We apply the model to the situation in which there is no pre-epidemic treatment, only treatment of latent and infective members of the population during the epidemic. In addition to the basic reproduction number 0
, we consider a control reproduction number c
. The control reproduction number c
is defined as the number of secondary infections caused by introducing a single infective into a susceptible population with control measures in place, and its value may be calculated from the parameters of the model.
The final size relation for the treatment model is
is determined by the model parameters, with ρT
if there is no treatment of infectives. An expression for the parameter ρT
is given in the electronic supplementary material. The final size relation for the treatment model is valid for all 0
. If c
<1, there is no epidemic for the treatment model, while if c
>1, there is an epidemic but eventually I
tends to zero. As an example, shows the contour lines of the number of doses (calculated from (10) in the electronic supplementary material and the number of doses used in the course of a treatment) as a function of the treatment parameters L
in a population of 999 susceptibles and one infective if 0
=1.5. The location of the curve c
=1 may be read from the figure; it runs approximately from the point (2.6,
7) to the point (7,
3.8) and is concave up. The curve c
roughly parallels the contour for 20 doses. As can be seen in , if c
≥1, estimates of the number of doses needed during an outbreak are very sensitive to parameter values. Note that the scale of the spacing of contour lines in changes for contours below 20 and contours above 20.
Figure 3 Total number of doses used in a population of 1000 individuals over the course of the outbreak as a function of the mean times to treatment and prophylaxis (in days), for 0=1.5, with S0=999 and I0=1.
We may estimate various combinations of treatment rates of latent and symptomatic members required to bring c
down below 1. As is shown in , this is feasible for 0
=1.5, but may not be possible for larger values of 0
. This possibility also depends on the availability of facilities to carry out the level of treatment required. In practice, epidemic management would probably include treatment of front line health care workers as protection and to assure the availability of care. This would be a policy decision and lies outside the model. Any estimate of the number of doses required would have to add the doses needed for such treatment.
Theoretically, the smallest reproduction number that can be achieved by treatment of infectives only is given by letting I
→∞ in the expression for c
. However, in practice, it is probably not possible to achieve a treatment rate I
>2, as this would correspond to a mean waiting time of only 12
h between developing symptoms and being treated.
If the incidence in the model is not mass action, the final size relation is an inequality, but dynamic simulations using standard incidence give numerical results very close to those given by the final size relation. Differences between the mass action and standard incidence functions are hardly noticeable in the case of interest where treatment can control the disease. We thus suggest that the above final size relation is a good approximation when general incidence functions are used.
Although the model is useful for comparing treatment strategies, a strong warning is in order. The number of doses of antiviral treatment needed as predicted by the model is essentially proportional to the number of members of the population infected during the epidemic. This number is very sensitive to the number of infectives introduced initially. Introducing two infectives instead of one initially would multiply the number of members infected during the epidemic and the number of doses required by 1.4. This means that since the number of initial infectives cannot be known in advance, the model cannot predict with any degree of reliability the number of doses which would be required in an epidemic. This underscores the importance of early identification of infectives to minimize the number of initial infectives in the system. However, the relative effectiveness of different management strategies is not affected by this critical dependence. We may also estimate the number of cases of disease per 1000 members of the population as a function of the treatment parameters, and this is shown in (as calculated from (11) in the electronic supplementary material but not including the index case).
Figure 4 Total cases as a function of the treatment rates, for 0=1.5, with S0=999 and I0=1.
A scenario considered in Gani et al. (2005)
is antiviral treatment of essentially all symptomatic infectives. We calculate that if 0
=1.5 and antiviral treatment is applied only to symptomatic infectives, a rate I
=0.4 per day would be required to bring the control reproduction number c
down to 1 and avert an epidemic, as can be seen in .
Another application of the treatment model is to the common annual vaccination program to protect against the strain of influenza thought to be the most likely to invade. It has also been suggested (Balicer et al. 2005
) that a possible response to an outbreak of a strain for which no specific vaccine has been developed as yet would be a program of treatment with a general antiviral as a stopgap until a specific vaccine can be produced. The model we have described can be applied to this limited drift situation as well, although the final size relation takes a more complicated form. With disease parameters as in Longini et al. (2004)
and vaccination that reduces susceptibility by 70% of a fraction γ
of the population before an epidemic and introduction of one infective into a total population of 1000 individuals, we obtain the results shown in for the number S∞
of members untreated but uninfected during the epidemic, the number ST∞
of members treated and uninfected during the epidemic and Nf
, the number of cases of disease during the epidemic. They indicate the benefits in reducing influenza cases of pre-epidemic vaccination of even a small fraction of the population. The fraction of the population that must be vaccinated to bring the reproduction number down to 1 is 0.28, but as shown in , a significant decrease in the number of cases of disease is achieved even if the reproduction number is not decreased to 1.
Fraction treated γ, untreated susceptibles S∞, treated susceptibles ST∞ and influenza cases Nf, with vaccination.