News

Formation and Growth of Co-Culture Tumour Spheroids: New Compartment-Based Mathematical Models and Experiments

by  Ryan J. Murphy (University of Melbourne), Gency Gunasingh (Frazer Institute, University of Queensland), Nikolas K. Haass (Frazer Institute, University of Queensland), Matthew J. Simpson (Queensland University of Technology)

Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and develop anti-cancer therapies. However, they are challenging to interpret as they are composed of two or more cell types that undergo multiple biological processes on overlapping timescales. In this study, we interpret new co-culture spheroid experimental data using Greenspan’s seminal monoculture model and multiple new and natural two-population extensions of Greenspan’s model. This allows us to reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. The mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types. Further, the new class of compartment-based mathematical models, which includes Greenspan-type models as a special case, provide opportunities for further mathematical and biological insights. 

Dr Ryan J. Murphy performed the mathematical and statistical modelling. Ms Gency Gunasingh performed the experimental work. Professor Nikolas K. Haass and Professor Matthew J. Simpson contributed equally.

VisualPDE: Rapid Interactive Simulations of Partial Differential Equations

by Benjamin J. Walker, Adam K. Townsend, Alexander K. Chudasama & Andrew L. Krause

Mathematical biology and other areas of science are employing increasingly complex models that take the form of partial differential equations. Such models can exhibit a rich set of behaviours, including those that defy intuition, such as diffusion-driven pattern formation. In this paper we present VisualPDE, a web-based tool enabling  real-time interactive exploration of such models. We feel that such interactive 'play' is an incredibly important and under-utilized way to develop intuition and build deep understanding of these models.

The paper opens by saying that a reader should simply go and play on the website, VisualPDE.com, themselves. It then outlines our rationale for developing this tool, some of the technical and design issues we faced, as well as some of the examples and use-cases we have already explored. The structure of this paper supplements the 'living' website with things we think a reader might find interesting, particularly around the wider context and technical aspects of designing the website. We hope it helps the wider community deepen our understanding of PDE models, and rethink how we teach and do research using mathematics more generally.

Image caption: This is an interactive simulation on the website which can also be viewed at https://visualpde.com/sim/?preset=BMB . It uses spatial heterogeneity to force a reaction-diffusion system to exhibit both a complex prepattern and emergent spot-like patterns in different parts of the domain.

Scalable Gromov-Wasserstein Based Comparison of Biological Time Series

by Natalia Kravtsova, Reginald L McGee and Adriana T Dawes

Read the paper

In this paper, we introduce a rigorous and powerful method for comparing time series data using a novel and computationally efficient modification of the Gromov-Wasserstein optimal transport distance. In brief, our method, which we denote GW$_{\tau}$, views each trajectory as a separate metric space and compares these metric spaces via optimal transport. This feature of our method makes it exceptionally flexible in the types and size of data sets that can be compared, including data sets that occur on different time scales, are missing measurements, or even lie in spaces of different dimensions. Its rigorously demonstrated properties show a clear increase in efficiency and accuracy over other methods and using a variety of data. Using our method, we show that averaging time series using recently proposed Fused Gromov-Wasserstein barycenters provides more reliable average trajectories compared to the most commonly used mean trajectories. Our easily implemented and fast GW$_{\tau}$ method can be applied to a wide range of time series data, from cell biology to ecology, and allows for new comparisons and quantifications that preserve key features in the data sets.

Natalia Kravtsova, a student author, led the research in this paper, including formal analysis, methodology, and visualization. Prof. McGee and Prof. Dawes provided supervision. All three authors contributed to the formulation and conceptualization of the research, and manuscript preparation.

Alys Clark (University of Auckland), Sara Loo (Johns Hopkins University), Fiona R. Macfarlane (University of St Andrews), and Thomas Woolley (Cardiff University).

1. People – Interviews with Dr Adrianne Jenner and Dr Michael Watson.
2. Editorial – on 'Research that is worth a thousand words: Visualising a conference' on the theme of conference illustration.
3. Featured Figures – Highlighting the research by early career researcher Chloé Colson and highlighting the most accessed paper from the Bulletin of Mathematical Biology August 2023 issue.

People

By Sara Loo

We interviewed Dr Adrianne Jenner, lecturer at Queensland University of Technology, find out more here.

We interviewed Dr Michael Watson, lecturer in Applied Mathematics at the University of New South Wales in Sydney, Australia and co-chair of the Cardiovascular Modelling subgroup, find out more here.

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Editorial

By Thomas Wooley

Research that is worth a thousand words: Visualising a conference

Conferences, workshops, and study groups serve as critical hubs for innovative discussions and interdisciplinary collaborations. Yet, the traditional format of these gatherings has remained steadfast – talks, slides and projectors. Unfortunately, the insights shared during these talks often prove to be as transient as a fading dream, slipping away quickly, leaving us with fleeting memories.

Taking notes during lectures can help, and if the speaker is willing, you can supplement them with presentation slides. However, more often than not, we find ourselves returning from these events with notebooks filled with intricate spider diagrams and hastily scribbled ideas that seem more at home on a conspiracy theorist's corkboard.

The most recent innovation has been recorded talks; you can literally relive the presentation. However, spotty connection issues, poor sound recording and non-existent video editing aside, surely, we must be able to make memory recall more… fun?

While lectures and presentations will always be the core of these gatherings, there's an untapped approach that could revolutionise the conference experience – conference illustration. Illustrations are not only visually captivating, making even the most mundane topics intriguing, but they also provide tangible outputs that can be used to showcase current and future work, satisfy funding requirements, and elevate audience engagement.

An illustrated talk by Maria Abou Chakra on the presentation The Climate Game.

My introduction to this practice occurred during the COVID-19 pandemic when Maria Abou Chakra's remarkable work began circulating on Twitter. I wholeheartedly encourage you to explore her talents in mathematical biology and artistry by following her on Twitter, @MariaAbouChakra, (yes, I refuse to call it X) and visiting her collection of sci-sketches on her website (http://individual.utoronto.ca/abouchakra/sci-sketchnotes/). Her work is both beautiful and highly informative. She truly is a renaissance woman with a foot both in the science and the aesthetic!

I've had the privilege of presenting for Maria at the "Modelling Cell Development and Regeneration Discussion Group," and I've consistently used her illustrations to convey my work to academic and lay audiences. They're undeniably more appealing than a slew of equations and technical jargon.

Maria’s illustration of my talk about putting stochasticity, or “noise”, in biological patterning systems.

• During a recent conversation with Maria, she explained how illustrating talks helped her retain information, focusing on the core concepts rather than minutiae. She generously shared some valuable tips for those aspiring to follow in her footsteps:

• 1)     Organize your sketch space, decide where elements like titles, conclusions, and key ideas should be placed. Preparation is key, as space can fill up quickly.

• 2)     Technology is not a prerequisite; a pen, pencil, and paper can be a great starting point.

• 3)     If you prefer technology, consider using software like Autodesk SketchBook (free) or Procreate.

• 4)     Familiarize yourself with the features and brushes of your chosen app.

• 5)     Practice. Skills are honed over time through effort and dedication.

However, for those without the time, skills, or inclination for such artistry, consider hiring an illustrator for your next conference. It's important to acknowledge that while these artists may not come cheap, their work holds immense value. In an era where AI produces "art", we must not underestimate the genuine skill of condensing and presenting information in a visually curated manner. We should be willing to compensate those with artistic talents, ensuring that such skills don't become lost, or devalued, in the face of automation.

If you're working with a tight conference budget, I understand that this may not be your top priority. However, if you have some flexibility in your finances, take a moment to explore the pool of talented illustrators available. You may be pleasantly surprised by the options at your disposal. Posting a message on Twitter, or Facebook could yield enthusiastic responses from skilled illustrators.

In a recent conference on interdisciplinary IVF challenges, https://thefertilitynetwork.wixsite.com/infer,  conducted with my colleague Dr Katerina Kaouri and funded by GW4, https://gw4.ac.uk/, we were fortunate to have some extra funds available. After evaluating our options, we decided to engage a local artist, Eleanor Beer (https://www.eleanorbeer.com/), as our conference illustrator.

Eleanor readily admitted she wasn't an IVF expert, but this was precisely the point. She focused on capturing the big ideas and overarching themes of the conference, not on mining details for her next research paper.

Eleanor Beer’s illustration of our recent “Interdisciplinary Challenges in IVF” conference.

Unsurprisingly, both the delegates and the organizing committee were thrilled with Eleanor's work. She provided us with a piece of art that we eagerly anticipate displaying in our department. It stands as a constant reminder of our funding success and the continuing scientific challenges that need to be addressed.

As we uncover the potential of incorporating illustrators into conferences, it becomes clear that their contributions have the power to revolutionize how we disseminate and absorb information. Their work transcends language barriers and kindles our scientific creativity. The benefits are substantial, and it is high time to recognize and embrace the visualization of conferences.

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Featured Figures

By Fiona Macfarlane

Early Career Feature - Chloé Colson, University of Oxford, UK

In this issue, we highlight research from Chloé Colson, a PhD student at the University of Oxford (UK) working with Philip Maini and Helen Byrne. We asked Chloé to tell us a little more about their paper `Investigating the Influence of Growth Arrest Mechanisms on Tumour Responses to Radiotherapy’:

Cancer is a heterogeneous disease, with tumours of the same type exhibiting large variation at the genotypic and phenotypic levels. These differences can have a significant influence on tumour sensitivity to treatment and, more generally, on patient prognosis. Improving our understanding of the mechanisms underpinning cancer is, therefore, essential for the development of effective patient-specific therapeutic protocols. In this paper, we aim to assess how two distinct mechanisms of growth control may affect tumour responses to radiotherapy (RT), an established cancer treatment used to treat more than 50% of cancer patients.

In previous work (Colson et al. 2022), we developed a novel ordinary differential equation model of solid tumour growth which distinguishes between growth arrest due to nutrient insufficiency, when cell proliferation and death rates balance, and due to contact inhibition, when the cell proliferation rate converges to zero, with no cell death. While it has been shown that both of these mechanisms can be simultaneously active in vitro in 2D monolayer and 3D spheroid assays (Helmlinger et al. 1997), most models of tumour growth only describe a single growth control mechanism. By considering both nutrient and space limited growth, our model exhibits three distinct regimes: nutrient limited (NL), space limited (SL) and bistable (BS), where both mechanisms of growth arrest coexist.

In the present work, we extend our tumour growth model to include time-dependent responses to RT and systematically study RT response in the three growth regimes introduced above. We construct three virtual populations of NL, SL and BS tumours, and, for each population, we initially consider tumour responses to a conventional fractionation schedule consisting of 5x2 Gy fractions per week for 8 weeks. We determine average responses and explore how values of key parameters (i.e., the tumour oxygen consumption rates (q₁, q₃) and the vascular volume) generate extreme (i.e., strongly positive and negative) behaviour. We find that tumour responses to RT are regime-dependent, with tumours in the SL cohort responding positively and tumours in the BS cohort responding poorly. We also identify the biological processes that may explain positive and negative treatment outcomes in each regime. For instance, as shown in the Figure, we find that increased RT efficacy for SL tumours may be due to limited tumour regrowth and/or RT cell kill. Finally, by studying the impact of the total dose and dosing frequency on tumour response, we elucidate how dosing strategies that maximise the reduction in tumour burden vary between regimes; higher doses applied at higher frequency are beneficial for SL tumours, whereas lower doses applied at lower frequency can be more effective for NL and BS tumours.

Subject to the validation of our findings with experimental data, we believe that our modelling framework has the potential to help guide patient-specific treatment protocol design and, thus, contribute to improving patient prognosis.

You can find out more about this interesting work here: https://link.springer.com/article/10.1007/s11538-023-01171-2

Most accessed article in the Bulletin of Mathematical Biology in August 2023

The article entitled “Could Mathematics be the Key to Unlocking the Mysteries of Multiple Sclerosis?” was the most accessed article in the August edition of the Bulletin of Mathematical Biology. This article was written by Georgia Weatherley, Robyn P. Araujo, Samantha J. Dando and Adrianne L. Jenner from the Queensland University of Technology.

In this paper, the authors review the existing mathematical efforts to understand multiple sclerosis (MS), a neuroimmunology disease affecting the brain and spine. The goal with this review was to highlight the opportunities for mathematicians to have major impact on MS, both in terms of diagnosis, prognosis and improving treatment design.

MS is a neurodegenerative disease where myelin, which surrounds and protects neurons in the brain and spine, is degraded by an overactive immune system. The loss of myelin causes a range of physical and cognitive impairments for which there is currently no cure. Existing mathematical models of MS, while limited in volume in comparison to diseases such as leukemia or malaria, are diverse and insightful. Modelling works range from non-spatial deterministic models (ODEs) to spatially deterministic models (PDEs) and spatially stochastic models (ABMs).

The authors summarise, to the best of their knowledge, all existing mathematical efforts to capture MS across the four major disease scales: population, systemic, CNS and molecular (cellular). As such, this review serves as a foundation for future modelling works in MS.

The modelling techniques developed by mathematical oncologists and immunologists are readily translatable to MS and could provide much needed answers to open problems in this complex, profoundly heterogeneous disease. This review is a call to arms for the mathematical biology community, complete with a list of open problems that could benefit from a mathematical approach.    

You can find out more about this interesting work here: https://link.springer.com/article/10.1007/s11538-023-01181-0

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Alys Clark (University of Auckland), Sara Loo (Johns Hopkins University), Fiona R. Macfarlane (University of St Andrews), and Thomas Woolley (Cardiff University).

1. News – updates from: 
   • SMB Subgroups
   • Past Conferences and Workshops

To see the articles in this issue, click the links at the above items.

Contributing Content

Issues of the newsletter are released four times per year in Spring, Summer, Autumn, and Winter. The newsletter serves the SMB community with news and updates, so please share it with your colleagues and contribute content to future issues.

If you have any suggestions for content or on how to improve the newsletter, please contact us at any time. We appreciate and welcome feedback and ideas from the community. The editors can be reached at newsletter@smb.org.

We have an upcoming newsletter editor position available, please get in touch if you would like more information on what is involved.

We hope you enjoy this issue of the newsletter!

Alys, Sara, Fiona, and Thomas
Editors, SMB Newsletter

News Section

By Alys Clark

SMB Subgroups Update

Mathematical Epidemiolgy (EPI) and Mathematical Oncology Group (ONCO)

The Mathematical Epidemiology and Mathematical Oncology subgroups are partnering to host a virtual mini-conference February 18-20 (12-4 p.m. EST/UTC-5). The mini-conference will include three plenary talks, contributed talks, a panel on opportunities at the intersection of mathematical epidemiology and oncology, and more! It will emphasize research advances at the intersection of mathematical oncology and infectious disease modeling. Registration and Abstract Submissions will open in December. For more details, join the MEPI and/or ONCO subgroups or contact MichaelRobert@vt.edu. The organizing committee includes: Jason George (Texas A&M), Meredith Greer (Bates College), Linh Huynh (Dartmouth), Harsh Jain (University of Minnesota-Duluth), and Michael Robert (Virginia Tech). 

Mathematical Oncology Group (ONCO)

We are pleased to announce that the ONCO subgroup will be organizing a minisymposium at the upcoming SMB meeting in Korea. This event will showcase the work of promising early-career  researchers who are making exciting and significant contributions to mathematical oncology. 

Thank you for your continued support, and we look forward to your participation in our upcoming events! Please also consider volunteering to serve as the next co-chair of our subgroup. The ONCO subgroup is co-chaired by Jason George (term ends in 2024), Linh Huynh (term ends in 2025), and Harsh Jain (term ends in 2024).

Mathematical Neuroscience Group

The Mathematical Neuroscience subgroup is pleased to introduce its new officers for the 2023-2025 term, as follows:

Chairperson: Yangyang Wang (Brandeis University, US, yangyangwang@brandeis.edu)

Vice-Chair: Chitaranjan Mahapatra (Paris Saclay University, France, c.mahapatra97@gmail.com)

Advisory Members: Hammed Fatoyinbo (Massey University, New Zealand, hammed@aims.edu.gh), Cheng Ly (Virginia Commonwealth University, US, cly@vcu.edu)

Conferences and Workshops

Reporting from the Atlantic Association for Research in the Mathematical Sciences and the Emerging Infectious Diseases Modelling (AARMS-EIDM) Summer School (19-31 Aug Newfoundland, Canada)

Submitted by Francisca Olajide

The Atlantic Association for Research in the Mathematical Sciences and the Emerging Infectious Diseases Modelling (AARMS-EIDM) Summer School, organized by Dr. Amy Hurford (Memorial University of Newfoundland and Labrador) took place at Bonne Bay Marine Station, in Norris Point on the breath-taking west coast of Newfoundland, Canada between August 19 and August 31, 2023. The summer school gave 39 participants the unique opportunity to enrol in two well-structured graduate-level mathematics courses, ”Mathematical Epidemiology" and “Data, Models, and Decision Support", both taught by seasoned professors, in a highly collaborative learning environment. 

Commencing the sessions, Dr. Amy Greer (University of Guelph) delivered an enlightening lecture on “Simple Epidemiological Models". As a young enthusiast in infectious disease modelling, I found the teaching explicit enough for anyone aiming to connect the domains of mathematics and epidemiology. In her lecture on “Host Heterogeneity", Dr. Greer highlighted heterogeneity in disease transmission and used the concept of the “WAIFW (Who-Acquires-Infection-From-Whom)" matrix to have participants see how incorporating heterogeneity in models helps to better understand disease dynamics and to estimate quantities such as age-dependent forces of infection.

During one of her sessions, Dr. Jane Heffernan (York University) presented a seemingly straightforward, yet thought-stimulating question: “Why make models?" Dr. Heffernan connected Fibonacci numbers, animal coats, and fractals to the notion of understanding patterns in the world around us. This connection highlighted an underlying rationale for making models. She further taught on in-host models, multi-pathogen models, and evolution, high-lighting how these kind of models can help us to better understand complexities in disease dynamics.

Dr. James Watmough (University of New Brunswick) took us further from models to forecasting. Dr. Watmough stated that forecasting is a scientific method; using the logistic growth equation, he demonstrated how to incorporate probabilistic components to quantify uncertainties in process and observation. Furthermore, he underscored the necessity of associating any valuable prediction with an assessment of its accuracy and reliability. In the context of infectious disease modelling, participants gained insight into addressing diverse sources of uncertainty that could arise from modelling disease dynamics, parameter selection, and making predictions.

Continuing from there, Dr. Amy Hurford delved into characterizing uncertainties. Dr. Hurford demonstrated the techniques for sensitivity analysis and uncertainty analysis, including Latin Hypercube Sampling and Monte Carlo simulations. She also covered the topic of “Decision Support", emphasizing the significant role of modelling in estimating the potential outcomes of decisions through relevant case studies.

Dr. Julien Arino gave an introduction to “Stochastic epidemiological models", illustrating why stochasticity matters by making a distinction between what the basic reproduction number conveys, in both deterministic and stochastic contexts. Dr. Arino demonstrated how to use the Gillespie algorithm to simulate Continous-time Markov chains (CTMC), a commonly used stochastic system. Additionally, he provided insights into the spatio-temporal spread of diseases, highlighting mobility as a key driver, using case studies of the Black Death and SARS-CoV-1.

The Summer School also featured talks from guest lecturers. Dr. Bouchra Nasri (One Health Modelling Network for Emerging Infections, Université de Montreal) discussed the vision for building a One Health data portal, based on a data source, documentation, and modelling approach to foster collaboration among researchers, public health agencies, and other relevant stakeholders. Dr. Steve Walker (McMaster University) discussed the Interna- tional Infectious Disease Data Archive, which integrates historical and publicly available incidence, mortality, and population data. Dr. Brenda Wilson (Memorial University of Newfoundland and Labrador) discussed decision-making in healthcare from both clinical and policy perspectives. In her words, “Not making a decision is an action", while stating that it is important to understand how to manage uncertainty and urgency. Dr. Edward Thommes (Sano  Pasteur, University of Guelph) took us through the process of building ensemble forecasts by combining individual forecasts, using a case study of seasonal influenza forecasts in Ontario.

The course content was extensive, and the instructors also did an excellent job integrating making models, handling uncertainties, and decision-making. They were very comfortable with questions from participants, which made the learning atmosphere even more warm and conducive for further discussions on class materials or research interests. Indeed, the summer school was an intensive 12 days of learning epidemiological concepts, advanced mathematical techniques, statistical methods, and computational skills, needed to holistically tackle infectious disease concerns.

Participants also had the opportunity to effectively apply tools and techniques learned in the summer school courses to collaboratively work on innovative projects aimed at advancing infectious disease modelling. This is particularly significant due to the diverse academic backgrounds of the participants, enabling them to cross-pollinate ideas, combine interdisciplinary insights, and effectively address infectious disease questions. Ultimately, this experience has better equipped participants with the skills and tools needed for infectious disease modelling to support decision-making in public health.

On the side, participants had the chance to explore some of the attractions in and around Norris point. While some participants hiked the 7:7 km out-and-back Tablelands trail, others did the 11.7 km Tablelands off-trail loop. The hike was a highlight of the summer school, as it allowed participants to bond and experience the geological diversity of the Tablelands trail. Some of the participants explored the Bonne Bay Marine Station Aquarium while the more adventurous ones went on the Western Brook Pond Tour. “The summer school was definitely a unique and enjoyable experience," remarked Qiuyi Su, a postdoctoral researcher from York University.

In an interview with CBC Newfoundland Morning, Dr. Amy Hurford stated that the choice of having the summer school at Bonne Bay Marine Station was to bring together and equip the next round of young infectious disease modelling researchers in an environment that allows for connection, collaboration, innovation, and community building. Indeed, from having dinner together to working on group projects and sharing research interests, participants built strong connections among themselves. “The summer school did not only teach us how to build models to solve infectious disease problems, but also how to connect, collaborate, and build relationships", wrote George Adu-Boahen, a Master's student from Memorial University of Newfoundland and Labrador, in his feedback.

Being a part of the summer school was an unforgettable experience, and we would like to express gratitude to all who made it possible. The summer school was supported by the Atlantic Association for Research in the Math ematical Sciences (AARMS), Mathematics for Public Health, the Canadian Network for Modelling Infectious Diseases, the One Health Modelling Network for Emerging Infections, Memorial University, and the Canadian Centre for Disease Modelling. A big thank you to Dr. Amy Hurford for organizing such an incredible event.

Further information about the summer school can be found here:

https://ahurford.github.io/aarms-summer-school/index.html

Reporting from the Workshop on Mathematical Perspectives on Immunobiology (11-14 Sept, Blagoevgrad, Bulgaria)

Submitted by Peter Rashkov (IMI-BAS)

The workshop was organized by the Institute of Mathematics and Informatics (Bulgarian Academy of Sciences) and was hosted by the South-West University’s Conference Centre Bachinovo in Blagoevgrad. The scientific programme included plenary talks, oral presentations, a poster session, and round-table discussion. The forum had a great turnout with 25 on-site and 4 online participants hailing from Algeria, Bulgaria, Cameroon, Canada, Czech Republic, Denmark, France, Germany, Great Britain, Hungary, Israel, Spain, and the United States. Nearly a third of the participants were PhD students or young researchers.

The plenary talks covered a wide range of scientific topics: overview of recent findings on virus evolution, systems pharmacology for optimized anticancer therapies, neutrophil dynamics in cancers, antiviral treatment for Sars-Cov-2, immunotherapy in cancer, and the role of inhibitory killer-cell immunoglobulin-like receptors in modulating T cell dynamics. This diversity reflects the wide applicability of mathematical models and methods in immunobiology. The round-table discussion focused on current problems in higher education in mathematical biology. Among the points raised were: fragmentation of research between mathematics, biology and computer science, general decline of interest in mathematics among high-school and university students, but also increased awareness of the importance of scientific and mathematical research during the Covid-19 pandemic among policymakers and the general public.

The Workshop’s organisers acknowledge the generous support from a SMB International Grant that enabled the participation of four PhD students at the workshop, and thank the Conference Centre’s staff for their hospitality. The gentle weather in Blagoevgrad and the pleasant atmosphere of the mountain forest and peaks in the background rounded off the event.

For more information, access the website: http://www.math.bas.bg/nummeth/workshop2023/

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Chemical Systems with Limit Cycles

by Radek Erban and Hye-Won Kang

Hilbert's 16th problem asks questions about the number of limit cycles that a planar polynomial system of ODEs can have. The solutions of such ODEs are two functions of time, x(t) and y(t), which can be plotted in the x-y plane as shown in the picture for an example which has four stable limit cycles, plotted using the black dashed lines as the four closed curves (periodic solutions). Other solutions of the same ODEs (plotted using different colours) all approach one of the limit cycles.

This example has been constructed using the approach developed in the paper, where we investigate limit cycles in chemical systems. Chemical systems with N=2 chemical species can be described by planar ODE systems. A lower bound on the maximum number of stable limit cycles in such chemical systems has been proven in the paper. This directly implies a lower bound on Hilbert's number H(n) denoting the maximum number of limit cycles for general planar n-degree polynomial ODE systems. In the paper, we also study more general systems with N>2 chemical species. We construct chemical systems with K stable limit cycles, where K can be arbitrarily large.

About the authors:

Radek Erban is a Professor of Mathematics at University of Oxford. Hye-Won Kang is an Associate Professor at University of Maryland. 

Coupling Mountain Pine Beetle and Forest Population Dynamics Predicts Transient Outbreaks that are Likely to Increase in Number with Climate Change

by Micah Brush and Mark A. Lewis

Read the paper

Mountain pine beetle (MPB) have spread well beyond their historical range, with destructive consequences for forests in Canada. We here present and analyze a new model that couples forest growth to MPB population dynamics to address long term questions about the risk of further spread and inform management strategies, particularly under climate change. This model captures key aspects of MPB biology, including a threshold for the number of beetles needed to overcome tree defenses and beetle aggregation that facilitates mass attacks. These mechanisms lead to a demographic Allee effect, which is known to be important in beetle population dynamics. We show that as forest resilience decreases, a fold bifurcation emerges and there is a stable fixed point with a non-zero MPB population. We derive conditions for the existence of this equilibrium. We then simulate biologically relevant scenarios and show that the beetle population approaches this equilibrium with transient boom and bust cycles with period related to the time of forest recovery. As forest resilience decreases, the Allee threshold also decreases. Thus, if host resilience decreases under climate change, for example under increased stress from drought, then the lower Allee threshold makes transient outbreaks more likely to occur in the future.

Image caption:

A graphical representation of the model dynamics. Pine beetles infest trees by mass attack, overcoming host defenses. They overwinter under the bark of the tree before emerging and dispersing the following summer. This is connected to a forest growth model where saplings grow everywhere there is available light before becoming susceptible to pine beetle attack. After they are infested, they die. Tree needles turn red the year following infestation, and are lost the following year, clearing room on the forest floor for new growth.

Could mathematics be the key to unlocking the mysteries of multiple sclerosis

by Georgia Weatherley, Robyn P Araujo, Samantha J Dando & Adrianne L Jenner

Read the paper

Our review of the existing mathematical models of multiple sclerosis (MS) is designed to highlight the opportunities for mathematics to unlock the mysteries of this disease. Rising in prevalence, MS is an autoimmune, neurodegenerative disease that results in the demyelination of nerve axons and causes physical and cognitive impairment. Existing mathematics models of MS, while limited in volume in comparison to diseases such as cancer and malaria, are diverse and insightful. They range from non-spatial and spatial deterministic models to agent-based models and other stochastic modelling techniques. So far, these models have successfully furthered our understanding of T cell responses, the underlying oscillatory dynamics of the disease, and potential treatment avenues. For mathematical oncologists and immunologists, MS presents a rich and rewarding opportunity to use transferable modelling techniques to shed light on this complex, profoundly heterogeneous disease. This review is a call to arms for the mathematical biology community, complete with a list of open problems that could benefit from a mathematical approach.

Georgia Weatherley is a PhD candidate at Queensland University of Technology, supervised by Adrianne Jenner and Robyn Araujo. This review was written and conceived in collaboration with Samantha Dando. All authors contributed to the writing and editing.

Investigating the influence of growth arrest mechanisms on tumour responses to radiotherapy

by Chole Colson, Philip K Maini & Helen M Byrne

Read the paper

In this paper, we investigate how nutrient and space limited mechanisms of growth control impact tumour responses to radiotherapy (RT). By distinguishing three growth regimes: nutrient limited (NL), space limited (SL) and bistable (BS), where both mechanisms of growth arrest are active, our study reveals qualitative differences between the RT responses of tumours in the monostable (i.e., NL and SL) regimes and the BS regime. In particular, NL and SL tumours have largely positive responses to treatment, while RT is deleterious for tumours in the BS regime. We find that the positive and negative responses observed in the monostable and BS regimes, respectively, may be underpinned by different biological mechanisms. Further, the RT dosing strategies that maximise the reduction in tumour burden also vary between regimes; higher doses applied at higher frequencies are beneficial for SL tumours, whereas lower doses applied at lower frequencies can prove more effective for NL and BS tumours. Subject to validating our findings against experimental data, we believe that our modelling framework may help guide patient-specific treatment protocol design and, thus, contribute to improving patient prognosis.

Chloé Colson is a PhD student, Philip K. Maini is a Professor of Mathematical Biology and director of the Wolfson Centre for Mathematical Biology, and Helen M. Byrne is a Professor of Mathematical Biology.