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Mathematical Biosciences & Engineering

2008 , Volume 5 , Issue 1

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A passivity-based stability criterion for a class of biochemical reaction networks
Murat Arcak and Eduardo D. Sontag
2008, 5(1): 1-19 doi: 10.3934/mbe.2008.5.1 +[Abstract](411) +[PDF](359.4KB)
Abstract:
This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. The main result determines global asymptotic stability of the network from the diag- onal stability of a dissipativity matrix which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encom- passes the secant criterion for cyclic networks presented in [1], and extends it to a general interconnection structure represented by a graph. The new stabil- ity test is illustrated on a mitogen-activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. The next problem addressed is the robustness of stability in the presence of di®usion terms. A compartmental model is used to represent the localization of the reactions, and conditions are presented under which stability is preserved despite the di®usion terms between the compartments.
Global stability analysis for SEIS models with n latent classes
Napoleon Bame, Samuel Bowong, Josepha Mbang, Gauthier Sallet and Jean-Jules Tewa
2008, 5(1): 20-33 doi: 10.3934/mbe.2008.5.20 +[Abstract](323) +[PDF](424.4KB)
Abstract:
We compute the basic reproduction ratio of a SEIS model with n classes of latent individuals and bilinear incidence.The system exhibits the traditional behaviour. We prove that if R0 ≤1, then the disease-free equilibrium is globally asymptotically stable on the nonnegative orthant and if R0 > 1, an endemic equilibrium exists and is globally asymptotically stable on the positive orthant.
A storage model with random release rate for modeling exposure to food contaminants
Patrice Bertail, Stéphan Clémençon and Jessica Tressou
2008, 5(1): 35-60 doi: 10.3934/mbe.2008.5.35 +[Abstract](352) +[PDF](466.4KB)
Abstract:
This paper presents the study of a continuous-time piecewisedeterministic Markov process for describing the temporal evolution of exposure to a given food contaminant. The quantity X of food contaminant present in the body evolves through its accumulation after repeated dietary intakes on the one hand, and the pharmacokinetics behavior of the chemical on the other hand. In the dynamic modeling considered here, the accumulation phenomenon is modeled by a simple marked point process with positive i.i.d. marks, and elimination in between intakes occurs at a random linear rate θX, randomness of the coefficient θ accounting for the variability of the elimination process due to metabolic factors. Via embedded chain analysis, ergodic properties of this extension of the standard compound Poisson dam with (deterministic) linear release rate are investigated, the latter being of crucial importance in describing the long-term behavior of the exposure process (Xt)t≥0 and assessing values such as the proportion of time the contaminant body burden is over a certain threshold. We also highlight the fact that the exposure process is generally not directly observable in practice and establish a validity framework for simulation-based statistical methods by coupling analysis. Eventually, applications to methyl mercury contamination data are considered.
A dominant predator, a predator, and a prey
Peter A. Braza
2008, 5(1): 61-73 doi: 10.3934/mbe.2008.5.61 +[Abstract](321) +[PDF](303.5KB)
Abstract:
A two-predator, one-prey model in which one predator interferes significantly with the other predator is analyzed. The dominant predator is harvested and the other predator has an alternative food source. The response functions used are Holling type II and they are predator-dependent and include the effects of interference. The analysis centers on bifurcation diagrams for various levels of interference in which the harvesting is the primary bifurcation parameter. There are different attractors for the high-interference and no- interference cases and these are discussed within an ecological context.
Self-organizing models of bacterial aggregation states
Manuela Caratozzolo, Santina Carnazza, Luigi Fortuna, Mattia Frasca, Salvatore Guglielmino, Giovanni Gurrieri and Giovanni Marletta
2008, 5(1): 75-83 doi: 10.3934/mbe.2008.5.75 +[Abstract](344) +[PDF](1388.4KB)
Abstract:
In this work, aggregation states of bacteria on engineered surfaces are investigated both from the experimental point of view and from the theo- retical one. The starting point of this work is a series of experiments carried out on abiotic surfaces in which bacteria adhere forming self-organized patterns. To reproduce the main characteristics of the phenomenon a model based on self-organization of a group of agents has been used. The agents represent bac- teria and are free to move on a given surface. On the basis of local rules they may adhere and then eventually form self-organized aggregates. Our numerical results demonstrate that few simple rules are able to explain the emergence of self-organized patterns. Depending on the parameters used, the model is able to reproduce the aggregation patterns observed under different experimental conditions and to predict the behavior of a culture of two bacterial species.
Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model
Guangrui Li, Ming Mei and Yau Shu Wong
2008, 5(1): 85-100 doi: 10.3934/mbe.2008.5.85 +[Abstract](338) +[PDF](2243.1KB)
Abstract:
The paper is devoted to the study of a time-delayed reaction- diffusion equation of age-structured single species population. Linear stability for this model was first presented by Gourley [4], when the time delay is small. Here, we extend the previous result to the nonlinear stability by using the technical weighted-energy method, when the initial perturbation around the wavefront decays to zero exponentially as x→-∞, but the initial perturbation can be arbitrarily large on other locations. The exponential convergent rate (in time) of the solution is obtained. Numerical simulations are carried out to confirm the theoretical results, and the traveling wavefronts with a large delay term in the model are reported.
The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model
Andrew L. Nevai and Richard R. Vance
2008, 5(1): 101-124 doi: 10.3934/mbe.2008.5.101 +[Abstract](298) +[PDF](450.0KB)
Abstract:
A global method of nullcline endpoint analysis is employed to de- termine the outcome of competition for sunlight between two hypothetical plant species with clonal growth form that differ solely in the height at which they place their leaves above the ground. This difference in vertical leaf placement, or canopy partitioning, produces species differences in sunlight energy capture and stem metabolic maintenance costs. The competitive interaction between these two species is analyzed by considering a special case of a canopy partitioning model (RR Vance and AL Nevai, J. Theor. Biol. 2007, 245:210-219; AL Nevai and RR Vance, J. Math. Biol. 2007, 55:105-145). Nullcline endpoint analysis is used to partition parameter space into regions within which either competitive exclusion or competitive coexistence occurs. The principal conclu- sion is that two clonal plant species which compete for sunlight and place their leaves at different heights above the ground but differ in no other way can, un- der suitable parameter values, experience stable coexistence even though they occupy an environment which varies neither over horizontal space nor through time.
Multiscale deformable registration of noisy medical images
Dana Paquin, Doron Levy and Lei Xing
2008, 5(1): 125-144 doi: 10.3934/mbe.2008.5.125 +[Abstract](277) +[PDF](3296.0KB)
Abstract:
Multiscale image registration techniques are presented for the reg- istration of medical images using deformable registration models. The tech- niques are particularly effective for registration problems in which one or both of the images to be registered contains significant levels of noise. A brief overview of existing deformable registration techniques is presented, and exper- iments using B-spline free-form deformation registration models demonstrate that ordinary deformable registration techniques fail to produce accurate re- sults in the presence of significant levels of noise. The hierarchical multiscale image decomposition described in E. Tadmor, S. Nezzar, and L. Vese's, ''A multiscale image representation using hierarchical (BV,L2) decompositions'' (Multiscale Modeling and Simulations, 2 (2004): 4, pp. 554-579) is reviewed, and multiscale image registration algorithms are developed based on the mul- tiscale decomposition. Accurate registration of noisy images is achieved by obtaining a hierarchical multiscale decomposition of the images and iteratively registering the resulting components. This approach enables a successful reg- istration of images that contain noise levels well beyond the level at which ordinary deformable registration fails. Numerous image registration experi- ments demonstrate the accuracy and efficiency of the multiscale registration techniques.
Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment
Oluwaseun Sharomi, Chandra N. Podder, Abba B. Gumel and Baojun Song
2008, 5(1): 145-174 doi: 10.3934/mbe.2008.5.145 +[Abstract](355) +[PDF](522.5KB)
Abstract:
This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two dis- eases. In the absence of TB infection, the model (HIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of back- ward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. The analysis of the respective reproduction thresholds shows that the use of a targeted HIV treatment (using anti-retroviral drugs) strategy can lead to effective control of HIV provided it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold. The full model, with both HIV and TB, is simu- lated to evaluate the impact of the various treatment strategies. It is shown that the HIV-only treatment strategy saves more cases of the mixed infection than the TB-only strategy. Further, for low treatment rates, the mixed-only strategy saves the least number of cases (of HIV, TB, and the mixed infection) in comparison to the other strategies. Thus, this study shows that if resources are limited, then targeting such resources to treating one of the diseases is more beneficial in reducing new cases of the mixed infection than targeting the mixed infection only diseases. Finally, the universal strategy saves more cases of the mixed infection than any of the other strategies.
Estimation of invasive pneumococcal disease dynamics parameters and the impact of conjugate vaccination in Australia
Karyn L. Sutton, H.T. Banks and Carlos Castillo-Chávez
2008, 5(1): 175-204 doi: 10.3934/mbe.2008.5.175 +[Abstract](428) +[PDF](426.3KB)
Abstract:
Pneumococcal diseases, or infections from the etiological agent Streptococcus pneumoniae, have long been a major cause of morbidity and mortality worldwide. Recent advances in the development of vaccines for these infections have raised questions concerning their widespread and/or long-term use. In this work, we use surveillance data collected by the Australian National Notifiable Diseases Surveillance system to estimate parameters in a mathemat- ical model of pneumococcal infection dynamics in a population with partial vaccination. The parameters obtained are of particular interest as they are not typically available in reported literature or measurable. The calibrated model is then used to assess the impact of the recent federally funded program that provides pneumococcal vaccines to large risk groups. The results presented here suggest the state of these infections may be changing in response to the programs, and warrants close quantitative monitoring.
Soliton-like excitation in a nonlinear model of DNA dynamics with viscosity
Conrad Bertrand Tabi, Alidou Mohamadou and Timoleon Crepin Kofane
2008, 5(1): 205-216 doi: 10.3934/mbe.2008.5.205 +[Abstract](287) +[PDF](259.8KB)
Abstract:
The study of solitary wave solutions is of prime significance for nonlinear physical systems. The Peyrard-Bishop model for DNA dynamics is generalized specifically to include the difference among bases pairs and vis- cosity. The small amplitude dynamics of the model is studied analytically and reduced to a discrete complex Ginzburg-Landau (DCGL) equation. Ex- act solutions of the obtained wave equation are obtained by the mean of the extended Jacobian elliptic function approach. These amplitude solutions are made of bubble solitons. The propagation of a soliton-like excitation in a DNA is then investigated through numerical integration of the motion equations. We show that discreteness can drastically change the soliton shape. The impact of viscosity as well as elasticity on DNA dynamic is also presented. The profile of solitary wave structures as well as the energy which is initially evenly distributed over the lattice are displayed for some fixed parameters.

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