Fig. 1.

Upper row: Decadic logarithm of the reduced peak time ${\tau }_{\mathrm{peak}}$ of the infected compartment versus $k$ and ${log}_{10}\left(\eta \right)$. (a) exact numerical solution of the SIR model, approximants (b) ${\tau }_{\mathrm{peak}}^{\mathrm{MT}}$ given by Eq. (14), (c) ${\tau }_{\mathrm{peak}}^{\mathrm{SK-I}}$ given by Eq. (19), (d) ${\tau }_{\mathrm{peak}}^{\mathrm{SK-II}}$ given by Eq. (20), and (e) ${\tau }_{\mathrm{peak}}^{\mathrm{CG}}$ given by Eq. (24). The dimensional peak time $t_{\mathrm{peak}}$ is obtained from ${\tau }_{\mathrm{peak}}$ upon dividing ${\tau }_{\mathrm{peak}}$ by the rate $a=\beta (s_0+i_0)$. A peak time exists only within the regime $\eta >1-k$. The corresponding relative deviations between approximant and exact solution are shown in the 2nd row (f)–(j). While the performance of the MT approximant is more accurate than 4.3% over the shown $k$-$\eta$-domain, the SK and CG approximants perform better than MT except within the regime of relatively large $\eta$ close to $1-k$. This is made more precise in Fig. 2. The color bar for all panels of the first row is shown in (a), the single color bar for the 2nd row is shown in (g). The latter goes from blue (high quality, $<1\text{\%}$ deviation from the exact solution) to yellow (low quality, $>5\text{\%}$ deviation).