Table 3.

Parameters of the model described in figure 1 and equation (3.1). The probability with which masking causes infections to be mild ($m$) is unknown. Our default value is chosen to be substantial so that for illustrative graphs constructed with fixed $m$ (figures 5, 6 in §5) the effect of mask-induced variolation is non-negligible. The recovery rates can be interpreted as the rates of ‘recovery or death’ since we do not explicitly model disease-induced mortality (cf. final paragraph of §3). The death rate μ refers to mortality from causes other than the focal disease. Note that mild illness is assumed to be associated with mild infectiousness. All birth rates were estimated for the years 2015–2020. We use a default latent period of T_lat = 3.7 days for all variants. The generation interval for an SEIR model is T_lat + T_inf [49, eqn (4.1)]. Setting γ_x = 1/T_gen,x in our model yields dynamics more similar to an SEIR version (cf. [49,52]), so it is a better approximation of the real world than an SIR version with 1/γ taken to be the observed mean infectious period. The transmission rate for severe infections (${\beta }_{\mathrm{s}}$) is set for each variant using equation (4.1) with $m=0$ and the associated ${\mathcal{R}}_0$ estimate listed in table 2. After specifying ${\beta }_{\mathrm{s}}$, we then set ${\beta }_{\mathrm{m}}=({\beta }_{\mathrm{m}}/{\beta }_{\mathrm{s}})\times {\beta }_{\mathrm{s}}$.

parameter	meaning	expression or default value
$m$	probability that an infected individual develops mild illness	0.6
${\beta }_{\mathrm{m}}$	transmission rate from mildly infectious individuals	equation (4.1)
${\beta }_{\mathrm{s}}$	transmission rate from severely infectious individuals	equation (4.1)
${\beta }_{\mathrm{m}}/{\beta }_{\mathrm{s}}$	ratio of transmission rates	1/2
${\gamma }_{\mathrm{m}}$	recovery rate from mild infections	1/Tgen,m
${\gamma }_{\mathrm{s}}$	recovery rate from severe infections	1/Tgen,s
ν	per capita annual birth rate [53]	0.0105 (Canada)
		0.0115 (UK)
		0.012 (USA)
μ	per capita annual death rate	ν
δ	rate of decay of immunity	1/Timm