2.1. Model and notation

We denote the number of elements in a set $\mathcal A$ by $|\mathcal A|$ . For $k \in \mathbb{N}$ , we use the notation $[k] = \{1,2, \ldots, k\}$ . We use $\mathbb{N}$ for the strictly positive integers and $\mathbb{N}_0 = \mathbb{N} \cup \{0\}$ for the nonnegative integers.

Unless specified otherwise, limits are for population size $n \to \infty$ . We say that an event occurs with high probability (w.h.p.) if the probability of the event converges to 1 as $n \to \infty$ . We adhere to the usual order notation, i.e. $f =O(g)$ means that $\limsup_{n \to \infty} |\,f(n)/g(n)| \lt \infty$ and $f =o(g)$ means that $\lim_{n \to \infty} |\,f(n)/g(n)| =0$ . In addition, for a sequence of random variables $\{X^{(n)}; n\in \mathbb{N}\}$ , we write $X^{(n)} = O_p(g)$ if $|X^{(n)}/g(n)|$ is bounded in probability and $X^{(n)}=o_p(g)$ if $X^{(n)}/g(n) \to 0$ in probability. See [Reference Janson, Luczak and Rucinski17, Section 1.2] for a discussion of this notation.

We consider a population of n individuals where $V^{(n)}$ denotes the set of all individuals. For some of our results, we consider a sequence of models in growing populations, i.e. for $n \to \infty$ . When no confusion is possible, we write $V= V^{(n)}$ . We assume that there are K types of individuals. For $i \in [K]$ , let $V_i = V_i^{(n)}$ be the set of vertices of type i and let $n_i = |V_i|$ be the number of vertices of type i.

Within this sequence of populations, we consider the spread of an infectious disease in which individuals are either susceptible or infected. We emphasise that individuals who have been infected will spend time in one or more exposed, infectious, recovered states; as will become clear, we are not restricting ourselves to an SI model with only susceptible and infectious individuals (see Remark 1 below and [Reference Diekmann, Heesterbeek and Britton11, p. 13] for more on this terminology). When individual $v \in V_i^{(n)}$ is infected, it makes contacts to different individuals of type j ( $i,j \in [K]$ ) according to a point process $\xi_v^j = \{\xi_v^j\,(t);\,t\geq 0\}$ on the interval $[0,\infty)$ . If there are more than $n_j$ points in the process then only the first $n_j$ points represent contacts. The time parameter in the definition of $\xi_v^j$ represents the time since infection of individual v, and corresponds to the length, or traversal time, in the graph representation of the epidemic. The processes $\xi_v^1, \xi_v^2, \ldots$ and $\xi_v^K$ may be dependent, and their joint distribution may depend on the type i of v. However, the point processes associated to different individuals are independent, i.e. the vectors of processes $\{(\xi_v^1, \ldots, \xi_v^K);v \in V\}$ are independent. For convenience, we introduce the vector of stochastic process $(\xi_{i1}, \xi_{i2},\ldots, \xi_{iK})$ , which is distributed as $(\xi_v^1, \ldots{,} \xi_v^K)$ for $v \in V_i,\,i \in [K]$ . We make the following assumptions.

Assumption 1

Assume that $n^{-1} n_i \to p_i>0$ for all $i \in [K]$ . Furthermore,

\begin{equation*} \max_{i,j\in [K]} \bigg|\frac{p_i}{p_j}-\frac{n_i}{n_j}\bigg|=O\bigg(\frac{1}{n}\bigg). \end{equation*}

Assumption 2

For all $i \in [K]$ , the distribution of $(\xi_{i1}, \xi_{i2},\ldots, \xi_{iK})$ is independent of population size n and, for all $i,j \in [K]$ ,

(1) \begin{equation} m_{ij}=\mathbb{E}[\xi_{ij}(\infty)]\lt \infty, \end{equation}

i.e. the expected number $m_{ij}$ of contacts that an individual of type i makes with individuals of type j is finite for all $i,j \in [K]$ . Furthermore, we assume that the sum $\xi_{i1}+ \xi_{i2}+\cdots+ \xi_{iK}$ does almost surely (a.s.) contain only jumps of size 1 and its distribution has no atoms.

Assumption 3

There exists a constant $\kappa \lt 1$ such that, for all $i \in [K]$ ,

\begin{align*} \mathbb{P}\Big(\max_{v \in V} \xi_v^i(\infty) \lt n^{\kappa}\Big) \to 1 \quad\text{as } n \to \infty. \end{align*}

For future reference, we let $M=\{m_{ij}\}_{i,j\in[K]}$ denote the matrix with the $m_{ij}$ (as defined in Assumption 2) as elements. Note that Assumption 3 is easily met, e.g. if, for all $v \in V$ and $i \in [K]$ , there exists $\varepsilon>0$ and $\ell_0 \in (0,\infty)$ , for which $\mathbb{P}(\xi_v^i(\infty)>\ell) \lt \ell^{-(1+\varepsilon)}$ for all $\ell>\ell_0$ , then, for $\kappa \in (1/(1+\varepsilon),1)$ ,

\begin{align*} \mathbb{P}\Big(\max_{v \in V} \xi_v^i(\infty) \lt n^{\kappa}\Big) &= \prod_{v \in V} [1-\mathbb{P}( \xi_v^i(\infty) \gt n^{\kappa})] \\ &\geq 1 - \sum_{v \in V}\mathbb{P}( \xi_v^i(\infty) > n^{\kappa}) \\ &\geq 1 - n \max_{j \in [K]}\mathbb{P}( \xi_{ji}(\infty) > n^{\kappa}) \\ & >1 -n \times n^{-(1+\varepsilon)\kappa} \\ & \to 1. \end{align*}

At the points of $\xi_v^j (\,j \in [K]$ ), v contacts an individual from $V_j$ . The individuals that are contacted are uniformly chosen without replacement. If v is of type j, we allow for v to be among the contacted individuals in $\xi_v^j$ . If the individual that is contacted is still susceptible at the time of the contact then it becomes infected. The contacted individuals are chosen without replacement for technical reasons (e.g. we are assured that (2) below is exact). If we assume that contacted individuals are chosen with replacement and that there are multiple contacts from v with v’ say, then only the first of those contacts is relevant and all other contacts to v’ may be ignored, and the corresponding points of the point process erased. For given n, we can then obtain the distribution of this ‘erased’ point process from the distribution of the vector of point processes $ (\xi_v^j\,;\,j \in [K]) $ and continue the analysis as presented in this paper.

Note that we may assign the point processes $\{\xi_v^j\,;\,v \in V,\, j \in [K]\}$ and decorate the points with the labels of the individuals these points represent contacts to, already before the epidemic starts. This allows us to create a new set of random variables $\{\eta(u,v)\,;\,u,v \in V\}$ . Assume that individual v is of type $j \in [K]$ . If, for some $u \in V$ , there is a point in $\xi_u^j$ with label v, then $\eta(u,v)$ takes the value of this point. If there is no such point in $\xi_u^j$ , set $\eta(u,v)= \infty$ . Observe that the distribution of $\{\eta(u,v)\,;\,u,v \in V\}$ depends on n in this construction, because the probability that v is chosen as a label is decreasing in n. Note that there is a broad class of models that satisfy Assumptions 1–3; see Remark 1 for an example of a specific class of epidemic models.

This construction provides us with a graph representation of the population and the epidemic on it. We construct the weighted random graph $G=(V,E)$ , with weights corresponding to lengths as follows. The edge set E consists of all directed pairs $ (u,v) \in V \times V $ , with $u \neq v$ . For an edge $ (u,v) \in V \times V $ from u to v, we say that u is the tail of (u,v) and v is its head. The length of edge (u,v) is given by $\eta(u,v)$ for all $ (u,v) \in E $ . For some of our arguments, we restrict to the weighted edge set $E' \subset E$ of all edges $ (u,v) \in E $ with finite length, i.e. $ (u,v) \in E' $ if and only if $\eta(u,v)\lt \infty$ . The corresponding random graph is denoted by $G'=(V, E')$ . For $i,j \in [K]$ , $u \in V_i$ , and $v \in V_j$ , we also introduce the random variable $\eta_{ij}$ , which is distributed as $\eta(u,v)$ conditioned on $\eta(u,v)\lt \infty$ (i.e. conditioned on $(u,v) \in E'$ ). Observe that

(2) \begin{align}\nonumber \mathbb{P}(\eta_{ij} \leq t) &= \mathbb{P}(\eta(u, v) \leq t\ \,\,|\, \ \,\eta(u, v) \lt \infty)\nonumber \\ &= \frac{\mathbb{P}(\eta(u,v) \leq t)}{\mathbb{P}(\eta(u,v) \lt \infty)} \nonumber \\ &=\frac{ \mathbb{E}[\xi_{ij}(t)]/{n_j} } {\mathbb{E}[\xi_{ij}(\infty)]/{n_j} } \nonumber \\ &= \frac{\mathbb{E}[\xi_{ij}(t)]}{m_{ij}}, \end{align}

which is independent of n. We mainly consider the random graph $G'=(V,E')$ . See Figure 1 for an example of Ǵ.

Figure 1.

An example of $G'=(V,E')$ with $n=8$ and $K=2$ . The vertices are labeled $a,b,\ldots,h$ . Vertices of type 1 are represented by circles and vertices of type 2 by boxes. Contacts made by vertices of type 1 are represented by solid directed edges and contacts made by vertices of type 2 by dashed directed edges. The numeric values next to the edges are their lengths, each length reflecting the time $\eta(u,v)$ between infection of the tail u up until the first contact with the head $v,\,u,v \in \{a,b,\ldots,h\}$ . In the figure we have placed nodes such that directed edges are pointing towards the right, but this is simply for convenience since there is no true spatial component in the underlying epidemic model.

An ordered set of distinct vertices $\pi =(v_1, v_2, \ldots, v_m)$ is a path in Ǵ if $(v_i,v_{i+1}) \in E'$ for all integers $i \in [m-1]$ . With some abuse of terminology, we sometimes refer to the set of edges connecting the vertices of $\pi$ as the path $\pi$ and speak of a path in É. The length of a path $\pi =(v_1, v_2, \ldots, v_m)$ is $\ell(\pi) = {\sum_{i=1}^{m-1} \eta(v_i,v_{i+1})}$ . Assumption 2 implies that all nonzero paths in É have different lengths with probability 1. Let $\Pi_{uv}$ be the set of all paths from u to v in É and define the (quasi) distance from u to v as $d(u,v)=\min_{\pi \in \Pi_{uv}}\ell(\pi)$ . As an example, in Figure 1 the distance from a to d is given by

\begin{align*} d(a,d) =\min(\eta(a,b)+\eta(b,d), \eta(a,c)+\eta(c,d))=1.8. \end{align*}

In general, $d(u,v)\neq d(v,u)$ if $u\neq v$ since the graph Ǵ is directed. Therefore, d is actually a quasidistance. We say that $d(u,v)=\infty$ if $\Pi_{uv} = \emptyset$ . Furthermore, $d(v,v)=0$ for all $v\in V$ .

Next we formulate an ‘irreducibility’ assumption.

An epidemic process is reproduced from Ǵ as follows. Let $V_{\text{init}}=V_{\text{init}}^{(n)}$ be the set of vertices that are initially infected. This set may be predetermined or randomly selected and satisfies the following assumption.

At time t, a vertex v is susceptible if $t\lt \sigma_v$ , exposed if $t \in [\sigma_v, \sigma_v + L_v)$ , while it is infectious if $t \in [\sigma_v + L_v, \sigma_v + L_v+ \iota_v)$ . Finally, v is recovered if $t \geq \sigma_v + L_v+ \iota_v$ . If $\mathbb{P}(L_v =0)=1$ then we are in the so-called SIR (susceptible $\to$ infectious $\to$ recovered) epidemics framework. In addition, if $\iota_v$ is exponentially distributed then the process is called a Markov SIR epidemic.

Finally, to conclude this section, we introduce the basic reproduction number $R_0$ . As this is possibly the most studied quantity in mathematical modelling of the spread of infectious diseases, no epidemic modelling paper would be complete without at least mentioning $R_0$ . In a single-type epidemic model $R_0$ is defined as the expected number of infectious contacts made by a newly infected individual in an otherwise susceptible population. The multi-type equivalent of $R_0$ is given by the dominant eigenvalue of the $K \times K$ matrix $M=\{m_{ij}\}_{i,j\in[K]}$ (see Assumption 2). The number $R_0$ is always real and strictly positive [Reference Diekmann, Heesterbeek and Britton11, Chapter 7], [Reference Jagers15, Chapter 4]. We say that the epidemic process is supercritical, critical, or subcritical if $R_0>1$ , $R_0=1$ , or $R_0\lt 1$ , respectively. As $n \to \infty$ , the epidemic becomes large with positive probability if and only if the process is supercritical [Reference Diekmann, Heesterbeek and Britton11]. In the remainder of the paper we assume that $R_0>1$ .

2.2. Main results

Since we assume that $R_0>1$ , a large outbreak occurs with positive probability and at the end of such a large outbreak, a positive fraction of the infected individuals is of type j w.h.p. However, the question that we are interested in is: what fraction of those individuals was infected by individuals of type i for $i,j\in[K]$ ? So, if we denote the set of infected individuals of type j by $\pmb\ell_j$ , and the subset of those individuals which are infected by individuals of type i by $\pmb\ell_j^i$ , then we are interested in $|\pmb\ell_j^i|/|\pmb\ell_j|$ . For example, in Figure 1, if a represents the initially infectious individual then there are two individuals of type 1 that are infected by an individual of type 1 (namely the individuals represented by vertices c and g) and one individual of type 1 that is infected by an individual of type 2 (namely the individual represented by vertex d). So, the fraction of individuals infected by an individual of type 1 among all infected individuals of type 1 is $\tfrac23$ (we exclude vertex a because it was initially infected, and not infected by another individual in the population). The question posed above leads to the main results that are formulated in Theorems 1 and 2 below. For our results, we need to define a multi-type branching process $\{{\mathcal Z}(t)\,;\,t \geq 0\}$ . The branching process is defined as follows. Let $Z^i(t)$ be the number of particles in the branching process at time t if the process is started by a single particle of type $i,\,i \in [K]$ . For $i,j \in [K]$ , particles of type j in ${\mathcal Z}(t)$ give birth to particles of type i according to a Poisson process with intensity ${p_i} \mathbb{E}[\xi_{ij}(\textrm{d} a)]/{p_j}$ , where a is the age of the particle of type j. All of these Poisson processes are independent.

The definition of the branching process ${\mathcal Z}(t)$ is such that we can apply theory from [Reference Iksanov and Meiners14]. In particular, there exists a Malthusian parameter $\alpha>0$ (which is specified in Section 3.2) and a random variable $W^i$ such that

\begin{equation*} \textrm{e}^{-\alpha t}Z^i(t) \to W^i \quad\text{a.s. as } t \to \infty \end{equation*}

and

\begin{equation*} \mathbb{P}(W^i \in (0, \infty))=1 -\mathbb{P}(W^i =0) = \mathbb{P}(Z^i(t) \to \infty). \end{equation*}

Let $\{W^i(r)\,;\,r \in \mathbb{N}\}$ be independent copies of $W^i$ . We now state our first main result.

Theorem 1

Conditioned on the occurrence of a large outbreak, the fraction of infected individuals of type j that are infected by an individual of type i during an outbreak in a population of size n converges in probability to $\rho_{ij}$ as $n\to \infty$ . Here $\sum_{i=1}^K\rho_{ij}=1$ , and

(3) \begin{equation} \rho_{ij}=\frac{1}{\mathbb{P}(Z^{\,j}(t) \to \infty)} \mathbb{E}\biggl[\frac{\sum_{r=1}^{X_{ij}}\textrm{e}^{-\alpha \tau_{ijr}} {W}^i(r)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} \textrm{e}^{-\alpha \tau_{kjr}} {W}^k(r)} {\bf 1}\bigg(\sum_{k =1}^K \sum_{r=1}^{X_{kj}} {W}^k(r) >0\bigg) \biggr], \end{equation}

where, for $k \in [K]$ and $r \in \mathbb{N}$ , the random variables $\tau_{kjr}$ are independent with distribution function $\mathbb{P}(\tau_{kjr} \lt a) = {\mathbb{E}[\xi_{kj}(a)]}/{m_{kj}}$ and $X_{kj}$ is Poisson distributed with expectation ${p_k m_{kj}}/{p_j}$ .

Note that, in general, it is hard to give a more explicit expression for $\rho_{ij}$ than (3). Often, there is no explicit description of the distribution of ${W}^k(r)$ . We are able to obtain some bounds for (3) for a special case of a two-type model with general distributions $(\xi_{i1}, \xi_{i2})$ , leading to Theorem 2.

Consider $K=2$ . Let ${m_{ji}^{(b)}}=(p_i/p_j) m_{ij},\,i,j=1,2$ , where $m_{ij}$ is defined by (1). From theory on multi-type supercritical branching processes [Reference Jagers15, Chapter 4] we know that $ (q_1, q_2) $ is the unique solution in $ (0, 1)^2 $ of

(4) \begin{align} x & = \sum_{k=0}^{\infty} \frac{(m_{11}^{(b)})^k}{k!}\textrm{e}^{-m_{11}^{(b)}} x^k \sum_{\ell=0}^{\infty} \frac{(m_{12}^{(b)})^{\ell}}{\ell!}\textrm{e}^{-m_{12}^{(b)}} y^{\ell} = \textrm{e}^{-[m_{11}^{(b)}(1-x)+ m_{12}^{(b)}(1-y)]},\end{align}

(5) \begin{align} y & = \sum_{k=0}^{\infty} \frac{(m_{21}^{(b)})^k}{k!}\textrm{e}^{-m_{21}^{(b)}} x^k \sum_{\ell=0}^{\infty} \frac{(m_{22}^{(b)})^{\ell}}{\ell!}\textrm{e}^{-m_{22}^{(b)}} y^{\ell} = \textrm{e}^{-[m_{21}^{(b)}(1-x)+ m_{22}^{(b)}(1-y)]}. \end{align}

Furthermore, $\tilde{q}_1$ is defined as the smallest positive solution of

(6) \begin{equation} x= \textrm{e}^{-m_{11}^{(b)}(1-x)}, \end{equation}

which might be 1 or strictly smaller than 1, depending on whether or not $p_1m_{11} = m_{11}^{(b)} \leq 1$ . The quantities $q_1, q_2$ , and $\tilde q_1$ relate to extinction probabilities of backward branching processes; see Section 4.3.

Theorem 2

Consider the two-type model with general distributions $(\xi_{i1}, \xi_{i2})$ as described above. In the limit as the population size $n\to\infty$ , for fixed $p_1$ , $m_{ij},\,i,j=1,2,$ and conditioned on the occurrence of a large outbreak, the fraction of ultimately infected vertices of type 2 that is infected by type 1 is bounded from below by

\begin{equation*} \rho_{21}^-=\bigg(1-\frac{m_{11}^{(b)}(\tilde q_1+q_1)}{2}\bigg)\frac{\tilde q_1 - q_1}{1-q_1}, \end{equation*}

and the fraction of ultimately infected vertices of type 1 that is infected by type 1 is bounded from above by

\begin{equation*} \rho_{11}^+={1-\rho_{21}^-=}1-\bigg(1-\frac{m_{11}^{(b)}(\tilde q_1+q_1)}{2}\bigg)\frac{\tilde q_1 - q_1}{1-q_1}. \end{equation*}

Moreover, by interchanging the roles of types 1 and 2, we obtain the lower bound $\rho_{12}^-$ and upper bound $\rho_{22}^+$ .

Through additional assumptions on the point process we obtain Corollary 1. Let ${(\xi_v^1,\xi_v^2),} v \in V$ , be obtained from a single marked point process $\xi_v$ . In the process $\xi_v$ the points get independently mark 1 with probability $p_1$ and mark 2 otherwise. Then the process $\xi_v^1$ consists of the points with mark 1, while $\xi_v^2$ consists of the points with mark 2. By construction of the process, the probability that an ultimately infected vertex of type j is infected by a vertex of type i is the same for $j=1$ and $j=2$ , i.e. $\rho_{i1}=\rho_{i2}=\rho_i,\,i=1,2$ . With some abuse of notation, we write $\xi_i$ for a point process distributed as $\xi_v$ for $v \in V_i,\,i = 1,2$ . Furthermore, note that $m_{ij}=\mathbb E[\xi_{ij}(\infty)]=p_j\mathbb E[\xi_i(\infty)]$ , i.e. we can write $m_{ij}=p_j \tilde m_i$ with $\tilde m_i=\mathbb E[\xi_i(\infty)]$ . Here $\tilde m_i$ can be interpreted as the expected number of secondary cases caused by a newly infected individual of type i in an otherwise susceptible population (one can think of the $\tilde m_i$ as the type-specific reproduction numbers). Then the basic reproduction number is $R_0=p_1\tilde m_1+p_2\tilde m_2$ , with $p_2=1-p_1$ . Indeed, a newly infected individual is of type i with probability $p_i$ and the expected number of secondary cases it produces is $\tilde m_i,\,i=1,2$ .

For this model, we can compute $\rho_1^-$ and $\rho_1^+$ , the lower and upper bounds for the fraction of the infected vertices that are infected by type-1 vertices, if the matrix M and $p_1$ are held fixed. For $i=1,2$ , we let $\tilde q_i$ be the smallest solution in (0,1] of

(7) \begin{equation} x= \textrm{e}^{-p_i \tilde m_{i}(1-x)}. \end{equation}

Note that, since $\textrm{e}^{-p_i \tilde m_{i}(1-x)}$ is the PGF of a Poisson distribution, $x=1$ is always a solution and there is an additional solution in (0, 1) if $p_im_i>1$ . Furthermore, we let q be the unique solution in (0, 1) of

(8) \begin{equation} x = \textrm{e}^{-(1-x)(p_1 \tilde m_{1}+ p_2 \tilde m_{2})}=\textrm{e}^{-(1-x)R_0}. \end{equation}

Note that we assume that $R_0=p_1\tilde m_{1}+ p_2\tilde m_{2}>1$ , so the unique solution $q\in(0, 1)$ exists. We can interpret $1-\tilde q_1$ (respectively $1-\tilde q_2$ ) as the fraction of the population that ultimately gets infected when only individuals of type 1 (respectively type 2) are able to transmit, conditional on a large outbreak. Furthermore, $1-q$ can be interpreted as the fraction of the population that ultimately gets infected, conditional on a large outbreak (or, conversely, q is the fraction of the population that remains susceptible throughout the epidemic). We note that it is possible that $\tilde{q}_1=1$ even if $R_0>1$ .

Corollary 1

Consider the two-type model that is described above. In the limit as the population size $n\to\infty$ , for fixed $p_1$ , $\tilde{m}_1$ , and $\tilde{m}_2$ and conditioned on the occurrence of a large outbreak, the fraction of ultimately infected vertices that is infected by type 1 vertices is bounded from above by

\begin{equation*} \rho_1^+ = 1-\bigg( 1 - \frac{p_1 \tilde m_1 (\tilde{q}_1 + q)}{2} \bigg) \frac{\tilde{q}_1-q}{1-q}. \end{equation*}

By interchanging the role of types 1 and 2, we also obtain the lower bound $\rho_1^-$ . Indeed, note that $\rho_1^-=1-\rho_2^+$ , where

\begin{equation*} \rho_2^+ = 1-\bigg( 1 - \frac{p_2 \tilde m_2 (\tilde{q}_2 + q)}{2} \bigg) \frac{\tilde{q}_2-q}{1-q}. \end{equation*}

3.1. The susceptibility process

In this subsection we use the idea of susceptibility sets [Reference Ball and Lyne4], [Reference Ball and Neal5], [Reference Ball, Sirl and Trapman6], which were originally introduced to analyse epidemics in discrete time or from a generation perspective. In order to answer questions relating to who-infected-whom, we need to extend the idea of susceptibility sets to a real-time perspective and construct the susceptibility set through a stochastic process that we call the susceptibility process.

We define the susceptibility process $\{\mathcal S^{(n)}_v(t)\,;\,t \geq 0\}$ as

\begin{equation*} \mathcal S^{(n)}_v(t) = \{u \in V\,;\, d(u,v) \leq t\}. \end{equation*}

Note that, for a given graph Ǵ, ${\mathcal S^{(n)}_v(t)}$ is nonrandom and nondecreasing in t. The susceptibility set of vertex v is given by ${\mathcal S^{(n)}_v = \lim_{t \to \infty} \mathcal S^{(n)}_v(t)}$ , i.e. the susceptibility set ${S_v^{(n)}}$ of v consists of all vertices $u\in V$ for which there is a path from u to v in Ǵ. As an illustration, in Figure 1, the susceptibility set of vertex f is given by ${\mathcal S^{(n)}_f }= \{a,b,c,d\}$ , while for $t=1.1$ , ${\mathcal S^{(n)}_f(1.1) }= \{c,d\}$ . Note that ${\mathcal S^{(n)}_v \cap V_ {\text{init}} }= \emptyset$ if and only if v remains uninfected throughout the epidemic, i.e. if and only if there is no path in Ǵ from $V_ {\text{init}}$ to v. Also, note that the susceptibility set may contain vertices of different types.

Define, for $a \geq 0$ and $j \in [K]$ ,

\begin{equation*} \mathcal S^{(n)}_v(t;a,j) = \{u \in \mathcal S^{(n)}_v(t)\cap V_j; d(u,v) > t- a\} = V_j\cap (\mathcal S^{(n)}_v(t)\setminus \mathcal S^{(n)}_v(t-a)). \end{equation*}

That is, $\mathcal S_v(t;a,j)$ consists of the vertices of type j in ${\mathcal S^{(n)}_v(t)}$ , that are not part of ${\mathcal S^{(n)}_v(t-a)}$ , or in other words, the vertices of type j that join v’s susceptibility process between times $t-a$ and t.

Below, we construct the random graph $G'=(V,E')$ simultaneously with and through the susceptibility process. In particular, we treat the susceptibility process as a random process. We can do this because in the construction of Ǵ presented in Section 2 we did not specify the order in which we create the edges of É. In this and the following section we choose a particular useful random order to create É.

Let $v \in V \setminus V_{\text{init}}$ be a randomly chosen vertex. We derive the susceptibility process $\{{\mathcal S^{(n)}_v(t)}\,;\,t \in (0,t^*)\}$ by constructing part of the random graph Ǵ around vertex v by means of the process of growing graphs $\{\hat{G}(\ell)\}= \{\hat{G}(\ell)\,;\,\ell \in \mathbb{N}_0\}$ in which vertices in the susceptibility process are ‘explored’ one at a time. We note that $\{\hat{G}(\ell)\}$ depends on v. Here $t^*= t^*(n)$ is a given time. We defer the specification of $t^*$ until (11). We also note that $t^*$ and $\ell$ are closely related. The process ${\{\hat{G}(\ell)\}}$ allows us to couple ${\{\mathcal S^{(n)}_v(t)\}}$ with an appropriate branching process. In this way, we can make the coupling between the susceptibility process and the backward branching process that is needed to prove Theorems 1 and 2 in Section 4.

Before we define the process $\{\hat{G}(\ell)\}$ around vertex v, we introduce some additional variables and terminology. For $\ell \in \mathbb{N}_0$ , $\hat{G}(\ell)$ is a 4-tuple:

\begin{equation*} \hat{G}(\ell)= ({\hat{V}}^{a}(\ell),{\hat{V}}^{p}(\ell),{\hat{V}}^{e}(\ell),\hat{E}(\ell)). \end{equation*}

Here $\hat E(\ell)$ denotes the edge set of $\hat{G}(\ell)$ . Vertices in $\hat{G}(\ell)$ can be ‘active’, ‘passive’, or ‘explored’. The sets of these vertices are denoted by ${\hat{V}}^{a}(\ell)$ , ${\hat{V}}^{p}(\ell)$ , and ${\hat{V}}^{e}(\ell)$ , respectively. The sets ${\hat{V}}^{a}(\ell)$ , ${\hat{V}}^{p}(\ell)$ , ${\hat{V}}^{e}(\ell)$ , and $\hat{E}(\ell)$ are defined in the construction below (where also their names will become apparent). Finally, before we explain the construction, we mention that throughout the process we may ‘flag’ the process (see step 4). This flagging plays a role when coupling $\{\hat{G}(\ell)\}$ with a branching process to represent the susceptibility process. The construction is as follows.

1. To set the initial conditions of the construction, let ${\hat{V}}^a(0) = \{v\}$ and ${\hat{V}}^e(0) = \emptyset$ . The edge set $\hat{E}(0)$ is created by assigning the point processes $(\xi_v^1,\ldots{,} \xi_v^K)$ to vertex v and then decorating the points of the point processes $\xi_v^i$ ( $i \in [K]$ ) with vertices in $V_k$ , and creating edges with tail v and heads being the decorations of the points of $ (\xi_v^1,\ldots, \xi_v^K) $ . Additionally, ${{\hat{V}}^p(0)}$ is the set of all heads of edges in ${\hat{E}(0)}$ that are in $V\setminus \{ v \}$ . For a discussion on the initial edges, see note (e) below.

2. For $\ell \in \mathbb{N}_0$ , assume that ${\hat{V}}^a(\ell)\neq \emptyset$ and that there exists a vertex in ${\hat{V}}^a(\ell)$ from which there is a path in $\hat{G}(\ell)$ to v of length at most $t^*$ . In step $\ell+1$ pick (according to some rule) one of the vertices from ${{\hat{V}}^a(\ell)}$ , from which there is a path in ${\hat{G}(\ell)}$ to v of length at most $t^*$ . Say that this vertex is $v' \in V_j$ . Move v’ to the set of explored vertices, i.e. ${{\hat{V}}^e(\ell+1) = {\hat{V}}^e(\ell) \cup \{v'\}}$ . For each $i \in [K]$ , assign to vʹ independently a binomial random number $x(v';i)$ with parameters $n_i$ and $m_{ij}/n_j$ .

3. The remainder of step $\ell+1$ is split into ${\sum_{i=1}^K x(v'\,;\,i)}$ substeps which we index by $\ell' \in {[\sum_{i=1}^K x(v'\,;\,i)]}$ . We introduce

   \begin{equation*} \bigg\{{G}^*(\ell,\ell')\,;\,\ell \in \mathbb{N}_0,\, \ell' \in \{0,1,\ldots, \sum_{i=1}^K x(v'\,;\,i)\}\bigg\}, \end{equation*}
   where
   \begin{equation*} {G}^*(\ell,\ell')= ({\hat{V}}^{*,a}(\ell,\ell'),{\hat{V}}^{*,p}(\ell,\ell'),{\hat{V}}^{*,e} (\ell,\ell'),\hat{E}^*(\ell,\ell')). \end{equation*}
   We set ${G}^*(\ell,0)=\hat{G}(\ell)$ and determine $G^*(\ell,\ell')$ for $\ell'>0$ using step 4.

4. In the $\ell'$ th substep of step $\ell+1$ pick uniformly a vertex from $V_i$ , where

   \begin{align*} i = \inf\bigg\{k \in [K]\,;\,\sum_{j=1}^{k}x(v',j) \geq \ell'\bigg\}. \end{align*}
   If we pick a vertex we have picked in one of the first $\ell'-1$ substeps of step $\ell+1$ then we say that the process is flagged. In that case we choose new vertices from $V_i$ until we obtain a vertex that was not chosen in the previous $\ell'-1$ substeps. This step is equivalent to picking the vertices without replacement. Note that if we pick one of the vertices picked in any of the $\ell$ completed steps, we do not flag the process yet (but we may still do so later, as described below). Say that the vertex that is picked is vʺ.

   • (4a) If $v'' \in {\hat{V}^{a}(\ell) \cup \hat{V}^{e}(\ell)}$ then nothing changes in the graph, i.e. ${G}^*(\ell+1,\ell')= {G}^*(\ell+1,\ell'-1)$ . This is because if $v'' \in {\hat{V}}^{a}(\ell) \cup {\hat{V}}^{e}(\ell)$ then we have already created the edges with tail vʺ.

   • (4b) If $v'' \in V \setminus {({\hat{V}}^{a}(\ell) \cup {\hat{V}}^{e}(\ell))}$ , assign to vʺ the vector of point processes ${(\xi_{v''}^{j'}},\, j' \in [K])$ . The distribution of $(\xi_{v''}^{j'},\, j' \in [K])$ is equal to the distribution of $(\xi_{ij'},\, j' \in [K])$ , given that $\xi_{ij}$ contains a vertex with label v’. Assign label v’ to a uniformly chosen point in $\xi_{v''}^{j}$ , and assign uniform labels without replacement from $V_j \setminus \{ v'\}$ to the other points in ${\xi_{v''}^{j}}$ and uniform labels from $V_{j'}$ to the points in ${\xi_{v''}^{j'}}$ for $j' \in [K]\setminus \{\,j\}$ .

     If none of the newly assigned labels correspond to vertices in ${{\hat{V}}^{e}(\ell)}$ then ${\hat{E}^*}(\ell+1,\ell')$ contains all edges in $\hat{E}^*(\ell+1,\ell'-1)$ plus the edges with tail vʺ and heads corresponding to the labels of the points in ${(\xi_{v''}^{j'},\, j' \in [K])}$ , with the obvious edge lengths. In addition, all heads of those edges which were not in ${\hat{V}}^{*,a}(\ell+1,\ell'-1)$ move to ${\hat{V}}^{*,p}(\ell+1,\ell')$ (if they were not in that set already). Furthermore, $v'' \in {\hat{V}}^{*,a}(\ell+1,\ell')$ .

     If any of the newly assigned labels correspond to vertices in ${\hat{V}}^{e}(\ell)$ then we flag the process and we return to the start of step 4. This last part of step (4b) is equivalent to conditioning on the event that there are no edges with tail vʺ and an already explored vertex as head.

   • (4c) Set ${\hat G(\ell+1)=\hat G(\ell+1, \sum_{i=1}^Kx(v',i))}$ .

5. Continue this process by increasing $\ell$ until there are no active vertices having a path of length less than $t^*$ towards v in $\hat{G}(\ell)$ . Say that $\ell^*$ is the smallest $\ell$ for which this is the case.

An example of $\hat{G}(\ell)$ for $\ell =0,1,2$ is given in Figure 2. We note that it might happen that several active and/or explored vertices are the tails of edges with the same passive vertex as a head in the construction. Furthermore, note that passive vertices are not necessarily part of the susceptibility set of v, since in the subgraphs of Ǵ presented in this figure there are no paths from the passive vertices to v. In step 4 of the construction above we do not exclude that a newly added edge with an active vertex as tail has another active vertex as head.

Figure 2.

An example of the construction of $\hat{G}(\ell)$ for $\ell =0$ (left), $\ell=1$ (middle), and $\ell =2$ (right). In each step one active vertex becomes explored. The edges with this vertex as head are identified, and if the tails of those edges were not yet part of the process $\{\hat{G}(\ell)\}$ before, those tails are active vertices. All edges with these active vertices as tails are identified. If the heads of those edges are newly added to the process then those heads are the passive vertices.

We note the following.

1. (a) From $\hat{G}(\ell^*)$ , we can obtain a realisation of É in the following way. For all $i \in [K]$ and all $u \in V_i \setminus ({\hat{V}}^a(\ell^*) \cup {\hat{V}}^e(\ell^*))$ , assign to u a vector of point processes $(\xi_{u}^{j},\, j \in [K])$ , which are distributed as $(\xi_{ij},\, j \in [K])$ . Then assign (without replacement) to the points in $\xi_u^j$ uniformly chosen vertices from $V_j$ . If, for any $j \in [K]$ , a vertex assigned to the points of $\xi_u^j$ is in ${\hat{V}}^e(\ell)$ then assign a new vector of point processes $(\xi_{u}^{j},\, j \in [K])$ , distributed as $(\xi_{ij},\, j \in [K])$ to u. If none of the vertices assigned to the points of $(\xi_{u}^{j},\, j \in [K])$ is in ${\hat{V}}^e(\ell)$ , then edges with u as tail and the vertices assigned to the points in $(\xi_{u}^{j},\, j \in [K])$ as heads are in É. The edge set É then consists of all edges in $\hat{E}(\ell^*)$ and all of the newly created edges. In remarks (b)–(g) below we use this recipe for the construction of É. In Section 4 we specify an order in which the edges with tails in $V_i \setminus ({\hat{V}}^a(\ell^*) \cup {\hat{V}}^e(\ell))$ are added to obtain the realisation of É.

2. (b) The edge set $\hat{E}(\ell)$ is a subset of É and contains all edges in $\hat{G}(\ell)$ .

3. (c) All edges in É with tails in ${\hat{V}}^{a}(\ell)$ are in $\hat{E}(\ell)$ , but there might still be edges in É with heads in ${\hat{V}}^{a}(\ell)$ that are not in $\hat{E}(\ell)$ .

4. (d) Every passive vertex is the head of an edge in $\hat{E}(\ell)$ , but the passive vertices themselves are not explored, and none of the edges in É with tail in ${\hat{V}}^{p}(\ell)$ are in $\hat{E}(\ell)$ .

5. (e) It might seem strange that we add the passive vertices and the edges from the active vertices to them to the susceptibility process. We need to do this because, when a vertex (v say) becomes active, we have information on the point processes $\xi_{v}^{j},\,j \in [K]$ , which in turn influences the probability that other active vertices become the head of an edge which has v as tail. By incorporating the passive vertices in the susceptibility process, we neutralise this problem.

6. All edges in É with heads or tails in ${\hat{V}}^{e}(\ell)$ are in $\hat{E}(\ell)$ .

7. The tails of edges in $\hat{E}(\ell)$ are in ${\hat{V}}^{a}(\ell) \cup {\hat{V}}^{e}(\ell)$ and their heads are in ${\hat{V}}^{a}(\ell) \cup {\hat{V}}^{p}(\ell) \cup {\hat{V}}^{e}(\ell)$ .

The construction of ${\{\hat{G}(\ell)\}}$ yields ${{\hat{V}}^e(\ell^*) = \mathcal S^{(n)}_v(t^*)}$ . Furthermore, if the process ${\{\hat G(\ell)\}}$ is not flagged until step $\ell^*$ then the construction of ${\hat{V}}^e(\ell^*)$ (and the distances from the vertices in this set to v) is equivalent to constructing a branching process $\{\tilde{\mathcal{Z}}^{(n)}(t)\,;\,t \geq 0\}$ up to time $t^*$ . In this branching process, particles of type j give birth to a binomial distributed number of particles of type i, $i \in [K]$ , where the parameters of the binomial-distributed random variable are $n_i$ and $ (n_j)^{-1}m_{ij} $ . The number of children of the different types of particles are independent. Furthermore, the ages of the mother particles of type j at birth of a child of type i are independent and have density $\mathbb{E}[\xi_{ij}(\textrm{d} a)]/m_{ij}$ .

3.2. The (backward) branching process

We create a multi-type branching process $\{\mathcal Z(t)\,;\,t \geq 0\}$ that can be coupled to ${\{\mathcal S^{(n)}_v(t)}\,;\,t \geq 0\}$ . The coupling is performed in Section 3.3. The branching process $\{\mathcal Z(t)\,;\,t \geq 0\}$ is constructed in such a way that the distribution of the corresponding tree-like graph is the same as that of $\{\mathcal S^{(n)}_v(t)\,;\,t \geq 0\}$ up to time $t^*$ with $t^*$ defined by (11). We leave out some of the details in the arguments, which can be filled in analogous to those given in [Reference Ball and Donnelly3], [Reference Ball, Sirl and Trapman7], and [Reference Barbour and Reinert8] for related models.

Without loss of generality, we assume that $v \in V_1$ . The multi-type (backward) branching process is as follows. The (single) ancestor is of type 1. Particles of type j give birth to particles of type i according to a Poisson process with intensity ${p_i} \mathbb{E}[\xi_{ij}(\textrm{d} a)]/{p_j}$ , where a is the age of the type-j particle, $i,j\in[K]$ . All Poisson processes are independent.

The branching process $\{\mathcal Z(t);t \geq 0\}$ is analysed using existing theory from [Reference Iksanov and Meiners14]. First, the mean offspring measure of this backward branching process is defined through

\begin{equation*}\mu^{(b)}_{ji}(\textrm{d} t)= \frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(\textrm{d} t)].\end{equation*}

Note that

\begin{equation*} m^{(b)}_{ji}= \int_0^{\infty} \mu^{(b)}_{ji}(\textrm{d} t)= \int_0^{\infty} \frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(\textrm{d} t)]= \frac{p_i}{p_j}m_{ij} \end{equation*}

is the expected number of children of type i of a particle of type j. Here $m_{ij}$ is given by (1) (and the corresponding matrix is M). Let ${M^{(b)}=\{m^{(b)}_{ji}\}_{j,i \in [K]}}$ . Straightforward matrix theory reveals that M and ${M^{(b)}}$ have the same dominant eigenvalue $R_0$ , which, by assumption, is strictly larger than 1, i.e. the branching process is supercritical. Define

\begin{equation*}\hat{m}^{(b)}_{ji}(x)= \int_{0}^{\infty} \textrm{e}^{-x t} \frac{p_i}{p_j} \mathbb{E}[\xi_{ij}(\textrm{d} t)], \end{equation*}

and let ${\hat{M}^{(b)}(x)}$ be the matrix with elements ${\hat{m}^{(b)}_{ji}(x)}$ . Finally, let $\alpha$ (the Malthusian parameter) be such that the dominant eigenvalue of ${\hat{M}^{(b)}(\alpha)}$ is equal to 1. Note that $K\lt \infty$ , all elements of ${M^{(b)}}$ are finite and the dominant eigenvalue $R_0$ of ${M^{(b)}}$ is real and larger than 1. Therefore, $\alpha$ exists and is positive.

We define the random vector $Z^i(t)$ as $Z^i(t) = (Z_1^i(t),Z_2^i(t),\ldots,Z_K^i(t))$ , where $Z_j^i(t)$ is the number of particles of type $j \in [K]$ in $\mathcal Z(t)$ if the process starts with one newborn particle of type i. With some abuse of notation, let $\sigma(x)$ be the time of birth of particle x in the branching process. Note that particles in the branching process $\mathcal Z(t)$ never die, although they might stop producing offspring.

We know that, for the Malthusian parameter $\alpha$ defined above,

(9) \begin{equation} \textrm{e}^{-\alpha t} Z^{i}(t) \to W^{i} =(W^{i}_1, W^{i}_2, \ldots{,} W^{i}_K) \quad\text{a.s. as } t\to \infty, \end{equation}

where $W^{i}$ is a random vector that has, with probability 1, strictly positive elements on the set ${\sum_{j=1}^K Z^{i}_j(t) }\to \infty$ as $t \to \infty$ [Reference Iksanov and Meiners14].

For particle $x \in \{\mathcal Z(t)\,;\,t \geq 0\}$ , define

\begin{equation*} \varphi_x(t) = \varphi_x(t\,;\,a) = {\bf 1}(t-\sigma(x)\lt a). \end{equation*}

Let ${\hat{Z}_j^{i}(t\,;\, a) = \sum_x \varphi_x(t\,;\,a)}$ , where the sum is taken over all particles of type j in $\{\mathcal Z(t)\,;\,t \geq 0\}$ , i.e. $\hat{Z}_j^{i}(t\,;\, a)$ is the number of particles of type j that have age less than a in the branching process at time t. Note that ${\hat{Z}_j^{i}(t; a)}$ is increasing in a.

If ${Z_j^{i}(t)} \to \infty$ for all $j \in [K]$ (i.e. if new particles keep on being born in the branching process) then

(10) \begin{equation} \frac{\hat{Z}_j^{i}(t\,;\, a)}{\sum_{j=1}^K Z_j^i(t)} \to c(a,j) \quad\text{a.s. as } t \to \infty, \end{equation}

where c(a, j) is a constant independent of i (see Theorem 2.7 of [Reference Iksanov and Meiners14]). For our purposes, we do not need to specify c(a, j) further.

Although the branching process is not dependent on n, we want to have some bound on the number of vertices born in the branching process as a function of n. This is used in the coupling in Section 3.3. We set

(11) \begin{equation} t^*= \frac{1-\kappa}{4\alpha}\log n \end{equation}

(where $\kappa {\lt 1}$ is as in Assumption 3). It follows by (9) that, for all $i,j \in [K]$ ,

\begin{equation*}\textrm{e}^{-\alpha t^*}Z^{i}_j(t^*) = n^{-(1-\kappa)/4} Z^{i}_j(t^*) \to W^i_j \in (0,\infty) \quad\text{a.s. as } n \to \infty \end{equation*}

on the survival set of $\{\mathcal Z(t)\,;\,t \geq 0\}$ . In particular, this implies that

(12) \begin{equation} Z^{i}_j(t^*) =o(n^{(1-\kappa)/3}) \quad\text{w.h.p.} \end{equation}

3.3. The coupling

Note that the only difference between the branching process ${\{\mathcal{\tilde Z}^{(n)}(t)\}}$ associated to the susceptibility process of Section 3.1 and the multi-type branching process $\{\mathcal Z(t)\}$ of Section 3.2 is the distribution of the number of particles of type i that a particle of type j gives birth to (binomially distributed with parameters $n_i$ and $m_{ij}/n_j$ and Poisson distributed with expectation $p_j/p_im_{ij}$ , respectively).

We know from [Reference Barbour, Holst and Janson9, Equation (1.23)] (see also [Reference Barbour and Reinert8]) that the total variation distance between a binomial distributed random variable with parameters $n_i$ and $ ({1}/{n_i})({n_i}/{n_j}) m_{ij} $ and a Poisson random variable with parameter $ ({n_i}/{n_j}) m_{ij} $ is $O({1}/{n_i})$ . Moreover, the total variation distance between a Poisson distributed random variable with parameter $ ({n_i}/{n_j}) m_{ij} $ and a Poisson distributed random variable with parameter $ ({p_i}/{p_j}) m_{ij} $ is $O(\sqrt{|{n_i}/{n_j}-{p_i}/{p_j}|}) = O({1}/{\sqrt{n}})$ [Reference Barbour, Holst and Janson9, Theorem 1.C] (see also [Reference Barbour and Reinert8]). Here we have also used Assumption 1. By the triangle inequality, this implies that the total variation distance between a Poisson distributed random variable with expectation $ ({p_i}/{p_j}) m_{ij} $ and a binomial-distributed random variable with parameters $n_i$ and $ ({1}/{n_i})({n_i}/{n_j}) m_{ij} $ is $O({1}/{\sqrt{n}})$ . Hence, as long as the number of particles born in any of the two branching processes is $o(\sqrt{n})$ , the two processes can be coupled w.h.p. In the remainder of this section we show that the coupling holds w.h.p. up to time $t^*$ , with $t^*=t^*(n)$ given by (11).

By construction, if the first $|{\hat{V}}^a(\ell^*)| + |{\hat{V}}^e(\ell^*)|$ vertices that we ‘try to include’ in $\{{\hat{V}}^{a}(\ell) \cup {\hat{V}}^{e}(\ell)\,;\,\ell \in \mathbb{N}_0\}$ are all different and none of the first $|{\hat{V}}^p(\ell^*)|$ that we ‘try to include’ in $\{{\hat{V}}^{p}(\ell)\,;\,\ell \in \mathbb{N}_0\}$ are in ${\hat{V}}^{a}(\ell^*) \cup {\hat{V}}^{e}(\ell^*)$ , then the process is not flagged.

The law of large numbers yields

(13) \begin{equation} |{\hat{V}}^a(\ell^*)| \leq 2 \max_{i,j \in [K]}\frac{n_i}{n_j}m_{ij} |{\hat{V}}^e(\ell^*)| \quad\text{w.h.p.} \end{equation}

Here we use the fact that the expected number of edges in É with any given vertex in V as head is bounded from above by $\max_{i,j \in [K]}({n_i}/{n_j})m_{ij}$ . Equation (Reference Grimmett and Stirzaker13) implies that

(14) \begin{equation} |{\hat{V}}^e(\ell^*)| + |{\hat{V}}^a(\ell^*)| = O(|{\hat{V}}^e(\ell^*)|) \quad\text{w.h.p.} \end{equation}

Using Assumption 3, we obtain

(15) \begin{equation} |{\hat{V}}^p(\ell^*)| \leq n^{\kappa} (|{\hat{V}}^a(\ell^*)|+ |{\hat{V}}^e(\ell^*)|) \quad\text{w.h.p.} \end{equation}

Combining inequalities (14) and (15) yields

(16) \begin{equation} |{\hat{V}}^p(\ell^*)| = O_p(n^{\kappa}|{\hat{V}}^e(\ell^*)|). \end{equation}

Since $\{\mathcal Z(t);t \geq 0\}$ can be coupled with $\{\mathcal S^{(n)}_v(t);t \geq 0\}$ w.h.p. until $o(\sqrt{n})$ particles are born in $\{\mathcal Z(t);t \geq 0\}$ , we obtain, by using (12), $|\mathcal Z(t^*)| = o_p(n^{(1-\kappa)/3})$ . Therefore, we also know that $|\mathcal Z(t^*)| = o_p(n^{1/2})$ and $|{\hat{V}}^e(\ell^*)| = o_p(n^{(1-\kappa)/3})$ . Combined with (16) this gives

\begin{equation*}|{\hat{V}}^e(\ell^*)| + |{\hat{V}}^a(\ell^*)| = o_p(n^{(1-\kappa)/3}) \end{equation*}

and

\begin{equation*}|{\hat{V}}^p(\ell^*)| = o(n^{(1+2\kappa)/3}) = o_p(n). \end{equation*}

Using birthday-problem-like arguments [Reference Grimmett and Stirzaker13, p. 24], the probability that the first $|{\hat{V}}^e(\ell^*)| + |{\hat{V}}^a(\ell^*)|$ activated vertices in $\{\hat{G}(\ell)\,;\, \ell \in \mathbb{N}_0\}$ are not all different is bounded from above by

\begin{equation*} \Big(\min_{j \in [K]}n_j\Big)^{-1}(|{\hat{V}}^e(\ell^*)| + |{\hat{V}}^a(\ell^*)|)^2= o_p(n^{-(1+2\kappa)/3}), \end{equation*}

while the probability that among the first $|{\hat{V}}^p(\ell^*)|$ vertices that we ‘try to include’ in $\{{\hat{V}}^{p}(\ell)\,;\,\ell \in \mathbb{N}_0\}$ there are vertices in ${\hat{V}}^{a}(\ell^*) \cup {\hat{V}}^{e}(\ell^*)$ is bounded from above by

\begin{align*} |{\hat{V}}^p(\ell^*)| \frac{|{\hat{V}}^e(\ell^*)| + |{\hat{V}}^a(\ell^*)|}{\min_{j \in [K]}n_j} &= o_p(n^{(1+2\kappa)/3}) \times o_p(n^{(1-\kappa)/3}) \times O(n^{-1}) \\ &=o_p(n^{(\kappa-1)/3}) \\ &=o_p(1). \end{align*}

We conclude that the probability that the process ${\{\hat{G}(\ell)\,;\, \ell \in \mathbb{N}_0\}}$ is flagged up to and including step $\ell^*$ goes to 0 as $n \to \infty$ , as desired. We summarise the main result of this section in the following lemma.

Lemma 3.1

There exists a probability space on which we can define the branching process $\{\mathcal Z(t)\,;\,t \geq 0\}$ and the susceptibility process $\{\mathcal S^{(n)}_v(t)\,;\,t \geq 0\}$ for all n, such that

\begin{equation*} \mathbb{P}(|\mathcal S^{(n)}_{v}(t\,;\,a,j)| = \hat{Z}_j^i(t\,;\, a) \text{ for all}\ t \leq t^*,\ a \in (0,t^*),\ \text{and}\ j \in [K]) \to 1, \end{equation*}

where $t^*= t^*(n)$ is given by (11).

4.1. Proof of Theorem 1

We are interested in the expected fraction of vertices infected by vertices of type i among the ultimately infected vertices of type j during a major outbreak. By exchangeability, this expected fraction is given by

(17) \begin{equation} \mathbb{P}(v\text{ is infected by a type-}i\ \text{vertex}\ \,\,|\, \ \, v\text{ is ultimately infected},\, v \in V_j). \end{equation}

Consider ${\mathcal S^{(n)}_v(t^*)}$ , where we also assign to each vertex $v' \in {\mathcal S^{(n)}_v(t^*)}$ the value $\sigma'_{v'} = d(v',v)$ . In Section 3.1 we constructed part of the graph Ǵ by looking backward in time. To analyse (17), we construct another part of the graph Ǵ, by looking forward in time and starting at $V_{\text{init}}$ . By Assumption 5, $\mathcal S^{(n)}_v(t^*)$ does not overlap with $V_{\text{init}}$ w.h.p. We condition on this event.

We construct the relevant part of Ǵ through a series of subgraphs $\{\tilde{G}(\ell); \ell \in \mathbb{N}_0\}$ as follows.

1. Construct the graph $\hat{G}(\ell^*)$ as in Section 3.1 and recall that the set of vertices ${\mathcal S^{(n)}_v(t^*)}$ is by construction equal to ${\hat{V}}^e(\ell^*)$ . Let $\tilde{G}(0)=(\tilde{V}(0),\tilde{E}(0))$ have vertex set $V(0) = V_{\text{init}} \cup {\hat{V}}^e(\ell^*) \cup {\hat{V}}^p(\ell^*)$ and edge set $\tilde{E}(0)$ , the subset of $\hat{E}(\ell^*)$ containing all edges with tails in ${\hat{V}}^e(\ell^*)$ . Note that $\tilde{E}(0)$ does not contain edges with tails in ${\hat{V}}^e(\ell^*)$ , i.e. $\tilde{E}(0)$ does not contain the edges in $\hat{G}(\ell^*)$ that have tails from which the distance to v exceeds $t^*$ .

2. The vertices in $V_{\text{init}}$ are active in $\tilde{V}(0)$ , while all other vertices are passive.

3. Assume that we know that $\tilde{G}(\ell)=(\tilde{V}(\ell),\tilde{E}(\ell))$ . Let $\tilde{\sigma}_v(\ell)$ be the distance from $V_{\text{init}}$ to v in $\tilde{G}(\ell)$ . If there is no path from $V_{\text{init}}$ to v in $\tilde{G}(\ell)$ then we set $\tilde{\sigma}_v(\ell) = \infty$ .

4. If there are active vertices in $\tilde{V}(\ell)$ with distance from $V_{\text{init}}$ less than $\tilde{\sigma}_v(\ell)-t^*$ , then we construct $\tilde{G}(\ell+1)=(\tilde{V}(\ell+1),\tilde{E}(\ell+1))$ as follows. Pick the active vertex with the lowest distance from $V_{\text{init}}$ in $\tilde{G}(\ell)$ (in the case of a tie, which occurs if there are still several active vertices in $V_{\text{init}}$ , make a uniform choice among those vertices). Say that this vertex is $u \in V_i$ . Then assign to u the point processes $ (\xi_u^1, \ldots{,} \xi_u^K) $ , having the correct distribution (see Section 2) and label the points of $\xi_u^j$ ( $j\in [K]$ ) with independent uniform vertices from $V_j$ without replacement.

   Create a preliminary graph $\tilde{G}^{\text{prel}}(\ell+1)$ consisting of $\tilde{G}(\ell)$ together with the vertices corresponding to the labels assigned to the points of $ (\xi_u^1, \ldots{,} \xi_u^K) $ , and the edges with tail u and as heads those vertices corresponding to the labels assigned to $ (\xi_u^1, \ldots{,} \xi_u^K) $ . Note that some of the heads of the newly added edges might be in ${\hat{V}}(\ell)$ . If the distance from u to v is at least $t^*$ in $\tilde{G}^{\text{prel}}(\ell+1)$ , then we set $\tilde{G}(\ell+1) = \tilde{G}^{\text{prel}}(\ell+1)$ and we make u passive. Furthermore, the other active vertices in ${\hat{V}}(\ell)$ together with all vertices from ${\hat{V}}(\ell+1)\setminus {\hat{V}}(\ell)$ are the active vertices of ${\hat{V}}(\ell+1)$ . Note that, by construction, vertices of $\mathcal S^{(n)}_v(t^*)$ never become active. If, on the other hand, the distance from u to v is less than $t^*$ in $\tilde{G}^{\text{prel}}(\ell+1)$ , then we set $\tilde{G}(\ell+1) = \tilde{G}(\ell)$ . This construction is equivalent to conditioning on the distance from u to v in $\tilde{G}(\ell+1)$ being larger than $t^*$ . This conditioning is necessary because all vertices from which the distance to v is less than $t^*$ are in $\mathcal S^{(n)}_v(t^*)$ and u was active in ${\hat{V}}(\ell)$ and therefore not in $\mathcal S^{(n)}_v(t^*)$ .

5. If there are no active vertices in $\tilde{G}(\ell)$ with distance from $V_{\text{init}}$ less than $\tilde{\sigma}_v(\ell)-t^*$ , then the shortest path from $V_{\text{init}}$ to v is the same in Ǵ (see the next step) as in $\tilde{G}(\ell)$ , and we set $\tilde{G}(k) = \tilde{G}(\ell)$ for all $k \geq \ell$ . Let

   \begin{align*} \ell^{{{\dagger}}} = \min\{\ell \in \mathbb{N}\,;\, \tilde{G}(k) = \tilde{G}(\ell)\ \text{for all}\ k \geq \ell\} \end{align*}
   be the index after which $\{\tilde{G}(\ell)\}$ does not change anymore.

6. Finally, we finish with the construction of Ǵ from $\tilde{G}(\ell^{{{\dagger}}})$ by assigning one by one an appropriate vector of point processes to the vertices in $V \setminus \tilde{V}(\ell^{{{\dagger}}})$ , i.e. for $u \in V_i \cap (V \setminus \tilde{V}(\ell^{{{\dagger}}}))$ assign to u a preliminary vector of point processes $ (\xi_u^1, \ldots{,} \xi_u^K) $ , which is distributed as $ (\xi_{i1}, \ldots{,} \xi_{iK}) $ and independent of the construction up to now. Assign labels to the points of $\xi_u^j,\,j \in [K]$ , which correspond to vertices in $V_j$ and create preliminary edges with u as tail and the vertices corresponding to the labels of the points of $ (\xi_u^1, \ldots{,}\xi_u^K) $ as the heads. If these preliminary edges make the distance from u to v less than $t^*$ , delete those edges, take a new realisation of $ (\xi_u^1, \ldots{,} \xi_u^K) $ , and continue as before, until the distance from u to v is at least $t^*$ . Then we make the preliminary edges part of É and continue by creating the edges with another vertex from $V \setminus \tilde{V}(\ell^{{{\dagger}}})$ as the tail.

   Note that in the construction of ${\mathcal S^{(n)}_v(t^*)}$ through $\hat{G}(\ell)$ we have created all edges with tails in ${\mathcal S^{(n)}_v(t^*)}$ , and we have identified all vertices in V from which the distance in Ǵ is at most $t^*$ . In the remainder of the construction of Ǵ we created all edges of É with tails not in ${\mathcal S^{(n)}_v(t^*)}$ . Here in this step we had to be careful in the construction because the newly created edges should be such that the distance from ${V \setminus \mathcal S^{(n)}_v(t^*)}$ to v should be at least $t^*$ .

Next, for the coupling, we analyse ${\mathcal S^{(n)}_v(t^*\,;\,a,j)}$ even further. Denote the vertices in $\{u \in V_i\,;\, (u,v) \in E'\}$ , i.e. the vertices in $V_i$ , which are in the first generation of the susceptibility set of v by $u_{i1},u_{i2}, \ldots, u_{i, X_{i}(v)}$ . Here

\begin{align*} X_{i}(v) = |\{u \in V_i\,;\, (u,v) \in E'\}| \end{align*}

is the number of vertices of type i in this first generation of the susceptibility set. Note that $X_{i}(v)$ is distributed as $X_{ij}$ , where $X_{ij}$ is binomially distributed with parameters $n_i$ and $m_{ij}/p_j$ . So, $X_{ij}$ converges in distribution to a Poisson-distributed random variable with expectation $ (p_i/p_j)m_{ij} $ . For $r \in [X_{i}(v)]$ , let $\tau_{ir}(v)= \eta(u_{ir},v)$ be the length of edge $ (u_{ir},v) \in E' $ . Here $\tau_{ir}(v)$ is distributed as $\tau_{ijr}$ with distribution function $\mathbb P(\tau_{ijr}\leq a)=\mathbb E[\xi_{ij}(a)]/m_{ij}$ . For convenience, we let $X_i=X_i(v)$ and $\tau_{ir}=\tau_{ir}(v)$ .

We consider the susceptibility processes of the vertices in $\{u_{ir}; i \in [K],\, r \in [X_{i}]\}$ up to distance $t^*$ from v separately, i.e. we consider the susceptibility set of $u_{ir}$ up to time $t^*- \tau_{ir}$ . Those susceptibility sets are (following the arguments in Section 3.2) w.h.p. not overlapping and independent. Also, note that

(18) \begin{equation} \mathcal S^{(n)}_{v}(t^*\,;\,a,j)= \{ v\} \cup \bigcup_{i =1}^K \bigcup_{r=1}^{X_{i}} \mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{ir}\,;\,a,j). \end{equation}

By Lemma 3.3, we know that

(19) \begin{equation} |\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{ir},a,j)| = \hat{Z}^i_j(t^*-\tau_{ir}\,;\, a) \quad\text{w.h.p.} \end{equation}

Furthermore, by (9) and (10), we obtain

(20) \begin{equation} e^{-\alpha t} \hat{Z}^{i}_j(t\,;\, a){\bf 1}(\hat{Z}^{i}(t) \to \infty) \to c(a,j) W^{i}{\bf 1}(W^i >0) \quad \text{a.s. as } t \to \infty. \end{equation}

Since the sets ${\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{ir})}$ are w.h.p. not overlapping (and do not contain v), it follows from (18) that

\begin{equation*} |\mathcal S^{(n)}_{v}(t^*\,;\,a,j)\setminus v|= \sum_{i =1}^K \sum_{r=1}^{X_{i}} |\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{ir}\,;\,a,j)| \quad \text{w.h.p.} \end{equation*}

Then, by (19) and (20), for all $i \in [K]$ and $r \in [m_i]$ ,

\begin{equation*} \textrm{e}^{-\alpha t^*} |\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{ir};a,j)| \to \textrm{e}^{-\alpha \tau_{ir}} c(a,j) W^{i}(r) \end{equation*}

in probability on ${\{\hat{Z}^{i}_j(t) \to \infty\}}$ as $n \to \infty$ . Here, the random variables $W^i(1), W^i(2), \ldots$ are independent copies of $W^i, \,i\in[K]$ . Note that $\{\hat{Z}^{i}(t) \to \infty\}$ implies that the branching process survives and new particles are born w.h.p. in the interval $[t^*/2,t^*]$ . If $\{\hat{Z}^{i}(t) \not\to \infty\}$ then there is a last birth in the process. Because $t^*/2 \to \infty$ as $n \to \infty$ , there is then no particle born in the interval $[t^*/2,t^*]$ w.h.p. It follows from the coupling arguments above that, as $n \to \infty$ ,

(21) \begin{equation} \mathbb{P}\bigg(\mathcal S^{(n)}_{v}(t^*) \neq \mathcal S^{(n)}_{v}\bigg(\frac{t^*}{2}\bigg)\bigg) \to \mathbb{P}(\hat{Z}^{i}(t) \to \infty). \end{equation}

We are interested in the fraction of vertices (possibly specified by type and age) at time $t^*$ in $\mathcal S^{(n)}_v(t^*)$ that are connected to v through a path of vertices in $\mathcal S^{(n)}_v(t^*)$ that include $u_{ir},\,i \in [K]$ and $r \in [X_{ij}]$ . That is, we want to analyse

\begin{equation*} \frac{|\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{1r}\,;\,a,j)|}{\sum_{j =1}^K \sum_{r=1}^{X_{j}} |\mathcal S^{(n)}_{u_{jr}}(t^*-\tau_{jr}\,;\,a,j)|}, \end{equation*}

on the set ${\{\sum_{j =1}^K \sum_{r=1}^{X_{j}} |\mathcal S^{(n)}_{u_{jr}}(t^*-\tau_{jr}\,;\,a,j)| \neq 0\}}$ . Lemma 3.3 allows us to couple the epidemic process with a branching processes $\{\mathcal Z(t)\,;\,t \geq 0\}$ such that, w.h.p.,

(22) \begin{equation} \frac{|\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{1r}\,;\,a,j)|}{\sum_{j =1}^K \sum_{r=1}^{X_{j}} |\mathcal S^{(n)}_{u_{jr}}(t^*-\tau_{jr}\,;\,a,j)|} = \frac{Z_j^{i,r}(t^*-\tau_{1r}\,;\,a)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} Z_j^{k,r}(t^*-\tau_{kr}\,;\,a)}, \end{equation}

where the ${Z_j^{i,r}(t\,;\,a)}$ are independent copies of ${Z_j^{i}(t\,;\,a),\,r \in \mathbb{N}}$ . Multiplying the numerator and denominator of (22) by $\textrm{e}^{-\alpha t^*}$ and using (20) we find that (22) is equal to

\begin{equation*} \frac{\textrm{e}^{-\alpha \tau_{ir}} (\textrm{e}^{-\alpha (t^*-\tau_{ir})}Z_j^{i,r}(t^*-\tau_{1r}\,;\,a))}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} \textrm{e}^{-\alpha \tau_{kr}} (\textrm{e}^{-\alpha (t^*-\tau_{kr})} Z_j^{k,r}(t^*-\tau_{kr};a)}, \end{equation*}

which, on the set ${\{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} Z_j^{k,r}(t^*-\tau_{kr}\,;\,a)} \to \infty\}$ , converges a.s. to

(23) \begin{align} &\frac{\textrm{e}^{-\alpha \tau_{ir}} c(a,j) W^i(r)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} \textrm{e}^{-\alpha \tau_{kr}} c(a,j) {W}^k(r)}{\bf 1}\bigg(\sum_{k =1}^K \sum_{r=1}^{X_{kj}} W^k(r)>0\bigg) \nonumber \\ &\qquad= \frac{\textrm{e}^{-\alpha \tau_{ir}} {W}^i(r)}{\sum_{k =1}^K \sum_{r=1}^{X_{kj}} \textrm{e}^{-\alpha \tau_{kr}} {W}^k(r)}{\bf 1}\bigg(\sum_{k =1}^K \sum_{r=1}^{X_{kj}} W^k(r)>0\bigg) \end{align}

as $n \to \infty$ (i.e. as $t^* \to \infty$ ). Note that the right-hand side of (23) does not depend on a or j. Using Remark 3, it follows that

\begin{align*} \mathbb{P}(v\text{ is infected by a type-}i\ \text{vertex}\ \,\,|\, \ \, v\text{ is ultimately infected}) \end{align*}

is equal to

(24) \begin{equation} \frac{1}{\mathbb{P}(\mathcal S^{(n)}_{v}(t^*) \neq \mathcal S^{(n)}_{v}({t^*}/{2}))} \mathbb{E}\bigg[\frac{\sum_{r=1}^{X_i} |\mathcal S^{(n)}_{u_{ir}}(t^*-\tau_{ir}\,;\,a,j)|}{\sum_{k =1}^K \sum_{r=1}^{X_k} |\mathcal S^{(n)}_{u_{kr}}(t^*-\tau_{kr}\,;\,a,j)|} {\bf 1}\bigg(\mathcal S^{(n)}_{v}(t^*) \neq \mathcal S^{(n)}_{v}\bigg(\frac{t^*}{2}\bigg)\bigg) \bigg]. \end{equation}

It follows from (21), (22), (23), and the bounded convergence theorem that (24) converges to $\rho_{ij}$ , with $\rho_{ij}$ given by (3). This completes the proof for Theorem 1.

4.2. Towards the proof of Theorem 2: bounds for the ${\rho}_{{ij}}$

Theorem 1 provides us with an expression for the asymptotic fractions $\rho_{ij}$ of infected individuals of type j that were infected by individuals of type i. However, often there is no explicit description of the distribution of ${W}^k(r)$ . In Theorem 2 we consider bounds for the $\rho_{ij}$ for a special class of models, as specified in Section 2. In this section we discuss how to use the epidemic random graph Ǵ to obtain the bounds for $\rho_{ij}$ . In most of the subsequent analysis we analyse the graph Ǵ without taking the lengths of edges into account.

Note that Assumption 2 guarantees that, with probability 1, all paths in É have different lengths. In order to obtain the maximum and minimum of the probability $\rho_{ij}$ for fixed mean offspring matrix M, we first investigate the susceptibility set of v ( $v \in V_j$ ) restricted to the graph $G'_{ij}=(V,E'_{ij})$ . We denote this susceptibility set by $\mathcal S_{v,ij}$ . Here $E'_{ij} \subset E'$ is the subset of É that consists of the edges that either have tail vertex in $V_i$ or head vertex not in $V_j$ , i.e. $E \setminus E'_{ij}$ is the set of edges with tails in $V \setminus V_i$ and heads in $V_j$ . In Figure 3, the set $\mathcal S_{v,11}$ consists of all vertices of type 1 (the circles).

Figure 3.

Illustration of (a part of) the susceptibility set of v in $G'=(V,E')$ . Vertices of type 1 are represented by circles and vertices of type 2 are represented by squares. The part of the susceptibility set illustrated in the figure is the set connected to v through paths of edges with heads of type 1.

Let ${\mathcal S^{j}_{v,ij}}= V_j \cap \mathcal S_{v,ij}$ . If $\mathcal S_{v,ij} \cap V_{\text{init}} = \emptyset$ , i.e. if there is no path from $V_{\text{init}}$ to v in $G'_{ij}$ , then, by the definition of the epidemic process, every vertex of type j in $\mathcal S_{v,ij}$ has the same probability to be the first vertex to be infected in the epidemic, i.e. $u={\mathop{\text{argmin}}_ {u' \in \mathcal S^j_{v,ij}}d(V_{\text{init}},u')}$ is uniform in ${\mathcal S^j_{v,ij}}$ .

Condition on $\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset$ . If $u =v$ and $\mathcal S_{v,ij}\cap V_{\text{init}} = \emptyset$ , then v is infected by a vertex that is not of type i. On the other hand, if $u \neq v$ or $\mathcal S_{v,ij} \cap V_{\text{init}} \neq \emptyset$ , then v might be infected by a type-i vertex. Hence,

\begin{align*} 1-\rho_{ij} &= \mathbb{P}(v\text{ is not infected by a type-}i\ \text{vertex}\ \,\,|\, \ \,\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset) \\ & \geq \mathbb{P}(u = v\ \,\,|\, \ \,\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset). \end{align*}

Now assume that $\mathbb{P}(\eta_{i',j'}> n^{-1}) =1$ for $i' \in [K]\setminus \{i\}$ and $j'=j$ , with j the type of v, and $\mathbb{P}(\eta_{i',j}\lt n^{-2}) =1$ otherwise. This implies that the length of any path with only edges in $E'_{ij}$ is less than any edge in $E'\setminus E'_{ij}$ . The assumptions guarantee that, for $u \neq v$ or $\mathcal S_{v,ij} \cap V_{\text{init}} \neq \emptyset$ , conditioned on $\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset$ , v is infected by a vertex of type i (the tail of the edge with head v in the shortest path in $E_{ij}'$ from u or $V_{\text{init}}$ to v). Therefore, for this model,

\begin{equation*} \mathbb{P}(v\text{ is not infected by a type-}i\ \text{vertex}\ \,\,|\, \ \,\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset) = \mathbb{P}(u = v\ \,\,|\, \ \,\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset). \end{equation*}

Hence, for a given distribution of É, models with $\mathbb{P}(\eta_{i',j'}> n^{-1}) =1$ for $i' \in [K]\setminus \{i\}$ and $j'=j$ and $\mathbb{P}(\eta_{i',j}\lt n^{-2}) =1$ otherwise, are among the models for which the fraction of the ultimately infected vertices of type j, infected by a type-i vertex is maximised.

Using similar arguments, we find that $\rho_{ij}$ , the fraction of ultimately infected vertices of type j that are infected by vertices of type i, is minimal if the edge lengths of edges with tail of type i and head of type j are much longer than the other edges. This will be used in Section 4.3.

4.3. Proofs of Theorem 2 and Corollary 1

Consider the model that was introduced in Section 2 with $K=2$ and general distributions $ (\xi_{i1}, \xi_{i2}) $ . Using the arguments from Section 4.2, we can compute the upper bounds $\rho_{11}^+$ and $\rho_{22}^+$ (and lower bounds $\rho_{21}^-$ and $\rho_{11}^-$ ) explicitly. In this subsection we consider the model in which edges from $V_1$ to $V_1$ are infinitesimally short and all other edges in É are relatively long. From Section 4.2 we know that this is the model for which the fraction of vertices of type 1 infected by type-1 vertices is maximised. For reasons of convenience, we assume that v is of type 1 with $\mathcal S_{v}\cap V_{\text{init}}\neq\emptyset$ , i.e. the susceptibility set of v has at least one of the initial infectives in it. We note that it is harder (if not impossible) to obtain an explicit expression for $\rho_{12}^+$ or $\rho_{21}^+$ . As will become clear in the computation below, the difficulty with $\rho_{12}^+$ or $\rho_{21}^+$ is that one would need to consider paths in $G_{12}'$ and $G_{21}'$ that contain vertices of both type 1 and type 2. In contrast, for computing $\rho_{11}^+$ , paths in $G_{11}'$ that end in $v^*$ contain only vertices of type 1.

If we ignore the lengths of edges in Ǵ in the general model introduced in Section 2, then the approximating (backward) branching process describing the generation-based growth of $\mathcal S_{v}(t)$ is defined through the following offspring distributions. The number of children of type i of a particle of type j is Poisson distributed with expectation ${m_{ji}^{(b)}} = ({p_i}/{p_j})m_{ij},\,i,j = 1,2$ . The numbers of offspring of different particles are independent.

Let $Y= |\mathcal S_{v,11}|$ . It is easily seen that Y is approximated by the size of a branching process with Poisson offspring distribution that has expectation ${m_{11}^{(b)}}$ . Then Y is Borel distributed with parameter ${m_{11}^{(b)}}$ , i.e. for $\ell \in \{1,2,\ldots\}$ ,

\begin{equation*}\mathbb{P}(Y=\ell) = \frac{(m_{11}^{(b)} \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} \ell}}{\ell!} \end{equation*}

(see [1]). If ${m_{11}^{(b)}}>1$ then $\mathbb{P}(Y=\infty) > 0$ . Standard results on Borel distributions [1] give, for ${m_{11}^{(b)}} \leq 1$ ,

(25) \begin{equation} \mathbb{E}\bigg[\frac{1}{Y}\bigg] = 1-\frac{m_{11}^{(b)}}{2}. \end{equation}

Define

\begin{equation*} \rho_{11} = \mathbb{P}(v\text{ is infected by a type-1 vertex}\ \,\,|\, \ \,\mathcal S_{v} \cap V_{\text{init}} \neq \emptyset) \end{equation*}

for $v \in V_1$ . From the arguments in Section 4.2 we know that

(26) \begin{equation} \rho_{21}^- = \mathbb{E}\bigg[\frac{1}{Y}\biggm| \mathcal S_{v} \cap V_{\text{init}} \neq \emptyset\bigg]. \end{equation}

The consistency relation $\rho_{11}+\rho_{21}=1$ then reveals that

(27) \begin{equation} \rho_{11}^+=1-\rho_{21}^- \end{equation}

is an upper bound for $\rho_{11}$ .

We use the following extinction probabilities in the backward branching process.

• $q_1$ : the probability that the backward branching process starting with a single type-1 particle goes extinct.

• $q_2$ : the probability that the backward branching process starting with a single type-2 particle goes extinct.

• $\tilde{q}_1$ : the probability that the backward branching process restricted to type-1 particles dies out, i.e. $\tilde{q}_1 = \mathbb{P}(Y\lt\infty)$ .

These probabilities are defined by (4), (5), and (6), and they follow from the theory on multi-type supercritical branching processes [Reference Jagers15, Chapter 4].

We distinguish between $m_{11}^{(b)} \leq 1$ and $m_{11}^{(b)}>1$ . This distinction is not necessary, but we believe that the argument becomes clearer by treating the case $\tilde{q}_1= 1$ separately. In particular, all major ideas of the proof are shown if $m_{11}^{(b)} \leq 1$ , while we avoid the need for the conditioning described in Section 4.3.2 below. Once the proof of this special case is given, the proof for the theorem conditioned on $m_{11}^{(b)}>1$ is straightforward.

4.3.2. The case ${m_{11}^{(b)} > 1}$

Assume that $m_{11}^{(b)} > 1$ . Then it is possible that the backward process restricted to particles of type 1 is already large, i.e. the approximating branching process with Poisson( $m_{11}^{(b)}$ ) offspring distribution survives. We call this event $\mathcal A_1$ . In that case the probability that vertex v is infected by a vertex of type 1 approaches 1 as the population size tends to infinity.

The other possibility is that the backward process restricted to particles of type 1 stays small. Call this event $\mathcal A_1^C$ , the complement of $\mathcal A_1$ . The probability of this event is ${\mathbb{P}(\mathcal A_1^C)}=\tilde{q}_1$ , where $\tilde{q}_1$ is the unique solution in (0, 1) of (7) (note that ${m_{11}^{(b)}}>1$ ensures that $\tilde q_1$ exists). From the theory of branching processes we know that, conditioned on $\mathcal A_1^C$ , the approximating branching process is still a branching process with Poisson-distributed offspring distribution, but now with offspring expectation $m_{11}^{(b)} \tilde{q}_1$ . We still condition on the event that the approximating branching process (including both types of particles) survives.

Conditioned on ${\mathcal A_1^C}$ , the total size of the backward process restricted to particles of type 1 is Borel distributed with parameter ${m_{11}^{(b)}} \tilde{q}_1$ . Using Bayes’ rule,

\begin{equation*} \mathbb{P}(Y=\ell \ \,\,|\, \ \, \mathcal A,\mathcal A_1^C) = \frac{\mathbb{P}(\mathcal A\ \,\,|\, \ \, Y=\ell,\mathcal A_1^C) \mathbb{P}(Y= \ell\ \,\,|\, \ \, \mathcal A_1^C)}{\mathbb{P}(\mathcal A\ \,\,|\, \ \,\mathcal A_1^C)}. \end{equation*}

Note that

\begin{equation*} \mathbb{P}(\mathcal A\ \,\,|\, \ \,\mathcal A_1^C) = \frac{ \mathbb{P}(\mathcal A,\mathcal A_1^C)}{ \mathbb{P}(\mathcal A_1^C)}= \frac{\mathbb{P}(\mathcal A_1^C)-\mathbb{P}(\mathcal A^C,\mathcal A_1^C)}{ \mathbb{P}(\mathcal A_1^C)}= \frac{\mathbb{P}(\mathcal A_1^C)-\mathbb{P}(\mathcal A^C)}{ \mathbb{P}(\mathcal A_1^C)}= \frac{\tilde{q}_1-q_1}{\tilde{q}_1}. \end{equation*}

Then the same arguments as those leading to (30) but with ${m_{11}^{(b)}}$ replaced by ${m_{11}^{(b)} \tilde{q}_1}$ yield

\begin{equation*} \mathbb{P}(\mathcal A\ \,\,|\, \ \, Y=\ell,\mathcal A_1^C) =1- q^{\ell} = 1-(q_1)^{\ell} \textrm{e}^{m_{11}^{(b)} (1-q_1)\ell} \end{equation*}

and

\begin{equation*} \mathbb{P}(Y= \ell\ \,\,|\, \ \,\mathcal A_1^C)= \frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!}. \end{equation*}

Combining these identities with (7) yields

\begin{align*} & \mathbb{P}(Y=\ell\ \,\,|\, \ \,\mathcal A,\mathcal A_1^C) \\ &\qquad= \frac{\tilde{q}_1}{\tilde{q}_1-q_1} (1-(q_1)^{\ell} \textrm{e}^{m_{11}^{(b)} (1-q_1)\ell})\bigg(\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!} \bigg) \\ &\qquad= \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \bigg(\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!} \bigg) - \frac{q_1 (\tilde{q}_1)^{\ell}}{\tilde{q}_1-q_1} \bigg(\frac{(m_{11}^{(b)} q_1 \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} (\tilde{q}_1+q_1-1) \ell}}{\ell!} \bigg) \\ &\qquad = \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \bigg(\frac{(m_{11}^{(b)} \tilde{q}_1 \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} \tilde{q}_1 \ell}}{\ell!} \bigg) - \frac{\tilde{q}_1}{\tilde{q}_1-q_1}\bigg(\frac{(m_{11}^{(b)} q_1 \ell)^{\ell-1} \textrm{e}^{-m_{11}^{(b)} q_1 \ell}}{\ell!} \bigg). \end{align*}

By standard results on Galton–Watson branching processes we know that ${m_{11}^{(b)}} \tilde{q}_1 \leq 1$ . Moreover, since $q_1 \leq \tilde{q}_1$ , ${m_{11}^{(b)}} q_1 \leq 1$ also holds. Then, using (25), we obtain

\begin{align*} \mathbb{E}[Y^{-1}\ \,\,|\, \ \,\mathcal A,\mathcal A_1^C] &= \frac{\tilde{q}_1}{\tilde{q}_1-q_1} \bigg(1-\frac{m_{11}^{(b)} \tilde{q}_1}{2}\bigg) - \frac{q_1}{\tilde{q}_1-q_1} \bigg(1-\frac{m_{11}^{(b)} q_1}{2}\bigg) \\ & = 1 - \frac{m_{11}^{(b)} (\tilde{q}_1 + q_1)}{2}. \end{align*}

This leads to

(31) \begin{align} \rho_{21}^- &= \mathbb{E}[Y^{-1}\ \,\,|\, \ \,\mathcal A]\nonumber \\ &= \mathbb{E}[Y^{-1}\ \,\,|\, \ \,\mathcal A,\mathcal A_1^C]\mathbb{P} (\mathcal A_1^C\ \,\,|\, \ \,\mathcal A) + \mathbb{E}[Y^{-1}\ \,\,|\, \ \,\mathcal A,\mathcal A_1] \mathbb{P}(\mathcal A_1\ \,\,|\, \ \,\mathcal A)\nonumber \\ &= \mathbb{E}[Y^{-1}\ \,\,|\, \ \,\mathcal A,\mathcal A_1^C]\frac{\mathbb{P} (\mathcal A_1^C,\mathcal A)}{\mathbb{P}(\mathcal A)} + 0 \nonumber \\ &= \mathbb{E}[Y^{-1}\ \,\,|\, \ \,\mathcal A,\mathcal A_1^C]\frac{\mathbb{P} (\mathcal A_1^C)-\mathbb{P}(\mathcal A_1^C,\mathcal A^C)} {\mathbb{P}(\mathcal A)}\nonumber \\ &= \bigg( 1 - \frac{m_{11}^{(b)} (\tilde{q}_1 + q_1)}{2} \bigg) \frac{\tilde{q}_1-q_1}{1-q_1}. \end{align}

Note that expression (31) is consistent with result (30) for ${m_{11}^{(b)}}\leq 1$ , where $\tilde{q}_1=1$ . Finally, using (27), we find the expression for $\rho_{11}^+$ :

(32) \begin{equation} \rho_{11}^+=1-\bigg( 1 - \frac{m_{11}^{(b)} (\tilde{q}_1 + q_1)}{2} \bigg) \frac{\tilde{q}_1-q_1}{1-q_1}. \end{equation}

This completes the proof of Theorem 2.

Proof of Corollary 1. In Corollary 1 we consider the special case that $ (\xi_{i1}, \xi_{i2}) $ is obtained from the independent labelling of a one-dimensional point process $\xi_i$ . One can assign the types (type 1 with probability $p_1$ or type 2 with probability $p_2=1-p_1$ ) independently to the points of this process. In this special case we have $\rho_{11}=\rho_1=\rho_{12}$ and $\rho_{21}=\rho_2=\rho_{22}$ . Furthermore, ${m_{ji}^{(b)}}=p_i \tilde m_j$ , with $\tilde m_j=\mathbb{E}[\xi_j(\infty)]$ the expected number of secondary cases generated by a newly infected type-j individual in an otherwise susceptible population, $i,j=1,2$ . Note that (32) reduces to the fraction $\rho_1^+$ of the ultimately infected vertices that are infected by type-1 vertices:

\begin{equation*} \rho_1^+ = 1-\bigg( 1 - \frac{p_1\tilde m_1(\tilde{q}_1 + q)}{2} \bigg) \frac{\tilde{q}_1-q}{1-q}, \end{equation*}

where $\tilde{q}_1$ and q are solutions of (7) and (8). This proves Corollary 1.