The modelling tools and analyses that have been described here could be valuable in real time to help obtain a picture of the unfolding of an epidemic during its onset phase. They rely on: (i) credible daily incidence data, including classification of cases into ‘imported’ versus ‘local’ infections, (ii) representative sampling of infection networks. We now discuss several general points, which we found to be of importance and could be significant for future work in analysing epidemics.
In the case study reported here we were fortunate to possess an excellent database, due to the fact that at the beginning of novel H1N1 epidemic strong efforts were taken by the Israeli health authorities to test every suspected case of influenza. It should be noted, however, that the application of the methods presented here does not require that the available incidence data be complete, so that they can be employed in contexts where detection is far from complete. For example, since the estimator (2.4) for the effective reproductive number is invariant to multiplication of all the values i(t) and i0(t) by a fixed number, it can be employed under the assumption that i(t) and i0(t) represent an unknown (but fixed) fraction of the real cases. Changes in the detection efforts during the period in question, resulting in a change in the fraction of cases detected, will lead to biases in the estimates. Similarly, estimates of the next-generation matrix β for the age group model, as performed here, will still be valid assuming that the available incidence data are only a representative sample of the real cases, on the assumption that the detection rate is identical among the different age groups. Differences in the detection rate among different age groups (for example, if members of some age groups are less likely to seek medical care) will lead to biases in the estimated next-generation matrix.
Our results highlight the importance of taking into account infected persons arriving into a region or country, only as infectors, rather than being included as part of the local infected population. Failure to do so will result in an overestimate of Re. For example, had all infections been regarded as local, the estimate for Re via equation (2.4) (taking i0(t) = 0), would have resulted in Re = 1.27 (95% bootstrap CI [1.17,1.37]). Alternatively, if the imported infectives had been removed from the data, this would have resulted in Re =1.26 (95% bootstrap CI [1.16,1.37]). These values should be compared with the value Re = 1.06 that we obtained by taking imported infectives into account in the correct way. Of course as the epidemic spreads the number of immigrant cases becomes negligible in comparison to the locally infected cases, so that the correction becomes insignificant. But as the example here shows, at the initiation of the epidemic, the imported infectives have a significant effect on the estimate of Re.
We developed a new method for combining both the age-specific incidence data and infection networks data in order to estimate the next-generation matrix. Applying this method to our data, a next-generation matrix was estimated, and using simulations, bootstrap CIs for the elements of this matrix were obtained. Our simulation study made evident that as the size of the sampled infections network increases, the estimated matrix β converges to the true one used for generating the simulations (i.e. the estimator is consistent). Thus the method proposed here for estimating the next-generation matrix, without making any ad hoc assumptions about its structure, could be of value in modelling future epidemics. Since obtaining an accurate estimate using this method depends on having infection network data, this provides a strong motivation for collecting data on who was infected by whom at the beginning of an epidemic. It should be noted, however, that the method presented here for employing the infection network in estimating the next-generation matrix, as well as the bootstrap CIs computed by simulations, depend on the assumption that the sampling of the real infection network is random. To the extent that this is not the case, for example if disproportionate fractions of infections among particular pairs of age groups are made in contexts in which they are more likely to be identified than others (for example infection in a school class as opposed to infection on a public bus), the available infection network, and hence the estimated next-generation matrix, will be biased. Our bootstrap CIs account for uncertainty stemming from random sampling, but not for uncertainty stemming from systematic biases in sampling.
In comparing the observed incidence curves in three age groups with simulations of the three age group model, certain deviations of the data from the model were noted in the young adult age group, which were hypothesized to be related to outbreaks among soldiers. This demonstrates an important role of modelling in the study of an epidemic: deviations between a fitted model and the observed data can alert us to significant factors which were neglected by the model, and which may need to be taken into consideration in mitigation efforts.
An inherent limitation of the data at the beginning of an epidemic is the fact that although it is possible to estimate Re
, these data are insufficient to estimate R0
separately. In other words it is impossible to know what the real reproduction number R0
is and what the fraction of susceptible S0
in the population is. Without knowledge of these quantities one cannot predict the unfolding of the epidemic at later stages and in particular its final size [40
]. In order to make such predictions it is necessary to find an independent method to estimate either R0
For example, serological tests of random sample of the population could potentially be used to estimate S0
. This suggests the need for increasing standard surveillance efforts by ensuring inclusion of basic serological testing of the population, performed at the beginning of the epidemic. Using estimates of S0
(which could vary in different age groups), together with the modelling approach presented here, one could project forward in time to estimate the course of the epidemic and study possible interventions through simulation.