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Mathematical Biosciences & Engineering

2004 , Volume 1 , Issue 1

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From the Editors
Yang Kuang , Ying-Cheng Lai and  Zhiming Zheng
2004, 1(1): i-ii doi: 10.3934/mbe.2004.1.1i +[Abstract](34) +[PDF](26.9KB)
It is easy to argue that we are entering a new golden era of science and engineering. The predominant approach of conducting timely and in-depth scienti c research is forever changed from cycles of simple experiment and theory to cycles that combine of experiment, theory, mathematical modeling, and analysis. The landscape of science is now clearly dominated by the peaks of biosciences. Integrative and interdisciplinary research in life sciences and engineering is no longer merely a slogan but aggressively and collectively pursued and practiced by researchers. This shift of paradigms greatly enhances the central importance of mathematical and computational modeling and analysis in scienti c and engineering endeavors. With this exciting backdrop in mind, the American Institute of Mathematical Sciences (AIMS) and Beihang University have decided to launch a joint international journal on mathematical biosciences and engineering. This journal will maximally serve the scienti c communities in both developed and developing countries.

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Critical role of nosocomial transmission in the Toronto SARS outbreak
Glenn Webb , Martin J. Blaser , Huaiping Zhu , Sten Ardal and  Jianhong Wu
2004, 1(1): 1-13 doi: 10.3934/mbe.2004.1.1 +[Abstract](54) +[PDF](243.4KB)
We develop a compartmental mathematical model to address the role of hospitals in severe acute respiratory syndrome (SARS) transmission dynamics, which partially explains the heterogeneity of the epidemic. Comparison of the e ffects of two major policies, strict hospital infection control procedures and community-wide quarantine measures, implemented in Toronto two weeks into the initial outbreak, shows that their combination is the key to short-term containment and that quarantine is the key to long-term containment.
Modeling and optimal regulation of erythropoiesis subject to benzene intoxication
H. T. Banks , Cammey E. Cole , Paul M. Schlosser and  Hien T. Tran
2004, 1(1): 15-48 doi: 10.3934/mbe.2004.1.15 +[Abstract](51) +[PDF](311.9KB)
Benzene (C6H6) is a highly flammable, colorless liquid. Ubiquitous exposures result from its presence in gasoline vapors, cigarette smoke, and industrial processes. Benzene increases the incidence of leukemia in humans when they are exposed to high doses for extended periods; however, leukemia risks in humans subjected to low exposures are uncertain. The exposure-dose- response relationship of benzene in humans is expected to be nonlinear because benzene undergoes a series of metabolic transformations, detoxifying and activating, resulting in various metabolites that exert toxic e ffects on the bone marrow.
    Since benzene is a known human leukemogen, the toxicity of benzene in the bone marrow is of most importance. And because blood cells are produced in the bone marrow, we investigated the eff ects of benzene on hematopoiesis (blood cell production and development). An age-structured model was used to examine the process of erythropoiesis, the development of red blood cells. This investigation proved the existence and uniqueness of the solution of the system of coupled partial and ordinary di fferential equations. In addition, we formulated an optimal control problem for the control of erythropoiesis and performed numerical simulations to compare the performance of the optimal feedback law and another feedback function based on the Hill function.
Chaotic spreading of epidemics in complex networks of excitable units
F. S. Vannucchi and  S. Boccaletti
2004, 1(1): 49-55 doi: 10.3934/mbe.2004.1.49 +[Abstract](47) +[PDF](809.4KB)
We explore the dynamics of an epidemiological disease spreading within a complex network of individuals. The local behavior of the epidemics is modelled by means of an excitable dynamics, and the individuals are connected in the network through a weighted small-world wiring. The global behavior of the epidemics can have stationary as well as chaotic states, depending upon the probability of substituting short-range with long-range interactions. We describe the bifurcation scenario leading to such latter states, and discuss the relevance of the observed chaotic dynamics for the description of the spreading mechanisms of epidemics inside complex networks.
A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence
Andrei Korobeinikov and  Philip K. Maini
2004, 1(1): 57-60 doi: 10.3934/mbe.2004.1.57 +[Abstract](96) +[PDF](88.4KB)
Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I^p S^q$ for the case $p \leq 1$ are constructed. Global stability of the models is thereby established.
On deriving lumped models for blood flow and pressure in the systemic arteries
Mette S. Olufsen and  Ali Nadim
2004, 1(1): 61-80 doi: 10.3934/mbe.2004.1.61 +[Abstract](41) +[PDF](217.4KB)
Windkessel and similar lumped models are often used to represent blood flow and pressure in systemic arteries. The windkessel model was originally developed by Stephen Hales (1733) and Otto Frank (1899) who used it to describe blood flow in the heart. In this paper we start with the one-dimensional axisymmetric Navier-Stokes equations for time-dependent blood flow in a rigid vessel to derive lumped models relating flow and pressure. This is done through Laplace transform and its inversion via residue theory. Upon keeping contributions from one, two, or more residues, we derive lumped models of successively higher order. We focus on zeroth, first and second order models and relate them to electrical circuit analogs, in which current is equivalent to flow and voltage to pressure. By incorporating e ffects of compliance through addition of capacitors, windkessel and related lumped models are obtained. Our results show that given the radius of a blood vessel, it is possible to determine the order of the model that would be appropriate for analyzing the flow and pressure in that vessel. For instance, in small rigid vessels ($R <$ 0.2 cm) it is adequate to use Poiseuille's law to express the relation between flow and pressure, whereas for large vessels it might be necessary to incorporate spatial dependence by using a one-dimensional model accounting for axial variations.
Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection
Benjamin H. Singer and  Denise E. Kirschner
2004, 1(1): 81-93 doi: 10.3934/mbe.2004.1.81 +[Abstract](66) +[PDF](366.3KB)
There is significant disagreement in the epidemiological literature regarding the extent to which reinfection of latently infected individuals contributes to the dynamics of tuberculosis (TB) epidemics. In this study we present an epidemiological model of Mycobacterium tuberculosis infection that includes the process of reinfection. Using analysis and numerical simulations, we observe the effect that varying levels of reinfection has on the qualitative dynamics of the TB epidemic. We examine cases of the model both with and without treatment of actively infected individuals. Next, we consider a variation of the model describing a heterogeneous population, stratified by susceptibility to TB infection. Results show that a threshold level of reinfection exists in all cases of the model. Beyond this threshold, the dynamics of the model are described by a backward bifurcation. Uncertainty analysis of the parameters shows that this threshold is too high to be attained in a realistic epidemic. However, we show that even for sub-threshold levels of reinfection, including reinfection in the model changes dynamic behavior of the model. In particular, when reinfection is present the basic reproductive number, $R_0$, does not accurately describe the severity of an epidemic.
Controlling a model for bone marrow dynamics in cancer chemotherapy
Urszula Ledzewicz and  Heinz Schättler
2004, 1(1): 95-110 doi: 10.3934/mbe.2004.1.95 +[Abstract](40) +[PDF](248.1KB)
This paper analyzes a mathematical model for the growth of bone marrow cells under cell-cycle-speci c cancer chemotherapy originally proposed by Fister and Panetta [8]. The model is formulated as an optimal control problem with control representing the drug dosage (respectively its eff ect) and objective of Bolza type depending on the control linearly, a so-called $L^1$-objective. We apply the Maximum Principle, followed by high-order necessary conditions for optimality of singular arcs and give sufficient conditions for optimality based on the method of characteristics. Singular controls are eliminated as candidates for optimality, and easily veri able conditions for strong local optimality of bang-bang controls are formulated in the form of transversality conditions at switching surfaces. Numerical simulations are given.
Turing instabilities and pattern formation in a benthic nutrient-microorganism system
Martin Baurmann , Wolfgang Ebenhöh and  Ulrike Feudel
2004, 1(1): 111-130 doi: 10.3934/mbe.2004.1.111 +[Abstract](56) +[PDF](1310.2KB)
In general, the distributions of nutrients and microorganisms in sediments show complex spatio-temporal patterns, which often cannot be explained as resulting exclusively from the temporal fluctuations of environmental conditions and the inhomogeneity of the studied sediment's material. We studied the dynamics of one population of microorganisms feeding on a nutrient in a simple model, taking into account that the considered bacteria can be in an active or in a dormant state. Using this model, we show that the formation of spatio-temporal patterns can be the consequence of the interaction between predation and transport processes. Employing the model on a two-dimensional vertical domain, we show by simulations which patterns can arise. Depending on the strength of bioirrigation, we observe stripes or "hot spots'' (or "cold spots'') with high (or low) microbiological activity. A detailed study regarding the effect of non-homogeneous (depth dependent) forcing by bioirrigation shows that different patterns can appear in different depths.
Coexistence in a metapopulation model with explicit local dynamics
Zhilan Feng , Robert Swihart , Yingfei Yi and  Huaiping Zhu
2004, 1(1): 131-145 doi: 10.3934/mbe.2004.1.131 +[Abstract](30) +[PDF](351.5KB)
Many patch-based metapopulation models assume that the local population within each patch is at its equilibrium and independent of changes in patch occupancy. We studied a metapopulation model that explicitly incorporates the local population dynamics of two competing species. The singular perturbation method is used to separate the fast dynamics of the local competition and the slow process of patch colonization and extinction. Our results show that the coupled system leads to more complex outcomes than simple patch models which do not include explicit local dynamics. We also discuss implications of the model for ecological systems in fragmented landscapes.
Biological computing with diffusion and excitable calcium stores
H. G. E. Hentschel , Alan Fine and  C. S. Pencea
2004, 1(1): 147-159 doi: 10.3934/mbe.2004.1.147 +[Abstract](33) +[PDF](498.0KB)
Intracellular signaling often employs excitable stores of calcium coupled by diffusion. We investigate the ability of various geometric configurations of such excitable stores to generate a complete set of logic gates for computation. We also describe how the mechanism of excitable calcium-induced calcium release can be used for constructing coincidence detectors for biological signals.
Statistical properties of dynamical chaos
Vadim S. Anishchenko , Tatjana E. Vadivasova , Galina I. Strelkova and  George A. Okrokvertskhov
2004, 1(1): 161-184 doi: 10.3934/mbe.2004.1.161 +[Abstract](30) +[PDF](638.9KB)
This study presents a survey of the results obtained by the authors on statistical description of dynamical chaos and the e ffect of noise on dynamical regimes. We deal with nearly hyperbolic and nonhyperbolic chaotic attractors and discuss methods of diagnosing the type of an attractor. We consider regularities of the relaxation to an invariant probability measure for diff erent types of attractors. We explore peculiarities of autocorrelation decay and of power spectrum shape and their interconnection with Lyapunov exponents, instantaneous phase di ffusion and the intensity of external noise. Numeric results are compared with experimental data.
Noise in ecosystems: A short review
B. Spagnolo , D. Valenti and  A. Fiasconaro
2004, 1(1): 185-211 doi: 10.3934/mbe.2004.1.185 +[Abstract](34) +[PDF](1485.9KB)
Noise, through its interaction with the nonlinearity of the living systems, can give rise to counter-intuitive phenomena such as stochastic resonance, noise-delayed extinction, temporal oscillations, and spatial patterns. In this paper we briefly review the noise-induced effects in three different ecosystems: (i) two competing species; (ii) three interacting species, one predator and two preys, and (iii) N-interacting species. The transient dynamics of these ecosystems are analyzed through generalized Lotka-Volterra equations in the presence of multiplicative noise, which models the interaction between the species and the environment. The interaction parameter between the species is random in cases (i) and (iii), and a periodical function, which accounts for the environmental temperature, in case (ii). We find noise-induced phenomena such as quasi-deterministic oscillations, stochastic resonance, noise-delayed extinction, and noise-induced pattern formation with non-monotonic behaviors of patterns areas and of the density correlation as a function of the multiplicative noise intensity. The asymptotic behavior of the time average of the$ i^{th}$ population when the ecosystem is composed of a great number of interacting species is obtained and the effect of the noise on the asymptotic probability distri- butions of the populations is discussed.

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