TABLE 1.

Summary of the all-or-none and leaky vaccine models and the assumptions for the ecological vaccine model. $N$ is the total population; $x$ denotes the proportion of the population vaccinated (assumed constant over time); $\varphi$ is the vaccine effect on susceptibility; $S_{ut}$ and $S_{vt}$ denote the number of unvaccinated and vaccinated susceptibles at time $t$; $Y_{ut}$ and $Y_{vt}$ denote new cases in time $t$ among unvaccinated and vaccinated; and ${\mathit{\lambda }}_t^{†}$ is a generic force of infection

	All-or-none	Leaky
Initial susceptible population
	$S_{u0}(\varphi )=(1-x)N$	$S_{u0}=(1-x)N$
	$S_{v0}(\varphi )=(1-\varphi )xN$	$S_{y0}=xN$
Force of infection
	${\mathit{\lambda }}_{ut}^{†}={\mathit{\lambda }}_t^{†}$	${\mathit{\lambda }}_{ut}^{†}={\mathit{\lambda }}_t^{†}$
	${\mathit{\lambda }}_{vt}^{†}={\mathit{\lambda }}_t^{†}$	${\mathit{\lambda }}_{vt}^{†}=(1-\varphi ){\mathit{\lambda }}_t^{†}$
Progression
$Y_{u,t+1}\mid {\mathit{\lambda }}_{ut}^{†}$	$\text{Bin}\left(S_{ut}(\varphi ),1-e^{-{\mathit{\lambda }}_t^{†}}\right)$	$\text{Bin}\left(S_{ut},1-e^{-{\mathit{\lambda }}_t^{†}}\right)$
$Y_{v,t+1}\mid {\mathit{\lambda }}_{vt}^{†}$	$\text{Bin}\left(S_{vt}(\varphi ),1-e^{-{\mathit{\lambda }}_t^{†}}\right)$	$\text{Bin}\left(S_{vt},1-e^{-(1-\varphi ){\mathit{\lambda }}_t^{†}}\right)$
Implied aggregate model
$Y_{t+1}\mid {\mathit{\lambda }}_t^{†}$	$\text{Bin}\left(S_t(\varphi ),1-e^{-{\mathit{\lambda }}_t^{†}}\right)$	Convolution of binomials
Simplifying assumptions
Poisson's approximate binomials
	$\text{Poi}\left(S_t(\varphi )\left(1-e^{-{\mathit{\lambda }}_t^{†}}\right)\right)$	$\text{Poi}\left(S_{ut}\left(1-e^{-{\mathit{\lambda }}_t^{†}}\right)+S_{vt}\left(1-e^{-(1-\varphi ){\mathit{\lambda }}_t^{†}}\right)\right)$
Taylor approximation
	$\text{Poi}\left(S_t(\varphi ){\mathit{\lambda }}_t^{†}\right)$	$\text{Poi}\left(\left(S_{ut}+(1-\varphi )S_{vt}\right){\mathit{\lambda }}_t^{†}\right)$
Negligible number of infections
	$S_t(\varphi )\approx (1-\varphi x)N$	$S_{ut}\approx (1-x)N,S_{vt}\approx xN$
Ecological vaccine model
$Y_{t+1}\mid {\mathit{\lambda }}_t^{†},\varphi \sim \text{Poisson}\left({\mathit{\lambda }}_t^{†}(1-\varphi x)N\right)$