7.4. Probability Generating Functions, Practical Class 4

AMSI 2026

The first 3 questions relate to Section 4.6SIR outbreak sizes in small populations.

1. An SIR outbreak occurs in a small closed population of size \(N\). Let \(q_\ell\) denote the probability that exactly \(\ell\) individuals (including the index case) are ever infected.

   (a) In the transmission network interpretation used, individuals are nodes and directed edges represent potential transmission. Explain why the final outbreak size is equal to the size of the out-component of the initially infected individual.

   (b) For a population of size \(N=3\), write down the system of equations that determines \(q_1,\ldots,q_3\) using the Ball-matrix approach. You do not need to solve the system.

2. Consider the SIR model with transmission rate \(\beta\) and recovery rate \(\gamma\), for which the offspring PGF is

   \[ \mu(x) = \frac{\gamma}{\beta + \gamma - \beta x}. \]

   (a) Compute \(\mu\left(\frac{M-1}{N-1}\right)\) and explain what this quantity represents in the context of small populations.

   (b) Using the definition of the Ball matrix, write an explicit expression for the coefficients \(c_{M,1}\) and \(c_{M,2}\) in terms of \(\mu\), \(M\), and \(N\).

   (c) The matrix defining the system for \(q_\ell\) is lower triangular. Why does this guarantee that the probabilities \(q_1,q_2,\ldots,q_N\) can be solved sequentially?

3. The text compares simulated outbreak size distributions with the theoretical probabilities \(q_\ell\).

   (a) For a fixed population size \(N=50\), describe how you expect the distribution of outbreak sizes to change as \(R_0=\beta/\gamma\) increases from below 1 to above 1.

   (b) Explain why, even when \(R_0>1\), there is still a substantial probability of small outbreaks in a small population.

   (c) Give one advantage and one limitation of the Ball matrix method for computing outbreak size distributions as \(N\) becomes large.

The next 2 questions relate to Section 4.7SIR disease in Heterogeneous Populations:

4. In the heterogeneous mean-field SIR model, individuals are grouped by contact rate \(k\in\{0,1,2,\ldots\}\), and we track \((S_k(t),I_k(t),R_k(t))\) for each \(k\). Define

   \[ \psi(x)=\sum_{k\ge 0} p_k x^k,\qquad \psi'(1)=\sum_{k\ge 0} k p_k, \]

   and

   \[ \pi_I(t)=\frac{\sum_{k\ge 0} k I_k(t)}{N\psi'(1)}. \]

   (a) Explain the infinite system of ODEs. What does each term in the equation represent?

   \[\begin{align*} \frac{d}{dt}S_k &= -\beta \pi_I k S_k\\ \frac{d}{dt}I_k &= \beta \pi_I k S_k-\gamma I_k,\\ \frac{d}{dt}R_k &= \gamma I_k, \end{align*}\]

   together with the definition of \(\pi_I(t)\).

   (b) Explain why the total combined transmission rate in the population is

   \[ \beta\sum_{\hat{k}\ge 0}\hat{k} I_{\hat{k}}, \]

   and why the rate at which individuals in class \(S_k\) become infected is

   \[ \beta \pi_I k S_k. \]

   (c) The text describes a “naive approach” that truncates the system by ignoring classes \(k>K\) for some cutoff \(K\). Explain why this can be inaccurate when \(\sum_{k>K} k^2 p_k\) is not small, and give an example (in words) of a degree distribution where large \(k\) cannot be neglected.

5. Using the reductions in the section, the dynamics can be written in terms of \((I(t),\pi_I(t),\theta(t))\) and \(\psi\).

   \[\begin{align*} \frac{d}{dt} I &= \beta(1-\rho)N\pi_I\theta\psi'(\theta)- \gamma I\\ \frac{d}{dt} \pi_I &= \beta \pi_I (1-\rho)\frac{ \theta \frac{d}{d\theta} (\theta\psi'(\theta))}{\psi'(1)} - \gamma \pi_I\\ \frac{d}{dt} \theta &= - \beta \pi_I\theta\\ S &= (1-\rho)N\psi(\theta)\\ R &= N-I-S \end{align*}\]

   (a) Focus on the \(d\pi_I/dt\) equation. Assuming that at early time \(\theta(t)\) remains approximately \(1\) for a long period, determine what sort of functional form \(\pi_I\) has and how it depends on \(\psi\) (and derivatives of \(\psi\)).

   (b) Consider two populations with the same mean contact rate \(\psi'(1)\) but different contact-rate distributions, one narrow and the other heavy-tailed (\(p_k\) decays slowly for large \(k\)). Qualitatively, how would you expect the early epidemic growth (via \(\pi_I\)) to differ between them?

   (c) How does this relate to the statement in the text about the significance of large \(k\) individuals?

The next question relates to Section 5.1Multivariate Probability Generating Functions

6. Let \((X_i,Y_i)\), \(i=1,2,\ldots\), be independent and identically distributed random pairs with joint PGF \(\mu(x,y)\), and let \(K\) be a non-negative integer-valued random variable with PGF \(\psi(x)\), independent of the pairs.

   (a) Using Theorem 5.1 find the PGF of of the joint distribution of

   \[ \left(\sum_{i=1}^K X_i,\;\sum_{i=1}^K Y_i\right) \]

   (b) Theorem 5.2 considers two joint PGFs \(\xi_1(x,y)\) and \(\xi_2(x,y)\) and a joint count PGF \(\psi(x,y)\). Explain in words what the theorem says about the joint PGF of

   \[ \sum_{i=1}^\ell (X_i,Y_i) + \sum_{i=1}^m (X_i',Y_i'), \]

   where \(\ell\) and \(m\) have joint PGF \(\psi(x,y)\).

   (c) Give a concrete example (described mathematically or in words) of a random sum that can be analysed using Theorem 5.2, identifying the roles of \(\xi_1\), \(\xi_2\), \(\ell\), and \(m\).

The final question relates to Section 5.4Stem Cells

7. In the two-type generation-based Galton-Watson process, let \(\xi_1(x,y)\) and \(\xi_2(x,y)\) be the offspring PGFs for a type-1 and type-2 individual respectively, where

   \[ \xi_1(x,y)=\sum_{j,k\ge 0}p_{j,k}x^jy^k, \qquad \xi_2(x,y)=\sum_{j,k\ge 0}q_{j,k}x^jy^k. \]

   Let \(\vec{\Phi}_g(x,y)=(\Phi_g(x,y|(1,0)),\,\Phi_g(x,y|(0,1)))\) denote the joint PGFs for the numbers of type-1 and type-2 individuals at generation \(g\) given a single initial individual of either type.

   (a) Write down the recursive “backward approach” update that expresses \(\vec{\Phi}_{g+1}(x,y)\) in terms of \(\vec{\Phi}_g(x,y)\) and the offspring PGFs.

   (b) Explain why \(\vec{\Phi}_0(x,y)=(x,y)\).

   (c) For a simple example where \(\xi_1(x,y)=x^2\) and \(\xi_2(x,y)=y\), compute \(\vec{\Phi}_1(x,y)\) and \(\vec{\Phi}_2(x,y)\) using the backward recursion.

8. The probability that both types are extinct by generation \(g\) given an initial type-1 individual is

   \[ \mathbb{P}[\text{extinct both types by }g | (1,0)] = \Phi_g(0,0 | (1,0)). \]

   Define \(\vec{\alpha}(g)=(\alpha_1(g),\alpha_2(g))=\vec{\Phi}_g(0,0)\).

   (a) Show that \(\vec{\alpha}(g)\) satisfies the iterative equation

   \[ \vec{\alpha}(g)=\big(\xi_1(\vec{\alpha}(g-1)),\;\xi_2(\vec{\alpha}(g-1))\big), \]

   with \(\vec{\alpha}(0)=(0,0)\).

   (b) Explain why the long–time extinction probabilities \((\alpha_1,\alpha_2)\) must satisfy

   \[ (\alpha_1,\alpha_2)=\big(\xi_1(\alpha_1,\alpha_2),\;\xi_2(\alpha_1,\alpha_2)\big). \]

   (c) Suppose \(\xi_1(x,y)=0.4x+0.6y\) and \(\xi_2(x,y)=0.5x^2+0.5y\). Find the fixed point(s) of the extinction equations in part (b). Which fixed point corresponds to the extinction probability?