Table 2.

Prior distributions and initial values (obtained from CIPHA data) used for model fits

(a) Prior distributions used for initialising model fits
Name	$\gamma$	Prior distributions			Initial values
		$\beta$	$\mathbf{I},{\mathbf{I}}_{\mathbf{ij}}$	${\mathbf{e}}_{\mathbf{ij}},{\mu }_{\mathbf{i}}$	$\mathbf{S},{\mathbf{S}}_{\mathbf{ij}},\mathbf{R},{\mathbf{R}}_{\mathbf{ij}}$
Core Model	1/7	$\text{Unif}[0,10]$	$\text{Unif}[0,100]$	$\text{Unif}[0,1]$	Table 2b
Sensitivity to $\gamma$	1/3	$\text{Unif}[0,10]$	$\text{Unif}[0,100]$	$\text{Unif}[0,1]$	Table 2b
Sensitivity to $\gamma$	1/11	$\text{Unif}[0,10]$	$\text{Unif}[0,100]$	$\text{Unif}[0,1]$	Table 2b
Double Removed	1/7	$\text{Unif}[0,10]$	$\text{Unif}[0,100]$	$\text{Unif}[0,1]$	Table 2c

(b) Initial values when assuming all infected from start of pandemic moved to removed class
S	${\mathbf{S}}_{\mathbf{1}\mathbf{A}}$	${\mathbf{S}}_{\mathbf{2}\mathbf{A}}$	${\mathbf{S}}_{\mathbf{1}\mathbf{P}}$	${\mathbf{S}}_{\mathbf{2}\mathbf{P}}$	R	${\mathbf{R}}_{\mathbf{1}\mathbf{A}}$	${\mathbf{R}}_{\mathbf{2}\mathbf{A}}$	${\mathbf{R}}_{\mathbf{1}\mathbf{P}}$	${\mathbf{R}}_{\mathbf{2}\mathbf{P}}$
1266979	478347	312594	74665	384481	70887	42715	20895	6738	26715

(c) Initial values when assuming infected individuals are underestimated by 50%
S	${\mathbf{S}}_{\mathbf{1}\mathbf{A}}$	${\mathbf{S}}_{\mathbf{2}\mathbf{A}}$	${\mathbf{S}}_{\mathbf{1}\mathbf{P}}$	${\mathbf{S}}_{\mathbf{2}\mathbf{P}}$	R	${\mathbf{R}}_{\mathbf{1}\mathbf{A}}$	${\mathbf{R}}_{\mathbf{2}\mathbf{A}}$	${\mathbf{R}}_{\mathbf{1}\mathbf{P}}$	${\mathbf{R}}_{\mathbf{2}\mathbf{P}}$
1196092	435632	291699	67927	357766	141774	85430	41790	13476	53430

Notation is as follows $\beta$: Transmission rate, $\gamma$: Recovery rate, $e_{\mathit{ij}}$: Effectiveness of i doses of vaccine j against infection, S: Unvaccinated susceptible individuals, $S_{\mathit{ij}}$: Susceptible individuals who have received i doses of vaccine j, I: Unvaccinated infected individuals, $I_{\mathit{ij}}$: Infected individuals who have received i doses of vaccine j, R: Unvaccinated removed individuals, $R_{\mathit{ij}}$: Removed individuals who have received i doses of vaccine j, and ${\mu }_i$: Infectiveness of an infected individual with i doses of either vaccine. For number of doses we have $i\in \{1,2\}$ and for vaccines $j\in \{A,P\}$ for Oxford-AstraZeneca and Pfizer-BioNTech respectively