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Mathematical Biosciences & Engineering

2013 , Volume 10 , Issue 2

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Using wavelet denoising and mathematical morphology in the segmentation technique applied to blood cells images
Macarena Boix and Begoña Cantó
2013, 10(2): 279-294 doi: 10.3934/mbe.2013.10.279 +[Abstract](652) +[PDF](544.5KB)
Abstract:
Accurate image segmentation is used in medical diagnosis since this technique is a noninvasive pre-processing step for biomedical treatment. In this work we present an efficient segmentation method for medical image analysis. In particular, with this method blood cells can be segmented. For that, we combine the wavelet transform with morphological operations. Moreover, the wavelet thresholding technique is used to eliminate the noise and prepare the image for suitable segmentation. In wavelet denoising we determine the best wavelet that shows a segmentation with the largest area in the cell. We study different wavelet families and we conclude that the wavelet db1 is the best and it can serve for posterior works on blood pathologies. The proposed method generates goods results when it is applied on several images. Finally, the proposed algorithm made in MatLab environment is verified for a selected blood cells.
Longitudinal displacement in viscoelastic arteries: A novel fluid-structure interaction computational model, and experimental validation
Martina Bukač and Sunčica Čanić
2013, 10(2): 295-318 doi: 10.3934/mbe.2013.10.295 +[Abstract](853) +[PDF](1581.1KB)
Abstract:
Recent in vivo studies, utilizing ultrasound contour and speckle tracking methods, have identified significant longitudinal displacements of the intima-media complex, and viscoelastic arterial wall properties over a cardiac cycle. Existing computational models that use thin structure approximations of arterial walls have so far been limited to models that capture only radial wall displacements. The purpose of this work is to present a simple fluid-struture interaction (FSI) model and a stable, partitioned numerical scheme, which capture both longitudinal and radial displacements, as well as viscoelastic arterial wall properties. To test the computational model, longitudinal displacement of the common carotid artery and of the stenosed coronary arteries were compared with experimental data found in literature, showing excellent agreement. We found that, unlike radial displacement, longitudinal displacement in stenotic lesions is highly dependent on the stenotic geometry. We also showed that longitudinal displacement in atherosclerotic arteries is smaller than in healthy arteries, which is in line with the recent in vivo measurements that associate plaque burden with reduced total longitudinal wall displacement.
    This work presents a first step in understanding the role of longitudinal displacement in physiology and pathophysiology of arterial wall mechanics using computer simulations.
Modeling of the kinetics of vitamin D$_3$ in osteoblastic cells
Robert P. Gilbert, Philippe Guyenne and Ying Liu
2013, 10(2): 319-344 doi: 10.3934/mbe.2013.10.319 +[Abstract](654) +[PDF](707.5KB)
Abstract:
A one-dimensional model for the transport of vitamin D$_3$ in an osteoblast cell is proposed, from its entry through the membrane to its activation of RANKL synthesis in the nucleus. In the membrane and cytoplasm, the transport of D$_3$ and RANKL is described by a diffusion process, while their interaction in the nucleus is modeled by a reaction-diffusion process. For the latter, an integral equation involving the boundary conditions, as well as an asymptotic solution in the regime of small concentrations, are derived. Numerical simulations are also performed to investigate the kinetics of D$_3$ and RANKL through the entire cell. Comparison between the asymptotics and numerics in the nucleus shows an excellent agreement. To our knowledge, this is the first time, albeit using a simple model, a description of the complete passage of D$_3$ through the cell membrane, the cytoplasm, into the cell nucleus, and finally the production of RANKL with its passage to the exterior of the cell, has been modeled.
Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey
Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca and José D. Flores
2013, 10(2): 345-367 doi: 10.3934/mbe.2013.10.345 +[Abstract](860) +[PDF](654.0KB)
Abstract:
The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation.
    We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions.
    Trajectories starting at one side of this separatrix curve have the equilibrium point $(0,0)$ as their $\omega $-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point $(K,0)$ in which the predators disappear and prey attains their carrying capacity.
    We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction.
    Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility
Andrey V. Melnik and Andrei Korobeinikov
2013, 10(2): 369-378 doi: 10.3934/mbe.2013.10.369 +[Abstract](842) +[PDF](331.3KB)
Abstract:
We consider global asymptotic properties for the SIR and SEIR age structured models for infectious diseases where the susceptibility depends on the age. Using the direct Lyapunov method with Volterra type Lyapunov functions, we establish conditions for the global stability of a unique endemic steady state and the infection-free steady state.
An extension of Gompertzian growth dynamics: Weibull and Fréchet models
J. Leonel Rocha and Sandra M. Aleixo
2013, 10(2): 379-398 doi: 10.3934/mbe.2013.10.379 +[Abstract](567) +[PDF](1784.7KB)
Abstract:
In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by $Beta^*(p,q)$, which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for $p = 2$, the investigation is extended to the extreme value models of Weibull and Fréchet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the $Beta^*(2,q)$ densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.
Competition of motile and immotile bacterial strains in a petri dish
Silogini Thanarajah and Hao Wang
2013, 10(2): 399-424 doi: 10.3934/mbe.2013.10.399 +[Abstract](766) +[PDF](715.2KB)
Abstract:
Bacterial competition is an important component in many practical applications such as plant roots colonization and medicine (especially in dental plaque). Bacterial motility has two types of mechanisms --- directed movement (chemotaxis) and undirected movement. We study undirected bacterial movement mathematically and numerically which is rarely considered in literature. To study bacterial competition in a petri dish, we modify and extend the model used in Wei et al. (2011) to obtain a group of more general and realistic PDE models. We explicitly consider the nutrients and incorporate two bacterial strains characterized by motility. We use different nutrient media such as agar and liquid in the theoretical framework to discuss the results of competition. The consistency of our numerical simulations and experimental data suggest the importance of modeling undirected motility in bacteria. In agar the motile strain has a higher total density than the immotile strain, while in liquid both strains have similar total densities. Furthermore, we find that in agar as bacterial motility increases, the extinction time of the motile bacteria decreases without competition but increases in competition. In addition, we show the existence of traveling-wave solutions mathematically and numerically.
Mathematical modelling and control of echinococcus in Qinghai province, China
Liumei Wu, Baojun Song, Wen Du and Jie Lou
2013, 10(2): 425-444 doi: 10.3934/mbe.2013.10.425 +[Abstract](948) +[PDF](882.5KB)
Abstract:
In this paper, two mathematical models, the baseline model and the intervention model, are proposed to study the transmission dynamics of echinococcus. A global forward bifurcation completely characterizes the dynamical behavior of the baseline model. That is, when the basic reproductive number is less than one, the disease-free equilibrium is asymptotically globally stable; when the number is greater than one, the endemic equilibrium is asymptotically globally stable. For the intervention model, however, the basic reproduction number alone is not enough to describe the dynamics, particularly for the case where the basic reproductive number is less then one. The emergence of a backward bifurcation enriches the dynamical behavior of the model. Applying these mathematical models to Qinghai Province, China, we found that the infection of echinococcus is in an endemic state. Furthermore, the model appears to be supportive of human interventions in order to change the landscape of echinococcus infection in this region.
Dynamics of an infectious diseases with media/psychology induced non-smooth incidence
Yanni Xiao, Tingting Zhao and Sanyi Tang
2013, 10(2): 445-461 doi: 10.3934/mbe.2013.10.445 +[Abstract](851) +[PDF](605.9KB)
Abstract:
This paper proposes and analyzes a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. The proposed models extend the classic models with media coverage by including a piecewise smooth incidence rate to represent that the reduction factor because of media coverage depends on both the number of cases and the rate of changes in case number. On the basis of properties of Lambert W function the implicitly defined model has been converted into a piecewise smooth system with explicit definition, and the global dynamic behavior is theoretically examined. The disease-free is globally asymptotically stable when a certain threshold is less than unity, while the endemic equilibrium is globally asymptotically stable for otherwise. The media/psychological impact although does not affect the epidemic threshold, delays the epidemic peak and results in a lower size of outbreak (or equilibrium level of infected individuals).
On latencies in malaria infections and their impact on the disease dynamics
Yanyu Xiao and Xingfu Zou
2013, 10(2): 463-481 doi: 10.3934/mbe.2013.10.463 +[Abstract](695) +[PDF](417.0KB)
Abstract:
In this paper, we modify the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. One novelty of our model is that we introduce two general probability functions ($P_1(t)$ and $P_2(t)$) to reflect the fact that the latencies differ from individuals to individuals. We justify the well-posedness of the new model, identify the basic reproduction number $\mathcal{R}_0$ for the model and analyze the dynamics of the model. We show that when $\mathcal{R}_0 <1$, the disease free equilibrium $E_0$ is globally asymptotically stable, meaning that the malaria disease will eventually die out; and if $\mathcal{R}_0 >1$, $E_0$ becomes unstable. When $\mathcal{R}_0 >1$, we consider two specific forms for $P_1(t)$ and $P_2(t)$: (i) $P_1(t)$ and $P_2(t)$ are both exponential functions; (ii) $P_1(t)$ and $P_2(t)$ are both step functions. For (i), the model reduces to an ODE system, and for (ii), the long term disease dynamics are governed by a DDE system. In both cases, we are able to show that when $\mathcal{R}_0>1$ then the disease will persist; moreover if there is no recovery ($\gamma_1=0$), then all admissible positive solutions will converge to the unique endemic equilibrium. A significant impact of the latencies is that they reduce the basic reproduction number, regardless of the forms of the distributions.
Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays
Zhaohui Yuan and Xingfu Zou
2013, 10(2): 483-498 doi: 10.3934/mbe.2013.10.483 +[Abstract](761) +[PDF](394.7KB)
Abstract:
We consider a mathematical model that describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is a modification of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function $f$ for virus infections, and (iii) a nonlinear removal rate function $h$ for infected cells. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we shown that the model has the threshold dynamics: if the basic reproduction number (BRN) is less than or equal to one, then the infection free equilibrium is globally asymptotically stable, meaning that HIV virus will be cleared; whereas if the BRN is larger than one, then there exist an infected equilibrium which is globally asymptotically stable, implying that the HIV-1 infection will persist in the host and the viral concentration will approach a positive constant level. This together with the dependence/independence of the BRN on $f$ and $h$ reveals the effect of the adoption of these nonlinear functions.

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