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

In this section, we include all non-pharmacological steps an individual can take to prevent an infectious contact, such as avoiding close contacts with people, wearing a mask and frequent hand washing. The effect of these measures is modeled by decreasing the rate at which an individual makes infectious contacts.

By practicing personal infection control and distancing, a susceptible person reduces their rate of making infectious contacts by a factor denoted λ_{S}. For example, a susceptible individual may chose to meet fewer people by going shopping less frequently and/or wear a P2 mask when in a public place and/or makes an effort to wash hands more frequently. If the combined effort of this changed personal behaviour by a susceptible reduces the mean number of infectious contacts, i.e. contacts sufficiently close for infection to occur if the contact is with a fully infectious person, by 30% (say) then λ_{S} = 0.7.

Similarly, by practicing personal infection control and distancing,, an infectious person reduces their rate of making contacts with susceptible people by a factor denoted lI. For example, suppose that when infected an individual continues the practice of meeting fewer people by going shopping less frequently and/or wearing a P2 mask when in a public place and/or making an effort to wash hands frequently. If the combined effort of this changed personal behaviour by an infective reduces the mean number of infectious contacts, i.e. contacts sufficiently close for infection to occur if the contact is with a fully susceptible person, by 30% (say) then λ_{I} = 0.7.

As infections occur essentially as a result of person-to-person contacts between susceptible and infectious individuals we model the combined effect of susceptible and infectious individuals practicing personal infection control and distancing by reducing the rate of infectious contacts occurring by a factor λ_{S} × λ_{I}.

We consider the case where people practice personal infection control and distancing only outside the household and where they practice this within the household as well.

Although the baseline and intervention epidemic curves depend on the form of the infectiousness function, the relative effectiveness of this intervention is similar for flat and peaked infectivity, so we show plots with flat infectivity only. Figure 4.3 shows the epidemic curve with (blue) and without (red) the intervention, assuming that all individuals reduce their rate of making contacts by 30% - that is, λ_{S} = λ_{I} = 0.7.

**Figure 4.3** The number of new cases per week in a population of 1 million households with *R*_{0} = 2.5 and 3.5 and with personal infection control and distancing (referred to as ‘distancing’) outside only or both inside and outside the household in the SEIR_{H} model. Each graph shows the median cases per day with (blue solid line) and without (red dotted line) distancing measures that reduce susceptibility and infectivity by 30%. The shaded region represents 90% of the stochastic simulations.

**Figure 4.4** A comparison of the effects of personal infection control and distancing and isolation in the SEIRH model. The left-hand plot compares the effective reproduction number as a function of the level of intervention, λ, for four interventions – isolating cases with flat infectivity (blue dashed line), isolating cases with peaked infectivity (black dot-dashed line), distancing outside the household with flat infectivity (green solid line) and distancing both inside and outside the household with flat infectivity (red dotted line). For personal infection control and distancing measures, λ = λ_{S } = λ_{I }, while for isolating cases, λ is the fraction of the infectious period before isolation. The right hand plot compares the epidemic curves for each of these interventions with λ chosen for each intervention so that the effective reproduction number is equal to 1.5.

These graphs show the cases where the distancing measures are practiced outside the household only, and where this is practiced both outside and inside the household for *R*_{0} = 2.5 and 3.5. These interventions are sufficient to eliminate the disease if *R*_{0} is 1.5, and reduce the effective reproduction number very close to λ if *R*0 = 2.5 and personal infection control and distancing is practiced within and outside the household.

We can also calculate the impact of these interventions on the effective reproduction number. Figure 4.4 compares the effective reproduction number for isolation and personal infection control and distancing as a function of the degree of intervention (λ).

In the case of personal distancing, this λ is λ_{S } and λ_{I }. In the case of isolation, λ is the fraction of the infectious period that is spent before isolation. We see that isolation requires a greater reduction in the intervention parameter (λ) to achieve the same decrease in *R*. The right hand figure shows the resulting epidemic curves if λ is set for each intervention to result in an effective *R* of 1.5. Although the reproduction numbers (and the total number of cases) are the same for all four curves, the epidemic peak is higher and occurs sooner under isolation than under personal infection control and distancing

It is clear that, for influenza, personal infection control and distancing has much greater potential to reduce transmission than does isolating cases soon after diagnosis. The effectiveness of personal distancing measures are not influenced by the form of the infectiousness function, and is more effective at delaying the peak of the outbreak. In contrast, isolating cases soon after diagnosis has minimal effect when the infectiousness function is peaked, and is much less effective at slowing the spread of disease.

It should be noted, however, that the extent to which personal infection control and distancing is practiced by uninfected and asymptomatic infected individuals may vary considerably between individuals and is likely to change according to the level of community concern.

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