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

5.6 The effect of antiviral drug use on the full dynamics of a local epidemic

Page last updated: April 2006

As mentioned, the advantage of assessing interventions with respect to R or q is that a considerable amount of community structure can be incorporated into the models with moderate effort. However, it is also necessary to look at the full calendar-time dynamics of any local epidemic, if it occurs. Specifically, this provides the total attack rate, the timing of the epidemic peak, the shape of the epidemic and can provide the number of courses of antiviral drugs used as a function of time. It is very time consuming to incorporate extensive community structure into models that describe the transmission dynamics and so here we use an SIR model to make calculations and present illustrations on how AV use affects the full dynamics.

An unusual feature of the use of AVs for individuals thought to have been exposed is that the individuals identified as a contact are administered AVs temporarily and then come off AVs. To accommodate this feature we developed a model depicted schematically in Figure 5.7. This model does not explicitly include a latent phase.
Schematic of the modified SIR-model.  Individuals are in one of five states: S, the susceptible population; Ip, individuals on prophlyaxis who became infected

Figure 5.7 Schematic of the modified SIR-model. Individuals are in one of five states: S, the susceptible population; Ip, individuals on prophlyaxis who became infected (“breakthrough” cases); Inp, individuals not on prophylaxis who became infected; Rp, recovered individuals who were on prophylaxis when infected; Rnp, recovered individuals who were not on prophylaxis when infected. An individual from any of these five states may also be classified as a contact, of which there are two possible types: Cp, individuals who have come into contact with an infectious individual and who have subsequently been administered antiviral drugs for prophylaxis; Cnp, individuals who have come into contact with an infectious individual and who have not received antiviral drugs. Healthy contacts lose their contact classification after three days.

Top of page
The model developed assumes the following:

i. On average, an infectious individual makes a meaningful contact with around 20 people over a period of a few days; see Edmunds et al. (1997).

ii. A proportion of these contacts are provided with antiviral drugs for prophylaxis. Data (see Edmunds et al., 1997) suggests that up to 85% of contacts should be traceable. We allow for a delay in administering the drugs. Typically we have set this at four days following the onset of symptoms in the case. Two days due to the delay in providing treatment to the initial infective and a couple of days to allow for contact tracing.

iii. A proportion of infectious individuals who were not provided antiviral drugs for prophylaxis receive them for treatment. We allow for a delay until the drug is administered. Typically, we have set this at two days from exposure. It is assumed that breakthrough cases (those becoming sick while receiving prophylaxis) will continue to use the antiviral drugs that they already have, as treatment.

iv. Infectiousness during the infectious period is assumed to be constant so that infectivity is flat.

v. The effect of providing AVs for treatment is to reduce infectiousness to 0.7 (et) times baseline infectiousness.

vi. The effect of providing AVs for prophylaxis is to reduce susceptibility to 0.2 (es) times baseline susceptibility, and, if infected, to reduce infectiousness to 0.4 (ei) times baseline infectiousness.

vii. The antiviral stockpile is 4 million doses. The population size is 20 million. Note, however, that the results apply similarly to a smaller population, e.g. Sydney, as long as the available number of AVs courses are reduced proportionately.

viii. An antiviral distribution program is only initiated once 10 new cases are observed on a single day. We examined the sensitivity to this delay.

The main outcomes from this model are illustrated in Figure 4.12 and Figure 4.13. The effect of treatment is to flatten the epidemic. It is noteworthy that this intervention becomes ineffective for reducing transmission when R0 is 1.7, as seen from Figure 4.12(b). The effect of targeted prophylaxis is markedly different. Prophylaxis has almost no effect on the final attack rate, but consistently delays the bulk of the epidemic for over 6 months. With an effective contact tracing program, delays of over 1 year may be achievable, even for higher values of R0 [see Figure 4.13(a) and Figure 4.13(b)]. For R0 of around 3.5, combined prophylaxis and treatment strategies can delay the peak of the epidemic by around six months.

We now examine some aspects of the model in more detail. A central feature of the model is that it allows for the tracing of contacts of infectives. A percentage of contacts are identified, traced and then provided with AVs for prophylaxis. Contacts (both those provided with AVs for prophylaxis and those not traced) may either develop disease or return to the general population.

The model allows for realistic "wastage" of antiviral drugs by acknowledging that many courses of drugs will be provided to contacts who, unknown at the time, will not actually become infected.

Top of page

Figure 4.12 and Figure 4.13 do not reveal some aspects of the model's outputs. While treatment is not an effective strategy on its own, it can play a significant role in delaying the onset to epidemic peak when used in conjunction with an established prophylaxis strategy. An example of this is seen by comparing Figure 5.8(c) with Figure 5.8(b). As mentioned above, the delay to the epidemic peak is reduced for higher values of R0.
Results presented in this section assume a flat infectiousness function. Under the assumption of a peaked infectiousness function, the delay to epidemic peak will be reduced for two reasons:
a. the doubling time is shorter under a peaked infectiousness function, as discussed in Section 2.4;
b. contact tracing is less effective with a peaked infectiousness function because infections occur earlier in the infectious period.
A thorough investigation of the effect of peaked infectivity will be considered in future work.

While provision of prophylaxis can provide a delay to the onset of the epidemic, it requires a significant effort on the part of the health system which must contact trace and then deliver the antiviral drugs to an ever increasing number of people over an extended period of time. Figure 5.9(a) shows that, even in a highly effective intervention, as in Figure 5.8(c), there is, in fact, exponential growth in the number of infectives. Figure 5.9(b) shows the doses per day required to be distributed. In fact the doses per day reach a level that may not be achievable in practice.
All three graphs show the curves for the susceptibles (S), the traced contacts (C), the infectives (I), the removals (R) and those receiving Oseltamivir (O) for the model with R0 = 1.7

Figure 5.8
(a) Baseline epidemic, i.e. without intervention. The epidemic peak occurs at 0.32 years.
(b) 40% of contacts are provided AVs for prophylaxis. The epidemic peak occurs at 0.76 years.
(c) 50% of non-prophylaxis cases are provided AVs for treatment and 40% of contacts are provided AVs for prophylaxis. The epidemic peak occurs at 1.2 years.
All three graphs show the curves for the susceptibles (S), the traced contacts (C), the infectives (I), the removals (R) and those receiving Oseltamivir (O) for the model with R0 = 1.7.

Both graphs assume parameters as in Figure 5.8(c)

Figure 5.9
(a) A zoom of the epidemic curve presented in Figure 5.8(c), using a log scale on the vertical axis. The epidemic grows exponentially from the very beginning. The first kink shows the correction in growth due to implementation of the interventions. The point at which the stockpile runs out is clearly visible as a kink in the epidemic curve just before the 12 month mark.
(b) Doses distributed per day for the intervention shown in Figure 5.8(c).
Both graphs assume parameters as in Figure 5.8(c); in particular, R 0 = 1.7.

Top of page

If the percentage of contacts traced and provided with prophylaxis is not maintained, the intervention will fail to delay the epidemic. The number of doses per day peaks at around 100,000 for a wide range of scenarios and interventions.

Protecting HCWs and essential services workers will be important during a pandemic. If the health system fails, other interventions such as contact tracing and provision of AVs for treatment and prophylaxis will be unsustainable.

Prophylaxis of HCWs and essential service people can be taken into account, at least partially, by including a constant drain on the AV stockpile. This method does not take into account the effect on dynamics from this intervention.

We assume that individuals spend half of their time taking prophylaxis and half of their time off work, based on a 6 week on, 6 week off rotation.

Table 5.2 is based on the 40% prophylaxis, 50% treatment scenario presented above, with R0=1.7. With no intervention, the epidemic peaks after 0.32 years and the attack rate is 70%. As the number of persons (HCWs, essential services) provided with AVs increases the benefit of prophylaxis and treatment for the population as a whole is eroded. Providing a small group of HCWs (25,000) with prophylaxis has only a small effect on the delay to peak, reducing it from 1.22 years to 1.20 years. If all 1 million essential service workers in Australia were provided with AVs for prophylaxis, the stockpile would be used up rapidly for virtually no benefit.

Number of persons undergoing non-targeted protection (N_HCW) Time to peak of epidemic (years)
0
1.22
25,000  
1.20
250,000
0.95
1,000,000
0.49

Table 5.2 Time of the peak of the epidemic for different levels of prophylaxis distributed to HCWs. The provision of AVs to essential services brings the time to epidemic peak back towards baseline. If a small group of dedicated influenza workers (25,000-50,000) are provided with AVs then the time to epidemic peak can be kept above 1 year. If AVs are distributed on-mass to essential services, they will be largely wasted and provide very little benefit to the population as a whole. Without a constant depletion term, the epidemic peaks at 1.22 years. For baseline (no intervention) the peak is at 0.32 years.

Table 5.2 shows that it is possible to protect a small influenza taskforce and still have an effective intervention. A quick calculation [maximum doses per day from Figure 5.9(b) divided by 25,000] shows that the peak load on the 25,000 strong workforce would be delivering around 5 courses of AVs per day (a couple of families), which seems reasonable. For most of the time, the load would be an order of magnitude or two lower than this.

If however, AVs are provided indiscriminately, their potential to curtail an epidemic will be lost. It must be kept in mind that once the AV stockpile runs out, if the number of susceptibles in the community is still large (as it will be) then the epidemic will run its natural course. Little or no benefit will have been gained by protecting the million or so essential services workers for a couple of months.

The calculation for Table 5.2 is rather crude. A more dynamic and detailed calculation is presented in Section 6.2 where the number of HCWs requiring protection is calculated from the current number of infectives in the population. Table 5.2 simply demonstrates that with careful planning, the lifetime of the stockpile is not compromised by protecting a subset of the HCWs.

Document download

This publication is available as a downloadable document.

Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease(PDF 873 KB)