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Research   /   Research Areas Mathematical Biology

Mathematical biology is expanding and developing rapidly as scientists in biological sciences turn from descriptive experiments to more quantitative experiments. The diversity and complexity of living organisms means there are vastly more challenges for mathematicians to explain and predict biological systems through modeling.

Mathematical biology is a broad topic that can cover a large range of length scales, from the submicron lengths of DNA polymers to the kilometer length scales of migration patterns of animal herds.

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Biofluid Mechanics

Developmental biology, microbiology, neuroscience, population dynamics.

Research within ESAM involves the development of mathematical models of interesting biological systems, the development of new analytical and computational methods to solve these models, and interaction with experimental groups to verify the validity of the investigation. Specific areas of current research include biofilms (an aggregation of bacteria on solid surfaces surrounded by gas or liquid), vesicle and cell dynamics, and the dynamics of aneurysms.

• Michael J. Miksis
• Petia Vlahovska

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One example of this is the study of cancer. New models based, for example, upon the latest gene expression and high throughout sequencing data are producing new insights into some of the causes underlying this many-faceted disease.

• William Kath

In analogy with the morphogenesis of our planet, while the details of the current shape of continents depends on time and historical contingency, the mechanisms that drive tectonic plates and the flows in the Earth's core are far more general. What are the underlying principles and mechanisms that drives the emergence of organismal form? This is the central question driving our research.

Our work explores two avenues of research: 1) Solving data-driven, inverse-problems that allow us to make measurements of physical forces and chemical kinetics that experiments do not give direct access, and 2) in close collaboration with experimentalists, we combine measurements made in their labs with ours to guide the development of mathematical models that are phenomenological in nature and formalize our intuition for how the physical properties of polymers, cells, and tissues emerges from, and constrains, the biological process of interest.

Madhav Mani

• Niall Mangan

This is perhaps the most pressing challenge in quantitative biology and biomedicine, and groups in ESAM are using tools from statistics, machine learning, and statistical physics to build data-driven mathematical models to address it.

Microorganisms are complex systems unto themselves. They can exhibit significant adaptability to their environment caused by stresses such as starvation, competition from other species, or exposure to antibiotics. They are capable of locomotion, cell-to-cell communication, and coordination of both offensive and defensive strategies. The behavior of individual cells is regulated by expression of genes, which are often modeled by systems of ordinary differential equations. Often these systems incorporate elements of stochasticity to account for the non-determinant nature of these systems.

Colonies of microorganisms, taken as an aggregate, can be modeled using continuum models in the form of reaction-diffusion systems. These systems track the flow of nutrients and products in a larger environment.

Understanding microbiological systems is of critical importance to society. Most people are familiar with common pathogens that cause disease, but may not know that bacteria are also used to help clean waste water, produce household chemicals, and even produce energy. Mathematical modeling can aid in improving treatment of disease to reduce the possibility of antibacterial resistance and also streamline biochemical processes to improve energy or biochemical production beneficial to society.

• David Chopp

Problems such as these are being investigated by faculty and students within ESAM to understand the working of brain structures involved in the formation of memory and the processing of visual and olfactory information.

• Hermann Riecke

Many problems modeling the evolution of populations involve several different physical effects. These include:

• Birth and death rates as functions of the populations
• Intra-species competition, i.e., when the birthrate decreases as the population increases, due to crowding and competition for scarce resources
• Inter-species competition, when two or more species compete for the same resources
• Diffusive spreading
• Non-local interactions, i.e., the evolution of a population at a location depends not just on local conditions but also on resources in a neighborhood of the location

• Alvin Bayliss
• Vladimir Volpert

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Department of Engineering Sciences and Applied Mathematics McCormick School of Engineering and Applied Science 2145 Sheridan Road, Room M426 Evanston, IL 60208 Phone: 847-491-3345 Fax: 847-491-2178 Email Department

• © 2017

Topics in Mathematical Biology

• Karl Peter Hadeler 0

Universität Tübingen, Tübingen, Germany

You can also search for this author in PubMed   Google Scholar

• Written by a pioneer and expert in Mathematical Biology
• Analyzes the impact of quiescent phases in biology with mathematical models
• Presents classical mathematical biology models in detail with a focus on quiescence
• Casts new light on excitability of steady states, epidemic outbreaks, survival of the fittest and many more topics
• Holds in store many gems for the readers

Part of the book series: Lecture Notes on Mathematical Modelling in the Life Sciences (LMML)

12k Accesses

24 Citations

• Table of contents

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Table of contents (8 chapters)

Front matter, coupling and quiescence.

• Karl-Peter Hadeler

Delay and Age

Lotka–volterra and replicator systems, homogeneous systems, epidemic models, coupled movements, traveling fronts, back matter.

This book analyzes the impact of quiescent phases on biological models. Quiescence arises, for example, when moving individuals stop moving, hunting predators take a rest, infected individuals are isolated, or cells enter the quiescent compartment of the cell cycle. In the first chapter of Topics in Mathematical Biology general principles about coupled and quiescent systems are derived, including results on shrinking periodic orbits and stabilization of oscillations via quiescence. In subsequent chapters classical biological models are presented in detail and challenged by the introduction of quiescence. These models include delay equations, demographic models, age structured models, Lotka-Volterra systems, replicator systems, genetic models, game theory, Nash equilibria, evolutionary stable strategies, ecological models, epidemiological models, random walks and reaction-diffusion models. In each case we find new and interesting results such as stability of fixed points and/or periodic orbits, excitability of steady states, epidemic outbreaks, survival of the fittest, and speeds of invading fronts. 

The textbook is intended for graduate students and researchers in mathematical biology who have a solid background in linear algebra, differential equations and dynamical systems. Readers can find gems of unexpected beauty within these pages, and those who knew K.P. (as he was often called) well will likely feel his presence and hear him speaking to them as they read.

• 35Q92, 37N25, 92Bxx
• Mathematical Biology
• Quiescent states
• Population dynamics
• Epidemic models
• Reaction-diffusion equations
• Stability and Bifurcations
• Travelling Fronts

“This advanced textbook is well-suited for graduate students and researchers in mathematical biology with a solid background in mathematics, particularly linear algebra, differential equations and dynamical systems, and the material is put on a rigorous mathematical basis.” (W. Huyer, Monatshefte für Mathematik, Vol. 192 (4), August, 2020)

Karl Peter Hadeler

Book Title : Topics in Mathematical Biology

Authors : Karl Peter Hadeler

Series Title : Lecture Notes on Mathematical Modelling in the Life Sciences

DOI : https://doi.org/10.1007/978-3-319-65621-2

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer International Publishing AG 2017

Softcover ISBN : 978-3-319-65620-5 Published: 22 January 2018

eBook ISBN : 978-3-319-65621-2 Published: 20 December 2017

Series ISSN : 2193-4789

Series E-ISSN : 2193-4797

Edition Number : 1

Number of Pages : XIV, 353

Number of Illustrations : 26 b/w illustrations, 2 illustrations in colour

Topics : Mathematical and Computational Biology

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Editor’s Picks: Mathematical Biology

This collection showcases articles published in PLOS ONE that apply mathematics to biological questions. As the selection shows, this field is very broad, spanning topics such as angiogenesis, morphogenesis, cell dynamics and interactions, metabolism modelling, biochemical processes, and touches on all scales of biological organization. In this Editor’s Picks, PLOS ONE Associate Editor Carla Pegoraro highlights some of the latest research on mathematical and physical biology. We welcome submissions in this field and hope to grow our collection over time.

Image Credit: 10.1371/journal.pone.0149254

• Biological Systems
• Growth and Morphology
• Cell Dynamics
• Metabolism Modelling
• Biological Processes
• Molecular Modelling

Although traditional models of epidemic spreading focus on the number of infected, susceptible and recovered individuals, a lot of…

In this paper, we propose to use a linear system with switching methodology for description and analysis of complex biological systems. We…

Natural microbial communities contain hundreds to thousands of interacting species. For this reason, computational simulations are playing…

Ultrasound has many uses, such as in medical imaging, monitoring of crystallization, characterization of emulsions and suspensions, and…

• Image credit 10.1371/journal.pone.0149674 PLOS ONE Fuzzy Stochastic Petri Nets for Modeling Biological Systems with Uncertain Kinetic Parameters February 24, 2016 Fei Liu, Monika Heiner, Ming Yang
• Image credit 10.1371/journal.pone.0182385 PLOS ONE Modeling and simulation of biological systems using SPICE language August 7, 2017 Morgan Madec, Christophe Lallement, Jacques Haiech

Growth curve measurements are commonly used in microbiology, while the use of microplate readers for such measurements provides better…

During development of biological organisms, multiple complex structures are formed. In many instances, these structures need to exhibit a…

A systematic understanding of the evolution and growth dynamics of invasive solid tumors in response to different chemotherapy strategies…

• Image credit 10.1371/journal.pone.0083962 PLOS ONE Computational Modeling of 3D Tumor Growth and Angiogenesis for Chemotherapy Evaluation January 3, 2014 Lei Tang, Anne L. van de Ven, Dongmin Guo, Vivi Andasari, Vittorio Cristini, King C. Li, Xiaobo Zhou
• Image credit 10.1371/journal.pone.0159478 PLOS ONE Computational Screening of Tip and Stalk Cell Behavior Proposes a Role for Apelin Signaling in Sprout Progression November 9, 2016 Margriet M. Palm, Marchien G. Dallinga, Erik van Dijk, Ingeborg Klaassen, Reinier O. Schlingemann, Roeland M. H. Merks
• Image credit 10.1371/journal.pone.0166655 PLOS ONE Normal Wound Healing and Tumor Angiogenesis as a Game of Competitive Inhibition December 9, 2016 Irina Kareva, Abdo Abou-Slaybi, Oliver Dodd, Olga Dashevsky, Giannoula Lakka Klement
• Image credit 10.1371/journal.pone.0099116 PLOS ONE CellFIT: A Cellular Force-Inference Toolkit Using Curvilinear Cell Boundaries June 12, 2014 G. Wayne Brodland, Jim H. Veldhuis, Steven Kim, Matthew Perrone, David Mashburn, M. Shane Hutson
• Image credit 10.1371/journal.pone.0093207 PLOS ONE How Does a Single Cell Know When the Liver Has Reached Its Correct Size? April 1, 2014 Nadine Hohmann, Wei Weiwei, Uta Dahmen, Olaf Dirsch, Andreas Deutsch, Anja Voss-Böhme
• Image credit 10.1371/journal.pone.0054802 PLOS ONE The Flux-Based PIN Allocation Mechanism Can Generate Either Canalyzed or Diffuse Distribution Patterns Depending on Geometry and Boundary Conditions January 28, 2013 Michael Luke Walker, Etienne Farcot, Jan Traas, Christophe Godin
• Image credit 10.1371/journal.pone.0127972 PLOS ONE Emergent Stratification in Solid Tumors Selects for Reduced Cohesion of Tumor Cells: A Multi-Cell, Virtual-Tissue Model of Tumor Evolution Using CompuCell3D June 17, 2015 Maciej H. Swat, Gilberto L. Thomas, Abbas Shirinifard, Sherry G. Clendenon, James A. Glazier
• Image credit 10.1371/journal.pone.0085585 PLOS ONE The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion January 22, 2014 Alexander Bucksch, Greg Turk, Joshua S. Weitz
• Image credit 10.1371/journal.pone.0149254 PLOS ONE Spiral and Rotor Patterns Produced by Fairy Ring Fungi March 2, 2016 Nathaniel Karst, David Dralle, Sally Thompson
• Image credit 10.1371/journal.pone.0174946 PLOS ONE Turing mechanism underlying a branching model for lung morphogenesis April 4, 2017 Hui Xu, Mingzhu Sun, Xin Zhao
• Image credit 10.1371/journal.pone.0191089 PLOS ONE Cell death as a trigger for morphogenesis March 22, 2018 Boris Aguilar, Ahmadreza Ghaffarizadeh, Christopher D. Johnson, Gregory J. Podgorski, Ilya Shmulevich, Nicholas S. Flann

Cell shape change is one of the driving forces of animal morphogenesis, and the model organism Caenorhabditis elegans has played a…

Tumor associated angiogenesis is the development of new blood vessels in response to proteins secreted by tumor cells. These new blood…

A model for hematopoiesis is presented that explicitly includes the erythrocyte, granulocyte, and thrombocyte lineages and their common…

Cancer Stem Cells (CSC), a subset of cancer cells resembling normal stem cells with self-renewal and asymmetric division capabilities, are…

Extracellular recordings of neuronal cells are frequently a part of in vitro and in vivo experimental studies as a means of monitoring…

The human skeleton undergoes constant remodeling throughout the lifetime. Processes occurring on microscopic and molecular scales degrade…

• Image credit 10.1371/journal.pone.0165570 PLOS ONE Modelling Chemotactic Motion of Cells in Biological Tissues October 31, 2016 Bakhtier Vasiev
• Image credit 10.1371/journal.pone.0065864 PLOS ONE A Mechanochemical Model of Cell Reorientation on Substrates under Cyclic Stretch June 6, 2013 Jin Qian, Haipei Liu, Yuan Lin, Weiqiu Chen, Huajian Gao
• Image credit 10.1371/journal.pone.0104969 PLOS ONE Directional Collective Cell Migration Emerges as a Property of Cell Interactions September 2, 2014 Mae L. Woods, Carlos Carmona-Fontaine, Chris P. Barnes, Iain D. Couzin, Roberto Mayor, Karen M. Page
• Image credit 10.1371/journal.pone.0111896 PLOS ONE A Three-Dimensional Computational Model of Collagen Network Mechanics November 11, 2014 Byoungkoo Lee, Xin Zhou, Kristin Riching, Kevin W. Eliceiri, Patricia J. Keely, Scott A. Guelcher, Alissa M. Weaver, Yi Jiang
• Image credit 10.1371/journal.pone.0193975 PLOS ONE Distinguishing cell shoving mechanisms March 12, 2018 Pingyu Nan, Darragh M. Walsh, Kerry A. Landman, Barry D. Hughes
• Image credit 10.1371/journal.pone.0198420 PLOS ONE Mathematical modelling of reversible transition between quiescence and proliferation June 1, 2018 Nishtha Pandey, P. K. Vinod
• Image credit 10.1371/journal.pone.0150578 PLOS ONE The Combined Effect of Hydrophobic Mismatch and Bilayer Local Bending on the Regulation of Mechanosensitive Ion Channels March 9, 2016 Omid Bavi, Manouchehr Vossoughi, Reza Naghdabadi, Yousef Jamali
• Image credit 10.1371/journal.pone.0161915 PLOS ONE Spheroid Formation of Hepatocarcinoma Cells in Microwells: Experiments and Monte Carlo Simulations August 29, 2016 Yan Wang, Myung Hee Kim, Seyed R. Tabaei, Jae Hyeok Park, Kyuhwan Na, Seok Chung, Vladimir P. Zhdanov, Nam-Joon Cho

Metabolic models can estimate intrinsic product yields for microbial factories, but such frameworks struggle to predict cell performance…

The chloroplast is a central part of plant cells, as this is the organelle where the photosynthesis, fixation of inorganic carbon, and…

Methanogenic archaea occupy a functionally important niche in the gut microbial ecosystem of mammals. Our purpose was to quantitatively…

• Image credit 10.1371/journal.pone.0208282 CC BY 4.0 PLOS ONE A quantitative model of nitrogen fixation in the presence of ammonium November 29, 2021

High-level behaviour of metabolic systems results from the properties of, and interactions between, numerous molecular components.…

Computational models of normal liver function and xenobiotic induced liver damage are increasingly being used to interpret in vitro and in…

• Image credit 10.1371/journal.pone.0064567 PLOS ONE Community Flux Balance Analysis for Microbial Consortia at Balanced Growth May 31, 2013 Ruchir A. Khandelwal, Brett G. Olivier, Wilfred F. M. Röling, Bas Teusink, Frank J. Bruggeman
• Image credit 10.1371/journal.pone.0162428 PLOS ONE A Liver-Centric Multiscale Modeling Framework for Xenobiotics September 16, 2016 James P. Sluka, Xiao Fu, Maciej Swat, Julio M. Belmonte, Alin Cosmanescu, Sherry G. Clendenon, John F. Wambaugh, James A. Glazier
• Image credit 10.1371/journal.pone.0103582 PLOS ONE Simulating Metabolism with Statistical Thermodynamics August 4, 2014 William R. Cannon
• Image credit 10.1371/journal.pone.0147651 PLOS ONE Simulating Serial-Target Antibacterial Drug Synergies Using Flux Balance Analysis January 28, 2016 Andrew S. Krueger, Christian Munck, Gautam Dantas, George M. Church, James Galagan, Joseph Lehár, Morten O. A. Sommer
• Image credit 10.1371/journal.pone.0182570 PLOS ONE Parametric studies of metabolic cooperativity in Escherichia coli colonies: Strain and geometric confinement effects August 18, 2017 Joseph R. Peterson, John A. Cole, Zaida Luthey-Schulten

The circadian rhythm functions as a master clock that regulates many physiological processes in humans including sleep, metabolism,…

Many animals are able to sense the earth’s magnetic field, including varieties of arthropods and members of all major vertebrate groups.…

Animal movement drives important ecological processes such as migration and the spread of infectious disease. Current approaches to…

Collective behaviour is of fundamental importance in the life sciences, where it appears at levels of biological complexity from single…

Collective phenomena are studied in a range of contexts—from controlling locust plagues to efficiently evacuating stadiums—but the central…

• Image credit 10.1371/journal.pone.0128111 PLOS ONE Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply June 18, 2015 Aimee M. Torres Rojas, Alejandro Meza Romero, Ignacio Pagonabarraga, Rui D. M. Travasso, Eugenia Corvera Poiré
• Image credit 10.1371/journal.pone.0153297 PLOS ONE A Simple, Realistic Stochastic Model of Gastric Emptying April 8, 2016 Jiraphat Yokrattanasak, Andrea De Gaetano, Simona Panunzi, Pairote Satiracoo, Wayne M. Lawton, Yongwimon Lenbury
• Image credit 10.1371/journal.pone.0145309 PLOS ONE A Spatially Continuous Model of Carbohydrate Digestion and Transport Processes in the Colon December 17, 2015 Arun S. Moorthy, Stephen P. J. Brooks, Martin Kalmokoff, Hermann J. Eberl

A method for predicting HIV drug resistance by using genotypes would greatly assist in selecting appropriate combinations of antiviral…

Molecular clustering at the plasma membrane has long been identified as a key process and is associated with regulating signalling…

Enzymes play a critical role in a wide array of industrial, medical, and research applications and with the recent explosion of genomic…

• Image credit 10.1371/journal.pone.0172072 PLOS ONE Accelerated flexible protein-ligand docking using Hamiltonian replica exchange with a repulsive biasing potential February 16, 2017 Katja Ostermeir, Martin Zacharias
• Image credit 10.1371/journal.pone.0176229 PLOS ONE Molecular simulations and Markov state modeling reveal the structural diversity and dynamics of a theophylline-binding RNA aptamer in its unbound state April 24, 2017 Becka M. Warfield, Peter C. Anderson

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Mathematical biology, group overview.

The Mathematical Biology Group at UBC is an interdisciplinary research group that applies mathematics in a wide range of biological fields including immunology, epidemiology, molecular, cell and developmental biology, electrophysiology and neuroscience, ecology, game theory and evolution. Specific areas of focus include protein interactions and protein mobility, regulation of gene expression, cell behaviour, tissue mechanics, immune responses, epidemic dynamics, bacterial cell division, the evolution of cooperation, game theory, speciation, and social aggregation (swarming). The group consists of a large group of core faculty members, some with cross appointments, and many affiliated faculty in mathematics and other departments, and a good number of postdoctoral researchers and graduate students. It is one of Canada’s largest and most established math biology groups.

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We teach undergraduate and graduate courses in mathematical biology. Graduate students receive degrees (Master's and PhD) as members of the UBC Institute of Applied Mathematics. We are always interested to hear from potential graduate students and postdoctoral fellows. We recruit from many different backgrounds, including mathematics, physics, chemistry, computer science, statistics, engineering and the biological sciences. We often supervise undergraduate thesis projects (for instance, from the Biophysics and Integrated Science programs) and take on undergraduate students as volunteers ( REX ) and through summer award programs ( NSERC USRA and WLI URA ).  

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Mathematical Biology

Lee Segel, an eminent applied mathematician and a founder of modern mathematical biology, observed that “mathematical biology sounds like a narrow specialty, but in fact it encompasses all of biology and most of mathematics”. Much current research is driven by the flood of “big data” (next-generation genomic and whole-transcriptome sequencing, dozens of environmental variables measured globally in nearly continuous time) and “small data” (e.g., individual ion channels in nerve cells, motion-tracking of insect wings) that open new phenomena to quantitative modeling, and by the increasing challenges of biosphere sustainability.

Mathematics plays an important role at all levels of biological organization and the full range is represented in CAM, including:

•     Identifying the structure and parameters of biochemical and genetic networks;
•     Identifying genes associated with complex diseases and differences among individuals;
•     How do insects walk, hover, and turn in flight;
•     Biomechanics of muscular tissue and effects of muscular degeneration;
•     How infectious diseases spread on contact networks;
•     How rapid evolution affects population and ecosystem dynamics;

    Finding improved solutions to conservation, environmental and sustainability problems such as nature reserve design, invasive species management, pest and disease management in agriculture, environmental remediation and carbon sequestration.

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Department of Mathematics

Mathematical biology.

Mathematical biology is a rich and diverse field at the intersection of mathematics and biology. Our faculty apply mathematical techniques to important biological problems in collaboration with students and faculty from several ecological and biological departments from across Oregon State University and the world. Research topics revolve around the mathematics of infectious diseases, deterministic and stochastic models of population dynamics, spatial ecology and genomics, among others.

Mathematical biology research faculty

Vrushali A. Bokil

Interim Dean – College of Science, Professor

Professor Bokil's general research interests are in applied mathematics, scientific computing, numerical analysis and mathematical biology. Her primary research interests are in the numerical solution of wave propagation problems. Specifically, she has conducted research on the numerical solution of Maxwell's equations using a variety of finite difference and finite element methods. Bokil is also working on several problems in mathematical ecology which involve the construction and analyses of deterministic and stochastic models for applications in population dynamics, epidemiology and spatial ecology.

View Vrushali's directory profile 

Benjamin D. Dalziel

Assistant Professor

I have broad interests in ecological and evolutionary dynamics, particularly related to population health, and to the maintenance of biodiversity. I am particularly interested in (i) the ecology and evolution of infectious diseases, especially the impact of host population structure on pathogen spread and diversification and (ii) how collective behavior affects trophic interactions and ecosystem stability. To address these questions I work with students and collaborators at the interface of mathematical models and empirical data.

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Patrick De Leenheer

My research interests are in mathematical biology and dynamical systems. I've worked on problems in ecology, epidemiology, bioreactors, chemical and biochemcial reaction networks, social networks, physics and graphs. Some of this work has been in collaboration with natural scientists. Most of the models I work on are examples of dynamical systems (ODEs, PDEs, linear and nonlinear maps,etc) and I'm interested in both theoretical and applied problems arising in this context. Occasionally, I use ideas and methods from control theory, especially when there is a need to modify or influence the behavior of a given dynamical system.

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Chad Giusti

Research focus: Topological neuroscience; computation and coding in neural systems; development of topological, algebraic, and geometric methods for scientific applications; topology of spaces of configurations and embeddings

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Yevgeniy Kovchegov

Yevgeniy Kovchegov works in the field of probability and stochastic processes. His research interests include the mathematical models of statistical mechanics, interacting particle systems, models of mathematical biology, stochastic self-similarity, and quantum computation. Kovchegov's work is centered around the following topics: self-similar random trees; extending the probabilistic coupling method; orthogonal polynomials in stochastic processes, probability and statistics; mixing times; quantum walks and quantum computation; chaos and fractals; applications of probability theory in biological systems, geomorphology, computer and wireless networks, network coding, and environmental sciences.

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Swati Patel

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Enrique A. Thomann

Thomann's research is primarily in problems in partial differential equations arising from fluid mechanics. He also collaborates with colleagues in other departments, as well as in the Mathematics Department, in the development of mathematical models to problems in Ecology, Oceanography, Hydrology and management of Natural Resources.

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Courses in mathematical biology.

• MTH 427/527 Introduction to Mathematical Biology
• MTH 428/528 Stochastic Elements in Mathematical Biology

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Editorial article, editorial: insights in mathematical biology 2022.

• Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UMR CNRS 6623, Besançon, France

Editorial on the Research Topic Insights in mathematical biology 2022

This Insights in Mathematical Biology Research Topic aimed to showcase some new developments and future research perspectives in the mathematical biology field. The three articles published in this topic are a very small sample of some of the current interests in mathematical biology, from the optimization of food production to the optimization of numerical simulations for complex mathematical models describing medical problems and the understanding of multi-scale molecular–cellular aspects in health and disease. These articles also emphasize some of the current approaches in mathematical biology that oppose single-scale vs. multi-scale models, simple vs. complex models, and fast vs. slow numerical simulations.

In the following, we briefly summarize these three papers.

O'Hare focused on maximizing food production in a sustainable manner. To investigate the competitive behavior between two different types of food producers (one where farmers harvest and make profit at the end of the season and a second one where farmers harvest and make profit continuously throughout the season), the author considered a simple individual-based framework. This framework was applied to 25 farms distributed on a spatial grid, which could interact with neighboring farms and produce food using either of the two food-producing systems. The interaction rule between the different farms was given by a simple algebraic equation that considered the distance between farms (which could belong to different farm networks: square grid, linear network, circular network, and random network) and the time since the farmers started to implement actions that could impact neighboring farms. The profit obtained by each farmer (calculated using an integral equation) depended on the running cost of each farmer to produce their crops, the price at which the crop is sold, and on the interaction rule between the different farms. The solution of this integral equation showed that the profit obtained through farming was calculated for all farms, and the results showed that profit depended on the farm's position in the landscape. Moreover, neighboring farms that produced similar crops could benefit from others' treatment of their crops, while neighboring farms that produced different crops had a negative impact on each other. The study also modeled the impact of connections between different farms (connections by a road or a river) and investigated the level of landscape pollution at various farms and how this was transported toward other connected farms. The toy individual-based model developed in this study to calculate the profit for each farmer (with its farm belonging to a spatially distributed network of farms) showed that even simple mathematical models could be used to generate new testable hypotheses for complex ecological problems. However, such toy models will have to be modified in the future to describe specific landscape conditions, specific food producers, specific pollutants (pesticides) use, and their dispersion characteristics to neighboring farms.

Egberts et al. focused on a different type of optimization: the optimization of numerical simulations in the context of a very complex mathematical model for wound healing. The model, described by nine non-linear partial differential equations, focused on the spatio-temporal interactions between different cell types (fibroblasts, myofibroblasts), collagen, and chemicals during post-burn tissue contraction. To improve the performance of the finite element numerical simulations of this 2D spatial model for cell- and tissue-scale dynamics of wound healing, the authors considered a feed-forward neural network (with two hidden layers) trained and then tested on 2D finite element simulations. The resulting optimized network was used to predict the wound boundary contraction and the total strain energy density. It was also suggested that the fast numerical simulations could be eventually integrated into medical practice through the development of apps that consider age-related parameters to predict wound contractures.

Hodgkinson et al. focused on a biological aspect underlying many medical problems: the molecular and cell-scale dynamics behind the regulation of cell cycle via a family of proteins called cyclin-dependent kinases and the p53 gene. The study presented two mathematical models described by integro-partial differential equations: the first one focused on the quantity of cyclins across cell cycle at the level of one single cell, and the second one focused on the role of cyclin, the p53 protein, and the mdm2 protein (which degrades p53) in cell-cycle dynamics at the level of whole-cell populations. In the first model, the variables describing the amount of S-cyclin and M-cyclin in a cell (where the S-cyclins and M-cyclins denoted the cyclins synthesized in the S and M phases of cell cycle) depended on time and space. In the second model, the variables describing the densities of cells in the various phases of cell cycle (S, G, M) depended on time, cyclin level synthesized in various phases, cell damage level, and p53 and mdm2 levels. Numerical simulations (using a finite difference scheme together with a mid-point integration method for integrals) with the first model showed oscillatory dynamics between the S-cyclin and M-cyclin states. Numerical simulations with the second model (using a finite difference scheme based on a predictor–corrector MacCormack scheme for time integration and central differences for the spatial gradients) showed the distributions of cell populations in S and M phases across the (cyclin, damage) space and the distribution of the cell population in the G phase across the (cyclin, damage) space, the (p53, damage) space, and the (p53, mdm2) space for various parameter values. Due to the complexity of this second model in [2] and the computational limitations on the number of dimensions that could be simulated simultaneously at a resolution that could allow enough accuracy, this second model did not include any spatial cell structure. This is an open problem that will have to be addressed in the future.

In summary, the articles in this Research Topic highlight some of the current open problems in cell biology, medicine, and ecology. All these articles also mention the issue of parameterizing these models (either simple or complex) with real single-scale and multi-scale data, which is mainly due to the lack of such data.

Author contributions

The author confirms being the sole contributor of this work and has approved it for publication.

RE acknowledges support from a French Agence Nationale de Recherche (ANR) grant number ANR-21-CE45-0025-01.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Keywords: developments in mathematical biology, optimization, food production, neural networks and finite element, structured cell populations

Citation: Eftimie R (2023) Editorial: Insights in mathematical biology 2022. Front. Appl. Math. Stat. 9:1197661. doi: 10.3389/fams.2023.1197661

Received: 31 March 2023; Accepted: 30 May 2023; Published: 13 June 2023.

Reviewed by:

Copyright © 2023 Eftimie. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Raluca Eftimie, raluca.eftimie@univ-fcomte.fr

This article is part of the Research Topic

Insights in Mathematical Biology 2022

Mathematical Biology Research Topics Ideas [MS PhD]

List of Research Topics and Ideas of Mathematical Biology for MS and Ph.D. Thesis.

• Collaborative Workshop for Women in Mathematical Biology
• Disentangling biology from mathematical necessity in twentieth-century gymnosperm resilience trends
• Mathematical Modeling in Biology. Part 1. Dynamic Models of Primary Photosynthesis Processes
• A Fractional Epidemic Model with Mittag-Leffler Kernel for COVID-19
• Mathematical analysis of the spread of SIQR model with Caputo fractional order derivative
• Preface of the special issue
• Biology, Physics and Nonlinear Science
• A mathematical model to examine the effect of quarantine on the spread of coronavirus
• An Analytical Scheme to Characterise the Mathematical Discourse of Biology Tasks
• Prospect for application of mathematical models in combination cancer treatments
• Predictions Based on Mathematical Model of COVID-19 in India: A Case Study of India
• Mathematical Approach to Assess the Severity of Road Accidents in Bangladesh Using a SEIR-type Model
• Fractional Mathematical Oncology: On the potential of non-integer order calculus applied to interdisciplinary models
• … of Mathematical Language Ability and Parental Supports on Students’ Academic Achievement in Secondary School Sciences (Physics, Chemistry and Biology) …
• Correction to: Mathematical Model of Chronic Dermal Wounds in Diabetes and Obesity
• Modelling Cell Invasion: A Review of What JD Murray and the Embryo Can Teach Us
• Controlling the Spatial Spread of a Xylella Epidemic
• A mathematical model for the cannabis epidemic in a South African province with a non-linear incidence rate
• Mathematical analysis for a time-delayed Leukemia model with treatment
• A systematic search for switch-like behavior in type II toxin–antitoxin systems
• Mathematical modeling of COVID-19 epidemic with effect of awareness programs
• Fitting Personalized Mechanistic Mathematical Models of Acute Myeloid Leukaemia to Clinical Patient Data.
• Awareness and Control of Smoking Habit through Education: A Mathematical Model
• A New Mathematical Model for Controlling Tumor Growth Based on Microenvironment Acidity and Oxygen Concentration
• Population models, mathematical epidemiology, and the COVID-19 pandemic
• The role of RdCVFL in a mathematical model of photoreceptor interactions
• A robust mathematical model of adaxial–abaxial patterning
• Gain and Loss of Function mutations in biological chemical reaction networks: a mathematical model with application to colorectal cancer cells
• Mathematical modeling of population dynamics based on recurrent equations: results and prospects. Part I
• Patient-Specific Modelling of Blood Coagulation
• Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic
• A novel mathematical model of heterogeneous cell proliferation
• On mathematical analysis of a discrete electrical lattice with nonlinear dispersion
• A Mathematical Modeling Study: Assessing Impact of Mismatch Between Influenza Vaccine Strains and Circulating Strains in Hajj
• Correction to: Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model
• Towards a mathematical understanding of colonization resistance
• Generalized Euler-Lotka equation for correlated cell divisions
• Bespoke Turing Systems
• Invading and receding sharp-fronted travelling waves
• Mathematical modeling of COVID-19 spreading with asymptomatic infected and interacting peoples
• Mathematical model of COVID-19 transmission in the presence of waning immunity
• Stochastic dynamics of hepatitis B epidemics
• On the Mathematical Modelling of Competitive Invasive Weed Dynamics
• Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics
• Dynamics of mathematical model interaction of living populations in making biogas as an alternative of renewable energy
• Competitive exclusion in a general multi-species chemostat model with stochastic perturbations
• Antibody Dynamics for Plasmodium vivax Malaria: A Mathematical Model
• Mathematical analysis of a two-strain disease model with amplification
• A competition model in the chemostat with allelopathy and substrate inhibition
• Harnessing Social Media in the Modelling of Pandemics—Challenges and Opportunities
• Mathematical modelling of proteome constraints within metabolism
• Are There Any Parameters Missing in the Mathematical Models Applied in the Process of Spreading COVID-19?
• Mathematical and simulation methods for deriving extinction thresholds in spatial and stochastic models of interacting agents
• A Two-Dimensional Model of Potassium Signaling and Oscillatory Growth in a Biofilm
• Mathematical modeling and mechanisms of pattern formation in ecological systems: a review
• Mathematical Analysis of A Tuberculosis-Lymphatic filariasis Co-infection Model
• Modeling LSD1-Mediated Tumor Stagnation
• Isolating Patterns in Open Reaction–Diffusion Systems
• A mathematical model for the population dynamics of Dengue multiserotype interactions
• Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics
• Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students
• Mathematical Transmission of HIV/AID with Early Treatment
• A Novel Mathematical Model for COVID-19 with Remedial Strategies
• 28 Models Later: Model Competition and the Zombie Apocalypse
• Collective Oscillations in Coupled-Cell Systems
• A Dynamic Model to Assess Human Papillomavirus Vaccination Strategies in a Heterosexual Population Combined with Men Who have Sex with Men
• Development of the mathematical model of catalytic oxidation-reduction reaction kinetics for wastewater treatment technology using vortex layer devices
• Mathematical Models of Morphogen Gradients and Growth Control
• Dilafruz Nishonova Mathematical Assessment of the Risk of Developing Dysfunctions of Autonomic and Thyroid Statuses from the Point of View of Evidence-Based …
• Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs
• Topological data analysis approaches to uncovering the timing of ring structure onset in filamentous networks
• Modelling the spread of atherosclerosis considering relapse and linear treatment
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• Reconstructing the Intrinsic Statistical Properties of Intermittent Locomotion Through Corrections for Boundary Effects
• A quasi-equilibrium reduced model of pancreatic insulin secretion
• Tools for time-course simulation in systems biology: a brief overview
• The structure of infinitesimal homeostasis in input–output networks
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• Exploring effects of DNA methylation and gene expression on pan-cancer drug response by mathematical models
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• Analysis of the Parametric Correlation in Mathematical Modeling of In Vitro Glioblastoma Evolution Using Copulas
• Systems Biology Analysis for Ewing Sarcoma
• Where to cut to delay a pandemic with minimum disruption? Mathematical analysis based on the SIS model
• Risk of sexually transmitted Zika virus in a cohort of economically disadvantaged urban residents
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• Law of mass action and saturation in SIR model with application to Coronavirus modelling
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• A primer on using mathematics to understand COVID-19 dynamics: Modeling, analysis and simulations
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• A Mathematical Model of Contact Tracing during the 2014–2016 West African Ebola Outbreak
• A fragmentation phenomenon for a nonenergetic optimal control problem: Optimization of the total population size in logistic diffusive models
• Mathematical and Statistical Analysis of Doubling Times to Investigate the Early Spread of Epidemics: Application to the COVID-19 Pandemic
• Mathematical Modeling of Retinal Degeneration: Aerobic Glycolysis in a Single Cone
• Establishment of mathematical model for minimization of human energy in a plastic moulding operation
• The Schrödinger-KdV equation of fractional order with Mittag-Leffler nonsingular kernel
• Reduced biomechanical models for precision-cut lung-slice stretching experiments
• Setting the basis of best practices and standards for curation and annotation of logical models in biology—highlights of the [BC] 2 2019 CoLoMoTo/SysMod Workshop
• Growth curve: an intelligent life history described by a mathematical model using the Practical Program for Forces Modeling spreadsheet
• Calcium-dependent insulin resistance in hepatocytes: mathematical model
• The discrete-time Kolmogorov systems with historic behavior
• Towards a replicator dynamics model of age structured populations
• Dimensionality affects extinction of bistable populations
• Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India
• Towards a Framework of Mathematical Competencies in China
• (Non) local logistic equations with Neumann conditions
• Within-host bacterial growth dynamics with both mutation and horizontal gene transfer
• Mathematical modeling of endothelial network
• Impact of Media-Induced Fear on the Control of COVID-19 Outbreak: A Mathematical Study
• Assisting K-12 Teachers to Make the Connection between Computational Thinking in Cybersecurity Unplugged Activities and Mathematical Mindset through A …
• Equation-based models of wound healing and collective cell migration
• [PS][PS] Density dependent transmission from process algebra models of disease spread
• Effects of global warming on pattern dynamics of vegetation: Wuwei in China as a case
• Mathematical Modeling and Optimal Control of Complex Epidemiological Networks
• The Historiography of Modern Evolutionary Biology
• Statistics of nascent and mature rna fluctuations in a stochastic model of transcriptional initiation, elongation, pausing, and termination
• Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media
• An efficient implementation of legofit software to infer demographic histories from population genetic data
• Modelling biological control dynamics of the banana weevil Cosmopolites sordidus (Germar) by generalist predators in a banana agroecosystem
• Reproducing country-wide COVID-19 dynamics can require the usage of a set of SIR systems
• Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis
• Separation of timescales for the seed bank diffusion and its jump-diffusion limit
• Mathematical modeling of multiple pathways in colorectal carcinogenesis using dynamical systems with Kronecker structure
• Density dependent diffusion models for the interaction of particle ensembles with boundaries
• On the effect of antiresorptive drugs on the bone remodeling of the mandible after dental implantation: a mathematical model
• Mathematical modeling with optimal control analysis of social media addiction
• On structural and practical identifiability
• Transmission dynamics of COVID-19 in Nepal: Mathematical model uncovering effective controls
• Optimising reactive disease management using spatially explicit models at the landscape scale
• S4: Detailed analysis of the mathematical model
• The role of children in the spread of COVID-19: Using household data from Bnei Brak, Israel, to estimate the relative susceptibility and infectivity of children
• Evolution of anisotropic diffusion in two-dimensional heterogeneous environments
• Mathematical knowledge of elementary education student teachers: variable analysis
• Exact maximal reduction of stochastic reaction networks by species lumping
• Mathematical modeling of ovarian follicle development: A population dynamics viewpoint
• Predicting the results of competition between two breast cancer lines grown in 3-D spheroid culture
• Spatial correlations constrain cellular lifespan and pattern formation in corneal epithelium homeostasis
• Quantum Mathematical Integrated Information Theory: From the Classical Version
• Mathematical modelling and numerical bifurcation analysis of inbreeding and interdisciplinarity dynamics in academia
• Dynamic modelling of covid-19 and the use of “Merah Putih” vaccination and herbal medicine treatment as optimal control strategies in Semarang city Indonesia
• Stability analysis of a novel Delay Differential Equation of HIV Infection of CD4 T-cells
• Mathematical modeling of cavity development in lung tuberculosis
• Correction to: Modeling the Effect of HIV/AIDS Stigma on HIV Infection Dynamics in Kenya
• Optimal Control for Tuberculosis with Exogenous Reinfection and Stigmatization
• Iterative methods for solving Riccati differential equations
• Validity of Mathematical Modelling Assessments for Interdisciplinary Learning
• Limit cycles and global dynamics of planar piecewise linear refracting systems of focus–focus type
• Mathematical modeling of plasticity and heterogeneity in EMT
• Biomedical Image Processing and Classification
• The Case for Biocalculus: Improving Student Understanding of the Utility Value of Mathematics to Biology and Affect toward Mathematics
• Non-random patterns of chytrid infections on phytoplankton host cells: mathematical and chemical ecology approaches
• Numerical approach to chaotic pattern formation in diffusive predator–prey system with Caputo fractional operator
• Mathematical Model of Development of an Innovative Education System
• Pattern formations of a delayed diffusive predator–prey model with predator harvesting and prey social behavior
• Computational method for approximating the behaviour of a triopoly: an application to the mobile telecommunications sector in Greece
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• Fractal-fractional order dynamical behavior of an HIV/AIDS epidemic mathematical model
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• [PS][PS] für Angewandte Analysis und Stochastik
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ABOUT THE CENTER

The Center for Mathematical Biology is the focal point for interdisciplinary research in mathematics and biology at the University of Pennsylvania. The research interests of the core members of Center range from ecology and evolutionary genetics to physiology and biophysics, on the one hand, and game theory, probability, partial differential equations and numerical analysis on the other.

The Simons Postdoctoral Fellows are a central part of the Center. The fellows are intellectually independent, but work together with core members. Short and long-term visitors are invited as speakers in a regular seminar series, as well as to participate in focused workshops on research topics of current interest.

Major funding for the Center comes from the Math + X grant awarded to the University of Pennsylvania through the Simons Foundation.

Mathematics

Fellowships, co-directors.

Dr. Mori works in mathematical biophysics and physiology. In particular, I am interested in how ionic electrodiffusion and fluid mechanics, and soft condensed matter physics more generally, shape physiological responses such as cell motility, cell polarization and (electrical) signal propagation. The study of such problems lead naturally to interesting and often novel problems in the analysis and numerical analysis of (partial) differential equations, which is also an important aspect of my research program.

Dr. Plotkin’s group uses mathematics and computation to study questions in evolutionary biology and ecology. Research in the group is concerned primarily with adaptation in populations. Related interests include the evolution of robustness and adaptability, the evolutionary ecology of viral populations, the evolution of cooperation, conflict, coalitions, and group decision-making.

CORE MEMBERS

Dr. Akçay studies the theory of social evolution in context ranging from microbial ecology to human cultural evolution. He uses a combination of mathematical modeling, agent-based simulations, and comparative analyses to understand the evolutionary dynamics of social phenomena across the tree of life. Within this broad area, his current interests include the co-evolution of social structure, dynamics of cumulative culture and ecological adaptation in human societies, evolution of social norms, and evolution and ecology of symbioses, among other topics. 

The Goulian lab is broadly interested in the regulatory circuits that enable bacteria to sense and respond to their environment. He develops simple mathematical models to understand how these systems function and to develop testable predictions to guide experiments. He is also interested in how regulatory circuits evolve and their natural variation within and between species.

Dr. Hynd’s primary interest is in partial differential equations which arise in physical and phenomenological models.  I’m particularly interested in problems involving geometry, probability and also especially optimization. I usually employ mathematical analysis in attempts to uncover properties of solutions. On occasion, I’ll also use numerical methods to approximate related quantities of interest.

Dr. Katifori’s group is generally interested in understanding the geometrical and topological principles governing the form and function of living organisms. They primarily focus on theoretical questions inspired by and related to biological transport networks such as the mammalian and plant vasculature. The group tries to understand how living flow networks function, how they develop, what determines their structure and to what extent evolution has driven them to optimality.

Dr, Kim is interested in models of development and evolution of development; geometry of data analysis; and, graphical models. He has worked on mathematical properties of tree-graph models for phylogenies, statistics of spatial processes and geometrical shape, geometrical representation of biological data, and developmental dynamics. He also carries out empirical research in single cell biology applied to problems in cell differentiation and cell phenotypes.

The Mathijssen lab is interested in exploring the physics of life: we combine experimental and theoretical techniques across the disciplines of physics and biology. Our main goals are to unravel the physics of pathogens, to design biomedical materials, and understand the collective functionality of living systems out of equilibrium. Recent themes include ultra-fast biology and hydrodynamic communication (Nature 2019), pathogen clearance in the airways (Nature Physics 2020), and bacterial contamination dynamics (Nature Communications 2019), and designing microrobotic active carpets (Advanced Science 2021).

Robin Pemantle does research in probability and combinatorics.Within probability, Pemantle works on a variety of discreteprobability models, e.g., recently, error correcting in noisy channels, invasion percolation, fractal trees, fault detection, random fitness landscapes. In combinatorics, Pemantle works on ACSV (analytic combinatoricsin several variables), an enumeration technique which has been applied to diverse areas such as random walks, quantum walks, lattice tilings and search trees.

Dr. Strain works in the field of mathematical analysis and studies partial differential equations. He has proven results on partial differential equations from diverse areas including fluid dynamics, kinetic theory, and materials science. Strain does research on problems involving local and global existence and uniqueness of solutions, large time sharp asymptotic behavior and convergence to equilibrium, finite time blow up, and ill-posedness of solutions. He has studied numerous physically motivated partial differential equations including the incompressible Navier-Stokes equations, the relativistic Euler system, the Muskat problem, the Boltzmann and Landau equation under Newtonian mechanics or special-relativity and the Vlasov equations.

Dr. Tishkoff studies genomic and phenotypic variation in ethnically diverse Africans. Her research combines field work, laboratory research, and computational methods to examine African population history, human adaptation, and the genetic basis of variable complex traits including disease risk.  She uses an integrative genomics approach, incorporating data from genomics, transcriptomics, epigenomics, metabolomics, and the gut microbiome to identify the role of genetics and environment on variable traits in human populations. 

[email protected]

Center for Mathematical Biology School of Arts & Sciences University of Pennsylvania Philadelphia, PA 19104 USA