remotes::install_github("QuartzSoftwareLLC/qaverted")
Use the qaverted function to run find the number of averted events based on the Averted Formula derived below.
| Variable |
Description |
| $A$ |
Events Averted |
| $E$ |
Current Vaccine Effectiveness |
| $E_n$ |
New Vaccine Effectiveness |
| $I_v$ |
Incidence in vaccinated population |
| $u$ |
Vaccine uptake |
$$I_v = \frac{(1 - E) * u * I}{1 - u + u * (1 - E)} $$
$$ A = (1 - \frac{1 - E_n}{1 - E})* I_v$$
qaverted(0.6, 0.7, 0.5, 1e6)
Using unvaccinated attack $A_{uv}$ and vaccinated attack rate $A_v$, we can calculate vaccine efficacy $E$.
$$E = \frac{A_{uv} - A_v}{A_{uv}}$$
Which can be simplified to
$$ E = 1 - \frac{A_v}{A_{uv}}$$
Attack rate ($A$) can be calculated from incidence ($I$) and population ($P$):
$$A = I / P$$
Furthermore vaccinated population ($P_v$) and unvaccinated population ($P_{uv}$) can be calculated using total population ($P$) and vaccine uptake $u$.
$$P_v = P * u$$
$$P_{uv} = P * (1 - u)$$
Consequently,
$$E = 1 - \frac{I_v}{P*u} / \frac{I_{uv}}{P (1 - u)}$$
$I_{uv}$ can be calculated as the difference between total incidence ($I$) and vaccinated incidence ($I_v$)
$$E = 1 - \frac{I_v}{P*u} / \frac{I - I_{v}}{P (1 - u)}$$
Solving for ($I_v$)
$$I_v = \frac{(1 - E) * u * I}{1 - u + u * (1 - E)} $$
After solving for ($I_v$) we calculate the number of cases in the vaccinated population if efficacy were 0 ($I_{E0}$)
$$ I_{E0} = \frac{I_v}{1 - E} $$
Next we calculate the number of predicted cases ($I_n$) at the new efficacy provided ($E_n$)
$$ I_n = I_{E0} * (1 - E_n) $$
Lastly we take the difference between ($I_n$) and ($I_v$) to calculate the change in incidence ($I_{\Delta}$)
$$I_{\Delta} = I_n - I_v$$
This simplifies further to
$$I_{\Delta} = I_{E0} * (1 - E_n) - I_v$$
Expanding $I_{E0}$
$$I_{\Delta} = \frac{I_v}{1 - E} * (1 - E_n) - I_v$$
Rearranging
$$I_{\Delta} = \frac{1 - E_n}{1 - E} * I_v - I_v$$
Factoring out $I_v$
$$I_{\Delta} = \frac{1 - E_n}{1 - E} * I_v - I_v$$
$$I_{\Delta} = (\frac{1 - E_n}{1 - E} - 1)* I_v$$
Lastly we can calculate events averted ($A$) by multiplying $I_\Delta$ by $-1$.
$$ A = - I_\Delta$$
$$ A = (1 - \frac{1 - E_n}{1 - E})* I_v$$
This leaves us with two final formulas required to calculate number averted ($A$)
$$ A = (1 - \frac{1 - E_n}{1 - E})* I_v$$
$$I_v = \frac{(1 - E) * u * I}{1 - u + u * (1 - E)} $$