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Mathematical Biosciences & Engineering

Open Access Articles

Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components
Feng Rao, Carlos Castillo-Chavez and Yun Kang
2018, 15(6): 1401-1423 doi: 10.3934/mbe.2018064 +[Abstract](5849) +[HTML](1561) +[PDF](1231.06KB)

This paper investigates the complex dynamics of a Harrison-type predator-prey model that incorporating: (1) A constant time delay in the functional response term of the predator growth equation; and (2) environmental noise in both prey and predator equations. We provide the rigorous results of our model including the dynamical behaviors of a positive solution and Hopf bifurcation. We also perform numerical simulations on the effects of delay or/and noise when the corresponding ODE model has an interior solution. Our theoretical and numerical results show that delay can either remain stability or destabilize the model; large noise could destabilize the model; and the combination of delay and noise could intensify the periodic instability of the model. Our results may provide us useful biological insights into population managements for prey-predator interaction models.

Modeling crowd dynamics through coarse-grained data analysis
Sebastien Motsch, Mehdi Moussaïd, Elsa G. Guillot, Mathieu Moreau, Julien Pettré, Guy Theraulaz, Cécile Appert-Rolland and Pierre Degond
2018, 15(6): 1271-1290 doi: 10.3934/mbe.2018059 +[Abstract](8055) +[HTML](1886) +[PDF](2444.92KB)

Understanding and predicting the collective behaviour of crowds is essential to improve the efficiency of pedestrian flows in urban areas and minimize the risks of accidents at mass events. We advocate for the development of crowd traffic management systems, whereby observations of crowds can be coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bi-directional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional flows, i.e. the relation between the pedestrian fluxes and densities. We design and run a laboratory experiments involving a total of 119 participants walking in opposite directions in a circular corridor and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. Finally, we propose a simple segregation strategy for enhancing the traffic efficiency, and use the BM model to determine the conditions under which this strategy would be beneficial. The BM model, therefore, could serve as a building block to develop on the fly prediction of crowd movements and help deploying real-time crowd optimization strategies.

A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis
József Z. Farkas, Gary T. Smith and Glenn F. Webb
2018, 15(5): 1203-1224 doi: 10.3934/mbe.2018055 +[Abstract](6234) +[HTML](1400) +[PDF](5861.18KB)

We quantify a recent five-category CT histogram based classification of ground glass opacities using a dynamic mathematical model for the spatial-temporal evolution of malignant nodules. Our mathematical model takes the form of a spatially structured partial differential equation with a logistic crowding term. We present the results of extensive simulations and validate our model using patient data obtained from clinical CT images from patients with benign and malignant lesions.

Optimal design for dynamical modeling of pest populations
H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass and Jay A. Rosenheim
2018, 15(4): 993-1010 doi: 10.3934/mbe.2018044 +[Abstract](5939) +[HTML](1486) +[PDF](1031.22KB)

We apply SE-optimal design methodology to investigate optimal data collection procedures as a first step in investigating information content in ecoinformatics data sets. To illustrate ideas we use a simple phenomenological citrus red mite population model for pest dynamics. First the optimal sampling distributions for a varying number of data points are determined. We then analyze these optimal distributions by comparing the standard errors of parameter estimates corresponding to each distribution. This allows us to investigate how many data are required to have confidence in model parameter estimates in order to employ dynamical modeling to infer population dynamics. Our results suggest that a field researcher should collect at least 12 data points at the optimal times. Data collected according to this procedure along with dynamical modeling will allow us to estimate population dynamics from presence/absence-based data sets through the development of a scaling relationship. These Likert-type data sets are commonly collected by agricultural pest management consultants and are increasingly being used in ecoinformatics studies. By applying mathematical modeling with the relationship scale from the new data, we can then explore important integrated pest management questions using past and future presence/absence data sets.

Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model
Stephen A. Gourley, Xiulan Lai, Junping Shi, Wendi Wang, Yanyu Xiao and Xingfu Zou
2018, 15(4): 1033-1054 doi: 10.3934/mbe.2018046 +[Abstract](6124) +[HTML](1262) +[PDF](675.6KB)

Lyme disease is transmitted via blacklegged ticks, the spatial spread of which is believed to be primarily via transport on white-tailed deer. In this paper, we develop a mathematical model to describe the spatial spread of blacklegged ticks due to deer dispersal. The model turns out to be a system of differential equations with a spatially non-local term accounting for the phenomenon that a questing female adult tick that attaches to a deer at one location may later drop to the ground, fully fed, at another location having been transported by the deer. We first justify the well-posedness of the model and analyze the stability of its steady states. We then explore the existence of traveling wave fronts connecting the extinction equilibrium with the positive equilibrium for the system. We derive an algebraic equation that determines a critical value $c^*$ which is at least a lower bound for the wave speed in the sense that, if $c < c^*$, there is no traveling wave front of speed $c$ connecting the extinction steady state to the positive steady state. Numerical simulations of the wave equations suggest that \begin{document}$c^*$\end{document} is the minimum wave speed. We also carry out some numerical simulations for the original spatial model system and the results seem to confirm that the actual spread rate of the tick population coincides with \begin{document}$c^*$\end{document}. We also explore the dependence of \begin{document}$c^*$\end{document} on the dispersion rate of the white tailed deer, by which one may evaluate the role of the deer's dispersion in the geographical spread of the ticks.

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](9272) +[HTML](1456) +[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](2823) +[HTML](1148) +[PDF](31.6KB)
Special issue on Erice ‘MathCompEpi 2015’ Proceedings
Alberto D'Onofrio, Paola Cerrai and Piero Manfredi
2018, 14(1): i-iv doi: 10.3934/mbe.201801i +[Abstract](7725) +[HTML](385) +[PDF](214.6KB)
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](8076) +[HTML](1188) +[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](7839) +[HTML](3138) +[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](8440) +[HTML](3011) +[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](3224) +[HTML](1291) +[PDF](72.3KB)
Mathematical methods in systems biology
Eugene Kashdan, Dominique Duncan, Andrew Parnell and Heinz Schättler
2016, 13(6): i-ii doi: 10.3934/mbe.201606i +[Abstract](3617) +[PDF](79.1KB)
The editors of this Special Issue of Mathematical Biosciences and Engineering were the organizers for the Third International Workshop "Mathematical Methods in System Biology" that took place on June 15-18, 2015 at the University College Dublin in Ireland. As stated in the workshop goals, we managed to attract a good mix of mathematicians and statisticians working on biological and medical applications with biologists and clinicians interested in presenting their challenging problems and looking to find mathematical and statistical tools for their solutions.

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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](5494) +[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](3606) +[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](3550) +[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](2230) +[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](3189) +[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](3182) +[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).
Application of ecological and mathematical theory to cancer: New challenges
Yangjin Kim, Avner Friedman, Eugene Kashdan, Urszula Ledzewicz and Chae-Ok Yun
2015, 12(6): i-iv doi: 10.3934/mbe.2015.12.6i +[Abstract](2455) +[PDF](120.7KB)
According to the World Health Organization, cancer is among the leading causes of morbidity and mortality worldwide. Despite enormous efforts of cancer researchers all around the world, the mechanisms underlying its origin, formation, progression, therapeutic cure or control are still not fully understood. Cancer is a complex, multi-scale process, in which genetic mutations occurring at a sub-cellular level manifest themselves as functional changes at the cellular and tissue scale.

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2018 Impact Factor: 1.313




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