Population-level study of disease distribution and determinants: incidence, prevalence, risk factors, outbreaks, intervention effects. Substrate of public health, pharmacovigilance, and health-policy decisions.
epidemiology
Incidence and prevalence
Incidence rate = new cases per person-time; prevalence = existing cases per population at a time point. Relation at steady state:…
Basic reproduction number R₀
Expected secondary cases from one primary case in a fully susceptible population. Outbreak grows if R₀ > 1; herd immunity threshold = 1 –…
SIR compartmental model
dS/dt = –βSI/N; dI/dt = βSI/N – γI; dR/dt = γI. Basic deterministic model of epidemics; R₀ = β/γ, final-size equation ln(S∞/S₀) = –R₀(1 –…
Case–control study
Retrospective design sampling on outcome: compare exposure histories of cases and controls. Estimates odds ratio OR = (a·d)/(b·c) with 2×2…
Cohort study
Prospective (or retrospective) follow-up of exposed vs unexposed groups; estimates incidence rate ratio directly. More valid for temporal…
Confounding and causal inference
A third variable associated with both exposure and outcome creates spurious association. Controlled via stratification, regression…
Herd immunity
Population-level protection of susceptibles when a sufficient fraction is immune — R_effective = R₀(1 – p) < 1 when p > 1 – 1/R₀. Enables…
Outbreak investigation
CDC 10-step method: confirm outbreak, define cases, find cases, describe (person/place/time), hypothesis generation, analytic study…
SIR R_0 = β/γ; R_eff = R_0·S/N; herd-immunity v_c = 1 − 1/R_0; dS/dt, dI/dt, dR/dt ODE
SIR basic reproduction number framework — the canonical compartmental-epidemic-model core quantity (Kermack-McKendrick 1927; Anderson & May…
Kermack-McKendrick final size: R_∞ = 1 − exp(−R_0·R_∞); transcendental fixed-point in R_∞
Kermack-McKendrick final-size framework — the canonical fixed-point equation relating the eventual cumulative attack rate R_∞ (fraction of…
Generation-interval MGF: M_T(s) = γ/(γ−s) (Exp(γ)); E[T]=1/γ; Var[T]=1/γ²
Generation-interval MGF framework — the canonical moment-generating-function description of the epidemiological generation time T (the…
SIR: R_0=β/γ; v_c=(β−γ)/β; at R_0=2: v_c=1/2; at R_0=4: v_c=3/4
Sympy-exact symbolic witness of the SIR basic reproduction number canonical forms and two integer-R_0 herd-immunity anchors. Setup: β, γ,…
KMK final-size: F=R_∞−1+exp(−R_0·R_∞)=0; dF/dR_∞|_0=1−R_0; outbreak iff R_0>1
Sympy-exact symbolic witness of the Kermack-McKendrick final-size equation implicit canonical form and the outbreak-threshold…
Generation MGF: M_T=γ/(γ−s); E[T]=1/γ; E[T²]=2/γ²; Var[T]=1/γ²
Sympy-exact symbolic witness of the generation-interval exponential MGF canonical form and its moment anchors. Setup: γ, s — both positive…
Snow cholera (1854)
J Snow 1854 Broad-Street pump cholera-mapping; foundational text in epidemiology.
Smoking (Doll-Hill 1950)
Doll-Hill 1950 smoking-lung-cancer case-control; modern long-term mortality cohorts.
Bradford-Hill (1965)
A Bradford-Hill 1965 9-criteria for causality; modern updated by Susser 1973 + epidemiologic-causation literature.
Framingham (1948)
Framingham Heart Study 1948-present; modern 3-generation + risk-score + biomarker integration.
Sero-surveillance (Ferguson 2020)
Modern serological surveillance UKB + CDC; foundational for COVID-19 + pandemic-tracking.
K-McK detail (1927)
Kermack-McKendrick 1927 SIR ODE foundational + 2-population + spatial-extensions Diekmann + 2020-COVID re-derivation.
Vaccine (Jenner 1796)
E Jenner 1796 cowpox-smallpox vaccination; modern mRNA-Pfizer-Moderna 2020 + 13.6B doses delivered + smallpox-eradication 1980.
Herd immunity (Fox 1971)
J Fox 1971 herd-immunity threshold 1-1/R0; modern measles 95% threshold + COVID-Omicron R0=10 immunity-evasion.
Contact tracing (Frost 1936)
W Frost 1936 contact-tracing chains-of-infection; modern digital-Bluetooth Apple-Google 2020 + privacy-preserving exposure-notif.
Doll-Hill (1950)
R Doll-A Hill 1950 lung-cancer-smoking case-control; modern 2010 lifetime-RR=25× + e-cigarette-vaping uncertainty.
Framingham (1948)
T Dawber 1948 + Kannel 1961 Framingham heart-study; modern risk-score CVD-prediction global standards 75 years cohort.