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+# biology and medicine
+
+biology resisted mathematization longer than physics. living systems are messy, stochastic, and deeply complex. but math is steadily becoming biology's most powerful tool — and in medicine, mathematical errors are measured in lives.
+
+## population dynamics
+
+the lotka-volterra equations model predator-prey interactions:
+- prey grow exponentially when predators are scarce
+- predators grow when prey are abundant
+- the system oscillates: more prey → more predators → fewer prey → fewer predators → more prey → ...
+
+this is a pair of coupled differential equations, and the oscillating solution explains real population cycles (like the famous lynx-hare cycle in canadian fur trapping records). the math predicts the qualitative behavior — boom and bust — without knowing anything about the specific animals.
+
+more sophisticated models handle competition, mutualism, migration, and age structure. conservation biology uses these to predict extinction risk and design nature reserves.
+
+## epidemiology
+
+the SIR model (susceptible → infected → recovered) is the foundation of epidemic modeling. it's three differential equations:
+- dS/dt = -βSI (susceptible people get infected at a rate proportional to contact with infected)
+- dI/dt = βSI - γI (infected people either infect others or recover)
+- dR/dt = γI (recovered people are immune)
+
+the basic reproduction number R₀ — how many people one infected person infects on average — determines whether an epidemic grows or dies out. R₀ > 1 means epidemic; R₀ < 1 means it fizzles.
+
+during COVID, everyone suddenly cared about these models. "flatten the curve" was a mathematical statement: reduce β (through masking, distancing) to keep the infection peak below hospital capacity. the math was simple; getting people to act on it was the hard part.
+
+## genetics and bioinformatics
+
+DNA is a string over a 4-letter alphabet (A, T, C, G). comparing DNA sequences is a string-matching problem. finding genes is a pattern-recognition problem. building evolutionary trees is a graph theory problem.
+
+the human genome project was as much a computational/mathematical achievement as a biological one. sequence alignment algorithms (like BLAST) use dynamic programming — a mathematical technique — to compare your DNA sequence against billions of known sequences in seconds.
+
+CRISPR guide RNA design, protein structure prediction (AlphaFold), and drug-target interaction modeling are all mathematical problems at their core.
+
+## EEG and brain signal processing
+
+in my research on anesthetics and brain monitoring, the raw data is EEG signals — electrical voltage measurements from electrodes on the scalp. the raw signal is a mess: brain activity plus muscle artifacts, eye blinks, electrical noise.
+
+the math pipeline:
+1. **fourier analysis** decomposes the signal into frequency bands (delta 0.5-4 Hz, theta 4-8 Hz, alpha 8-13 Hz, beta 13-30 Hz, gamma 30+ Hz)
+2. **filtering** removes artifacts and noise
+3. **feature extraction** computes statistics (power spectral density, coherence between channels, entropy measures)
+4. **classification** — in our case, a CNN trained on spectrogram images to detect depth of anesthesia
+
+the goal: can we tell from brain signals alone how deeply anesthetized a patient is? too light and they might wake up during surgery. too deep and you risk complications. the math turns a subjective clinical judgment into an objective measurement — exactly the [counting and measurement](/wiki/immediate/counting-and-measurement) problem, but for consciousness.
+
+## medical statistics
+
+clinical trials are [probability](/wiki/immediate/probability-in-daily-life) in its highest-stakes application. does this drug work, or did we get lucky with our sample? p-values, confidence intervals, randomization, blinding — the entire machinery of evidence-based medicine is statistical.
+
+and the errors are consequential. p-hacking (running many statistical tests until one comes out significant) has contributed to a replication crisis across biomedical research. base rate neglect in diagnostic testing (see [probability](/wiki/immediate/probability-in-daily-life)) leads to unnecessary procedures and missed diagnoses. getting the math right literally saves lives.
+
+## the deep point
+
+biology is where math meets the messiest, most complex systems we know. the models are always dramatically simplified — a cell is not a differential equation, and a brain is not a neural network. but the simplifications reveal structure that would be invisible otherwise. the SIR model doesn't capture every detail of epidemic spread, but it explains *why* epidemics have the shape they do. that explanatory power — seeing the pattern through the noise — is what math brings to biology.
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