# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2007 , Volume 4 , Issue 3

Select all articles

Export/Reference:

2007, 4(3): 373-402 doi: 10.3934/mbe.2007.4.373 +[Abstract](1322) +[PDF](516.5KB)
Abstract:
An approach to modeling the impact of disturbances in an agricultural production network is presented. A stochastic model and its approximate deterministic model for averages over sample paths of the stochastic system are developed. Simulations, sensitivity and generalized sensitivity analyses are given. Finally, it is shown how diseases may be introduced into the network and corresponding simulations are discussed.
2007, 4(3): 403-430 doi: 10.3934/mbe.2007.4.403 +[Abstract](1103) +[PDF](390.3KB)
Abstract:
We develop a theory for sensitivity with respect to parameters in a convex subset of a topological vector space of dynamical systems in a Banach space. Specific motivating examples for probability measure dependent differential, partial differential and delay differential equations are given. Schemes that approximate the measures in the Prohorov sense are illustrated with numerical simulations for distributed delay differential equations.
2007, 4(3): 431-456 doi: 10.3934/mbe.2007.4.431 +[Abstract](1261) +[PDF](1918.7KB)
Abstract:
In this paper the dynamics of a tritrophic food chain (resource, consumer, top predator) is investigated, with particular attention not only to equilibrium states but also to cyclic behaviours that the system may exhibit. The analysis is performed in terms of two bifurcation parameters, denoted by $p$ and $q$, which measure the efficiencies of the interaction processes. The persistence of the system is discussed, characterizing in the $(p,q)$ plane the regions of existence and stability of biologically significant steady states and those of existence of limit cycles. The bifurcations occurring are discussed, and their implications with reference to biological control problems are considered. Examples of the rich dynamics exhibited by the model, including a chaotic regime, are described.
2007, 4(3): 457-470 doi: 10.3934/mbe.2007.4.457 +[Abstract](1410) +[PDF](642.4KB)
Abstract:
At the outset of an influenza pandemic, early estimates of the number of secondary cases generated by a primary influenza case (reproduction number, $R$) and its associated uncertainty can help determine the intensity of interventions necessary for control. Using a compartmental model and hospital notification data of the first two waves of the Spanish flu pandemic in Geneva, Switzerland in 1918, we estimate the reproduction number from the early phase of the pandemic waves. For the spring and fall pandemic waves, we estimate reproduction numbers of $1.57$ ($95\%$ CI: $1.45$, $1.70$) and $3.10$ ($2.81$, $3.39$), respectively, from the initial epidemic phase comprising the first $10$ epidemic days of the corresponding wave. Estimates of the variance of our point estimates of $R$ were computed via a parametric bootstrap. We compare these estimates with others obtained using different observation windows to provide insight into how early into an epidemic the reproduction number can be estimated.
2007, 4(3): 471-488 doi: 10.3934/mbe.2007.4.471 +[Abstract](1091) +[PDF](275.3KB)
Abstract:
Cells use a signal transduction mechanism to regulate certain metabolic pathways. In this paper, the regulatory mechanism is analyzed mathematically. For this analysis, a mathematical model for the pathways is first established using a system of differential equations. Then the linear stability, controllability, and observability of the system are investigated. We show that the linearized system is controllable and observable, and that the real parts of all eigenvalues of the linearized system are nonpositive using Routh's stability criterion. Controllability and observability are structural properties of a dynamical system. Thus our results may explain why the metabolic pathways can be controlled and regulated. Finally observer-based and proportional output feedback controllers are designed to regulate the end product to its desired level. Applications to the regulation of blood glucose levels are discussed.
2007, 4(3): 489-504 doi: 10.3934/mbe.2007.4.489 +[Abstract](1368) +[PDF](254.5KB)
Abstract:
This study involves the mathematical modelling of long-term HIV dynamics. The proposed model is able to predict the entire trajectory of the disease: initial viremia in the early weeks of the infection, latency, and progression to AIDS; a range spanning approximately ten years. The model outcomes were compared to clinical data and significant agreement was achieved. The formulated model considers all important population compartments including macrophages, latently-infected CD4+ T-cells, and cytotoxic T-lymphocytes (CTLs), an attempt which in many respects is novel in the area of HIV modelling. The ranges of the model parameters and initial conditions were obtained from literature, and their values were determined in this work directly by fitting published clinical data. Furthermore, the simulation results emphasize the importance of macrophages in HIV infection and progression to AIDS and show a clear correlation between the level of CTLs and HIV progression. The ability of the model to correlate analytical data gives credibility to its predictions, a fact that will be exploited in future research in modelling immunological and pharmacological avenues of treatment.
2007, 4(3): 505-522 doi: 10.3934/mbe.2007.4.505 +[Abstract](1132) +[PDF](349.8KB)
Abstract:
We describe several population models exposed to a mild life-long sexually transmitted disease, i.e. without significant increased mortality among infected individuals and providing no immunity/recovery. We then modify these models to include groups isolated from sexual contact and analyze their potential effect on the dynamics of the population. We are interested in how the isolated class may curb the growth of the infected group while keeping the healthy population at acceptable levels.
2007, 4(3): 523-530 doi: 10.3934/mbe.2007.4.523 +[Abstract](1234) +[PDF](468.0KB)
Abstract:
We study the feasibility of transferring data in an optical device by using a limited number of parallel channels. By applying a spatially localized correcting term to the evolution of a liquid crystal light valve in its spatio--temporal chaotic regime, we are able to restore the dynamics to a specified target pattern. The system is controlled in a finite time. The number and position of pinning points needed to attain control is also investigated.
2007, 4(3): 531-552 doi: 10.3934/mbe.2007.4.531 +[Abstract](1331) +[PDF](519.9KB)
Abstract:
Among the features of real immune responses that occur when antigens invade a body are two remarkable features. One is that the number of antibodies produced in the secondary invasion by identical antigens is more than 10 times larger than in the primary invasion. The other is that more effective antibodies, which are produced by somatic hypermutation during the immune response, can neutralize the antigens more quickly. This phenomenon is called ''affinity maturation''.
In this paper, we try to reproduce these features by dynamical system models and present possible factors to realize them. Further, we present a model in which the memory of the antigen invasion is realized without immune memory cells.
2007, 4(3): 553-563 doi: 10.3934/mbe.2007.4.553 +[Abstract](1434) +[PDF](211.7KB)
Abstract:
We compare four SIR-style models describing behavioral or immunological disease resistance that may be both partial and temporary in parameter regions feasible for interpandemic influenza. For the models studied, backward bifurcations and bistability may occur in contexts where resistance is due to behavior change, but they do not occur when resistance originates from an immune response. Care must be exercised to ensure that modeling assumptions about resistance are consistent with the biological mechanisms under study.

2018  Impact Factor: 1.313