The Mathematics of Infectious Diseases

TLDR

Mathematical models of infectious disease spread, including age‑structured, heterogeneous, and spatially structured variants, have been extensively analyzed and applied to specific diseases. The paper reviews threshold theorems based on R₀, σ, and R for classic SIR epidemic and endemic models. New expressions for R₀ in MSEIR and SEIR endemic models yield comparable results, and R₀ and σ values are estimated for diseases such as measles in Niger and pertussis in the United States.

Abstract

Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.