Contact matrices
A contact matrix records how frequently people in different population groups interact. It allows an epidemic model to replace homogeneous mixing with structured mixing.
Matrix definition
Suppose the population is divided into groups indexed by \(i\) and \(j\). Let
\[C=(c_{ij}),\]where \(c_{ij}\) is the average number of transmission-relevant contacts made per unit time by one person in group \(i\) with people in group \(j\).
A three-group example
| Children | Adults | Older adults | |
|---|---|---|---|
| Children | 8 | 3 | 1 |
| Adults | 3 | 6 | 2 |
| Older adults | 1 | 2 | 3 |
The first row says that one child has, on average, 8 relevant contacts with children, 3 with adults and 1 with older adults per unit time. Larger diagonal values indicate stronger within-group mixing, often called assortative mixing.
From contacts to infection pressure
If group \(j\) contains \(I_j\) infectious people out of \(N_j\), then \(I_j/N_j\) is the infectious fraction. Thus
\[c_{ij}\frac{I_j}{N_j}\]is the approximate rate at which a person in group \(i\) contacts infectious people from group \(j\).
Force of infection
Let \(q_i\) be a transmission/susceptibility factor for recipients in group \(i\). A common force of infection is
\[\boxed{\lambda_i=q_i\sum_j c_{ij}\frac{I_j}{N_j}}.\]| Quantity | Meaning |
|---|---|
| \(c_{ij}\) | contact rate from group \(i\) with group \(j\) |
| \(I_j/N_j\) | infectious proportion of source group \(j\) |
| \(q_i\) | factor converting infectious contacts into infection hazard for group \(i\) |
| \(\lambda_i\) | instantaneous per-capita infection rate for susceptible people in group \(i\) |
Numerical example
For adults, suppose
\[c_{21}=3,\qquad c_{22}=6,\qquad c_{23}=2,\]and infectious proportions are \(0.10\), \(0.05\), and \(0.02\). If \(q_2=0.04\), then
\[\lambda_2=0.04\left[3(0.10)+6(0.05)+2(0.02)\right]=0.0256\text{ day}^{-1}.\]Using the matrix in an epidemic model
For an age-structured SIR model,
\[\frac{dS_i}{dt}=-\lambda_iS_i,\]\[\frac{dI_i}{dt}=\lambda_iS_i-\gamma_iI_i,\]\[\frac{dR_i}{dt}=\gamma_iI_i.\]The groups are coupled because each \(\lambda_i\) depends on infection in all source groups.
Why contact matrices need not be symmetric
Per-person rates \(c_{ij}\) and \(c_{ji}\) can differ when group sizes differ. If total cross-group contacts are reciprocal, a common consistency condition is
\[\boxed{N_i c_{ij}=N_j c_{ji}}.\]Survey matrices often require balancing so that this reciprocity condition is approximately satisfied.
Setting-specific matrices
Contacts can be separated into home, school, work and other settings:
\[C=C^{\mathrm{home}}+C^{\mathrm{school}}+C^{\mathrm{work}}+C^{\mathrm{other}}.\]This allows an intervention to change only the relevant part of the mixing structure, such as reducing school contacts without changing household contacts.
Contact matrices and \(R_0\)
In structured models, contact patterns contribute to a next-generation matrix. The basic reproduction number is then
\[\boxed{R_0=\rho(K)},\]where \(\rho(K)\) is the spectral radius of the next-generation matrix.
Important limitations
Recorded social contacts are not automatically equivalent to transmission opportunities. Contact definitions, reporting bias, group sizes, susceptibility, infectiousness and setting all affect how a matrix should be used.