Developing next generation matrices from existing data
13th May 2020
The estimates of transmission rates in each sector (i.e., home, school, work and other locations) and the overall reproduction number are derived from baseline estimates of the daily, age-specific contact rates between individuals of different age groups. These contact rates are provided by the analysis in Prem et al.(1) where data from population-based contact diaries in eight European countries were projected to generate contact intensities for 144 other countries using Bayesian modelling techniques. The inferred values 𝑐𝑎𝑎′ give the number of (pre-COVID-19) typical daily contacts an individual of age 𝑎′ makes with an individual of age 𝑎. In the dataset, age bands are separated into 5 year age groups and contacts are further divided into four locations: work, home, school and other.
To estimate the transmission capacity associated with these contacts we convert the contact intensity matrices to next-generation matrices, 𝐾, whose elements, 𝑘𝑎𝑎′, give the number of new infections of age 𝑎 generated by individuals of age 𝑎′. As a first step, we compute an unscaled next-generation matrix 𝐾̅ by weighting the elements of the contact matrix 𝑐𝑎𝑎′ by the age-dependent relative susceptibility (𝜎𝑎) and infectivity (𝛽𝑎) of individuals in the population and the distribution of susceptible (𝑠𝑎) and total (𝑛𝑎) individuals in each age group. In particular, the elements, 𝑘̅
𝑎𝑎′, of the unscaled next-generation matrix (NGM), 𝐾̅, are given by
Here 𝜎𝑎 is the relative susceptibility to infection for an individual in age group 𝑎 and 𝛽𝑎 is their corresponding transmissibility once infected. Since the population is entirely susceptible upon first introduction of the infection such that 𝑠𝑎=𝑛𝑎.
For symmetry, we assume that the age-dependent susceptibility and transmissibility profiles are equal equivalent, i.e., 𝜎𝑎=𝛽𝑎, and are given by the following parametric equation:
where 𝜎rel is approximately equal to the relative susceptibility between individuals in the youngest (<5) and those in the oldest (>80) age groups. In the following analysis we assume baseline values of 𝜎min = 0.1, 𝑏 = 0.3 and 𝑐 = 27.
We choose values to match the proportion of each age group infected in China (the country used to calibrate the model) and then applied the calibrated values to Australian mixing matrices.
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