# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

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## Mathematical Biosciences & Engineering

2011 , Volume 8 , Issue 3

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2011, 8(3): 659-676 doi: 10.3934/mbe.2011.8.659 +[Abstract](91) +[PDF](794.8KB)
Abstract:
A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary state to be a spatially distributed evolutionary stable state is presented and illustrated with examples.
2011, 8(3): 677-687 doi: 10.3934/mbe.2011.8.677 +[Abstract](95) +[PDF](427.1KB)
Abstract:
We consider an SEIR epidemic model with vertical transmission introduced by Li, Smith and Wang, [23], and apply optimal control theory to assess the effects of vaccination strategies on the model dynamics. The strategy is chosen to minimize the total number of infectious individuals and the cost associated with vaccination. We derive the optimality system and solve it numerically. The theoretical findings are then used to simulate a vaccination campaign for rubella in China.
2011, 8(3): 689-694 doi: 10.3934/mbe.2011.8.689 +[Abstract](92) +[PDF](266.2KB)
Abstract:
The global stability for a delayed HIV-1 infection model is investigated. It is shown that the global dynamics of the system can be completely determined by the reproduction number, and the chronic infected equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [S. A. Gourley,Y. Kuang and J.D.Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2(2008), 140-153].
2011, 8(3): 695-709 doi: 10.3934/mbe.2011.8.695 +[Abstract](51) +[PDF](835.6KB)
Abstract:
Spread of tuberculosis (TB) due to the immigration from some developing countries with high TB incidence to developed countries poses an increasing challenge in the global TB control. Here a simple compartmental TB model with constant immigration, early and late latency is developed in order to investigate the impact of new immigrants with latent TB on the overall TB incidence, and to compare the difference contributed by different proportions of latently-infected new immigrants with high or low risk to develop active TB shortly after arrival. The global dynamics of the system is completely classified, numerical simulations are carried out for different scenarios, and potential applications to public health policy are discussed.
2011, 8(3): 711-722 doi: 10.3934/mbe.2011.8.711 +[Abstract](104) +[PDF](401.1KB)
Abstract:
An $S$-$I_c$-$I$-$R$ epidemic model is investigated for infectious diseases that can be transmitted through carriers, infected individuals who are contagious but do not show any disease symptoms. Mathematical analysis is carried out that completely determines the global dynamics of the model. The impacts of disease carriers on the transmission dynamics are discussed through the basic reproduction number and through numerical simulations.
2011, 8(3): 723-732 doi: 10.3934/mbe.2011.8.723 +[Abstract](112) +[PDF](313.8KB)
Abstract:
The regular nutritional intake of an expectant mother clearly affects the weight development of the fetus. Assuming the growth of the fetus follows a deterministic growth law, like a logistic equation, albeit dependent on the nutritional intake, the ideal solution is usually determined by the birth-weight being pre-assigned, for example, as a percentage of the mother's average weight. This problem can then be specified as an optimal control problem with the daily intake as the control, which appears in a Michaelis-Menten relationship, for which there are well-developed procedures to follow. The best solution is determined by requiring minimum total intake under which the preassigned birth weight is reached. The algorithm has been generalized to the case where the fetal weight depends in a detailed way on the cumulative intake, suitably discounted according to the history. The optimality system is derived and then solved numerically using an iterative method for the specific values of parameter. The procedure is generic and can be adapted to any growth law and any parameterisation obtained by the detailed physiology.
Shu Liao and Jin Wang
2011, 8(3): 733-752 doi: 10.3934/mbe.2011.8.733 +[Abstract](83) +[PDF](565.3KB)
Abstract:
In this paper, we conduct a dynamical analysis of the deterministic cholera model proposed in [9]. We study the stability of both the disease-free and endemic equilibria so as to explore the complex epidemic and endemic dynamics of the disease. We demonstrate a real-world application of this model by investigating the recent cholera outbreak in Zimbabwe. Meanwhile, we present numerical simulation results to verify the analytical predictions.
2011, 8(3): 753-768 doi: 10.3934/mbe.2011.8.753 +[Abstract](106) +[PDF](357.2KB)
Abstract:
A simple SEIR model for malaria transmission dynamics is formulated as our baseline model. The metamorphic stages in the mosquito population are then included and a simple stage-structured mosquito population model is introduced, where the mosquito population is divided into two classes, with all three aquatic stages in one class and all adults in the other class, to keep the model tractable in mathematical analysis. After a brief investigation of this simple stage-structured mosquito model, it is incorporated into the baseline model to formulate a stage-structured malaria model. A basic analysis for the stage-structured malaria model is provided and it is shown that a theoretical framework can be built up for further studies on the impact of environmental or climate change on the malaria transmission. It is also shown that both the baseline and the stage-structured malaria models undergo backward bifurcations.
2011, 8(3): 769-783 doi: 10.3934/mbe.2011.8.769 +[Abstract](74) +[PDF](370.8KB)
Abstract:
We present a stock-effort dynamical model of a fishery subdivided into fishing zones. The stock corresponds to a fish population moving between these zones, on which they are harvested by fishing fleets. We consider a linear chain of identical fishing zones. Fish movements between the zones, as well as vessels displacements, are assumed to take place at a faster time scale than the variation of the stock and the change of the fleet size. The vessels movements between the fishing areas are assumed to be stock dependent, i.e. the larger the stock density is in a zone the more vessels tends to remain in it. We take advantage of these two time scales to derive a reduced model governing the dynamics of the total harvested stock and the total fishing effort. Under some assumption, we obtain either a stable equilibrium or a stable limit cycle which involves large cyclic variations of the total fish stock and fishing effort. We show that there exists an optimal number of fishing zones that maximizes the total catch at equilibrium. We discuss the results in relation to fish aggregating devices (FADs) fisheries.
2011, 8(3): 785-806 doi: 10.3934/mbe.2011.8.785 +[Abstract](73) +[PDF](649.7KB)
Abstract:
Left Ventricular Assist Devices (LVADs) are implantable mechanical pumps that temporarily aid the function of the left ventricle. The use of LVADs has been associated with thrombus formation next to the aortic valve and close to the anastomosis region, especially in patients in which the native cardiac function is negligible and the aortic valve remains closed. Stagnation points and recirculation zones have been implicated as the main fluid dynamics factors contributing to thrombus formation. The purpose of the present study was to develop and use computer simulations based on a fluid-structure interaction (FSI) solver to study flow conditions corresponding to different strategies in LVAD ascending aortic anastomosis providing a scenario with the lowest likelihood of thrombus formation. A novel FSI algorithm was developed to deal with the presence of multiple structures corresponding to different elastic properties of the native aorta and of the LVAD cannula. A sensitivity analysis of different variables was performed to assess their impact of flow conditions potentially leading to thrombus formation. It was found that the location of the anastomosis closest to the aortic valve (within 4 cm away from the valve) and at the angle of 30$^\circ$ minimizes the likelihood of thrombus formation. Furthermore, it was shown that the rigidity of the dacron anastomosis cannula plays almost no role in generating pathological conditions downstream from the anastomosis. Additionally, the flow analysis presented in this manuscript indicates that compliance of the cardiovascular tissue acts as a natural inhibitor of pathological flow conditions conducive to thrombus formation and should not be neglected in computer simulations.
2011, 8(3): 807-825 doi: 10.3934/mbe.2011.8.807 +[Abstract](101) +[PDF](497.5KB)
Abstract:
This paper extends the work of Salceanu and Smith [12, 13] where Lyapunov exponents were used to obtain conditions for uniform persistence in a class of dissipative discrete-time dynamical systems on the positive orthant of $\mathbb{R}^m$, generated by maps. Here a unified approach is taken, for both discrete and continuous time, and the dissipativity assumption is relaxed. Sufficient conditions are given for compact subsets of an invariant part of the boundary of $\mathbb{R}^m_+$ to be robust uniform weak repellers. These conditions require Lyapunov exponents be positive on such sets. It is shown how this leads to robust uniform persistence. The results apply to the investigation of robust uniform persistence of the disease in host populations, as shown in an application.
2011, 8(3): 827-840 doi: 10.3934/mbe.2011.8.827 +[Abstract](99) +[PDF](480.3KB)
Abstract:
In this paper, we consider a competition model between $n$ species in a chemostat including both monotone and non-monotone growth functions, distinct removal rates and variable yields. We show that only the species with the lowest break-even concentration survives, provided that additional technical conditions on the growth functions and yields are satisfied. We construct a Lyapunov function which reduces to the Lyapunov function used by S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760-763] in the Monod case when the growth functions are of Michaelis-Menten type and the yields are constant. Various applications are given including linear, quadratic and cubic yields.
2011, 8(3): 841-860 doi: 10.3934/mbe.2011.8.841 +[Abstract](94) +[PDF](361.5KB)
Abstract:
The replicability of an oncolytic virus is measured by its burst size. The burst size is the number of new viruses coming out from a lysis of an infected tumor cell. Some clinical evidences show that the burst size of an oncolytic virus is a defining parameter for the success of virotherapy. This article analyzes a basic mathematical model that includes burst size for oncolytic virotherapy. The analysis of the model shows that there are two threshold values of the burst size: below the first threshold, the tumor always grows to its maximum (carrying capacity) size; while passing this threshold, there is a locally stable positive equilibrium solution appearing through transcritical bifurcation; while at or above the second threshold, there exits one or three families of periodic solutions arising from Hopf bifurcations. The study suggests that the tumor load can drop to a undetectable level either during the oscillation or when the burst size is large enough.
2011, 8(3): 861-873 doi: 10.3934/mbe.2011.8.861 +[Abstract](63) +[PDF](702.7KB)
Abstract:
Mutations of the SCN5A gene can significantly alter the function of cardiac myocyte sodium channels leading to increased risk of ventricular arrhythmia. Over the past decade, detailed Markov models of the action potential of cardiac cells have been developed. In such models, the effects of a drug can be treated as alterations in on- and off rates between open and inactivated states on one hand, and blocked states on the other hand. Our aim is to compute the rates specifying a drug in order to: (a) restore the steady-state open probability of the mutant channel to that of normal wild type channels; and (b) minimize the difference between whole cell currents in drugged mutant and wild type cells. The difference in the electrochemical state vector of the cell can be measured in a norm taking all components and their dynamical properties into account. Measured with this norm, the difference between the state of the mutant and wild-type cell was reduced by a factor of 36 after the drug was introduced and by factors of 4 over mexitiline and 25 over lidocaine. The results suggest the potential to synthesize more effective drugs based on mechanisms of action of existing compounds.
2011, 8(3): 875-888 doi: 10.3934/mbe.2011.8.875 +[Abstract](88) +[PDF](385.2KB)
Abstract:
In this paper, based on an SEIR epidemiological model with distributed delays to account for varying infectivity, we introduce a vaccination compartment, leading to an SVEIR model. By employing direct Lyapunov method and LaSalle's invariance principle, we construct appropriate functionals that integrate over past states to establish global asymptotic stability conditions, which are completely determined by the basic reproduction number $\mathcal{R}_0^V$. More precisely, it is shown that, if $\mathcal{R}_0^V\leq 1$, then the disease free equilibrium is globally asymptotically stable; if $\mathcal{R}_0^V > 1$, then there exists a unique endemic equilibrium which is globally asymptotically stable. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before acquiring immunity can be neglected, this condition would be satisfied and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.

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