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Mathematical Biosciences & Engineering

2008 , Volume 5 , Issue 3

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Input-dependent wave propagations in asymmetric cellular automata: Possible behaviors of feed-forward loop in biological reaction network
Akinori Awazu
2008, 5(3): 419-427 doi: 10.3934/mbe.2008.5.419 +[Abstract](2573) +[PDF](249.6KB)
Dynamical aspects of the asymmetric cellular automata were in- vestigated to consider the signaling processes in biological systems. As a meta- model of the cascade of feed-forward loop type network motifs in biological reaction networks, we consider the one dimensional asymmetric cellular au- tomata where the state of each cell is controlled by a trio of cells, the cell itself, the nearest upstream cell and the next nearest upstream cell. Through the systematic simulations, some novel input-dependent wave propagations were found in certain asymmetric CA, which may be useful for the signaling pro- cesses like the distinction, the filtering and the memory of external stimuli.
The effect of patterns of infectiousness on epidemic size
Luis F. Gordillo, Stephen A. Marion and Priscilla E. Greenwood
2008, 5(3): 429-435 doi: 10.3934/mbe.2008.5.429 +[Abstract](2707) +[PDF](186.1KB)
In the course of an infectious disease in a population, each in fected individual presents a different pattern of progress through the disease, producing a corresponding pattern of infectiousness. We postulate a stochastic infectiousness process for each individual with an almost surely finite integral, or total infectiousness. Individuals also have different contact rates. We show that the distribution of the final epidemic size depends only on the contact rates and the integrated infectiousness. As a particular case, zero infectiousness on an initial time interval corresponds to a period of latency, which does not affect the final epidemic size in general stochastic and deterministic epidemic models, as is well known from the literature.
Existence of multiple-stable equilibria for a multi-drug-resistant model of mycobacterium tuberculosis
Abba B. Gumel and Baojun Song
2008, 5(3): 437-455 doi: 10.3934/mbe.2008.5.437 +[Abstract](2575) +[PDF](5611.1KB)
The resurgence of multi-drug-resistant tuberculosis in some parts of Europe and North America calls for a mathematical study to assess the impact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministic compartmental model for the transmission dynamics of two strains of tubercu- losis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The model allows for the assessment of the treatment of people infected with the wild strain. The qualitative analysis of the model reveals the following. The model has a disease-free equilibrium, which is locally asymptotically stable if a cer- tain threshold, known as the effective reproduction number, is less than unity. Further, the model undergoes a backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium. One of the main nov- elties of this study is the numerical illustration of tri-stable equilibria, where the disease-free equilibrium coexists with two stable endemic equilibrium when the aforementioned threshold is less than unity, and a bi-stable setup, involving two stable endemic equilibria, when the effective reproduction number is greater than one. This, to our knowledge, is the first time such dynamical features have been observed in TB dynamics. Finally, it is shown that the backward bifurcation phenomenon in this model arises due to the exogenous re-infection property of tuberculosis.
Mathematical analysis of a HIV model with frequency dependence and viral diversity
Shingo Iwami, Shinji Nakaoka and Yasuhiro Takeuchi
2008, 5(3): 457-476 doi: 10.3934/mbe.2008.5.457 +[Abstract](2405) +[PDF](349.4KB)
We consider the effect of viral diversity on the human immune sys- tem with the frequency dependent proliferation rate of CTLs and the elimina- tion rate of infected cells by CTLs. The model has very complex mathematical structures such as limit cycle, quasi-periodic attractors, chaotic attractors, and so on. To understand the complexity we investigate the global behavior of the model and demonstrate the existence and stability conditions of the equilibria. Further we give some theoretical considerations obtained by our mathematical model to HIV infection.
Permanence for two-species Lotka-Volterra cooperative systems with delays
Guichen Lu and Zhengyi Lu
2008, 5(3): 477-484 doi: 10.3934/mbe.2008.5.477 +[Abstract](2851) +[PDF](182.5KB)
In this paper, a two-species Lotka-Volterra cooperative delay sys- tem is considered, and the relationships between the delays and the permanence are obtained. Some sufficient conditions for the permanence under the assumption of smallness of the delays are obtained. Two examples are given to illustrate the theorems.
Modelling the immunopathogenesis of HIV-1 infection and the effect of multidrug therapy: The role of fusion inhibitors in HAART
Gesham Magombedze, Winston Garira and Eddie Mwenje
2008, 5(3): 485-504 doi: 10.3934/mbe.2008.5.485 +[Abstract](2961) +[PDF](1158.1KB)
There is currently tremendous effort being directed at developing potent, highly active antiretroviral therapies that can effectively control HIV- 1 infection without the need for continuous, lifelong use of these drugs. In the ongoing search for powerful antiretroviral agents that can affect sustained control for HIV infection, mathematical models can help in assessing both the correlates of protective immunity and the clinical role of a given drug regimen as well as in understanding the efficacy of drug therapies administered at different stages of the disease. In this study, we develop a new mathematical model of the immuno-pathogenesis of HIV-1 infection, which we use to assess virological responses to both intracellular and extracellular antiretroviral drugs. We first develop a basic mathematical model of the immuno-pathogenesis of HIV-1 infection that incorporates three distinct stages in the infection cycle of HIV-1: entry of HIV-1 into the cytoplasm of CD4+ T cells, transcription of HIV-1 RNA to DNA within CD4+ T cells, and production of HIV-1 viral particles within CD4+ T cells. Then we extend the basic model to incorporate the effect of three major categories of anti-HIV-1 drugs: fusion/entry inhibitors (FIs), reverse transcriptase inhibitors (RTIs), and protease inhibitors (PIs). Model analysis establishes that the actual drug efficacy of FIs, γ and of PIs, κ is the same as their effective efficacies while the effective drug efficacy for the RTIs, γ εis dependent on the rate of transcription of the HIV-1 RNA to DNA, and the lifespan of infected CD4+ T cells where virions have only entered the cytoplasm and that this effective efficacy is less than the actual efficacy, ε. Our studies suggest that, of the three anti-HIV drug categories (FIs, RTIs, and PIs), any drug combination of two drugs that includes RTIs is the weakest in the control of HIV-1 infection.
A deterministic model of schistosomiasis with spatial structure
Fabio Augusto Milner and Ruijun Zhao
2008, 5(3): 505-522 doi: 10.3934/mbe.2008.5.505 +[Abstract](2399) +[PDF](760.6KB)
It has been observed in several settings that schistosomiasis is less prevalent in segments of river with fast current than in those with slow current. Some believe that this can be attributed to flush-away of intermediate host snails. However, free-swimming parasite larvae are very active in searching for suitable hosts, which indicates that the flush-away of larvae may also be very important. In this paper, the authors establish a model with spatial structure that characterizes the density change of parasites following the flush-away of larvae. It is shown that the reproductive number, which is an indicator of prevalence of parasitism, is a decreasing function of the river current velocity. Moreover, numerical simulations suggest that the mean parasite load is low when the velocity of river current flow is sufficiently high.
Modeling the rapid spread of avian influenza (H5N1) in India
Arni S.R. Srinivasa Rao
2008, 5(3): 523-537 doi: 10.3934/mbe.2008.5.523 +[Abstract](2351) +[PDF](269.8KB)
Controlling the spread of avian bird flu has become a challenging tasks for Indian agriculture and health administrators. After the first evidence and control of the virus in 2006, the virus attacked five states by January 2008. Based on the evidence of rapid spread of the avian bird flu type H5N1 among the Indian states of Maharashtra, Manipur, andWest Bengal, and in the partially affected states of Gujarat and Madhya Pradesh, a model is developed to understand the spread of the virus among birds and the effect of control measures on the dynamics of its spread. We predict that, in the absence of control measures, the total number of infected birds in West Bengal within ten and twenty days after initial discovery of infection were 780,000 and 2.1 million, respectively. When interventions are introduced, these values would have ranged from 65,000 to 225,000 after ten days and from 16,000 to 190,000 after twenty days. We show that the farm and market birds constitute the major proportion of total infected birds, followed by domestic birds and wild birds in West Bengal, where a severe epidemic hit recently. Culling 600,000 birds in ten days might have reduced the current epidemic before it spread extensively. Further studies on appropriate transmission parameters, contact rates of birds, population sizes of poultry and farms are helpful for planning.
Biological control of the chemostat with nonmonotonic response and different removal rates
Alain Rapaport and Jérôme Harmand
2008, 5(3): 539-547 doi: 10.3934/mbe.2008.5.539 +[Abstract](2321) +[PDF](213.5KB)
We show the global stabilization of the chemostat with nonmonotonic growth, adding a new species as a ''biological'' control, in presence of different removal rates for each species. This result is obtained by an extension of the Competitive Exclusion Principle in the chemostat, for the case of two species with different removal rates and at least one nonmonotonic response.
An extension of the Beretta-Kuang model of viral diseases
Ivo Siekmann, Horst Malchow and Ezio Venturino
2008, 5(3): 549-565 doi: 10.3934/mbe.2008.5.549 +[Abstract](2244) +[PDF](1414.3KB)
A model for the complete life cycle of marine viruses is presented. The Beretta-Kuang model introduces an explicit equation for viral particles but the replication process of viral particles in their hosts is not considered. The extended model keeps the structure of the original model. This makes it possible to estimate the growth parameters of the viruses for a given parametrisation of the Beretta-Kuang model.
Stabilization due to predator interference: comparison of different analysis approaches
G.A.K. van Voorn, D. Stiefs, T. Gross, B. W. Kooi, Ulrike Feudel and S.A.L.M. Kooijman
2008, 5(3): 567-583 doi: 10.3934/mbe.2008.5.567 +[Abstract](3176) +[PDF](717.6KB)
We study the influence of the particular form of the functional response in two-dimensional predator-prey models with respect to the stability of the nontrivial equilibrium. This equilibrium is stable between its appearance at a transcritical bifurcation and its destabilization at a Hopf bifurcation, giving rise to periodic behavior. Based on local bifurcation analysis, we introduce a classification of stabilizing effects. The classical Rosenzweig-MacArthur model can be classified as weakly stabilizing, undergoing the paradox of enrichment, while the well known Beddington-DeAngelis model can be classified as strongly stabilizing. Under certain conditions we obtain a complete stabilization, resulting in an avoidance of limit cycles. Both models, in their conventional formulation, are compared to a generalized, steady-state independent two-dimensional version of these models, based on a previously developed normalization method. We show explicitly how conventional and generalized models are related and how to interpret the results from the rather abstract stability analysis of generalized models.
Calculation of $R_0$ for age-of-infection models
Christine K. Yang and Fred Brauer
2008, 5(3): 585-599 doi: 10.3934/mbe.2008.5.585 +[Abstract](2640) +[PDF](210.7KB)
We consider age-of-infection epidemic models to describe multiple- stage epidemic models, including treatment. We derive an expression for the basic reproduction number $R_0$ in terms of the distributions of periods of stay in the various compartments. We find that, in the model without treatment, $R_0$ depends only on the mean periods in compartments, and not on the form of the distributions. In treatment models, $R_0$ depends on the form of the dis- tributions of stay in infective compartments from which members are removed for treatment, but the dependence for treatment compartments is only on the mean stay in the compartments. The results give a considerable simplification in the calculation of the basic reproduction number.

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