Strongly heterogeneous transmission of COVID-19 in mainland China: local and regional variation

Yuke Wang; Peter F.M. Teunis

Adopting terminology of Teunis et al. [14] a transmission probability matrix V can be defined where element v i,j is the probability that subject i was infected by another subject j; v i is a vector of transmission probabilities linking case i to any other case. The total number of observed subjects is n. Elements of V can be estimated by utilizing a distance kernel κ i,j (X i,j |i ← j), that defines a pairwise likelihood that subject i was infected by subject j. The distribution of the serial interval ( Figure 1a ): the distance in time between pairs of cases defines a practical distance kernel, translating the time intervals between symptom onsets in any two cases into a likelihood that these cases were linked as a transmission pair [19] . As outlined in [14] , the elements of the transmission probability matrix V may be estimated in a Markov chain Monte Carlo procedure. The elements of the transmission matrix are subject to constraints. Diagonal elements must be zero (subjects cannot infect themselves) and rows must add to 1 (unless the parent of the corresponding subject was not observed). Additional constraints may be imposed, for instance by preventing links between subjects known to have not been in contact. A mask matrix M (n × n, like V ) may be defined, with elements 1 where links are admissible, and 0 where they are not. This mask M may be applied to the matrix of kernels κ i,j () by elementwise multiplication. Elements of V representing pairs of subjects with inadmissible links are thus excluded: the corresponding v i,j are set to zero, and they are not updated in MCMC estimation. The mask M can be used to define contacts: whenever subject i is known to have been in contact with another subject j element m i,j = 1. And perhaps more importantly: when subject i is known to not have had contact with subject k then m i,k = 0.