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# The reproduction number $R_t$ in structured and nonstructured populations

• Using daily counts of newly infected individuals, Wallinga and Teunis (WT) introduced a conceptually simple method to estimate the number of secondary cases per primary case ($R_t$) for a given day. The method requires an estimate of the generation interval probability density function (pdf), which specifies the probabilities for the times between symptom onset in a primary case and symptom onset in a corresponding secondary case. Other methods to estimate $R_t$ are based on explicit models such as the SIR model; therefore, one might expect the WT method to be more robust to departures from SIR-type behavior. This paper uses simulated data to compare the quality of daily $R_t$ estimates based on a SIR model to those using the WT method for both structured (classical SIR assumptions are violated) and nonstructured (classical SIR assumptions hold) populations. By using detailed simulations that record the infection day of each new infection and the donor-recipient identities, the true $R_t$ and the generation interval pdf is known with negligible error. We find that the generation interval pdf is time dependent in all cases, which agrees with recent results reported elsewhere. We also find that the WT method performs essentially the same in the structured populations (except for a spatial network) as it does in the nonstructured population. And, the WT method does as well or better than a SIR-model based method in three of the four structured populations. Therefore, even if the contact patterns are heterogeneous as in the structured populations evaluated here, the WT method provides reasonable estimates of $R_t$, as does the SIR method.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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