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Mathematical Biosciences & Engineering

2012 , Volume 9 , Issue 1

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Nonlinear stochastic Markov processes and modeling uncertainty in populations
H.Thomas Banks and  Shuhua Hu
2012, 9(1): 1-25 doi: 10.3934/mbe.2012.9.1 +[Abstract](48) +[PDF](433.9KB)
We consider an alternative approach to the use of nonlinear stochastic Markov processes (which have a Fokker-Planck or Forward Kolmogorov representation for density) in modeling uncertainty in populations. These alternate formulations, which involve imposing probabilistic structures on a family of deterministic dynamical systems, are shown to yield pointwise equivalent population densities. Moreover, these alternate formulations lead to fast efficient calculations in inverse problems as well as in forward simulations. Here we derive a class of stochastic formulations for which such an alternate representation is readily found.
The implications of model formulation when transitioning from spatial to landscape ecology
Robert Stephen Cantrell , Chris Cosner and  William F. Fagan
2012, 9(1): 27-60 doi: 10.3934/mbe.2012.9.27 +[Abstract](42) +[PDF](395.8KB)
In this article we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable habitat to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).
An evaluation of dynamic outlet boundary conditions in a 1D fluid dynamics model
Rachel Clipp and  Brooke Steele
2012, 9(1): 61-74 doi: 10.3934/mbe.2012.9.61 +[Abstract](33) +[PDF](478.8KB)
When modeling the cardiovascular system, the use of boundary conditions that closely represent the interaction between the region of interest and the surrounding vessels and organs will result in more accurate predictions. An often overlooked feature of outlet boundary conditions is the dynamics associated with regulation of the distribution of pressure and flow. This study implements a dynamic impedance outlet boundary condition in a one-dimensional fluid dynamics model using the pulmonary vasculature and respiration (feedback mechanism) as an example of a dynamic system. The dynamic boundary condition was successfully implemented and the pressure and flow were predicted for an entire respiration cycle. The cardiac cycles at maximal expiration and inspiration were predicted with a root mean square error of $0.61$ and $0.59$ mm Hg, respectively.
Nonlinear functional response parameter estimation in a stochastic predator-prey model
Gianni Gilioli , Sara Pasquali and  Fabrizio Ruggeri
2012, 9(1): 75-96 doi: 10.3934/mbe.2012.9.75 +[Abstract](54) +[PDF](696.7KB)
Parameter estimation for the functional response of predator-prey systems is a critical methodological problem in population ecology. In this paper we consider a stochastic predator-prey system with non-linear Ivlev functional response and propose a method for model parameter estimation based on time series of field data. We tackle the problem of parameter estimation using a Bayesian approach relying on a Markov Chain Monte Carlo algorithm. The efficiency of the method is tested on a set of simulated data. Then, the method is applied to a predator-prey system of importance for Integrated Pest Management and biological control, the pest mite Tetranychus urticae and the predatory mite Phytoseiulus persimilis. The model is estimated on a dataset obtained from a field survey. Finally, the estimated model is used to forecast predator-prey dynamics in similar fields, with slightly different initial conditions.
Impact of discontinuous treatments on disease dynamics in an SIR epidemic model
Zhenyuan Guo , Lihong Huang and  Xingfu Zou
2012, 9(1): 97-110 doi: 10.3934/mbe.2012.9.97 +[Abstract](68) +[PDF](382.0KB)
We consider an SIR epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we are able to determine the basic reproduction number $\mathcal{R}_0$, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that in the case that an equilibrium is asymptotically stable, the convergence to the equilibrium can actually be achieved in finite time, and we can estimate this time in terms of the model parameters, initial sub-populations and the initial treatment strength. This suggests that from the view point of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we use to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on non-smooth analysis.
Threshold dynamics for a Tuberculosis model with seasonality
Xinli Hu
2012, 9(1): 111-122 doi: 10.3934/mbe.2012.9.111 +[Abstract](43) +[PDF](392.1KB)
In this paper, we investigate a SEILR tuberculosis model incorporating the effect of seasonal fluctuation, where the loss of sight class is considered. The basic reproduction number $R_{0}$ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if $R_{0}<1$, and there exists at least one positive periodic solution and the disease is uniformly persistent if $R_{0}>1$. Numerical simulations are provided to illustrate analytical results.
A statistical approach to the use of control entropy identifies differences in constraints of gait in highly trained versus untrained runners
Rana D. Parshad , Stephen J. McGregor , Michael A. Busa , Joseph D. Skufca and  Erik Bollt
2012, 9(1): 123-145 doi: 10.3934/mbe.2012.9.123 +[Abstract](25) +[PDF](875.7KB)
Control entropy (CE) is a complexity analysis suitable for dynamic, non-stationary conditions which allows the inference of the control effort of a dynamical system generating the signal [4]. These characteristics make CE a highly relevant time varying quantity relevant to the dynamic physiological responses associated with running. Using High Resolution Accelerometry (HRA) signals we evaluate here constraints of running gait, from two different groups of runners, highly trained collegiate and untrained runners. To this end, we further develop the control entropy (CE) statistic to allow for group analysis to examine the non-linear characteristics of movement patterns in highly trained runners with those of untrained runners, to gain insight regarding gaits that are optimal for running. Specifically, CE develops response time series of individuals descriptive of the control effort; a group analysis of these shapes developed here uses Karhunen Loeve Analysis (KL) modes of these time series which are compared between groups by application of a Hotelling $T^{2}$ test to these group response shapes. We find that differences in the shape of the CE response exist within groups, between axes for untrained runners (vertical vs anterior-posterior and mediolateral vs anterior-posterior) and trained runners (mediolateral vs anterior-posterior). Also shape differences exist between groups by axes (vertical vs mediolateral). Further, the CE, as a whole, was higher in each axis in trained vs untrained runners. These results indicate that the approach can provide unique insight regarding the differing constraints on running gait in highly trained and untrained runners when running under dynamic conditions. Further, the final point indicates trained runners are less constrained than untrained runners across all running speeds.
Assessing the effect of non-pharmaceutical interventions on containing an emerging disease
Zi Sang , Zhipeng Qiu , Xiefei Yan and  Yun Zou
2012, 9(1): 147-164 doi: 10.3934/mbe.2012.9.147 +[Abstract](30) +[PDF](238.2KB)
Non-pharmaceutical interventions, such as quarantine, isolation and entry screening, are usually the primary public health measures to control the spread of an emerging infectious disease through a human population. This paper proposes a multi-regional deterministic compartmental model to assess the effectiveness and implications of non-pharmaceutical interventions. The reproduction number is determined as the spectral radius of a nonnegative matrix product. Comparisons are made using the reproduction number, epidemic peaks and cumulative number of infections and mortality as indexes. Simulation results show that quarantine of suspected cases and isolation of cases with symptom are effective in reducing disease burden for multiple regions. Using entry screening strategy leads to a moderate time delay for epidemic peaks, but is of no help for preventing an epidemic breaking out. The study further shows that isolation strategy is always the best choice in the presence or absence of stringent hygiene precautions and should be given priority in combating an emerging epidemic.
Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes
Cruz Vargas-De-León
2012, 9(1): 165-174 doi: 10.3934/mbe.2012.9.165 +[Abstract](37) +[PDF](344.0KB)
A delayed vector-bias model for malaria transmission with incubation period in mosquitoes is studied. The delay $\tau$ corresponds to the time necessary for a latently infected vector to become an infectious vector. We prove that the global stability is completely determined by the threshold parameter, $R_0(\tau)$. If $R_0(\tau)\leq1$, the disease-free equilibrium is globally asymptotically stable. If $R_0(\tau)>1$ a unique endemic equilibrium exists and is globally asymptotically stable. We apply our results to Ross-MacDonald malaria models with an incubation period (extrinsic or intrinsic).
A quasi-lumped model for the peripheral distortion of the arterial pulse
Panagiotes A. Voltairas , Antonios Charalambopoulos , Dimitrios I. Fotiadis and  Lambros K. Michalis
2012, 9(1): 175-198 doi: 10.3934/mbe.2012.9.175 +[Abstract](29) +[PDF](588.5KB)
As blood circulates through the arterial tree, the flow and pressure pulse distort. Principal factors to this distortion are reflections form arterial bifurcations and the viscous character of the flow of the blood. Both of them are expounded in the literature and included in our analysis. The nonlinearities of inertial effects are usually taken into account in numerical simulations, based on Navier-Stokes like equations. Nevertheless, there isn't any qualitative, analytical formula, which examines the role of blood's inertia on the distortion of the pulse. We derive such an analytical nonlinear formula. It emanates from a generalized Bernoulli's equation for an an-harmonic, linear, viscoelastic, Maxwell fluid flow in a linear, viscoelastic, Kelvin-Voigt, thin, cylindrical vessel. We report that close to the heart, convection effects related to the change in the magnitude of the velocity of blood dominate the alteration of the shape of the pressure pulse, while at remote sites of the vascular tree, convection of vorticity, related to the change in the direction of the velocity of blood with respect to a mean axial flow, prevails. A quantitative comparison between the an-harmonic theory and related pressure measurements is also performed.
Effect of branchings on blood flow in the system of human coronary arteries
Benchawan Wiwatanapataphee , Yong Hong Wu , Thanongchai Siriapisith and  Buraskorn Nuntadilok
2012, 9(1): 199-214 doi: 10.3934/mbe.2012.9.199 +[Abstract](35) +[PDF](1169.5KB)
In this work, we investigate the behavior of the pulsatile blood flow in the system of human coronary arteries. Blood is modeled as an incompressible non-Newtonian fluid. The transient phenomena of blood flow through the coronary system are simulated by solving the three dimensional unsteady state Navier-Stokes equations and continuity equation. Distributions of velocity, pressure and wall shear stresses are determined in the system under pulsatile conditions on the boundaries. Effect of branching vessel on the flow problem is investigated. The numerical results show that blood pressure in the system with branching vessels of coronary arteries is lower than the one in the system with no branch. The magnitude of wall shear stresses rises at the bifurcation.

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