Table 2.

Common function and variable names. When we use a PGF for the number of susceptible individuals, active infections, and/or completed infections x and s correspond to susceptible individuals, y and i to active infections, and z and r to completed infections.

Function/variable name	Interpretation
$$\begin{gathered} f\left(x\right)={\sum }_i{p}_i{x}^i \\ g\left(x\right)={\sum }_i{q}_i{x}^i \end{gathered}$$	Arbitrary PGFs.
$$\begin{gathered} \mu \left(y\right)={\sum }_i{p}_i{y}^i \\ \hat{\mu }\left(y\right)=\left(\beta y^2+\gamma \right)/\left(\beta +\gamma \right) \\ \hat{\mu }\left(y,z\right)=\left(\beta y^2+\gamma z\right)/\left(\beta +\gamma \right) \end{gathered}$$	Without hats: The PGF for the offspring distribution in discrete time. With hats: The PGF for the outcome of an unknown event in a continuous-time Markovian outbreak: y accounts for active infections and z accounts for completed infections.
α, ${\alpha }_g$, $\alpha \left(t\right)$	Probability of either eventual extinction, extinction by generation g, or by time t in an infinite population.
$$\begin{gathered} {\text{Φ}}_g\left(y\right)={\sum }_i{\phi }_i\left(g\right)y^i \\ \text{Φ}\left(y,t\right)={\sum }_i{\phi }_i\left(t\right)y^i \end{gathered}$$	PGF for the number of active infections in generation g or at time t in an infinite population.
$$\begin{gathered} {\text{Ω}}_{\infty }\left(z\right)={\sum }_{r<\infty }{\omega }_r{z}^r+{\omega }_{\infty }{z}^{\infty } \\ {\text{Ω}}_g\left(z\right)={\sum }_r{\omega }_r\left(g\right)z^r \\ \text{Ω}\left(z,t\right)={\sum }_r{\omega }_r\left(t\right)z^r \end{gathered}$$	The PGF for the distribution of completed infections at the end of a small outbreak, in generation g, or at time t in an infinite population. If ${\mathrm{ℛ}}_0>1$, then one of the terms in the expansion of ${\text{Ω}}_{\infty }\left(z\right)$ is ${\omega }_{\infty }{z}^{\infty }$ where ${\omega }_{\infty }$ is the probability of an epidemic.
$$\begin{gathered} {\text{Π}}_g\left(y,z\right)={\sum }_{i,r}{\pi }_{i,r}\left(g\right)y^i{z}^r \\ \text{Π}\left(y,z,t\right)={\sum }_{i,r}{\pi }_{i,r}\left(t\right)y^i{z}^r \end{gathered}$$	The PGF for the joint distribution of current infections and completed infections either at generation g or time t in an infinite population.
$\text{Ξ}\left(x,y,t\right)={\sum }_{s,i}{\xi }_{s,i}\left(t\right)x^s{y}^i$	The PGF for the joint distribution of susceptibles and current infections at time t in a finite population of size N (used for continuous time only). In the SIR case we can infer the number recovered from this and the total population size.
$\chi \left(x\right)={\sum }_i{p}_i{x}^i$	PGF for the ‘‘ancestor distribution’’, analogous to the offspring distribution.
$\psi \left(x\right)={\sum }_{\kappa }P\left(\kappa \right)x^{\kappa }$	PGF for the distribution of susceptibility for the continuous time model where rate of receiving transmission is proportional to κ.
β, γ	The individual transmission and recovery rates for the Markovian continuous time model.