The mathematics of infectious diseases.

*(English)*Zbl 0993.92033Summary: 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.