2.1. Definition of the network model

Consider a population of size n (assumed to be large) having a network structure in which each individual belongs to exactly one household and exactly one workplace. To simplify things we assume all households have size $h\ge 2$ and all workplaces have size w, where we make the additional assumption that $w=dh$ for some integer $d\ge 1$ (so the size of a workplace is a multiple of the size of a household, and unless $d=1$ , the workplaces are larger than the households).

The parameter $\theta$ determining the amount of overlap (or rather the lack of overlap) is defined as follows. First we let each workplace consist of d households, so all individuals of the same household are also in the same workplace, implying a complete overlap. Then each individual independently decides whether to be a mover with probability $\theta$ or a remainer with the remaining probability $1-\theta$ . The workplace spots of movers are made vacant, and all movers are distributed uniformly at random among the vacant spots in the different workplaces, thus creating new workplace constellations, but all workplaces still have size $w=dh$ . As a consequence, the larger is $\theta$ , the fewer pairs of household members who are also work colleagues.

In summary, the parameters of the network (beside the overall population size n, which must be a multiple of w) are h, the household size, $w=dh$ , the workplace size, and $\theta$ , the probability of being a mover (the tunable overlap can hence be defined as $1-\theta$ ).

2.2. Definition of the epidemic model on the network

The epidemic is a stochastic continuous-time SIR model in which each infected individual is infectious for an independent random duration I, the length of the infectious period (possibly preceded by a latent period having an arbitrary distribution). During the infectious period an individual makes three types of infectious contact at times of independent Poisson processes: at rate $\beta_\textrm{H}$ the individuals make infectious contact with other members of their household, each time choosing which individual independently and uniformly at random, implying the rate of infectious contact with a specific individual equals $\beta_\textrm{H}/(h-1)$ . In addition, the infectious individual makes contact with workplace individuals at rate $\beta_\textrm{W}$ , each time with an independent, uniformly selected individual of the same workplace. Finally, the infectious individual makes global contacts at rate $\beta_\textrm{G}$ , each time with an individual chosen independently and uniformly from the entire population. An infectious contact with a susceptible individual results in the latter getting infected (such individuals starting their infectious contact processes), whereas contacts with already infected people have no effect. After their infectious period is over, the individual recovers and becomes immune, thus playing no further role in the epidemic.

The epidemic is initiated by one randomly selected individual being externally infected and the remaining individuals being susceptible. The epidemic runs its course until there are no infectious individuals present in the community, when it stops. Each individual is then either recovered or susceptible, and the random distribution of how the recovered and susceptible individuals are distributed in the community defines the final outcome. The final number infected is denoted Z and is referred to as the final size.

For ease of analysis, we assume that each global contact is with an individual chosen uniformly from the entire population, including also the infector itself. Thus a global contact may be with an individual in the infector’s own household or workplace. This assumption represents no loss of generality and has no impact on our asymptotic results.

The focus of this paper is solely on the final outcome (size) of the epidemic, and since the infectious period distributions and the Poisson processes governing infectious contacts are independent of time, the results are invariant to an arbitrary exposed (latent) period (see [Reference Ludwig23] and [Reference Pellis, Ferguson and Fraser27]). Therefore, in the following we focus on SIR epidemic models, with all results carrying over to corresponding SEIR (susceptible $\to$ exposed $\to$ infective $\to$ recovered) epidemic models.

2.3. Parameterisation

Without loss of generality we assume ${\mathbb E}[I]=1$ (meaning that time is measured in units of the mean infectious period). Our focus is on h and w being fixed and letting $n\to\infty$ . If w becomes large with n then the probability of an individual making multiple workplace infectious contacts with a given individual is small and tends to 0 as $w \to \infty$ . In this case the epidemic resembles the household epidemic model in terms of probability of a major outbreak and final size of the epidemic. Similarly, if both h and w grow with n, it becomes increasingly unlikely an infective makes multiple contacts with any member of the population, and the epidemic closely resembles a homogeneously mixing epidemic in terms of probability of a major outbreak and final size of the epidemic.

The epidemic parameters are the three infectious contact rates $\beta_\textrm{H}$ , $\beta_\textrm{W}$ , and $\beta_\textrm{G}$ , and the infectious period distribution I (assumed to have mean ${\mathbb E}[I]=1$ ). Later we reparameterise the three infection rates to $\beta=\beta_\textrm{H}+\beta_\textrm{W}+\beta_\textrm{G}$ , $\pi_\textrm{G}=\beta_\textrm{G}/(\beta_\textrm{H}+\beta_\textrm{W}+\beta_\textrm{G})$ , and $\pi_{\textrm{H}|\textrm{G}^\textrm{c}}=\beta_\textrm{H}/(\beta_\textrm{H}+\beta_\textrm{W})$ , thus being the overall contact rate, the fraction of the overall contact rate from global contacts, and the fraction of the remaining (local) contacts that are in households, respectively.

2.4. Notation

We use $\stackrel{\textrm{D}}{\longrightarrow}$ and $\stackrel{\textrm{p}}{\longrightarrow}$ to denote convergence in distribution and convergence in probability, respectively, and $\stackrel{\textrm{D}}{=}$ to denote equal in distribution. For a random variable, X say, taking values in the positive integers, $f_X$ denotes its probability-generating function (PGF), i.e. $f_X(s)={\mathbb E}[s^X]$ ( $0 \le s \le 1$ ). For any real-valued random variable Y, $\mu_Y$ denotes its mean and $\phi_Y$ denotes its Laplace transform, so $\phi_Y(\nu)={\mathbb E}[\!\exp\!(\!-\!\nu Y)]$ $(\nu \ge 0)$ . For a positive integer n and $p \in [0,1]$ , $\textrm{Bin}(n,p)$ denotes a binomial random variable with n trials and success probability p. For $\lambda\gt0$ , $\textrm{Exp}(\lambda)$ denotes an exponential random variable with rate $\lambda$ and hence mean $\lambda^{-1}$ . For $x \in \mathbb{R}$ , $\lfloor x \rfloor$ denotes the usual floor function, so $\lfloor x \rfloor$ is the greatest integer less than or equal to x. For a non-negative random variable X, we write $Y\sim \textrm{Po}(X)$ when Y has a mixed Poisson distribution with parameter distributed as X, i.e. when ${\mathbb P}(Y=k) = {\mathbb E}[X^k \textrm{e}^{-X} / k!]$ for $k=0,1,2,\ldots$ .

5.1. Introduction

Let $\mathcal{E}^{(n)}$ denote the epidemic model defined in Section 2. When $\pi_\textrm{G}\gt0$ , the epidemic model $\mathcal{E}^{(n)}$ is a special case of the general framework of Ball and Neal [Reference Ball and Neal10] for SIR epidemics in populations with two levels of mixing. We first outline that general framework in Subsection 5.2, introducing key concepts of local infectious clumps, local susceptibility sets, and the severity of a local infectious clump, and explain heuristically how the limiting severity of a local infectious clump as $n \to \infty$ is related to the event of a major outbreak and the limiting size of a local susceptibility set is related to the final outcome of a major outbreak. In the present model, the structure of local infectious clumps and local susceptibility sets is far more complicated than in the examples considered in [Reference Ball and Neal10], and consequently proofs and calculations of limiting major outbreak probability and final size are considerably more difficult. Local infectious clumps and local susceptibility sets for $\mathcal{E}^{(n)}$ are analysed in Subsection 5.3. Crucial to this analysis are complexes, defined in Subsection 5.3.1, which are associated with workplaces but contain also households of movers who moved out of the workplace. The analysis of the limiting local infectious clump as $n \to \infty$ is simpler when $I \equiv 1$ . This is treated in Subsection 5.3.2, with the extension to the case of general I and calculation of the limiting major outbreak probability $\rho$ being given in Subsection 5.3.3. Local susceptibility sets are analysed in Subsection 5.3.4, where calculations of the threshold parameter $R_*$ and the limiting final size of a major outbreak z are described. Numerical implementation of the calculation of $R_*$ , z, and $\rho$ is considered briefly in Subsection 5.4. Finally, proofs of Theorems 3.1 and 3.2 for $\pi_\textrm{G}\gt0$ are given in Subsections 5.5 and 5.6, respectively.

5.2. General framework

We derive the properties of the model by using the general framework of Ball and Neal [Reference Ball and Neal10] for SIR epidemics in populations with two levels of mixing. We first outline the key points of that framework in the current setting. Label the individuals in the population $1, 2, \ldots, n$ and let $\mathcal{N}=\{1,2,\ldots, n\}$ . (The precise labelling is not important at present.) We construct on $\mathcal{N}$ a random directed graph $\mathcal{G}^{(n)}$ of potential local (i.e. household or workplace) infectious contacts as follows. For each $(i,j) \in \mathcal{N}^2$ , with $i \ne j$ , there is a directed edge from i to j in $\mathcal{G}^{(n)}$ if and only if i, if infected, makes infectious contact with j. More precisely, let $I_i$ denote the length of i’s infectious period. Then, given $I_i$ , there is a directed edge from i to each individual in i’s household independently with probability $1-\exp\!(\!-\!\beta_\textrm{H}' I_i)$ , where $\beta_\textrm{H}'=\beta_\textrm{H}/(h-1)$ , and a directed edge from i to each individual in i’s workplace independently with probability $1-\exp\!(\!-\!\beta_\textrm{W}' I_i)$ , where $\beta_\textrm{W}'=\beta_\textrm{W}/(w-1)$ . If an individual belongs to both i’s household and i’s workplace, then the directed edge is present with probability $1-\exp\!(\!-\!(\beta_\textrm{H}'+\beta_\textrm{W}')I_i)$ .

We use $\mathcal{G}^{(n)}$ to define for each $i \in \mathcal{N}$ its local infectious clump and local susceptibility set as follows. For $(i,j) \in \mathcal{N}^2$ , write $i \leadsto j$ if and only if there is a chain of directed edges from i to j, with the convention that $i \leadsto i$ . For $i \in \mathcal{N}$ , let $\mathcal{C}^{(n)}_i=\{\,j \in \mathcal{N}\;:\;i \leadsto j\}$ and $\mathcal{S}^{(n)}_i=\{\,j \in \mathcal{N}\;:\; j \leadsto i\}$ be i’s local infectious clump and local susceptibility set, respectively. Let $\mathcal{E}^{(n)}$ denote the epidemic model described in Subsection 2.2. A realisation of the final outcome of $\mathcal{E}^{(n)}$ can be constructed using $I_1, I_2, \ldots, I_n$ and $\mathcal{G}^{(n)}$ by attaching to each $i \in \mathcal{N}$ an independent Poisson process having rate $\beta_\textrm{G}$ giving the times individuals make global contacts if they are infectious. (As in Subsection 2.2, each global contact is with an individual chosen independently and uniformly from $\mathcal{N}$ .) Observe that if i is infected by a global contact in $\mathcal{E}^{(n)}$ , then each member of $\mathcal{C}^{(n)}_i$ is necessarily infected in $\mathcal{E}^{(n)}$ . Similarly, if i is an initial susceptible individual in $\mathcal{E}^{(n)}$ , then i is infected in $\mathcal{E}^{(n)}$ if and only if a member of $\mathcal{S}^{(n)}_i$ is infected by a global contact or $\mathcal{S}^{(n)}_i$ contains the initial infective.

For $i \in \mathcal{N}$ , let $C^{(n)}_i=|\mathcal{C}^{(n)}_i|$ and $S^{(n)}_i=|\mathcal{S}^{(n)}_i|$ be the sizes of i’s local infectious clump and local susceptibility set, respectively, and let $A^{(n)}_i=\sum_{j \in \mathcal{C}^{(n)}_i} I_j$ be the severity of i’s local infectious clump. Note that $C^{(n)}_1, C^{(n)}_2, \ldots, C^{(n)}_n$ are identically distributed, with distribution that depends on n (unless $\theta=0$ ), but they are not independent. A similar comment holds for $S^{(n)}_1, S^{(n)}_2, \ldots, S^{(n)}_n$ and $A^{(n)}_1, A^{(n)}_2, \ldots, A^{(n)}_n$ . Let $C^{(n)}, S^{(n)}$ , and $A^{(n)}$ be random variables distributed as $C^{(n)}_1$ , $S^{(n)}_1$ , and $A^{(n)}_1$ , respectively. We show below that there exist random variables C, S, and A such that

(5.1) \begin{equation}C^{(n)} \stackrel{\textrm{D}}{\longrightarrow} C, \quad S^{(n)} \stackrel{\textrm{D}}{\longrightarrow} S, \quad \text{and} \quad A^{(n)} \stackrel{\textrm{D}}{\longrightarrow} A \quad \text{as } n \to \infty.\end{equation}

The distributions of C, S, and A are defective, with a mass at infinity, if the epidemic $\mathcal{E}^{(n)}$ in the absence of global infection (i.e. with $\beta_\textrm{G}=0$ ) is supercritical in the limit $n \to \infty$ .

Suppose that ${\mathbb P}(C \lt \infty)=1$ , and recall that the epidemic $\mathcal{E}^{(n)}$ is initiated by a single individual being externally infected. Then, for large n, the early stages of $\mathcal{E}^{(n)}$ can be approximated by a branching process of local infectious clumps, since the probability that clumps arising from distinct global contacts have a non-empty intersection is small. Let $R^{(n)}$ be the number of global contacts that emanate from a typical local infectious clump. Then $R^{(n)}\sim\textrm{Po}(\beta_\textrm{G} A^{(n)})$ , i.e. $R^{(n)}$ is distributed as a mixed Poisson distribution with random mean $\beta_\textrm{G} A^{(n)}$ . Thus, ${\mathbb E}[R^{(n)}]=\beta_\textrm{G} {\mathbb E}[A^{(n)}]$ . Moreover, since $I_j$ is independent of the event $\{\,j \in \mathcal{C}^{(n)}_i\}$ and ${\mathbb E}[I_j]=1$ ,

\begin{align*}{\mathbb E}\big[A^{(n)}_i\big]&={\mathbb E}\left[\sum_{j=1}^n I_j 1_{\{\,j \in \mathcal{C}^{(n)}_i\}}\right] \\[5pt] &= \sum_{i=1}^n {\mathbb E}[I_j] {\mathbb P}\big(\,j \in \mathcal{C}^{(n)}_i\big)=\sum_{j=1}^n {\mathbb E}\Big[1_{\{\,j \in \mathcal{C}^{(n)}_i\}}\Big]={\mathbb E}\big[C^{(n)}_i\big].\end{align*}

Hence, ${\mathbb E}[R^{(n)}]=\beta_\textrm{G} {\mathbb E}[C^{(n)}]$ . Now $R^{(n)} \stackrel{\textrm{D}}{\longrightarrow} R$ as $n \to \infty$ , where $R \sim \textrm{Po}(\beta_\textrm{G} A)$ , so the process of local infectious clumps in $\mathcal{E}^{(n)}$ can be approximated by a Galton–Watson branching process $\mathcal{B}$ , having a single ancestor and offspring distribution R.

Let $\xi$ be the extinction probability of $\mathcal{B}$ . Then $\xi$ is given by the smallest solution in [0, 1] of $f_R(s)=s$ . Conditioning on A and noting that the PGF of $\textrm{Po}(X)$ is ${\mathbb E}[\!\exp\!(\!-\!(1-s)X]$ , we have

(5.2) \begin{equation}f_R(s)={\mathbb E}\left[{\mathbb E}[s^R|A]\right]={\mathbb E}[\!\exp\!(\!-\!\beta_\textrm{G} A(1-s))]=\phi_A(\beta_\textrm{G}(1-s)).\end{equation}

Further, $\xi\lt1$ if and only if $R_*\gt1$ , where

(5.3) \begin{equation}R_* = {\mathbb E}[R] = \beta_\textrm{G} {\mathbb E}[A]= \beta_\textrm{G} {\mathbb E}[C].\end{equation}

Thus, $R_*$ serves as a threshold parameter for the epidemic $\mathcal{E}^{(n)}$ , in that as $n \to \infty$ a major outbreak occurs with non-zero probability if and only if $R_*\gt1$ . Note that $\rho$ in Theorem 3.1 is given by $1-\xi$ .

The above assumes that ${\mathbb P}(C \lt \infty)=1$ . If ${\mathbb P}(C \lt \infty)\lt1$ then the branching process $\mathcal{B}$ is defined as above but now $R_*=\infty$ and an individual may have infinitely many offspring. The branching process $\mathcal{B}$ can still be used to approximate the early stages of $\mathcal{E}^{(n)}$ for large n, since an individual having infinitely many offspring in $\mathcal{B}$ necessarily corresponds to a major outbreak in $\mathcal{E}^{(n)}$ . The above heuristic argument is made rigorous in the proof of Theorem 3.1 in Subsection 5.5.

Suppose that n is large and a major outbreak occurs. Let z be the fraction of the population that is infected. Since ${\mathbb E}[I]=1$ , by the law of large numbers, the total severity of the epidemic is approximately nz (the mean severity of nz individuals), so the total number of global contacts made during the epidemic is approximately distributed as $\textrm{Po}(\beta_\textrm{G} nz)$ . Since each global contact is with an individual chosen independently and uniformly from the population, the numbers of global contacts made with distinct individuals are approximately independent $\textrm{Po}(\beta_\textrm{G} z)$ random variables. Hence, the probability a given individual avoids global contact throughout the epidemic is approximately

(5.4) \begin{equation}\pi=\exp\!(\!-\!\beta_\textrm{G} z).\end{equation}

The probability an individual chosen uniformly at random from the population is infected by the epidemic is given by z. Recall that such an individual, if initially susceptible, avoids infection from the epidemic if and only if (i) all members of its susceptibility set avoid global infection and (ii) its susceptibility set does not contain the initial infective. The former event has asymptotic probability 0 if $S=\infty$ , and the latter event has asymptotic probability 1 if $S\lt\infty$ . For large n, individuals avoid global infection approximately independently, so $1-z \approx f_S(\pi)={\mathbb E}[\pi^S]$ . In Subsection 5.6, we show that these approximations are exact in the limit as $n \to \infty$ , so, with use of (5.4), z satisfies

(5.5) \begin{equation}1-z=f_S\big(\textrm{e}^{-\beta_\textrm{G} z}\big).\end{equation}

Note that $s=1-z$ satisfies the equation governing the extinction probability of a Galton–Watson process having offspring distribution $\textrm{Po}(\beta_\textrm{G} S)$ . Further, by exchangeability

(5.6) \begin{equation}{\mathbb E}\big[C^{(n)}\big] =\frac{1}{n}{\mathbb E}\left[\sum_{i=1}^n \sum_{j=1}^n 1_{\{i \leadsto j\}}\right]=\frac{1}{n}{\mathbb E}\left[\sum_{j=1}^n \sum_{i=1}^n 1_{\{\,j \leadsto i\}}\right]= {\mathbb E}\big[S^{(n)}\big],\end{equation}

so letting $n \to \infty$ suggests $\beta_\textrm{G} {\mathbb E}[S]=\beta_\textrm{G} {\mathbb E}[C]=R_*$ . (See Appendix A for a formal justification for the present model.) It follows from standard branching process theory that if $R_* \le 1$ then $z=0$ is the only solution of (5.5), while if $R_*\gt1$ , there is a unique second solution in (0, 1]. When $R_*\gt1$ , this second solution gives the fraction of the population that is infected by a major outbreak in the limit $n \to \infty$ .

In Subsection 5.3.4, we show that S, the size of a typical local susceptibility set in the limit $n \to \infty$ , is distributed as the total progeny of a three-type branching process. The PGF of S is given in (5.15), and $R_*$ is given in (5.18). When $I \equiv 1$ , the possible directed edges in $\mathcal{G}^{(n)}$ are present independently, with probability $1-\exp\!(\!-\!\beta_\textrm{H}')$ if the two individuals are in the same household but not the same workplace, $1-\exp\!(\!-\!\beta_\textrm{W}')$ if they are in the same workplace but not the same household, and $1-\exp\!(-(\beta_\textrm{H}'+\beta_\textrm{W}'))$ if they are in both the same household and the same workplace. It follows that $C^{(n)} \stackrel{\textrm{D}}{=} S^{(n)}$ , so $C \stackrel{\textrm{D}}{=} S$ . Note that in this case $A^{(n)}=C^{(n)}$ , whence $A \stackrel{\textrm{D}}{=} C \stackrel{\textrm{D}}{=} S$ , and (5.2) and (5.5) imply that $\rho=z$ . When I is non-constant, C is distributed as the total progeny of a $3(h+w-2)$ -type branching process and A as the sum of the lifetimes of all individuals in that branching process. The derivation of the Laplace transform $\phi_A(\nu)$ in that case is outlined in Subsection 5.3.3.

Although this section is concerned with the case $\pi_\textrm{G}\gt0$ , we note here that when $\pi_\textrm{G}=0$ , we have $\rho={\mathbb P}(C=\infty)$ and $z={\mathbb P}(S=\infty)$ . Further, the threshold parameter $R_\textrm{L}$ determines whether the above-mentioned three-type branching process is subcritical, critical, or supercritical.

5.3. Asymptotic properties of local infectious clumps and susceptibility sets

5.3.1. Complexes

To apply the theory outlined in Subsection 5.2, we need to obtain the distribution of the random variables C, A, and S, describing the size and severity of a typical local infectious clump and the size of a typical local susceptibility set, respectively, in the limit as $n \to \infty$ . For this purpose, we first define a complex. There are $m = n/w$ complexes in the network, where m is the number of workplaces. Each complex is derived from a workplace containing d households with individuals belonging to one complex or two complexes depending on whether they are a remainer (belong to a single complex) or a mover (belongs to two complexes). We label the workplaces $1,2,\ldots, m$ , and associate the ith complex with the ith workplace. There are three types of individuals in the ith complex as follows.

• • Movers who start in workplace i (their household belongs to workplace i) but move to a different workplace.

• • Remainers who start in workplace i (their household belongs to workplace i) and remain in that workplace.

• • Movers who start in a different workplace (their household is part of a different workplace) and move into workplace i.

We consider complex i. We label the households in complex i, $(i,1), (i,2), \ldots, (i,d)$ . Then complex i comprises $2 d+1$ groups of individuals. These are as follows.

• • Group $2j-1$ $(\,j=1,2,\ldots,d)$ . These are the individuals in household (i, j) who remain in workplace i.

• • Group 2j $(\,j=1,2,\ldots,d)$ . These are the individuals in household (i, j) who move to a different workplace.

• • Group $2d+1$ . The individuals who move into workplace i.

Therefore in complex i, household-based infections occur in household (i, j) among the individuals in groups $2j-1$ and 2j and workplace-based infections involve groups $j=1,3,\ldots, 2d+1$ (the odd-numbered groups). We note that group $2j-1$ $(\,j=1,2,\ldots,d)$ can be empty if all members of household (i, j) move to different workplaces. We still associate the household (i, j) with workplace i.

In Figure 4 we illustrate how complexes and the population are constructed using an example where $h=d=2$ and $m=4$ , giving $w=4$ and $n =16$ .

Figure 4.

Illustration of the construction of a population of $n=16$ individuals divided into four workplaces, labelled A, B, C, and D, of size $4(\!=w)$ , with each workplace consisting of $2(\!=d)$ households of size $2(\!=h)$ . Red squares denote workplaces and green ellipses denote households. (a) The initial workplaces with solid circles representing remainers and open circles representing movers with movers numbered. There are eight movers and eight remainers. (b) The construction of complexes. Open squares represent the places created in a workplace by movers, and the number in the square denotes the mover who fills the place. Note that in complexes A and D there are households consisting completely of movers with the household residing outside the workplace. (c) The final population structure created from the complexes.

When $\theta=0$ , everyone is a remainer, the complexes do not intersect, and each complex consists of a workplace that is partitioned into d households. The distributions of $C^{(n)}$ , $A^{(n)}$ , and $S^{(n)}$ are each independent of n, so $C^{(n)} \stackrel{\textrm{D}}{=} C$ , $A^{(n)} \stackrel{\textrm{D}}{=} A$ , and $S^{(n)} \stackrel{\textrm{D}}{=} S$ for all n. The threshold parameter $R_*$ and the Laplace transform $\phi_A(\nu)$ can be computed, at least in principle, with the use of exact results for the final outcome of stochastic multitype SIR epidemics given in [Reference Ball2, Section 3] or [Reference Picard and Lefèvre30, Section 4]. The probability mass function (PMF) of S can be computed in principle with the use of Lemma 4.1 in [Reference Ball3]. In practice, these computations are possible only for suitably small d. In the following, we assume that $\theta\gt0$ .

5.3.2. Local infectious clumps

Suppose first that the infectious period I is non-random, so ${\mathbb P}(I=1)=1$ . We construct the local infectious clump of a typical individual, $i^*$ say, on a generation basis by considering local epidemic spread within successive complexes. Note that apart from the initial generation, a local epidemic enters a complex through one of two ways: either a mover in one of groups $2, 4, \ldots, 2d$ is infected in their workplace or a mover in group $2d+1$ is infected in their household. The initial individual $i^*$ can be any member of a complex.

We observe that the groups $1,3,\ldots, 2d-1$ are exchangeable, as are the groups $2,4,\ldots, 2d$ . Therefore we need to consider only the behaviour of the local spread in a complex in the cases where the initial infective in a complex is in group 1, 2, or $2d+1$ . Let R denote that the initial infective is in group 1 (or $3,5,\ldots,2d-1$ ); that is, the individual is a remainer. Let H denote that the initial infective is in group 2 (or $4,6,\ldots,2d$ ); that is, the individual belongs to complex i through their household. Let W denote that the initial infective is in group $2d+1$ ; that is, the individual belongs to complex i through the workplace.

The initial infective in a complex being of type X is informative about the structure of the complex. For example, if $X=W$ then we know that there is at least one mover in the workplace of the complex. Without loss of generality, if the initial individual is of type R or type H we assign that individual to group 1 or group 2, respectively. Let $M_1 \sim \textrm{Bin} (h-1,\theta)$ , and for $j=2,3,\ldots,d$ let $M_j \sim \textrm{Bin} (h, \theta)$ . The number of individuals in groups $2j-1$ and 2j $(\,j=2,3,\ldots,d)$ are $h-M_j$ and $M_j$ , respectively, regardless of the type of the initial infective. If $X=R$ then the number of individuals in groups 1, 2, and $2d+1$ are $h-M_1$ , $M_1$ , and $\sum_{l=1}^d M_l$ , respectively; otherwise there are $h-1-M_1$ , $1+M_1$ , and $1+\sum_{l=1}^d M_l$ individuals in groups 1, 2, and $2d+1$ , respectively.

If n is large, the probability that, in the early stages of the construction of an infectious clump, there is a complex that contains movers who either moved into the complex from the same workplace or moved out of their original workplace to the same workplace is small. Thus, we approximate the size $C^{(n)}$ of the clump by assuming that for each complex all the members from groups $2,4,\ldots, 2d$ are in different workplaces and all members of group $2d+1$ are in different households. Further, we assume that apart from the initial infective in a complex, all movers in a complex have not been involved previously in the construction of the clump. Under these assumptions, $C^{(n)}$ is approximated by the total progeny of a three-type branching process, denoted by $\mathcal{B}_C$ , which we now describe.

Consider the spread of a local epidemic within a typical complex defined as above. For $X, Y=R, H, W$ , let $Z_{XY}$ be defined as follows. Let $Z_{XR}$ denote the number of remainer individuals (excluding the initial infective if $X=R$ ) infected in a complex when there is a single initial infective in group type X. Let $Z_{XH}$ denote the number of individuals in group $2d+1$ (excluding the initial infective if $X=W$ ) infected in the complex when there is a single infective in group type X. Individuals infected in group $2d+1$ will be infected in the workplace and will start an epidemic in a new complex through their household. In other words, in that other complex they will belong to group $2, 4, \ldots, 2d$ . Similarly, let $Z_{XW}$ denote the number of individuals in groups $2, 4, \ldots, 2d$ (excluding the initial infective if $X=H$ ) infected in the complex when there is a single infective in group type X. Individuals infected in groups $2, 4, \ldots, 2d$ will be infected in the household and will start an epidemic in a new complex through their workplace. In other words, in that other complex they will belong to group $2d+1$ .

If $i^*$ , the initial member of the local infectious clump, is a remainer then the initial individual in $\mathcal{B}_C$ has type R, and its offspring is distributed as $(Z_{RR}, Z_{RH}, Z_{RW})$ . If $i^*$ is a mover then $i^*$ belongs to two complexes, and its offspring is distributed as $(Z_{HR}, Z_{HH}, Z_{HW})+(Z_{WR}, Z_{WH}, Z_{WW})$ , where the two random vectors are independent since I is non-random. All subsequent individuals in the branching process will be of type H or type W, and their offspring are distributed as $(Z_{HR}, Z_{HH}, Z_{HW})$ or $(Z_{WR}, Z_{WH}, Z_{WW})$ , respectively. Note that individuals in the branching process $\mathcal{B}_C$ correspond to complexes in the local infectious clump.

Consider the branching process $\hat{\mathcal{B}}_C$ , in which there is a single ancestor, who is either of type H or of type W, and the offspring distributions of type-H and type-W individuals are $(Z_{HR}, Z_{HH}, Z_{HW})$ and $(Z_{WR}, Z_{WH}, Z_{WW})$ , respectively. Thus, type-R individuals may exist in $\hat{\mathcal{B}}_C$ but they have no offspring. Let $\hat{Z}_H$ be the total progeny of $\hat{\mathcal{B}}_C$ (i.e. the total number of individuals infected and not the total number of complexes infected), counting all types and including the ancestor, given that the ancestor is of type H, and define $\hat{Z}_W$ similarly.

In addition, for $X=R, H, W$ and $0 \le s_1, s_2, s_3 \le 1$ , let

\[g_X(s_1, s_2, s_3)={\mathbb E}\big[s_1^{Z_{XR}}s_2^{Z_{XH}}s_3^{Z_{XW}}\big].\]

By taking expectations with respect to the offspring of the ancestor, we have

(5.7) \begin{equation}f_{\hat{Z}_H}(s)=s g_H\big(s, f_{\hat{Z}_H}(s), f_{\hat{Z}_W}(s)\big) \quad \text{and} \quad f_{\hat{Z}_W}(s)=s g_W\big(s, f_{\hat{Z}_H}(s), f_{\hat{Z}_W}(s)\big).\end{equation}

For each $s \in [0,1]$ , the equations (5.7) determine $f_{\hat{Z}_H}(s)$ and $f_{\hat{Z}_W}(s)$ .

Let C be the total progeny of $\mathcal{B}_C$ , counting all types and including the ancestor, given that the ancestor is a mover with probability $\theta$ , otherwise it is a remainer. Then, conditioning on the type of the ancestor and the spread within its complex(es), we have

(5.8) \begin{equation}f_C(s)=(1-\theta)sg_R\big(s, f_{\hat{Z}_H}(s), f_{\hat{Z}_W}(s)\big)+\theta g_H\big(s, f_{\hat{Z}_H}(s), f_{\hat{Z}_W}(s)\big)f_{\hat{Z}_W}(s).\end{equation}

(Note, with the use of (5.7), that for $s\gt0$ the second term on the right-hand side of (5.8) may be expressed as $\theta s^{-1}f_{\hat{Z}_H}(s)f_{\hat{Z}_W}(s)$ ; we use the form in (5.8) as it is also valid when $s=0$ .)

Lemma 5.1. For the model with a constant infectious period,

\begin{equation*}C^{(n)} \stackrel{\textrm{D}}{\longrightarrow} C \quad \text{as } n \to \infty.\end{equation*}

Proof. We prove the lemma by coupling realisations of the local infectious clump $\mathcal{C}^{(n)}_{i^*}$ of an individual $i^*$ in $\mathcal{E}^{(n)}$ and the branching process $\mathcal{B}_C$ , both of which are constructed on a generation basis, taking complexes in the same order. We start with a realisation of $\mathcal{C}^{(n)}_{i^*}$ and use the local spread within its complexes to give a realisation of $\mathcal{B}_C$ until in $\mathcal{C}^{(n)}_{i^*}$ we encounter a complex which contains a mover whose original workplace is the same as that of another mover used previously in the construction of $\mathcal{C}^{(n)}_{i^*}$ . Let $D^{(n)}$ be the number of complexes used in $\mathcal{C}^{(n)}_{i^*}$ before we encounter such a complex, where $D^{(n)}=\infty$ if no such complex is encountered. As soon as such a complex is encountered, the offsprings of all subsequent individuals in $\mathcal{B}_C$ are sampled independently of $\mathcal{C}^{(n)}_{i^*}$ from their appropriate distributions.

Note that for $k=1,2,\ldots$ , if $C^{(n)}_{i^*}=k$ and $D^{(n)}\gt k+1$ then $C^{(n)}_{i^*}=C$ , since in the construction of $\mathcal{C}^{(n)}_{i^*}$ each new complex necessarily contains at least one new individual. Hence, for $k=1,2,\ldots$ ,

\begin{align*}{\mathbb P}\big(C^{(n)} \le k\big)&={\mathbb P}\big(C^{(n)} \le k, D^{(n)}\gt k+1\big)+{\mathbb P}\big(C^{(n)} \le k, D^{(n)} \le k+1\big)\nonumber\\[5pt] &={\mathbb P}\big(C \le k, D^{(n)}\gt k+1\big)+{\mathbb P}\big(C^{(n)} \le k, D^{(n)} \le k+1\big).\end{align*}

In addition,

\begin{equation*}{\mathbb P}(C \le k)={\mathbb P}\big(C \le k, D^{(n)}\gt k+1\big)+{\mathbb P}\big(C \le k, D^{(n)} \le k+1\big).\end{equation*}

It follows that

\begin{equation*}|{\mathbb P}\big(C^{(n)} \le k\big)-{\mathbb P}(C \le k)| \le {\mathbb P}\big(D^{(n)} \le k+1\big).\end{equation*}

Recall that $\theta\gt0$ . Let $T^{(n)}$ be the total number of movers in $\mathcal{E}^{(n)}$ . A simple application of Chebyshev’s inequality yields ${\mathbb P}(T^{(n)}\lt\theta n/2) \to 0$ as $n \to \infty$ . The maximum size of a complex is $2w=2h d$ , so the number of movers in the first $k+1$ complexes is at most $2w(k+1)$ . Given $T^{(n)} \ge \theta n/2$ , the probability that any two given distinct movers come from the same original workplace is bounded above by $(w-1)/(\frac{1}{2}\theta n-1)$ . Similarly, the probability a given mover moves back to their original workplace is bounded above by $w/(\frac{1}{2}\theta n)$ . Thus,

\begin{align*}{\mathbb P}\big(D^{(n)} \le k+1\big) &\le {\mathbb P}\big(T^{(n)}\lt\theta n/2\big)+\binom{2w(k+1)}{2} \frac{w-1}{\frac{1}{2}\theta n-1} {\mathbb P}\big(T^{(n)}\gt\theta n/2\big) \\[5pt] & \quad + 2 w (k+1) \frac{w}{\frac{1}{2}\theta n} {\mathbb P}\big(T^{(n)}\gt\theta n/2\big) \to 0\end{align*}

as $n \to \infty$ . Hence, for $k=1,2,\ldots$ ,

\[\lim_{n \to \infty} {\mathbb P}\big(C^{(n)} \le k\big) = {\mathbb P}(C \le k),\]

and the lemma follows.

When the infectious period I is random, the local infectious clump $\mathcal{C}^{(n)}_{i^*}$ can still be constructed on a generation basis and coupled to a limiting process as in Lemma 5.1. However, the limiting process is not a branching process, since the infectious periods of movers infected in a within-complex epidemic (i.e. individuals who subsequently start epidemics in new complexes) are not independent of the size of that within-complex epidemic. The limiting process can be made a branching process by further typing, for example by also typing infected movers by the number of individuals they infect in their household/workplace in the next complex, as we describe in Subsection 5.3.3.

5.3.3. Major outbreak probability

Recall from Subsection 5.2 that the major outbreak probability $\rho=1-\xi$ , where $\xi$ is the smallest root in [0, 1] of $f_R(s)=s$ , with $f_R(s)$ being given by (5.2). Suppose first that $I \equiv 1$ . Then, as noted in Subsection 5.2, we have that $A \stackrel{\textrm{D}}{=} C$ and $\rho=z$ . We describe the computation of z (for the case of general I) in Subsection 5.3.4, thus enabling $\rho$ to be calculated. Alternatively, substituting $A \stackrel{\textrm{D}}{=} C$ into (5.2) yields $f_R(s)=f_C(\textrm{e}^{-\beta_\textrm{G}(1-s)})$ , so $f_R(s)$ can be obtained with the use of (5.8). We outline below how the distribution of the limiting severity A can be obtained for general I in a constructive manner, from which $\rho$ can be obtained with the use of (5.2). The key building block is the epidemic within a complex and in particular the joint distribution of the severity of the within-complex epidemic and the number of infectious contacts outside the complex made by movers infected within the complex.

For $X=W,H$ and $l=1,2,\ldots, w-1$ , let $A^{X,l}$ be the severity, excluding the initial infective, in a complex, where the initial infective in the complex is a mover who belongs to the complex only through its workplace if $X=W$ (household if $X=H$ ) and the initial infective infects l individuals within the complex (either in the workplace or in the household depending on whether $X=W$ or $X=H$ ). Note that if we know the number of infectious contacts the initial infective makes within a complex then the size of the epidemic in the complex is conditionally independent of the initial infective’s infectious period. In addition, for $Y=W,H$ and $k=1,2,\ldots, w-1$ , let $Z^{X,l}_{Y,k}$ denote the number of type-(Y, k) movers infected in the complex with an initial infective of type (X, l). A type-(Y, k) individual is a mover who belongs to another complex through their workplace if $Y=W$ (household if $Y=H$ ) and infects k individuals within their other complex. We are interested in the joint distribution of $(A^{X,l}, \mathbf{Z}^{X,l})$ , where $\mathbf{Z}^{X,l} = (Z^{X,l}_{H,1},\ldots, Z^{X,l}_{H,h-1}, Z^{X,l}_{W,1}, \ldots, Z^{X,l}_{W,w-1})$ is a random vector of length $w+h-2$ .

For $X=W,H$ and $l=1,2,\ldots, w-1$ , let $\hat{A}^{X,l}$ be the severity (sum of infectious periods), excluding the initial infective, in a branch of the infectious clump, where the initial infective in a branch is a mover who belongs to the initial complex in a branch only through its workplace if $X=W$ (household if $X=H$ ) and the initial infective infects l individuals within the complex (either in the workplace or in the household depending on whether $X=W$ or $X=H$ ). A branch of an infectious clump starts with a mover individual who begins a complex epidemic and consists of all subsequent complex epidemics emanating from the initial infectious complex in the branch. By the asymptotic branching process of the growth of an infectious clump it follows that

(5.9) \begin{align} \hat{A}^{X,l} \stackrel{\textrm{D}}{=} A^{X,l} + \sum_{(Y,k)} \sum_{i=1}^{Z^{X,l}_{Y,k}} \hat{A}^{Y,k,i},\end{align}

where $\hat{A}^{Y,k,1}, \hat{A}^{Y,k,2}, \ldots$ are independent and identically distributed (i.i.d.) copies of $\hat{A}^{Y,k}$ and sums are equal to 0 if vacuous.

For a given complex structure, i.e. sizes of groups $1,2, \ldots, 2d+1$ , $\mathbf{Z}^{X,l}$ is a vector of final state random variables defined on a multitype SIR epidemic, and the joint Laplace transform generating function of $(A^{X,l},\mathbf{Z}^{X,l})$ can be computed with the use of Theorem 5.1 in [Reference Ball and O’Neill11] or Theorem 4.2 in [Reference Ball3]. The unconditional joint Laplace transform generating function of $(A^{X,l}, \mathbf{Z}^{X,l})$ can then be obtained by taking expectations with respect to the complex structure. For fixed $\nu \ge 0$ , a set of $w+h-2$ non-linear equations which determine $(\phi_{\hat{A}^{H,1}}(\nu), \ldots, \phi_{\hat{A}^{H,h-1}}(\nu), \phi_{\hat{A}^{W,1}}(\nu), \ldots, \phi_{\hat{A}^{W,w-1}}(\nu))$ can then be derived with the use of (5.9), enabling, at least in principle, the $\phi_{\hat{A}^{X,l}}(\nu)$ to be computed.

Consider the initial individual in the infectious clump where the individual is a mover. Let I denote its infectious period, and let $Q_\textrm{H}$ and $Q_\textrm{W}$ denote the number of individuals the initial infective infects within their household and workplace, respectively. Then $Q_\textrm{H} | I \sim \textrm{Bin} (h-1, 1- \exp\!(-I \beta_\textrm{H} /(h-1)))$ and $Q_\textrm{W} | I \sim \textrm{Bin} (w-1, 1- \exp\!(-I \beta_\textrm{W}/(w-1)))$ , where $Q_\textrm{H}$ and $Q_\textrm{W}$ are conditionally independent given I. Let $\tilde{A}_\textrm{M}$ denote the severity of an infectious clump where the initial infective is a mover. Conditional on the triple $(I,Q_\textrm{H}, Q_\textrm{W})$ ,

(5.10) \begin{align} \tilde{A}_\textrm{M} \stackrel{\textrm{D}}{=} I + \hat{A}^{H,Q_\textrm{H}} + \hat{A}^{W,Q_\textrm{W}},\end{align}

with I, $\hat{A}^{H,Q_\textrm{H}}$ , and $\hat{A}^{W,Q_\textrm{W}}$ conditionally independent given $Q_\textrm{H}$ and $Q_\textrm{W}$ . (The initial infective is responsible for instigating two branches, originating from each of the complexes to which it belongs.) By conditioning first on I and then on $(Q_\textrm{H}, Q_\textrm{W})$ , we can compute the Laplace transform $\phi_{\tilde{A}_\textrm{M}}(\nu)$ using (5.10).

Consider now the case when the initial infective in an infectious clump is a remainer. Then the initial infective belongs to only one complex; in all subsequent complexes in the clump, the initial infective in the complex is a mover, but in this first complex it is a remainer. Let $A^{R,(\,j,l)}$ denote the severity, excluding the initial infective, in a complex, where the initial infective in the complex is a remainer who makes contact with j of its housemates and l of its workplace colleagues. Note that some individuals can belong to both the same household and the same workplace as the initial infective, so the number of individuals infected by the initial infective can be less than $j+l$ . For $Y=W,H$ and $k=1,2,\ldots, w-1$ , let $Z^{R,(\,j,l)}_{Y,k}$ denote the number of type-(Y, k) movers infected in the complex by an initial infective of type (R, j, l). Let $\tilde{A}_\textrm{R}$ denote the severity of an infectious clump where the initial infective is a remainer. Conditional on the triple $(I,Q_\textrm{H},Q_\textrm{W})$ ,

(5.11) \begin{align} \tilde{A}_\textrm{R} \stackrel{\textrm{D}}{=} I + A^{R,(Q_\textrm{H},Q_\textrm{W})} + \sum_{(Y,k)} \sum_{i=1}^{Z^{R,(Q_\textrm{H},Q_\textrm{W})}_{Y,k}} \hat{A}^{Y,k,i},\end{align}

where $ \hat{A}^{Y,k,1}, \hat{A}^{Y,k,2}, \ldots$ are i.i.d. copies of $ \hat{A}^{Y,k}$ and are independent of $(I,A^{R,(Q_\textrm{H},Q_\textrm{W})})$ given $Z^{R,(Q_\textrm{H},Q_\textrm{W})}$ . For each (j, l), the joint Laplace transform generating function of $(A^{R,(\,j,l)},\mathbf{Z}^{R,(\,j,l)})$ can be computed similarly to that of $(A^{X,l}, \mathbf{Z}^{X,l})$ above, whence $\phi_{\tilde{A}_\textrm{R}}(\nu)$ can be computed with the use of (5.11).

Finally we note that A is a mixture of $\tilde{A}_\textrm{M}$ and $\tilde{A}_\textrm{R}$ with mixing probabilities $\theta$ and $1-\theta$ , so

(5.12) \begin{equation}\phi_A(\nu)=\theta \phi_{\tilde{A}_\textrm{M}}(\nu)+(1-\theta)\phi_{\tilde{A}_\textrm{R}}(\nu),\end{equation}

and with the use of (5.2), $\rho=1-\xi$ , where $\xi$ is the smallest root of $\phi_A(\beta_\textrm{G}(1-s))=s$ in [0, 1].

5.3.4. Local susceptibility sets

The local susceptibility set $\mathcal{S}^{(n)}_{i^*}$ of an individual $i^*$ in $\mathcal{E}^{(n)}$ can be constructed on a generation basis in an fashion analogous to the construction of $\mathcal{C}^{(n)}_{i^*}$ in Subsection 5.3.2. We first construct the susceptibility set of $i^*$ within the one or two complexes to which $i^*$ belongs. We then construct the susceptibility sets of all movers (apart from $i^*$ if it is a mover) in that susceptibility set in their other complex, and so on. A result equivalent to Lemma 5.1 holds. Moreover, the limiting process is necessarily a branching process, even if I is random, since when we consider the susceptibility, $\mathcal{S}_{C,j^*}$ say, of a single individual $j^*$ within a complex C, the distribution of $\mathcal{S}_{C,j^*}$ does not depend on $I_{j^*}$ .

An argument similar to the proof of Lemma 5.1 shows that $S^{(n)} \stackrel{\textrm{D}}{\longrightarrow} S$ as $n \to \infty$ , where S is the total progeny of a three-type branching process, denoted by $\mathcal{B}_S$ , which is defined analogously to $\mathcal{B}_C$ in Subsection 5.3.2. For $X, Y=R, H, W$ , let $Z_{XY}^S$ be defined similarly to $Z_{XY}$ in Subsection 5.3.2 but for the susceptibility set of an individual within a typical complex. Thus, in the above notation, $Z_{XY}^S$ is the number of type-Y individuals in $\mathcal{S}_{C,j^*}\setminus \{\,j^*\}$ , given $j^*$ is of type X. For $X=H, W$ , let $\hat{Z}_X^S$ be the total number of individuals, including the ancestor, in the branching process $\hat{\mathcal{B}}_S$ that is defined analogously to $\hat{\mathcal{B}}_C$ in the obvious fashion. For $X=R, H, W$ and $0 \le s_1, s_2, s_3 \le 1$ , let

(5.13) \begin{equation}g_X^S(s_1, s_2, s_3)={\mathbb E}\Big[s_1^{Z_{XR}^S}s_2^{Z_{XH}^S}s_3^{Z_{XW}^S}\Big].\end{equation}

Then, cf. (5.7), for each $s \in [0,1]$ , the PGFs $f_{\hat{Z}_H^S}(s)$ and $f_{\hat{Z}_W^S}(s)$ of ${\hat{Z}_H^S}$ and ${\hat{Z}_W^S}$ , respectively, are determined by

(5.14) \begin{equation}f_{\hat{Z}_H^S}(s)=s g_H^S\Big(s, f_{\hat{Z}_H^S}(s), f_{\hat{Z}_W^S}(s)\Big) \quad \text{and} \quad f_{\hat{Z}_W^S}(s)=s g_W^S\Big(s, f_{\hat{Z}_H^S}(s), f_{\hat{Z}_W^S}(s)\Big).\end{equation}

Let S be the total progeny of $\mathcal{B}_S$ . Then, cf. (5.8),

(5.15) \begin{equation}f_S(s)=(1-\theta)sg_R^S\Big(s, f_{\hat{Z}_H^S}(s), f_{\hat{Z}_W^S}(s)\Big)+\theta g_H^S\Big(s, f_{\hat{Z}_H^S}(s), f_{\hat{Z}_W^S}(s)\Big)f_{\hat{Z}_W^S}(s).\end{equation}

Recall from (5.3) that $R_*=\beta_\textrm{G} \mu_S$ , since $\mu_C=\mu_S$ (see Appendix A). Conditioning on the first generation of $\hat{\mathcal{B}}_S$ or suitable differentiation of (5.14) yields

\[\begin{pmatrix}\mu_{\hat{Z}_H^S}\\[5pt] \mu_{\hat{Z}_W^S}\end{pmatrix}=\begin{pmatrix}1+\mu_{Z_{HR}^S}\\[5pt] 1+\mu_{Z_{WR}^S}\end{pmatrix}+\mathbf{M}^S\begin{pmatrix}\mu_{\hat{Z}_H^S}\\[5pt] \mu_{\hat{Z}_W^S}\end{pmatrix}\!,\]

where

(5.16) \begin{align} \mathbf{M}^S = \begin{pmatrix} \mu_{Z_{HH}^S}\;\;\;\;\;\; & \mu_{Z_{HW}^S} \\[5pt] \mu_{Z_{WH}^S}\;\;\;\;\;\; & \mu_{Z_{WW}^S} \end{pmatrix}\!.\end{align}

Let $\zeta^S$ be the maximum eigenvalue (Perron–Frobenius root) of $\mathbf{M}^S$ . In Subsection 6.4 we show that $\zeta^S$ serves as a threshold parameter for the epidemic with $\pi_\textrm{G}=0$ , i.e. for the epidemic with only local infection, so we also denote $\zeta^S$ by $R_\textrm{L}$ . Thus,

(5.17) \begin{align}R_\textrm{L} (= \zeta^S) =\frac{\mu_{Z_{HH}^S}+\mu_{Z_{WW}^S}+\sqrt{\Big(\mu_{Z_{HH}^S}-\mu_{Z_{WW}^S}\Big)^2+4 \mu_{Z_{HW}^S} \mu_{Z_{WH}^S}}}{2}.\end{align}

Observe that $\mu_{\hat{Z}_H^S}=\mu_{\hat{Z}_W^S}=\infty$ if and only if $R_\textrm{L} \ge 1$ ; otherwise

\[\begin{pmatrix}\mu_{\hat{Z}_H^S}\\[5pt] \mu_{\hat{Z}_W^S}\end{pmatrix}=(I-\mathbf{M}^S)^{-1}\begin{pmatrix}1+\mu_{Z_{HR}^S}\\[5pt] 1+\mu_{Z_{WR}^S}\end{pmatrix}\!.\]

Differentiation of (5.15) and use of $R_*=\beta_\textrm{G} \mu_S$ then yields

(5.18) \begin{align}R_*=\beta_\textrm{G}\left[ 1-2\theta +(1-\theta)\Big(\mu_{Z_{RR}^S}+\mu_{Z_{RH}^S} \mu_{\hat{Z}_H^S}+\mu_{Z_{RW}^S} \mu_{\hat{Z}_W^S}\Big)+\theta\Big(\mu_{\hat{Z}_H^S}+\mu_{\hat{Z}_W^S}\Big) \right]\!.\end{align}

Note that $R_*\lt \infty$ if and only if $R_\textrm{L}\lt1$ .

5.4. Numerical computation

We outline computation of $R_*$ and z (and hence also $\rho$ when $I \equiv 1$ ). Computation of $\rho$ when I is non-constant is indicated in Subsection 5.3.3, but its implementation is computationally more intensive and intricate, owing to the appreciably larger dimension of the approximation branching process $\mathcal{B}_C$ .

For $X=R, H, W$ , let ${\boldsymbol{Z}}_X^S=(Z_{XR}^S, Z_{XH}^S,Z_{XW}^S)$ . Note that ${\boldsymbol{Z}}_X^S$ has finite support, so its joint PGF $g_X^S(s_1, s_2, s_3)$ and mean vector are easily computed from its joint PMF. Thus, given the joint PMFs of ${\boldsymbol{Z}}_X^S$ $(X=R ,H, W)$ , the PGF $f_S(s)$ is easily computed with the use of (5.14) and (5.15), enabling the limiting final size z to be computed with the use of (5.5), and $R_*$ is easily computed with the use of (5.18). We outline how the joint PMF of ${\boldsymbol{Z}}_R^S$ may be computed; the joint PMFs of ${\boldsymbol{Z}}_H^S$ and ${\boldsymbol{Z}}_W^S$ may be computed similarly.

Recall the binomially distributed independent random variables $M_1, M_2, \ldots, M_d$ defined in Subsection 5.3.2. Consider the susceptibility set, $\mathcal{S}_{C,j^*}$ say, of a single individual $j^*$ within a typical complex. Given $j^*$ is a remainer, the number of individuals in the complex in groups $2j-1$ and 2j $(\,j=1,2,\ldots, d)$ are $h-M_j$ and $M_j$ , respectively, and the number of individuals in group $2d+1$ is $\sum_{i=1}^d M_i$ . Let ${\boldsymbol{S}}^C_R=(S^C_{R,1},S^C_{R,2}, \ldots, S^C_{R,2d+1})$ , where $S^C_{R,i}$ is the number of group-i individuals in the complex susceptibility set $\mathcal{S}_{C,j^*}$ , excluding $j^*$ if $i=1$ . Let ${\boldsymbol{M}}=(M_1, M_2, \ldots, M_d)$ and ${\boldsymbol{m}}$ denote a realisation of ${\boldsymbol{M}}$ . For each possible ${\boldsymbol{m}}$ , the joint PMF of ${\boldsymbol{S}}^C_R|{\boldsymbol{M}}={\boldsymbol{m}}$ may be computed with the use of Lemma 4.1 in [Reference Ball3]. The joint PMF of ${\boldsymbol{S}}^C_R$ may then be computed with the use of the law of total probability, and the joint PMF of ${\boldsymbol{Z}}_R^S$ follows easily.

Let $\boldsymbol{\ell}=(\ell_1, \ell_2, \ldots, \ell_{2d+1})$ be a possible realisation of ${\boldsymbol{S}}^C_R$ . With the use of Lemma 4.1 in [Reference Ball3], ${\mathbb P}({\boldsymbol{S}}^C_R=\boldsymbol{\ell}|{\boldsymbol{M}}={\boldsymbol{m}})$ takes the form

\[{\mathbb P}\big({\boldsymbol{S}}^C_R=\boldsymbol{\ell}|{\boldsymbol{M}}={\boldsymbol{m}}\big)=\alpha_R({\boldsymbol{m}},\boldsymbol{\ell}) \beta_R(\boldsymbol{\ell}),\]

where $\beta_R(\boldsymbol{\ell})$ is defined in terms of multivariate Gontcharoff polynomials (see [Reference Lefèvre and Picard22]). The $\alpha_R({\boldsymbol{m}},\boldsymbol{\ell})$ are readily computed but the $\beta_R(\boldsymbol{\ell})$ need to be obtained by solving a triangular system of linear equations, which is numerically prohibitive unless d and n are suitably small. However, note that the $\beta_R(\boldsymbol{\ell})$ are independent of ${\boldsymbol{m}}$ and hence need computing only once.

5.5. Proof of Theorem 3.1 when $\pi_\textrm{G}\gt0$

We prove here a version of Theorem 3.1 in which $\log n$ is replaced by $\kappa_n=\log \log n$ . The result for $\log n $ follows from the proof of Theorem 3.2 in Subsection 5.6. We use the construction of the final size of $\mathcal{E}^{(n)}$ via local infectious clumps outlined in Subsection 5.2. Let $Z^{(n)}$ denote the final size of $\mathcal{E}^{(n)}$ and let $\check{Z}^{(n)}$ be the final size of a modified epidemic, $\check{\mathcal{E}}^{(n)}$ , in which the size of local infectious clumps is capped at $\kappa_n$ . The latter can be achieved by growing the local infectious clumps in real time and stopping the growth of a clump if and when its size reaches $\kappa_n$ . Thus, in $\check{\mathcal{E}}^{(n)}$ , the local infectious clump of individual i is $\check{\mathcal{C}}^{(n)}_i$ , where $\check{\mathcal{C}}^{(n)}_i \subseteq \mathcal{C}^{(n)}_i$ and $\check{\mathcal{C}}^{(n)}_i$ has size

\[\check{C}^{(n)}_i=\begin{cases}C^{(n)}_i & \text{ if } C^{(n)}_i \le \kappa_n,\\[5pt] \lfloor \kappa_n \rfloor& \text { if } C^{(n)}_i \gt \kappa_n.\end{cases}\]

Let $\mathcal{B}^{(n)}$ be the branching process of local infected clumps defined analogously to the branching process $\mathcal{B}$ in Subsection 5.2, except (C, A, R) is distributed as $(C^{(n)}, A^{(n)}, R^{(n)})$ . Let $Z^{(n)}_I$ be the total number of infected individuals in $\mathcal{B}^{(n)}$ , i.e. the sum of the sizes of the local infectious clumps of all individuals in $\mathcal{B}^{(n)}$ . Let $\check{\mathcal{B}}^{(n)}$ be a modified version of $\mathcal{B}^{(n)}$ in which the size of local infectious clumps is capped at $\kappa_n$ and let $\check{Z}^{(n)}_I$ be the total number of infected individuals in $\check{\mathcal{B}}^{(n)}$ . Observe that

(5.19) \begin{align}{\mathbb P}\big(Z^{(n)} \le \kappa_n\big)={\mathbb P}\big(\check{Z}^{(n)} \le \kappa_n\big) \quad\text{and}\quad {\mathbb P}\big(Z^{(n)}_I \le \kappa_n\big)={\mathbb P}\big(\check{Z}^{(n)}_I \le \kappa_n\big).\end{align}

We construct coupled realisations of $\check{\mathcal{E}}^{(n)}$ and $\check{\mathcal{B}}^{(n)}$ as follows. Let $\tilde{\mathcal{G}}^{(n)}$ be the random graph with vertex set $\mathcal{N}$ in which for distinct $i, j\in \mathcal{N}$ there is an edge between i and j if and only if i and j belong to the same complex. For $(i,j) \in \mathcal{N}^2$ , let $\rho(i,j)$ be the length of the shortest path between i and j in $\tilde{\mathcal{G}}^{(n)}$ , with $\rho(i,i)=0$ . Let $J^{(n)}_1, J^{(n)}_2, \ldots$ be i.i.d., each uniformly distributed on $\mathcal{N}$ . The initial infective in $\check{\mathcal{E}}^{(n)}$ is individual $J^{(n)}_1$ , and for $k=1,2, \ldots$ , the kth global contact in $\check{\mathcal{E}}^{(n)}$ is with individual $J^{(n)}_k$ . Let

\[F_n=\bigcap_{i=1}^{\lfloor \kappa_n \rfloor-1}\bigcap_{j=i+1}^{\lfloor \kappa_n \rfloor} \big\{\rho\big(J^{(n)}_i,J^{(n)}_j\big)\gt2\kappa_n\big\}.\]

If $F_n$ occurs then $\check{\mathcal{C}}^{(n)}_{J^{(n)}_k}$ $(k=1,2, \ldots, \lfloor \kappa_n \rfloor)$ are independent, so $(\check{C}^{(n)}_{J^{(n)}_k},\check{R}^{(n)}_{J^{(n)}_k})$ $(k=1,2, \ldots, \lfloor \kappa_n \rfloor)$ can be used to give the clump sizes and offsprings of the first $\lfloor \kappa_n \rfloor$ individuals in $\check{\mathcal{B}}^{(n)}$ . (Here, $\check{R}^{(n)}_i$ is the number of global contacts that emanate from $\check{\mathcal{C}}^{(n)}_i$ .) Let $F_n^\textrm{c}$ denote the complement of $F_n$ . If $F_n^\textrm{c}$ occurs then we construct a realisation of $\check{\mathcal{B}}^{(n)}$ independently of $\check{\mathcal{E}}^{(n)}$ . Now

\begin{align*}{\mathbb P}\big(\check{Z}^{(n)} \le \kappa_n\big)={\mathbb P}\big(\check{Z}^{(n)} \le \kappa_n, F_n\big)+{\mathbb P}\big(\check{Z}^{(n)} \le \kappa_n, F_n^\textrm{c}\big)\end{align*}

and

\begin{align*}{\mathbb P}\big(\check{Z}^{(n)}_I \le \kappa_n\big)={\mathbb P}\big(\check{Z}^{(n)}_I \le \kappa_n, F_n\big)+{\mathbb P}\big(\check{Z}^{(n)}_I \le \kappa_n, F_n^\textrm{c}\big).\end{align*}

By construction, if $F_n$ occurs then $\check{Z}^{(n)} \le \kappa_n$ if and only if $\check{Z}^{(n)}_I \le \kappa_n$ , so

\begin{align*}\big|{\mathbb P}\big(\check{Z}^{(n)} \le \kappa_n\big)-{\mathbb P}\big(\check{Z}^{(n)}_I \le \kappa_n\big)\big| \le {\mathbb P}\big(F_n^\textrm{c}\big).\end{align*}

Note that, since each individual belongs to one or two complexes and each complex contains at most 2w individuals, for $K=1,2,\ldots$ , there are at most

(5.20) \begin{align}H(K)=2\sum_{i=1}^K (2w)^i \le 4(2w)^K\end{align}

individuals within $\rho$ -distance K of a given individual, so

\begin{align*}{\mathbb P}\big(F_n^\textrm{c}\big) \le \frac{\kappa_n(\kappa_n-1)4(2w)^{2\kappa_n}}{n} \to 0 \quad \text{as }n \to \infty.\end{align*}

Thus, recalling (5.19), we have

(5.21) \begin{align}\lim_{n \to \infty} \big|{\mathbb P}\big(Z^{(n)} \le \kappa_n\big)-{\mathbb P}\big(Z^{(n)}_I \le \kappa_n\big)\big|=0.\end{align}

Recall that $\xi$ is the extinction probability of $\mathcal{B}$ , and let $\xi_n$ be the extinction probability of $\mathcal{B}^{(n)}$ . Now $R^{(n)} \stackrel{\textrm{D}}{\longrightarrow} R$ as $n \to \infty$ , so $\lim_{n \to \infty}\xi_n=\xi$ (see [Reference Britton, Janson and Martin-Löf17], Lemma 4.1). Hence,

(5.22) \begin{align}\limsup_{n \to \infty} {\mathbb P}\big(Z^{(n)}_I \le \kappa_n\big) \le \limsup_{n \to \infty} {\mathbb P}\big(Z^{(n)}_I\lt\infty\big)=\limsup_{n \to \infty}\xi_n=\xi.\end{align}

In addition, for fixed $k=1,2,\ldots$ ,

(5.23) \begin{align}\liminf_{n \to \infty} {\mathbb P}\big(Z^{(n)}_I \le \kappa_n\big) \ge \liminf_{n \to \infty} {\mathbb P}\big(Z^{(n)}_I \le k\big)={\mathbb P}(Z_I \le k),\end{align}

where $Z_I$ is the total number of individuals infected in $\mathcal{B}$ . Now ${\mathbb P}(Z_I \le k) \to \xi$ as $k \to \infty$ , so letting $k \uparrow \infty$ in (5.23) yields

\begin{align*}\liminf_{n \to \infty}{\mathbb P}\big(Z^{(n)}_I \le \kappa_n\big) \ge \xi,\end{align*}

which together with (5.22) implies $\lim_{n \to \infty} {\mathbb P}(Z^{(n)}_I \le \kappa_n)= \xi$ . Thus, with the use of (5.21),

\begin{align*}\lim_{n \to \infty} {\mathbb P}\big(Z^{(n)} \le \kappa_n\big)=\xi,\end{align*}

whence $\lim_{n \to \infty} {\mathbb P}(Z^{(n)} \gt \kappa_n)=\rho=1-\xi$ , which completes the proof of Theorem 3.1, with $\log n$ replaced by $\kappa_n$ .

Note that an argument similar to the above shows that

\begin{align*}\lim_{n \to \infty}{\mathbb P}\big(Z^{(n)} \le k\big)={\mathbb P}(Z_I \le k)\qquad(k=1,2,\ldots),\end{align*}

so $Z^{(n)} \stackrel{\textrm{D}}{\longrightarrow} Z_I$ as $n \to \infty$ . The random variable $Z_I$ is defective, with a mass at infinity, if $R_*\gt1$ .

5.6. Proof of Theorem 3.2 when $\pi_\textrm{G}\gt0$

To prove Theorem 3.2, we use the adaptation of the embedding technique of Scalia-Tomba [Reference Scalia-Tomba31, Reference Scalia-Tomba32] to the general two levels of mixing setting (see [Reference Ball and Neal10], Subsection 6.2, and [Reference Ball and Sirl13], Subsection 3.4.2).

Let $I_1, I_2, \ldots, I_n$ and the random directed graph $\mathcal{G}^{(n)}$ be as in Subsection 5.2, and let $L_1, L_2, \ldots, L_n$ be an independent sequence of i.i.d. exponential random variables, each having rate $\beta_\textrm{G}$ . Given, $I_1, I_2, \ldots, I_n$ , $\mathcal{G}^{(n)}$ and $L_1, L_2, \ldots, L_n$ , we construct for each $t \ge 0$ a realisation of the final outcome of an epidemic $\tilde{\mathcal{E}}^{(n)}(t)$ as follows. Note that in this construction t refers to global infectious pressure and not time. We determine first who is infected globally; specifically, for $i \in \mathcal{N}$ , individual i is infected globally if and only if $L_i \le t$ . We then use the random directed graph $\mathcal{G}^{(n)}$ to determine who is subsequently infected locally. Specifically, for $i \in \mathcal{N}$ , individual i is infected locally if and only if at least one member of i’s local susceptibility set $\mathcal{S}^{(n)}_i$ is infected globally. For $i \in \mathcal{N}$ , let

\begin{align*}\chi^{(n)}_i(t)&=1_{\{i \text{ is infected in } \tilde{\mathcal{E}}^{(n)}(t)\}}\\[5pt] &=\begin{cases}1 & \text{ if } \min_{j \in \mathcal{S}^{(n)}_i} L_j \le t,\\[5pt] 0 & \text { otherwise } .\end{cases}\end{align*}

Let

\begin{align*}R^{(n)}_{\bullet}(t)= \sum_{i=1}^n \chi^{(n)}_i(t)\quad\text{and}\quad A^{(n)}_{\bullet}(t)= \sum_{i=1}^n I_i \chi^{(n)}_i(t)\end{align*}

be respectively the total size and severity of the epidemic $\tilde{\mathcal{E}}^{(n)}(t)$ .

To connect with the epidemic $\mathcal{E}^{(n)}$ , consider an epidemic, $\hat{\mathcal{E}}^{(n)}$ say, that is initiated by exposing the population to $T^{(n)}_0$ units of global infectious pressure, so each individual is exposed to $\bar{T}^{(n)}_0=n^{-1}T^{(n)}_0$ units of global infectious pressure. These $T^{(n)}_0$ units of global infectious pressure may infect some individuals globally, and the consequent local spread will lead to $A^{(n)}_{\bullet}(\bar{T}^{(n)}_0)$ further units of global infectious pressure. Thus, the population is now exposed to $T^{(n)}_0+A^{(n)}_{\bullet}(\bar{T}^{(n)}_0)$ units of global infectious pressure, and the process can be continued in the obvious fashion. For $k=0,1,\ldots$ , let

\begin{align*}\bar{T}^{(n)}_{k+1}=\bar{T}^{(n)}_0+\bar{A}^{(n)}_{\bullet}\big(\bar{T}^{(n)}_k\big),\end{align*}

where $\bar{A}^{(n)}_{\bullet}(t)=n^{-1}A^{(n)}_{\bullet}(t)$ . Then $k_*^{(n)}=\min\{k\;:\;\bar{T}^{(n)}_{k+1}=\bar{T}^{(n)}_k\}$ is well defined as the population is finite. Let $T^{(n)}_{\infty}=T^{(n)}_{k_*^{(n)}}$ and $\bar{T}^{(n)}_{\infty}=n^{-1}T^{(n)}_{\infty}$ . Note that

(5.24) \begin{align}\bar{T}^{(n)}_{\infty}=\min\big\{t \ge 0\;:\; t=\bar{T}^{(n)}_0+\bar{A}^{(n)}_{\bullet}(t)\big\}.\end{align}

The total size and severity of the epidemic $\hat{\mathcal{E}}^{(n)}$ are given by $R^{(n)}_{\bullet}(\bar{T}^{(n)}_{\infty})$ and $A^{(n)}_{\bullet}(\bar{T}^{(n)}_{\infty})$ , respectively.

The epidemics $\mathcal{E}^{(n)}$ and $\hat{\mathcal{E}}^{(n)}$ have different laws, since $\mathcal{E}^{(n)}$ is initiated by a single individual chosen uniformly at random from the population being externally infected and the number of initial infectives in $\hat{\mathcal{E}}^{(n)}$ , $\hat{Z}^{(n)}_\textrm{init}$ say, is distributed as $R^{(n)}_{\bullet}(\bar{T}^{(n)}_0)$ . However, owing to the lack-of-memory property of the exponential distribution, $\mathcal{E}^{(n)}$ has the same law as $\hat{\mathcal{E}}^{(n)}$ conditioned upon $\hat{Z}^{(n)}_\textrm{init}=1$ . Suppose that ${\mathbb P}(T^{(n)}_0=1)=1$ . Then $\hat{Z}^{(n)}_\textrm{init} \sim \text{Bin}(n, 1-\exp\!(\!-\!\beta_\textrm{G}/n))$ , so

\begin{align*}{\mathbb P}\big(\hat{Z}^{(n)}_\textrm{init}=1\big) &=n \exp\!(\!-\!(n-1)\beta_\textrm{G}/n)(1-\exp\!(\!-\!\beta_\textrm{G}/n)) \\[5pt] & \to \beta_\textrm{G} \exp\!(\!-\!\beta_\textrm{G}) \quad \text{as } n \to \infty.\end{align*}

Thus, $\liminf_{n \to \infty}{\mathbb P}(\hat{Z}^{(n)}_\textrm{init}=1)\gt0$ and (see [Reference Janson20]) it follows that convergence in probability results for $\hat{\mathcal{E}}^{(n)}$ hold also for $\hat{\mathcal{E}}^{(n)}|\hat{Z}^{(n)}_\textrm{init}=1$ , and hence for $\mathcal{E}^{(n)}$ .

Recall the random variable S defined in (5.1) in Subsection 5.2, describing the size of a typical local susceptibility set in the limit $n \to \infty$ . For $t \ge 0$ , let $r(t)=1-f_S(\textrm{e}^{-\beta_\textrm{G} t})$ and $\bar{R}^{(n)}_{\bullet}(t)=n^{-1}R^{(n)}_{\bullet}(t)$ .

Lemma 5.2. For any $t_0\gt0$ ,

(5.25) \begin{align}\!\!\sup\nolimits_{t_0 \le t \lt \infty} \big|\bar{R}^{(n)}_{\bullet}(t)-r(t)\big| \stackrel{\textrm{p}}{\longrightarrow} 0 \quad\textit{and}\quad \!\sup\nolimits_{t_0 \le t\lt \infty} \big|\bar{A}^{(n)}_{\bullet}(t)-r(t)\big| \stackrel{\textrm{p}}{\longrightarrow} 0 \quad \textit{as } n \to \infty.\end{align}

Proof. We prove the result for $\bar{A}^{(n)}_{\bullet}(t)$ . The result for $\bar{R}^{(n)}_{\bullet}(t)$ is proved similarly. Recall that we assume ${\mathbb E}[I]=1$ .

For $t \ge 0$ , let

\begin{align*}a^{(n)}(t)={\mathbb E}\big[\bar{A}^{(n)}_{\bullet}(t)\big]&={\mathbb E}\big[I_1 \chi^{(n)}_1(t)\big]\\[5pt] &={\mathbb E}\big[I_1\big(1-\exp\!\big(\!-\!\beta_\textrm{G} t S^{(n)}_1\big)\big)\big]\\[5pt] &=1-{\mathbb E}\big[\!\exp\!\big(\!-\!\beta_\textrm{G} t S^{(n)}_1\big)\big],\end{align*}

since $I_1$ and $S^{(n)}_1$ are independent. We show below that for each $t \in (0, \infty)$ , $\bar{A}^{(n)}_{\bullet}(t)-a^{(n)}(t) \stackrel{\textrm{p}}{\longrightarrow} 0$ as $n \to \infty$ . Now $a^{(n)}(t) \to r(t)$ as $n \to \infty$ , since $S^{(n)}_1 \stackrel{\textrm{D}}{\longrightarrow} S$ as $n \to \infty$ and $0 \le \exp\!(\!-\!\beta_\textrm{G} t S^{(n)}_1) \le 1$ for all n. Thus, $\bar{A}^{(n)}_{\bullet}(t) \stackrel{\textrm{p}}{\longrightarrow} r(t)$ as $n \to \infty$ . This convergence holds also for $t=\infty$ , since $\bar{A}^{(n)}_{\bullet}(\infty)=n^{-1}\sum_{i=1}^n I_i \stackrel{\textrm{p}}{\longrightarrow} 1=r(\infty)$ as $n \to \infty$ by the weak law of large numbers. The random function $\bar{A}^{(n)}_{\bullet}(t)$ is non-decreasing in t on $[t_0, \infty]$ , and r(t) is continuous and increasing on $[t_0,\infty]$ . The second result in (5.25) then follows by an argument similar to the proof of Lemma 5.1 in Ball and Lyne [Reference Ball and Lyne8]; see [Reference Ball and Sirl13], p. 187.

To complete the proof, we show that for each $t \in (0, \infty)$ , $\bar{A}^{(n)}_{\bullet}(t)-a^{(n)}(t) \stackrel{\textrm{p}}{\longrightarrow} 0$ as $n \to \infty$ . Note that since ${\mathbb E}[\bar{A}^{(n)}_{\bullet}(t)]=a^{(n)}(t)\to r(t)$ as $n \to \infty$ , it suffices to show that, for all $\epsilon\gt0$ ,

(5.26) \begin{align}{\mathbb P}\big(\bar{A}^{(n)}_{\bullet}(t)\lt a^{(n)}(t)-\epsilon\big) \to 0\quad \text{as } n \to \infty.\end{align}

(If there exist $\epsilon, \delta \gt0$ such that ${\mathbb P}(\bar{A}^{(n)}_{\bullet}(t) \ge a^{(n)}(t)+\epsilon)\gt\delta$ for arbitrary large n, then (5.26) implies that ${\mathbb E}[\bar{A}^{(n)}_{\bullet}(t)]\not\to r(t)$ as $n \to \infty$ .)

Let $\epsilon\gt0$ and $t \in [t_0, \infty)$ be arbitrary but fixed. For $m\gt0$ , let

\begin{align*}A^{(n,m)}_{\bullet}(t)=\sum_{i=1}^n \min(I_i,m) \chi^{(n)}_i(t) \quad\text{and}\quad \bar{A}^{(n,m)}_{\bullet}(t)=n^{-1} A^{(n,m)}_{\bullet}(t).\end{align*}

Now ${\mathbb E}[\bar{A}^{(n,m)}_{\bullet}(t)] \uparrow a^{(n)}(t)$ as $m \uparrow \infty$ , by the monotone convergence theorem, so choose m sufficiently large that ${\mathbb E}[\bar{A}^{(n,m)}_{\bullet}(t)]\gt a^{(n)}(t)-\frac{\epsilon}{2}$ . Then, with the use of Chebyshev’s inequality,

\begin{align*}{\mathbb P}\big(\bar{A}^{(n)}_{\bullet}(t)\lt a^{(n)}(t)-\epsilon\big) &\le {\mathbb P}\big(\bar{A}^{(n,m)}_{\bullet}(t)\lt a^{(n)}(t)-\epsilon\big)\\[5pt] &\le {\mathbb P}\big(\bar{A}^{(n,m)}_{\bullet}(t)-{\mathbb E}\big[\bar{A}^{(n,m)}_{\bullet}(t)\big]\lt-\epsilon/2\big)\\[5pt] &\le \frac{\textrm{var}\big(\bar{A}^{(n,m)}_{\bullet}(t)\big)}{\epsilon^2/4}=4\frac{\textrm{var}\big(A^{(n,m)}_{\bullet}(t)\big)}{n^2 \epsilon^2}.\end{align*}

Thus, it suffices to show that

(5.27) \begin{align}\lim_{n \to \infty} n^{-2}\textrm{var}\big(A^{(n,m)}_{\bullet}(t)\big)=0.\end{align}

Let $\tilde{\chi}^{(n)}_i(t)=1-\chi^{(n)}_i(t)$ and $\tilde{A}^{(n,m)}_{\bullet}(t)=\sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t)$ , where $\tilde{I}_i=\min(I_i,m)$ . Let $\sigma_{\tilde{I}}^2=\textrm{var}(\tilde{I}_1)$ . Noting that $A^{(n,m)}_{\bullet}(t)+\tilde{A}^{(n,m)}_{\bullet}(t)=\sum_{i=1}^n \tilde{I}_i$ , we have

\begin{align*}n\sigma_{\tilde{I}}^2=\textrm{var}\big(A^{(n,m)}_{\bullet}(t)\big)+\textrm{var}\big(\tilde{A}^{(n,m)}_{\bullet}(t)\big)+2\textrm{cov}\big(A^{(n,m)}_{\bullet}(t),\tilde{A}^{(n,m)}_{\bullet}(t)\big).\end{align*}

Further,

\begin{align*}\textrm{cov}\big(A^{(n,m)}_{\bullet}(t),\tilde{A}^{(n,m)}_{\bullet}(t)\big)&=\textrm{cov}\left(\sum_{i=1}^n \tilde{I}_i\big(1-\tilde{\chi}^{(n)}_i(t)\big), \sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t) \right)\\[5pt] &= \textrm{cov}\left(\sum_{i=1}^n \tilde{I}_i, \sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t)\right)-\textrm{var}\big(\tilde{A}^{(n,m)}_{\bullet}(t)\big),\end{align*}

so (5.27) holds if

(5.28) \begin{align}\lim_{n \to \infty} n^{-2} \textrm{var}\big(\tilde{A}^{(n,m)}_{\bullet} (t)\big)=0\quad \text{and}\quad \lim_{n \to \infty} n^{-2} \textrm{cov}\left(\sum_{i=1}^n \tilde{I}_i, \sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t)\right)=0.\end{align}

For $t \ge 0$ , let $\hat{A}^{(n,m)}_{\bullet}(t)=\sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t)1_{\{S^{(n)}_i \le \kappa_n\}}$ , where $\kappa_n=\log \log n$ as in Subsection 5.5. Then, using the independence of $\tilde{I}_i$ and $S^{(n)}_i$ , and recalling that ${\mathbb E}[I]=1$ (so ${\mathbb E}[\tilde{I}_1]\lt1$ ), we have

(5.29) \begin{align}\big|{\mathbb E}\big[\tilde{A}^{(n,m)}_{\bullet}(t)\big]-{\mathbb E}\big[\hat{A}^{(n,m)}_{\bullet}(t)\big]\big|&={\mathbb E}\left[\sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t)1_{\{S^{(n)}_i \gt \kappa_n\}}\right]\\[5pt] &=\sum_{i=1}^n {\mathbb E} \left[\tilde{I}_i \exp\!\big(\!-\!\beta_\textrm{G} t S^{(n)}_i\big)1_{\{S^{(n)}_i \gt \kappa_n\}}\right]\nonumber\\[5pt] &\le ng_n(t),\nonumber\end{align}

where $ g_n(t) = \exp\!(\!-\!\beta_\textrm{G} t \kappa_n)$ . Further,

\begin{align*}{\mathbb E}\big[\tilde{A}^{(n,m)}_{\bullet}(t)^2\big]&={\mathbb E}\left[\sum_{i=1}^n \tilde{I}_i \tilde{\chi}^{(n)}_i(t) \sum_{j=1}^n \tilde{I}_j \tilde{\chi}^{(n)}_j(t)\right]\\[5pt] &=\sum_{i=1}^n \sum_{j=1}^n {\mathbb E}\left[\tilde{I}_i \tilde{I}_j \exp\!\big(\!-\!\beta_\textrm{G} t \big|\mathcal{S}^{(n)}_i \cup \mathcal{S}^{(n)}_j\big|\big)\right]\!.\end{align*}

Thus,

(5.30) \begin{align}\big| {\mathbb E}\big[&\tilde{A}^{(n,m)}_{\bullet}(t)^2\big]-{\mathbb E}\big[\hat{A}^{(n,m)}_{\bullet}(t)^2\big]\big|\nonumber \\[5pt] &=\sum_{i=1}^n \sum_{j=1}^n{\mathbb E}\Big[\tilde{I}_i\tilde{I}_j\exp\!\big(\!-\!\beta_\textrm{G} t |\mathcal{S}^{(n)}_i \cup \mathcal{S}^{(n)}_j|\big)\Big(1-1_{\{S^{(n)}_i \le \kappa_n\}}1_{\{S^{(n)}_j \le \kappa_n\}}\Big)\Big]\nonumber\\[5pt] &\le \sum_{i=1}^n \sum_{j=1}^n{\mathbb E}\Big[\tilde{I}_i\tilde{I}_j\exp\!\big(\!-\!\beta_\textrm{G} t |\mathcal{S}^{(n)}_i \cup \mathcal{S}^{(n)}_j|\big)\Big(1_{\{S^{(n)}_i \gt \kappa_n\}}+1_{\{S^{(n)}_j \gt \kappa_n\}}\Big)\Big]\nonumber\\[5pt] &\le 2g_n(t)\sum_{i=1}^n \sum_{j=1}^n {\mathbb E}[\tilde{I}_i \tilde{I}_j]\le 2g_n(t)\big[n\sigma_{\tilde{I}}^2+n^2\big].\end{align}

It follows from (5.29) and (5.30) that

(5.31) \begin{align}\lim_{n \to \infty} n^{-2} \textrm{var}\big(\tilde{A}^{(n,m)}_{\bullet} (t)\big)=0 \iff \lim_{n \to \infty} n^{-2} \textrm{var}\big(\hat{A}^{(n,m)}_{\bullet} (t)\big)=0,\end{align}

since $g_n(t) \to 0$ as $n \to \infty$ .

Note that, by exchangeability,

(5.32) \begin{align}n^{-2} \textrm{var}\big(\hat{A}^{(n,m)}_{\bullet} (t)\big)&=n^{-2} \sum_{i=1}^n \sum_{j=1}^n \textrm{cov}\Big(\tilde{I}_i \tilde{\chi}^{(n)}_i(t)1_{\{S^{(n)}_i \le \kappa_n\}}, \tilde{I}_j \tilde{\chi}^{(n)}_j(t)1_{\{S^{(n)}_j \le \kappa_n\}} \Big)\nonumber\\[5pt] &=n^{-1}\sum_{j=1}^n \textrm{cov}\Big(\tilde{I}_1 \tilde{\chi}^{(n)}_1(t)1_{\{S^{(n)}_1 \le \kappa_n\}}, \tilde{I}_j \tilde{\chi}^{(n)}_j(t)1_{\{S^{(n)}_j \le \kappa_n\}} \Big)\nonumber\\[5pt] &=\textrm{cov}\left(\tilde{I}_1 \tilde{\chi}^{(n)}_1(t)1_{\{S^{(n)}_1 \le \kappa_n\}}, \tilde{I}_J \tilde{\chi}^{(n)}_J(t)1_{\{S^{(n)}_J \le \kappa_n\}} \right)\!,\end{align}

where J is chosen uniformly at random from $\mathcal{N}$ .

Recall from Subsection 5.5 the graph $\tilde{\mathcal{G}}^{(n)}$ and the associated distance function $\rho(i,j)$ . Let $X^{(n)}_i(t)=\tilde{I}_i \tilde{\chi}^{(n)}_i(t)1_{\{S^{(n)}_i \le \kappa_n\}}$ $(i \in \mathcal{N})$ and $U^{(n)}=1_{\{\rho(1,J)\le 3 \kappa_n\}}$ . Then

\begin{align*}\textrm{cov}\big(X^{(n)}_1(t),X^{(n)}_J(t)\big)&={\mathbb E}\big[\textrm{cov}\big( X^{(n)}_1(t),X^{(n)}_J(t) \mid U^{(n)}\big)\big]\\[5pt] &\quad+\textrm{cov}\big({\mathbb E}\big[X^{(n)}_1(t)\mid U^{(n)} \big], {\mathbb E}\big[X^{(n)}_J(t)\mid U^{(n)} \big]\big).\end{align*}

Let $p_n={\mathbb P}(U^{(n)}=1)$ and note that $p_n \le \frac{H(3\kappa_n)}{n}$ , where $H(K)\le 4(2w)^K$ ; see (5.20). Hence, $p_n \to 0$ as $n \to \infty$ . Further, $X^{(n)}_1(t)$ and $X^{(n)}_J(t)$ are independent if $U^{(n)}=0$ , since $\{i \in \mathcal{N}\;:\; \rho(1,i) \le \kappa_n+1\}$ and $\{\,j \in \mathcal{N}\;:\; \rho(\,j,J) \le \kappa_n+1\}$ are disjoint. Thus,

\begin{align*}\big|{\mathbb E}\big[\textrm{cov}\big( X^{(n)}_1(t),X^{(n)}_J(t) \mid U^{(n)}\big)\big]\big| &\le p_n \big|\textrm{cov}\big( X^{(n)}_1(t),X^{(n)}_J(t) \mid U^{(n)}=1\big)\big|\\[5pt] &\le p_n {\mathbb E}[\tilde{I}^2] \to 0\quad \text{as } n \to \infty.\end{align*}

In addition,

\begin{align*}&\big|\textrm{cov}\big({\mathbb E}\big[X^{(n)}_1(t)\mid U^{(n)} \big], {\mathbb E}\big[X^{(n)}_J(t)\mid U^{(n)} \big]\big)\big|\\[5pt] & \quad=p_n(1-p_n)\big({\mathbb E}\big[X^{(n)}_J(t)\mid U^{(n)}=1 \big]-{\mathbb E}\big[X^{(n)}_J(t)\mid U^{(n)}=0 \big]\big)^2\\[5pt] & \quad \le p_n(1-p_n)({\mathbb E}[\tilde{I}_1])^2 \to 0\quad \text{as } n \to \infty.\end{align*}

Thus, $\textrm{cov}(X^{(n)}_1(t),X^{(n)}_J(t)) \to 0$ as $n \to \infty$ , so the first condition in (5.28) is satisfied with the use of (5.31) and (5.32). The second condition in (5.28) is proved similarly, so the details are left to the reader.

Suppose that ${\mathbb P}(T^{(n)}_0=1)=1$ and that $R_*\gt1$ . Note that the equation $r(t)=t$ is the same equation satisfied by z in (5.5) in Subsection 5.2. Now $r(0)=0$ , $r'(0)=R_*\gt1$ and $r''(t)\lt0$ for $t \ge 0$ . Thus, r(t) is concave on $[0, \infty)$ . Further, $r(\infty)=1$ , so there exists a unique $z \in (0,\infty)$ satisfying $r(z)=z$ and $r'(z)\lt1$ . (If $r'(z)=1$ then $r(z)=\int_0^{z} r'(x)\,\textrm{d}x\gt \int_0^{z}1 \,\textrm{d}x=z$ , contradicting $r(z)=z$ .) It follows from (5.24) and the second result in (5.25) that $\min(\bar{T}^{(n)}_{\infty}, |\bar{T}^{(n)}_{\infty}-z|) \stackrel{\textrm{p}}{\longrightarrow} 0$ as $n \to \infty$ , and hence from the first result in (5.25) that

(5.33) \begin{align}\min\big(\bar{R}^{(n)}_{\bullet}\big(\bar{T}^{(n)}_{\infty}\big),|\bar{R}^{(n)}_{\bullet}\big(\bar{T}^{(n)}_{\infty}\big)-z|\big) \stackrel{\textrm{p}}{\longrightarrow} 0 \quad\text{as } n \to \infty.\end{align}

Recall that $\bar{R}^{(n)}_{\bullet}(\bar{T}^{(n)}_{\infty})=n^{-1}R^{(n)}_{\bullet}(\bar{T}^{(n)}_{\infty})$ , where $R^{(n)}_{\bullet}(\bar{T}^{(n)}_{\infty})$ is the total size of the epidemic $\hat{\mathcal{E}}^{(n)}$ , and that convergence in probability results that hold for $\hat{\mathcal{E}}^{(n)}$ hold also for $\mathcal{E}^{(n)}$ . Let $\bar{Z}^{(n)}=n^{-1}Z^{(n)}$ , where $Z^{(n)}$ is the total size of the epidemic $\mathcal{E}^{(n)}$ . It follows from the above that $\bar{Z}^{(n)} \stackrel{\textrm{p}}{\longrightarrow} \bar{Z}$ as $n \to \infty$ , where the random variable $\bar{Z}$ has support $\{0, z\}$ . All that remains is to determine ${\mathbb P}(\bar{Z}=0)$ . Using Theorem 3.1, with $\log n$ replaced by $\kappa_n$ , we have

(5.34) \begin{align}\liminf_{n \to \infty} {\mathbb P}\bigg(\bar{Z}^{(n)} \le \frac{z}{2}\bigg) \ge \liminf_{n \to \infty} {\mathbb P}\big(Z^{(n)} \le \kappa_n\big)=\xi.\end{align}

To obtain an upper bound, first note we have using (5.33) that, for any $\epsilon \in (0,z/2)$ ,

(5.35) \begin{align}\limsup_{n \to \infty}{\mathbb P}\bigg(\bar{Z}^{(n)} \le \frac{z}{2}\bigg)= \limsup_{n \to \infty}{\mathbb P}\big(\bar{Z}^{(n)} \le \epsilon\big).\end{align}

For $K=1,2,\ldots$ , let $\check{\mathcal{E}}^{(n)}_K$ be a modification of the epidemic $\mathcal{E}^{(n)}$ in which the size of all local infectious clumps is capped at K and let $\check{Z}^{(n)}_K$ be the total size of $\check{\mathcal{E}}^{(n)}_K$ . Clearly,

(5.36) \begin{align}{\mathbb P}\big(\bar{Z}^{(n)} \le \epsilon\big) \le {\mathbb P}\big(\check{Z}^{(n)}_K \le n \epsilon\big).\end{align}

To derive an upper bound for ${\mathbb P}(\check{Z}^{(n)}_K \le n \epsilon)$ , we extend a technique which dates back to [Reference Whittle33]. A realisation of $\check{\mathcal{E}}^{(n)}_K$ can be constructed in a fashion analogous to that of $\check{\mathcal{E}}^{(n)}$ in Subsection 5.5, by using a sequence $J^{(n)}_1, J^{(n)}_2, \ldots$ of i.i.d. random variables that are uniformly distributed on $\mathcal{N}$ to determine the individuals contacted by successive global contacts. A lower-bound process may be obtained by deleting any global contact (and ignoring the consequent local infectious clump) that is with an individual whose $\rho$ -distance from any previously globally contacted individual (including the initial infective) is less than or equal to 2K. While the total size of $\check{\mathcal{E}}^{(n)}_K$ is $\le \epsilon n$ , the probability that a global contact is deleted is bounded above by $\frac{\epsilon n H(2K)}{n} \le 4\epsilon (2w)^{2K}$ , where the function H is defined in (5.20). Thus,

(5.37) \begin{align}{\mathbb P}\big(\check{Z}^{(n)}_K \le n \epsilon\big) \le {\mathbb P}\big(\check{Z}^{(n)}_{I,K, \epsilon} \le n \epsilon\big),\end{align}

where $\check{Z}^{(n)}_{I,K, \epsilon}$ is the total number of individuals infected in the branching process $\check{\mathcal{B}}^{(n)}_{K, \epsilon}$ , whose law is similar to that of $\mathcal{B}^{(n)}$ , except the size of local infectious clumps is capped at K and each individual (i.e. local infectious clump) is deleted at birth independently with probability $\min(4\epsilon (2w)^{2K},1)$ . Let $\mathcal{B}_{K, \epsilon}$ be an analogous branching process derived from $\mathcal{B}$ . Let $\xi_n(K, \epsilon)$ and $\xi(K, \epsilon)$ be the extinction probabilities of $\check{\mathcal{B}}^{(n)}_{K, \epsilon}$ and $\mathcal{B}_{K, \epsilon}$ , respectively, and note that $\lim_{n \to \infty} \xi_n(K, \epsilon)=\xi(K, \epsilon)$ . Using (5.35), (5.36), and (5.37), we have

(5.38) \begin{align}\limsup_{n \to \infty}{\mathbb P}\bigg(\bar{Z}^{(n)} \le \frac{z}{2}\bigg)\le \limsup_{n \to \infty} {\mathbb P}\big(\check{Z}^{(n)}_{I,K, \epsilon} \le n \epsilon\big) \le \limsup_{n \to \infty} \xi_n(K, \epsilon) =\xi(K, \epsilon).\end{align}

Now $\xi(K, \epsilon) \downarrow \xi(K)$ as $\epsilon \downarrow 0$ , where $\xi(K)$ is the extinction probability for the branching process derived from $\mathcal{B}$ in which clumps are capped at size K, and $\xi(K) \downarrow \xi$ as $ K \uparrow \infty$ . Thus, (5.34) and (5.38) yield

\begin{align*}\lim_{n \to \infty}{\mathbb P}\bigg(\bar{Z}^{(n)} \le \frac{z}{2}\bigg)=\xi,\end{align*}

whence ${\mathbb P}(\bar{Z}=0)=\xi=1-{\mathbb P}(\bar{Z}=z)$ .

Recall from the end of Subsection 5.5 that $Z^{(n)} \stackrel{\textrm{D}}{\longrightarrow} Z_I$ as $n \to \infty$ . It follows easily from the above that if $(\upsilon_n)$ is any sequence of non-negative integers satisfying $\upsilon_n \to \infty$ and $n^{-1}\upsilon_n \to 0$ as $n \to \infty$ , then

\begin{align*}\lim_{n \to \infty}{\mathbb P}\big(Z^{(n)} \ge \upsilon_n\big)=1-\xi,\end{align*}

and if $R_*\gt1$ ,

\begin{align*}\bar{Z}^{(n)}|Z^{(n)} \ge \upsilon_n \stackrel{\textrm{p}}{\longrightarrow} z \quad \text{as } n \to \infty.\end{align*}

6.1. Introduction and overview

In this section we consider the special case where $\pi_\textrm{G} =0$ , so that all infection is local through the households and workplaces. Throughout this section we assume that $\theta \gt0$ , since otherwise the complexes and workplaces coincide, and in the absence of global infection the epidemic is restricted to its original workplace. We prove Theorems 3.1 and 3.2 concerning the probability, $\rho$ , and final size, z, of a major outbreak hold in this setting. Note that $\rho=0$ unless the local infectious clumps and local susceptibility sets are supercritical, i.e. unless $R_\textrm{L}\gt1$ , in which case z and $\rho$ are given by (6.6) and (6.19), respectively.

Let $i^\ast$ denote the initial infective. We couple the construction of the local infections clump for individual $i^\ast$ , $\mathcal{C}^{(n)}_{i^\ast}$ , to a forward branching process $\mathcal{B}_C$ . Then we show that

\begin{align*} {\mathbb P} \big(C^{(n)}\gt \log n\big) \rightarrow \rho \quad \mbox{as } n \rightarrow \infty,\end{align*}

where $C^{(n)}= |\mathcal{C}^{(n)}_{i^\ast}|$ and $\rho$ is the non-extinction probability of $\mathcal{B}_C$ given by (6.19). Furthermore, we show that there exists $\delta^\prime \gt0$ such that

(6.1) \begin{align} {\mathbb P} \big(C^{(n)}\gt \delta^\prime n |C^{(n)}\gt \log n\big) \rightarrow 1 \quad \mbox{as } n \rightarrow \infty.\end{align}

In the absence of global infection, $Z^{(n)} =C^{(n)}$ .

Similarly, we couple the construction of the local infections susceptibility set, $\mathcal{S}^{(n)}$ , to a backward branching process $\mathcal{B}_S$ . We show that

(6.2) \begin{align} {\mathbb P} \big(S^{(n)} \gt \log n\big) \rightarrow z \quad \mbox{as } n \rightarrow \infty,\end{align}

where z is the non-extinction probability of $\mathcal{B}_S$ given by (6.6). Consider a typical initially susceptible individual $j^\ast$ , with local susceptibility set $\mathcal{S}^{(n)}_{j^\ast}$ having size $S^{(n)}_{j^\ast} \sim S^{(n)}$ . If $S^{(n)}_{j^\ast} \lt \log n$ then with high probability (i.e. with probability tending to 1 as $n \to \infty$ ), $\mathcal{S}^{(n)}_{j^\ast}$ does not contain $i^\ast$ , so $j^\ast$ is not infected by the epidemic. Suppose instead that $S^{(n)}_{j^\ast} \ge \log n$ . Then in the event of a major outbreak, it follows from (6.1) that $C^{(n)}\gt \delta^\prime n$ with high probability, and consequently $\mathcal{S}^{(n)}_{j^\ast} \cap \mathcal{C}^{(n)}$ is non-empty with high probability and $j^\ast$ is infected by the epidemic. The events $S^{(n)}_{j_1^\ast} \lt \log n$ and $S^{(n)}_{j_2^\ast} \lt \log n$ , where $j_1^\ast$ and $j_2^\ast$ are chosen uniformly at random from the initial susceptible individuals, are asymptotically independent, and consequently

(6.3) \begin{align} \frac{Z^{(n)}}{n} | Z^{(n)} \gt \log n \stackrel{\textrm{p}}{\longrightarrow} z \quad \mbox{as } n \to \infty.\end{align}

This is the intuition underlying the proof. Rather than considering whether $S^{(n)}_{j^\ast} \lt \log n$ , it is convenient to consider whether $\mathcal{S}^{(n)}_{j^\ast}$ goes extinct before generation $\ell_n$ , where $\ell_n = \lceil a \log \log n \rceil$ for suitable choice of $a \gt0$ (see [Reference Ball, Sirl and Trapman14]).

The remainder of this section is organised as follows. In Subsection 6.2, we derive some asymptotic properties of local infectious clumps and susceptibility sets, focusing mainly on the latter. In Subsection 6.3, we consider the final outcome of the epidemic. We show first in Lemma 6.2 that the fraction of the population whose local susceptibility set reaches generation $\ell_n$ converges in probability to z. Then, in Lemma 6.3, we consider the special case when $I \equiv 1$ and show that, conditional upon the occurrence of a major outbreak, the difference between that fraction and $n^{-1}Z^{(n)}$ converges in probability to 0 as $n \to \infty$ , leading to (6.3), i.e. Theorem 3.2 in the case $\pi_\textrm{G} =0$ and $I \equiv 1$ . Finally, in Subsection 6.4, we extend the proof to the case of general I.

6.2. Asymptotic properties of local infectious clumps and susceptibility sets

For some $a \gt0$ , let $\ell_n = \lceil a \log \log n \rceil$ , where further requirements on a will be defined as required. We start by coupling the construction of $\mathcal{C}^{(n)}$ to $\mathcal{B}_C$ and the construction of $\mathcal{S}^{(n)}$ to $\mathcal{B}_S$ over the first $\ell_n$ generations, where the initial individual is generation 0. Since the maximum size of a complex is 2w (all individuals in the workplace are movers), we have that the number of individuals whose shortest path from a randomly selected individual has length at most $\ell_n$ is bounded above by

(6.4) \begin{align} \sum_{i=0}^{\ell_n} 2 (2w)^i = 2 \frac{[2w]^{\ell_n+1} -1}{2w -1},\end{align}

where the factor 2 comes from mover individuals belonging to two complexes. For any $\epsilon \gt0$ , the right-hand side of (6.4) is less than $n^\epsilon$ for all sufficiently large n. The total number of movers in the population is $\textrm{Bin} (n, \theta)$ , so the proportion of the population who are movers is approximately $\theta$ . Given any single complex contains at most 2w movers it is straightforward using the birthday problem (see, for example, [Reference Ball and Donnelly7], p. 4) to show that with probability tending to 1, distinct complexes are used in the first $\ell_n$ generations of $\mathcal{S}^{(n)}$ or $\mathcal{C}^{(n)}$ . In other words, the growth of the population from a randomly selected individual is, with high probability, a tree-like configuration of complexes over the first $\ell_n$ generations (cf. the proof of Lemma 5.1). This can be taken further. Let $\mathcal{K}^{(n)} = \mathcal{K}^{(n)}_S \cup \mathcal{K}^{(n)}_C$ be a set of $K = K_S + K_C \in \mathbb{N} $ randomly selected individuals in a population of size n, where for the $K_S$ $(K_C)$ individuals in $\mathcal{K}^{(n)}_S$ $( \mathcal{K}^{(n)}_C)$ we construct the susceptibility set (clump) to generation $\ell_n$ . Then with probability tending to 1, no complex is used more than once in the construction of any of the susceptibility sets or clumps for the first $\ell_n$ generations.

To obtain z we use a construction of a local susceptibility set similar to the local infectious clump given in Subsection 5.3.2. In Subsection 5.3.2 and Lemma 5.1, the focus is on the PMF of $C^{(n)}$ and its limit C. In the absence of global infection, we simply need to consider whether or not the limiting branching process, $\mathcal{B}_S$ , goes extinct.

The construction of $\mathcal{B}_S$ given in Subsection 5.3.4 can again be used with $\mathbf{Z}^S_X = (Z^S_{XH}, Z^S_{XW})$ being the number of new complex susceptibility sets that are added to a local susceptibility set (offspring in the branching process) from a type-X complex susceptibility set. Let $j^*$ denote the initial individual for whom we are constructing a local susceptibility set. If $j^*$ is a mover then they belong to two complexes, and their offspring distribution is $\mathbf{Z}^S_H + \mathbf{Z}^S_W$ , where the two random vectors are independent. If $j^*$ is a remainer, then without loss of generality they belong to group 1, and $\mathbf{Z}_R^S = (Z^S_{RH}, Z^S_{RW})$ gives the number of movers in the complex who belong to the susceptibility set of $j^*$ . Note that, beyond the identity (mover or remainer) of the initial individual, we are not interested in any remainers who belong to the complex susceptibility set as they play no role in the extinction probability.

For $X=R, H, W$ and $\mathbf{s} \in [0,1]^2$ , let

\begin{align*} \bar{g}_X^S (\mathbf{s}) = {\mathbb E} \left[ s_1^{Z_{XH}^S} s_2^{Z_{XW}^S} \right]\!.\end{align*}

Let $\eta^S_H$ $(\eta^S_W)$ denote the probability that the branching process $\mathcal{B}_S$ goes extinct from a single type-H (type-W) individual. Then ${\eta}^S = (\eta^S_H, \eta^S_W)$ is the smallest solution in $[0,1]^2$ of

(6.5) \begin{align} \mbox{$\eta$}^S = \big( \bar{g}^S_H \big(\mbox{$\eta$}^S\big), \bar{g}^S_W \big(\mbox{$\eta$}^S\big) \big),\end{align}

and z, the probability that $\mathcal{B}_S$ does not go extinct, satisfies

(6.6) \begin{align} 1-z = (1-\theta) \bar{g}_R^S (\mbox{$\eta$}^S) + \theta \eta^S_H \eta^S_W.\end{align}

For $i=1,2,\ldots$ , let $\mathbf{V}_i^S = (V_{H,i}^S, V_{W,i}^S)$ denote the number of individuals in the ith generation of $\mathcal{B}_S$ . Similarly, let $\mathbf{V}_i^{S^{(n)}} = (V_{H,i}^{S^{(n)}} , V_{W,i}^{S^{(n)}} )$ denote the number of individuals in the ith generation of $\mathcal{S}^{(n)}$ . Since with probability tending to 1, $\mathcal{S}^{(n)}$ can be coupled to $\mathcal{B}_S$ to coincide over the first $\ell_n$ generations, we have that

\begin{align*} {\mathbb P} \big( \mathbf{V}_{\ell_n}^{S^{(n)}} \neq \mathbf{V}_{\ell_n}^S\big) \rightarrow 0 \quad \mbox{as } n \rightarrow \infty.\end{align*}

We use properties of branching processes to explore $\mathbf{V}_{\ell_n}^S$ .

The mean offspring matrix, $\mathbf{M}^S$ , given in (5.16), with Perron–Frobenius eigenvalue $\zeta^S (=R_\textrm{L})$ , determines the behaviour of the branching process $\mathcal{B}_S$ . Note that $\eta^S=(1,1)$ if $\zeta^S\le 1$ , and $\eta^S$ is the unique solution of (6.5) if $\zeta^S \gt 1$ . Hence, $z \gt0$ if and only if $\zeta^S \gt1$ . In addition, for $0 \lt \theta \lt1$ , all elements of $\mathbf{M}^S$ are non-zero and the branching process $\mathcal{B}_S$ is aperiodic. Therefore, by (8.2) in [Reference Mode24], p. 19, there exists a random variable $W^S$ with ${\mathbb P}(W^S=0) =1-z$ and a continuous distribution on $(0,\infty)$ such that

(6.7) \begin{align} \lim_{k \rightarrow \infty} \frac{1}{[\zeta^S]^k} \mathbf{V}^S_k = W^S \mathbf{v}^S \quad \mbox{almost surely},\end{align}

where $\mathbf{v}^S \gt \mathbf{0}$ is the normalised left eigenvector of $\mathbf{M}^S$ corresponding to the Perron–Frobenius root $\zeta^S$ . For $\theta=1$ , the branching process $\mathcal{B}_S$ has period 2, and the arguments are easily modified by letting $\ell_n = \lceil a \log \log n \rceil+1$ if $\lceil a \log \log n \rceil$ is odd and using single-type branching process theory.

Lemma 6.1. Suppose that $\zeta^S \gt1$ . Let $c^S = \log \zeta^S \gt0$ and $a= 5 /(2 c^S)$ , and recall that $\ell_n = \lceil a \log \log n \rceil$ . Then

(6.8) \begin{align} {\mathbb P} \big([\!\log n]^2 \lt \mathbf{V}^{S^{(n)}}_{\ell_n}\lt [\!\log n]^3 \big) \rightarrow z \quad \mbox{\textit{as} } n \rightarrow \infty,\end{align}

where z satisfies (6.6).

Proof. Note that

\begin{align*} [\zeta^S]^{\ell_n} = \exp (c^S \lceil a \log \log n \rceil) \approx \exp \!\left( \log [\!\log n]^{c^S a} \right) = [\!\log n]^{c^S a} . \end{align*}

Therefore,

\begin{align*} \mathbf{V}^S_{\ell_n} \approx [\!\log n]^{c^S a} W^S \mathbf{v}^S, \end{align*}

where $W^S=0$ if only and if the branching process goes extinct. Therefore, for any $0 \lt d_0 \lt c^S a \lt d_1$ , conditional on non-extinction of the branching process, $\mathcal{B}_S$ , we have that for all sufficiently large n, the number of individuals in generation $\ell_n$ lies between $[\!\log n]^{d_0} $ and $[\!\log n]^{d_1} $ . Since $a c^S = 2.5$ , (6.8) follows immediately.

It is straightforward to adapt the arguments used in the proof of Lemma 6.1 to prove (6.2). Let $0 \lt p_0 \leq 1$ . Then using (6.7), we have that ${\mathbb P}( \mathbf{V}^S_{\lceil p_0\ell_n \rceil} \neq \mathbf{0}) \rightarrow z$ as $n \rightarrow \infty$ . For sufficiently small $p_0$ , the number of individuals in the first $p_0 \ell_n$ generations is less than $[\!\log n]$ with probability tending to 1 as $n \rightarrow \infty$ . However, if the susceptibility set reaches generation $\lceil p_0\ell_n \rceil$ without going extinct, it will, with probability tending to 1, reach generation $\ell_n$ without going extinct, as the number of individuals in generation $\lceil p_0\ell_n \rceil$ tends to infinity as $n \rightarrow \infty$ .

6.3. Size of major outbreaks

For $n=1,2,\ldots$ and $j=1,2,\ldots, n$ , let $\check{S}_j^{(n)}$ denote the number of individuals in generation $\ell_n$ of the susceptibility set of individual j. In particular, we are interested in whether or not $\check{S}_j^{(n)} =0$ , i.e. whether or not the susceptibility set goes extinct by generation $\ell_n$ . Let

\begin{align*} \check{Z}^{(n)} = \sum_{j=1}^n 1_{\{ \check{S}_j^{(n)} \neq 0\}}\end{align*}

denote the number of susceptibility sets which do not go extinct by generation $\ell_n$ .

Lemma 6.2. Let z satisfy (6.6). Then

(6.9) \begin{align} \frac{\check{Z}^{(n)}}{n} \stackrel{\textrm{p}}{\longrightarrow} z \quad \mbox{\textit{as} } n \rightarrow \infty.\end{align}

Proof. With the use of Markov’s inequality, (6.9) holds if for $m=1,2$ ,

(6.10) \begin{align} {\mathbb E} \left[ \left( \frac{\check{Z}^{(n)}}{n} \right)^m \right] \rightarrow z^m \quad \mbox{as } n \rightarrow \infty.\end{align}

Since the individuals in the population are exchangeable, we have that

\begin{align*} {\mathbb E} \left[ \frac{\check{Z}^{(n)}}{n} \right] = {\mathbb P} \big( \check{S}_1^{(n)} \neq 0 \big) \rightarrow z \quad \mbox{as } n \rightarrow \infty.\end{align*}

Similarly, we have that

(6.11) \begin{align} {\mathbb E} \left[ \left(\frac{\check{Z}^{(n)}}{n} \right)^2 \right] = \frac{1}{n} {\mathbb P} \big( \check{S}_1^{(n)} \neq 0 \big) + \frac{n-1}{n} {\mathbb P} \big( \check{S}_1^{(n)} \neq 0, \check{S}_2^{(n)} \neq 0\big),\end{align}

where individuals 1 and 2 denote two individuals selected uniformly at random from the population.

Let $E_n$ be the event that two randomly selected individuals 1 and 2 are within a distance $2 \ell_n$ of each other, under the metric $\rho$ defined just after (5.19). With the use of (6.4) it is straightforward to show that, for any $\epsilon \gt0$ , the number of individuals within a distance $2 \ell_n$ of individual 1 is at most $n^\epsilon$ , for all sufficiently large n. Hence, ${\mathbb P}(E_n) \rightarrow 0$ as $n \rightarrow \infty$ . Conditional upon $E_n^\textrm{c}$ , the growths of the susceptibility sets of individuals 1 and 2 over the first $\ell_n$ generations are independent. (These susceptibility sets, restricted to $\ell_n$ generations, cannot overlap since otherwise there exists $i \in \mathcal{N}$ such that $\rho(1,i) \le \ell_n$ and $\rho(i,2) \le \ell_n$ , whence $\rho(1,2) \le 2\ell_n$ , which is a contradiction given the occurrence of $E_n^\textrm{c}$ .) Therefore,

(6.12) \begin{align} & {\mathbb P} \big( \check{S}_1^{(n)} \neq 0, \check{S}_2^{(n)} \neq 0\big) \nonumber \\[5pt] & \quad = {\mathbb P} \big( \check{S}_1^{(n)} \neq 0, \check{S}_2^{(n)} \neq 0, E_n\big) + {\mathbb P} \big(E_n^\textrm{c} \big) \prod_{j=1}^2 {\mathbb P} \big(\check{S}_j^{(n)} \neq 0|E_n^\textrm{c} \big) \nonumber \\[5pt] & \quad \rightarrow z^2 \quad \mbox{as } n \rightarrow \infty.\end{align}

The lemma follows since (6.12) and (6.11) imply (6.10) holds for $m=2$ .

In Lemma 6.3 we show, conditional on $Z^{(n)} (\!= C^{(n)})\gt \log n$ , i.e. a major epidemic outbreak has occurred, that those infected in the epidemic consist mainly of those individuals whose susceptibility set does not go extinct in $\ell_n$ generations. We prove the lemma under the assumption that $I\equiv 1$ , a constant infectious period. In this case the constructions of infectious clumps and susceptibility sets are identical, and $\mathbf{V}^{C^{(n)}}_{\ell_n} \stackrel{\textrm{D}}{=} \mathbf{V}^{S^{(n)}}_{\ell_n}$ , where $\mathbf{V}^{C^{(n)}}_{\ell_n}$ is the number of individuals in the $\ell_n$ th generation of the infectious clump. Observe that Lemma 6.3 yields Theorem 3.2 with $\pi_\textrm{G}=0$ (recall $R_\textrm{L}=\zeta^S$ ). In Subsection 6.4, we show how the arguments are adjusted to general I.

Lemma 6.3. Suppose that $I \equiv 1$ and $\zeta^S \gt 1$ , and let z satisfy (6.6). Then

(6.13) \begin{align}\frac{C^{(n)}}{n} \; | \; C^{(n)}\gt \log n = \frac{Z^{(n)}}{n} \; | \; Z^{(n)} \gt \log n \stackrel{\textrm{p}}{\longrightarrow} z \quad \mbox{\textit{as} } n \rightarrow \infty.\end{align}

Proof. Note that $C^{(n)} \stackrel{\textrm{D}}{=} S^{(n)}$ , since $I \equiv 1$ , so (6.1) implies $\lim_{n \to \infty} {\mathbb P}(C^{(n)} \gt \log n)=z$ , where $z\gt0$ as $\zeta^S \gt 1$ . Hence, Lemma 6.2 implies

(6.14) \begin{align}\frac{\check{Z}^{(n)}}{n} \; | \; C^{(n)}\gt \log n \stackrel{\textrm{p}}{\longrightarrow} z \quad \mbox{as } n \rightarrow \infty.\end{align}

Thus, to prove (6.13), we show that

\begin{align*} \frac{\check{Z}^{(n)}}{n} - \frac{C^{(n)}}{n} \; | \; C^{(n)}\gt \log n \stackrel{\textrm{p}}{\longrightarrow} 0 \quad \mbox{as } n \rightarrow \infty, \end{align*}

with the lemma then following immediately from $C^{(n)}= Z^{(n)}$ and (6.14).

By Markov’s inequality, for any $\epsilon \gt0$ ,

(6.15) \begin{align} {\mathbb P} \left( \left|\frac{\check{Z}^{(n)}}{n} - \frac{C^{(n)}}{n} \right| \gt \epsilon \; | \; C^{(n)}\gt \log n \right) \leq \frac{1}{\epsilon} {\mathbb E} \left[ \left|\frac{\check{Z}^{(n)}}{n} - \frac{C^{(n)}}{n} \right| \; | \; C^{(n)}\gt \log n\right]\!.\end{align}

We can bound the expectation on the right-hand side of (6.15) by

(6.16) \begin{align} & {\mathbb E} \left[ \left|\frac{\check{Z}^{(n)}}{n} - \frac{C^{(n)}}{n} \right| \; | \; C^{(n)}\gt \log n\right] \nonumber \\[5pt] & \quad = {\mathbb E} \left[ \frac{1}{n} \left| \sum_{j=1}^n \Big( 1_{\{\check{S}_j^{(n)} \neq 0 \}} - 1_{\{ i^\ast \in \mathcal{S}_j^{(n)} \}} \Big) \right| \; | \; C^{(n)}\gt \log n\right] \nonumber \\[5pt] & \quad \leq \frac{1}{n} \sum_{j=1}^n {\mathbb E} \left[ \left| 1_{\{\check{S}_j^{(n)} \neq 0 \}} - 1_{\{ i^\ast \in \mathcal{S}_j^{(n)} \}} \right| \; | \; C^{(n)}\gt \log n \right] \nonumber \\[5pt] & \quad = {\mathbb P} \left( 1_{\{\check{S}_1^{(n)} \neq 0 \}} \neq 1_{\{ i^\ast \in \mathcal{S}_1^{(n)} \}} \; | \; C^{(n)}\gt \log n \right)\!,\end{align}

where the final equality in (6.16) follows from the exchangeability of individuals.

Let $\check{E}_n$ be the event that the length of the shortest path between individuals 1 and $i^\ast$ is less than or equal to $\ell_n$ . Then ${\mathbb P} (\check{E}_n) \rightarrow 0$ as $n \rightarrow \infty$ and

\begin{align*} {\mathbb P} \big( \check{S}_1^{(n)} = 0, i^\ast \in \mathcal{S}_1^{(n)} \; | \; \check{E}_n^\textrm{c} \big) =0.\end{align*}

Hence, ${\mathbb P} ( \check{S}_1^{(n)} = 0, i^\ast \in \mathcal{S}_1^{(n)} | C^{(n)}\gt \log n ) \rightarrow 0$ as $n \rightarrow \infty$ .

We complete the proof of Lemma 6.3 by showing that

(6.17) \begin{align} {\mathbb P} \big( \check{S}_1^{(n)} \neq 0, i^\ast \not\in \mathcal{S}_1^{(n)} | C^{(n)}\gt \log n \big) \rightarrow 0 \quad \mbox{as } n \rightarrow \infty.\end{align}

Following Lemma 6.1 and the subsequent arguments, for $a=5/(2c^S)$ and $\ell_n = \lceil a \log \log n \rceil$ , we have that

\begin{align*} {\mathbb P} \left(\left. \mathbf{V}^{C^{(n)}}_{\ell_n} \gt [\!\log n]^2 \right| C^{(n)}\gt \log n \right) \rightarrow 1 \quad \mbox{as } n \rightarrow \infty. \end{align*}

From generation $\ell_n$ we couple the growth of the infectious clump to a lower-bound branching process. As $n \to \infty$ , the probability that the total number of movers who belong to complexes only through a household (workplace) exceeds $\theta n/2$ tends to 1. The number of movers who belong to a given complex only through a household (workplace) is at most w. Therefore, conditional on the number of movers exceeding $\theta n/2$ , the probability of selecting a given complex, when selecting a mover who belongs to a complex through a household (workplace) at random, is at most $2 w/\theta n$ . It follows that if an infectious clump comprises K complexes then the probability of attempting to add an existing complex from within the clump to the clump is at most $K \times 2 w/\theta n$ .

Fix $0 \lt \delta \lt [\zeta^S -1]/\zeta^S$ and let $\delta^\prime = \delta \theta/2 w$ . Construct $\mathcal{C}^{(n)}_{i^\ast}$ from generation $\ell_n$ onwards until $\delta^\prime n$ complexes have been added to $\mathcal{C}^{(n)}_{i^\ast}$ as follows. Consider the growth of the infectious clump from each of the $\mathbf{V}^{C^{(n)}}_{\ell_n} \gt [\!\log n]^2$ individuals in generation $\ell_n$ in turn. We choose complexes to add to the infectious clump according to the number of movers in the complex. While fewer than $\delta^\prime n$ complexes have been added to $\mathcal{C}^{(n)}_{i^\ast}$ , the probability we attempt to add an existing complex to the clump is less than $\delta$ . We construct a lower-bound branching process, $\mathcal{B}^{\textrm{L}}_\delta$ , where the addition of each complex is culled independently with probability $\delta$ . Since $[1 -\delta] \zeta^S \gt1$ , the extinction probability, $z_{L,\delta}$ , of $\mathcal{B}^L_\delta$ is less than 1. It follows that

\begin{align*} {\mathbb P} \big( C^{(n)}\gt \delta^\prime n | C^{(n)}\gt \log n \big) \rightarrow 1 \quad \mbox{as } n \rightarrow \infty.\end{align*}

Given that $P ( \check{S}_1^{(n)} \gt[\!\log n]^2 |\check{S}_1^{(n)} \neq 0) \rightarrow 1$ as $n \rightarrow \infty$ , we have, using the theorem of total probability, that

(6.18) \begin{align} & {\mathbb P} \big( i^\ast \not\in \mathcal{S}_1^{(n)} | \check{S}_1^{(n)} \neq 0,C^{(n)}\gt \log n \big) \nonumber \\[5pt] & \quad \leq {\mathbb P} \big( i^\ast \not\in \mathcal{S}_1^{(n)} | \check{S}_1^{(n)} \gt[\!\log n]^2,C^{(n)}\gt \delta^\prime n \big) \nonumber \\[5pt] &\quad + {\mathbb P} \big(\check{S}_1^{(n)} \leq[\!\log n]^2| \check{S}_1^{(n)} \neq 0,C^{(n)}\gt \log n \big) \nonumber \\[5pt] &\quad + {\mathbb P} \big( C^{(n)}\leq \delta^\prime n | \check{S}_1^{(n)} \neq 0, C^{(n)}\gt \log n \big).\end{align}

Using an argument similar to (6.12), we have that ${\mathbb P}( \check{S}_1^{(n)} \neq 0,C^{(n)}\gt \log n) \rightarrow z^2$ as $n \rightarrow \infty$ . It is then straightforward to show that the last two terms on the right-hand side of (6.18) converge to 0 as $n \rightarrow \infty$ . Finally, as $n \rightarrow \infty$ ,

\begin{align*}{\mathbb P} \big( i^\ast \not\in \mathcal{S}_1^{(n)} | \check{S}_1^{(n)} \gt[\!\log n]^2,C^{(n)}\gt \delta^\prime n \big) \leq \left[ 1 - \frac{[\!\log n]^2}{n} \right]^{\delta^\prime n} \rightarrow 0, \end{align*}

with (6.17) following immediately, completing the proof of the lemma.