Age structure
Age-structured epidemic models divide the population into age groups so that contact patterns, susceptibility, infectiousness and disease outcomes can differ between groups.
Age-specific compartments
For age group \(i\), an SIR model uses
\[S_i(t),\qquad I_i(t),\qquad R_i(t),\]with
\[N_i=S_i+I_i+R_i.\]The groups are not independent because infectious people in one group can infect susceptible people in another.
The contact matrix
Let \(c_{ij}\) be the average rate at which one person in recipient group \(i\) has transmission-relevant contacts with people in source group \(j\).
| Group 1 | Group 2 | Group 3 | |
|---|---|---|---|
| Group 1 | \(c_{11}\) | \(c_{12}\) | \(c_{13}\) |
| Group 2 | \(c_{21}\) | \(c_{22}\) | \(c_{23}\) |
| Group 3 | \(c_{31}\) | \(c_{32}\) | \(c_{33}\) |
Always state the row-column convention because different sources may use the opposite orientation.
Force of infection
The force of infection \(\lambda_i(t)\) is the instantaneous per-capita infection rate experienced by a susceptible individual in group \(i\).
A simple form is
\[\boxed{\lambda_i(t)=q_i\sum_j c_{ij}\frac{I_j(t)}{N_j}},\]where \(q_i\) is a transmission/susceptibility factor for a susceptible person in group \(i\).
| Quantity | Meaning |
|---|---|
| \(c_{ij}\) | contact rate from recipient group \(i\) with source group \(j\) |
| \(I_j/N_j\) | infectious fraction of source group \(j\) |
| \(q_i\) | baseline probability or transmission factor converting infectious contacts into infection hazard |
| \(\lambda_i\) | per-capita infection rate for susceptible people in group \(i\) |
Age-specific susceptibility and infectiousness
If recipient susceptibility and source infectiousness differ by age, keep the baseline transmission factor explicit. A consistent extension is
\[\boxed{\lambda_i=q\,s_i\sum_j c_{ij}\,\tau_j\frac{I_j}{N_j}},\]where \(q\) is the baseline transmission probability per relevant contact, \(s_i\) is relative susceptibility of recipient group \(i\), and \(\tau_j\) is relative infectiousness of source group \(j\).
Age-structured SIR equations
Once \(\lambda_i\) is defined,
\[\frac{dS_i}{dt}=-\lambda_iS_i,\]\[\frac{dI_i}{dt}=\lambda_iS_i-\gamma_iI_i,\]\[\frac{dR_i}{dt}=\gamma_iI_i.\]The groups are coupled through \(\lambda_i\), which contains infectious prevalence from all source groups.
Age-structured SEIR models
If a latent period is important, add an exposed compartment:
\[S_i\longrightarrow E_i\longrightarrow I_i\longrightarrow R_i.\]The infection flow is \(\lambda_iS_i\), progression is typically \(\sigma_iE_i\), and recovery/removal is \(\gamma_iI_i\).
Age structure and severe outcomes
Transmission and clinical burden are different questions. If \(h_i\) is the probability that an infection in age group \(i\) requires hospital care, then for \(C_i\) infections a simple expected number of hospitalisations is
\[H_i=h_iC_i,\qquad H=\sum_i H_i.\]A group can therefore contribute relatively little to transmission while experiencing a large share of severe outcomes.
Age-targeted interventions
School closure can change contact-matrix entries involving children; working from home can alter adult contact entries; vaccination can change susceptibility, infectiousness or progression in selected groups; and protection of high-risk groups can reduce exposure or severe outcomes.
Age structure and \(R_0\)
With multiple interacting groups, transmission pathways are organised in a next-generation matrix \(K\). Its entry \(K_{ij}\) represents the expected number of new infections in group \(i\) generated by one infected individual associated with group \(j\), under the disease-free conditions used to construct the matrix.
\[\boxed{R_0=\rho(K)},\]where \(\rho(K)\) is the spectral radius. Thus \(R_0\) summarises transmission across all interacting groups rather than coming from one ratio such as \(\beta/\gamma\).
Contact data and reciprocity
Contact matrices are usually estimated from surveys or other data. Recorded contacts may require adjustment for population sizes and reciprocity. Under a reciprocal-contact interpretation, total contacts from group \(i\) to group \(j\) should be compatible with total contacts reported in the reverse direction.
Choosing age groups
Narrow age bands provide more detail but create more compartments and parameters. Broad groups are easier to interpret but can hide important heterogeneity. The grouping should match the biological question and available data.