Mathematical Models Push New Frontiers in Decoding Biology’s Deepest Complexities

Mathematics has quietly become one of the most powerful tools in modern biology, and a new collection of peer-reviewed studies shows just how far that partnership has evolved. A special Research Topic published in Frontiers in Applied Mathematics and Statistics, titled New frontiers in the application of mathematics to biological sciences, brings together eight articles that use advanced mathematical and computational frameworks to tackle problems ranging from parasitic disease transmission to the neural control of human movement. Guest editors Ramoshweu Solomon Lebelo of Vaal University of Technology, Saheed Ojo Akindeinde of Botswana International University of Science and Technology, and Appanah Rao Appadu of the University of Western Cape coordinated the collection, which was open for submissions from April 2025 to 31 March 2026. Of nineteen manuscripts received, eight survived extensive peer review: seven research articles and one opinion piece.

The driving motivation behind the collection is a recognition that biological systems are rarely simple enough to be captured by a single modeling paradigm. Complexity in living systems emerges from many sources at once: biological scale, spatial structure, memory effects, non-linear interactions, regulatory networks, and incomplete or truncated information. When biological mechanisms are expressed in mathematical terms, models can reveal interactions that cannot be observed directly, identify key parameters, and help researchers explore possible interventions and management strategies before committing resources to costly experiments. The editors argue that progress in computational mathematics, dynamical systems, optimization, fractional calculus, and network theory is steadily expanding the range of biological questions that can be treated quantitatively, and the eight accepted articles illustrate that expansion in striking detail.

The first major contribution, from Maphiri and colleagues, tackles Taenia solium cysticercosis with a coupled multiscale model that links parasite dynamics within individual human and pig hosts to population-level transmission processes. The model covers the entire disease lifecycle and makes the connection between within-host and between-host processes explicit. Its analysis shows that the basic reproduction number, the fundamental quantity governing whether an infection can persist, depends on parameters at both biological scales. Sensitivity analysis identifies within-host developmental rates and between-host transmission parameters as the main factors responsible for disease persistence, and simulation studies using a non-standard finite difference approximation demonstrate the mutual influence between individual parasite burdens and population-level infection patterns. The work underscores a central theme of the collection: biological phenomena are often not understandable when processes are examined at a single level in isolation, and interventions at one scale can ripple through to outcomes at another.

Where the cysticercosis study emphasizes biological scale, a second article by Oh emphasizes spatial transport. It examines the spread of Cassava Mosaic Disease using a vector-host model that accounts for transmission mediated by whiteflies, coupling plant infection dynamics with vector advection-diffusion and including a class of protected plants to represent partial cross-protection. The model derives an explicit invasion threshold, characterizes the minimal invasion speed and the effective transport parameters for the coupled host-vector system, and its numerical traveling-wave calculations agree closely with the predicted invasion speeds within the parameter range considered. The findings indicate that increasing vector mortality, improving the efficacy of protection, and enhancing roguing, the removal of infected plants, can prevent an invasion or restrict its spatial spread. The analysis also flags practical field-level factors such as infected planting material, wind variability, and planting and harvesting cycles that may affect implementation.

A third study shifts the methodological ground entirely, moving from differential equations to probabilistic state-based modeling. Padi and colleagues introduce an Interdependent Emission State’s Hidden Markov Model, or IES-HMM, to study COVID-19 prevalence across six southern Indian states. Unlike traditional hidden Markov models, where observed states do not depend on hidden processes, this formulation 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 describing the model’s statistical features. They propose that the approach could support optimization models for healthcare management during disease treatment, illustrating how stochastic modeling can complement deterministic methods when researchers must connect unobserved and observed disease data.

Ecological pattern formation takes center stage in a study by Appadu and colleagues, who investigate a one-dimensional diffusive predator-prey model featuring a Beddington-DeAngelis functional response, a formulation in which predator feeding efficiency declines at high prey density. The team derives the conditions for various bifurcations, including Turing instability, the mechanism behind spontaneous spatial pattern formation, and Hopf bifurcation, which produces oscillatory behavior. Using the Forward Time Central Space numerical scheme, they generate prey and predator density profiles as functions of both time and space at these bifurcation points, and they calculate numerical convergence rates that agree with theoretically expected values. This careful verification distinguishes the work from earlier studies that provided only limited numerical detail, and it demonstrates that computer simulations can support, but not replace, rigorous mathematical analysis when analytical solutions are unavailable.

Memory effects, a hallmark of addiction, are the focus of a contribution by Akindeinde and colleagues, who construct a spatially heterogeneous time-fractional reaction and diffusion model describing the coupled dynamics of cocaine and heroin abuse. The persistence of addiction, delayed behavioral responses, and relapse are captured through Caputo fractional derivatives, which encode memory directly into the equations. To handle spatial variability, the authors employ both smoothly changing and multi-patch diffusion profiles. They prove the existence, uniqueness, positivity, and boundedness of solutions and obtain global stability conditions using Lyapunov-based functionals. To overcome the computational burden of fractional reaction-diffusion models, they develop a fractional version of the Parker-Sochacki method, an explicit and computationally efficient approximation that reduces the large amount of global memory usually associated with fractional-order models. Their numerical experiments show that lower fractional orders lead to longer persistence of substance abuse, while heterogeneous diffusion produces strongly non-uniform spatial distributions.

The collection then moves beyond differential equations altogether. Takane and colleagues present a Boolean-based logical digraph for representing and analyzing regulatory interactions in biological networks. The framework extends traditional excitation-inhibition representations by using Boolean logic to encode a broader variety of regulatory interactions and establishes a bidirectional relationship between regulatory networks, Boolean transfer functions, and logical digraphs. The resulting representation captures information on network connectivity, dynamics, limit cycles, and attractors, the stable long-term states toward which a system evolves. Applied to the neural network responsible for oscillatory swimming in the mollusc Tritonia, the framework shows how the spectral properties of the associated state matrix yield mathematical information about the dynamics expressed by attractors and limit cycles. Openly available software routines make the framework accessible for analyzing and visualizing biological regulatory networks.

Statistical estimation receives its own spotlight in a study by Basheer and colleagues, who address a persistent computational problem in truncated Poisson regression: missing zero counts can bias and destabilize standard parameter estimation. The team proposes a meta-heuristic method based on the Black Kite Optimization algorithm to optimize the likelihood function, sidestepping the limitations of conventional iterative estimation methods. Evaluated through extensive Monte Carlo simulations and a practical example based on real data across various sample sizes and covariate configurations, the proposed estimator achieved a reduction in mean squared error compared with conventional estimators. The result positions meta-heuristic optimization as a practical computational tool for truncated count-data models with applications in healthcare and epidemiology, and it highlights the increasingly interdisciplinary character of mathematical methods in the biological sciences.

The volume closes with an opinion piece by White and colleagues that pushes the discussion toward the interface of mechanics, control theory, and neuroscience. The authors argue that the principles of Lagrangian and Hamiltonian higher-derivative mechanics should be incorporated into Optimal Feedback Control Theory as a means of examining the constraints involved in human voluntary movement. As a conceptual contribution rather than a numerical or theoretical study, it extends the range of mathematical systems that can be applied to neural control and biological motor systems, and it points toward promising areas for further theoretical development. Taken together, the eight articles demonstrate that mathematical modeling is useful not merely for describing biological phenomena but for finding mechanisms, measuring interactions, testing ideas, and investigating ways to manage and control living systems, a frontier the editors hope will inspire continued collaboration across mathematics, computation, and the life sciences.

Subject of Research: Applications of advanced mathematical and computational modeling methods to biological and ecological systems

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

Article References: Lebelo, R. S., Akindeinde, S. O., & Appadu, A. R. (2026). Editorial: New frontiers in the application of mathematics to biological sciences. Frontiers in Applied Mathematics and Statistics, 12, Article 1986520. https://doi.org/10.3389/fams.2026.1986520

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1986520

Keywords: mathematical biology, mathematical modeling, disease transmission, fractional calculus, reaction-diffusion, predator-prey dynamics, hidden Markov models, biological networks, optimization, neural control, multiscale modeling, numerical methods