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Mathematical Biosciences & Engineering

2007 , Volume 4 , Issue 2

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Comparison between stochastic and deterministic selection-mutation models
Azmy S. Ackleh and Shuhua Hu
2007, 4(2): 133-157 doi: 10.3934/mbe.2007.4.133 +[Abstract](52) +[PDF](642.2KB)
We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection-mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, unlike the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.
A final size relation for epidemic models
Julien Arino, Fred Brauer, P. van den Driessche, James Watmough and Jianhong Wu
2007, 4(2): 159-175 doi: 10.3934/mbe.2007.4.159 +[Abstract](87) +[PDF](213.5KB)
A final size relation is derived for a general class of epidemic models, including models with multiple susceptible classes. The derivation depends on an explicit formula for the basic reproduction number of a general class of disease transmission models, which is extended to calculate the basic reproduction number in models with vertical transmission. Applications are given to specific models for influenza and SARS.
Theoretical models for chronotherapy: Periodic perturbations in funnel chaos type
Juvencio Alberto Betancourt-Mar and José Manuel Nieto-Villar
2007, 4(2): 177-186 doi: 10.3934/mbe.2007.4.177 +[Abstract](55) +[PDF](250.9KB)
In this work, the Räossler system is used as a model for chrono- therapy. We applied a periodic perturbation to the y variable to take the Rössler system from a chaotic behavior to a simple periodic one, varying the period and amplitude of forcing. Two types of chaos were considered, spiral and funnel chaos. As a result, the periodical windows reduced their areas as the funnel chaos character increased in the system. Funnel chaos, in this chrono- therapy model, could be considered as a later state of a dynamical disease, more irregular and difficult to suppress.
An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems
Chichia Chiu and Jui-Ling Yu
2007, 4(2): 187-203 doi: 10.3934/mbe.2007.4.187 +[Abstract](63) +[PDF](576.3KB)
Reaction-diffusion-chemotaxis systems have proven to be fairly accurate mathematical models for many pattern formation problems in chemistry and biology. These systems are important for computer simulations of patterns, parameter estimations as well as analysis of the biological systems. To solve reaction-diffusion-chemotaxis systems, efficient and reliable numerical algorithms are essential for pattern generations. In this paper, a general reaction-diffusion-chemotaxis system is considered for specific numerical issues of pattern simulations. We propose a fully explicit discretization combined with a variable optimal time step strategy for solving the reactiondiffusion- chemotaxis system. Theorems about stability and convergence of the algorithm are given to show that the algorithm is highly stable and efficient. Numerical experiment results on a model problem are given for comparison with other numerical methods. Simulations on two real biological experiments will also be shown.
Modeling diseases with latency and relapse
P. van den Driessche, Lin Wang and Xingfu Zou
2007, 4(2): 205-219 doi: 10.3934/mbe.2007.4.205 +[Abstract](88) +[PDF](285.7KB)
A general mathematical model for a disease with an exposed (latent) period and relapse is proposed. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. For this model with a general probability of remaining in the exposed class, the basic reproduction number $\R_0$ is identified and its threshold property is discussed. In particular, the disease-free equilibrium is proved to be globally asymptotically stable if $\R_0<1$. If the probability of remaining in the exposed class is assumed to be negatively exponentially distributed, then $\R_0=1$ is a sharp threshold between disease extinction and endemic disease. A delay differential equation system is obtained if the probability function is assumed to be a step-function. For this system, the endemic equilibrium is locally asymptotically stable if $\R_0>1$, and the disease is shown to be uniformly persistent with the infective population size either approaching or oscillating about the endemic level. Numerical simulations (for parameters appropriate for bovine tuberculosis in cattle) with $\mathcal{R}_0>1$ indicate that solutions tend to this endemic state.
Evolutionary dynamics of prey-predator systems with Holling type II functional response
Jian Zu, Wendi Wang and Bo Zu
2007, 4(2): 221-237 doi: 10.3934/mbe.2007.4.221 +[Abstract](64) +[PDF](261.6KB)
This paper considers the coevolution of phenotypes in a community comprising the populations of predators and prey. The evolutionary dynamics is constructed from a stochastic process of mutation and selection. We investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. It is shown that branching in the prey can induce secondary branching in the predators. Furthermore, it is shown that the evolutionary dynamics admits a stable limit cycle. The evolutionary cycle is a likely outcome of the process, which requires higher evolutionary speed of prey than of predators. It is also found that different evolutionary rates and conversion efficiencies can influence the lengths of evolutionary cycles.
A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse
Wenxiang Liu, Thomas Hillen and H. I. Freedman
2007, 4(2): 239-259 doi: 10.3934/mbe.2007.4.239 +[Abstract](81) +[PDF](302.7KB)
In this paper we use a mathematical model to study the effect of an $M$-phase specific drug on the development of cancer, including the resting phase $G_0$ and the immune response. The cell cycle of cancer cells is split into the mitotic phase (M-phase), the quiescent phase ($G_0$-phase) and the interphase ($G_1,\ S,\ G_2$ phases). We include a time delay for the passage through the interphase, and we assume that the immune cells interact with all cancer cells. We study analytically and numerically the stability of the cancer-free equilibrium and its dependence on the model parameters. We find that quiescent cells can escape the $M$-phase drug. The dynamics of the $G_0$ phase dictates the dynamics of cancer as a whole. Moreover, we find oscillations through a Hopf bifurcation. Finally, we use the model to discuss the efficiency of cell synchronization before treatment (synchronization method).
The role of delays in innate and adaptive immunity to intracellular bacterial infection
Simeone Marino, Edoardo Beretta and Denise E. Kirschner
2007, 4(2): 261-286 doi: 10.3934/mbe.2007.4.261 +[Abstract](63) +[PDF](1705.2KB)
The immune response in humans is complex and multi-fold. Initially an innate response attempts to clear any invasion by microbes. If it fails to clear or contain the pathogen, an adaptive response follows that is specific for the microbe and in most cases is successful at eliminating the pathogen. In previous work we developed a delay differential equations (DDEs) model of the innate and adaptive immune response to intracellular bacteria infection. We addressed the relevance of known delays in each of these responses by exploring different kernel and delay functions and tested how each affected infection outcome. Our results indicated how local stability properties for the two infection outcomes, namely a boundary equilibrium and an interior positive equilibrium, were completely dependent on the delays for innate immunity and independent of the delays for adaptive immunity. In the present work we have three goals. The first is to extend the previous model to account for direct bacterial killing by adaptive immunity. This reflects, for example, active killing by a class of cells known as macrophages, and will allow us to determine the relevance of delays for adaptive immunity. We present analytical results in this setting. Second, we implement a heuristic argument to investigate the existence of stability switches for the positive equilibrium in the manifold defined by the two delays. Third, we apply a novel analysis in the setting of DDEs known as uncertainty and sensitivity analysis. This allows us to evaluate completely the role of all parameters in the model. This includes identifying effects of stability switch parameters on infection outcome.
Subthreshold coexistence of strains: the impact of vaccination and mutation
Maia Martcheva, Mimmo Iannelli and Xue-Zhi Li
2007, 4(2): 287-317 doi: 10.3934/mbe.2007.4.287 +[Abstract](65) +[PDF](305.8KB)
We consider a model for a disease with two competing strains and vaccination. The vaccine provides complete protection against one of the strains (strain 2) but only partial protection against the other (strain 1). The partial protection leads to existence of subthreshold equilibria of strain 1. If the first strain mutates into the second, there are subthreshold coexistence equilibria when both vaccine-dependent reproduction numbers are below one. Thus, a vaccine that is specific toward the second strain and that, in absence of other strains, should be able to eliminate the second strain by reducing its reproduction number below one, cannot do so because it provides only partial protection to another strain that mutates into the second strain.
On the stability of periodic solutions in the perturbed chemostat
Frédéric Mazenc, Michael Malisoff and Patrick D. Leenheer
2007, 4(2): 319-338 doi: 10.3934/mbe.2007.4.319 +[Abstract](119) +[PDF](273.7KB)
We study the chemostat model for one species competing for one nutrient using a Lyapunov-type analysis. We design the dilution rate function so that all solutions of the chemostat converge to a prescribed periodic solution. In terms of chemostat biology, this means that no matter what positive initial levels for the species concentration and nutrient are selected, the long-term species concentration and substrate levels closely approximate a prescribed oscillatory behavior. This is significant because it reproduces the realistic ecological situation where the species and substrate concentrations oscillate. We show that the stability is maintained when the model is augmented by additional species that are being driven to extinction. We also give an input-to-state stability result for the chemostat-tracking equations for cases where there are small perturbations acting on the dilution rate and initial concentration. This means that the long-term species concentration and substrate behavior enjoys a highly desirable robustness property, since it continues to approximate the prescribed oscillation up to a small error when there are small unexpected changes in the dilution rate function.
A finite element method for growth in biological development
Cornel M. Murea and H. G. E. Hentschel
2007, 4(2): 339-353 doi: 10.3934/mbe.2007.4.339 +[Abstract](77) +[PDF](373.6KB)
We describe finite element simulations of limb growth based on Stokes flow models with a nonzero divergence representing growth due to nutrients in the early stages of limb bud development. We introduce a ''tissue pressure'' whose spatial derivatives yield the growth velocity in the limb and our explicit time advancing algorithm for such tissue flows is described in detail. The limb boundary is approached by spline functions to compute the curvature and the unit outward normal vector. At each time step, a mixed-hybrid finite element problem is solved, where the condition that the velocity is strictly normal to the limb boundary is treated by a Lagrange multiplier technique. Numerical results are presented.
Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter
Sun Yi, Patrick W. Nelson and A. Galip Ulsoy
2007, 4(2): 355-368 doi: 10.3934/mbe.2007.4.355 +[Abstract](112) +[PDF](258.5KB)
In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

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