Table 4.

A summary of our results for application of PGFs to the continuous-time disease process. We assume individuals transmit with rate β and recover with rate γ. The functions $\hat{\mu }\left(y\right)=\left(\beta y^2+\gamma \right)/\left(\beta +\gamma \right)$ and $\hat{\mu }\left(y,z\right)=\left(\beta y^2+\gamma z\right)/\left(\beta +\gamma \right)$ are given in System (14).

Question	Section	Solution
Probability of eventual extinction α given a single introduced infection.	3.1	$\alpha =\text{min}\left(1,\gamma /\beta \right)$
Probability of extinction by time t, $\alpha \left(t\right)$.	3.1.1	$\alpha \left(t\right)$ where $\dot{\alpha }=\left(\beta +\gamma \right)[\hat{\mu }\left(\alpha \right)-\alpha ]$ and $\alpha \left(0\right)=0$.
PGF of the distribution of number of infected individuals at time t (assuming one infection at time 0).	3.2	$\text{Φ}\left(y,t\right)$ where $\text{Φ}\left(y,0\right)=y$ and $\text{Φ}$ solves either $\frac{\partial }{\partial t}\text{Φ}=\left(\beta +\gamma \right)[\hat{\mu }\left(y\right)-y]\frac{\partial }{\partial y}\phi$ or $\frac{\partial }{\partial t}\text{Φ}=\left(\beta +\gamma \right)[\hat{\mu }\left(\text{Φ}\right)-\text{Φ}].$
PGF of the number of completed cases at time t.	3.4	$\text{Ω}\left(z,t\right)$ where $\text{Ω}\left(z,0\right)=1$ and $\text{Ω}$ solves $\frac{\partial }{\partial t}\text{Ω}=\left(\beta +\gamma \right)[\hat{\mu }\left(\text{Ω},z\right)-\text{Ω}]$
PGF of the joint distribution of the number of current and completed cases at time t (assuming one infection at time 0).	3.3	$\text{Π}\left(y,z,t\right)$ where $\text{Π}\left(y,z,0\right)=y$ and $\text{Π}$ solves either $\frac{\partial }{\partial t}\text{Π}=\left(\beta +\gamma \right)[\hat{\mu }\left(y,z\right)-y]\frac{\partial }{\partial y}\text{Π}$ or $\frac{\partial }{\partial t}\text{Π}=\left(\beta +\gamma \right)[\hat{\mu }\left(\text{Π},z\right)-\text{Π}].$
PGF of the final size distribution.	3.4	${\text{Ω}}_{\infty }\left(z\right)={\text{lim}}_{t\to \infty }\text{Ω}\left(z,t\right)$. This also solves ${\text{Ω}}_{\infty }\left(z\right)=\hat{\mu }\left({\text{Ω}}_{\infty }\left(z\right),z\right)$. If epidemics are possible this has a discontinuity at $\left|z\right|=1$.
Probability an outbreak infects exactly j individuals	3.4	$\frac{1}{j}\frac{{\beta }^{j-1}{\gamma }^j}{{\left(\beta +\gamma \right)}^{2j-1}}\left(\begin{array}{c}2j-2\\ j-1\end{array}\right)$.
PGF for the joint distribution of the number susceptible and infected at time t for SIS dynamics in a population of size N.	3.5.1	$\text{Ξ}\left(x,y,t\right)$ where $\text{Ξ}$ solves $\frac{\partial }{\partial t}\text{Ξ}=\frac{\beta }{N}\left(y^2-xy\right)\frac{\partial }{\partial x}\frac{\partial }{\partial y}\text{Ξ}+\gamma \left(x-y\right)\frac{\partial }{\partial y}\text{Ξ}$
PGF for the joint distribution of the number susceptible and infected at time t for SIR dynamics in a population of size N.	3.5.2	$\text{Ξ}\left(x,y,t\right)$ where $\text{Ξ}$ solves $\frac{\partial }{\partial t}\text{Ξ}=\frac{\beta }{N}\left(y^2-xy\right)\frac{\partial }{\partial x}\frac{\partial }{\partial y}\text{Ξ}+\gamma \left(1-y\right)\frac{\partial }{\partial y}\text{Ξ}$