Table 3.

Model parameters estimated by Bayesian inference based on daily deaths until 1 July 2020, assuming that after first lockdown ${\mathcal{R}}_{\mathrm{c}}(t)$ begins a gradual return to the baseline ${\mathcal{R}}_0$ at a fixed rate (corresponding to $T_2=120$ days in this case). Model selection based on maximum log-likelihood (LL) and Akaike information criterion (AIC). Best fitting models have lower AIC scores (bold). Infection fatality ratio, $\varphi =0.9\%$. Herd immunity threshold ($\mathcal{H}$) calculated from ${\mathcal{R}}_0$ and CV using formulas (14) or (22), as appropriate. $T_0$ and $T_2$ parameterise linear reduction and increase in transmissibility, respectively, before and after first lockdown (larger T$\iff$ lower slope; Fig. 6).

	Heterogeneous susceptibility		Heterogeneous connectivity		Homogeneous
	Median	$95\%$ CI	Median	$95\%$ CI	Median	$95\%$ CI
Common parameters
$c_1$a	$0.3133$	$(0.2989,0.3277)$	$0.3182$	$(0.3002,0.3419)$	$0.2358$	$(0.2291,0.2419)$
$\eta$b	15	$\{15\}$	15	$\{13,14,15\}$	17	$\{17\}$
$\nu$c	$2.521$	$(2.369,2.653)$	$1.822$	$(1.681,2.011)$	0	–
England
$T_0$	$-14.79$	$(-14.991,-13.922)$	$-14.815$	$(-14.994,-13.952)$	$-11.51$	$(-13.00,-10.17)$
${\mathcal{R}}_0$	$3.438$	$(3.380,3.470)$	$3.451$	$(3.383,3.533)$	$3.017$	$(2.940,3.109)$
$\mathcal{H}$	$15.46\%$	$(14.06\%,17.15\%)$	$14.97\%$	$(12.55\%,17.29\%)$	$66.85\%$	$(65.99\%,67.84\%)$
Scotland
$T_0$	$-2.253$	$(-3.033,-2.010)$	$-2.229$	$(-2.930,-2.007)$	$-4.407$	$(-5.651,-4.017)$
${\mathcal{R}}_0$	$3.308$	$(3.263,3.358)$	$3.319$	$(3.268,3.388)$	$3.234$	$(3.194,3.302)$
$\mathcal{H}$	$15.01\%$	$(13.68\%,16.74\%)$	$14.53\%$	$(12.22\%,16.77\%)$	$69.08\%$	$(68.69\%,69.71\%)$
Model selection
LL	$-951.7$		$-960.5$		$-3308$
AIC	$\mathbf{1917}$		1935		6627

^aTransmissibility reduction due to lockdown 1.

^bDifference between mean-time-to-death and mean-time-to-recovery (sampled from a continuous interval and reduced to the nearest integer before entering the model).

^cCoefficient of variation (CV).