Mathematical Biosciences Research Articles

Existence and stability criteria for global synchrony and for synchrony in two alternating clusters of pulse-coupled oscillators updated to include conduction delays

Phase Response Curves (PRCs) have been useful in determining and analyzing various phase-locking modes in networks of oscillators under pulse-coupling assumptions, as reviewed in Mathematical Biosciences, 226:77–96, 2010. Here, we update that review to include progress since 2010 on pulse coupled oscillators with conduction delays. We then present original results that extend the derivation of the criteria for stability of global synchrony in networks of pulse-coupled oscillators to include conduction delays. We also incorporate conduction delays to extend previous studies that showed how an alternating firing pattern between two synchronized clusters could enforce within-cluster synchrony, even for clusters unable to synchronize themselves in isolation. To obtain these results, we used self-connected neurons to represent clusters. These results greatly extend the applicability of the stability analyses to networks of pulse-coupled oscillators since conduction delays are ubiquitous and strongly impact the stability of synchrony. Although these analyses only strictly apply to identical oscillators with identical connections to other oscillators, the principles are general and suggest how to promote or impede synchrony in physiological networks of neurons, for example. Heterogeneity can be interpreted as a form of frozen noise, and approximate synchrony can be sustained despite heterogeneity. The pulse-coupled oscillator model can not only be used to describe biological neuronal networks but also cardiac pacemakers, lasers, fireflies, artificial neural networks, social self-organization, and wireless sensor networks.

A Research Introduction for Early College Students: Based on the OSU–HBCU Pilot Project

We joined an initiative to engage faculty and undergraduate students in research by scheduling regular early college-level mathematics research presentations with pre- and post-lecture exercises. While our program featured six talks that were predominantly in the mathematical biosciences, the concept of creating themed topics for presentations can be extended to other areas of mathematical research, and the length of the program can also be scaled. This article discusses three of the presentations, including QR codes to access their recordings. The purpose of this article is to present the materials of the talks from this initiative and how faculty members can use the materials for potential explorations with students. The hope is the audience may use these, as presented, for study with students, with minimal start-up investment and to use this as a model for future undergraduate research efforts.

Existence and Stability Criteria for Global Synchrony and for Synchrony in two Alternating Clusters of Pulse-Coupled Oscillators Updated to Include Conduction Delays.

37N25, 39A06, 39A30, 92B25, 92C20.

Persistence in repeated games encourages the evolution of spite

Social behavior is divided into four types: altruism, spite, mutualism, and selfishness. The former two are costly to the actor; therefore, from the perspective of natural selection, their existence can be regarded as mysterious. One potential setup which encourages the evolution of altruism and spite is repeated interaction. Players can behave conditionally based on their opponent's previous actions in the repeated interaction. On the one hand, the retaliatory strategy (who behaves altruistically when their opponent behaved altruistically and behaves non-altruistically when the opponent player behaved non-altruistically) is likely to evolve when players choose altruistic or selfish behavior in each round. On the other hand, the anti-retaliatory strategy (who is spiteful when the opponent was not spiteful and is not spiteful when the opponent player was spiteful) is likely to evolve when players opt for spiteful or mutualistic behavior in each round. These successful conditional behaviors can be favored by natural selection. Here, we notice that information on opponent players’ actions is not always available. When there is no such information, players cannot determine their behavior according to their opponent's action. By investigating the case of altruism, a previous study (Kurokawa, 2017, Mathematical Biosciences, 286, 94–103) found that persistent altruistic strategies, which choose the same action as the own previous action, are favored by natural selection. How, then, should a spiteful conditional strategy behave when the player does not know what their opponent did? By studying the repeated game, we find that persistent spiteful strategies, which choose the same action as the own previous action, are favored by natural selection. Altruism and spite differ concerning whether retaliatory or anti-retaliatory strategies are favored by natural selection; however, they are identical concerning whether persistent strategies are favored by natural selection.

Retraction notice to "A video images-aware knowledge extraction method for intelligent healthcare management of basketball players" [Mathematical Biosciences and Engineering 20(2) (2023) 1919-1937

Retraction notice to "A video images-aware knowledge extraction method for intelligent healthcare management of basketball players" [Mathematical Biosciences and Engineering 20(2) (2023) 1919-1937

Retraction notice to "A novel architecture design for artificial intelligence-assisted culture conservation management system" [Mathematical Biosciences and Engineering 20(6) (2023) 9693-9711

Retraction notice to "A novel architecture design for artificial intelligence-assisted culture conservation management system" [Mathematical Biosciences and Engineering 20(6) (2023) 9693-9711

Retraction notice to "Kinesin family member 15 can promote the proliferation of glioblastoma" [Mathematical Biosciences and Engineering 19(8) (2022) 8259-8272

Retraction notice to "Kinesin family member 15 can promote the proliferation of glioblastoma" [Mathematical Biosciences and Engineering 19(8) (2022) 8259-8272

A circulant inertia three Hopfield neuron system: dynamics, offset boosting, multistability and simple microcontroller- based practical implementation

This work investigates the dynamics and implementation of a circulant inertia three Hopefield neuron model with each neuron activated by a non-monotonic Crespi function. Owing its source to the work previously done by Song and co-authors [Song et al (2019) Mathematical Biosciences and Engineering 16. 6406–6426], we propose a network made up of three neurons connected cyclically. We demonstrate that the model is capable of the coexistence of two, three, four, six, seven, eight and ten attractors basing on different initial states. The system is dissipative and presents fifteen unstable equilibrium points for a given rank of parameters. Accordingly, we demonstrate the Hopf bifurcation in the model when the bifurcation parameter is the first synaptic weight. Moreover, using bifurcation diagrams, Maximum Lyapunov Exponent diagram, phase portraits, two parameters Lyapunov diagrams, double-sided Poincaré section and basin of attraction, intriguing phenomena have been revealed such as hysteresis, coexistence of parallel branches of bifurcation, antimonotonicity and transient chaos to name a few. A number of coexisting attractors have been developed by the new network which can be used to build sophisticated cryptosystem or to explain the possible tasks of a brain in normal or abnormal cases. To verify the feasibility of the model, a microcontroller-based implementation has been used to demonstrate the period-doubling route to chaos obtained numerically.

Convergence of a diffuse interface Poisson-Boltzmann (PB) model to the sharp interface PB model: A unified regularization formulation

Both the sharp interface and diffuse interface Poisson-Boltzmann (PB) models have been developed in the literature for studying electrostatic interaction between a solute molecule and its surrounding solvent environment. In the mathematical analysis and numerical computation for these PB models, a significant challenge is due to singular charge sources in terms of Dirac delta distributions. Recently, based on various regularization schemes for the sharp interface PB equation, the first regularization method for the diffuse interface PB model has been developed in [S. Wang, E. Alexov, and S. Zhao, Mathematical Biosciences and Engineering, 18, 1370–1405, (2021)] for analytically treating the singular charges. This work concerns with the convergence of a diffuse interface PB model to the sharp interface PB model, as the diffused Gaussian-convolution surface (GCS) approaches to the sharp solvent accessible surface (SAS). Due to the limitation in numerical algorithm and mesh resolution, such a convergence is impossible to be verified numerically. Through analyzing the weak solution for the regularized PB equations, the convergences for both the reaction-field potential and electrostatic free energy are rigorously proved in this work. Moreover, this study provides a unified regularization for both sharp interface and diffuse interface PB models, and clarifies the connection between this unified formulation and the existing regularizations. This lays a theoretical foundation to develop regularization for more complicated PB models.

Dynamic publication media with the COPASI R Connector (CoRC)

In this article we show how dynamic publication media and the COPASI R Connector (CoRC) can be combined in a natural and synergistic way to communicate (biochemical) models. Dynamic publication media are becoming a popular tool for authors to effectively compose and publish their work. They are built from templates and the final documents are created dynamically. In addition, they can also be interactive. Working with dynamic publication media is made easy with the programming environment R via its integration with tools such as R Markdown, Jupyter and Shiny. Additionally, the COmplex PAthway SImulator COPASI (http://www.copasi.org), a widely used biochemical modelling toolkit, is available in R through the use of the COPASI R Connector (CoRC, https://jpahle.github.io/CoRC). Models are a common tool in the mathematical biosciences, in particular kinetic models of biochemical networks in (computational) systems biology.We focus on three application areas of dynamic publication media and CoRC: Documentation (reproducible workflows), Teaching (creating self-paced lessons) and Science Communication (immersive and engaging presentation). To illustrate these, we created six dynamic document examples in the form of R Markdown and Jupyter notebooks, hosted on the platforms GitHub, shinyapps.io, Google Colaboratory.Having code and output in one place, creating documents in template-form and the option of interactivity make the combination of dynamic documents and CoRC a versatile tool.All our example documents are freely available at https://jpahle.github.io/DynamiCoRC under the Creative Commons BY 4.0 licence.

Correction to: Heart Disease and Stroke Statistics-2019 Update: A Report From the American Heart Association.

Correction to: Heart Disease and Stroke Statistics-2019 Update: A Report From the American Heart Association.

Modeling the effect of temperature variability on malaria control strategies

In this study, a non-autonomous (temperature dependent) and autonomous (temperature independent) models for the transmission dynamics of malaria in a population are designed and rigorously analysed. The models are used to assess the impact of temperature changes on various control strategies. The autonomous model is shown to exhibit the phenomenon of backward bifurcation, where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium when the associated reproduction number is less than unity. This phenomenon is shown to arise due to the presence of imperfect vaccines and disease-induced mortality rate. Threshold quantities (such as the basic offspring number, vaccination and host type reproduction numbers) and their interpretations for the models are presented. Conditions for local asymptotic stability of the disease-free solutions are computed. Sensitivity analysis using temperature data obtained from Kwazulu Natal Province of South Africa [K. Okuneye and A.B. Gumel. Mathematical Biosciences 287 (2017) 72–92] is used to assess the parameters that have the most influence on malaria transmission. The effect of various control strategies (bed nets, adulticides and vaccination) were assessed via numerical simulations.

Multi-scale dynamics of infectious diseases

To address the challenge of multiscale dynamics of infectious diseases, the Mathematical Biosciences Institute organized a workshop at The Ohio State University to bring together scientists from a variety of disciplines to share expertise gained through looking at infectious diseases across different scales. The researchers at the workshop, held in April 2018, were specifically looking at three model systems: foot-and-mouth disease, vector-borne diseases and enteric diseases. Although every multiscale model must be necessarily derived from a multiscale system, not every multiscale system has to lead to multiscale models. These three model systems seem to have produced a variety of both multiscale and integrated single-scale mechanistic models that have developed their own strengths and particular challenges. Here, we present papers from some of the workshop participants to show the breadth of the field.

Using the stochastic Galerkin method as a predictive tool during an epidemic

The ability to accurately predict the course of an epidemic is extremely important. This article looks at an influenza outbreak that spread through a small boarding school. Predictions are made on multiple days throughout the epidemic using the stochastic Galerkin method to consider a range of plausible values for the parameters. These predictions are then compared to known data points. Predictions made before the peak of the epidemic had much larger variances compared to predictions made after the peak of the epidemic.
 
 References B. M. Chen-Charpentier, J. C. Cortes, J. V. Romero, and M. D. Rosello. Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation, 219(9):4208 4218, 2013. doi:10.1016/j.amc.2012.11.007 B. M. Chen-Charpentier and D. Stanescu. Epidemic models with random coefficients. Mathematical and Computer Modelling, 52:1004 1010, 2010. doi:10.1016/j.mcm.2010.01.014 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with individualised parameter distributions. In Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C160C176, August 2016. doi:10.21914/anziamj.v57i0.10394 D. B. Harman and P. R. Johnston. Applying the stochastic galerkin method to epidemic models with uncertainty in the parameters. Mathematical Biosciences, 277:25 37, 2016. doi:10.1016/j.mbs.2016.03.012 D. B. Harman and P. R. Johnston. Boarding house: find border. 2019. doi:10.6084/m9.figshare.7699844.v1 D. B. Harman and P. R. Johnston. SIR uniform equations. 2 2019. doi:10.6084/m9.figshare.7692392.v1 H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4):599653, 2000. doi:10.1137/S0036144500371907 R.I. Hickson and M.G. Roberts. How population heterogeneity in susceptibility and infectivity influences epidemic dynamics. Journal of Theoretical Biology, 350(0):70 80, 2014. doi:10.1016/j.jtbi.2014.01.014 W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115(772):700721, August 1927. doi:10.1098/rspa.1927.0118 M. G. Roberts. A two-strain epidemic model with uncertainty in the interaction. The ANZIAM Journal, 54:108115, 10 2012. doi:10.1017/S1446181112000326 M. G. Roberts. Epidemic models with uncertainty in the reproduction number. Journal of Mathematical Biology, 66(7):14631474, 2013. doi:10.1007/s00285-012-0540-y F. Santonja and B. Chen-Charpentier. Uncertainty quantification in simulations of epidemics using polynomial chaos. Computational and Mathematical Methods in Medicine, 2012:742086, 2012. doi:10.1155/2012/742086 Communicable Disease Surveillance Centre (Public Health Laboratory Service) and Communicable Diseases (Scotland) Unit. Influenza in a boarding school. BMJ, 1(6112):587, 1978. doi:10.1136/bmj.1.6112.586 G. Strang. Linear Algebra and Its Applications. Thomson, Brooks/Cole, 2006. D. Xiu. Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, 2010.

Zeroing neural networks: A survey

Using neural networks to handle intractability problems and solve complex computation equations is becoming common practices in academia and industry. It has been shown that, although complicated, these problems can be formulated as a set of equations and the key is to find the zeros of them. Zeroing neural networks (ZNN), as a class of neural networks particularly dedicated to find zeros of equations, have played an indispensable role in the online solution of time-varying problem in the past years and many fruitful research outcomes have been reported in the literatures. The aim of this paper is to provide a comprehensive survey of the research on ZNNs, including continuous-time and discrete-time ZNN models for various problems solving as well as their applications in motion planning and control of redundant manipulators, tracking control of chaotic systems, or even populations control in mathematical biosciences. By considering the fact that real-time performance is highly demanded for time-varying problems in practice, stability and convergence analyses of different continuous-time ZNN models are reviewed in detail in a unified way. For the case of discrete-time problems solving, the procedures on how to discretize a continuous-time ZNN model and the techniques on how to obtain an accuracy solution are summarized. Concluding remarks and future directions of ZNN are pointed out and discussed.

STOCHASTIC ANALYSIS OF A DETERMINISTIC AND SEASONALLY-FORCED SEI MODEL FOR IMPROVED DISEASE SPREAD SIMULATION

The geographic distribution of different viruses has developed widely, giving rise to an escalating number of cases during the past two decades. The deterministic Susceptible, Exposed, Infectious (SEI) models can demonstrate the spatio-temporal dynamics of the diseases and have been used extensively in modern mathematical and mechano-biological simulations. This article presents a functional technique to model the stochastic effects and seasonal forcing in a reliable manner by satisfying the Lipschitz criteria. We have emphasized that the graphical portrayal can prove to be a powerful tool to demonstrate the stability analysis of the deterministic as well as the stochastic modeling. Emphasis is made on the dynamical effects of the force of infection. Such analysis based on the parametric sweep can prove to be helpful in predicting the disease spread in urban as well as rural areas and should be of interest to mathematical biosciences researchers.

Some Novel Exponential and Complex Structural Properties of the Fisher Equation Arising in Mathematical Bioscience

In this manuscript, we consider the Bernoulli sub-equation function method for obtaining new exponential and complex prototype structures to the Fisher Equation arising in Mathematical biosciences. We obtain new results by using the technique for new properties of model and for more understanding of properties of model. We plot two- and three-dimensional surfaces of the results by using Wolfram Mathematica 9. At the end of this manuscript, we submit a conclusion in the comprehensive manner.

Preface.

This volume was inspired by the topics presented at the international conference "Micro and Macro Systems in Life Sciences" which was held on Jun 8-12, 2015 in Będlewo, Poland. System biology is an approach which tries to understand how micro systems, at the molecular and cellular levels, affect macro systems such as organs, tissue and populations. Thus it is not surprising that a major theme of this volume evolves around cancer and its treatment. Articles on this topic include models for tumor induced angiogenesis, without and with delays, metastatic niche of the bone marrow, drug resistance and metronomic chemotherapy, and virotherapy of glioma. Methods range from dynamical systems to optimal control. Another well represented topic of this volume is mathematical modeling in epidemiology. Mathematical approaches to modeling and control of more specific diseases like malaria, Ebola or human papillomavirus are discussed as well as a more general approaches to the SEIR, and even more general class of models in epidemiology, by using the tools of optimal control and optimization. The volume also brings up challenges in mathematical modeling of other diseases such as tuberculosis. Partial differential equations combined with numerical approaches are becoming important tools in modeling not only tumor growth and treatment, but also other diseases, such as fibrosis of the liver, and atherosclerosis and its associated blood flow dynamics, and our volume presents a state of the art approach on these topics. Understanding mathematics behind the cell motion, appearance of the special patterns in various cell populations, and age structured mutations are among topics addressed inour volume. A spatio-temporal models of synthetic genetic oscillators brings the analysis to the gene level which is the focus of much of current biological research. Mathematics can help biologists to explain the collective behavior of bacterial, a topic that is also presented here. Finally some more across the discipline topics are being addresses, which can appear as a challenge in studying problems in systems biology on all, macro, meso and micro levels. They include numerical approaches to stochastic wave equation arising in modeling Brownian motion, discrete velocity models, many particle approximations as well as very important aspect on the connection between discrete measurement and the construction of the models for various phenomena, particularly the one involving delays. With the variety of biological topics and their mathematical approaches we very much hope that the reader of the Mathematical Biosciences and Engineering will find this volume interesting and inspirational for their own research.

Mathematical methods in systems biology.

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.

Travelling Wave Solutions for a Reaction-Diffusion Models of Cell Growth

Recently, Harko and Mak considered two mathematical reaction-diffusion models of gioblastoma growth (Mathematical Biosciences and Engineering 12 41-69 (2015)). They obtained exact travelling wave solutions for particular functional choices of the diffusion term D(c) and the reaction/growth term Q(c), where c represents the cell concentration. Here, alternative choices of the diffusion function and the reaction/growth function are considered which lead to exact travelling wave solutions based upon the Lambert W function. These novel solutions offer alternative but exactly solvable models for cell growth complementing those previously presented.