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MBE is bimonthly, focusing on new developments in the fastgrowing fields of mathematical biosciences and bioengineering. MBE is now online only.
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.
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TOP 10 Most Read Articles in MBE, January 2017
1 
Dynamical Models of Tuberculosis and Their Applications
Volume 1, Number 2, Pages: 361  404, 2004
Carlos CastilloChavez
and Baojun Song
Abstract
Full Text
Related Articles
The reemergence of tuberculosis (TB) from the 1980s to the early
1990s instigated extensive researches on the mechanisms behind the
transmission dynamics of TB epidemics. This article provides a
detailed review of the work on the dynamics and control of TB. The
earliest mathematical models describing the TB dynamics appeared in
the 1960s and focused on the prediction and control strategies using
simulation approaches. Most recently developed models not only pay
attention to simulations but also take care of dynamical analysis
using modern knowledge of dynamical systems. Questions addressed by
these models mainly concentrate on TB control strategies, optimal
vaccination policies, approaches toward the elimination of TB in the
U.S.A., TB coinfection with HIV/AIDS, drugresistant TB, responses
of the immune system, impacts of demography, the role of public
transportation systems, and the impact of contact patterns. Model
formulations involve a variety of mathematical areas, such as ODEs
(Ordinary Differential Equations) (both autonomous and
nonautonomous systems), PDEs (Partial Differential Equations),
system of difference equations, system of integrodifferential
equations, Markov chain model, and simulation models.

2 
The estimation of the effective reproductive number from disease outbreak data
Volume 6, Number 2, Pages: 261  282, 2009
Ariel CintrónArias,
Carlos CastilloChávez,
Luís M. A. Bettencourt,
Alun L. Lloyd
and H. T. Banks
Abstract
Full Text
Related Articles
We consider a single outbreak susceptibleinfectedrecovered (SIR)
model and corresponding estimation procedures for the
effective reproductive number $\mathcal{R}(t)$. We discuss the
estimation of the underlying SIR parameters with a
generalized least squares (GLS) estimation
technique. We do this in the context of appropriate statistical
models for the measurement process. We use asymptotic statistical
theories to derive the mean and variance of the limiting
(Gaussian) sampling distribution and to perform post statistical
analysis of the inverse problems. We illustrate the ideas and
pitfalls (e.g., large condition numbers on the corresponding
Fisher information matrix) with both synthetic and influenza
incidence data sets.

3 
An application of queuing theory to SIS and SEIS epidemic models
Volume 7, Number 4, Pages: 809  823, 2010
Carlos M. HernándezSuárez,
Carlos CastilloChavez,
Osval Montesinos López
and Karla HernándezCuevas
Abstract
References
Full Text
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In this work we consider every individual of a population to be a server whose state can be either busy (infected) or idle (susceptible). This server approach allows to consider a general distribution for the duration of the infectious state, instead of being restricted to exponential distributions. In order to achieve this we first derive new approximations to quasistationary distribution (QSD) of SIS (Susceptible Infected Susceptible) and SEIS (Susceptible Latent Infected Susceptible) stochastic epidemic models. We give an expression that relates the basic reproductive number, $R_0$ and the server utilization, $\rho$.

4 
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.

5 
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
Full Text
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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.

6 
Raves, clubs and ecstasy: the impact of peer pressure
Volume 3, Number 1, Pages: 249  266, 2005
Baojun Song,
Melissa CastilloGarsow,
Karen R. RíosSoto,
Marcin Mejran,
Leilani Henso
and Carlos CastilloChavez
Abstract
Full Text
Related Articles
Ecstasy has gained popularity among young adults who frequent raves and nightclubs. The Drug Enforcement Administration reported a 500 percent increase in the use of ecstasy between 1993 and 1998. The number of ecstasy users kept growing until 2002, years after a national public education initiative against ecstasy use was launched. In this study, a system of differential equations is used to model the peerdriven dynamics of ecstasy use. It is found that backward bifurcations describe situations when sufficient peer pressure can cause an epidemic of ecstasy use. Furthermore, factors that have the greatest influence on ecstasy use as predicted by the model are highlighted. The effect of education is also explored, and the results of simulations are shown to illustrate some possible outcomes.

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
Full Text
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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 
Modeling some properties of circadian rhythms
Volume 11, Number 2, Pages: 317  330, 2013
Miguel LaraAparicio,
Carolina BarrigaMontoya,
Pablo PadillaLongoria
and Beatriz FuentesPardo
Abstract
References
Full Text
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Mathematical models have been very useful in biological research. From the
interaction of biology and mathematics, new problems have emerged that have
generated advances in the theory, suggested further experimental work and
motivated plausible conjectures. From our perspective, it is absolutely
necessary to incorporate modeling tools in the study of circadian rhythms
and that without a solid mathematical framework a real understanding of them
will not be possible. Our interest is to study the main process underlying
the synchronization in the pacemaker of a circadian system: these
mechanisms should be conserved in all living beings. Indeed, from an
evolutionary perspective, it seems reasonable to assume that either they
have a common origin or that they emerge from similar selection
circumstances. We propose a general framework to understand the emergence of
synchronization as a robust characteristic of some cooperative systems of
nonlinear coupled oscillators. In a first approximation to the problem we
vary the topology of the network and the strength of the interactions among
oscillators. In order to study the emergent dynamics, we carried out some
numerical computations. The results are consistent with experiments reported
in the literature. Finally, we proposed a theoretical framework to study the
phenomenon of synchronization in the context of circadian rhythms: the
dissipative synchronization of nonautonomous dynamical systems.

9 
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
Full Text
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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.

10 
A note on the use of optimal control on a discrete time model of influenza dynamics
Volume 8, Number 1, Pages: 183  197, 2011
Paula A. GonzálezParra,
Sunmi Lee,
Leticia Velázquez
and Carlos CastilloChavez
Abstract
References
Full Text
Related Articles
A discrete time Susceptible  Asymptomatic  Infectious  Treated  Recovered (SAITR) model is introduced in the context of influenza transmission.
We evaluate the potential effect of control measures such as social distancing and antiviral treatment on the dynamics of a single outbreak.
Optimal control theory is applied to identify the best way of reducing morbidity and mortality at a minimal cost. The problem is solved by using a discrete version of Pontryagin's maximum principle. Numerical results show that dual strategies have stronger impact in the reduction of the final epidemic size.

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