Quantitative Methods

A growing collection of bite-sized references covering the calculus, linear algebra, probability, and statistics that underpin quantitative work in infectious disease ecology, evolution, and epidemiology. Each page pairs the core idea and notation with a worked example and runnable code in R, Python, and Julia for simulation and intuition.

Foundations & notation

• The Language of Mathematics — number systems, symbols, and how to read an equation
• Mathematical Notation — symbols, sums, products, and LaTeX
• Functions and Graphs — common functions and how to plot them
• Exponentials and Logarithms — rules, the number $e$, and log identities

Working with data

• Manipulating Data Frames — dplyr, data.table, pandas, Polars, and DataFrames.jl
• Graphing Data — matching charts to questions and the grammar of graphics

Sequences, series & limits

• Sequences — notation, monotonicity, and boundedness
• Limits — convergence, properties, and L’Hôpital’s rule
• Series — arithmetic, geometric, and power series
• Taylor and Maclaurin Series — polynomial approximation

Differentiation

• Derivatives — the instantaneous rate of change
• Common Derivatives — a reference table
• Product Rule
• Quotient Rule
• Chain Rule
• Partial Derivatives
• The Gradient

Integration

• Integrals — area under a curve and the Fundamental Theorem
• Common Integrals — a reference table
• u-Substitution
• Integration by Parts

Optimization

• Optimization and Critical Points — maxima, minima, and convexity

Convexity & inequalities

• Jensen’s Inequality and Nonlinear Averaging
• The Legendre Transform — convex conjugates

Linear algebra

• Matrix and Vector Notation
• Matrix Operations
• Inverse, Determinant, and Rank
• Eigenvalues and Eigenvectors
• Jacobians

Probability & statistics

• Statistical Inference — population, parameter, sample, estimate
• Probability Basics
• Random Variables — pmf, pdf, and the CDF
• Common Distributions: An Overview
  • Normal · Binomial · Poisson · Exponential · t
• Expected Value
• Moment Generating Functions
• Measures of Center — mean, median, quantiles
• Measures of Variability — variance, SD, standard error
• Sampling Distributions
• The Law of Large Numbers
• The Central Limit Theorem
• Markov Chains — transition matrices and stationary distributions
• Branching Processes — Galton–Watson, extinction, and outbreaks
• Random Walks and Brownian Motion — diffusive scaling and the Wiener process
• Copulas — separating dependence from the marginals via Sklar’s theorem
• Maximum Likelihood Estimation
• Moment Matching — method of moments and distributional approximation
• Monotonic Transformations
• Hypothesis Testing
• p-Values
• Confidence Intervals
• Permutation Tests
• Diagnostic Testing and Screening — sensitivity, specificity, PPV, and ROC
• Proper Scoring Rules — Brier, log score, CRPS, and forecast calibration

Regression & generalized linear models

• Linear Regression
• Logistic Regression
• Generalized Linear Models — GLMs and Poisson regression
• Hierarchical (Multilevel) Models — partial pooling and shrinkage

Bayesian inference

• Bayesian Inference — priors, likelihood, and the posterior
• Markov Chain Monte Carlo

Gaussian processes & spatial statistics

Gaussian processes:

• Gaussian Processes — distributions over functions and GP regression
• Covariance Functions and the Matérn Family — kernels, smoothness, and lengthscale
• Hilbert-Space Approximations for Gaussian Processes — fast basis-function GPs

Geostatistics & areal models:

• Kriging and Geostatistics — variograms and best linear unbiased prediction
• Spatial Point Processes — Poisson, Cox, and log-Gaussian Cox processes
• Areal Models: CAR, ICAR, and BYM — disease mapping on a neighborhood graph
• Bayesian Spatial Models with INLA — fast latent-Gaussian inference and the SPDE approach
• Distances on a Sphere: Haversine and Beyond — great-circle vs Euclidean distance

Survival analysis

• Survival Analysis — Kaplan–Meier, hazards, censoring
• Cox Proportional Hazards Regression

Experimental & study design

• Experimental Design — experimental vs observational, sources of bias
• Factorial Designs — main effects and interactions
• Fractional Factorial Designs — partial designs, aliasing, resolution
• Optimal Experimental Design — D-, A-, and I-optimality
• Response Surface Methodology — optimizing over continuous factors
• Latin Hypercube Sampling — space-filling designs for computer experiments
• Global Sensitivity Analysis — Sobol indices and Morris screening
• Survey Sampling — SRS, stratified, cluster, and weighting

Causal inference

• Causal Inference — confounding, counterfactuals, and Simpson’s paradox
• Instrumental Variables — estimating causal effects under confounding
• Mendelian Randomization — genetic variants as instruments

Statistical & population genetics

Population-genetics foundations:

• Hardy–Weinberg Equilibrium — genotype frequencies and the χ² test
• Linkage Disequilibrium — $D$, $D'$, and $r^2$
• Genetic Drift and the Wright–Fisher Model
• Selection and Mutation–Selection Balance
• Population Structure and F_ST
• The Coalescent — genealogies backward in time

Association & complex traits:

• Genome-Wide Association Studies
• Multiple Testing and False Discovery Rate
• Population Stratification and PCA Control
• Heritability and Variance Components
• Quantitative Genetics and the Breeder’s Equation
• Polygenic Scores

Molecular evolution:

• Detecting Selection with dN/dS
• The Molecular Clock and Phylodynamics

Evolutionary dynamics

• Evolutionary Game Theory — ESS and replicator dynamics
• The Evolution of Cooperation — the Prisoner’s Dilemma and Nowak’s five rules
• Adaptive Dynamics and the Evolution of Virulence
• The Price Equation and Evolutionary Epidemiology — selection, transmission, and within-host change as an exact identity

Population & community ecology

Single-species dynamics:

• Exponential and Logistic Growth
• Discrete-Time Models and the Logistic Map
• Structured Population Models — Leslie matrices
• Metapopulations and the Levins Model
• Asynchrony and the Inflationary Effect — how spatiotemporal variation and dispersal inflate abundance and persistence

Species interactions & stability:

• Lotka–Volterra Predator–Prey Dynamics
• Competition and Coexistence
• The Community Matrix and Stability

Biodiversity & community structure:

• Diversity Indices — Shannon, Simpson, Hill numbers
• Species-Abundance Distributions and Neutral Theory

Dynamical-systems & epidemic-dynamics toolkit:

• Equilibria and Linear Stability — nullclines and phase planes
• Bifurcations — thresholds and tipping points
• The Next-Generation Matrix and R₀

Spatial dynamics & pattern formation:

• Spatial Diffusion and the Heat Equation — random movement and $\sqrt{t}$ spreading
• Reaction–Diffusion and Spatial Spread — the Fisher–KPP wave
• Turing Patterns — diffusion-driven pattern formation
• Spatial Moment Equations — mean density and spatial covariance from a stochastic individual-based model

Networks

• Networks and Graphs — adjacency matrices, degree, and structure
• Centrality and Node Importance — degree, betweenness, eigenvector centrality
• Random-Graph Models — Erdős–Rényi, scale-free, small-world
• Networks in Ecology and Epidemiology — food webs and transmission networks

Pharmacokinetics & pharmacodynamics

• Pharmacokinetics: Compartment Models — ADME, clearance, half-life, AUC
• Pharmacodynamics: Dose–Response — the Emax/Hill model
• Antimicrobial PK/PD — MIC and the PK/PD indices

Epidemic modeling

• Compartmental Models — the SIR model and $R_0$
• Density-Dependent and Frequency-Dependent Transmission — how the contact rate scales with host density, and the critical density threshold
• SEIR and Compartmental Extensions — latent classes, waning, demography
• Vector-Borne Disease Models — the Ross–Macdonald framework
• Stochastic Epidemics and the Gillespie Algorithm
• The Effective Reproduction Number and Forecasting
• Fitting Dynamic Models to Data — calibration and identifiability