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

The role of antiviral drugs is the theme of Section 5. However, as antiviral drugs also play a role in reducing transmission we present some of the results for the effect of using antiviral drugs here. The model used to derive the graphs in Figure 4.12 and 4.13 is described in greater detail in Section 5.4. Note that the model used for these calculations is a deterministic model, which means that it does not take chance fluctuations into account. In particular, this means is that it will indicate elimination only when the intervention is adequate to bring *R* below 1. The probability (q) of achieving elimination by chance, during the early stages, when *R* exceeds 1 is dealt with in Section 5.3.

Figure 4.12 and Figure 4.13 illustrate that the use of antiviral drugs for treatment affects transmission quite differently than their use for prophylaxis. Treatment use tends to flatten the epidemic curve while prophylaxis use delays the epidemic with little change to its shape when it occurs. For higher values of the reproduction number, treatment strategies become ineffective, failing to significantly reduce the overall attack rate and the height of the epidemic peak. In contrast, prophylaxis strategies continue to be effective in delaying the onset of the epidemic, although the ensuing epidemic remains of about the same size. Use of antiviral drugs for prophylaxis is an effective way of postponing the peak of the epidemic and may, depending on the value of *R*, provide enough time for a vaccine to be developed and distributed.

**Figure 4.12** Epidemic curves as a function of treatment.

**(a)** As the proportion of cases treated (ψ) increases, the epidemic curve is reduced in size, but the peak is delayed only marginally, except for extremely high treatment coverage scenarios. The base-line attack rate is 50% (R ≈ 1.4) and the infectious period is 6 days.

**(b)** As for **(a)**, but for a baseline attack rate of 70% (R ≈ 1.7). Treatment is ineffective in either reducing the attack rate or delaying the epidemic.

**Figure 4.13** Epidemic curves as a function of prophylaxis. As the proportion provided with prophylaxis (ε) increases, the epidemic curve is delayed significantly, while its peak size is only marginally reduced. The graph is shown for ε ≤ 0.5, because above this value the intervention delays the epidemic for a very long time. The infectious period is 6 days.

** (a)** Baseline attack rate of 50% (R ≈ 1.4). **(b)** Baseline attack rate of 70% (R ≈ 1.7).

Further results based on this model are given in Section 5.4.

Figure 4.14 shows the effect of combining household-based quarantine and prophylaxis measures in the case where

*R*

_{0}=1.5. The model used for calculation is the same as that described in Section 4.8, with the inclusion of prophylaxis (using the default parameters for effectiveness given in Appendix B).

Early intervention is very effective, and a high level of quarantine compliance, or deployment of prophylaxis is able to suppress the epidemic entirely if *R*_{0} is 1.5. Note that in the case of only 20% compliance with quarantine, even though the epidemic grows exponentially, the size of the epidemic at 60 days is vastly reduced, from 66,500 to 8,100 (Figure 4.14).

Intervention at 3 days after infection is able to suppress the epidemic very well, though a combination of a high level of compliance and prophylaxis are required for complete suppression, or the epidemic continues at a low level.

If intervention cannot be implemented before day 4 after infection, no intervention is able to suppress the epidemic entirely, though a combination of prophylaxis and high level of quarantine compliance greatly reduces the number of cases in the first 60 days.

It may seem infeasible to quarantine infective households early enough to have an effect. The top graph of Figure 4.14 shows that successfully quarantining only 20% of households at day 2 after infection could usefully slow the growth of the epidemic in its early stages.

**Figure 4.14** The effect of quarantining households on the first 60 days of the epidemic for four levels of compliance and the effect of prophylactic use of antiviral drugs (AVs) for household members. Parameter values: *R*_{0}=1.5, peaked infectiousness function.

The black dashed line (the top curve) is equivalent to the situation where there is no intervention.

**Top graph:** Intervention occurs two days after the primary case is infected

**Middle graph: ** Intervention occurs three days after the primary case is infected

**Bottom graph: ** Intervention occurs four days after the primary case is infected

I_{60} = total number infected on day 60

The order (top to bottom) of the curves in each graph are as in the legend.

These results are relatively sensitive to the largest household size used in the model. In general, if smaller households are used, both the quarantine and prophylaxis interventions are less effective (results not shown). This is because (according to a Reed-Frost model), a substantial fraction of the transmission occurs within larger households. As a result, much of the benefit could be gained by targeting larger households.

The calculations suggest that one should

aim to quarantine and prophylax within 2-3 days of the infection of the primary case, i.e. within one day of onset of symptoms in the primary case;

give priority to prophylaxis and quarantine of larger households.

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