• transmission tree: directed acyclic graph representing transmission events of an outbreak

• transmission chain: path from a case to a descending case

Different transmission contexts need different responses.

The number of possible transmission trees grows very fast:

• for 10 cases with unique onsets, \(\sim 3,500,000\) trees

• for 60 cases with unique onsets, \(\sim 8^{81}\) trees (more or less the estimated number of atoms in the universe)

• mutations accumulate in the pathogen genome along the transmission chains

• can be used to reconstruct transmission trees

For most diseases, whole genome sequences alone are not sufficient for reconstructing transmission trees.

• data (e.g. dates of symptom onset) restrict possible trees

• combine different types of data to identify a small set of plausible trees

Original outbreaker model used serial interval and genetic data to reconstruct transmission events.

Combines generation time, incubation period, and dates of onset.

Dates of symptom onset (\(t_i\)), generation time (\(w()\)), incubation period (\(f()\)), number of generations (\(\kappa_i\)).

\[ p(t_i | T_i^{inf}) \times p(T_i^{inf} | T_{\alpha_i}^{inf}, \kappa_i) = f(t_i - T_i^{inf}) \times w^{(\kappa_i)}(T_i^{inf} - T_{\alpha_i}^{inf}) \]

• \(\kappa_i\): number of generations between case \(i\) and its more recent sampled "ancestor"; \(\pi\) case reporting probability

Geometric distribution:

\[ p(\kappa_i | \pi) = (1 - \pi^{})^{\kappa_i - 1} \times \pi \]

• \(d()\): number of mutations between 2 sequences; \(s_i\), \(s_{\alpha_i}\): sequences of infectee their infector; \(L\): genome length; \(\mu\): mutation rate per generation of infection

\[ p(s_i | s_{\alpha_i} \mu) = \mu^{d(s_i, s_{\alpha_i})} \times (1 - \mu)^{(\kappa_i L - d(s_i, s_{\alpha_i}))} \]

Relies on: contact reporting probability (\(\epsilon\)) and probability of contact between non-transmission pairs (\(\lambda\)).

\(\alpha_i = j\): \(p(c_{i,j} = 1 ) = \epsilon\) ; \(p(c_{i,j} = 0) = 1 - \epsilon\)

\(\alpha_i \neq j\): \(p(c_{i,j} = 1) = \lambda \epsilon\) ; \(p(c_{i,j} = 0) = (1 - \lambda) + \lambda (1 - \epsilon)\)