Heterogeneous mixing
Heterogeneous mixing means that people or groups do not all have the same contact patterns or transmission opportunities.
Homogeneous mixing as the baseline
In a simple SIR model, infection is often represented by
\[\beta\frac{SI}{N}.\]This assumes that every susceptible individual experiences the same average infectious environment.
Sources of heterogeneity
| Source | Possible effect |
|---|---|
| age | different within- and between-age contact patterns |
| household | repeated close contact within homes |
| occupation | different numbers and types of contacts |
| behaviour | different social activity and risk |
| geography | more mixing within nearby locations |
| network position | some individuals connect many others |
Group-based mixing
Suppose the population is divided into groups and \(c_{ij}\) is the contact rate from recipient group \(i\) with source group \(j\). A simple force of infection is
\[\boxed{\lambda_i=q_i\sum_j c_{ij}\frac{I_j}{N_j}},\]where \(q_i\) converts infectious contacts into infection hazard for susceptible people in group \(i\).
This lets different groups experience different infection pressure while preserving a clear contact-matrix interpretation.
Heterogeneous susceptibility and infectiousness
Contact patterns are only one source of heterogeneity. If susceptibility and infectiousness are represented by relative, dimensionless modifiers, keep the baseline transmission factor explicit:
\[\boxed{\lambda_i=q\,s_i\sum_j c_{ij}\,\tau_j\frac{I_j}{N_j}},\]| Symbol | Meaning |
|---|---|
| \(q\) | baseline transmission probability or transmission factor per relevant infectious contact |
| \(s_i\) | relative susceptibility of recipient group \(i\) |
| \(\tau_j\) | relative infectiousness of source group \(j\) |
| \(c_{ij}\) | contact rate between recipient group \(i\) and source group \(j\) |
Individual-level heterogeneity
Some differences are better represented with a contact network. Individuals are nodes and contacts are edges. A person with many connections has high degree and may have more opportunities both to acquire and to transmit infection.
High degree does not guarantee a large outbreak from one person because transmission remains stochastic, but highly connected individuals can contribute disproportionately on average.
Why population averages can mislead
Two populations can have the same average number of contacts while distributing those contacts very differently. Concentrating contacts in a smaller group can change early growth, depletion of susceptibility and intervention effects.
Targeted interventions
If one group contributes disproportionately to transmission, changing that group's contacts, susceptibility or infectiousness can have a larger effect than the same proportional intervention applied to a low-contact group.
Heterogeneity and \(R_0\)
In structured populations, transmission pathways are collected in a next-generation matrix \(K\). The basic reproduction number is
\[\boxed{R_0=\rho(K)},\]where \(\rho(K)\) is the spectral radius. This summarises all group-to-group transmission pathways rather than one population-average contact rate.
Ways to represent heterogeneous mixing
| Approach | What it represents |
|---|---|
| multiple compartments | groups with different parameters |
| contact matrices | different mixing rates within and between groups |
| networks | individual contact structure |
| spatial models | location-dependent mixing and movement |