15  Vaccine performance

15.1 Vaccine efficacy

Definition 15.1 Efficacy is the percentage reduction in the disease rate (or infection risk (Shim & Galvani, 2012)) among vaccinated individuals compared to similar unvaccinated individuals (Verani et al., 2017).

15.2 Vaccine effectiveness

Definition 15.2 Effectiveness is the percentage reduction in the disease rate (or infection risk (Shim & Galvani, 2012)) among vaccinated individuals compared to unvaccinated individuals, similar to efficacy, but in real-world conditions (Verani et al., 2017).

Vaccine effectiveness is categorized into 4 types:

  1. Direct
  2. Indirect
  3. Total
  4. Overall

Let \(R_v\) be the infection risk in the vaccinated group and \(R_u\) be the infection risk in the unvaccinated group.

The absolute reduction in infection risk due to vaccination is \(R_u - R_v\).

Vaccine efficacy or effectiveness \(VE\) is the percentage reduction in infection risk, so we divide the absolute reduction by the infection risk of the unvaccinated group.

\[\begin{align} VE &= \frac{R_u - R_v}{R_u} \\ &= 1 - \frac{R_v}{R_u} \\ &= 1 - \text{Relative risk} \end{align}\]

Let have an example:

  • In the control group, the infection risk is \(\frac{4}{8}\) as 4 out of 8 people are infected.
  • In the vaccinated group, the infection risk is \(\frac{1}{8}\) as 1 person is infected among 8 people.

\[VE = \frac{\frac{4}{8} - \frac{1}{8}}{\frac{4}{8}} = \frac{4 - 1}{4} = \frac{3}{4} = 0.75\]

While 4 people get infected without vaccination, only 1 would get infected if they were vaccinated. The vaccine can prevent 3 of these infections. So the vaccine efficacy is \(\frac{3}{4}\) or 75%.

15.3 Measure vaccine effectiveness

15.3.1 Screening method

  1. We have data of all or a random sample of disease cases over a given period in a population, from which we know the proportion vaccinated (\(PCV\)) of these cases.

  2. We also know the proportion of the population vaccinated (\(PPV\)).

Then the vaccine effectiveness (\(VE\)) is calculated as:

\[\begin{align} VE &= 1 - \frac{\frac{PCV}{1 - PCV}}{\frac{PPV}{1 - PPV}} = 1 - \frac{PCV}{1 - PCV} \times \frac{1 - PPV}{PPV} \\ &= \frac{PPV(1 - PCV) - PCV(1 - PPV)}{PPV(1 - PCV)} \\ &= \frac{PPV - PPV \times PCV - PCV + PCV \times PPV}{PPV(1 - PCV)} \\ &= \frac{PPV - PCV}{PPV(1 - PCV)} \end{align}\]

In which:

  • \(\frac{PCV}{1 - PCV}\) is the odd of vaccination in disease cases group.
  • \(\frac{PPV}{1 - PPV}\) is the odd of vaccination in the population.
  • \(\frac{\frac{PCV}{1 - PCV}}{\frac{PPV}{1 - PPV}} = \frac{PCV}{1 - PCV} \times \frac{1 - PPV}{PPV}\) is the odds ratio of vaccination between 2 groups (cases vs population).
birth_cohort <- c(1980:1986)
cases <- c(82, 98, 180, 177, 112, 140, 151)
vac <- c(5, 9, 28, 37, 22, 27, 27)
cov <- c(70, 70.9, 76, 81, 83.7, 84.5, 83.1)
df <- data.frame(birth_cohort, cases, vac, cov)
df
df$pcv <- df$vac/df$cases
df$ppv <- df$cov / 100
df$rr <- (df$pcv / (1 - df$pcv)) / (df$ppv / (1 - df$ppv))
lm(rr ~ birth_cohort, data = df)

Call:
lm(formula = rr ~ birth_cohort, data = df)

Coefficients:
 (Intercept)  birth_cohort  
   -3.029905      0.001551  
1 - exp(-3.029905)
[1] 0.9516798

15.3.2 Sensitivity analysis

ve_screening <- function(pcv, ppv) {
  1 - (pcv / (1 - pcv)) / (ppv / (1 - ppv))
}

pcv <- seq(0.01, 0.99, 0.01)
ppv <- seq(0.01, 0.999, 0.01)
df <- expand.grid(pcv, ppv)
colnames(df) <- c("pcv", "ppv")

df$ve <- ve_screening(pcv = df$pcv, ppv = df$ppv)

# 
df <- df[df$ppv >= df$pcv,]
Code 
ggplot(df, aes(x = pcv, y = ppv, z = ve)) +
  geom_contour_filled() +
  scale_x_continuous(expand = c(0, 0)) +
  scale_y_continuous(expand = c(0, 0)) +
  labs(x = "PCV", y = "PPV", fill = "Vaccine effectiveness") +
  theme_classic()

Code 
plot_ly(
  df,
  x = ~ve,
  y = ~pcv,
  z = ~ppv,
  intensity = ~ve,
  type = "mesh3d",
  hovertext = paste0(
    "<b>ppv:</b> ",
    df$ppv,
    "<br>",
    "<b>pcv:</b> ",
    df$pcv,
    "<br>",
    "<b>ve</b> ",
    df$ve,
    "<br>"
  ),
  hoverinfo = "text",
  showscale = F
)

15.4 Vaccine impact

Definition 15.3 Impact quantifies the reduction in disease rate (either percentage decline or absolute change in disease rate) at the population level after vaccine introduction. It is determined by: (1) vaccine effectiveness, (2) vaccine coverage and (3) herd immunity (Verani et al., 2017).