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MBE focuses on new developments in the fastgrowing fields of mathematical biosciences and bioengineering.
Authors will be granted full access to all MBE publications for one year.
Areas covered include general mathematical methods and their applications in biology, medical sciences and bioengineering with an emphasis on work related to mathematical modeling, nonlinear and stochastic dynamics.
The editorial board of MBE is strongly committed to promoting cuttingedge integrative and interdisciplinary research bridging mathematics, life sciences and engineering.
 Publishes 6 issues a year in February, April, June, August, October and December.
 Publishes online only.
 Indexed in Science Citation IndexExpanded, PubMed, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences (CC/PC&ES), INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
 Archived in Portico and CLOCKSS.
 MBE is a publication of the American Institute of Mathematical Sciences. All rights reserved.

TOP 10 Most Read Articles in MBE, June 2017
1 
A partial differential equation model of metastasized prostatic cancer
Volume 10, Number 3, Pages: 591  608, 2013
Avner Friedman
and Harsh Vardhan Jain
Abstract
References
Full Text
Related Articles
Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castrationresistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.

2 
Mathematical modelling of tuberculosis epidemics
Volume 6, Number 2, Pages: 209  237, 2009
Juan Pablo Aparicio
and Carlos CastilloChávez
Abstract
Full Text
Related Articles
The strengths and limitations of using homogeneous mixing and
heterogeneous mixing epidemic models are explored in the context
of the transmission dynamics of tuberculosis. The focus is on
three types of models: a standard incidence homogeneous mixing
model, a nonhomogeneous mixing model that incorporates
'household' contacts, and an agestructured model. The models are
parameterized using demographic and epidemiological data and the
patterns generated from these models are compared. Furthermore,
the effects of population growth, stochasticity, clustering of
contacts, and age structure on disease dynamics are explored. This
framework is used to asses the possible causes for the observed
historical decline of tuberculosis notifications.

3 
Time variations in the generation time of an infectious disease:
Implications for sampling to appropriately quantify transmission
potential
Volume 7, Number 4, Pages: 851  869, 2010
Hiroshi Nishiura
Abstract
References
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Although the generation time of an infectious disease plays a key role in estimating its transmission potential, the impact of the sampling time of generation times on the estimation procedure has yet to be clarified. The present study defines the period and cohort generation times, both of which are timeinhomogeneous, as a function of the infection time of secondary and primary cases, respectively. By means of analytical and numerical approaches, it is shown that the period generation time increases with calendar time, whereas the cohort generation time decreases as the incidence increases. The initial growth phase of an epidemic of Asian influenza A (H2N2) in the Netherlands in 1957 was reanalyzed, and estimates of the basic reproduction number, $R_0$, from the LotkaEuler equation were examined. It was found that the sampling time of generation time during the course of the epidemic introduced a timeeffect to the estimate of $R_0$. Other historical data of a primary pneumonic plague in Manchuria in 1911 were also examined to help illustrate the empirical evidence of the period generation time. If the serial intervals, which eventually determine the generation times, are sampled during the course of an epidemic, direct application of the sampled generationtime distribution to the LotkaEuler equation leads to a biased estimate of $R_0$. An appropriate quantification of the transmission potential requires the estimation of the cohort generation time during the initial growth phase of an epidemic or adjustment of the timeeffect (e.g., adjustment of the growth rate of the epidemic during the sampling time) on the period generation time. A similar issue also applies to the estimation of the effective reproduction number as a function of calendar time. Mathematical properties of the generation time distribution in a heterogeneously mixing population need to be clarified further.

4 
A Simple Epidemic Model with Surprising Dynamics
Volume 2, Number 1, Pages: 133  152, 2004
F. Berezovskaya,
G. Karev,
Baojun Song
and Carlos CastilloChavez
Abstract
Full Text
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A simple model incorporating demographic and epidemiological
processes is explored. Four reparameterized quantities the basic
demographic reproductive number ($\R_d$), the basic epidemiological
reproductive number ($\R_0$), the ratio ($\nu$) between the average
life spans of susceptible and infective class, and the relative
fecundity of infectives ($\theta$), are utilized in qualitative
analysis. Mathematically, nonanalytic vector fields are handled by
blowup transformations to carry out a complete and global dynamical
analysis. A family of homoclinics is found, suggesting that a
disease outbreak would be ignited by a tiny number of infectious
individuals.

5 
Parametrization of the attainable set for a nonlinear control model of a biochemical process
Volume 10, Number 4, Pages: 1067  1094, 2013
Ellina Grigorieva,
Evgenii Khailov
and Andrei Korobeinikov
Abstract
References
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In this paper, we study a threedimensional nonlinear model of a controllable reaction
$ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecified
nonlinear function. A model of this type describes a variety of reallife processes in chemical
kinetics and biology; in this paper our particular interests is in its application to waste
water biotreatment. For this control model, we analytically study the corresponding attainable
set and parameterize it by the moments of switching of piecewise constant control functions.
This allows us to visualize the attainable sets using a numerical procedure.
These analytical results generalize the earlier findings, which were obtained for a trilinear
reaction rate (which corresponds to the law of mass action) and reported
in [18,19], to the case of a general rate of reaction. These results allow to
reduce the problem of constructing the optimal control to a straightforward constrained finite
dimensional optimization problem.

6 
Effect of branchings on blood flow in the system of human coronary arteries
Volume 9, Number 1, Pages: 199  214, 2011
Benchawan Wiwatanapataphee,
Yong Hong Wu,
Thanongchai Siriapisith
and Buraskorn Nuntadilok
Abstract
References
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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 nonNewtonian fluid. The transient phenomena of
blood flow through the coronary system are simulated by solving the
three dimensional unsteady state NavierStokes 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.

7 
The impact of vaccines and vaccinations: Challenges and opportunities for modelers
Volume 8, Number 1, Pages: 77  93, 2011
Roy Curtiss III
Abstract
References
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This review focuses on how infectious diseases and their prevention and control by development of vaccines and widespread vaccination has shaped evolution of human civilization and of the animals and plants that humans depend on for food, labor and companionship. After describing major infectious diseases and the current status for control by vaccination, the barriers to infection and the attributes of innate and acquired immunity contributing to control are discussed. The evolution in types of vaccines is presented in the context of developing technologies and in improving adjuvants to engender enhanced vaccine efficacy. The special concerns and needs in vaccine design and development are discussed in dealing with epidemics/pandemics with special emphasis on influenza and current global problems in vaccine delivery.

8 
Regulation of Th1/Th2 cells in asthma development: A mathematical model
Volume 10, Number 4, Pages: 1095  1133, 2013
Yangjin Kim,
Seongwon Lee,
YouSun Kim,
Sean Lawler,
Yong Song Gho,
YoonKeun Kim
and Hyung Ju Hwang
Abstract
References
Full Text
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Airway exposure levels of lipopolysaccharide (LPS) determine type I versus type II
helper T cell induced experimental asthma. While high LPS levels induce Th1dominant
responses, low LPS levels derive Th2 cell induced
asthma. The present paper develops a mathematical model of asthma development which
focuses on the relative balance of Th1 and Th2 cell induced asthma. In the present work
we represent
the complex network of interactions between cells and molecules by a mathematical model.
The model describes the behaviors of cells (Th0, Th1, Th2 and macrophages)
and regulatory molecules (IFN$\gamma$, IL4, IL12, TNFα) in response to
high, intermediate, and low levels of LPS.
The simulations show how
variations in the levels of injected LPS
affect the development of
Th1 or Th2 cell responses through differential cytokine induction.
The model also predicts the coexistence of
these two types of response
under certain biochemical and biomechanical conditions in the microenvironment.

9 
Shearthinning effects of hemodynamics in patientspecific cerebral aneurysms
Volume 10, Number 3, Pages: 649  665, 2013
Alberto Gambaruto,
João Janela,
Alexandra Moura
and Adélia Sequeira
Abstract
References
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Two different generalized Newtonian mathematical models for blood flow, derived for the same experimental data, are compared, together with the Newtonian model, in three different anatomically realistic geometries of saccular cerebral aneurysms obtained from rotational CTA. The geometries differ in size of the aneurysm and the existence or not of side branches within the aneurysm.
Results show that the differences between the two generalized Newtonian mathematical models are smaller than the differences between these and the Newtonian solution, in both steady and unsteady simulations.

10 
Model for hepatitis C virus transmissions
Volume 10, Number 4, Pages: 1045  1065, 2013
Elamin H. Elbasha
Abstract
References
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Hepatitis C virus (HCV) is a leading cause of chronic liver disease. This
paper presents a deterministic model for HCV infection transmission and uses
the model to assess the potential impact of antiviral therapy. The model is
based on the susceptibleinfectiveremovedsusceptible (SIRS) compartmental
structure with chronic primary infection and possibility of reinfection.
Important epidemiologic thresholds such as the basic and control
reproduction numbers and a measure of treatment impact are derived. We find
that if the control reproduction number is greater than unity, there is a
locally unstable infectionfree equilibrium and a unique, globally
asymptotically stable endemic equilibrium. If the control reproduction
number is less than unity, the infectionfree equilibrium is globally
asymptotically stable, and HCV will be eliminated. Numerical simulations
suggest that, besides the parameters that determine the basic reproduction
number, reinfection plays an important role in HCV transmissions and
magnitude of the public health impact of antiviral therapy. Further,
treatment regimens with better efficacy holds great promise for lowering the
public health burden of HCV disease.

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