1. Number of events averted by vaccination (NAE)

The number of events averted by vaccination at time \(t\) is calculated as:

\[NAE_t = e_{A(t)} \frac{VC_t VE_t}{(1 - VC_t VE_t)}\]

with:

• \(e_{A(t)}\): Number of events at time \(t\)
• \(VC_t\): Vaccine coverage at time \(t\)
• \(VE_t\): Vaccine effectiveness at time \(t\)

2. Number of avertable events considering an increase in final coverage (NAbE)

The number of averted events with alpha parameter at time \(t\):

\[NAbE_t(\alpha) = e_{A(t)} \frac{VC(\alpha)_t VE_t}{(1 - VC(\alpha)_t VE_t)}\]

with:

• \(e_{A(t)}\): Number of events at time \(t\)
• \(VC(\alpha)_t\): Hypothetical vaccine coverage during time \(t\) if the vaccine coverage was increased by a factor of alpha parameter
• \(VE_t\): Vaccine effectiveness at time \(t\)

\(VC(\alpha)_t\) is computed as:

\[VC_t(\alpha) = \sum_{i=1}^{t} (VC_t - VC_{t-1}) \left(1 + \frac{\alpha}{VC_T}\right)\]

with:

• \(VC_t\): Vaccine coverage at time \(t\) (\(VC_0 = 0\))
• \(VC_{t-1}\): Vaccine coverage at time \(t-1\)
• \(\alpha\): Alpha parameter
• \(VC_T\): Maximum vaccine coverage

3. Number Needed to Vaccinate (NNV)

The number needed to vaccinate at time \(t\) is calculated as:

\[NNV_t = \frac{1}{R_{B(t)} VE_t}\]

where \[R_{B(t)} = \frac{e_{A(t)} + NAE_t}{N_e}\]

with:

• \(VE_t\): Vaccine effectiveness at time \(t\)
• \(e_{A(t)}\): Number of events at time \(t\)
• \(NAE_t\): Number of events averted by vaccination at time \(t\)
• \(N_t\): Population at risk at time \(t\)