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Mathematical Biosciences & Engineering

Open Access Articles

A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity
Ping Yan
2018, 15(1): 275-298 doi: 10.3934/mbe.2018012 +[Abstract](157) +[HTML](38) +[PDF](878.3KB)

For an intervention against the spread of communicable diseases, the idealized situation is when individuals fully comply with the intervention and the exposure to the infectious agent is comparable across all individuals. Some level of non-compliance is likely where the intervention is widely implemented. The focus is on a more accurate view of its effects population-wide. A frailty model is applied. Qualitative analysis, in mathematical terms, reveals how large variability in compliance renders the intervention less effective. This finding sharpens our vague, intuitive and empirical notions. An effective reproduction number in the presence of frailty is defined and is shown to be invariant with respect to the time-scale of disease progression. This makes the results in this paper valid for a wide spectrum of acute and chronic infectious diseases. Quantitative analysis by comparing numerical results shows that they are also robust with respect to assumptions on disease progression structure and distributions, such as with or without the latent period and the assumed distributions of latent and infectious periods.

Daozhou Gao, Shigui Ruan and Jifa Jiang
2017, 14(5-6): i-ii doi: 10.3934/mbe.201705i +[Abstract](62) +[HTML](20) +[PDF](31.6KB)
Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis
Andrea Pugliese, Abba B. Gumel, Fabio A. Milner and Jorge X. Velasco-Hernandez
2018, 15(1): 125-140 doi: 10.3934/mbe.2018005 +[Abstract](216) +[HTML](36) +[PDF](642.3KB)

Three deterministic Kermack-McKendrick-type models for studying the transmission dynamics of an infection in a two-sex closed population are analyzed here. In each model it is assumed that infection can be transmitted through heterosexual contacts, and that there is a higher probability of transmission from one sex to the other than vice versa. The study is focused on understanding whether and how this bias in transmission reflects in sex differences in final attack ratios (i.e. the fraction of individuals of each sex that eventually gets infected). In the first model, where the other two transmission modes are not considered, the attack ratios (fractions of the population of each sex that will eventually be infected) can be obtained as solutions of a system of two nonlinear equations, that has a unique solution if the net reproduction number exceeds unity. It is also shown that the ratio of attack ratios depends solely on the ratio of gender-specific susceptibilities and on the basic reproductive number of the epidemic \begin{document}$ \mathcal{R}_0 $\end{document}, and that the gender-specific final attack-ratio is biased in the same direction as the gender-specific susceptibilities. The second model allows also for infection transmission through direct, non-sexual, contacts. In this case too, an analytical expression is derived from which the attack ratios can be obtained. The qualitative results are similar to those obtained for the previous model, but another important parameter for determining the value of the ratio between the attack ratios in the two sexes is obtained, the relative weight of direct vs. heterosexual transmission (namely, ρ). Quantitatively, the ratio of final attack ratios generally will not exceed 1.5, if non-sexual transmission accounts for most transmission events (ρ ≥ 0.6) and the ratio of gender-specific susceptibilities is not too large (say, 5 at most).

The third model considers vector-borne, instead of direct transmission. In this case, we were not able to find an analytical expression for the final attack ratios, but used instead numerical simulations. The results on final attack ratios are actually quite similar to those obtained with the second model. It is interesting to note that transient patterns can differ from final attack ratios, as new cases will tend to occur more often in the more susceptible sex, while later depletion of susceptibles may bias the ratio in the opposite direction.

The analysis of these simple models, despite their lack of realism, can help in providing insight into, and assessment of, the potential role of gender-specific transmission in infections with multiple modes of transmission, such as Zika virus (ZIKV), by gauging what can be expected to be seen from epidemiological reports of new cases, disease incidence and seroprevalence surveys.

Mathematical analysis of a weather-driven model for the population ecology of mosquitoes
Kamaldeen Okuneye, Ahmed Abdelrazec and Abba B. Gumel
2018, 15(1): 57-93 doi: 10.3934/mbe.2018003 +[Abstract](307) +[HTML](53) +[PDF](905.5KB)

A new deterministic model for the population biology of immature and mature mosquitoes is designed and used to assess the impact of temperature and rainfall on the abundance of mosquitoes in a community. The trivial equilibrium of the model is globally-asymptotically stable when the associated vectorial reproduction number $({\mathcal R}_0)$ is less than unity. In the absence of density-dependence mortality in the larval stage, the autonomous version of the model has a unique and globally-asymptotically stable non-trivial equilibrium whenever $1 < {\mathcal R}_0 < {\mathcal R}_0^C$ (this equilibrium bifurcates into a limit cycle, via a Hopf bifurcation at ${\mathcal R}_0={\mathcal R}_0^C$). Numerical simulations of the weather-driven model, using temperature and rainfall data from three cities in Sub-Saharan Africa (Kwazulu Natal, South Africa; Lagos, Nigeria; and Nairobi, Kenya), show peak mosquito abundance occurring in the cities when the mean monthly temperature and rainfall values lie in the ranges $[22 -25]^{0}$C, $[98 -121]$ mm; $[24 -27]^{0}$C, $[113 -255]$ mm and $[20.5 -21.5]^{0}$C, $[70 -120]$ mm, respectively (thus, mosquito control efforts should be intensified in these cities during the periods when the respective suitable weather ranges are recorded).

Machine learning of swimming data via wisdom of crowd and regression analysis
Jiang Xie, Junfu Xu, Celine Nie and Qing Nie
2017, 14(2): 511-527 doi: 10.3934/mbe.2017031 +[Abstract](378) +[HTML](80) +[PDF](928.1KB)

Every performance, in an officially sanctioned meet, by a registered USA swimmer is recorded into an online database with times dating back to 1980. For the first time, statistical analysis and machine learning methods are systematically applied to 4,022,631 swim records. In this study, we investigate performance features for all strokes as a function of age and gender. The variances in performance of males and females for different ages and strokes were studied, and the correlations of performances for different ages were estimated using the Pearson correlation. Regression analysis show the performance trends for both males and females at different ages and suggest critical ages for peak training. Moreover, we assess twelve popular machine learning methods to predict or classify swimmer performance. Each method exhibited different strengths or weaknesses in different cases, indicating no one method could predict well for all strokes. To address this problem, we propose a new method by combining multiple inference methods to derive Wisdom of Crowd Classifier (WoCC). Our simulation experiments demonstrate that the WoCC is a consistent method with better overall prediction accuracy. Our study reveals several new age-dependent trends in swimming and provides an accurate method for classifying and predicting swimming times.

Avner Friedman, Mirosław Lachowicz, Urszula Ledzewicz, Monika Joanna Piotrowska and Zuzanna Szymanska
2017, 14(1): i-i doi: 10.3934/mbe.201701i +[Abstract](109) +[HTML](25) +[PDF](72.3KB)
A posterior probability approach for gene regulatory network inference in genetic perturbation data
William Chad Young, Adrian E. Raftery and Ka Yee Yeung
2016, 13(6): 1241-1251 doi: 10.3934/mbe.2016041 +[Abstract](212) +[PDF](540.8KB)
Inferring gene regulatory networks is an important problem in systems biology. However, these networks can be hard to infer from experimental data because of the inherent variability in biological data as well as the large number of genes involved. We propose a fast, simple method for inferring regulatory relationships between genes from knockdown experiments in the NIH LINCS dataset by calculating posterior probabilities, incorporating prior information. We show that the method is able to find previously identified edges from TRANSFAC and JASPAR and discuss the merits and limitations of this approach.
Modeling eating behaviors: The role of environment and positive food association learning via a Ratatouille effect
Anarina L. Murillo, Muntaser Safan, Carlos Castillo-Chavez, Elizabeth D. Capaldi Phillips and Devina Wadhera
2016, 13(4): 841-855 doi: 10.3934/mbe.2016020 +[Abstract](71) +[PDF](495.2KB)
Eating behaviors among a large population of children are studied as a dynamic process driven by nonlinear interactions in the sociocultural school environment. The impact of food association learning on diet dynamics, inspired by a pilot study conducted among Arizona children in Pre-Kindergarten to 8th grades, is used to build simple population-level learning models. Qualitatively, mathematical studies are used to highlight the possible ramifications of instruction, learning in nutrition, and health at the community level. Model results suggest that nutrition education programs at the population-level have minimal impact on improving eating behaviors, findings that agree with prior field studies. Hence, the incorporation of food association learning may be a better strategy for creating resilient communities of healthy and non-healthy eaters. A Ratatouille effect can be observed when food association learners become food preference learners, a potential sustainable behavioral change, which in turn, may impact the overall distribution of healthy eaters. In short, this work evaluates the effectiveness of population-level intervention strategies and the importance of institutionalizing nutrition programs that factor in economical, social, cultural, and environmental elements that mesh well with the norms and values in the community.
A two-strain TB model with multiple latent stages
Azizeh Jabbari, Carlos Castillo-Chavez, Fereshteh Nazari, Baojun Song and Hossein Kheiri
2016, 13(4): 741-785 doi: 10.3934/mbe.2016017 +[Abstract](60) +[PDF](1301.6KB)
A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving ``arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below $1$.
Susanne Ditlevsen and Petr Lansky
2016, 13(3): i-i doi: 10.3934/mbe.201600i +[Abstract](48) +[PDF](86.7KB)
This Special Issue of Mathematical Biosciences and Engineering contains 11 selected papers presented at the Neural Coding 2014 workshop. The workshop was held in the royal city of Versailles in France, October 6-10, 2014. This was the 11th of a series of international workshops on this subject, the first held in Prague (1995), then Versailles (1997), Osaka (1999), Plymouth (2001), Aulla (2003), Marburg (2005), Montevideo (2007), Tainan (2009), Limassol (2010), and again in Prague (2012). Also selected papers from Prague were published as a special issue of Mathematical Biosciences and Engineering and in this way a tradition was started. Similarly to the previous workshops, this was a single track multidisciplinary event bringing together experimental and computational neuroscientists. The Neural Coding Workshops are traditionally biennial symposia. They are relatively small in size, interdisciplinary with major emphasis on the search for common principles in neural coding. The workshop was conceived to bring together scientists from different disciplines for an in-depth discussion of mathematical model-building and computational strategies. Further information on the meeting can be found at the NC2014 website at The meeting was supported by French National Institute for Agricultural Research, the world's leading institution in this field.
    Understanding how the brain processes information is one of the most challenging subjects in neuroscience. The papers presented in this special issue show a small corner of the huge diversity of this field, and illustrate how scientists with different backgrounds approach this vast subject. The diversity of disciplines engaged in these investigations is remarkable: biologists, mathematicians, physicists, psychologists, computer scientists, and statisticians, all have original tools and ideas by which to try to elucidate the underlying mechanisms. In this issue, emphasis is put on mathematical modeling of single neurons. A variety of problems in computational neuroscience accompanied with a rich diversity of mathematical tools and approaches are presented. We hope it will inspire and challenge the readers in their own research.
    We would like to thank the authors for their valuable contributions and the referees for their priceless effort of reviewing the manuscripts. Finally, we would like to thank Yang Kuang for supporting us and making this publication possible.
Seasonality and the effectiveness of mass vaccination
Dennis L. Chao and Dobromir T. Dimitrov
2016, 13(2): 249-259 doi: 10.3934/mbe.2015001 +[Abstract](63) +[PDF](448.3KB)
Many infectious diseases have seasonal outbreaks, which may be driven by cyclical environmental conditions (e.g., an annual rainy season) or human behavior (e.g., school calendars or seasonal migration). If a pathogen is only transmissible for a limited period of time each year, then seasonal outbreaks could infect fewer individuals than expected given the pathogen's in-season transmissibility. Influenza, with its short serial interval and long season, probably spreads throughout a population until a substantial fraction of susceptible individuals are infected. Dengue, with a long serial interval and shorter season, may be constrained by its short transmission season rather than the depletion of susceptibles. Using mathematical modeling, we show that mass vaccination is most efficient, in terms of infections prevented per vaccine administered, at high levels of coverage for pathogens that have relatively long epidemic seasons, like influenza, and at low levels of coverage for pathogens with short epidemic seasons, like dengue. Therefore, the length of a pathogen's epidemic season may need to be considered when evaluating the costs and benefits of vaccination programs.
Algebraic and topological indices of molecular pathway networks in human cancers
Peter Hinow, Edward A. Rietman, Sara Ibrahim Omar and Jack A. Tuszyński
2015, 12(6): 1289-1302 doi: 10.3934/mbe.2015.12.1289 +[Abstract](73) +[PDF](478.1KB)
Protein-protein interaction networks associated with diseases have gained prominence as an area of research. We investigate algebraic and topological indices for protein-protein interaction networks of 11 human cancers derived from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database. We find a strong correlation between relative automorphism group sizes and topological network complexities on the one hand and five year survival probabilities on the other hand. Moreover, we identify several protein families (e.g. PIK, ITG, AKT families) that are repeated motifs in many of the cancer pathways. Interestingly, these sources of symmetry are often central rather than peripheral. Our results can aide in identification of promising targets for anti-cancer drugs. Beyond that, we provide a unifying framework to study protein-protein interaction networks of families of related diseases (e.g. neurodegenerative diseases, viral diseases, substance abuse disorders).
Hybrid models of cell and tissue dynamics in tumor growth
Yangjin Kim and Hans G. Othmer
2015, 12(6): 1141-1156 doi: 10.3934/mbe.2015.12.1141 +[Abstract](95) +[PDF](3437.4KB)
Hybrid models of tumor growth, in which some regions are described at the cell level and others at the continuum level, provide a flexible description that allows alterations of cell-level properties and detailed descriptions of the interaction with the tumor environment, yet retain the computational advantages of continuum models where appropriate. We review aspects of the general approach and discuss applications to breast cancer and glioblastoma.
Mathematically modeling the biological properties of gliomas: A review
Nikolay L. Martirosyan, Erica M. Rutter, Wyatt L. Ramey, Eric J. Kostelich, Yang Kuang and Mark C. Preul
2015, 12(4): 879-905 doi: 10.3934/mbe.2015.12.879 +[Abstract](88) +[PDF](6066.4KB)
Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumor's aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.
Traveling bands for the Keller-Segel model with population growth
Shangbing Ai and Zhian Wang
2015, 12(4): 717-737 doi: 10.3934/mbe.2015.12.717 +[Abstract](70) +[PDF](551.1KB)
This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.
A nosocomial epidemic model with infection of patients due to contaminated rooms
Cameron Browne and Glenn F. Webb
2015, 12(4): 761-787 doi: 10.3934/mbe.2015.12.761 +[Abstract](73) +[PDF](2782.2KB)
A model of epidemic bacterial infections in hospitals is developed. The model incorporates the infection of patients and the contamination of healthcare workers due to environmental causes. The model is analyzed with respect to the asymptotic behavior of solutions. The model is interpreted to provide insight for controlling these nosocomial epidemics.
The evolutionary dynamics of a population model with a strong Allee effect
Jim M. Cushing
2015, 12(4): 643-660 doi: 10.3934/mbe.2015.12.643 +[Abstract](90) +[PDF](464.5KB)
An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).
Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions
Lin Wang, James Watmough and Fang Yu
2015, 12(4): 699-715 doi: 10.3934/mbe.2015.12.699 +[Abstract](78) +[PDF](2913.5KB)
In this paper, we study a diffusive plant-herbivore system with homogeneous and nonhomogeneous Dirichlet boundary conditions. Stability of spatially homogeneous steady states is established. We also derive conditions ensuring the occurrence of Hopf bifurcation and steady state bifurcation. Interesting transient spatio-temporal behaviors including oscillations in one or both of space and time are observed through numerical simulations.
Peter Hinow, Pierre Magal and Shigui Ruan
2015, 12(4): i-iv doi: 10.3934/mbe.2015.12.4i +[Abstract](62) +[PDF](115.3KB)
This special issue is dedicated to the 70th birthday of Glenn F. Webb. The topics of the 12 articles appearing in this special issue include evolutionary dynamics of population growth, spatio-temporal dynamics in reaction-diffusion biological models, transmission dynamics of infectious diseases, modeling of antibiotic-resistant bacteria in hospitals, analysis of Prion models, age-structured models in ecology and epidemiology, modeling of immune response to infections, modeling of cancer growth, etc. These topics partially represent the broad areas of Glenn's research interest.

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Optimal design for parameter estimation in EEG problems in a 3D multilayered domain
H. T. Banks, D. Rubio, N. Saintier and M. I. Troparevsky
2015, 12(4): 739-760 doi: 10.3934/mbe.2015.12.739 +[Abstract](100) +[PDF](773.3KB)
The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical $D$-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.

2016  Impact Factor: 1.035




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