Final size of an epidemic — final

final_size calculates the final size of an epidemic outbreak in a population with heterogeneous mixing, and with heterogeneous susceptibility to infection such as that conferred by an immunisation programme.

Usage

final_size(
  r0,
  contact_matrix,
  demography_vector,
  susceptibility,
  p_susceptibility,
  solver = c("iterative", "newton"),
  control = list()
)

Arguments

r0: The basic reproductive number \(R_0\) of the disease.
contact_matrix: Social contact matrix. Entry \(m_{ij}\) gives average number of contacts in group \(i\) reported by participants in group \(j\)
demography_vector: Demography vector. Entry \(v_{i}\) gives proportion of total population in group \(i\) (model will normalise if needed).
susceptibility: A matrix giving the susceptibility of individuals in demographic group \(i\) and risk group \(k\).
p_susceptibility: A matrix giving the probability that an individual in demography group \(i\) is in risk (or susceptibility) group \(k\). Each row represents the overall distribution of individuals in demographic group \(i\) across risk groups, and each row must sum to 1.0.
solver: Which solver to use. Options are "iterative" (default) or "newton", for the iterative solver, or the Newton solver, respectively. Special conditions apply when using the Newton solver, see the control argument.
control: A list of named solver options, see Solver options.

Value

A data.frame of the proportion of infected individuals, within each demography group and susceptibility group combination. If the demography groups and susceptibility groups are named, these names are added to relevant columns. If the groups are not named, synthetic names are added of the form demo_grp_<i>, for each demographic group \(i\).

Solver options

The control argument accepts a list of solver options, with the iterative solver taking two extra arguments than the Newton solver. This is an optional argument, and default options are used within the solver functions if an argument is missing. Arguments provided override the solver defaults.

Common options

iterations: The number of iterations over which to solve for the final size, unless the error is below the solver tolerance. Default = 10000.
tolerance: The solver tolerance; solving for final size ends when the error drops below this tolerance. Defaults to set 1e-6. Larger tolerance values are likely to lead to inaccurate final size estimates.

Iterative solver options

step_rate: The solver step rate. Defaults to 1.9 as a value found to work well.
adapt_step: Boolean, whether the solver step rate should be changed based on the solver error. Defaults to TRUE.

Examples

# load example POLYMOD data included in the package
data(polymod_uk)
r0 <- 2.0
contact_matrix <- polymod_uk$contact_matrix
demography_vector <- polymod_uk$demography_vector

# define the number of age and susceptibility groups
n_demo_grps <- length(demography_vector)
n_risk_grps <- 3

# In this example, all risk groups from all age groups are fully
# susceptible
susceptibility <- matrix(
  data = 1, nrow = n_demo_grps, ncol = n_risk_grps
)

p_susceptibility <- matrix(
  data = 1, nrow = n_demo_grps, ncol = n_risk_grps
)
# p_susceptibility rows must sum to 1.0
p_susceptibility <- p_susceptibility / rowSums(p_susceptibility)

# using default arguments for `solver` and `control`
final_size(
  r0 = r0,
  contact_matrix = contact_matrix,
  demography_vector = demography_vector,
  susceptibility = susceptibility,
  p_susceptibility = p_susceptibility
)
#>   demo_grp   susc_grp susceptibility p_infected
#> 1   [0,20) susc_grp_1              1  0.8532885
#> 2  [20,40) susc_grp_1              1  0.7974557
#> 3      40+ susc_grp_1              1  0.6880834
#> 4   [0,20) susc_grp_2              1  0.8532885
#> 5  [20,40) susc_grp_2              1  0.7974557
#> 6      40+ susc_grp_2              1  0.6880834
#> 7   [0,20) susc_grp_3              1  0.8532885
#> 8  [20,40) susc_grp_3              1  0.7974557
#> 9      40+ susc_grp_3              1  0.6880834

# using manually specified solver settings for the iterative solver
control <- list(
  iterations = 100,
  tolerance = 1e-3,
  step_rate = 1.9,
  adapt_step = TRUE
)

final_size(
  r0 = r0,
  contact_matrix = contact_matrix,
  demography_vector = demography_vector,
  susceptibility = susceptibility,
  p_susceptibility = p_susceptibility,
  solver = "iterative",
  control = control
)
#>   demo_grp   susc_grp susceptibility p_infected
#> 1   [0,20) susc_grp_1              1  0.8532715
#> 2  [20,40) susc_grp_1              1  0.7974597
#> 3      40+ susc_grp_1              1  0.6880988
#> 4   [0,20) susc_grp_2              1  0.8532715
#> 5  [20,40) susc_grp_2              1  0.7974597
#> 6      40+ susc_grp_2              1  0.6880988
#> 7   [0,20) susc_grp_3              1  0.8532715
#> 8  [20,40) susc_grp_3              1  0.7974597
#> 9      40+ susc_grp_3              1  0.6880988

# manual settings for the newton solver
control <- list(
  iterations = 100,
  tolerance = 1e-3
)

final_size(
  r0 = r0,
  contact_matrix = contact_matrix,
  demography_vector = demography_vector,
  susceptibility = susceptibility,
  p_susceptibility = p_susceptibility,
  solver = "newton",
  control = control
)
#>   demo_grp   susc_grp susceptibility p_infected
#> 1   [0,20) susc_grp_1              1  0.9999876
#> 2  [20,40) susc_grp_1              1  0.9999873
#> 3      40+ susc_grp_1              1  0.9999867
#> 4   [0,20) susc_grp_2              1  0.9999876
#> 5  [20,40) susc_grp_2              1  0.9999873
#> 6      40+ susc_grp_2              1  0.9999867
#> 7   [0,20) susc_grp_3              1  0.9999876
#> 8  [20,40) susc_grp_3              1  0.9999873
#> 9      40+ susc_grp_3              1  0.9999867