessary information to predict the tumor growth until it reaches a stationary state. This fractal dimension is distorted by the effects of external fluctuations. The model predicts a phenomenon which indicates stochastic resonance when the multiplicative and the additive noise are correlated.
2008, 5(2): 315-335 doi: 10.3934/mbe.2008.5.315 +[Abstract](602) +[PDF](463.3KB)
Abstract:
This paper is concerned with an indefinite weight linear eigenvalue problem in cylindrical domains. We investigate the minimization of the positive principal eigenvalue under the constraint that the weight is bounded by a positive and a negative constant and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. Both our analysis and numerical simulations for rectangular domains indicate that there exists a threshold value such that if the total weight is below this threshold value, then the optimal favorable region is a circular-type domain at one of the four corners, and a strip at the one end with shorter edge otherwise.
2008, 5(2): 337-354 doi: 10.3934/mbe.2008.5.337 +[Abstract](576) +[PDF](363.1KB)
Abstract:
We consider an age-structured model of a harvested population. This model is a discrete-time system that includes a nonlinear stock-recruitment relationship. Our purpose is to estimate the stock state. To achieve this goal, we built an observer, which is an auxiliary system that uses the total number of fish caught over each season and gives a dynamical estimation of the number of fish by age class. We analyse the convergence of the observer and we show that the error estimation tends to zero with exponential speed if a condition on the fishing effort is satisfied. Moreover the constructed observer (dynamical estimator) does not depend on the poorly understood stock-recruitment relationship. This study shows how some tools from nonlinear control theory can help to deal with the state estimation problem in the field of renewable resource management.
2008, 5(2): 355-388 doi: 10.3934/mbe.2008.5.355 +[Abstract](493) +[PDF](2348.5KB)
Abstract:
The CompuCell3D modeling environment provides a convenient platform for biofilm simulations using the Glazier-Graner-Hogeweg (GGH) model, a cell-oriented framework designed to simulate growth and pattern formation due to biological cells' behaviors. We show how to develop such a simulation, based on the hybrid (continuum-discrete) model of Picioreanu, van Loosdrecht, and Heijnen (PLH), simulate the growth of a single-species bacterial biofilm, and study the roles of cell-cell and cell-field interactions in determining biofilm morphology. In our simulations, which generalize the PLH model by treating cells as spatially extended, deformable bodies, differential adhesion between cells, and their competition for a substrate (nutrient), suffice to produce a fingering instability that generates the finger shapes of biofilms. Our results agree with most features of the PLH model, although our inclu- sion of cell adhesion, which is difficult to implement using other modeling approaches, results in slightly different patterns. Our simulations thus pro- vide the groundwork for simulations of medically and industrially important multispecies biofilms.
2008, 5(2): 389-402 doi: 10.3934/mbe.2008.5.389 +[Abstract](667) +[PDF](235.7KB)
Abstract:
A new SEIR model with distributed infinite delay is derived when the infectivity depends on the age of infection. The basic reproduction number R0, which is a threshold quantity for the stability of equilibria, is calculated. If $R_0$ < 1, then the disease-free equilibrium is globally asymptotically stable and this is the only equilibrium. On the contrary, if $R_0$ > 1, then an endemic equilibrium appears which is locally asymptotically stable. Applying a perma- nence theorem for infinite dimensional systems, we obtain that the disease is always present when $R_0$ > 1.
2008, 5(2): 403-418 doi: 10.3934/mbe.2008.5.403 +[Abstract](553) +[PDF](219.3KB)
Abstract:
HIV transmission process involves a long incubation and infection period, and the transmission rate varies greatly with infection stage. Conse- quently, modeling analysis based on the assumption of a constant transmission rate during the entire infection period yields an inaccurate description of HIV transmission dynamics and long-term projections. Here we develop a general framework of mathematical modeling that takes into account this heterogeneity of transmission rate and permits rigorous estimation of important parameters using a regression analysis of the twenty-year reported HIV infection data in China. Despite the large variation in this statistical data attributable to the knowledge of HIV, surveillance efforts, and uncertain events, and although the reported data counts individuals who might have been infected many years ago, our analysis shows that the model structured on infection age can assist us in extracting from this data set very useful information about transmission trends and about effectiveness of various control measures.

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