Estimating the effective reproduction number for heterogeneous models using incidence data

Abstract

The effective reproduction number, $R\left(t\right)$ , plays a key role in the study of infectious diseases, indicating the current average number of new infections caused by an infected individual in an epidemic process. Estimation methods for the time evolution of $R\left(t\right)$ , using incidence data, rely on the generation interval distribution, g(τ), which is usually obtained from empirical data or theoretical studies using simple epidemic models. However, for systems that present heterogeneity, either on the host population or in the expression of the disease, there is a lack of data and of a suitable general methodology to obtain g(τ). In this work, we use mathematical models to bridge this gap. We present a general methodology for obtaining explicit expressions of the reproduction numbers and the generation interval distributions, within and between model sub-compartments provided by an arbitrary compartmental model. Additionally, we present the appropriate expressions to evaluate those reproduction numbers using incidence data. To highlight the relevance of such methodology, we apply it to the spread of COVID-19 in municipalities of the state of Rio de Janeiro, Brazil. Using two meta-population models, we estimate the reproduction numbers and the contributions of each municipality in the generation of cases in all others.

Keywords: COVID-19; effective reproduction number; mathematical models; meta-population models.

Figure 1.

Comparison between SIR and SEIIR outputs. SIR results in blue and SEIIR ones in orange. The bar graph (a) compares the SIR and SEIIR ${\overline{\mathcal{R}}}_i$ time averages for all 11 municipalities. In (b), for each municipality, estimations for the total number of exported infections that are reported for both models are displayed with the total commuter movement in that city. The names of the municipalities are abbreviated using acronyms: Rio de Janeiro (RJ), Duque de Caxias (DdC), Nova Iguaçu (NI), São João de Meriti (SJdM), Niterói (Nt), São Gonçalo (SG), Belford Roxo (BR), Nilópolis (Ns), Mesquita (Mq), Queimados (Q), Magé (Ma).

Figure 2.

Influence on number of infections caused by a municipality on another. The heat map captures the time-averaged influence on the number of infections caused by a municipality on another as a fraction of the total number of daily infections, ${\mathcal{T}}_{ij}/{\mathcal{B}}_i$.

Figure 3.

Visualization of mutual municipality influences. Here, we provide a visualization of the most relevant results from figure 2. Only non-autochthonous influences above 5% were considered. The thickness of the lines connecting municipalities is proportional to the number of infections that one generates on the other. The colour of each line represents the municipality responsible for generating the infections.

Figure 4.

Reconstruction of the time series and reproduction numbers for SG (a,c) and SJdM (b,d). Black dots represent, in (a,b), the daily new infections and, in (c,d), the sum of all the reproduction numbers related to new infection on the municipality, i.e. $\sum _j{\mathcal{R}}_{ij}$. Blue dots indicate, in (a,b), the amount of new infections generated in the respective cities due to commute flow with RJ and, in (c,d), the reproduction number related to the new infections in the municipality due to RJ. Red dots correspond, in (a,b), to the number of new infections due to Nt in SG (a), and due to NI in SJdM (b), and, in (c,d), to the reproduction number related to the new infections in the municipality due to Nt in SG (c), and due to NI in SJdM (d). In (a,b), the grey lines contouring the blue shaded areas indicate the number of new infections of that given municipality.