Estimation of country-level basic reproductive ratios for novel Coronavirus (COVID-19) using synthetic contact matrices

Joe Hilton; Matt J Keeling

We assume that we have a population divided into K discrete age classes C 1 , ...C K . Suppose we have a set of outbreak data in the form x = (x 1 , ..., x K ), where x i is the cumulative number of cases so far in age class C i , expressed as a fraction of all the cases, so that i x i = 1. Denote by k i,j the expected number of contacts with individuals of age class C i made per day by a single individual of age class C j . The matrix K = (k i,j ) will be asymmetric, with the ith row corresponding to an average individual of each age class's contacts with age class C i , and the ith column corresponding to the contacts made per day by an average individual of age class C i . Let p i be the conditional probability that a susceptible individual in age class C i becomes infected, given that they have had contact with an infectious individual. We assume that this probability depends only on the age class of the recipient, and not on that of the infector, i.e. we have age-dependent susceptibility but homogeneous infectivity. The expected number of infections generated in age class C i in a single day by an infectious indvidual in age class C j is given by p i k i,j . Defining z i = p i γ −1 , the expected number of cases in age class C i generated by a single infectious individual in age class C j over its entire infectious period is given by