Calibration And Beta

The stochastic SEIRS and SEIHRD-family models derive a baseline transmission coefficient \(\beta\) from the requested \(R_0\) and the next-generation matrix.

Contact Matrix

The age-stratified contact matrix \(C\) replaces the single contact-rate scalar used in a homogeneous-mixing model. An element \(C_{ij}\) is the average number of daily contacts that a person in group \(i\) (contact initiator) has with people in group \(j\) (contactee).

The matrix can be asymmetric. Children may have frequent contact with adult teachers and caregivers even though many adults have little contact with children. The repository’s state matrices use the combined home, work, school, and community setting from the Epydemix implementation of Mistry 2021.

Changing the contact matrix changes the transmission coefficient required to produce the same requested \(R_0\).

Next-Generation Matrix

Let:

\[K = \beta S C \operatorname{diag}(w)\]

where:

  • \(S\) is a diagonal relative susceptibility matrix

  • \(C\) is the contact matrix

  • \(w\) is an age-specific infectiousness-duration weight

The simulator solves:

\[\rho(K) = R_0\]

where \(\rho()\) is the spectral radius or dominant eigenvalue, resulting in a scalar. Then solving for \(\beta\) we get:

\[\beta = \frac{R_0}{\rho(S C \operatorname{diag}(w))}.\]

For basic SEIRS:

\[w_i = \frac{1}{\gamma}.\]

If susceptibility and infectious duration are equal across ages and the spectral radius of \(C\) is 1, this reduces to the familiar homogeneous result:

\[\beta = R_0\gamma.\]

For SEIHRD, \(w\) combines the probability of entering the asymptomatic or symptomatic path, time spent in IA, IP, and IS, and each compartment’s relative infectiousness. This page uses \(\eta\) for relative infectiousness so that \(\rho(\cdot)\) remains reserved for the spectral-radius function:

\[w_j = (1-p_{A,j}) \left( \eta_{IP}d_{IP} + d_{IS} \right) + p_{A,j}\eta_{IA}d_{IA}.\]

Here \(p_{A,j}\) is prop_E_to_IA for age group \(j\), the \(d\) terms are mean infectious durations, \(\eta_{IP}\) is the code parameter rel_inf_IP_to_IS, and \(\eta_{IA}\) is rel_inf_IA_to_IS. The simulator’s travel parameter remains named rho in input files and code; it is unrelated to either these \(\eta\) multipliers or the spectral-radius function \(\rho(\cdot)\).

The simulator logs both the estimated beta and the \(R_0\) re-derived from the resulting next-generation matrix.

For intervention scenarios such as antiviral treatment, baseline \(\beta\) should be calibrated to untreated natural history. The intervention then changes realized transmission during simulation by moving people into treated compartments with lower infectiousness.

SEAITRD is currently a special case: it uses the configured beta = R0 / beta_scale relationship rather than the general next-generation-matrix estimator.

Deterministic SEIRS currently uses the simplified beta = R0 * gamma / spectral_radius(C) form. This is equivalent to the general expression when relative susceptibility and infectious duration are uniform across age groups.