Calculating the final size of an epidemic

When analysing an epidemic-prone infection, such as pandemic influenza, it is important to understand how many infections there could be in total. The overall number of infections that occur during such an epidemic, is called the ‘final epidemic size’. The expected final size of an epidemic can be calculated using methods detailed in Andreasen (2011), Miller (2012), Kucharski et al. (2014), and Bidari et al. (2016), and which are implemented in the finalsize package.

An advantage of the final size approach is that it is only necessary to define pathogen transmissibility and population susceptibility to calculate the epidemic size, rather than requiring estimates of additional time-dependent processes like the duration of infectiousness or delay from infection to onset of symptoms. So finalsize will be particularly relevant for questions where understanding the overall size of an epidemic is more important than estimating its shape.

Use case

An epidemic is underway. We want to know how many individuals we would expect to be infected in total for a given level of transmission and population susceptibility: this is the cumulative sum of all infections, or the final size of the epidemic.

What we have

  1. An estimate of the infection’s basic reproduction number \(R_0\);
  2. An estimate of the population size;
  3. An estimate of the susceptibility of the population to the infection.

What we assume

  1. That the infection dynamics can be captured using a Susceptible-Infectious-Recovered (SIR) or Susceptible-Exposed-Infectious-Recovered (SEIR) model, where individuals who have been infected acquire immunity against subsequent infection, at least for the remaining duration of the epidemic.
# load finalsize

Defining a value for \(R_0\)

A number of statistical methods can be used to estimate the \(R_0\) of an epidemic in its early stages from available data. These are not discussed here, but some examples are given in the episoap package.

Instead, this example considers a infection with an \(R_0\) of 1.5, similar to that which could potentially be observed for pandemic influenza.

# define r0 as 1.5
r0 <- 1.5

Getting population estimates

Population estimates at the country scale are relatively easy to get from trusted data aggregators such as Our World in Data. More detailed breakdowns of population estimates at the sub-national scale may be available from their respective national governments. Here, we use an estimate for the U.K. population size of about 67 million.

# get UK population size
uk_pop <- 67 * 1e6

This initial example assumes uniform mixing, i.e., that all individuals in the population have a similar number of social contacts. This can be modelled in the form of a single-element contact matrix, which must be divided by the population size.

# prepare contact matrix
contact_matrix <- matrix(1.0) / uk_pop

Social contacts are well known to be non-uniform, with age being a strong influence on how many contacts a person has and, moreover, on the ages of their contacts. A relatively simple example is that of children of school-going age, who typically have more social contacts than the elderly, and most of whose social contacts are with other schoolchildren.

The “Modelling heterogeneous contacts” vignette explores how this can be incorporated into final epidemic size calculations using finalsize.

Modelling population susceptibility

In this initial example, the population is assumed to be fully susceptible to infection. This is modelled in the form of a matrix with a single element, called susceptibility.

# all individuals are fully susceptible
susceptibility <- matrix(1.0)

Since all individuals are fully susceptible, the break-up of the population into susceptibility groups can be represented as another single-element matrix, called p_susceptibility.

# all individuals are in the single, high-susceptibility group
p_susceptibility <- matrix(1.0)

Susceptibility to infection is well known to vary due to a number of factors, including age, prior exposure to the pathogen, or immunisation due to a vaccination campaign.

The “Modelling heterogeneous susceptibility” vignette explores how variation in susceptibility within and between demographic groups can be incorporated into final epidemic size calculations using finalsize.

Running final_size

The final size of the epidemic in the population can then be calculated using the only function in the package, final_size(). This example allows the function to fall back on the default options for the arguments solver ("iterative") and control (an empty list).

# calculate final size
final_size_data <- final_size(
  r0 = r0,
  contact_matrix = contact_matrix,
  demography_vector = uk_pop,
  susceptibility = susceptibility,
  p_susceptibility = p_susceptibility

# view the output data frame
#>     demo_grp   susc_grp susceptibility p_infected
#> 1 demo_grp_1 susc_grp_1              1  0.5828132

This is the final epidemic size without accounting for heterogeneity in social contacts by age or other factors, and without accounting for variation in susceptibility to infection between or within demographic groups.

This value, of about 58% of the population infected, is easily converted to a count, and suggests that about 39 million people would be infected over the course of this epidemic.

A short-cut for homogeneous populations

The example above explains the steps that go into final size calculation. For one very specific use-case, wherein the population is assumed to have homogeneous mixing (i.e., groups do not differ in their social contacts), and homogeneous and full susceptibility to infection, the final size calculation only depends on the \(R_0\) of the epidemic.

In this case, the final size calculation can be simplified to final_size(r0), with all other parameters taking their default values, which assume homogeneous social contacts and susceptibility.

#>     demo_grp   susc_grp susceptibility p_infected
#> 1 demo_grp_1 susc_grp_1              1  0.5828132


Andreasen, Viggo. 2011. “The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number.” Bulletin of Mathematical Biology 73 (10): 2305–21.
Bidari, Subekshya, Xinying Chen, Daniel Peters, Dylanger Pittman, and Péter L. Simon. 2016. “Solvability of Implicit Final Size Equations for SIR Epidemic Models.” Mathematical Biosciences 282 (December): 181–90.
Kucharski, Adam J., Kin O. Kwok, Vivian W. I. Wei, Benjamin J. Cowling, Jonathan M. Read, Justin Lessler, Derek A. Cummings, and Steven Riley. 2014. “The contribution of social behaviour to the transmission of influenza A in a human population.” PLoS pathogens 10 (6): e1004206.
Miller, Joel C. 2012. “A Note on the Derivation of Epidemic Final Sizes.” Bulletin of Mathematical Biology 74 (9): 2125–41.