Contact matrices
A contact matrix describes how frequently individuals in different population groups interact.
Matrix form
\[C=(c_{ij}),\]where \(c_{ij}\) is typically the average rate of relevant contacts made by an individual in group \(i\) with individuals in group \(j\).
Transmission
In an age-structured model, a force of infection may be constructed as
\[\lambda_i=\beta\sum_j c_{ij}\frac{I_j}{N_j}.\]Thus infection pressure on group \(i\) depends on infectious prevalence across groups and the contact pattern linking them.
Interpretation
Contact matrices allow mixing to be structured rather than homogeneous and can be changed to represent interventions affecting particular settings or groups.
Key idea. A contact matrix is the mathematical bridge between population mixing patterns and group-specific transmission.