Editorial: New frontiers in the application of mathematics to biological sciences

Abstract

Mathematical modelling is necessary to understand complex biological systems, since it provides quantitative means to investigate disease transmission, population dynamics, ecological interactions, biological networks, and physiological processes. When biological mechanisms are expressed in mathematical terms, models can reveal interactions that cannot be observed directly, identify key parameters, and to assist in expanding exploration of possible interventions and management strategies. Increasing complexity, together with the variety of biological systems, continues to drive the development of models and computational techniques that address spatial structure, multiscale interactions, memory effects, nonlinear dynamics, and regulatory mechanisms.The fact that infectious diseases appear and reappear, along with the difficulties involved in ecological conservation, population management, and biological control, shows how valuable these methods are. Mathematical models offer a cost-effective way to study complex biological processes and evaluate possible outcomes across different scenarios. Progress in areas such as computational mathematics, dynamical systems, optimization, fractional calculus, and network theory is expanding the range of biological questions that can be treated quantitatively.Following the success of the Research Topic Modelling and Numerical Modelling with Differential Equations in Mathematical Biology, Medicine and the Environment [2] and its second volume [3], the present Research Topic, New Frontiers in the Application of Mathematics to Biological Sciences, has been established in order to highlight innovative mathematical and computational methods applied to biological and ecological systems. It called for contributions in fields including dynamical systems, integro-partial differential equations, population genetics and growth, bioinformatics, heartbeat pattern modelling, optimal control theory, evolutionary game theory, and fractional-order calculus. The broader objective was to show how mathematical analysis, numerical computation, optimization, and modelling can be combined in order to advance our understanding of complex biological phenomena.The novelty of this Research Topic rests not only in applying mathematics to biological problems, but also in integrating complementary mathematical frameworks to address the many-layered complexity in biological systems. This complexity emerges from diverse sources, including biological scale, spatial structure, memory, nonlinear interactions, regulatory networks, and incomplete or truncated information. The contributions illustrate that these problems often cannot be sufficiently addressed within a single modelling paradigm. Accordingly, this Research Topic assembles differential-equation models, multiscale and reaction-diffusion formulations, fractional differential calculus, bifurcation and invasion analysis, graph-theoretic representations, statistical estimation, and optimisation-based methods. Collectively, these approaches define a more extensive methodological frontier in mathematical biology: advancing mathematical and computational structures flexi-ble enough to capture biological complexity and maintaining analytical structure and computational manageability.Three topic editors coordinated the Research Topic, which was open for submissions from April 2025 to 31 March 2026. Nineteen manuscripts were received and underwent extensive peer review. Following this process, eight papers were accepted for publication, comprising seven research articles and one opinion article. Collectively, these contributions illustrate the breadth of state-of-the-art mathematical approaches to biological sciences, ranging from multiscale and reaction-diffusion models to spatial invasion, probabilistic and state-based modeling, biological network analysis, statistical estimation, and mathematical systems for neural control.The articles in this volume show a development from biological processes across scales and spatial domains, using mathematical descriptions of nonlinear phenomena and memory, to more general computational representations and methodological models of biological systems. The discussion begins with research that directly examines spatial or biologicalscale structure, such as the multiscale model of Taenia solium cysticercosis and the spatial vector-host model of Cassava Mosaic Disease, followed by a probabilistic approach to COVID-19 disease prevalence based on an Interdependent Emission State's Hidden Markov model. It then turns to mathematical models of nonlinear ecological and behavioural dynamics, illustrated by the diffusive predator-prey model and the time-fractional cocaine-heroin model. The scope then shifts from differential-equation descriptions to other mathematical representations and computational methods, such as the Boolean logical digraph for biological regulatory networks and the meta-heuristic estimation of truncated Poisson models. The opinion piece on higher-derivative mechanics and neural control also pushes the discussion further toward the growing interaction among mechanics, control theory, and neuroscience. Maphiri et al. created a coupled multiscale model of the dynamics of Taenia solium cysticercosis, accounting for within-host parasite dynamics in both humans and pigs as well as population-level transmission processes. It covers the entire disease lifecycle and clearly shows the connection between within-host and between-host processes. Their analysis indicates that the basic reproduction number is influenced by parameters at both scales, and the sensitivity analysis identifies the within-host developmental rates and the between-host transmission parameters as the main factors responsible for the persistence of the disease. Simulation studies using a nonstandard finite difference approximation yield further insights and demonstrate the mutual influence between the within-host parasite burden and the population-level infection patterns. The study highlights the importance of integrated approaches which at the same time take into account individual hosts, livestock management, and environmental contamination.The paper is of particular importance to the field of multiscale mathematical biology since it demonstrates that biological phenomena are not always understandable when the various processes are examined at a single level in isolation. The model achieves this by making a clear connection between within-host and population-level dynamics, thus providing a framework for assessing how processes and interventions at one biological level can influence outcomes at another. The study therefore emphasizes a key viewpoint within the Research Topic, namely the need for mathematical models that maintain the important biological connections across different scales.Although the earlier work focuses on biological scales, Oh presents a different kind of complexity: that of spatial transport. The study looks at the spatial spread of Cassava Mosaic Disease by using a vector-host model which takes into account transmission mediated by whiteflies. The model connects the dynamics of plant infection with vector advection-diffusion and includes a class of protected plants to account for partial cross-protection. It derives an explicit invasion threshold and characterises the minimal invasion speed and the effective transport parameters for the coupled host-vector system. Numerical travelling-wave calculations closely agree with the predicted invasion speeds within the parameter range under consideration. The findings indicate that increasing vector mortality, improving the efficacy of protection, and enhancing roguing can prevent an invasion or restrict the spatial spread. The analysis also points out realistic factors, such as infected planting material, wind variability, and the planting and harvesting cycles, which may affect the implementation at the field level.The study shows that it is important to include spatial transport when constructing mathematical models of plant disease; in addition to allowing an evaluation of how long infection remains, the suggested framework also provides a way of measuring how fast an invasion can spread over agricultural areas. By combining invasion theory with effective transport parameters and numerical travelling waves, the approach gives a comprehensive means of mathematically managing vector-borne plant diseases.Oh's research studies disease invasion with a spatial vector-host model, but Tirupathi et al. uses a statistical method to look at disease prevalence. The authors introduce an Interdependent Emission State's Hidden Markov Model (IES-HMM) to study COVID-19 in six southern Indian states. Unlike traditional models where observed states do not depend on hidden processes, this model assumes that emission states are influenced by both invisible and emission states. The authors use conditional probabilities to calculate the likelihood of three-day sequences and build mathematical relationships to describe the model's statistical features. They also propose that this approach could help create optimization models for healthcare management during disease treatment.The study expands the mathematical treatment of the Research Topic by shifting from differential-equation models of disease dynamics towards probabilistic, state-based models of disease prevalence. The authors illustrate how the invisible and emission states can be linked together in order to connect unobserved and observed disease data; their examination of the statistical features and of the conditional sequence probabilities shows that stochastic modelling can complement deterministic methods when studying biological and epidemiological systems.This study on the predator-prey model changes the emphasis from spatial invasion to pattern formation and temporal oscillations. Appadu et al. investigate a one-dimensional diffusive predator-prey model featuring a Beddington-DeAngelis functional response. They derive the conditions for various bifurcations, such as Turing instability and Hopf bifurcation. By means of the Forward Time Central Space Scheme, they produce the prey and predator density profiles as functions of both time and space at these bifurcation points. The authors also calculate the numerical convergence rates. This work is different from that of Song et al. [1] in that it includes a separate theoretical analysis and a detailed numerical methodology. In contrast to Song et al. [1], which gives only limited details regarding the numerical methods, this study verifies its results by computing the numerical convergence rate in time and showing that it agrees with the theoretically expected results. As a whole, the research combines bifurcation analysis with computational validation in order to improve the understanding of spatial ecological mechanisms. This article highlights the complementary functions of dynamical-systems analysis and numerical computation in the investigation of spatial ecological models. In particular, comparing the theoretical and numerical convergence rates is an essential step in the evaluation process when analytical solutions are not available. The study links together pattern formation, bifurcation theory, and computational validation, demonstrating that computer simulations can support but not replace rigorous mathematical analysis.Another source of complexity, though related, is memory. Akindeinde et al. create a spatially heterogeneous time-fractional reaction and diffusion model in order to describe the coupled dynamics of cocaine and heroin abuse. In this model, the effects of memory associated with the persistence of addiction, delayed behavioural responses, and relapse are taken into account by means of Caputo fractional derivatives. To deal with spatial variability, the authors use both smoothly changing and multi-patch diffusion profiles. They prove the existence, uniqueness, positivity, and boundedness of the solutions and obtain the conditions for global stability by means of appropriate Lyapunov-based functionals. In order to overcome the computing difficulties involved in fractional reaction-diffusion models, they develop a fractional version of the Parker-Sochacki method which gives an explicit and computationally efficient approximation while reducing the large amount of global memory usually linked with fractional-order models. The numerical experiments indicate that lower fractional orders lead to a longer persistence of substance abuse, while heterogeneous diffusion results in strong nonuniform spatial distributions.The study shows that fractional-order calculus and spatial heterogeneity can be combined in order to account for two important features of substance-use dynamics: memorydependent persistence and regional variation. A significant methodological contribution lies in the fact that a detailed analysis is linked to an efficient numerical method for the resulting fractional reaction-diffusion model. As a result, this study demonstrates that progress in numerical analysis enables the examination of ever more complex biological and behavioural models.The work then goes beyond the use of differential equations. Takane et al. present a Boolean-based logical digraph for the representation and analysis of regulatory interactions in biological networks. This framework extends the traditional excitation-inhibition representations by using Boolean logic to encode a broader variety of regulatory interactions and sets up a bidirectional relationship between regulatory networks, Boolean transfer functions, and logical digraphs. The resulting representation includes information on the network's connectivity, dynamics, limit cycles, and attractors. When applied to the neural network which is responsible for oscillatory swimming in the mollusc Tritonia, the framework shows how the spectral properties of the associated state matrix can yield mathematical information concerning the dynamics expressed by attractors and limit cycles. The fact that open software routines are available also helps to enable the use of the framework for the analysis and visualisation of biological regulatory networks.The study extends the mathematical description of biological regulatory systems by linking logical descriptions to graph-theoretical and spectral properties. Specifically, the bidirectional transition between a regulatory network and its logical digraph offers a means of moving from a biological interpretation to mathematical analysis. The use of this approach with the Tritonia neural network shows the potential of the framework for analysing complex regulatory structures and for obtaining information on their dynamical behaviour.The trend extends from dynamical representations to statistical estimation. Basheer et al. consider a computational issue in truncated Poisson regression, since missing zero counts can lead to bias and inefficiency in standard parameter estimation. They suggest using a meta-heuristic method based on the Black Kite Optimization