Comparing vaccination strategies

Last updated on 2025-03-05 | Edit this page

Estimated time: 70 minutes

Overview

Questions

  • What are the direct and indirect effects of vaccination?
  • What are the benefits of targeted vaccination programs?
  • What happens if we combine vaccination and NPIs?

Objectives

  • Understand the potential impact of vaccination programs
  • Implement vaccination programs and NPI measures simultaneously using {epidemics}

Prerequisite

Learners should familiarise themselves with following concept dependencies before working through this tutorial:

Outbreak response : Intervention types.

Introduction

Vaccine programs can be used to help control epidemics via direct and indirect protection from infection. We can use mathematical models to make predictions about the potential impact of different vaccination strategies.

Depending on how infections spread, targeting specific risk groups for vaccination may be a more effective strategy than vaccinating everyone. When vaccine programs are not immediately available to be rolled out, either from a capacity or development perspective, we may also want to know how non-pharmaceutical interventions (NPIs) can be used to control the epidemic in the mean time, as well as after a vaccine program has started.

In this tutorial we will compare different vaccination strategies using models from {epidemics}. We will use the following R packages:

R 

library(ggplot2)
library(epidemics)
library(tidyverse)
library(dplyr)

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Take a look to the Contributing.md file for more writing guidelines.

Direct and indirect effects of vaccination

Vaccination programs using infection-blocking vaccines have two benefits:

  1. Reducing individual risk of infection (direct effect of vaccination)
  2. Reducing onward contribution to transmission (indirect effect of vaccination).

We will illustrate this using results from model_default() in {epidemics}. Using the contact matrix and parameters from the COVID-19 outbreak example in Modelling interventions we will investigate the impact of two vaccination programs on the number of infections. We will assume that vaccination programs start on day 0 and continue to be in place for 150 days. We will assume all age groups are vaccinated at the same rate in each program as follows :

  • vaccination program 01 : vaccination rate 0.001
  • vaccination program 02 : vaccination rate 0.01.

R 

# prepare vaccination objects
vaccinate_01 <- vaccination(
  name = "vaccinate all",
  time_begin = matrix(0, nrow(contact_matrix)),
  time_end = matrix(150, nrow(contact_matrix)),
  nu = matrix(c(0.001, 0.001, 0.001))
)

vaccinate_02 <- vaccination(
  name = "vaccinate all",
  time_begin = matrix(0, nrow(contact_matrix)),
  time_end = matrix(150, nrow(contact_matrix)),
  nu = matrix(c(0.01, 0.01, 0.01))
)

We see the intuitive result that the higher the vaccination rate, the more people that are vaccinated.

To understand the indirect effect of vaccinations, we want to know the effect that vaccination has on transmission, and hence the rate at new infections occur. We will use the function new_infections() in {epidemics} to calculate the number of new infections over time for the different vaccination programs.

The inputs required are :

  • data : the model output,
  • compartments_from_susceptible : this is an optional input, but in our case needed. We don’t want the number of people vaccinated to be counted as new infections, so we need to specify the name of the model compartment where individuals transition out from susceptible (in this example vaccinated),
  • by_group : should the results be calculated for each demographic group separately.

R 

vaccinate_01_infections <- new_infections(output_vaccinate_01,
                                          compartments_from_susceptible =
                                            "vaccinated",
                                          by_group = FALSE)

We see that after accounting for the number of people vaccinated, there are fewer new infections when you have a higher vaccination rate. This is because there are fewer people susceptible to infection, and therefore fewer people who can become infected and contribute to onward infection.

This indirect effect of vaccination programs is key to successfully controlling and eradicating infections.

To evaluate the impact of the vaccination programs, we are often interested in both the peak size (i.e. healthcare pressure at single point in time) as well as overall epidemic size (i.e. cumulative number of infections).

We can find the cumulative sum using the R function cumsum() and use purr::map_dfr() to loop over a list of new infection data frames. We can see the difference in infection numbers is by several orders of magnitude.

R 

# create function that returns the intervention type and cumulative sum for
# given infections
find_cumsum <- function(infections) {
  return(data.frame(intervention_type = unique(infections$intervention_type),
                    cumulative_sum = tail(cumsum(infections$new_infections),
                                          n = 1)))
}

# create list of interventions
interventions <- lst(baseline_infections,
                     vaccinate_01_infections,
                     vaccinate_02_infections)

# apply function to each data frame in the list
map_dfr(interventions, find_cumsum)

OUTPUT 

  intervention_type cumulative_sum
1          baseline    53478985.93
2   vaccinate 0.001    44803253.91
3    vaccinate 0.01       34550.48

Herd immunity threshold

There exists a target threshold of vaccination based on the basic reproduction number \(R_0\) to achieve herd immunity .

The proportion of the population (\(p\)) that needs to be immune to achieve herd immunity is:

\[p = 1- \frac{1}{R_0}. \]

For details of the mathematical derivation check out this Plus Magazine article.

Targeted vaccination

For infections that have a higher burden in different risk groups, targeted vaccination programs can be utilised to control infection. For example, if there is heterogeneity in contacts by age, targeting different age groups for vaccination will result in different disease trajectories.

We will use the same contact matrix as above from Modelling interventions :

R 

contact_matrix

OUTPUT 

                 age.group
contact.age.group    [0,15)  [15,65)       65+
          [0,15)  6.8461538 1.655174 0.5892799
          [15,65) 6.0737213 9.169207 4.1223575
          65+     0.5258855 1.002545 1.7142857

There is higher levels of mixing within the 0-15 and 15-65 age group, than in the 65+ age a group so we expect that targeting different age groups will result in different disease trajectories.

To show the effect of targeted vaccination, we will compare the following scenarios:

  • vaccinating all age groups at a rate 0.001,
  • vaccinating ages 0-15 only (group 1) at a rate 0.003,
  • vaccinating ages 15-65 only (group 2) at a rate 0.003,
  • vaccinating ages 65+ only (group 3) at a rate 0.003.

Vaccinating group 3 only results in the highest epidemic peak size. Targeting group 2 results in the smallest epidemic peak size. Whereas targeting all age groups pushes the epidemic peak time later. As before, let’s compare the cumulative number of infections under the different interventions:

R 

interventions_targetted <- lst(baseline_infections,
                               vaccinate_01_infections,
                               vaccinate_group_1_infections,
                               vaccinate_group_2_infections,
                               vaccinate_group_3_infections)

# apply function to each data frame in the list
map_dfr(interventions_targetted, find_cumsum)

OUTPUT 

  intervention_type cumulative_sum
1          baseline       53478986
2   vaccinate 0.001       44803254
3 vaccinate group 1       50429226
4 vaccinate group 2       42035053
5 vaccinate group 3       51634221

Age specific infection-fatality-risk

Targeting specific age groups can reduce the impact of infections that have age specific risk of mortality. To illustrate this, we can define an age specific infection-fatality-risk (IFR) :

R 

ifr <- c(0.00001, 0.001, 0.4)
names(ifr) <- rownames(contact_matrix)

To convert infections to deaths, we will need new infections by age group, so we call new_infections() with by_group = TRUE and then multiple the new infections by the IFR for that age group:

R 

vaccinate_group_1_age <- new_infections(output_vaccinate_group_1,
                                        compartments_from_susceptible =
                                          "vaccinated", by_group = TRUE)

vaccinate_group_1_deaths <-
  1:3 %>%
  map_dfr(function(x) vaccinate_group_1_age %>%
            filter(demography_group == names(ifr)[x]) %>%
            mutate(deaths = new_infections * ifr[x]))

Vaccinating all age groups has the greatest impact on deaths in the eldest age group (over 65 years of age), but we also see that vaccinating group 2 reduces the deaths in the eldest age group. Targeting younger groups can the number of deaths in the eldest age group (over 65 years of age).

R 

baseline_deaths$intervention_type  <- "baseline"
vaccinate_01_deaths$intervention_type  <- "vaccinate 0.001"
vaccinate_group_1_deaths$intervention_type  <- "vaccinate group 1"
vaccinate_group_2_deaths$intervention_type  <- "vaccinate group 2"
vaccinate_group_3_deaths$intervention_type  <- "vaccinate group 3"

output_deaths <- rbind(baseline_deaths,
                       vaccinate_01_deaths,
                       vaccinate_group_1_deaths,
                       vaccinate_group_2_deaths,
                       vaccinate_group_3_deaths)

output_deaths %>%
  filter(demography_group == "65+") %>%
  ggplot() +
  ggtitle("Deaths") +
  geom_line(
            aes(time,
              deaths,
              colour = intervention_type,
              linetype = demography_group
            ),
            linewidth = 1) +
  theme_bw() +
  labs(
    x = "Simulation time (days)",
    y = "Individuals"
  )

Vaccination versus NPIs

Modelling is a useful tool for understanding the potential impact of the timing and different combinations of control measures. In an outbreak scenario, vaccines can be used alongside other control measures. A recent study by Barnsley et al. (2024) used modelling to show the impact the COVID-19 vaccine could have had alongside NPIs if the vaccine had been developed within 100 days of the pathogen threat being recognised. We will use this study as inspiration to show the impact of a vaccine versus NPI implemented at different times.

The NPI we will consider is temporarily closing schools, an intervention that has been used as a control measure for seasonal influenza (Cowling et al. 2008). Specifically, we will model a school closure that will reduce the contacts between school aged children (aged 0-15) by 0.5, and will cause a small reduction (0.01) in the contacts between adults (aged 15 and over). For the vaccine program, we assume an equal vaccination rate of 0.01 in each age group.

We define early implementation as the first day of the simulation (i.e. when there are less than 100 infections in total), and late implementation as 50 days after the start of the simulation (i.e. when there are approximately 50,000 infections in total). We assume the control measures are in place for 100 days after they start.

The combinations we will consider are :

  • early implementation of closing schools,
  • early implementation of a vaccine,
  • late implementation of a vaccine,
  • early implementation of closing schools and late implementation of a vaccine.

The code below details how the interventions are simulated:

R 

early_start <- 0
late_start <- 50
duration <- 100
time_end <- 200

# close schools early
close_schools_early <- intervention(
  name = "School closure",
  type = "contacts",
  time_begin = early_start,
  time_end = early_start + duration,
  reduction = matrix(c(0.5, 0.01, 0.01))
)

# vaccination late
vacc_late <- vaccination(
  name = "vaccinate late",
  time_begin = matrix(late_start, nrow(contact_matrix)),
  time_end = matrix(late_start + duration, nrow(contact_matrix)),
  nu = matrix(c(0.01, 0.01, 0.01))
)

# npis started early + vaccination late
output_npi_early_vacc_late <- model_default(
  population = uk_population,
  transmission_rate = transmission_rate,
  infectiousness_rate = infectiousness_rate,
  recovery_rate = recovery_rate,
  vaccination = vacc_late,
  intervention = list(contacts = close_schools_early),
  time_end = time_end, increment = 1.0
)

Early implementation of NPIs (early npi) delays the timing of the peak of infectious individuals but does not decrease the magnitude by much once the measure is lifted. Early implementation of the vaccine (early vacc) is the most effective control for reducing the peak number of infectious individuals, because it reduces the size of the susceptible group rather than just temporarily keeping infections away from it. Having an NPI in place before the vaccine is implemented (early npi, late vacc) is more effective than simply just implementing the vaccine late (late vacc).

Lifting NPI measures

Investigate the impact lifting the after implementing the vaccine. Adapt the code below to lift the NPI after 20, 50 and 100 days

R 

duration_20 <- 20
duration_50 <- 50

close_schools_early_20 <- intervention(
  name = "School closure",
  type = "contacts",
  time_begin = early_start,
  time_end = early_start + duration_20,
  reduction = matrix(c(0.5, 0.01, 0.01))
)

close_schools_early_50 <- intervention(
  name = "School closure",
  type = "contacts",
  time_begin = early_start,
  time_end = early_start + duration_50,
  reduction = matrix(c(0.5, 0.01, 0.01))
)

output_npi_early_vacc_late_20 <- model_default(
  population = uk_population,
  transmission_rate = transmission_rate,
  infectiousness_rate = infectiousness_rate,
  recovery_rate = recovery_rate,
  vaccination = vacc_late,
  intervention = list(contacts = close_schools_early_20),
  time_end = 300, increment = 1.0
)

output_npi_early_vacc_late_50 <- model_default(
  population = uk_population,
  transmission_rate = transmission_rate,
  infectiousness_rate = infectiousness_rate,
  recovery_rate = recovery_rate,
  vaccination = vacc_late,
  intervention = list(contacts = close_schools_early_50),
  time_end = 300, increment = 1.0
)

output_npi_early_vacc_late_20$intervention_type <- "20 days"
output_npi_early_vacc_late_50$intervention_type <- "50 days"
output_npi_early_vacc_late$intervention_type <- "100 days"

output_npis <- rbind(output_npi_early_vacc_late_20,
                     output_npi_early_vacc_late_50,
                     output_npi_early_vacc_late)

output_npis %>%
  filter(compartment == "infectious") %>%
  ggplot() +
  aes(
    x = time,
    y = value,
    color = intervention_type,
    linetype = intervention_type
  ) +
  stat_summary(
    fun = "sum",
    geom = "line",
    linewidth = 1
  ) +
  scale_y_continuous(
    labels = scales::comma
  ) +
  geom_vline(
    xintercept = c(
      early_start,
      early_start + duration
    ),
    linetype = 2
  ) +
  geom_vline(
    xintercept = c(
      late_start,
      late_start + duration
    ),
    linetype = 3
  ) +
  theme_bw() +
  labs(
    x = "Simulation time (days)",
    y = "Individuals"
  )

Summary

Using model generated disease trajectories we have investigated the potential impact of different vaccination strategies. When infection has group specific risk, targeted vaccination programs may be more effective in reducing epidemic peak. NPIs can be used alongside vaccine development to delay epidemic peaks before vaccines are readily available.

We used a SEIR model with a vaccination class, but if we modelled immunity to infection then we could investigate the impact of a vaccination program if it was implemented very late in the epidemic (i.e. after post-infection immunity had accumulated). See for example, Baguelin et al. 2010

In this tutorial we have investigated the potential effect of infection blocking vaccines, but we can use models to explore other effects of vaccines including imperfect vaccines. The disease trajectories generated in this tutorial can be used to calculate measures such as DALYs averted as part of health economic analyses.

Key Points

  • Herd immunity is a indirect effect of vaccination programs
  • Targeted vaccination programs have benefits when there is heterogeneity in contacts
  • The timing of implementation of vaccination programs and NPIs can result in very different disease trajectories