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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, May 2016
1 
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
Related Articles
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.

2 
Mathematical modelling of tuberculosis epidemics
Volume 6, Number 2, Pages: 209  237, 2009
Juan Pablo Aparicio
and Carlos CastilloChávez
Abstract
Full Text
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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 
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.

5 
Dynamical Models of Tuberculosis and Their Applications
Volume 1, Number 2, Pages: 361  404, 2004
Carlos CastilloChavez
and Baojun Song
Abstract
Full Text
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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.

6 
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
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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.

7 
Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics
Volume 8, Number 1, Pages: 199  222, 2011
Karen R. RíosSoto,
Baojun Song
and Carlos CastilloChavez
Abstract
References
Full Text
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The recent H1N1 ("swine flu") pandemic and recent H5N1 ("avian flu") outbreaks have brought increased attention to the study of the role of animal populations as reservoirs for pathogens that could invade human populations. It is believed that pigs acquired flu strains from birds and humans, acting as a mixing vessel in generating new influenza viruses. Assessing the role of animal reservoirs, particularly reservoirs involving highly mobile populations (like migratory birds), on disease dispersal and persistence is of interests to a wide range of researchers including public health experts and evolutionary biologists. This paper studies the interactions between transient and resident bird populations and their role on dispersal and persistence. A metapopulation framework based on a system of nonlinear ordinary differential equations is used to study the transmission dynamics and control of avian diseases. Simplified versions of mathematical models involving a limited number of migratory and resident bird populations are analyzed. Epidemiological time scales and singular perturbation methods are used to reduce the dimensionality of the model. Our results show that mixing of bird populations (involving residents and migratory birds) play an important role on the patterns of disease spread.

8 
Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza
Volume 8, Number 1, Pages: 141  170, 2011
Olivia Prosper,
Omar Saucedo,
Doria Thompson,
Griselle TorresGarcia,
Xiaohong Wang
and Carlos CastilloChavez
Abstract
References
Full Text
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The lessons learned from the 20092010 H1N1 influenza pandemic, as it moves out of the limelight, should not be underestimated, particularly since the probability of novel influenza epidemics in the near future is not negligible and the potential consequences might be huge. Hence, as the world, particularly the industrialized world, responded to the potentially devastating effects of this novel AH1N1 strain with substantial resources, reminders of the recurrent loss of life from a well established foe, seasonal influenza, could not be ignored. The uncertainties associated with the reported and expected levels of morbidity and mortality with this novel AH1N1 live in a backdrop of $36,000$ deaths, over 200,000 hospitalizations, and millions of infections (20% of the population) attributed to seasonal influenza in the USA alone, each year. So, as the Northern Hemisphere braced for the possibility of a potentially "lethal" second wave of the novel AH1N1 without a vaccine ready to mitigate its impact, questions of who should be vaccinated first if a vaccine became available, came to the forefront of the discussion. Uncertainty grew as we learned that the vaccine, once available, would be unevenly distributed around the world. Nations capable of acquiring large vaccine supplies soon became aware that those who could pay would have to compete for a limited vaccine stockpile. The challenges faced by nations dealing jointly with seasonal and novel AH1N1 cocirculating strains under limited resources, that is, those with no access to novel AH1N1 vaccine supplies, limited access to the seasonal influenza vaccine, and limited access to antivirals (like Tamiflu) are explored in this study. One and twostrain models are introduced to mimic the influenza dynamics of a single and cocirculating strains, in the context of a single epidemic outbreak. Optimal control theory is used to identify and evaluate the "best" control policies. The controls account for the cost associated with social distancing and antiviral treatment policies. The optimal policies identified might have, if implemented, a substantial impact on the novel H1N1 and seasonal influenza cocirculating dynamics. Specifically, the implementation of antiviral treatment might reduce the number of influenza cases by up to 60% under a reasonable seasonal vaccination strategy, but only by up to 37% when the seasonal vaccine is not available. Optimal social distancing policies alone can be as effective as the combination of multiple policies, reducing the total number of influenza cases by more than 99% within a single outbreak, an unrealistic but theoretically possible outcome for isolated populations with limited resources.

9 
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
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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, $\rho$.

10 
Multiple outbreaks
for the same pandemic: Local transportation and social distancing
explain the different "waves" of AH1N1pdm cases observed in
México during 2009
Volume 8, Number 1, Pages: 21  48, 2011
Marco Arieli HerreraValdez,
Maytee CruzAponte
and Carlos CastilloChavez
Abstract
References
Full Text
Related Articles
Influenza outbreaks have been of relatively limited historical
interest in México. The 2009 influenza pandemic not only
changed México's health priorities but also brought to the
forefront some of the strengths and weaknesses of México's
epidemiological surveillance and public health system. A year
later, México's data show an epidemic pattern characterized by
three "waves''. The reasons this threewave patterns are theoretically
investigated via models that incorporate México's general trends
of land transportation, public health measures, and the regular
opening and closing of schools during 2009. The role of vaccination
is also studied taking into account delays in access and limitations
in the total and daily numbers of vaccines available. The research
in this article supports the view that the thee epidemic "waves" are
the result of the synergistic interactions of three factors:
regional movement patterns of Mexicans, the impact and effectiveness
of dramatic social distancing measures imposed during the first
outbreak, and the summer release of school children followed by
their subsequent return to classes in the fall. The three "waves"
cannot be explained by the transportation patterns alone
but only through the combination of transport patterns and changes
in contact rates due to the use of explicit or scheduled social
distancing measures. The research identifies possible vaccination
schemes that account for the school calendar and whose effectiveness
are enhanced by social distancing measures. The limited impact of
the late arrival of the vaccine is also analyzed.

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