Numerical analysis & modelling in the life sciences: a topical collection in honor of Ezio Venturino

We consider a spatial generalization of evolutionary game theory in which strategies are distributed over a spatial array of sites. We assume that the strategy corresponding to a given site has local interactions with the strategies sitting on neighbouring sites, and that the strategies change if neighbouring strategies are doing better. After briefly setting the stage with a formal definition of spatial evolutionary game theory, we consider the spatial extension of the Hawk-Dove game, and we show that the results are qualitatively different from those obtained from classical evolutionary game theory. For example, the proportion of Hawks in the population is in general lower in the spatial game than in the classical one. We also consider spatial generalizations of the extensions of the Hawk-Dove game obtained by including strategies such as Retaliator and Bully. Here, too, the results from the spatial game are very different from the classical results. In particular, with space Retaliator is a much more successful strategy than one would expect from classical considerations. This suggests that, in general, spatial structure may facilitate the evolution of strategies such as Retaliator, which do not necessarily prosper classically, and which are reminiscent of the \`nice', \`provokable' and `forgiving' strategies which seem to play a central role in the evolution of cooperation. The results indicate that including spatial structure in evolutionary game theory is a fruitful extension.

In order to develop a better understanding of the evolutionary dynamics of HIV drug resistance, it is necessary to quantify accurately the in vivo fitness costs of resistance mutations. However, the reliable estimation of such fitness costs is riddled with both theoretical and experimental difficulties. Experimental fitness assays typically suffer from the shortcoming that they are based on in vitro data. Fitness estimates based on the mathematical analysis of in vivo data, however, are often questionable because the underlying assumptions are not fulfilled. In particular, the assumption that the replication rate of the virus population is constant in time is frequently grossly violated. By extending recent work of Marée and colleagues, we present here a new approach that corrects for time-dependent viral replication in time-series data for growth competition of mutants. This approach allows a reliable estimation of the relative replicative capacity (with confidence intervals) of two competing virus variants growing within the same patient, using longitudinal data for the total plasma virus load, the relative frequency of the two variants and the death rate of infected cells. We assess the accuracy of our method using computer-generated data. An implementation of the developed method is freely accessible on the Web (http://www.eco.ethz.ch/fitness.html).

We describe in this paper a computer program for automated mathematical modelling and simulation of robotic dynamic systems using fuzzy logic techniques, genetic algorithms and fractal theory. The computer program combines soft computing (SC) techniques with mathematical methods and can be considered an intelligent system for the domain of modelling and simulation of robotic systems. This domain is quite complex because robotic systems can be viewed as nonlinear dynamical systems, and it is a well known fact that even very simple nonlinear dynamical systems can exhibit chaotic behavior. The computer program simulates the reasoning of a human expert in the process of developing mathematical models of robotic dynamic systems (RDS). The program contains the knowledge of the human experts expressed as fuzzy rules (in the knowledge base) for mathematical modelling and simulation (MMS) of robotic systems. The computer program uses efficiently SC techniques and fractal theory for MMS of RDS. Mathematical modelling and simulation of robotic systems is very important because it can help in the control of an actual system or in the design of a new system using the results of the simulations.

Advances in quantitative biology data collection and analysis across scales (molecular, cellular, organismal, and ecological) have transformed how we understand, categorize, and predict complex biological systems. This surge of quantitative data creates an opportunity to apply, develop, and evaluate mathematical models of biological systems and explore novel methods of analysis. Simultaneously, thanks to increased computational power, mathematicians, engineers and physical scientists have developed sophisticated models of biological systems at different scales. Novel modeling schemes can offer deeper understanding of principles in biology, but there is still a disconnect between modeling and experimental biology that limits our ability to fully realize the integration of mathematical modeling and biology. In this work, we explore the urgent need to expand the use of existing mathematical models across biological scales, develop models that are robust to biological heterogeneity, harness feedback loops within the iterative modeling process, and nurture a cultural shift towards interdisciplinary and cross-field interactions. Better integration of biological experimentation and robust mathematical modeling will transform our ability to understand and predict complex biological systems.

We describe in this paper a computer program for Mathematical Modelling and Simulation of Robotic Dynamic Systems using Fuzzy Logic Techniques and Fractal Theory. The computer program combines Artificial Intelligence (AI) techniques with mathematical methods and can be considered an Intelligent Tutoring System (ITS) for the domain of modelling and simulation of robotic systems. This domain is quite complex because robotic systems can be viewed as non-linear dynamical systems, and it is a well known fact that even very simple non-linear dynamical systems can exhibit chaotic behavior. The computer program simulates the reasoning of a human expert in the process of teaching how to develop mathematical models of Robotic Dynamic Systems (RDS). The program contains the knowledge of the human experts expressed as fuzzy rules (in the knowledge base) for Mathematical Modelling and Simulation (MMS) of robotic systems. The ITS also contains knowledge about teaching methodologies for this domain (in the knowledge base). The ITS uses efficiently AI techniques to teach MMS of RDS, and also to monitor the learning process of students of this domain. Mathematical Modelling and Simulation of Robotic Systems is very important because it can help in the control of an actual system or in the design of a new system using the results of the simulations.

Automated mathematical modelling, simulation and behavior identification of robotic dynamic systems using a new fuzzy-fractal-genetic approach

Mathematical models can be useful to understand the dynamics of the epidemic and how control measures may affect potential future trajectories. I will describe some models of Covid-19 dynamics in New Zealand, how they have been calibrated to data, and some of the insights they can provide. Michael Plank is a Professor in the School of Mathematics and Statistics at the University of Canterbury in New Zealand and Principal Investigator at Te Pūnaha Matatini, New Zealand's Centre of Research Excellence in Complex Systems and Data Analytics. He obtained his BSc(Hons) in Mathematics from the University of Bristol in 2000 and his PhD in Applied Mathematics from the University of Leeds in 2003. He started at the University of Canterbury as a postdoctoral research fellow in 2004 and as a permanent academic staff member in 2006. Professor Plank is an expert in mathematical modelling of complex biological and social systems at multiple scales, from intracellular signalling and collective cell behaviour, through to large ecosystem dynamics. His research is application-driven and focuses on mechanistic mathematical and stochastic models that capture emergent behaviour and offer qualitative insight into underlying mechanisms. His areas of expertise include ecological and social networks, population dynamics, epidemiological models, size-structured marine ecosystems, collective cell behaviour, and intracellular dynamics. His research draws on numerous fields in applied mathematics including stochastic processes, integro and partial differential equations, dynamical systems, spatial moment dynamics, statistical modelling, and parameter inference. His research has funded been funded by industry and government, by Te Pūnaha Matatini and by grants from the Marsden Fund and Australian Research Council. Professor Plank has served in various governance and editorial roles. He is currently President of the NZ Branch of Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) and a member of the University of Canterbury's Academic Board. He is an Editorial Board member for the ANZIAM Journal. He has served as a panellist for the Marsden Fund and Performance-Based Research Fund and represents ANZIAM at the International Council for Industrial and Applied Mathematics.

Since 2002 we have offered an undergraduate major in Mathematical Biology at Harvey Mudd College. The major was developed and is administered jointly by the mathematics and biology faculty. In this paper we describe the major, courses, and faculty and student research and discuss some of the challenges and opportunities we have experienced.

Sufficient Conditions for Multiple Niche Polymorphism

Введение. В статье обоснована необходимость применения основных логических операций как стороны мыслительной деятельности в контексте математического моделирования технических систем и их элементов. Подчеркнута нормативность технических знаний и технического труда, следствием которой является нормативность процесса математического моделирования технических систем и их элементов. Показаны психолого-педагогические основания математического моделирования технических систем и их элементов в рамках политехнического обучения. Цель статьи – обосновать вопросы практического применения логических приемов познания, способствующих развитию аналитико-синтетической учебной деятельности студентов в контексте математического моделирования технических систем и их элементов. Материалы и методы. Основными методами исследования явились теоретический анализ основных логических операций как стороны мыслительной деятельности, подходов к вопросам преподавания математического моделирования и применения информационных технологий, частная методика преподавания математического моделирования в техническом вузе. Результаты. Выявлена система критериальных оценок основных логических операций как стороны мыслительной деятельности студентов, показано их содержание, проведено наблюдение за одной из учебных групп студентов, обучающихся по направлению подготовки 221700 «Стандартизация и метрология», выполнен анализ индивидуализированных уровневых показателей аналитико-синтетической учебной деятельности студентов в контексте математического моделирования технических систем и их элементов. Обсуждение. Разработка предположения о структурировании логических приемов познания и их влиянии на развитие аналитико-синтетической учебной деятельности студентов в контексте математического моделирования технических систем и их элементов привела к выявлению средств и способов учебных действий конкретной целевой направленности. Заслуживает внимания дальнейший поиск психолого-педагогических оснований переноса математических знаний, способов действий к исследованию технических систем и их элементов с применением информационных технологий. Заключение. Полученные результаты свидетельствуют о том, что развитие аналитико-синтетической деятельности студентов становится возможным при сочетании определенных условий: структурно-логической последовательности изучения естественнонаучных дисциплин, наличия строго описанных компетенций профессиональной направленности, разработки системы критериальных оценок аналитико-синтетической учебной деятельности студентов в контексте математического моделирования и их содержательной проработки, готовности и приверженности студентов воспринимать системность предметного содержания изучаемой дисциплины. Анализ результативности применения системы логических приемов аналитико-синтетической учебной деятельности студентов в контексте математического моделирования показал, что, с одной стороны, по отдельным знаниевым и процессуальным процедурам объективно не существует четких разделительных границ, с другой стороны – владение студентами отдельными абстрактными и практическими нормами, а мера и эффективность их применения способствуют развитию представлений о системности предметного содержания дисциплины. Introduction. The paper substantiates the necessity of applying basic logical operations as aspects of mental activities to the context of mathematical modeling of technological systems and their elements. The emphasized normative character of technical knowledge and labor leads to the standardization of the process of mathematical modeling of technological systems and their elements. The paper also presents psychological and pedagogical foundations of mathematical modeling of technological systems and their elements as part of polytechnic training. Materials and Methods. The research is based on theoretical analysis of the following: basic logical operations as aspects of the mental activity, the approach to teaching mathematical modeling and the use of IT solutions in teaching, individual methods of teaching mathematical modeling as part of technical education. Results. The paper defines the criterion score system in assessing basic logical operations as parts of students’ mental activity. The components of the criterion score system are described. The research also presents the observations of a group of students whose specialty is «Standardization and Metrology». The observations are followed by the analysis of students’ indices of their analytic and synthetic activities as learners in the context of mathematical modeling of technological systems and their elements. Discussion. The hypothesis about structuring cognitive processes and the influence of cognitive logic on students’ analytic and synthetic activities in the process of mathematical modeling of technological systems and their elements unraveled various means and instruments of performing specifically targeted educational activities. The opportunities of the further research are mentioned: psychological and pedagogic bases of transferring mathematical knowledge and modus operandi to exploring technological systems and the elements of the latter with the use of IT solutions require further scientific attention. Conclusion. The results of the research confirm that development of students’ analytic and synthetic activities becomes possible due to the combination of the following conditions: structural and logical sequence of actions in studying natural science subjects, existence of the detailed inventory of professional orientation competencies, developed criterion score system in assessing basic logical operations as parts of students’ mental activity and their readiness to perceive the complexity of the subject under study. The analysis showed that, on the one hand, there are no strict boundaries between specific knowledge procedures. On the other hand, the use of certain theoretical and practical standards develop students’ idea of the complexity of the content area of the subject.

Mathematical modeling of physiological systems: An essential tool for discovery

The biological process and molecular functions involved in the cancer progression remain difficult to understand for biologists and clinical doctors. Recent developments in high-throughput technologies urge the systems biology to achieve more precise models for complex diseases. Computational and mathematical models are gradually being used to help us understand the omics data produced by high-throughput experimental techniques. The use of computational models in systems biology allows us to explore the pathogenesis of complex diseases, improve our understanding of the latent molecular mechanisms, and promote treatment strategy optimization and new drug discovery. Currently, it is urgent to bridge the gap between the developments of high-throughput technologies and systemic modeling of the biological process in cancer research. In this review, we firstly studied several typical mathematical modeling approaches of biological systems in different scales and deeply analyzed their characteristics, advantages, applications, and limitations. Next, three potential research directions in systems modeling were summarized. To conclude, this review provides an update of important solutions using computational modeling approaches in systems biology.

The article considers a bicomplex calculation for calculating the invariant power supply systems based on renewable energy sources. Modern energy supply systems based on renewable energy sources have non-linear systems with complex transients and possible critical and chaotic regimes. The study of structures of hypernumerical systems, their features, methods of calculation and approximation of the elementary functions of a hypercomplex variable allows to effectively apply such systems in mathematical modelling of invariant power supply systems based on renewable energy sources. In some cases, the use of hypernumerical systems makes it possible to replace the original problem with an equivalent one, that is to build a bicomplex solution model. The system of complex numbers was considered as the initial system. With recurrent doubling of the system, hypernumerical systems of different dimensions with different properties were obtained, which made it possible to assign different values to the products of imaginary units. It is proved that the introduction of additional conditions of commutativity and associativity, which apply to real numbers and imaginary units, allows to specify the choice of a hypernumerical system. In the analysis of nonstationary processes of invariant systems and the study of the possibilities of hypernumerical systems, the expediency of choosing a bicomplex calculation method in mathematical modelling of systems with multiple modulation is substantiated. The method of bicomplex representation involves direct and inverse bicomplex transformation, which allows obtaining an analytically complete solution for the analysis of an invariant power supply system based on renewable energy sources. Examples of the use of bicomplex integral transformation for the analysis of systems with multiple modulation are considered. The application of the hypercomplex calculus apparatus for the transformation of systems of differential equations is proposed to simplify or compress them into one equation. It is shown that the use of hypercomplex calculus allows to significantly reduce the amount of processed information without reducing the informativeness of the mathematical model. The proposed formulation of tasks in a hypercomplex view allowed to compress the processing information and obtain a compact vortex for the output signal.

A new fractional analysis on the interaction of HIV with [formula omitted] T-cells

Integrated modelling of biological systems is challenged by composing components with sufficient kinetic data and components with insufficient kinetic data or components built only using experts’ experience and knowledge. Fuzzy continuous Petri nets (FCPNs) combine continuous Petri nets with fuzzy inference systems, and thus offer an hybrid uncertain/certain approach to integrated modelling of such biological systems with uncertainties. In this paper, we give a formal definition and a corresponding simulation algorithm of FCPNs, and briefly introduce the FCPN tool that we have developed for implementing FCPNs. We then present a methodology and workflow utilizing FCPNs to achieve hybrid (uncertain/certain) modelling of biological systems illustrated with a case study of the Mercaptopurine metabolic pathway. We hope this research will promote the wider application of FCPNs and address the uncertain/certain integrated modelling challenge in the systems biology area.