Brownian snails with removal:
epidemics in diffusing populations

Numerous mathematical models have been introduced to describe the spread of a disease around a population. Such models may be deterministic or stochastic, or a mixture of each; they may incorporate a range of factors including susceptibility, infectivity, recovery, and removal; the population members (termed ‘particles’) may be distributed about some given space; and so on. We propose two models in which (i) the particles move randomly about the space that they inhabit, (ii) infection may be passed between particles that are sufficiently close to one another, and (iii) after the elapse of a random time since infection, a particle is removed from the process. These models differ from that of Beckman, Dinan, Durrett, Huo, and Junge [3] through the introduction of the permanent ‘removal’ of particles, and this new feature brings a significant new difficulty to the analysis.

We shall concentrate mostly on the case in which the particles inhabit ${\mathbb{R}}^{d}$ where $d\geq 2$. Here is a concrete example of the processes studied here.

• (a)

  Particles are initially distributed in ${\mathbb{R}}^{d}$ in the manner of a rate-$\lambda$ Poisson process conditioned to contain a point at the origin $0$.

• (b)

  Particles move randomly within ${\mathbb{R}}^{d}$ according to independent Brownian motions with variance-parameter $\sigma^{2}$.

• (c)

  At time $0$ the particle at the origin (the initial ‘infective’) suffers from an infectious disease, which may be passed to others when sufficiently close.

• (d)

  When two particles, labelled $P$ and $P^{\prime}$, are within a given distance $\delta$, and $P$ is already infected, then particle $P^{\prime}$ becomes infected.

• (e)

  Each particle is infected for a total period of time having the exponential distribution with parameter $\alpha\in[0,\infty)$, and is then permanently removed.

The fundamental question is to determine for which vectors $(\lambda,\delta,\sigma,\alpha)$ it is the case that (with strictly positive probability) infinitely many particles become infected. For simplicity, we shall assume henceforth that

We shall generally assume $\alpha>0$. In the special case $\alpha=0$, (studied in [3]) a particle once infected remains infected forever, and the subsequent analysis is greatly facilitated by a property of monotonicity that is absent in the more challenging case $\alpha>0$ considered in the current work.

Two protocols for movement feature in this article.

1. A.

   Delayed diffusion model. The initial infective starts to move at time $0$, and all other particles remain stationary until they are infected, at which times they begin to move.

2. B.

   Diffusion model. All particles begin to move at time $0$.

The main difficulty in studying these models arises from the fact that particles are permanently removed after a (random) period of infectivity. This introduces a potential non-monotonicity into the model, namely that the presence of infected particles may hinder the growth of the process through the creation of islands of ‘removed’ particles that can act as barriers to the further spread of infection. A related situation (but without the movement of particles) was considered by Kuulasma [21] in a discrete setting, and the methods derived there are useful in our Section 3.7 (see also Alves et al. [1, p. 4]). This issue may be overcome for the delayed diffusion model, but remains problematic in the case of the diffusion process.

Let $I$ denote the set of particles that are ever infected, and

We say the process

Let $\lambda_{\text{\rm c}}$ denote the critical value of $\lambda$ for the disk (or ‘Boolean’) percolation model with radius $1$ on ${\mathbb{R}}^{d}$ (see, for example, [25]). It is immediate for both models above that

since in that case the disease spreads instantaneously to the percolation cluster $C$ containing the initial infective, and in addition we have ${\mathbb{P}}_{\lambda,\alpha}(|C|=\infty)>0$.

We write $\theta_{\text{\rm d}}$ (respectively, $\theta_{\text{\rm dd}}$) for the function $\theta$ of (1.2) in the case of the diffusion model (respectively, delayed diffusion model). The following two theorems are proved in Sections 3 and 4 as special cases of results for more general epidemic models than those given above.

The delayed diffusion model has no phase transition when $d=1$; see Theorems 3.2 and 3.6.

For the diffusion model, we have no proof of survival for $d\geq 2$ and small positive $\alpha$ (that is, that $\theta_{\text{\rm d}}(\lambda,\alpha)>0$ for some $\lambda<\lambda_{\text{\rm c}}$ and $\alpha>0$), and neither does the current work answer the question of whether or not survival ever occurs when $d=1$. See Section 4.3. The above theorems are proved using a perturbative argument, and thus fall short of the assertion that $\underline{\lambda}=\lambda_{\text{\rm c}}$.

The methods of proof may be made quantitative, leading to bounds for the numerical values of the critical points $\alpha_{\text{\rm c}}$. Such bounds are far from precise, and therefore we do not explore them here. Our basic estimates for the growth of infection hold if the intensity $\lambda$ of the Poisson process is non-constant so long as it is bounded uniformly between two strictly positive constants. The existence of the subcritical phase may be proved for more general diffusions than Brownian motion.

The related literature is somewhat ramified, and a spread of related problems have been studied by various teams. We mention a selection of papers but do not attempt a full review, and we concentrate on works associated with ${\mathbb{R}}^{d}$ rather than with trees or complete graphs.

The delayed diffusion model may be viewed as a continuous-time version of the ‘frog’ random walk process studied in Alves et al. [1, 2], Ramirez and Sidoravicius [29], Fontes et al.  [7], Benjamini et al. [4], and Hoffman, Johnson, and Junge [15, 16]. See Popov [28] for an early review. Kesten and Sidoravicius [17, 18, 19] considered a variant of the frog model as a model for infection, both with and without recuperation (that is, when infected frogs recover and become available for reinfection—see also Section 4.3 of the current work). The paper of Beckman et al. [3] is devoted to the delayed diffusion model without removal (that is, with $\alpha=0$). Peres et al.  [27] studied three geometric properties of a Poissonian/Brownian cloud of particles, in work inspired in part by the dynamic Boolean percolation model of van den Berg et al. [5]. Related work has appeared in Gracar and Stauffer [9].

A number of authors have considered the frog model with recuperation under the title ‘activated random walks’. The reader is referred to the review by Rolla [30], and for recent work to Stauffer and Taggi [33] and Rolla et al. [31].

We write ${\mathbb{Z}}_{0}=\{0,1,2,\dots\}$ and $1_{A}$ (or $1(A)$) for the indicator function of an event or set $A$. Let $S(r)$ denote the closed $r$-ball of ${\mathbb{R}}^{d}$ with centre at the origin, and $S=S(1)$. The $d$-dimensional Lebesgue measure of a set $A$ is written $|A|_{d}$, and the Euclidean norm $\|\cdot\|_{d}$. The radius of $M\subseteq{\mathbb{R}}^{d}$ is defined by

We abbreviate ${\mathbb{P}}_{\lambda,\alpha}$ (respectively, ${\mathbb{E}}_{\lambda,\alpha}$) to the generic notation ${\mathbb{P}}$ (respectively, ${\mathbb{E}}$).

The contents of this paper are as follows. The two models are defined in Section 2 with a degree of generality that includes general diffusions and a more general process of infection. The delayed diffusion model is studied in Section 3, and the diffusion model in Section 4. Theorem 1.1 (respectively, Theorem 1.2) is contained within Theorem 3.1 (respectively, Theorem 4.1).

This introduction closes with a short account of some of the principal remaining open problems. For concreteness, we restrict ourselves to the Brownian models of Section 1.2 without further reference to the random-walk versions of these models, and the general models of Section 2.1. This section is positioned here despite the fact that it refers sometimes to versions of the models that have not yet been fully introduced (see Section 2).

• A.

  For the Brownian delayed diffusion model, show that the critical value $\alpha_{\text{\rm c}}(\lambda)$ of Theorem 1.1 satisfies $\alpha_{\text{\rm c}}(\lambda)<\infty$ whenever $\lambda<\lambda_{\text{\rm c}}$.

• B.

  When $d\geq 2$, prove survival in the Brownian diffusion model for some $\lambda<\lambda_{\text{\rm c}}$ and small $\alpha>0$. More specifically, show that $\theta_{\text{\rm d}}(\lambda,\alpha)>0$ for suitable $\lambda$ and $\alpha$.

• C.

  Having resolved problem B, show the existence of a critical value $\alpha_{\text{\rm c}}=\alpha_{\text{\rm c}}(\lambda)$ for the Brownian diffusion model such that survival occurs when $\alpha<\alpha_{\text{\rm c}}$ and not when $\alpha>\alpha_{\text{\rm c}}$. Furthermore, identify the set of $\lambda$ such that $\alpha_{\text{\rm c}}(\lambda)<\infty$.

• D.

  Decide whether or not survival can ever occur for the Brownian diffusion model in one dimension.

• E.

  In either model, prove a shape theorem for the set of particles that are either infected or removed at time $t$.

Let $d\geq 1$. A diffusion process in ${\mathbb{R}}^{d}$ is a solution $\zeta$ to the stochastic differential equation

where $W$ is a standard Brownian motion in ${\mathbb{R}}^{d}$. (We may write either $W_{t}$ or $W(t)$.) For definiteness, we shall assume that: $\zeta(0)=0$; $\zeta$ has continuous sample paths; the instantaneous drift vector $a$ and variance matrix $\sigma$ are locally Lipschitz continuous. We do not allow $a$, $\sigma$ to be time-dependent. We call the process ‘Brownian’ if $\zeta$ is a standard Brownian motion, which is to say that $a$ is the zero vector and $\sigma$ is the identity matrix.

Let $\zeta$ be such a diffusion, and let $(\zeta_{i}:i\in{\mathbb{Z}}_{0})$ be independent copies of $\zeta$. Let $\alpha\in(0,\infty)$, $\rho\in[0,\infty)$, and let $\mu:{\mathbb{R}}^{d}\to[0,\infty)$ be integrable with

We call $\mu$ radially decreasing if

Let $\Pi=(X_{0}=0,X_{1},X_{2},\dots)$ be a Poisson process on ${\mathbb{R}}^{d}$ (conditioned to possess a point at the origin $0$) with constant intensity $\lambda\in(0,\infty)$. At time $0$, particles with label-set ${\mathcal{P}}=\{P_{0},P_{1},P_{2},\dots\}$ are placed at the respective points $X_{0}=0,X_{1},X_{2},\dots$. We may refer to a particle $P_{i}$ by either its index $i$ or its initial position $X_{i}$.

We describe the process of infection in a somewhat informal manner (see also Section 2.4). For $i\in{\mathbb{Z}}_{0}$, at any given time $t$ particle $P_{i}$ is in one of three states S (susceptible), I (infected), and R (removed). Thus the state space is $\Omega=\Pi\times\{\text{\rm S},\text{\rm I},\text{\rm R}\}^{{\mathbb{Z}}_{0}}$, and we write $\omega(t)=(\omega_{i}(t):i\in{\mathbb{Z}}_{0})$ for the state of the process at time $t$. Let $S_{t}$ (respectively, $I_{t}$, $R_{t}$) be the set of particles in state S (respectively, I, R) at time $t$. We take

so that $I_{0}=\{P_{0}\}$ and $S_{0}={\mathcal{P}}\setminus\{P_{0}\}$. The only particle-transitions that may occur are S $\to$ I and I $\to$ R. The transitions $\text{\rm S}\to\text{\rm I}$ occur at rates that depend on the locations of the currently infected particles.

We shall refer to the above (in conjunction with the specific infection assumptions of Sections 2.2 or 2.3) as the general model. When $a\equiv 0$ and $\sigma\equiv 1$ in (2.1) (or, more generally, $\sigma$ is constant), we shall refer to it as the Brownian model. We shall prove the existence of a subcritical phase (characterized by the absence of survival) for the general model subject to weak conditions. Our proof of survival for the delayed diffusion model is for the Brownian model alone. Estimates for the volume of the sausage generated by $\zeta$ play roles in the calculations, and it may be that, in this regard or another, the behaviour of a general model is richer than that of its Brownian version.

Each particle $P_{j}$ is stationary if and only if it is in state S. If it becomes infected (at some time $B_{j}$, see (2.5)), henceforth it follows the diffusion $X_{j}+\zeta_{j}$. We write

for the position of $P_{j}$ at time $t$.

We describe next the rate at which a given particle $P$ infects another particle $P^{\prime}$. The function $\mu$, given above, encapsulates the spatial aspect of the infection process, and a parameter $\rho\in(0,\infty)$ represents its intensity,

• ($\text{\rm S}\to\text{\rm I}$)

  Let $t>0$, and let $P_{j}$ be a particle that is in state S at all times $s<t$. Each $P_{i}\in I_{t}$ (with $i\neq j$) infects $P_{j}$ at rate $\rho\mu(X_{j}-\pi_{i}(t))$. The aggregate rate at which $P_{j}$ becomes infected is

  (2.4)

  $$\sum_{i\in I_{t},\,i\neq j}\rho\mu(X_{j}-\pi_{i}(t)).$$

• ($\text{\rm I}\to\text{\rm R}$)

  An infected particle is removed at rate $\alpha$.

Transitions of other types are not permitted. We take the sample path $\omega=(\omega(t):t\geq 0)$ to be pointwise right-continuous, which is to say that, for $i\in{\mathbb{Z}}_{0}$, the function $\omega_{i}(\cdot)$ is right-continuous. The infection time $B_{j}$ of particle $P_{j}$ is given by

The infection rates $\rho\mu(X_{j}-\pi_{i}(t))$ of (2.4) are finite, and hence infections take place at a.s. distinct times. We may thus speak of $P_{j}$ as being ‘directly infected’ by $P_{i}$. We speak of a point $z\in\Pi$ as being directly infected by a point $y\in\Pi$ when the associated particles have that property. If $P_{j}$ is infected directly by $P_{i}$, we call $P_{j}$ a child of $P_{i}$, and $P_{i}$ the parent of $P_{j}$.

Following its infection, particle $P_{i}$ remains infected for a further random time $T_{i}$, called the lifetime of $P_{i}$, and is then removed. The times $T_{i}$ are random variables with the exponential distribution with parameter $\alpha>0$, and are independent of one another and of the $X_{j}$ and $\zeta_{j}$.

In the above version of the delayed diffusion model, $\rho$ is assumed finite. When $\rho=\infty$, we shall consider only situations in which

In this situation, a susceptible particle $P_{j}$ becomes infected at the earliest instant that it belongs to $\pi_{i}(t)+M$ for some $P_{i}\in I_{t}$, $i\neq j$. This happens when either (i) $P_{i}$ infects $P_{j}$ as $P_{i}$ diffuses around ${\mathbb{R}}^{d}$ post-infection, or (ii) at the moment $B_{i}$ of infection of $P_{i}$, particle $P_{j}$ is infected instantaneously by virtue of the fact that $X_{j}\in\pi_{i}(B_{i})+M$. These two situations are investigated slightly more fully in the following definition of ‘direct infection’.

The role of the Boolean model of continuum percolation becomes clear when $\rho=\infty$, and we illustrate this, subject to the simplifying assumption that $M$ is symmetric in the sense that $x\in M$ if and only if $-x\in M$. Let $\Pi=(X_{i}:i\in{\mathbb{Z}}_{0})$ be a Poisson process in ${\mathbb{R}}^{d}$ with constant intensity $\lambda$, and declare two points $X_{i}$, $X_{j}$ to be adjacent if and only if $X_{j}-X_{i}\in M$. This adjacency relation generates a graph $G$ with vertex-set $\Pi$. In the delayed diffusion process on the set $\Pi$, entire clusters of the percolation process are infected simultaneously.

Since there can be many (even infinitely many) simultaneous infections at the same time instant when $\rho=\infty$, the notion of ‘direct infection’ requires amplification. For $j\neq 0$, we say that $P_{j}$ is directly infected by $P_{0}$ if $P_{j}$ is in state S at all times $s<Y_{j}$, where $Y_{j}=\inf\{t\geq 0:X_{j}\in\zeta_{0}(t)+M\}$, and in addition $Y_{j}<T_{0}$. We make a similar definition, as follows, for direct infections by $P_{i}$ with $i\neq 0$. Let $i\neq 0$ and $j\neq 0,i$.

• (a)

  We say that $P_{j}$ is dynamically infected by $P_{i}$ if the following holds. Particle $P_{i}$ (respectively, $P_{j}$) is in state I (respectively, state S) at all times $Y_{i,j}-\epsilon$ for $\epsilon\in(0,\epsilon_{0})$ and some $\epsilon_{0}>0$, where $Y_{i,j}=\inf\{t\geq B_{i}:X_{j}\in X_{i}+\zeta_{i}(t)+M\}$, and in addition $Y_{i,j}-B_{i}\leq T_{i}$.

• (b)

  We say that $P_{j}$ is instantaneously infected by $P_{i}$ if the following holds. There exist $n\geq 1$ and $k,i_{1},i_{2},\dots,i_{n}=i,i_{n+1}=j$ such that $i_{1}$ is dynamically infected by $k$ (at time $B_{i_{1}}$) and

  $$X_{i_{m+1}}\in T_{m}\setminus T_{m-1},\qquad m=1,2,\dots,n,$$

  where

  $$T_{0}:=\varnothing,\quad T_{m}=\bigcup_{r=1}^{m}(X_{r}+M).$$

Condition (a) corresponds to infection through movement of $P_{i}$, and (b) corresponds to instantaneous infection at the moment of infection of $P_{i}$. We say that $P_{j}$ is directly infected by $P_{i}$ if it is infected by $P_{i}$ either dynamically or instantaneously.

Certain events of probability $0$ are overlooked in the above informal description including, for example, the event of being dynamically infected by two or more particles, and the event of being instantaneously infected by an infinite chain $(i_{r})$ but by no finite chain.

In either case $\rho<\infty$ or $\rho=\infty$, we write $\theta_{\text{\rm dd}}(\lambda,\rho,\alpha)$ for the probability that infinitely many particles are infected. For concreteness, we note our special interest in the case in which:

• (a)

  $\zeta$ is a standard Brownian motion,

• (b)

  $\mu=1_{S}$ with $S$ the closed unit ball of ${\mathbb{R}}^{d}$.

The diffusion model differs from the delayed diffusion model of Section 2.2 in that all particles begin to move at time $t=0$. The location of $P_{j}$ at time $t$ is $X_{j}+\zeta_{j}(t)$, and the transition rates are given as follows. First, suppose $\rho\in(0,\infty)$.

• ($\text{\rm S}\to\text{\rm I}$)

  Let $t>0$, and let $P_{j}$ be susceptible at all times $s<t$. Each $P_{i}\in I_{t}$ (with $i\neq j$) infects $P_{j}$ at rate $\rho\mu(X_{j}+\zeta_{j}(t)-X_{i}-\zeta_{i}(t))$. The aggregate rate at which $P_{j}$ becomes infected is

  (2.7)

  $$\sum_{i\in I_{t},\,i\neq j}\rho\mu\bigl{(}X_{j}+\zeta_{j}(t)-X_{i}-\zeta_{i}(t)\bigr{)}.$$

• ($\text{\rm I}\to\text{\rm R}$)

  An infected particle is removed at rate $\alpha$.

As in Section 2.2, we may allow $\rho=\infty$ and $\mu=1_{M}$ with $M$ compact. In either case $\rho<\infty$ or $\rho=\infty$ we write $\theta_{\text{\rm d}}(\lambda,\rho,\alpha)$ for the probability that infinitely many particles are infected.

We shall not investigate the formal construction of the above processes as strong Markov processes with right-continuous sample paths. The interested reader may refer to the related model involving random walks on ${\mathbb{Z}}^{d}$ (rather than diffusions or Brownian motions on ${\mathbb{R}}^{d}$) with $\alpha=0$, as considered in some depth by Kesten and Sidoravicius in [17] and developed for the process with ‘recuperation’ in their sequel [18].

Instead, we sketch briefly how such processes may be built around a triple $(\Pi,\boldsymbol{\zeta},\boldsymbol{\delta})$, where $\Pi$ is a rate-$\lambda$ Poisson process of initial positions, $\boldsymbol{\zeta}=(\zeta_{i}:i\in{\mathbb{Z}}_{0})$ is a family of independent copies of the diffusion $\zeta$, and $\boldsymbol{\delta}=(\delta_{i}:i\in{\mathbb{Z}}_{0})$ is a family of independent rate-$\alpha$ Poisson processes on $(0,\infty)$, that are independent of the pair $(\Pi,\boldsymbol{\zeta})$. We place particles $P_{i}$ at the points of $\Pi$, and $P_{i}$ deviates from its initial point according to $\zeta_{i}$. An initial infected particle $P_{0}$ is placed at the origin at time $0$, and it diffuses according to $\zeta_{0}$. The infection is communicated according to the appropriate rules (either delayed or not, and either (i) via the pair $(\rho,\mu)$ with $\rho<\infty$, or (ii) with $\rho=\infty$ and $\mu=1_{S}$). After infection, $P_{i}$ is removed at the next occurrence of the Poisson process $\delta_{i}$.

The above construction is straightforward so long as there exist, at any given time a.s., only finitely many simultaneous infections. In the two models with $\rho<\infty$, simultaneous infections can occur only after the earliest time $T_{\infty}$ at which there exist infinitely many infected particles. It is a consequence of Proposition 3.8(c) that ${\mathbb{P}}(T_{\infty}<\infty)=0$ for the general delayed diffusion model with $\rho<\infty$.

The issue is slightly more complex when $\rho=\infty$ and $\mu=1_{S}$, since infinitely many simultaneous infections may take place in the supercritical phase of the percolation process of moving disks. In this case, we assume invariably that $\lambda<\lambda_{\text{\rm c}}$, so that there is no percolation of disks at any fixed time, and indeed it was shown in [5] that, a.s., there is no percolation at all times. Therefore, there exist, a.s., only finitely many simultaneous infections at any given instant. Bounds for the growth of generation sizes are found at (4.4) and (4.21).

We consider the Brownian delayed diffusion model of Section 2.2, and we adopt the notation of that section. Recall the critical point $\lambda_{\text{\rm c}}$ of the Boolean continuum percolation on ${\mathbb{R}}^{d}$ in which a closed unit ball is placed at each point of a rate-$\lambda$ Poisson process. Let $\theta_{\text{\rm dd}}(\lambda,\rho,\alpha)$ be the probability that the process survives.

This theorem extends Theorem 1.1. Its proof is found in Sections 3.2–3.7.

The situation is different in one dimension, where it turns out that the Brownian model has no phase transition. The proof of the following theorem, in a version valid for the general delayed diffusion model, may be found in Section 3.3.

Theorems 3.1 and 3.2 are stated for the case of a single initial infective. The proofs are valid also with a finite number of initial infectives distributed at the points of some arbitrary subset $I_{0}$ of ${\mathbb{R}}^{d}$. By the proof of the forthcoming Proposition 3.4, the set $I$ of ultimately infected particles is stochastically increasing in $I_{0}$.

Consider the delayed diffusion model with $d\geq 2$. Suppose that either $\rho\in(0,\infty)$ with $\mu$ as in (2.2), or $\rho=\infty$ and

where $S$ is the closed unit ball with centre at the origin. It turns out that the set of infected particles may be considered as a type of percolation model on the random set $\Pi$. This observation will be useful in exploring the phases of the former model.

The proof of the main result of this section, Proposition 3.3, is motivated in part by work of Kuulasmaa [21] where a certain epidemic model was studied via a related percolation process (a similar argument is implicit in [1, p. 4]). Recall the initial placements $\Pi=(X_{0}=0,X_{1},X_{2},\dots)$ of particles $P_{i}$, with law denoted P (and corresponding expectation E); we condition on $\Pi$.

Fix $i\geq 0$, and consider the following infection process. The particle $P_{i}$ is the unique initially infected particle, and it diffuses according to $\zeta_{i}$ and has lifetime $T_{i}$. All other particles $P_{j}$, $j\neq i$, are kept stationary for all time at their respective locations $X_{j}$. As $P_{i}$ moves around ${\mathbb{R}}^{d}$, it infects other particles in the usual way; newly infected particles are permitted neither to move nor to infect others. Let $J_{i}$ be the (random) set of particles infected by $P_{i}$ in this process.

Let $\tau_{i,j}\in(0,\infty]$ be the time of the first infection by $P_{i}$ of $P_{j}$, assuming that $P_{i}$ is never removed. Write $i\to j$ if $\tau_{i,j}<T_{i}$, which is to say that this infection takes place before $P_{i}$ is removed. Thus,

Suppose first that $\rho<\infty$. Given $(\Pi,\zeta_{i},T_{i})$, the vector $\vec{\tau}_{i}=(\tau_{i,j}:j\neq i)$ contains conditionally independent random variables with respective distribution functions

and

When $\rho=\infty$, we have that

the first hitting time of $X_{j}-X_{i}$ by the radius-$1$ sausage of $\zeta_{i}$. As above, we write $i\to j$ if $\tau_{i,j}<T_{i}$, with $J_{i}$ and $\vec{\tau}_{i}$ given accordingly.

One may thus construct sets $J_{i}$ for all $i\geq 0$; given $\Pi$, the set $J_{i}$ depends only on $(\zeta_{i},T_{i})$, and therefore the $J_{i}$ are conditionally independent given $\Pi$. The sets $\{J_{i}:i\geq 0\}$ generate a directed graph $\vec{G}=\vec{G}_{\Pi}$ with vertex-set ${\mathbb{Z}}_{0}$ and directed edge-set $\vec{E}=\{[i,j\rangle:i\to j\}$. Write $\vec{I}$ for the set of vertices $k$ of $\vec{G}$ such that there exists a directed path of $\vec{G}$ from $0$ to $k$. To the edges of $\vec{G}$ we attach random labels, with edge $[i,j\rangle$ receiving the label $\tau_{i,j}$.

From the vector $(\vec{\tau}_{i},T_{i}:i\geq 0)$, we can construct a copy of the general delayed diffusion process by allowing an infection by $P_{i}$ of $P_{j}$ whenever $i\to j$ and in addition $P_{j}$ has not been infected previously by another particle. Let $I$ denote the set of ultimately infected particles in this coupled process.

We turn our attention to the Brownian case. By rescaling in space/time, we obtain the following. The full parameter-set of the process is $\{\lambda,\rho,\alpha,\mu,\sigma\}$, where $\sigma$ is the standard-deviation parameter of the Brownian motion, and we shall sometimes write $\theta_{\text{\rm dd}}(\lambda,\rho,\alpha,\mu,\sigma)$ accordingly.

It was stated in Theorem 3.2 that the Brownian model with $\mu=1_{S}$ never survives in one dimension. We state and prove a version of this ‘no survival’ theorem for the general delayed diffusion model of Section 2.2, subject to a weak condition which includes the Brownian model.

Throughout this section, $T_{\alpha}$ denotes a random variable having the exponential distribution with parameter $\alpha$, assumed to be independent of all other random variables involved in the models. A typical diffusion is denoted $\zeta$, and we write $M_{t}=\sup\{|\zeta(s)|:s\in[0,t]\}$.

It follows that. for all $\alpha$, there is no survival if

These two conditions (that ${\mathbb{E}}|\zeta(T_{\alpha})|<\infty$ and ${\mathbb{E}}(M_{T_{\alpha}})<\infty$) are equivalent when $\zeta$ is Brownian motion (see, for example, [12, Thm 13.4.6]), and indeed they hold for all $\alpha$ in the Brownian case. This implies Theorem 3.2. The proof of Theorem 3.6 has some similarity to that of [1, Thm 1.1].