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Mathematical Biosciences & Engineering

2007 , Volume 4 , Issue 4

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Global stability of equilibria in a tick-borne disease model
Shangbing Ai
2007, 4(4): 567-572 doi: 10.3934/mbe.2007.4.567 +[Abstract](31) +[PDF](106.8KB)
In this short note we establish global stability results for a four-dimensional nonlinear system that was developed in modeling a tick-borne disease by H.D. Gaff and L.J. Gross (Bull. Math. Biol., 69 (2007), 265--288) where local stability results were obtained. These results provide the parameter ranges for controlling long-term population and disease dynamics.
Evaluating growth measures in an immigration process subject to binomial and geometric catastrophes
Jesus R. Artalejo , A. Economou and  M.J. Lopez-Herrero
2007, 4(4): 573-594 doi: 10.3934/mbe.2007.4.573 +[Abstract](41) +[PDF](95.9KB)
Populations are often subject to the effect of catastrophic events that cause mass removal. In particular, metapopulation models, epidemics, and migratory flows provide practical examples of populations subject to disasters (e.g., habitat destruction, environmental catastrophes). Many stochastic models have been developed to explain the behavior of these populations. Most of the reported results concern the measures of the risk of extinction and the distribution of the population size in the case of total catastrophes where all individuals in the population are removed simultaneously. In this paper, we investigate the basic immigration process subject to binomial and geometric catastrophes; that is, the population size is reduced according to a binomial or a geometric law. We carry out an extensive analysis including first extinction time, number of individuals removed, survival time of a tagged individual, and maximum population size reached between two consecutive extinctions. Many explicit expressions are derived for these system descriptors, and some emphasis is put to show that some of them deserve extra attention.
On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China
Nicolas Bacaër , Xamxinur Abdurahman , Jianli Ye and  Pierre Auger
2007, 4(4): 595-607 doi: 10.3934/mbe.2007.4.595 +[Abstract](56) +[PDF](241.6KB)
Heterogeneity in sexual behavior is known to play an important role in the spread of HIV. In 1986, a mathematical model based on ordinary differential equations was introduced to take into account the distribution of sexual activity. Assuming proportionate mixing, it was shown that the basic reproduction number $R_0$ determining the epidemic threshold was proportional to $M+V/M$, where $M$ is the mean and $V$ the variance of the distribution. In the present paper, we notice that this theoretical distribution is different from the one obtained in behavioral surveys for the number of sexual partnerships over a period of length $\tau$. The latter is a ''mixed Poisson distribution'' whose mean $m$ and variance $v$ are such that $M=m/\tau$ and $V=(v-m)/\tau^2$. So $M+V/M=(m+v/m-1)/\tau$. This way, we improve the link between theory and data for sexual activity models of HIV/AIDS epidemics. As an example, we consider data concerning sex workers and their clients in Yunnan, China, and find an upper bound for the geometric mean of the transmission probabilities per partnership in this context.
Improved integrate-and-fire model for RSA
Michele Barbi , Angelo Di Garbo and  Rita Balocchi
2007, 4(4): 609-615 doi: 10.3934/mbe.2007.4.609 +[Abstract](45) +[PDF](263.2KB)
A simple operational model of heart rate variability is described, accounting in particular for the respiratory sinus arrhythmia, and is fitted to some interbeat interval sequences recorded from normal subjects at rest. The model performance is evaluated using a test based on the nonlinear prediction approach. Moreover, a short comparative account of two similar models described in the literature is given.
A model of activity-dependent changes in dendritic spine density and spine structure
S. M. Crook , M. Dur-e-Ahmad and  S. M. Baer
2007, 4(4): 617-631 doi: 10.3934/mbe.2007.4.617 +[Abstract](27) +[PDF](1441.7KB)
Recent evidence indicates that the morphology and density of dendritic spines are regulated during synaptic plasticity. See, for instance, a review by Hayashi and Majewska [9]. In this work, we extend previous modeling studies [27] by combining a model for activity-dependent spine density with one for calcium-mediated spine stem restructuring. The model is based on the standard dimensionless cable equation, which represents the change in the membrane potential in a passive dendrite. Additional equations characterize the change in spine density along the dendrite, the current balance equation for an individual spine head, the change in calcium concentration in the spine head, and the dynamics of spine stem resistance. We use computational studies to investigate the changes in spine density and structure for differing synaptic inputs and demonstrate the effects of these changes on the input-output properties of the dendritic branch. Moderate amounts of high-frequency synaptic activation to dendritic spines result in an increase in spine stem resistance that is correlated with spine stem elongation. In addition, the spine density increases both inside and outside the input region. The model is formulated so that this long-term potentiation-inducing stimulus eventually leads to structural stability. In contrast, a prolonged low-frequency stimulation paradigm that would typically induce long-term depression results in a decrease in stem resistance (correlated with stem shortening) and an eventual decrease in spine density.
Optimal control on hybrid ODE Systems with application to a tick disease model
Wandi Ding
2007, 4(4): 633-659 doi: 10.3934/mbe.2007.4.633 +[Abstract](34) +[PDF](726.4KB)
We are considering an optimal control problem for a type of hybrid system involving ordinary differential equations and a discrete time feature. One state variable has dynamics in only one season of the year and has a jump condition to obtain the initial condition for that corresponding season in the next year. The other state variable has continuous dynamics. Given a general objective functional, existence, necessary conditions and uniqueness for an optimal control are established. We apply our approach to a tick-transmitted disease model with age structure in which the tick dynamics changes seasonally while hosts have continuous dynamics. The goal is to maximize disease-free ticks and minimize infected ticks through an optimal control strategy of treatment with acaricide. Numerical examples are given to illustrate the results.
Modelling periodic oscillations during somitogenesis
Peng Feng and  Menaka Navaratna
2007, 4(4): 661-673 doi: 10.3934/mbe.2007.4.661 +[Abstract](31) +[PDF](446.7KB)
We consider a model of genetic network that has been previously presented by J. Lewis. This model takes the form of delay differential equations with two delays. We give conditions for the local stability of the non-trivial steady state. We investigate the condition underwhich stability is lost and oscillations occur. In particular, we show that when the ratio of the time delays passes a threshold, sustained oscillations occur through a Hopf bifurcation. Through numerical simulations, we further investigate the ways in which various parameters influence the period and the amplitude of the oscillations. In conclusion, we discuss the implications of our results.
Final and peak epidemic sizes for SEIR models with quarantine and isolation
Z. Feng
2007, 4(4): 675-686 doi: 10.3934/mbe.2007.4.675 +[Abstract](56) +[PDF](294.3KB)
Two $SEIR$ models with quarantine and isolation are considered, in which the latent and infectious periods are assumed to have an exponential and gamma distribution, respectively. Previous studies have suggested (based on numerical observations) that a gamma distribution model (GDM) tends to predict a larger epidemic peak value and shorter duration than an exponential distribution model (EDM). By deriving analytic formulas for the maximum and final epidemic sizes of the two models, we demonstrate that either GDM or EDM may predict a larger epidemic peak or final epidemic size, depending on control measures. These formulas are helpful not only for understanding how model assumptions may affect the predictions, but also for confirming that it is important to assume realistic distributions of latent and infectious periods when the model is used for public health policy making.
Mesoscopic model for tumor growth
Elena Izquierdo-Kulich and  José Manuel Nieto-Villar
2007, 4(4): 687-698 doi: 10.3934/mbe.2007.4.687 +[Abstract](21) +[PDF](359.7KB)
In this work, we propose a mesoscopic model for tumor growth to improve our understanding of the origin of the heterogeneity of tumor cells. In this sense, this stochastic formalism allows us to not only to reproduce but also explain the experimental results presented by Brú. A significant aspect found by the model is related to the predicted values for $\beta$ growth exponent, which capture a basic characteristic of the critical surface growth dynamics. According to the model, the value for growth exponent is between 0,25 and 0,5, which includes the value proposed by Kadar-Parisi-Zhang universality class (0,33) and the value proposed by Brú (0,375) related to the molecular beam epitaxy (MBE) universality class. This result suggests that the tumor dynamics are too complex to be associated to a particular universality class.
Global analysis of discrete-time SI and SIS epidemic models
Jianquan Li , Zhien Ma and  Fred Brauer
2007, 4(4): 699-710 doi: 10.3934/mbe.2007.4.699 +[Abstract](49) +[PDF](173.5KB)
Discrete-time SI and SIS models formulated as the discretization of a continuous-time model may exhibit behavior different from that of the continuous-time model such as period-doubling and chaotic behavior unless the step size in the model is sufficiently small. Some new discrete-time SI and SIS epidemic models with vital dynamics are formulated and analyzed. These new models do not exhibit period doubling and chaotic behavior and are thus better approximations to continuous models. However, their reproduction numbers and therefore their asymptotic behavior can differ somewhat from that of the corresponding continuous-time model.
Hybrid multiscale landmark and deformable image registration
Dana Paquin , Doron Levy and  Lei Xing
2007, 4(4): 711-737 doi: 10.3934/mbe.2007.4.711 +[Abstract](35) +[PDF](1867.3KB)
An image registration technique is presented for the registration of medical images using a hybrid combination of coarse-scale landmark and B-splines deformable registration techniques. The technique is particularly effective for registration problems in which the images to be registered contain large localized deformations. A brief overview of landmark and deformable registration techniques is presented. The hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and L. Vese, A multiscale image representation using hierarchical $(BV,L^2)$ decompositions, Multiscale Modeling and Simulations, vol. 2, no. 4, pp. 554--579, 2004, is reviewed, and an image registration algorithm is developed based on combining the multiscale decomposition with landmark and deformable techniques. Successful registration of medical images is achieved by first obtaining a hierarchical multiscale decomposition of the images and then using landmark-based registration to register the resulting coarse scales. Corresponding bony structure landmarks are easily identified in the coarse scales, which contain only the large shapes and main features of the image. This registration is then fine tuned by using the resulting transformation as the starting point to deformably register the original images with each other using an iterated multiscale B-splines deformable registration technique. The accuracy and efficiency of the hybrid technique is demonstrated with several image registration case studies in two and three dimensions. Additionally, the hybrid technique is shown to be very robust with respect to the location of landmarks and presence of noise.
The utility of preemptive mass influenza vaccination in controlling a SARS outbreak during flu season
Qingling Zeng , Kamran Khan , Jianhong Wu and  Huaiping Zhu
2007, 4(4): 739-754 doi: 10.3934/mbe.2007.4.739 +[Abstract](23) +[PDF](250.9KB)
During flu season, respiratory infections can cause non-specific influenza-like-illnesses (ILIs) in up to one-half of the general population. If a future SARS outbreak were to coincide with flu season, it would become exceptionally difficult to distinguish SARS rapidly and accurately from other ILIs, given the non-specific clinical presentation of SARS and the current lack of a widely available, rapid, diagnostic test. We construct a deterministic compartmental model to examine the potential impact of preemptive mass influenza vaccination on SARS containment during a hypothetical SARS outbreak coinciding with a peak flu season. Our model was developed based upon the events of the 2003 SARS outbreak in Toronto, Canada. The relationship of different vaccination rates for influenza and the corresponding required quarantine rates for individuals who are exposed to SARS was analyzed and simulated under different assumptions. The study revealed that a campaign of mass influenza vaccination prior to the onset of flu season could aid the containment of a future SARS outbreak by decreasing the total number of persons with ILIs presenting to the health-care system, and consequently decreasing nosocomial transmission of SARS in persons under investigation for the disease.
Letter to the editors
Jian Zu , Wendi Wang and  Bo Zu
2007, 4(4): 755-755 doi: 10.3934/mbe.2007.4.755 +[Abstract](37) +[PDF](73.5KB)
Our paper Evolutionary dynamics of prey-predator systems with Holling type II functional response, recently published in MBE (pp. 221-237, vol. 4, 2007), heavily draws on earlier work by Michael Doebeli and Ulf Dieckmann:
    Doebeli, M & Dieckmann, U. (2000). Evolutionary branching and sympatric speciation caused by different types of ecological interactions. American Naturalist 156: S77-S101, cited as reference [11] in our paper.
    Dieckmann, U., Marrow, P. & Law, R. (1995). Evolutionary cycling in predator-prey interactions: population dynamics and the Red Queen. Journal of Theoretical Biology 176: 91-102, cited as reference [8] in our paper.

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