Table 5.

A summary of our results for application of PGFs to the final size and large-time dynamics of SIR disease. The PGFs χ and ψ encode the heterogeneity in susceptibility. The PGF χ is the PGF of the ancestor distribution (an ancestor of u is any individual who, if infected, would infect u). The PGF $\psi \left(x\right)={\sum }_{\kappa }p\left(\kappa \right)x^{\kappa }$ encodes the distribution of the contact rates.

Question	Section	Solution
Final size relation for an SIR epidemic assuming a vanishingly small fraction ρ randomly infected initially with $\rho N\gg 1$.	4.2	$r\left(\infty \right)=1-\chi \left(1-r\left(\infty \right)\right)$. [For standard assumptions, including the usual continuous-time assumptions, $\chi \left(x\right)=e^{-{\mathrm{ℛ}}_0\left(1-x\right)}$.]
Discrete-time number susceptible, infected, or recovered in a population with homogeneous susceptibility and given ${\mathrm{ℛ}}_0$, assuming an initial fraction ρ is randomly infected with $\rho N\gg 1$.	4.3	For $g>0$:$$\begin{gathered} S\left(g\right)=N\left(1-\rho \right)e^{-{\mathcal{R}}_0\left(1-S\left(g-1\right)/N\right)} \\ I\left(g\right)=N-S\left(g\right)-R\left(g\right) \\ R\left(g\right)=R\left(g-1\right)+I\left(g-1\right) \end{gathered}$$ with the initial condition $S\left(0\right)=\left(1-\rho \right)N$, $I\left(0\right)=\rho N$, and $R\left(0\right)=0$.
Discrete-time number susceptible, infected, or recovered in a population with heterogeneous susceptibility for SIR disease after g generations with an initial fraction ρ randomly infected where $\rho N\gg 1$.	4.3	For $g>0$:$$\begin{gathered} S\left(g\right)=N\left(1-\rho \right)\chi \left(S\left(g-1\right)/N\right) \\ I\left(g\right)=N-S\left(g\right)-R\left(g\right) \\ R\left(g\right)=R\left(g-1\right)+I\left(g-1\right) \end{gathered}$$ with the initial condition $S\left(0\right)=\left(1-\rho \right)N$, $I\left(0\right)=\rho N$, and $R\left(0\right)=0$.
Continuous time number susceptible, infected, or recovered for SIR disease as a function of time with an initial fraction ρ randomly infected where $\rho N\gg 1$. Assumes u receives infection at rate $\beta I{\kappa }_u/N\langle K\rangle$	4.4	For $t>0$:$\begin{array}{l}S\left(t\right)=\left(1-\rho \right)N\psi \left(\theta \left(t\right)\right)\\ I\left(t\right)=N-S\left(t\right)-R\left(t\right)\\ R\left(t\right)=\frac{\gamma N\langle K\rangle }{\beta }\ln \theta \left(t\right)\\ \dot{\theta }\left(t\right)=-\frac{\beta }{N\langle K\rangle }I\theta \left(t\right)\end{array}$ with the initial condition $\theta \left(0\right)=1$.