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

A major reason for the successful elimination of SARS was the early isolation of diagnosed cases. This intervention is likely to be much less effective for influenza since transmission can occur prior to showing symptoms, and the infectious period is generally much shorter. For the SEIR model, isolation at day four of their infection, two days after the time when symptoms start (as was eventually achieved for SARS in Singapore), leaves the infective with an effective reproduction number of 0.6**R*_{0 }for the flat infectiousness function, and with 0.86**R*_{0} for the peaked infectiousness function.

This intervention is sufficient to eliminate infection if *R*_{0} is 1.5 and infectivity is flat, but if infectivity is peaked, a local epidemic can still occur. Figure 4.1 shows the epidemic curves produced using the SEIR_{H} model with isolation (blue) and without isolation (red) for *R*_{0} values of 2.5 and 3.5, assuming flat or peaked infectivity. The epidemics take off sooner when infectivity is peaked as more infections take place early in the case’s infectious period. For the same reason, isolating individuals early has much more effect if infectivity is flat than if it is peaked. Similar results for the relative effect of these measures for flat infectivity are obtained when using the SEIR model.

We also observe that the effectiveness of isolation depends critically on the mean time from infection until isolation. Figure 4.2 shows the probability that a single case will start an outbreak that takes off and the effective reproduction number according to the days from infection until isolation, when each case is isolated in the SEIR model. The left-hand plots show the probability that an outbreak takes off, and the right-hand plots show the reproduction number. As mentioned above, if individuals are isolated after 4 days, this is sufficient to prevent a major outbreak if *R*_{0} = 1.5 and the infectivity is flat. If infectivity is peaked, there is about a 40% chance of a major outbreak. In order to ensure that a major outbreak does not occur for *R*_{0} values of 2.5 and 3.5, individuals would need to be isolated between 2 and 3 days after infection, and perhaps sooner if isolation is the only intervention used. As individuals will only start to show symptoms at this time, this strategy is unlikely to be practical.

**Figure 4.1**The number of new cases per week in the SEIR

_{H}model with a population of 1 million households with

*R*

_{0}= 2.5 and 3.5 and with flat and peaked infectivity. Each graph shows the median number of cases per day with (blue solid line) and without (red dotted line) isolating cases two days after onset of symptoms. The shaded region represents 90% of the stochastic simulations.

We have indicated that isolation by itself tends to be an effective intervention when

*R*is near 1. While this means that it is not very effective by itself, it can be an effective additional intervention when other interventions have succeeded in bringing

*R*close to 1.

**Figure 4.2**The probability that a single infected individual starts an outbreak that takes off and the effective reproduction number of the SEIR model depending on the time from infection until isolation of cases. The top plots assume that the infectivity function is flat, and the lower plots assume that the infectivity function is peaked.

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