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Mathematical Biosciences & Engineering

2011 , Volume 8 , Issue 4

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Controlling malaria with indoor residual spraying in spatially heterogenous environments
Mo'tassem Al-Arydah and  Robert Smith?
2011, 8(4): 889-914 doi: 10.3934/mbe.2011.8.889 +[Abstract](60) +[PDF](1028.3KB)
Indoor residual spraying – spraying insecticide inside houses to kill mosquitoes – has been one of the most effective methods of disease control ever devised, being responsible for the near-eradication of malaria from the world in the third quarter of the twentieth century and saving tens of millions of lives. However, with malaria resurgence currently underway, it has received relatively little attention, been applied only in select physical locations and not always at regular intervals. We extend a time-dependent model of malaria spraying to include spatial heterogeneity and address the following research questions: 1. What are the effects of spraying in different geographical areas? 2. How do the results depend upon the regularity of spraying? 3. Can we alter our control strategies to account for asymmetric phenomena such as wind? We use impulsive partial differential equation models to derive thresholds for malaria control when spraying occurs uniformly, within an interior disc or under asymmetric advection effects. Spatial heterogeneity results in an increase in the necessary frequency of spraying, but control is still achievable.
A mathematical model of the Purkinje-Muscle Junctions
Adnane Azzouzi , Yves Coudière , Rodolphe Turpault and  Nejib Zemzemi
2011, 8(4): 915-930 doi: 10.3934/mbe.2011.8.915 +[Abstract](41) +[PDF](715.0KB)
This paper is devoted to the construction of a mathematical model of the His-Purkinje tree and the Purkinje-Muscle Junctions (PMJ). A simple numerical scheme is proposed in order to perform some simple numerical experiments.
An SEIR epidemic model with constant latency time and infectious period
Edoardo Beretta and  Dimitri Breda
2011, 8(4): 931-952 doi: 10.3934/mbe.2011.8.931 +[Abstract](45) +[PDF](433.6KB)
We present a two delays SEIR epidemic model with a saturation incidence rate. One delay is the time taken by the infected individuals to become infectious (i.e. capable to infect a susceptible individual), the second delay is the time taken by an infectious individual to be removed from the infection. By iterative schemes and the comparison principle, we provide global attractivity results for both the equilibria, i.e. the disease-free equilibrium $\mathbf{E}_{0}$ and the positive equilibrium $\mathbf{E}_{+}$, which exists iff the basic reproduction number $\mathcal{R}_{0}$ is larger than one. If $\mathcal{R}_{0}>1$ we also provide a permanence result for the model solutions. Finally we prove that the two delays are harmless in the sense that, by the analysis of the characteristic equations, which result to be polynomial trascendental equations with polynomial coefficients dependent upon both delays, we confirm all the standard properties of an epidemic model: $\mathbf{E}_{0}$ is locally asymptotically stable for $\mathcal{R}% _{0}<1$ and unstable for $\mathcal{R}_{0}>1$, while if $\mathcal{R}_{0}>1$ then $\mathbf{E}_{+}$ is always asymptotically stable.
Effects of spatial structure and diffusion on the performances of the chemostat
Ihab Haidar , Alain Rapaport and  Frédéric Gérard
2011, 8(4): 953-971 doi: 10.3934/mbe.2011.8.953 +[Abstract](54) +[PDF](463.5KB)
Given hydric capacity and nutrient flow of a chemostat-like system, we analyse the influence of a spatial structure on the output concentrations at steady-state. Three configurations are compared: perfectly-mixed, serial and parallel with diffusion rate. We show the existence of a threshold on the input concentration of nutrient for which the benefits of the serial and parallel configurations over the perfectly-mixed one are reversed. In addition, we show that the dependency of the output concentrations on the diffusion rate can be non-monotonic, and give precise conditions for the diffusion effect to be advantageous. The study encompasses dead-zone models.
A mathematical model for cellular immunology of tuberculosis
Eduardo Ibarguen-Mondragon , Lourdes Esteva and  Leslie Chávez-Galán
2011, 8(4): 973-986 doi: 10.3934/mbe.2011.8.973 +[Abstract](47) +[PDF](343.6KB)
Tuberculosis (TB) is a global emergency. The World Health Organization reports about 9.2 million new infections each year, with an average of 1.7 million people killed by the disease. The causative agent is Mycobacterium tuberculosis (Mtb), whose main target are the macrophages, important immune system cells. Macrophages and T cell populations are the main responsible for fighting the pathogen. A better understanding of the interaction between Mtb, macrophages and T cells will contribute to the design of strategies to control TB. The purpose of this study is to evaluate the impact of the response of T cells and macrophages in the control of Mtb. To this end, we propose a system of ordinary differential equations to model the interaction among non-infected macrophages, infected macrophages, T cells and Mtb bacilli. Model analysis reveals the existence of two equilibrium states, infection-free equilibrium and the endemically infected equilibrium which can represent a state of latent or active infection, depending on the amount of bacteria.
Morphogenesis and aggressiveness of cervix carcinoma
Elena Izquierdo-Kulich , Margarita Amigó de Quesada , Carlos Manuel Pérez-Amor and  José Manuel Nieto-Villar
2011, 8(4): 987-997 doi: 10.3934/mbe.2011.8.987 +[Abstract](44) +[PDF](682.1KB)
A mathematical model was obtained to describe the relation between the tissue morphology of cervix carcinoma and both dynamic processes of mitosis and apoptosis, and an expression to quantify the tumor aggressiveness, which in this context is associated with the tumor growth rate. The proposed model was applied to Stage III cervix carcinoma in vivo studies. In this study we found that the apoptosis rate was signicantly smaller in the tumor tissues and both the mitosis rate and aggressiveness index decrease with Stage III patients’ age. These quantitative results correspond to observed behavior in clinical and genetics studies.
The Within-Host dynamics of malaria infection with immune response
Yilong Li , Shigui Ruan and  Dongmei Xiao
2011, 8(4): 999-1018 doi: 10.3934/mbe.2011.8.999 +[Abstract](46) +[PDF](1094.3KB)
Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value $R_0$, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if $R_0<1$. If $R_0>1$, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.
Global asymptotic properties of staged models with multiple progression pathways for infectious diseases
Andrey V. Melnik and  Andrei Korobeinikov
2011, 8(4): 1019-1034 doi: 10.3934/mbe.2011.8.1019 +[Abstract](31) +[PDF](451.3KB)
We consider global asymptotic properties of compartment staged-progression models for infectious diseases with long infectious period, where there are multiple alternative disease progression pathways and branching. For example, these models reflect cases when there is considerable difference in virulence, or when only a part of the infected individuals undergoes a treatment whereas the rest remains untreated. Using the direct Lyapunov method, we establish sufficient and necessary conditions for the existence and global stability of a unique endemic equilibrium state, and for the stability of an infection-free equilibrium state.
Mathematical analysis and numerical simulation of a model of morphogenesis
Ana I. Muñoz and  José Ignacio Tello
2011, 8(4): 1035-1059 doi: 10.3934/mbe.2011.8.1035 +[Abstract](38) +[PDF](513.0KB)
We consider a simple mathematical model of distribution of morphogens (signaling molecules responsible for the differentiation of cells and the creation of tissue patterns). The mathematical model is a particular case of the model proposed by Lander, Nie and Wan in 2006 and similar to the model presented in Lander, Nie, Vargas and Wan 2005. The model consists of a system of three equations: a PDE of parabolic type with dynamical boundary conditions modelling the distribution of free morphogens and two ODEs describing the evolution of bound and free receptors. Three biological processes are taken into account: diffusion, degradation and reversible binding. We study the stationary solutions and the evolution problem. Numerical simulations show the behavior of the solution depending on the values of the parameters.
A mathematical model of the compression of a spinal disc
Matthias Ngwa and  Ephraim Agyingi
2011, 8(4): 1061-1083 doi: 10.3934/mbe.2011.8.1061 +[Abstract](34) +[PDF](423.3KB)
A model is developed of the stress-strain response of an intervertebral disc to axial compression. This is based on a balance of increased intradiscal pressure, resulting from the compression of the disc, and the restraining forces generated by the collagen fibres within the annulus fibrosus. A formula is derived for predicting the loading force on a disc once the nucleus pressure is known. Measured material values of L3 and L4 discs are used to make quantitative predictions. The results compare reasonably well with experimental results.
Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies
Rakesh Pilkar , Erik M. Bollt and  Charles Robinson
2011, 8(4): 1085-1097 doi: 10.3934/mbe.2011.8.1085 +[Abstract](27) +[PDF](12920.4KB)
Bursts of 2.5mm horizontal sinusoidal anterior-posterior oscillations of sequentially varying frequencies (0.25 to 1.25 Hz) are applied to the base of support to study postural control. The Empirical Mode Decomposition (EMD) algorithm decomposes the Center of Pressure (CoP) data (5 young, 4 mature adults) into Intrinsic Mode Functions (IMFs). Hilbert transforms are applied to produce each IMF’s time-frequency spectrum. The most dominant mode in total energy indicates a sway ramble with a frequency content below 0.1 Hz. Other modes illustrate that the stimulus frequencies produce a ‘locked-in’ behavior of CoP with platform position signal. The combined Hilbert Spectrum of these modes shows that this phase-lock behavior of APCoP is more apparent for 0.5, 0.625, 0.75 and 1 Hz perturbation intervals. The instantaneous energy profiles of the modes depict significant energy changes during the stimulus intervals in case of lock-in. The EMD technique provides the means to visualize the multiple oscillatory modes present in the APCoP signal with their time scale dependent on the signals’s successive extrema. As a result, the extracted oscillatory modes clearly show the time instances when the subject’s APCoP clearly synchronizes with the provided sinusoidal platform stimulus and when it does not.
Dynamics of a delay Schistosomiasis model in snail infections
Chunhua Shan , Hongjun Gao and  Huaiping Zhu
2011, 8(4): 1099-1115 doi: 10.3934/mbe.2011.8.1099 +[Abstract](48) +[PDF](673.1KB)
In this paper we modify and study a system of delay differential equations model proposed by Nåsell and Hirsch (1973) for the transmission dynamics of schistosomiasis. The modified stochastic version of MacDonald’s model takes into account the time delay for the transmission of infection. We carry out bifurcation studies of the model. The saddle-node bifurcation of the model suggests that the transmission and spread of schistosomiasis is initial size dependent. The existence of a Hopf bifurcation due to the delay indicates that the transmission can be periodic.
A dynamic model describing heterotrophic culture of chorella and its stability analysis
Yan Zhang , Wanbiao Ma , Hai Yan and  Yasuhiro Takeuchi
2011, 8(4): 1117-1133 doi: 10.3934/mbe.2011.8.1117 +[Abstract](26) +[PDF](422.0KB)
Chlorella is an important species of microorganism, which includes about 10 species. Chlorella USTB01 is a strain of microalga which is isolated from Qinghe River in Beijing and has strong ability in the utilization of organic compounds and was identified as Chlorella sp. (H. Yan etal, Isolation and heterotrophic culture of Chlorella sp., J. Univ. Sci. Tech. Beijing, 2005, 27:408-412). In this paper, based on the standard Chemostat models and the experimental data on the heterotrophic culture of Chlorella USTB01, a dynamic model governed by differential equations with three variables (Chlorella, carbon source and nitrogen source) is proposed. For the model, there always exists a boundary equilibrium, i.e. Chlorella-free equilibrium. Furthermore, under additional conditions, the model also has the positive equilibria, i.e., the equilibira for which Chlorella, carbon source and nitrogen source are coexistent. Then, local and global asymptotic stability of the equilibria of the model have been discussed. Finally, the parameters in the model are determined according to the experimental data, and numerical simulations are given. The numerical simulations show that the trajectories of the model fit the trends of the experimental data well.
Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization
Zhenzhen Zheng , Ching-Shan Chou , Tau-Mu Yi and  Qing Nie
2011, 8(4): 1135-1168 doi: 10.3934/mbe.2011.8.1135 +[Abstract](36) +[PDF](1284.5KB)
Cell polarization, in which substances previously uniformly distributed become asymmetric due to external or/and internal stimulation, is a fundamental process underlying cell mobility, cell division, and other polarized functions. The yeast cell S. cerevisiae has been a model system to study cell polarization. During mating, yeast cells sense shallow external spatial gradients and respond by creating steeper internal gradients of protein aligned with the external cue. The complex spatial dynamics during yeast mating polarization consists of positive feedback, degradation, global negative feedback control, and cooperative effects in protein synthesis. Understanding such complex regulations and interactions is critical to studying many important characteristics in cell polarization including signal amplification, tracking dynamic signals, and potential trade-off between achieving both objectives in a robust fashion. In this paper, we study some of these questions by analyzing several models with different spatial complexity: two compartments, three compartments, and continuum in space. The step-wise approach allows detailed characterization of properties of the steady state of the system, providing more insights for biological regulations during cell polarization. For cases without membrane diffusion, our study reveals that increasing the number of spatial compartments results in an increase in the number of steady-state solutions, in particular, the number of stable steady-state solutions, with the continuum models possessing infinitely many steady-state solutions. Through both analysis and simulations, we find that stronger positive feedback, reduced diffusion, and a shallower ligand gradient all result in more steady-state solutions, although most of these are not optimally aligned with the gradient. We explore in the different settings the relationship between the number of steady-state solutions and the extent and accuracy of the polarization. Taken together these results furnish a detailed description of the factors that influence the tradeoff between a single correctly aligned but poorly polarized stable steady-state solution versus multiple more highly polarized stable steady-state solutions that may be incorrectly aligned with the external gradient.

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