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**The type-reproduction number \(T\) in models for infectious disease control.**
*(English)*
Zbl 1124.92043

Summary: A ubiquitous quantity in epidemic modelling is the basic reproduction number \(R_0\). This became so popular in the 1990s that “All you need to know is \(R_0\)!” became a familiar catch-phrase. The value of \(R_0\) defines, among other things, the control effort needed to eliminate the infection from a homogeneous host population, but can be misleading when applied to a heterogeneous population for the same purpose.

We have defined the type-reproduction number \(T\) for an infectious disease, and shown that this not only has the required threshold behaviour, but also correctly determines the critical control effort for heterogeneous populations. The two quantities coincide for homogeneous populations. We further develop the new threshold quantity as an indicator of control effort required in a system where multiple types of individuals are recognised when control targets a specific type.

We have defined the type-reproduction number \(T\) for an infectious disease, and shown that this not only has the required threshold behaviour, but also correctly determines the critical control effort for heterogeneous populations. The two quantities coincide for homogeneous populations. We further develop the new threshold quantity as an indicator of control effort required in a system where multiple types of individuals are recognised when control targets a specific type.

### MSC:

92D30 | Epidemiology |

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\textit{J. A. P. Heesterbeek} and \textit{M. G. Roberts}, Math. Biosci. 206, No. 1, 3--10 (2007; Zbl 1124.92043)

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### References:

[1] | Roberts, M. G.; Heesterbeek, J. A.P., A new method to estimate the effort required to control an infectious disease, Proceedings of the Royal Society London, Series B, 270, 1359 (2003) |

[2] | Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28, 365 (1990) · Zbl 0726.92018 |

[3] | Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0997.92505 |

[4] | Heesterbeek, J. A.P., A brief history of \(R_0\) and a recipe for its calculation, Acta Biotheoretica, 50, 189 (2002) |

[5] | Caswell, H., Matrix Population Models: Construction, Analysis and Interpretation (2001), Sinauer Associates: Sinauer Associates Sunderland |

[6] | Haydon, D. T.; Chase-Topping, M.; Shaw, D. J.; Matthews, L.; Friar, J. K.; Wilesmith, J.; Woolhouse, M. E.J., The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak, Proceedings of the Royal Society London, Series B, 270, 121 (2003) |

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