When mathematics meets medicine: from neuroscience to epidemiology and oncology
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Sofia Passos
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Mathematics and medicine may appear to belong to different intellectual worlds. Yet increasingly, the boundary between those sciences is dissolving. Modern medicine relies not only on the empirical study of anatomy and clinical trials, but also on equations, computational models, and mathematical frameworks that help scientists interpret the biological complexity. From the structure of neural networks to the spread of infectious diseases and the growth of tumors, mathematical thinking is becoming an essential tool in understanding and hence treating human disease.
Mathematics In the Study of Alzheimer´s Disease
The brain is one of the most complex systems known to science, and understanding how such a system processes information requires a constant enhancement of our techniques, and we can do those upgrades through new mathematical tools capable of analyzing large networks and dynamic signals.
Furthermore, mathematical analysis plays a very significant role in studying neurodegenerative diseases. In recent years, we saw a variety of techniques emerging and making it possible to diagnose Alzheimer’s Disease in much earlier stages, and supporting behavioral and therapeutic interventions on time to delay the symptoms. One of the most prominent examples of those diagnostic tools is fractal analysis, a mathematical concept borrowed from geometry, a tool that maps the brain in models that repeat themselves infinitely and allows us to best distinguish any interferences in that repetition.
Fig. 1 demonstrates some of the forms used and some equations that are implemented to understand the repetition:
When we map the brain in models like the Koch Snowflake and Sierpinski triangle (Fig. 1, A), using this concept, we create a dimension considered the norm of a healthy brain, and if anything affects the normal fractality, like the harmful accumulation of protein (the earliest phenomena of Alzheimer’s disease). In this sense, the integration of mathematics with neuroscience presents an advancement in our understanding of Alzheimer’s disease, allowing us to identify alterations in brain patterns well before traditional methods like MRI can detect changes.
Modeling the Spread of Disease
Epidemiology, the study of how diseases spread through populations, is perhaps one of the most visible and traditional intersections between mathematics and medicine. Infectious disease outbreaks can be modeled using systems of differential equations.
Describe how individuals move between epidemiological states such as susceptible, infected, and recovered.
A classic framework is the SIR model. The idea behind it is surprisingly intuitive: the model divides people into three categories—those who are susceptible to the disease, those who are infectious, and those who have recovered or been removed from the chain of transmission—helping researchers predict how quickly a disease may spread and evaluate the potential impact of public health interventions such as vaccination or social distancing.
This model captures a key feature of epidemics: infections tend to grow rapidly at the beginning. When one infected person transmits a pathogen to several others, the number of cases can increase exponentially. Epidemiologists therefore focus on the epidemic growth rate, which describes how quickly infections rise; the reproductive number, the average number of secondary infections caused by one infected individual in a fully susceptible population; and the generation interval, the time between when a person becomes infected and when that person infects someone else.
Mathematical models in epidemiology provide a framework for translating raw data into meaningful insight, which is crucial for interpreting early epidemics. When a new disease emerges, we rarely know its reproductive number immediately. Instead, scientists observe how quickly case numbers rise and use this information to estimate how contagious the disease might be, allowing public health systems to begin preparing responses as early as possible.
What makes this particularly fascinating is that the model relies on relatively simple mathematics. By representing populations with just three compartments and describing the transitions between them, scientists can capture patterns that emerge across entire societies. In this way, mathematics becomes more than an abstract discipline—it becomes a practical tool for understanding urgent challenges in medicine.
Mathematics in the Fight Against Cancer
Another emerging intersection between mathematics and medicine appears in mathematical oncology, a field that applies mathematical models to understand how cancer develops, grows, and responds to treatment. While cancer is fundamentally a biological disease, it also behaves like a dynamic system, where populations of cells grow, compete for resources, and evolve.
Mathematical oncology uses tools such as differential equations and computational simulations to represent tumor growth and the interactions between cancer cells, healthy tissue, and the immune system. These models help researchers explore questions that are difficult to answer experimentally, such as how tumors develop resistance to therapy or how different treatment strategies might influence tumor evolution.
An important idea in this field is that tumors can be understood as evolving ecosystems composed of many different cell populations. Mathematical models can describe how these populations compete and change over time, helping scientists predict how a tumor may respond to treatment. Increasingly, these models are being used to support clinical decision-making, combining patient data with simulations to design more personalized treatment strategies.
A Shared Language for Complex Systems
Together, these examples illustrate how mathematics is becoming an essential partner in modern medicine. Whether revealing hidden patterns in the brain, predicting the spread of infectious diseases, or modeling the complex evolution of tumors, mathematical tools
allow scientists to interpret biological systems in new and powerful ways. As data and computational methods continue to advance, the collaboration between mathematics and medicine will likely become even more important, helping researchers not only understand disease more deeply but also develop more precise and effective treatments.
Citations
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