r0_conversions: Converts between epidemiological parameters related to $R_0$

Description

Converts between $R_0$ and the transmission rate $\lambda$, or calculates the effective reproduction number $R_\text{eff}$ for a population, while accounting for population characteristics including demographic heterogeneity in social contacts, heterogeneity in the demographic distribution, and heterogeneity in susceptibility to infection.

Uses the R0 ($R_0$), contact matrix ($C$), population ($N$), and infectious period ($\gamma$) to calculate the transmission rate using the following equation. $$\lambda = R_0 / ({Max}(EV(C)) \gamma)$$ where $EV(C)$ denotes the eigenvalues of the matrix $C$ which is calculated from the social contacts matrix scaled by the number of individuals in each demographic and susceptibility group in the population.

Usage

lambda_to_r0(
  lambda,
  contact_matrix,
  demography_vector,
  susceptibility,
  p_susceptibility,
  infectious_period = 1.8
)
r0_to_lambda(
  r0,
  contact_matrix,
  demography_vector,
  susceptibility,
  p_susceptibility,
  infectious_period = 1.8
)
r_eff(r0, contact_matrix, demography_vector, susceptibility, p_susceptibility)

Value

Returns a single number for the calculated value.

Arguments

lambda: The transmission rate of the disease, also called the 'force of infection' ($\lambda$). This is different from the effective transmission rate ($\beta$).
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 $i$.
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.
infectious_period: Duration of the infectious period in days. Default value is 1.8 days.
r0: The basic reproductive number $R_0$ of the infection.

Details

Given the transmission rate ($\lambda$), social contacts matrix ($c$), demography ($N$), the distribution $P$ of each demographic group $i$ into susceptibility groups $S$, and the infectious period ($\gamma$)

r_eff() calculates the effective reproductive number;
lamda_to_r0() calculates the $R_0$ from the transmission rate as $$R_0 = {Max}(EV(C)) \times \lambda \gamma$$
r0_to_lambda() calculates the transmission rate as $$\lambda = R_0 / ({Max}(EV(C)) \gamma)$$ Note that this is also called the 'force of infection' and is different from the effective transmission rate often denoted $\beta$.

Here, $EV(C)$ denotes the eigenvalues of the matrix $C$ which is calculated from the social contacts matrix scaled by the number of individuals in each demographic and susceptibility group in the population.

Examples

Run this code

#### Prepare data ####
# Get example dataset and prepare contact matrix and demography
data(polymod_uk)
contact_matrix <- polymod_uk$contact_matrix
demography_vector <- polymod_uk$demography_vector

# define lambda
lambda <- 0.3

# define infectious period of 5 days
infectious_period <- 5
# define the number of age and susceptibility groups
n_demo_grps <- length(demography_vector)
n_risk_grps <- 3

# In this example, risk varies across groups
susceptibility <- matrix(
  data = c(0.5, 0.7, 1.0), nrow = n_demo_grps, ncol = n_risk_grps
)

# risk does not vary within groups
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)

#### Effective R ####
r0 <- 2.0
r_eff(
  r0 = r0,
  contact_matrix = contact_matrix,
  demography_vector = demography_vector,
  susceptibility = susceptibility,
  p_susceptibility = p_susceptibility
)

#### Transmission rate to R0 ####
lambda_to_r0(
  lambda, contact_matrix, demography_vector,
  susceptibility, p_susceptibility,
  infectious_period
)

#### R0 to Transmission rate ####
r0 <- 1.5
r0_to_lambda(
  r0, contact_matrix, demography_vector,
  susceptibility, p_susceptibility,
  infectious_period
)

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