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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 \(\sigma\), 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.

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.

### MSC:

92D30 | Epidemiology |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

34C23 | Bifurcation theory for ordinary differential equations |

35Q80 | Applications of PDE in areas other than physics (MSC2000) |

35B32 | Bifurcations in context of PDEs |