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Mathematics
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Applications of Differential Equations and Mathematical Modelling in Mathematical Biology
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Applications of Differential Equations and Mathematical Modelling in Mathematical Biology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E3: Mathematical Biology".

Deadline for manuscript submissions: 30 June 2026 | Viewed by 1508

Special Issue Editors

Prof. Dr. Cristina Januário

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Guest Editor
1. Department of Mathematics, Instituto Superior de Engenharia de Lisboa—ISEL, Rua Conselheiro Emídio Navarro 1, 1949-014 Lisbon, Portugal
2. Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal
Interests: dynamical systems; chaos theory; differential equations; applied mathematics
Prof. Dr. Jorge Duarte

E-Mail Website
Guest Editor
1. ISEL-Engineering Superior Institute of Lisbon, Department of Mathematics, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisboa, Portugal
2. Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Interests: differential equations; nonlinear dynamics; complex systems; chaos; mathematical biology

Special Issue Information

Dear Colleagues,

The use of mathematics for modelling is essential in many scientific areas. The complexity of real-world systems and the growing availability of biological data make the close interaction between scientists of different disciplines crucial for realism and significance of the obtained theoretical results. Mathematical biology is a particularly vibrant field, utilising mathematical, statistical and computer-based methods to investigate a wide range of biological problems in areas such as epidemiology, virology, complex ecosystems, genetics, cancer dynamics, and many others. In this Special Issue, the theory of differential equations stands out as indispensable for modelling and understanding biological phenomena, regarding its conceptual richness and applicability. By disclosing the compelling aspects of combining mathematics with real-world problems, this Special Issue is likely to attract a wide audience by its importance and interdisciplinarity.

Prof. Dr. Cristina Januário
Prof. Dr. Jorge Duarte
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • differential equations
  • mathematical biology
  • nonlinear models

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Published Papers (2 papers)

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Research

18 pages, 604 KB  
Article
Making Chaos Out of COVID-19 Testing
by Bo Deng, Jorge Duarte, Cristina Januário and Chayu Yang
Mathematics 2026, 14(2), 306; https://doi.org/10.3390/math14020306 - 15 Jan 2026
Viewed by 378
Abstract
Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of [...] Read more.
Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium (E1). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (λu>λs), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article
(This article belongs to the Special Issue Applications of Differential Equations and Mathematical Modelling in Mathematical Biology)
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29 pages, 5273 KB  
Article
Ion Channel Memory Drives Cardiac Early Afterdepolarizations in Fractional Models
by Noemi Zeraick Monteiro, Rodrigo Weber dos Santos and Sandro Rodrigues Mazorche
Mathematics 2025, 13(10), 1585; https://doi.org/10.3390/math13101585 - 12 May 2025
Cited by 1 | Viewed by 788
Abstract
Understanding how past factors influence ion channel kinetics is essential for understanding complex phenomena in cardiac electrophysiology, such as early afterdepolarizations (EADs), which are abnormal depolarizations during the action potential plateau associated with life-threatening arrhythmias. We developed a mathematical framework that extends Hodgkin-Huxley [...] Read more.
Understanding how past factors influence ion channel kinetics is essential for understanding complex phenomena in cardiac electrophysiology, such as early afterdepolarizations (EADs), which are abnormal depolarizations during the action potential plateau associated with life-threatening arrhythmias. We developed a mathematical framework that extends Hodgkin-Huxley type equations with gamma Mittag-Leffler distributed delays, using tools from Fractional Calculus. Traditional memoryless two-variable models fail to reproduce EADs. Our approach modifies FitzHugh-Nagumo, Mitchell-Schaeffer, and Karma cardiac models, enabling the generation of EADs in each of them. We analyze the emergence of these oscillations by discussing the fractional parameters and the mean and variance of the memory kernels. Stability observations are also presented. Full article
(This article belongs to the Special Issue Applications of Differential Equations and Mathematical Modelling in Mathematical Biology)
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