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The mathematics of infectious diseases. (English) Zbl 0993.92033

Summary: 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 σ, 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 σ 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.


MSC:
92D30Epidemiology
34C60Qualitative investigation and simulation of models (ODE)
34C23Bifurcation (ODE)
35Q80Appl. of PDE in areas other than physics (MSC2000)
35B32Bifurcation (PDE)