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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, December 2016
1 
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.

2 
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.

3 
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.

4 
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, $\rho$.

5 
Differential impact of sickle cell trait on symptomatic and asymptomatic malaria
Volume 9, Number 4, Pages: 877  898, 2012
Eunha Shim,
Zhilan Feng
and Carlos CastilloChavez
Abstract
References
Full Text
Related Articles
Individuals who carry the sickle cell trait ($S$gene) have a greatly reduced risk of experiencing symptomatic malaria infections. However, previous studies suggest
that the sickle cell trait does not protect against acquiring asymptomatic malaria infections, although the proportion of symptomatic infections is up to $50\%$ in areas where malaria is endemic. To examine the differential impact of the sickle cell trait on symptomatic and asymptomatic malaria, we developed a mathematical model of malaria transmission that incorporates the evolutionary dynamics of $S$gene frequency. Our model indicates that the fitness of sickle cell trait is likely to increase with the proportion of symptomatic malaria infections. Our model also shows that control efforts aimed at diminishing the burden of symptomatic malaria are not likely to eradicate malaria in endemic areas, due to the increase in the relative prevalence of asymptomatic infection, the reservoir of malaria. Furthermore, when the prevalence of symptomatic malaria is reduced, both the fitness and frequency of the $S$gene may decrease. In turn, a decreased frequency of the $S$gene may eventually increase the overall prevalence of both symptomatic and asymptomatic malaria. Therefore, the control of symptomatic malaria might result in evolutionary repercussions, despite shortterm epidemiological benefits.

6 
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
Full Text
Related Articles
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.

7 
Dynamics of a predatorprey system with prey subject to Allee effects and disease
Volume 11, Number 4, Pages: 877  918, 2014
Yun Kang,
Sourav Kumar Sasmal,
Amiya Ranjan Bhowmick
and Joydev Chattopadhyay
Abstract
References
Full Text
Related Articles
In this article, we propose a general predatorprey system where prey is subject to Allee effects and disease with the following unique features: (i) Allee effects built in the reproduction process of prey where infected prey (Iclass) has no contribution; (ii) Consuming infected prey would contribute less or negatively to the growth rate of predator (Pclass) in comparison to the consumption of susceptible prey (Sclass). We provide basic dynamical properties for this general model and perform the detailed analysis on a concrete model (SIPAllee Model) as well as its corresponding model in the absence of Allee effects (SIPnoAllee Model); we obtain the complete dynamics of both models: (a) SIPAllee Model may have only one attractor (extinction of all species), two attractors (bistability either induced by small values of reproduction number of both disease and predator or induced by competition exclusion), or three attractors (tristability); (b) SIPnoAllee Model may have either one attractor (only Sclass survives or the persistence of S and Iclass or the persistence of S and Pclass) or two attractors (bistability with the persistence of S and Iclass or the persistence of S and Pclass). One of the most interesting findings is that neither models can support the coexistence of all three S, I, Pclass. This is caused by the assumption (ii), whose biological implications are that I and Pclass are at exploitative competition for Sclass whereas Iclass cannot be superior and Pclass cannot gain significantly from its consumption of Iclass. In addition, the comparison study between the dynamics of SIPAllee Model and SIPnoAllee Model lead to the following conclusions: 1) In the presence of Allee effects, species are prone to extinction and initial condition plays an important role on the surviving of prey as well as its corresponding predator; 2) In the presence of Allee effects, disease may be able to save prey from the predationdriven extinction and leads to the coexistence of S and Iclass while predator can not save the diseasedriven extinction. All these findings may have potential applications in conservation biology.

8 
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
Related Articles
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.

9 
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.

10 
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
Related Articles
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.

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