Estimating the basic reproduction number at the beginning of an outbreak

We compare several popular methods of estimating the basic reproduction number, R0, focusing on the early stages of an epidemic, and assuming weekly reports of new infecteds. We study the situation when data is generated by one of three standard epidemiological compartmental models: SIR, SEIR, and SEAIR; and examine the sensitivity of the estimators to the model structure. As some methods are developed assuming specific epidemiological models, our work adds a study of their performance in both a well-specified (data generating model and method model are the same) and miss-specified (data generating model and method model differ) settings. We also study R0 estimation using Canadian COVID-19 case report data. In this study we focus on examples of influenza and COVID-19, though the general approach is easily extendable to other scenarios. Our simulation study reveals that some estimation methods tend to work better than others, however, no singular best method was clearly detected. In the discussion, we provide recommendations for practitioners based on our results.


SIR model
Least squares estimation for the IDEA method From (2), our objective function is Let η = log R 0 and ξ = log(1 + d) and note that both of these transfromations are monotone increasing. Next, we take partial derivatives of Q = Q(η, ξ) with respect to η and ξ.
Finally, we minimize Q by setting ∂Q ∂η = 0 and ∂Q ∂ξ = 0. Solving these equations, we obtain In the last step, we solve for η = log R 0 and thus find Posterior distributions First, we calculate the posterior distributions of each of the elements of θ. For ease of exposition, in each calculation we drop the subscript on the scale parameter k in the priors.
Hence the posterior distribution for β is gamma with shape parameter α + m k and scale parameter k + Hence the posterior distribution for σ is gamma with shape parameter α + m k and scale parameter k + Hence the posterior distribution for σ is gamma with shape parameter α + m k and scale parameter k +

I(t)dt
Hence the posterior distribution for γ is gamma with shape parameter α + m k and scale parameter k + t k τ I 1 I(t)dt.

Sensitivity to Prior
As mentioned previously, the joint prior distribution of the unknown rate parameters θ is made up of independent gamma distributions given by Γ(α, k) with mean k/α. In the main text, we assume that α is the same for the parameters β, σ, ρ, γ, while k varies and if appropriate will be denoted by k β , k σ , k ρ , k γ . In the simulations we took these to be α = 1 and k β = k σ = 3, k ρ = 2, k γ = 5. The prior distribution on −τ I 1 is exponential with rate one, and this is independent from the θ vector. In Figure 8, we compare the results in the main text with results repeating the method with a different prior distribution for the SIR/SEIR/SEAIR data assuming SIR/SEIR/SEAIR models respectively. The modified prior for the comparison is k β = 9/4, k γ = 3. These were chosen as alternative reasonable parameters for the influenza. The plots show that there was very little change between the two versions. Comparison of the fullBayes method for SIR, SEIR, and SEAIR data with two different prior distributions: same as main text is on the right and the modified version is on the left. The inflection point for the epidemic is marked in blue, and the true R 0 for the data is marked as a horizontal red line.