Approximation of the infection-age-structured SIR model by the conventional SIR model of infectious disease epidemiology

Abstract

During the SARS-CoV-2 pandemic, the effective reproduction number (R-eff) has frequently been used to describe the course of the pandemic. Analytical properties of R-eff are rarely studied. We analytically examine how and under which conditions the conventional susceptible-infected-removed (SIR) model (without infection age) serves as an approximation to the infection-age-structured SIR model. Special emphasis is given to the role of R-eff, which is an implicit parameter in the infection-age-structured SIR model and an explicit parameter in the approximation. The analytical findings are illustrated by a simulation study about an hypothetical intervention during a SARS-CoV-2 outbreak and by historical data from an influenza outbreak in Prussian army camps in the region of Arnsberg (Germany), 1918-1919.

Keywords: Lexis diagram; SARS-CoV-2; Spanish flu; effective reproduction number; influenza; net reproduction number.

Figure 1

Reproduction number $\mathcal{R}$ (ordinate) over calendar time t (abscissa, in days) in the simulation [cf. Figure 5 in Supporting Information S1 of Brinks et al. (6)].

Figure 2

Number of infected people $I$ over calendar time $t$ in the simulation. The blue curve corresponds to the exact solution (Equation 1) while the black curve is the approximation via Equation 20.

Figure 3

Approximate effective reproduction number $\mathcal{R}$ after 9 September 1918 (in days) for an influenza outbreak in Prussian army camps.

Figure 4

Number of infected people $I$ (ordinate) over calendar time $t$ in Prussian army camps in the region of Arnsberg (Germany). The blue curve corresponds to estimated numbers according to Nishiura (12) and the black curve is the approximation via Equation 20.