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Mathematical Biosciences & Engineering

2014 , Volume 11 , Issue 3

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Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology
Edward J. Allen
2014, 11(3): 403-425 doi: 10.3934/mbe.2014.11.403 +[Abstract](268) +[PDF](766.7KB)
Abstract:
Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.
Model validation for a noninvasive arterial stenosis detection problem
H. Thomas Banks, Shuhua Hu, Zackary R. Kenz, Carola Kruse, Simon Shaw, John Whiteman, Mark P. Brewin, Stephen E. Greenwald and Malcolm J. Birch
2014, 11(3): 427-448 doi: 10.3934/mbe.2014.11.427 +[Abstract](364) +[PDF](624.7KB)
Abstract:
A current thrust in medical research is the development of a non-invasive method for detection, localization, and characterization of an arterial stenosis (a blockage or partial blockage in an artery). A method has been proposed to detect shear waves in the chest cavity which have been generated by disturbances in the blood flow resulting from a stenosis. In order to develop this methodology further, we use one-dimensional shear wave experimental data from novel acoustic phantoms to validate a corresponding viscoelastic mathematical model. We estimate model parameters which give a good fit (in a sense to be precisely defined) to the experimental data, and use asymptotic error theory to provide confidence intervals for parameter estimates. Finally, since a robust error model is necessary for accurate parameter estimates and confidence analysis, we include a comparison of absolute and relative models for measurement error.
The global stability of an SIRS model with infection age
Yuming Chen, Junyuan Yang and Fengqin Zhang
2014, 11(3): 449-469 doi: 10.3934/mbe.2014.11.449 +[Abstract](284) +[PDF](482.8KB)
Abstract:
Infection age is an important factor affecting the transmission of infectious diseases. In this paper, we consider an SIRS model with infection age, which is described by a mixed system of ordinary differential equations and partial differential equations. The expression of the basic reproduction number $\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then the model only has the disease-free equilibrium, while if $\mathscr{R}_0>1$ then besides the disease-free equilibrium the model also has an endemic equilibrium. Moreover, if $\mathscr{R}_0<1$ then the disease-free equilibrium is globally asymptotically stable otherwise it is unstable; if $\mathscr{R}_0>1$ then the endemic equilibrium is globally asymptotically stable under additional conditions. The local stability is established through linearization. The global stability of the disease-free equilibrium is shown by applying the fluctuation lemma and that of the endemic equilibrium is proved by employing Lyapunov functionals. The theoretical results are illustrated with numerical simulations.
A metapopulation model for sylvatic T. cruzi transmission with vector migration
Britnee Crawford and Christopher Kribs-Zaleta
2014, 11(3): 471-509 doi: 10.3934/mbe.2014.11.471 +[Abstract](310) +[PDF](579.0KB)
Abstract:
This study presents a metapopulation model for the sylvatic transmission of Trypanosoma cruzi, the etiological agent of Chagas' disease, across multiple geographical regions and multiple overlapping host-vector transmission cycles. Classical qualitative analysis of the model and several submodels focuses on the parasite's basic reproductive number, illustrating how vector migration across patches and multiple transmission routes to hosts (including vertical transmission) determine the infection's persistence in each cycle. Numerical results focus on trends in endemic [equilibrium] persistence levels as functions of vector migration rates, and highlight the significance of the different epidemiological characteristics of transmission in each of the three regions.
Optimal sterile insect release for area-wide integrated pest management in a density regulated pest population
Luis F. Gordillo
2014, 11(3): 511-521 doi: 10.3934/mbe.2014.11.511 +[Abstract](268) +[PDF](337.5KB)
Abstract:
To determine optimal sterile insect release policies in area-wide integrated pest management is a challenge that users of this pest control method inevitably confront. In this note we provide approximations to best policies of release through the use of simulated annealing. The discrete time model for the population dynamics includes the effects of sterile insect release and density dependence in the pest population. Spatial movement is introduced through integrodifference equations, which allow the use of the stochastic search in cases where movement is described through arbitrary dispersal kernels. As a byproduct of the computations, an assessment of appropriate control zone sizes is possible.
Effect of intraocular pressure on the hemodynamics of the central retinal artery: A mathematical model
Giovanna Guidoboni, Alon Harris, Lucia Carichino, Yoel Arieli and Brent A. Siesky
2014, 11(3): 523-546 doi: 10.3934/mbe.2014.11.523 +[Abstract](304) +[PDF](1226.7KB)
Abstract:
Retinal hemodynamics plays a crucial role in the pathophysiology of several ocular diseases. There are clear evidences that the hemodynamics of the central retinal artery (CRA) is strongly affected by the level of intraocular pressure (IOP), which is the pressure inside the eye globe. However, the mechanisms through which this occurs are still elusive. The main goal of this paper is to develop a mathematical model that combines the mechanical action of IOP and the blood flow in the CRA to elucidate the mechanisms through which IOP elevation affects the CRA hemodynamics. Our model suggests that the development of radial compressive regions in the lamina cribrosa (a collagen structure in the optic nerve pierced by the CRA approximately in its center) might be responsible for the clinically-observed blood velocity reduction in the CRA following IOP elevation. The predictions of the mathematical model are in very good agreement with experimental and clinical data. Our model also identifies radius and thickness of the lamina cribrosa as major factors affecting the IOP-CRA relationship, suggesting that anatomical differences among individuals might lead to different hemodynamic responses to IOP elevation.
A model of optimal dosing of antibiotic treatment in biofilm
Mudassar Imran and Hal L. Smith
2014, 11(3): 547-571 doi: 10.3934/mbe.2014.11.547 +[Abstract](365) +[PDF](960.3KB)
Abstract:
Biofilms are heterogeneous matrix enclosed micro-colonies of bacteria mostly found on moist surfaces. Biofilm formation is the primary cause of several persistent infections found in humans. We derive a mathematical model of biofilm and surrounding fluid dynamics to investigate the effect of a periodic dose of antibiotic on elimination of microbial population from biofilm. The growth rate of bacteria in biofilm is taken as Monod type for the limiting nutrient. The pharmacodynamics function is taken to be dependent both on limiting nutrient and antibiotic concentration. Assuming that flow rate of fluid compartment is large enough, we reduce the six dimensional model to a three dimensional model. Mathematically rigorous results are derived providing sufficient conditions for treatment success. Persistence theory is used to derive conditions under which the periodic solution for treatment failure is obtained. We also discuss the phenomenon of bi-stability where both infection-free state and infection state are locally stable when antibiotic dosing is marginal. In addition, we derive the optimal antibiotic application protocols for different scenarios using control theory and show that such treatments ensure bacteria elimination for a wide variety of cases. The results show that bacteria are successfully eliminated if the discrete treatment is given at an early stage in the infection or if the optimal protocol is adopted. Finally, we examine factors which if changed can result in treatment success of the previously treatment failure cases for the non-optimal technique.
A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system
Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri and Diego P. Ruiz
2014, 11(3): 573-597 doi: 10.3934/mbe.2014.11.573 +[Abstract](221) +[PDF](531.1KB)
Abstract:
Functional response estimation and population tracking in predator-prey systems are critical problems in ecology. In this paper we consider a stochastic predator-prey system with a Lotka-Volterra functional response and propose a particle filtering method for: (a) estimating the behavioral parameter representing the rate of effective search per predator in the functional response and (b) forecasting the population biomass using field data. In particular, the proposed technique combines a sequential Monte Carlo sampling scheme for tracking the time-varying biomass with the analytical integration of the unknown behavioral parameter. In order to assess the performance of the method, we show results for both synthetic and observed data collected in an acarine predator-prey system, namely the pest mite Tetranychus urticae and the predatory mite Phytoseiulus persimilis.
Modeling and analysis of the San Francisco City Clinic Cohort (SFCCC) HIV-epidemic including treatment
Brandy Rapatski and Juan Tolosa
2014, 11(3): 599-619 doi: 10.3934/mbe.2014.11.599 +[Abstract](251) +[PDF](495.6KB)
Abstract:
We investigate two HIV/AIDS epidemic models. The first model represents the early San Francisco men having sex with men (MSM) epidemic. We use data from the San Francisco City Clinic Cohort Study (SFCCC), documenting the onset of HIV in San Francisco (1978-1984). The second model is a ``what-if'' scenario model including testing and treatment in the SFCCC epidemic. We use compartmental, population-level models, described by systems of ordinary differential equations. We find the basic reproductive number $R_0$ for each system, and we prove that if $R_0<1$, the system has only the disease-free equilibrium (DFE) which is locally and globally stable, whereas if $R_0>1$, the DFE is unstable. In addition, when $R_0>1$, both systems have a unique endemic equilibrium (EE). We show that treatment alone would not have stopped the San Francisco MSM epidemic, but would have significantly reduced its impact.
Modeling the endocrine control of vitellogenin production in female rainbow trout
Kaitlin Sundling, Gheorghe Craciun, Irvin Schultz, Sharon Hook, James Nagler, Tim Cavileer, Joseph Verducci, Yushi Liu, Jonghan Kim and William Hayton
2014, 11(3): 621-639 doi: 10.3934/mbe.2014.11.621 +[Abstract](206) +[PDF](1154.2KB)
Abstract:
The rainbow trout endocrine system is sensitive to changes in annual day length, which is likely the principal environmental cue controlling its reproductive cycle. This study focuses on the endocrine regulation of vitellogenin (Vg) protein synthesis, which is the major egg yolk precursor in this fish species. We present a model of Vg production in female rainbow trout which incorporates a biological pathway beginning with sex steroid estradiol-17β levels in the plasma and concluding with Vg secretion by the liver and sequestration in the oocytes. Numerical simulation results based on this model are compared with experimental data for estrogen receptor mRNA, Vg mRNA, and Vg in the plasma from female rainbow trout over a normal annual reproductive cycle. We also analyze the response of the model to parameter changes. The model is subsequently tested against experimental data from female trout under a compressed photoperiod regime. Comparison of numerical and experimental results suggests the possibility of a time-dependent change in oocyte Vg uptake rate. This model is part of a larger effort that is developing a mathematical description of the endocrine control of reproduction in female rainbow trout. We anticipate that these mathematical and computational models will play an important role in future regulatory toxicity assessments and in the prediction of ecological risk.
Global stability of an age-structured cholera model
Jianxin Yang, Zhipeng Qiu and Xue-Zhi Li
2014, 11(3): 641-665 doi: 10.3934/mbe.2014.11.641 +[Abstract](244) +[PDF](469.5KB)
Abstract:
In this paper, an age-structured epidemic model is formulated to describe the transmission dynamics of cholera. The PDE model incorporates direct and indirect transmission pathways, infection-age-dependent infectivity and variable periods of infectiousness. Under some suitable assumptions, the PDE model can be reduced to the multi-stage models investigated in the literature. By using the method of Lyapunov function, we established the dynamical properties of the PDE model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number $\mathcal R_0$: if $\mathcal R_0 < 1$ the cholera dies out, and if $\mathcal R_0 >1$ the disease will persist at the endemic equilibrium. Then the global results obtained for multi-stage models are extended to the general continuous age model.
Mathematical modeling of Glassy-winged sharpshooter population
Jeong-Mi Yoon, Volodymyr Hrynkiv, Lisa Morano, Anh Tuan Nguyen, Sara Wilder and Forrest Mitchell
2014, 11(3): 667-677 doi: 10.3934/mbe.2014.11.667 +[Abstract](221) +[PDF](543.8KB)
Abstract:
Pierce's disease (PD) is a fatal disease of grapevines which results from an infection by the plant pathogen Xyllela fastidiosa. This bacterium grows in the xylem (water-conducting) vessels of the plant blocking movement of water. PD can kill vines in one year and poses a serious threat to both the California and the expanding Texas wine industries. Bacteria are vectored from one vine to the next by a number of xylem feeding insect species. Of these, the Glassy-winged Sharpshooter (GWSS) is considered to be the primary xylem feeding insect in Texas vineyards. An extensive database of the xylem-feeding population frequencies was collected by USDA-APHIS for Texas vineyards over multiple years. This project focused on a subset of data, GWSS frequencies within 25 vineyards in Edwards Plateau located in central Texas. The proposed model investigates the natural population dynamics and the decline in GWSS, likely the result of pest management campaigns on the insects within the region. The model is a delay Gompertz differential equation with harvesting and immigration terms, and we use the data to estimate the model parameters.

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