22  Model vaccination

NoteAcknowledgement

This chapter is based on the teachings of my PhD supervisor, Dr. Marc Choisy, and a lecture given by Asst Prof. Hannah Clapham at MIDSEA Summer School 2024, for which I am deeply grateful.

To model vaccination, we need to decide 2 things:

  1. What is the vaccine effect? Does it protect against infection, severe disease, transmission, or a combination?

  2. How do we model imperfect protection?

22.1 Vaccine effects

22.1.1 Prevent infection

Vaccine prevents the pathogen from establishing an infection in the body. A vaccinated person is less likely to get infected.

Assuming a perfect vaccine against infection, the model can be designed as follows:

Vaccinated people \(V\) remain uninfected in their lifetime and do not participate in disease transmission.

You can simplify this model by moving susceptible people to the \(R\) compartment because they also do not participate in disease transmission.

22.1.2 Prevent severe disease

Vaccine reduces the severity of the disease if the vaccinated person does get infected. A vaccinated individual still get infected but experiences a much milder form of the disease.

Assuming a perfect vaccine against severe disease, the model can be designed as follows:

The infected population is split into a proportion of \(p_s\) for severe cases and \(1 - p_s\) for mild cases. Vaccinated people \(S_v\) still get infected, but will only experience mild disease.

22.1.3 Prevent transmission

Vaccine reduces the ability of a vaccinated individual, who becomes infected, to transmit the pathogen to others. A vaccinated person still get infected, still get sick, but they are less likely to spread the disease to others.

Assuming a perfect vaccine against transmission, the model can be designed as follows:

\(I\) being the only source of infection that contributes to the force of infection. Vaccinated people \(S_v\) still get infected and become \(I_v\), but they do not transmit the disease.

22.2 Imperfect protections

What is meant by a vaccine with effectiveness of 80% protection against a specific clinical endpoint? The mechanism of vaccine action is typically modelled in two ways:

22.2.1 All-or-nothing model

All-or-nothing (AoN, also called the polarized (Park et al., 2023), or take) model assumes that among vaccinated individuals, a proportion \(VE_P\) are completely protected, while the remaining fraction \(1 - VE_P\) remains completely unprotected (Park et al., 2023; Zachreson et al., 2023).

An effectiveness of 80% here implies that among vaccinated people, 80% are completely protected, and 20% receive no protection (World Health Organization, 2013).

Assumming a standard SIR model, vaccine protects against infection with vaccine effectiveness \(VE_P\), \(\rho\) represents the vaccination rate:

  • Among the susceptible \(S\), \(\rho S\) are vaccinated.
  • Among those \(\rho S\) who are vaccinated, \(\rho VE_P S\) are completely protected and go directly to \(R_v\), the remaining is \(\rho (1 - VE_P) S\).
  • The remaining \(\rho (1 - VE_P) S\) are completely unprotected, and will be infected with force of infection \(\lambda(t)\) just like those who are unvaccinated.
Figure 22.1: All-or-nothing vaccination model (Park et al., 2023)

Below is a per-time-step simulation of Figure 22.1. Each dot is one person. Press Step to advance the model by one \(dt\), Play to run it continuously. At every step, transitions are sampled from the same rates that drive the ODE: \(\lambda(t) = \beta(I + I_v)/N\) moves susceptibles right, \(\gamma\) moves infecteds right, and \(\rho\) pulls \(S\) down — splitting \(\rho VE_P\) along the diagonal straight to \(R_v\) from the rest into \(S_v\). The floating “+N” badges show how many people took each pathway in that step, and the chart traces every compartment over time.

\[\begin{align} \frac{dS}{dt} &= -\lambda(t) S - \rho S \\ \frac{dI}{dt} &= \lambda(t) S - \gamma I \\ \frac{dR}{dt} &= \gamma I \\ \frac{dS_v}{dt} &= \rho (1 - VE_P) S - \lambda(t) S_v \\ \frac{dI_v}{dt} &= \lambda(t) S_v - \gamma I_v \\ \frac{dR_v}{dt} &= \rho VE_P S + \gamma I_v \end{align}\]

22.2.2 Leaky model

Leaky (or degree) model assumes that all vaccinated individuals are partially protected (Zachreson et al., 2023).

An effectiveness of 80% here implies that all vaccinated people have the endpoint of interest reduced by 80% compared to non-vaccinees.

The assumption that no vaccinated people is totally or permanently protected implies one or both of the following (World Health Organization, 2013):

  • No amount (titre) of the immune marker is totally protective or, if it is, no individual can maintain that titre for a long period (because of waning or transient immunosuppression)
  • The degree of protection is a function of the level of the immune marker - the simplest explanation being that protection is a function of both the level of the immune marker and the challenge dose.

Assumming a standard SIR model, vaccine protects against infection with vaccine effectiveness \(VE_L\), \(\rho\) represents the vaccination rate.

  • Among susceptible \(S\), \(\rho S\) are vaccinated.
  • Among those \(\rho S\) who are vaccinated, force of infection \(\lambda(t)\) is reduced by a factor of \(1 - VE_L\).
Figure 22.2: Leaky vaccination model (Park et al., 2023)

\[\begin{align} \frac{dS}{dt} &= -\lambda(t) S - \rho S \\ \frac{dI}{dt} &= \lambda(t) S - \gamma I \\ \frac{dR}{dt} &= \gamma I \\ \frac{dS_v}{dt} &= \rho S - (1 - VE_L) \lambda(t) S_v \\ \frac{dI_v}{dt} &= (1 - VE_L) \lambda(t) S_v - \gamma I_v \\ \frac{dR_v}{dt} &= \gamma I_v \end{align}\]

22.3 Mixed effects

Imperfect vaccines often have a combination of effects: they can protect against infection, severe disease, and transmission at the same time.

Assuming a vaccine with leaky effectiveness \(VE_L\) against infection and severe disease, the model can be designed as follows:

  • Among susceptible \(S\), \(\rho S\) are vaccinated.
  • Among those \(\rho S\) who are vaccinated, force of infection \(\lambda(t)\) is reduced by a factor of \(1 - VE_L\).
  • Vaccine reduces the proportion of severe disease from \(p_s\) to \(p_s'\).

22.4 Multiple doses

Some vaccines require multiple doses, like measles (2 doses) or rotavirus (2 doses), with varying effectiveness per dose. Below is the DynaMICE model (Verguet et al., 2015) for the 2-dose measles vaccine, assuming AoN effectiveness as in Figure 22.1 for both doses.

22.5 Code

22.5.1 A perfect vaccine against infection

library(odin)
library(tidyr)
library(ggplot2)

pfvac_ode <- odin({
  # Derivatives
  deriv(S) <- -beta * S * I - rho * S
  deriv(I) <- beta * S * I - gamma * I
  deriv(R) <- gamma * I
  deriv(V) <- rho * S
  
  # Initial conditions
  initial(S) <- N_init - I_init
  initial(I) <- I_init
  initial(R) <- R_init
  initial(V) <- V_init
  
  # Parameters and initial values
  beta <- user(8.77e-8)
  gamma <- user(0.2)
  rho <- user(0.005)
  N_init <- user(5700000)
  I_init <- user(1)
  R_init <- user(0)
  V_init <- user(0)
})
# Initialize model
pfvac_mod <- pfvac_ode$new()
# How long to run
times <- seq(0,300)
# Run the model
pred <- data.frame(pfvac_mod$run(times))
Code 
df_plot <- pivot_longer(pred, cols = S:V, names_to = "comp", values_to = "n")

ggplot(df_plot, aes(x = t, y = n, color = comp)) +
  geom_line(linewidth = 1.2) +
  scale_color_brewer(palette = "PuOr", breaks = c("S", "I", "R", "V")) +
  labs(color = NULL, y = NULL, x = "Time") +
  theme_minimal() +
  theme(legend.position = "bottom")

22.5.2 AoN model against infection

22.5.3 Leaky model against infection