## Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease

The infectiousness function quantifies how infectious an individual is in terms of the time since infection. We focus mainly on two forms of this function. In the first the individual is latent for the first day and has an infectious period of five days. The infectiousness during the infectious period is constant, as in Elveback *et al.* (1976). This *flat* infectiousness function is shown in Figure 2.1(a). The second infectiousness function is depicted in Figure 2.1(b). It is motivated by viral shedding data and the estimate given in Ferguson *et al.* (2005). We refer to this as the peaked infectiousness function. For both scenarios we assume that the incubation period is two days [Hayden *et al.* (1998)].

Both *R* and the shape of the infectiousness function influence the dynamics of the epidemic. To illustrate their effects we show in Table 2.1 the typical progress of an epidemic during the exponential growth phase in terms of *R* and the two forms of infectiousness functions displayed in Figure 2.1. Note that both infectiousness functions result in the same eventual attack rate for a given *R*_{0}, but the epidemic curve is steeper for the peaked infectiousness function.

**Figure 2.1** The flat and peaked infectiousness functions. The corresponding *R* for these infectiousness functions is the same when they have the same area under the curve. The two curves depicted above correspond to *R* = 1 in an SEIR model.

R_{0} = 1.5 |
R_{0} = 2.5 |
R_{0} = 3.5 |
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Infectiousness function |
Flat | Peaked | Flat | Peaked | Flat | Peaked |

Cases on day 0 | 20 | 20 | 20 | 20 | 20 | 20 |

Cases on day 10 | 50 | 82 | 179 | 573 | 437 | 2300 |

Cases on day 20 | 128 | 335 | 1600 | 16,000 | 9500 | 260,000 |

Cases on day 30 | 324 | 1400 | 14,000 | 460,000 | 200,000 | 1.3 mil* |

Doubling time (days) | 7.5 | 4.9 | 3.2 | 2.1 | 2.2 | 1.46 |

Eventual attack rate (%) | 58 | 58 | 89 | 89 | 97 | 97 |

**Table 2.1**Number of infectious cases on days 10, 20 and 30 for the deterministic SEIR model starting with 20 initial infectious cases. The doubling times during this exponential growth phase and the percentage of the population eventually infected (the

*attack rate*) are also given.

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