\section{Introduction and setting}\label{sec_intro}
\input{Sec_Intro}

\section{Chase-escape on the dynamic graph}\label{sec_results_WKP}
\input{Sec_Results_WKP}


\section{Chase-escape on the connectivity graph}\label{sec_results_FPP}
\input{Sec_Results_FFP}



\section{Proofs}\label{sec_proofs}
\input{Sec_Proof_well_def}
\input{Sec_Proofs_Sup}
\input{Sec_Proofs_Sub}
\input{Sec_Proof_Exp}


\section*{Acknowledgements}
This work was funded by Orange Labs S.A.~under the project ID CRE K07151, the German Research Foundation under Germany's Excellence Strategy MATH+: The Berlin Mathematics Research Center, EXC-2046/1 project ID 390685689, and the German Leibniz Association via the Leibniz Competition 2020.
















\subsection{Street systems}\label{route_sec}
As a first layer of the system, we consider a {\em stationary random planar segment process} 
$$S=\bigcup_{i\in I} \{S_i\}\subset \mathbb R^2$$
where each $S_i$ is a line segment in $\mathbb R^2$ of finite length. The line segments may intersect and we call each maximal unintersected interval in $S$ a {\em street}. The endpoints of each street are called {\em crossings} (even if they are dead-ends). In order to have something specific in mind, $S$ can could be for example some planar tessellation process like the {\em Poisson--Voronoi tessellation (PVT)} or the {\em Poisson--Delaunay tessellation (PDT)}, see Figure~\ref{Fig_Streets} for an illustration. 
\begin{figure}[!htpb]
\centering
\input{Picture/pic1_Streetsystem_paper2}
\caption{Realization of a street system given by a Poisson--Voronoi tessellation.}
\label{Fig_Streets}
\end{figure}
We think of $S$ as to be an urban street system and note that for example the PVT indeed shares some common characteristics with medieval city topologies, see~\cite{courtat2012promenade}. 

In order to control spatial dependencies in $S$, we will almost exclusively work with systems that satisfy a quantitative mixing property called {\em stabilization}~\cite{stab1, stab2, stab3}. Additionally, we will assume that the system is connected in large regions with high probability and call this {\em asymptotic essential connectedness}~\cite{hirsch2019continuum,jahnel2022phase}. More precisely, let us 
define the square with side length $n\ge1$ centered at $x \in \mathbb R^2$ by
$$Q_n(x) = x + [-n/2, n/2]^2,$$ 
and abbreviate $Q_n = Q_n(o)$ and denote by $\mathrm{dist}(\varphi, \psi) = \inf\{|x-y|:\, x\in\varphi, y\in\psi\}$ the distance between sets $\varphi, \psi\subset\mathbb R^2$.
\begin{definition}[Stabilization and asymptotic essential connectedness]	\label{stabDef}
A stationary random segment process $S$ is called \emph{stabilizing} if there exists a random field of \textit{stabilization radii} $R = \{R_x\}_{x\in\mathbb R^2}$ that is measurable with respect to $S$ and $(S, R)$ is jointly stationary. Moreover, for $R(Q_n)=\sup_{y\in Q_n \cap\, \mathbb Q^2}R_y$ we have $\lim_{n\uparrow\infty}\P(R(Q_n) < n) = 1$ and for all $n \ge 1$, the random variables
 $$\Big\{f(S\cap Q_n(x))\mathbbmss{1}\{R(Q_n(x)) < n\}\Big\}_{x \in \varphi}$$
are independent for all non-negative bounded measurable functions $f$ and finite $\varphi \subset \mathbb R^2$ with $\mathrm{dist}(x, \varphi \setminus \{x\}) > 3n$ for all $x \in \varphi$.

Furthermore, $S$ is called {\em asymptotically essentially connected} if it is stabilizing and for all sufficiently large $n \ge 1$, whenever $R(Q_{2n})<n/2$, we have that $|S\cap Q_n|>0$ and $S\cap Q_n$ is contained in one of the connected components of $S\cap Q_{2n}$.
\end{definition}
We note that PVTs and PDTs are stabilizing~\cite{hirsch2019continuum} but Manhattan grids or (rectangular) Poisson-line tessellations are not stabilizing.

Finally, in what follows, we will always assume that the expected number of streets and the expected number of crossings in the unit square is finite and that the intensity of the street system is normalized, i.e., $\mathbb E[|S\cap Q|]=1$, where $|\cdot |$ denotes the total edge length.

\subsection{Mobile devices}\label{dev_sec}
As a second layer of our model we represent {\em initial positions of  devices} via a {\em Cox point processes} $X^\lambda = \{X_i\}_{i\ge1}$. That is, $X^\lambda$ is a Poisson point process with a random intensity measure
$$\Lambda_{S}(A) = \lambda |S\cap A|,\qquad\qquad A\subset \mathbb R^2 \text{ measurable},$$
where $\lambda\ge 0$ is a parameter that represents the expected number of devices per unit of street length, see Figure~\ref{Fig_Urban_Points}. 
\begin{figure}[!htpb]
\centering
\input{Picture/pic2_Streetsystem_and_user_paper2}
\caption{Realization of initial device positions (blue) confined to a street system given by a Poisson--Voronoi tessellation.}
\label{Fig_Urban_Points}
\end{figure}
Note that if $S$ is stationary so is $X^\lambda$.

Having defined the initial positions of the devices via a Cox point process on the street system, we now present our model for the device mobility. We start by considering the probability kernel
$$
\kappa^S(x, {\rm d} y),\qquad\qquad x\in S,
$$
which depend on $S$, and assume that the {\em support} of $\kappa^S(x,{\mathrm d} y)$, defined via
$${\mathrm{supp}}(\kappa^S(x,{\mathrm d} y))=\{y\in S\colon \kappa^S(x,B_{\varepsilon}(y))>0\text{ for all }{\varepsilon}>0\},$$
is contained in $S$ for any $x\in S$. We say that $\kappa$ has (uniformly) {\em bounded support} if there exists $K>0$ such
that $$|{\mathrm{supp}}(\kappa^S(x,{\mathrm d} y))|<K\qquad\qquad \text{almost all }S,$$
where, with a slight abuse of notation $|\cdot|$ denotes the Lebesgue measure on $\mathbb R^2$. 
Moreover, we will always assume that $\kappa$ is {\em translation covariant}, i.e.,
$$\kappa^S(x,A)=\kappa^{S-x}(o,A-x)\qquad\qquad \text{almost all }S, \text{ measurable }A\subset \mathbb R^2,\,   x\in \mathbb R^2.$$
The kernel $\kappa^S(x,{\mathrm d} y)$ serves as a {\em waypoint kernel}~\cite{bettHart} and an example is given by the uniform distribution on $S\cap B_L(x)$, see~\cite{HJCW22}, where $B_L(x)\subset\mathbb R^2$ denotes the disc of radius $L$ centered at $x\in \mathbb R^2$.
Furthermore, we require that the kernel is {\em finitely dependent}, i.e., that there is a constant $H>0$ such that for almost-all $S$ and $x\in S$, $\kappa^S(x,{\mathrm d} y)=\kappa^{S'}(x,{\mathrm d} y)$ for all $S'$ with $S\cap B_H(x)=S'\cap B_H(x)$.

It will become useful to consider waypoint kernels that allow for very small displacements. In order to formalize this, we call the kernel {\em $c$-well behaved} if
$$
B_c(x) \cap S \subset \text{supp}\big(\kappa^S(x, {\mathrm d} y)\big)\qquad\qquad\text{almost all }S,\, x\in S,
$$
where $\kappa^S(x,y)$ is the density 
and say that it is {\em well behaved} if it is $c$-well behaved for some $c > 0$.

Then, each device $X_i\in X^\lambda$ picks independently a {\em target location} $Y_i\in S$ according to $\kappa^S(X_i,{\mathrm d} y)$. Further, we equip each device $X_i\in X^\lambda$ with an i.i.d. velocity $V_i$, drawn from the distribution $\mu$ for which we assume that $0<v_{\rm min}\le v_{\rm max}<\infty$, where $v_{\rm min}:=\sup\{v\colon \mu([v,\infty))=1\}$ and $v_{\rm max}:=\inf\{v\colon \mu([0,v])=1\}$. Then, the path of the device $X_i$ is defined iteratively as follows. $X_i$ moves with speed $V_i$ to $Y_i$ along the shortest route in $S$ that connects $X_i$ and $Y_i$. If there are multiple shortest routes, the devices chooses one such route independently and uniformly at random. It then immediately returns to its starting position with the same velocity along the same shortest path. This procedure is then repeated indefinitely, see Figure~\ref{Fig_Mobil_Points} for an illustration.  
\begin{figure}[!htpb]
\centering
\input{Picture/pic3_Mobility_paper2.tex}
\caption{Realization of initial device positions (dotted blue) confined to a street system given by a Poisson--Voronoi tessellation and their respective positions at a fixed positive time (blue), with arrows indicating the corresponding displacement.}
\label{Fig_Mobil_Points}
\end{figure}

\subsection{Connectivity of mobile devices}\label{con_sec}
By a slight abuse of notation, we also denote by $X_i=(X_{i,t})_{t\ge 0}$ the trajectory of device $X_i$ in $S$. For any pair of devices $X_i$ and $X_j$ we then consider the set of {\em contact times}
$$
Z(X_i,X_j)=\{t\ge 0\colon |X_{i,t}-X_{j,t}|<r\text{ and } X_{i,t}, X_{j,t}\text{ are on the same street}\},
$$
as the times where the devices are on the same street and $r-$ close together, where $r>0$ is another parameter in the model, the {\em connectivity threshold}. 
Let us note that the constraint that contact times require the devices to be on the same street can be seen as a very strict {\em shadowing} assumption~\cite{BEEG21}. 

Next, we assume that any pair of devices $X_i$ and $X_j$ has an iid {\em infection time} $\rho(X_i,X_j)$, drawn from the distribution $\varrho$ on $(0,\infty)$. Then, we say that a {\em transmission from $X_i$ to $X_j$ is possible at time $t\ge 0$} if
$$[t-\rho(X_i,X_j),t]\subset Z(X_i,X_j),$$
see Figure~\ref{Fig5} for an illustration of the resulting space-time connectivity network.

\begin{figure}[!htpb]
\centering
\hspace{-7pt}
\includegraphics[scale = 0.53]{Picture_for_arXive/pix_connection_graph_Z/pic4_user_connection_paper2_0-1.png}
\includegraphics[scale = 0.53]{Picture_for_arXive/pix_connection_graph_Z/pic4_user_connection_paper2_1-1.png}
\includegraphics[scale = 0.53]{Picture_for_arXive/pix_connection_graph_Z/pic4_user_connection_paper2_2-1.png}

\vspace{2pt}
\includegraphics[scale = 0.53]{Picture_for_arXive/pix_connection_graph_Z/pic4_user_connection_paper2_3-1.png}
\includegraphics[scale = 0.53]{Picture_for_arXive/pix_connection_graph_Z/pic4_user_connection_paper2_4-1.png}
\includegraphics[scale = 0.53]{Picture_for_arXive/pix_connection_graph_Z/pic4_user_connection_paper2_5-1.png}
\caption{Realization of initial device positions (blue) on a street system given by a Poisson--Voronoi tessellation (red, only drawn in the initial picture, visually suppressed otherwise). Between two devices $X_i$ and $X_j$ an edge is drawn if $t\in Z(X_i,X_j)$ for $t=0,1,\dots,5$. We highlight that the total number of edges in this sequence of realizations stays roughly constant, however the edges become longer. This is due to the fact that we plot initial positions, but devices move and hence, over time, connect to more distant devices.
}
\label{Fig5}
\end{figure}

Let us note that the introduction of random transition times is a substantial generalization compared to the setup in~\cite{HJCW22}, where only the case $\varrho=\delta_\rho$ is considered for some fixed $\rho\ge0$. 

\subsection{Degree of devices}\label{well_sec}
In order to avoid scenarios in which a single device can transmit to infinitely many devices in the infinite time horizon that we consider, let us finally introduce a convenient condition for the {\em local connectedness} of the connectivity graph. For this, let $\ell_S(x,y)\subset \mathbb R^2$ be the shortest path between $x$ and $y$ on $S$ and write 
$$\ell_S(x)=\sup\{|\ell_S(x,y)|\colon y\in{\mathrm{supp}}(\kappa^S(x,{\mathrm d} y))\}+r/2$$
for the length of the shortest path starting in $x$ towards any reachable target on $S$ plus half the connectivity range. Then, consider the following {\em Cox--Boolean model with geostatistical markings} in which $X_i,X_j\in X^\lambda$ are connected whenever
$$
|X_i-X_j|\le \ell_S(X_i)+\ell_S(X_j)
$$
and note that this can be seen as a Boolean model in which the (random) discs, associated to each Cox point, depend on the underlying environment $S$. 
In particular, if $X_i$ can transmit to $X_j$, there also exists an (undirected) edge between $X_i$ and $X_j$ in the Cox--Boolean model with geostatistical markings. Using this, for $X_i\in X^\lambda$, we define its degree 
$$\deg(X_i)= \#\{X_j \in X^\lambda \setminus X_i \colon |X_i-X_j|\le \ell_S(X_i)+\ell_S(X_j)\},$$
which serves as an upper bound for the degree of $X_i$ in the original model.  
Now, we call the model {\em locally connected} if 
\begin{align}\label{locconn}
\P(\exists X_i\in X^\lambda\text{ such that }\deg (X_i)=\infty)=0
\end{align}
and, in the sequel, we will always assume that our system is locally connected. Let us also mention that, in order to ease notation, we use generic symbols $\P$ and $\mathbb E$ for the distribution and expectation of our model, even under changing parameters. 

The following proposition establishes a condition under which the network is locally connected. 
\begin{proposition}[Local connectedness]\label{locconn_def}
If $\kappa$ has bounded support, $S$ is exponentially stabilizing and $|S\cap Q_1|$ has exponential moments, then the network is locally connected.
\end{proposition}
The proof rests on a contradiction that follows from a first-moment method for the expected degree of a typical point using the good control on stabilization and moment properties of the street system. We present the details in Section~\ref{proof_locconn}. Let us note that the conditions are satisfied for our standard examples for street systems given by PVT and PDT, since they are exponentially stabilizing~\cite{hirsch2019continuum} and have exponential moments, see~\cite{JaTo19}. Also, as can be seen from the proof, the moment condition can be substantially relaxed.  

Having defined now our system of moving devices in an urban street system and their connectivity relations, in the following section, we introduce a {\em malware} into the network that propagates like an {\em infection} starting from a single typical device.   

\subsection{Regimes for absence of percolation}
Our first result verifies that indeed, insufficiently many devices and a too slow speed lead to absence of percolation. 
\begin{theorem}\label{thm_lowintense}
For any $\kappa$ and all $\gamma,r,\rho\ge 0$ the following holds:
\begin{enumerate}
\item[(1)] For all $\lambda\ge 0$ and $\mu$ we have that $T_{\mathrm c}(\kappa,\gamma,\lambda,\mu,\rho,r)>0$,
\item[(2)] For all $T\ge 0$ and $\mu$ we have that $\lambda_{\mathrm c}(\kappa,T,\gamma,\mu,\rho,r)>0$ and 
\item[(3)] for all $T,\lambda\ge 0$ we have that $v_{\mathrm c}(\kappa,T,\gamma,\lambda,\rho,r)>0$. 
\end{enumerate}
\end{theorem}
\begin{proof}
Each device can travel at most to a distance $Tv_{\rm max}$ from its starting position. Hence, any connected component of $g_{\kappa,T,\gamma,\mu,\rho,r}(X^\lambda)$ is contained in a connected component of the Gilbert graph $g_{2Tv_{\rm max}}(X^\lambda)$. In the Gilbert graph, any pair of points is connected if and only if their mutual distance is smaller than $2Tv$. But $g_{2Tv_{\rm max}}(X^\lambda)$ percolates with probability zero for all sufficiently small $T$, $v_{\rm max}$ or $\lambda$, see, for example~\cite{wias8}. 
\end{proof}

Let us mention that the comparison to the Gilbert graph, as described in the proof, allows us to derive first rough upper bounds for $\lambda_{\mathrm c}(\kappa,T,\gamma,\mu,\rho,r)$ and $v_{\mathrm c}(\kappa,T,\gamma,\lambda,\rho,r)$ using the numerical analysis provided in earlier reports, for example~\cite{CRE1} and the associated research papers~\cite{wias9,wias8}. 

\medskip
Next, note that $r_{\mathrm c}$ can very well be zero, since devices with non-parallel movement on a mutual edge share the same position exactly once for $r=0$. However, connections between devices that move together in the same direction can not exchange messages if $r=0$. For the consequences of this observation regarding the critical connectivity threshold, see Theorem~\ref{Thm_Rho_1} below. Let us note that, in principle, absence of percolation for $g_{\kappa,T,\gamma,\mu,\rho,0}(X^\lambda)$ could still imply $r_{\mathrm c}(\kappa,T,\gamma,\lambda,\mu,\rho)=0$. This would be the case if $g_{\kappa,T,\gamma,\mu,\rho,r}(X^\lambda)$ percolates for all positive $r$, which indicates that the parameters $(\kappa,T,\gamma,\lambda,\mu,\rho,0)$ are critical for percolation, rendering a mathematical analysis extremely hard.

\medskip
In order to see absence of percolation for large speeds $v_{\rm min}$, first note that any street $s$ such that $2|s|/v_{\rm min}<\rho$ can not be used to exchange messages. However, every street can be used for traveling. More precisely, in order to establish a connection, any device has to remain at least a time of $\rho$ on some street and during that time travels at least a distance of $\rho v_{\rm min}$. As devices reverse their direction when they reach either their destination or their home, the minimal length of a street such that communication on it is possible is given by $\rho v_{\rm min}/2$. Although staying on a street of length smaller than $\rho v_{\rm min}/2$ is possible if both the initial position and the destination are on that street, those devices can only connect to the same type of devices that have both their home and destination on the same street. In that way, only local connective components are formed, we will ignore this possibility in the following considerations.

We have the following result on absence of percolation for sufficiently high speeds. 
\begin{theorem}\label{thm_largespeeds}
We consider street systems of Poisson--Voronoi or Poisson--Delaunay type, which are in particular exponentially stabilizing and asymptotically essentially connected.  
Then, for any $\kappa$ with bounded support and all $T,\gamma,\lambda,r\ge 0$ and all $\rho>0$ we have that $v^{\mathrm c}(\kappa,T,\gamma,\lambda,\rho,r)<\infty$. 
\end{theorem}
\begin{proof}[Idea for the proof]
Note that, as $v_{\rm min}$ tends to infinity, any device $X_i\in S$ eventually reaches its target location $Y_i\in S$ before time $T$.
In particular, it is very unlikely to see a path from $X_i$ to any of its possible target locations that reaches very far away from $X_i$ in $\mathbb R^2$, and these shortest paths can be determined independently of the device process. Additionally, only devices that have a sufficiently long street on a shortest path to at least one of their possible target locations can contribute to the connectivity, and again this can be determined independently of the device process. However, as $v$ tends to infinity, existence of these long streets becomes also unlikely, leading to a thinning of the devices and eventually to a subcritical regime. 

Formally, we dominate our percolation model by a subcritical {\em geostatistical Cox--Boolean model}, see also~\cite{AT17,SRPD04}, where the (random) environment determines both, the random radii associated to the points, and a sufficiently strong thinning of the Cox points. 

We present the complete proof in Section~\ref{Sec_Proof_Sub}.
\end{proof}

We still have to deal with the connection-time parameter $\rho$. For this, note that we have a trivial bound for the connection time, given by $\rho_{\mathrm c}\le T/2$ since for $\rho> T/2$, in order to establish a connection on at least two different streets, the device has to stay on one street for at least $T/2$ time, leaving not enough time to form connections on any different street. Therefore, any device can only have connections to other devices that also spend the majority of their time on the same street.  As a consequence, each street forms its local connection cluster and communication between different clusters is impossible, which implies absence of percolation.
However, this trivial bound can be substantially improved, see below. Let us first make a statement on the absence of percolation for any positive $\rho$. 
\begin{theorem}\label{Thm_Rho_1}
For any $\kappa$ and all $T,\gamma,\lambda\ge 0$ we have that $\rho_{\mathrm c}(\kappa,T,\gamma,\lambda,\mu,0)=0$.
\end{theorem}
\begin{proof}
As indicated above, connections between devices that move together in the same direction on a street, can not exchange messages if $r=0$. Additionally, if $r=0$, devices in non-parallel movement can never have a contact time that includes intervals of positive length, and hence $g_{\kappa,T,\gamma,\mu,\rho,0}(X^\lambda)$ does not percolate with positive probability for any $\rho>0$. 
\end{proof} 

In order to provide potentially better upper bounds for the critical connection time $\rho_{\mathrm c}$ consider the following percolation model based on the street system. 
For all $S$ first denote by $S^{a}$ the thinned street system consisting of all streets in $S$ with length not smaller than $a\ge 0$. Second, we connect any pair of vertices that are endpoints of any of the long streets (with length not shorter than $a$) if and only if the length of the shortest path along the streets between them does not exceed $b\ge 0$. The resulting graph $S^{a,b}$ now consists of long streets and special edges between some of the endpoints of the long streets, see Figure~\ref{Fig_long_edge_percolation} for an illustration. 
\begin{figure}[h]
\centering
\input{Picture/pic6_long_edge_percolation_2.tex}
\caption{Realization of a street system given by a Poisson--Voronoi tessellation (red). Streets of length greater than $\rho v/2$ are drawn thick. Their endpoints (blue) are connected if their graph distance (indicated in dashed blue lines) is less than $v(T-2\rho)^+$. The blue vertices, combined with the thick red and thick blue edges form the percolation graph  $S^{v\rho/2,v(T-2\rho)^+}$.}
\label{Fig_long_edge_percolation}
\end{figure}
We say that $S^{a,b}$ percolates, if $S^{a,b}$ contains an infinitely-long component of connected streets and edges. Note that $\rho\mapsto \mathbbmss{1}\{S^{v_{\rm min}\rho/2,v_{\rm max}(T-2\rho)^+}\text{ percolates}\}$ is decreasing since we see less surviving streets and also less connections between endpoints of long streets. Here $(a)^+=\max\{a,0\}$. Hence, we can define a unique critical threshold via 
\begin{align*}
\rho_{\mathrm c}'(T,\gamma,\mu)&:=\sup\{\rho>0\colon \P(S^{v_{\rm min}\rho/2,v_{\rm max}(T-2\rho)^+}\text{ percolates})>0\}.
\end{align*}
In principle we could have that $\rho_{\mathrm c}'(T,\gamma,\mu)>T/2$ since $\P(S^{v_{\rm min}T/2,0}\text{ percolates})$ may be positive for sufficiently small $v_{\rm min}$, see Theorem~\ref{Thm_Rho_2} below for a related result. However, in this case, the reference model $S^{a,b}$ does not provide better bounds. We have the following theorem.
\begin{theorem}\label{Thm_Rho_1}
For any $\kappa$ and $\mu$ as well as all $T,\gamma,\lambda,r\ge 0$ we have that $\rho_{\mathrm c}(\kappa,T,\gamma,\lambda,\mu,r)\le \min\{T/2,\rho_{\mathrm c}'(T,\gamma,\mu)\}$.
\end{theorem}
Let us mention that $v\mapsto \mathbbmss{1}\{S^{v\rho/2,v(T-2\rho)^+}\text{ percolates}\}$ is not monotone since, for example, large velocities lead to more streets being eliminated, while also leading to the possibility to travel larger distances between remaining long streets.
Even for $T$ being much larger than $2\rho$, it is this interplay between the possibility to connect on sufficiently long streets and the ability to bridge between long streets (using all streets), that makes this percolation picture interesting. 
\begin{proof}
The central idea is that, even in a best-case scenario, where the device intensity is extremely large and the vicinity threshold is very small, in order for devices to form connections on at least two different streets, we must have that $2\rho+b/v\le T$, where $b$ is the graph distance between two endpoints of streets that are sufficiently long, and $v$ is a speed in the support of the speed distribution $\mu$. On the other hand, if every device can form a connection only on one single street, then the connected components in the original graph can only consist of either individual devices (that are unable to connect to any other device) or precisely those (finitely-many) devices associated to the street on which they form their connection. In particular, the initial position of a device does not have to lie on the street on which it establishes its connection. Hence, if $S^{v_{\rm min}\rho/2,v_{\rm max}(T-2\rho)^+}$ does not percolate, then also our system cannot percolate.
\end{proof} 

In order to complement Theorem~\ref{Thm_Rho_1}, as a final result for this section on absence of percolation, let us isolate a regime of fast speeds such that $\rho_{\mathrm c}'(T,\gamma,\delta_v)<T/2$. Note that this is in general a nontrivial task since $v\mapsto \mathbbmss{1}\{S^{v\rho/2,v(T-2\rho)^+}\text{ percolates}\}$ is not monotone. Let us denote by 
\begin{align*}
a_{\rm c}(b,\gamma)&:=\sup\{a>0\colon \P(S^{a,b}\text{ percolates})>0\}
\end{align*}
the largest street-thinning parameter such that we see percolation with positive probability in the presence of additional edges between endpoints of long streets. 
We further write $a_{\rm c}(\gamma):=\lim_{b\downarrow0}a_{\rm c}(b,\gamma)$, where we note that the limit is in fact an infimum since $b\mapsto a_{\rm c}(b,\gamma)$ is decreasing. Then we have the following result. 
\begin{theorem}\label{Thm_Rho_2}
For any $\gamma>0$ we have $a_{\rm c}(\gamma)<\infty$. Further, for any $\kappa$, $T,\gamma,\lambda,r\ge 0$ and $v>4 a_{\rm c}(\gamma)/T$  it holds that $\rho_{\mathrm c}'(T,\gamma,\delta_v)< T/2$.
\end{theorem}
In particular, the conditions presented in Theorem~\ref{Thm_Rho_2} imply that $\rho_{\mathrm c}(\kappa,T,\gamma,\lambda,\delta_v,r)\le \rho_{\mathrm c}'(T,\gamma,\delta_v)$ and analyzing absence of percolation for the auxiliary model  $S^{a,b}$ indeed provides better bounds. 
\begin{proof}
The proof for the finiteness of $a_{\rm c}(\gamma)$ rests on stabilization arguments and will be presented in Section~\ref{Sec_Proof_Sub_2}. 
Under the assumption that $a_{\rm c}(\gamma)<\infty$, note that, since $v>4 a_{\rm c}(\gamma)/T$, there exist $\varepsilon,b>0$ such that $v(T/2-\varepsilon)/2>a_{\rm c}(b,\gamma)$. In particular, we have that $\P(S^{v(T/2-\varepsilon)/2,b'}\text{ percolates})=0$ for all $b'<b$. 
Hence, for $\rho>T/2-\varepsilon$ and $\rho>T/2-b/(2v)$ we have also that $0=\P(S^{v\rho/2,v(T-2\rho)}\text{ percolates})$. 
Thus, $\rho_{\mathrm c}'(T,\gamma,\delta_v)< T/2$, which finishes this part of the proof. 
\end{proof} 
\subsection{Regimes for percolation}
In order to see percolation we have a variety of parameters for possible adjustments. In view of Theorem~\ref{thm_lowintense}, note that for too small device intensity or speed, we can never see percolation. Hence, we assume the speed to be positive, i.e., $v_{\rm min}>0$ and naturally also the time horizon to be positive. In order to avoid complications coming from streets that are too short to support communications, in the first step, let us suppose that $\rho=0$ and prove existence of percolation for large device intensities. Let us present a condition for the waypoint kernel that relaxes isotropy around the starting position. 
We say that a kernel $\kappa$ is {\em $c$-well behaved} if for almost-all $S$ and all $x\in S$, if
\begin{align*}
B_c(x) \cap S \subset \text{supp}(\kappa^S(x, {\mathrm d} y)).
\end{align*}
Further, we say that $\kappa$ is {\em well behaved} if $\kappa$ is $c$-well behaved for some $c > 0$.
Let us note that both $\kappa'$ and $\kappa''$ are $R$-well behaved. 
Here is our first result for the immediate connectivity $\rho=0$. 
\begin{theorem}\label{thm_largeintense}
Let $\kappa$ be well behaved and translation-covariant. Then, for all $v_{\rm min},T,\gamma>0$ and $r\ge 0$ we have that $\lambda_{\mathrm c}(\kappa,T,\gamma,\mu,0,r)<\infty$. 
\end{theorem}
Let us mention that this statement is in fact true under the assumption that $\mu\neq\delta_0$, as can be seen from the proof. 

\begin{proof}[Idea for the proof]
In order to prove this theorem we will construct a coupling of the original process to an auxiliary process of open streets such that existence of an unbounded component of open streets implies percolation of the original model. Here, we say that a street is \textit{open} if devices on that street can connect to devices on all its neighboring streets and also inter-connect on that street. In particular, any infinite connected component of open streets contains an infinite connected component of devices. 
However, as $\lambda$ increases, the probability for a street to be open converges to one, resulting in a percolation regime for open streets. 
We present the complete proof in Section~\ref{Sec_Proof_Sup}. 
\end{proof}

We have now proved the existence of a percolation regime for $\rho = 0$, if $\lambda$ is sufficiently large. In order to move beyond this setting, note that for $\rho>0$, very short streets $s$ with $|s|<\rho v_{\rm min}/2$ can never be used to establish a connection, however, they can be used for traveling. 
As a consequence, for the case where $\rho > 0$ we need stronger assumptions on $\mu$ and $r$ in order to guarantee that $\lambda_{\mathrm c}(\kappa,T,\gamma,\mu,\rho,r)<\infty$.

Again, we want to compare our original model to a simpler model on the street system, which we describe now. For $a>0$ and $p \in (0,1]$ we denote by $S^{a,L,p}$ a street system generated by a two-step thinning procedure performed on $S$. That is, we eliminate each street in $S$ that has length less than $a$ or length larger than $L$. Afterwards, each remaining street is eliminated independently with probability $p$. 
We say that $S^{a,L,p}$ percolates, if it contains an infinite connected component of streets. Let us define a critical thinning parameter,
$$
a_{\rm c}(L,p,\gamma):=\sup\{a>0\colon \P(S^{a,L,p}\text{ percolates})>0\},
$$
and note that $L\mapsto a_{\rm c}(L,p,\gamma)$ is increasing where as $p\mapsto a_{\rm c}(L,p,\gamma)$ is decreasing. Hence, we can define $a_{\rm c}(\gamma):=\lim_{L\uparrow\infty,\, p\downarrow 0}a_{\rm c}(L,p,\gamma)$. In principle, $a_{\rm c}(\gamma)$ could very well be zero, however, in Theorem~\ref{thm_twicethinstreet} below, we show that indeed, $a_{\rm c}(\gamma)>0$ for all $\gamma>0$. 

With these definitions, we are in the position to state the following result.
\begin{theorem}\label{thm-percolation-rho-positive}
Let $\kappa$ be translation-covariant and $c$-well behaved for some $c>2\rho v_{\rm min}$. Further let $v_{\rm min},\gamma,\rho>0$, $r>\rho v_{\rm min}$ and $T>2\rho$. Then, if $2\rho v_{\rm min}<a_{\rm c}(\gamma)$, then $\lambda_{\mathrm c}(\kappa,T,\gamma,\mu,\rho,r)<\infty$.
\end{theorem}
Again, let us mention that the statement of the theorem is in fact true under the assumption that $\mu\neq\delta_0$, as can be seen from the proof. Let us further mention that in fact, in the proof, we derive an upper bound for $\lambda_{\mathrm c}$ given by some $\lambda_{\mathrm c}'(\kappa,\gamma,\mu,\rho,r)<\infty$ that is independent of $T$, as long as $T>2\rho$. This in particular implies that for $\lambda>\lambda_{\mathrm c}'(\kappa,\gamma,\mu,\rho,r)$  we have that $T_{\mathrm c}(\kappa,\gamma,\mu,\lambda,\rho,r)\le 2\rho<\infty$. 

\begin{proof}[Idea for the proof]
Similar to the proof of Theorem~\ref{thm_largeintense}, the main argument is a comparison of our model to a street system in which we thin out very short streets, very large streets and all individual streets independently. The thinning of short streets becomes necessary, since for positive $\rho$, short streets can never be used to establish connections. The independent thinning comes from the fact that, even for large device intensities, a street has good self-contained connectivity properties only with high probability. In order to get uniform control on this probability, we have to disregard very long streets. We present the complete proof in Section~\ref{Sec_Proof_Sup_2}. 
\end{proof}


The previous theorem heavily rests on the assumption that $S^{a, L, p}$ percolates. In the following theorem we will show that, indeed, there are regimes where it percolates. 
\begin{theorem}\label{thm_twicethinstreet}
For all asymptotically-essentially connected street system we have that $a_{\rm c}(\gamma)>0$. 
\end{theorem}
\begin{proof}[Idea for the proof]
The proof rests on a comparison with a finitely-dependent site percolation process on $\mathbb Z^2$, leveraging stabilization and asymptotic-essential-connectedness arguments. We present the complete proof in Section~\ref{Sec_Proof_Sup_3}. 
\end{proof}
\subsection{Proof of Proposition~\ref{Prop_FPP_fixed_T}}\label{sec_Prop4-1}
\begin{proof}[Proof of Proposition~\ref{Prop_FPP_fixed_T}]
First note that by the assumption of supercriticality there is a positive probability that the typical device is connected to infinity within $g_{\rho,r}(S,X^\lambda)$ via at least one path of susceptible devices, i.e., in the absence of white knights. However, in case that $T_{\rm I}\le T_{\rm W}$, any white knight that is connected to the path can only eliminate the infection after the infected device has already passed the infection towards the subsequent susceptible device, leading to an unstoppable sequence of infections along the path, independent of the choice of ${\lambda_{\rm W}}$.

On the other hand, in case $T_{\rm I}>T_{\rm W}$, it suffices to consider the model conditioned on the event $\{X_o\leftrightsquigarrow\infty\}$ that $X_o$ is in the infinite cluster of $g_{\rho,r}(S,X^\lambda)$. 
Let us denote by $D$ the graph distance of $X_o$ to the set of white knight and note that $D<\infty$ almost surely as ${\lambda_{\rm W}}>0$. 
Then, closest white knight $Y_i$ experiences an infection attempt at time $DT_{\rm I}$. Note that, at this time, any infected device has a distance to $Y_i$ given by at most $2D$. This comes from the fact that the transmission times are deterministic and there is a one-to-one correspondence between graph distance to the typical device and elapsed time. However, there exists $N\in \mathbb N$ such that $NT_{\rm I}>(N+2D)T_{\rm W}$ and therefore the infection can reach at most a graph distance of $N+D$. Indeed, any device $X_i$ at graph distance $N+D$ from $X_o$, is in contact with $Y_i$ via $N+2D$ edges connecting infected or patched devices. But, since the patching is faster than the infection, for this $N$, on the joint path between $X_o$, $Y_i$ and $X_i$, the patch will have caught up with the infection before time $(N+D)T_{\rm I}$ which is strictly smaller than $NT_{\rm I}$.
\end{proof}

\subsection{Proof of Theorem~\ref{Thm_Exp_Sur} }\label{proofThm_Exp_Sur}

\begin{proof}[Proof of Theorem~\ref{Thm_Exp_Sur}]
As ${\varrho_{\rm W,min}} > 0$, we can choose $b$ large enough such that ${\varrho^b_{\rm I,min}}<{\varrho_{\rm W,min}}$.
Furthermore, we can choose $\mu'$ with $v'_{\mathrm{min}}$ such that
$v_{\mathrm{min}}\rho<v'_{\mathrm{min}}{\varrho^b_{\rm I,min}}<\min(a_{\rm c}/2,r,c/2)$. 
Now $v'_{\mathrm{min}}$ and ${\varrho^b_{\rm I,min}}$ satisfy the Conditions of Theorem~\ref{Thm_Survival}.
A close inspection of the proof of 
Theorem~\ref{Thm_Survival} reveals that, under the conditions in this theorem, there exists an infinite sequence of crossing devices, boundary devices and devices connecting boundary devices $\{X_{n_i}\}_{i\in \mathbb N}$, connecting the typical device to infinity, in the dynamical model considered in Theorem~\ref{Thm_Survival}.
Let us argue that this given sequence is also connected in $g_{\rho,r}(S, X^\lambda)$. Indeed, as the chain of devices between boundary devices is strongly localized, they are connected for all $t\ge 0$, especially for a continuous time-window of length $\rho$.
Now, the crossing devices are localized in such a way, that they travel at least a distance of $v'_{\mathrm{min}}{\varrho^b_{\rm I,min}}$ on their respective streets. As $v_{\mathrm{min}}\rho<v'_{\mathrm{min}}{\varrho^b_{\rm I,min}}$ the bridging devices can connect to their respective boundary devices. Therefore, the infection can propagate in $g_{\rho,r}(S, X^\lambda)$ along the given sequence to infinity, which finishes the proof.
\end{proof}

\subsection{Proof of Theorem~\ref{Thm_Exp_Sur_2} }\label{proofThm_Exp_Sur_2}
\begin{proof}[Proof of Theorem~\ref{Thm_Exp_Sur_2}]
Using similar arguments as in the analysis of chase-escape dynamics on the Gilbert graph in~\cite{hinsen2020phase}, we call a device $X_i\in V^{M,p}$ {\em good} if, in case of an infection, the infection is transmitted to all neighbors of $X_i$ before any neighbor of $X_i$ is able to patch it. If there is an infinite path of {\em good} devices, every device on this path is guaranteed to become infected and hence, if there exists an infinite component of good devices with positive probability, this implies survival of the infection also with positive probability.
Now, if a device $X_i$ has at most $M$ neighbors, the probability to be good is bounded by
\begin{align}\nonumber
\P\Big(\min_{X_j \in N(X_i)} {\rho_{\rm W}}(X_j,X_i) > \max_{X_j \in X^\lambda \cap N(X_i)} {\rho_{\rm I}}(X_i,X_j)\Big) &\ge \P\Big(\min_{i \in \{1,\dots,M\}} \rho_{{\rm W},i} > \max_{i \in \{1,\dots, M\}} \rho_{{\rm I},i}^b\Big)
\\\label{bound_1}
&=\P\Big(\min_{i \in \{1,\dots,M\}} \rho_{{\rm W},i} > b^{-1}\max_{i \in \{1,\dots, M\}} \rho_{{\rm I},i}\Big),
\end{align}
independently of $X_i$, where the $\rho_{{\rm I},i}^b$, and respectively the $\rho_{{\rm W},i}$, are iid random variables with distribution ${\varrho^b_{\rm I}}$ and $\varrho_{\rm W}$.

Now, by assumption, we can choose $M$ and $p$ such that  $g^{M,p}_{\rho,r}(S,X^\lambda)$ percolates with positive probability.
Hence, as ${\varrho_{\rm W}}$ had no atom at zero
we can choose $b=b(M)$ sufficiently large such that the bound~\eqref{bound_1} is larger than $p$. 

Note that the bound is uniform over the devices, since the infection and patch times are independent by definition, and therefore we can couple the connectivity graph to an independent thinning with parameter $p$ that dominates the process of good devices. 
As a consequence, $g^{M,p}_{\rho,r}(S,X^\lambda)$ is a subgraph of the thinned graph of good devices with at most $M$ neighbors. Hence, we have established percolation of good devices with positive probability as desired. 
\end{proof}


\subsection{Proof of Theorem~\ref{Thm_Exp_Ext} }\label{ProofThm_Exp_Ext}
Our strategy is again to employ the arguments from Section~\ref{Proof_Thm_Extinction} and Section~\ref{Proof_thm_InOut}. That is, we realize that it suffices to show absence of percolation in the geostatistical Cox--Boolean model
$$\mathcal{C}=\bigcup_{X_i\in X^{\lambda,{\rm th}}}B_{\ell_S(X_i)}(X_i),$$
where, 
$$X^{\lambda,{\rm th}}=\{X_i\in X^\lambda\colon \min_{X_j \in N_\lambda(X_i)} {\rho_{\rm I}}(X_i,X_j) < \min_{X_j \in N_{\lambda_{\rm W}}(X_i)} {\rho_{\rm W}}(X_j,X_i)\},$$
with $N_\lambda(X_i)$ the set of neighbors of $X_i$ in $g_{\rho,r}(S,X^\lambda)$ and $N_{{\lambda_{\rm W}}}(X_i)$ the set of neighbors of $X_i$ in $g_{\rho,r}(S,X_i\cup Y^{\lambda_{\rm W}})$. Indeed, devices in $X^\lambda\setminus X^{\lambda,{\rm th}}$ are patched before they can transmit the infection and can therefore not contribute to the dissipation of the infection.

Again, the thinning procedure depends on the entire point cloud, but, we can still employ Lemma~\ref{Lem_A} and Lemma~\ref{Lem_F} from above and only need to replace Lemma~\ref{Lem_C} by the following statement, which is presented in terms of $\P^*$, the Palm version of $\P$.
\begin{lemma}\label{Lem_C3}
There exists $c>0$ such that 
$$\P\big(G(o,\alpha)\big)\le c \alpha^2\P^*\big(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) < \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\big)$$ 
and 
$$\limsup_{{\lambda_{\rm W}}\uparrow\infty}\P^*\big(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) < \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\big)=0.$$ 
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{Lem_C3}]
Again, by the Slivnyak--Mecke formula and translation invariance we can estimate
\begin{align*}
\P\big(G(o,\alpha)\big)&\le \P\big(X^{\lambda,{\rm th}}(B_{10\alpha})>0\big)\le \mathbb E\big[X^{\lambda,{\rm th}}(B_{10\alpha})\big]\\
&\le \lambda 10^2\alpha^2\P^*\big(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) < \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\big).
\end{align*}
Now, 
\begin{align}\label{Est_1}
\P^*\big(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) < \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\big)\le \mathbb E^*\big[\P({\rho_{\rm W}}>{\varrho_{\rm I,min}})^{N_{\lambda_{\rm W}}(o)}\big],
\end{align}
where ${\rho_{\rm W}}$ is a random variable distributed according to ${\varrho_{\rm W}}$. Note that $\P({\rho_{\rm W}}>{\varrho_{\rm I,min}})<1$ by our assumptions on ${\varrho_{\rm W,min}}$. Further, the expression on the right-hand side of~\eqref{Est_1} can be further bounded from above by considering only those white knights that mimic the behavior of the typical device at the origin in the sense that they start close to the origin and have a target location close to the target location of the typical device. Then, by dominated convergence, the right-hand side of~\eqref{Est_1} tends to zero as ${\lambda_{\rm W}}$ tends to infinity and this finishes the proof.
\end{proof}
As now the Lemma is proven this concludes the proof of Theorem~\ref{Thm_Exp_Ext}.

\begin{proof}[Proof of Theorem~\ref{thm_InOut} Part 1]
For the proof we can follow the same line of arguments as used for the proof of Theorem~\ref{Thm_Extinction} above, only replacing Lemma~\ref{Lem_C} by Lemma~\ref{Lem_C2} and subsequently adjusting the arguments. We need to prove existence of ${\lambda_{\rm W,c}}<\infty$ such that for all ${\lambda_{\rm W}}>{\lambda_{\rm W,c}}$ there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that $\lim_{\alpha\uparrow\infty}\P(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb R^2))> X^{\lambda,{\rm th}}(B_{8\alpha}))=0$  for all $v<v_{c}(\lambda,{\lambda_{\rm W}})$. Referencing to~\cite{HJCW22}[Lemma III.2, Lemma III.4 and III.5] as well as to Lemma~\ref{Lem_A} and the initial argument presented in Theorem~\ref{Thm_Extinction} it suffices to note that, with the help of Lemma~\ref{Lem_C2}, there exists ${\lambda_{\rm W,c}}<\infty$ such that for all ${\lambda_{\rm W}}>{\lambda_{\rm W,c}}$ there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that for all $v<v_{c}(\lambda,{\lambda_{\rm W}})$ 
$$
\P^*\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^S(o,{\mathrm d} y))\text{ and } s\in\ell_S(o,y)\text{ such that $s$ is open}\big)<\frac 1 2 (100c\alpha_{\rm c})^{-2},
$$
where $\alpha_{\rm c}$ is defined as in the proof of Theorem~\ref{Thm_Extinction}. Then, using the same definitions for $f$ and $g$, the result follows by an application of Lemma~\ref{Lem_F}. 
\end{proof}
\subsection{Proof Proposition~\ref{locconn_def}}\label{proof_locconn}
\begin{proof}[Proof Proposition~\ref{locconn_def}]
First, let us assume that $\P(\exists X_i\in X^\lambda\text{ such that }\deg (X_i)=\infty)>0$. Then, by continuity of measures, there exists $R>0$ such that $\P\big(\exists X_i\in X^\lambda\cap Q_{R}\text{ such that}\deg (X_i)=\infty\big)>0$ and hence, by translation invariance, also $\P\big(\exists X_i\in X^\lambda\cap Q_1\text{ such that}\deg (X_i)=\infty\big)>0$. But the last inequality implies that $\mathbb E[\sum_{X_i \in X^\lambda \cap Q_1}\deg(X_i)]= \infty$. However, in order to derive a contradiction, we can use the Mecke–-Slivnyak Theorem twice to see that
\begin{align*}
    \mathbb E\Big[\sum_{X_i \in X^\lambda \cap Q_1}\deg(X_i)\Big] 
    &\le \mathbb E\Big[\sum_{X_i \in X^\lambda \cap Q_1} \sum_{X_j \in X^\lambda \setminus X_i}\mathbbmss{1}\{|X_i-X_j| \le \ell_S(X_i) + \ell_S(X_j)\}\Big]
    \\
    &=\lambda^2\mathbb E\Big[\int_{S\cap Q_1} {\mathrm d} x\int_{S }{\mathrm d} y \mathbbmss{1}\{|x-y| \le \ell_S(x) + \ell_S(y)\}\Big].
\end{align*}
Next, we bound the maximal reach $\ell_S$ by the street length in a stabilized region. For this, let 
$$K:= \sup\{|x-y|\colon x\in S, y\in{\mathrm{supp}}(\kappa^S(x,{\mathrm d} y))\}<\infty$$ denote the maximal reach of the kernel $\kappa$, which exists due to the assumption of a uniformly bounded support of $\kappa$. Using this, we can bound the length of any shortest path starting in $x\in S$ to any $y \in S\cap B_K(x)$, by the total street length of the box in which there is a connection between $x$ and $y$ due to asymptotic-essential connectedness. More precisely, note that if  $R(Q_{2n}(x)) < n/2$, then 
\begin{align*}
    \ell_S(x) \le |S\cap Q_{2n}(x)|+r/2,
\end{align*}
for any $n>2K$, and we can define
\begin{align*}
    N(x) := \min_{n\in \mathbb N, n>K} R(Q_{2n+1}(x))<n/2.
\end{align*}
In particular, for all $y \in Q_1(z)$ we have $\ell_S(y) \le |S\cap Q_{2N(z)+1}(z)|+r/2$ and hence
\begin{align*}
\mathbb E&\Big[\int_{S\cap Q_1} {\mathrm d} x\int_{S }{\mathrm d} y \mathbbmss{1}\{|x-y| \le \ell_S(x) + \ell_S(y)\}\Big]\\
&\le\mathbb E\Big[\sum_{n > K} \mathbbmss{1}\{N(o) = n\} |S\cap Q_1| \int_S {\mathrm d} y \mathbbmss{1} \{|y| \le \ell_S(y) + 1/2+r/2 + |S\cap Q_{2n+1}|\} \Big]\\
&\le\mathbb E\Big[\sum_{n > K}\sum_{m > K}\sum_{z\in \mathbb Z^2} \mathbbmss{1}\{N(o) = n\}\mathbbmss{1}\{N(z) = m\} |S\cap Q_1| \int_{S\cap Q_1(z)} {\mathrm d} y \mathbbmss{1} \{|y| \le \ell(y) + 1/2+r/2 + |S\cap Q_{2n+1}|\} \Big]\\
&\le\sum_{n > K}\sum_{m > K}\sum_{z\in \mathbb Z^2}\mathbb E\Big[ \mathbbmss{1}\{ N(o) = n\}\mathbbmss{1}\{N(z) = m\} |S\cap Q_1| |S\cap Q_1(z)| \mathbbmss{1} \{|z| \le |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\} \Big],
\end{align*}
where we used $|y|\le|z|+ 1/2$ for $y \in Q_1(z)$ in order to eliminate the dependence of the integrals on $x$ and $y$ respectively and thus derive a uniform bound for the integrals.
It remains to show that this bound is integrable.
By H\"older's inequality we can bound each summand by
\begin{align*}
    \P\big(N(o) = n\big)^{1/5}&\P\big(N(o) = m\big)^{1/5} \mathbb E\big[|S\cap Q_1|^5\big]^{2/5} \P\big(|z| \le |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\big)^{1/5}\\
    &\le C_1\exp\big(-c_1(n+m)\big)\P\big(|z| \le |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\big)^{1/5}
\end{align*}
for some constants $C_1,c_1>0$ due to the existence of exponential moments for $|S\cap Q_1|$ as well as the assumption of exponential stabilization. 
Now, the last factor can be further bounded by Markov's inequality as
\begin{align*}
    \P\big(|z| \le   |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\big)^{1/5}
    &\le |z|^{-3}\mathbb E\big[\big(|S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\big)^{15}\big]^{1/5}
    \\
    &\le |z|^{-3}\big(\mathbb E\big[\big(3|S\cap Q_{2m+1}|\big)^{15}] + (3+3r)^{15} + \mathbb E[(3|S\cap Q_{2n+1}|\big)^{15}]\big)^{1/5}
    \\
    &\le |z|^{-3}\big((3(2m+1)^2+3(2n+1)^2)^{15}\mathbb E\big[\big(|S\cap Q_{1}|\big)^{15}] + (3+3r)^{15}\big)^{1/5}
    \\
    &\le C_2 (mn)^{c_2}|z|^{-3},
\end{align*}
for some constants $C_2,c_2>0$. Finally, we can use $|z|>|z|_{\infty}$, where $|\cdot|_\infty$ denotes the $\ell_\infty$-norm, combined with the fact that for $k\in \mathbb N_0$ we have $\sum_{z\in Z}\mathbbmss{1}\{|z|_{\infty}=k\} \le 4k+1$, to see that
\begin{align*}
\sum_{n > K}\sum_{m > K}\sum_{k\in \mathbb N_0}(4k+1)C_2(mn)^{c_2}C_1\exp\big(-c_1(n+m)\big)k^{-3}<\infty,
\end{align*}
and hence the proof is finished. 
\end{proof}
\subsection{Proof of Theorem~\ref{Thm_Extinction} }\label{Proof_Thm_Extinction}
For the proof we adopt a multiscale argument in the spirit of the proof of existence of subcritical regimes for the Poisson--Boolean model with random radii in~\cite{G08}. We note that this approach has been also applied in the setting of Cox--Boolean models with random radii in~\cite{jahnel2022phase}. More specifically we adapt the arguments for absence of percolation in the connectivity graph for sufficiently large velocities presented in~\cite{HJCW22}. Roughly speaking, we prove that a Cox--Boolean model in which any two devices, that could transmit the infection somewhere on their path and are sufficiently close, are connected, becomes subcritical if the white-knight intensity is sufficiently large. 

First, we call a street $s=(c_1,c_2)$ {\em blocked for (device) $x$} if either $|s|<v_{\rm min}{\varrho_{\rm I,min}}/2$ or if there exist two white knights $Y_1,Y_2\in Y^{{\lambda_{\rm W}}}$ such that for $i\in \{1,2\}$ we have that ${\varrho_{\rm I,min}}>{\rho_{\rm W}}(Y_i,x)$, $Y_{i,t}\in s$ and $|c_i-Y_{i,t}|<r$ for all $t\ge 0$.
Otherwise we call $s$ {\em unblocked for $x$}.
Then, we consider the thinned process 
$$X^{\lambda,{\rm th}}=\{X_i\in X^\lambda\colon \text{there exists }y\in{\mathrm{supp}}(\kappa^S(X_i,{\mathrm d} y))\text{ and } s\in\ell_S(X_i,y)\text{ such that $s$ is unblocked for $X_i$}\}.$$
In words, for devices in $X^\lambda\setminus X^{\lambda,{\rm th}}$ every possible path only contains streets that are either to short to perform any transmission of the infection or that contain a white knight that patches the device before it can transmit the infection on the street. Let us note that here we use that $r>v_{\rm max}{\varrho_{\rm W,min}}$ is large enough to allow a successful patch.  Using these definitions, we note that, if the infection survives globally, then this implies that the geostatistical Cox--Boolean model
$$\mathcal{C}=\bigcup_{X_i\in X^{\lambda,{\rm th}}}B_{\ell_S(X_i)}(X_i),$$
contains an unbounded component. Hence, in order to prove absence of global survival, it suffices to prove absence of percolation in this Cox--Boolean model, which is also well defined by the local-connectedness assumption.
Let $\mathcal{C}_x(V)$ denote the connected component containing $x$ of the geostatistical Cox--Boolean model, based on points in $V\subset\mathbb R^2$. Then, we define for any $x\in \mathbb R^2$ and $\alpha>0$ the event
\begin{align*}
G(x,\alpha)&=\big\{\mathcal{C}_x(B_{10\alpha}(x))\not\subset B_{8\alpha}(x)\big\}
\end{align*}
that the cluster of $x$, only using points in $B_{10\alpha}(x)$, reaches beyond $B_{8\alpha}(x)$. Consider the set of $\alpha$-local points
\begin{align*}
A_S(\alpha)=\{x\in S\colon \ell_S(x)<\alpha \},
\end{align*}
let $S^*$ denote the Palm version of $S$ and recall the stabilization radii $R_y$ of the street system. Recall that $R(V)=\sup_{y\in V\cap\, \mathbb Q^2}R_y$ for any $V\subset\mathbb R^2$. Then, we can employ the following key lemma that establishes a scaling relation in the model.
\begin{lemma}[{\cite[Lemma III.1]{HJCW22}}]\label{Lem_A}
Consider the geostatistical Cox--Boolean model. Then, there exists a constant $c>0$ such that for all $\alpha>0$, we have 
\begin{align*}
\P\big(G(o,10\alpha)\big)&\le c\P \big(G(o,\alpha)\big)^2+c\alpha^2\P\big(o\in A^{\rm c}_{S^*}(\alpha)\big)+c\P\big(R(Q_{10\alpha})\ge \alpha\big)
\end{align*}
\end{lemma}
In the next lemma we deviate from~\cite[Proof of Theorem II.3]{HJCW22}. We show that the local percolation probability becomes zero for large white-knight intensities. 
\begin{lemma}\label{Lem_C}
There exists $c>0$ such that 
$$\P\big(G(o,\alpha)\big)\le c \alpha^2\P\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^{S^*}(o,{\mathrm d} y))\text{ and at least one unblocked }s\in\ell_{S^*}(o,y)\big)$$ 
and $\P\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^{S^*}(o,{\mathrm d} y))\text{ and at least one unblocked }s\in\ell_{S^*}(o,y)\big)$ tends to zero as ${\lambda_{\rm W}}$ tends to infinity. 
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{Lem_C}]
Note that 
\begin{align*}
\P\big(G(o,\alpha)\big)&\le \P\big(X^{\lambda,{\rm th}}(B_{10\alpha})>0\big)\le \mathbb E\big[X^{\lambda,{\rm th}}(B_{10\alpha})\big]\\
&\le \lambda\mathbb E\Big[\int_{S\cap B_{10\alpha}}{\mathrm d} x\P\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^S(x,{\mathrm d} y))\text{ and } s\in\ell_S(x,y)\text{ such that $s$ is unblocked for $x$}\big)\Big]\\
&\le \lambda 10^2\alpha^2\P\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^{S^*}(o,{\mathrm d} y))\text{ and } s\in\ell_{S^*}(o,y)\text{ such that $s$ is unblocked for $o$}\big)
\end{align*}
But, by dominated convergence $\P\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^{S^*}(o,{\mathrm d} y))\text{ and } s\in\ell_{S^*}(o,y)\text{ such that $s$ is unblocked for $o$}\big)$ tends to zero as ${\lambda_{\rm W}}$ tends to infinity and hence we arrive at the desired result. 
\end{proof}
As a final input for the proof, we recall the following essential result about convergence properties of functions satisfying some scaling inequality. 
\begin{lemma}[{\cite[Lemma 3.7]{G08}}]\label{Lem_F}
Let $f$ and $g$ be two bounded measurable functions from $[1,\infty]$ to $[0,\infty)$. Additionally, let $f$ be bounded by $1/2$ on $[1,10]$, $g$ be bounded by $1/4$ on $[1,\infty]$ and assume 
\begin{align*} 
f (\alpha) \le f (\alpha/10)^2 + g(\alpha),\qquad\text{for all }\alpha\ge 10.
\end{align*}
Then, $\lim_{\alpha\uparrow\infty}g(\alpha)=0$ implies that $\lim_{\alpha\uparrow\infty}f(\alpha)=0$. 
\end{lemma}
Now we are in the position to prove the theorem.
\begin{proof}[Proof of Theorem~\ref{Thm_Extinction}]
Recall that $S$ is assumed to be a PVT or PDT. In order to prove that ${\lambda_{\rm W,c}}<\infty$, it suffices to show that $\lim_{\alpha\uparrow\infty}\P(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb R^2))> X^{\lambda,{\rm th}}(B_{8\alpha}))=0$ for all sufficiently large ${\lambda_{\rm W}}$. But this is true if $\lim_{\alpha\uparrow\infty}\P\big(G(0,\alpha)\big)=0$, since by~\cite{HJCW22}[Lemma III.4 and III.5], we have that 
\begin{align*}
\P\big(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb R^2))> X^{\lambda,{\rm th}}(B_{8\alpha})\big)&\le \P\big(G(o,\alpha)\big)+\lambda\mathbb E\Big[\int_{S}{\mathrm d} x\mathbbmss{1}\{10\alpha\le |x|\le 9\alpha+\ell_S(x)\}\Big], 
\end{align*}
where $\mathbb E\Big[\int_{S}{\mathrm d} x\mathbbmss{1}\{10\alpha\le |x|\le 9\alpha+\ell_S(x)\}\Big]$ tends to zero as $\alpha$ tends to infinity. Now, in order to show $\lim_{\alpha\uparrow\infty}\P\big(G(0,\alpha)\big)=0$ we apply Lemma~\ref{Lem_A}, Lemma~\ref{Lem_C} and Lemma~\ref{Lem_F} for proper choices of $f$ and $g$. 

For this, note that, since we assume stabilization, in Lemma~\ref{Lem_A} we have that $\P(R(Q_{10\alpha})\ge \alpha)$ tends to zero as $\alpha$ tends to infinity and the same is true for $ \alpha ^2\P\big(o\in A^{\rm c}_{S^*}(\alpha)\big)$ by~\cite{HJCW22}[Lemma III.2]. Hence, we can define 
\begin{align*}
\alpha_1&:=\inf\{s\ge 1\colon \phi_1(\alpha)=\alpha ^2\P\big(o\in A^{\rm c}_{S^*}\big)<(8c^2)^{-1}\text{ for all }\alpha\ge s\}\\
\alpha_2&:=\inf\{s\ge 1\colon \phi_2(\alpha)=\P\big(R(Q_{10\alpha})\ge \alpha\big)<(8c^2)^{-1}\text{ for all }\alpha\ge s\},
\end{align*}
and set $\alpha_{\rm c}=\alpha_1\vee\alpha_2$ and note that $\alpha_{\rm c}<\infty$. Further, we let
\begin{align*}
{\lambda_{\rm W,c}}&=\inf\{{\lambda_{\rm W}}\ge 1\colon \P\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^{S^*}(o,{\mathrm d} y))\text{ and at least one unblocked }s\in\ell_{S^*}(o,y)\big)<\frac 1 2 (100c\alpha_{\rm c})^{-2}\},
\end{align*} 
where ${\lambda_{\rm W,c}}<\infty$ by Lemma~\ref{Lem_C}. Then, we denote
\begin{align*}
f(\alpha)=c\P\big(G(o,10\alpha_{\rm c}\alpha)\big)\quad&\text{ and }\quad g(\alpha)=c^2\big(\phi_1(\alpha_{\rm c}\alpha)+\phi_2(\alpha_{\rm c}\alpha)\big)
\end{align*}
and hence, using again Lemma~\ref{Lem_C}, we have that 
\begin{align*}
f(\alpha)&\le 1/2,\qquad\text{ for all }1\le \alpha\le10\text{ and }{\lambda_{\rm W}}>{\lambda_{\rm W,c}},
\end{align*}
and by definition,
\begin{align*}
g(\alpha)&\le 1/4,\qquad\text{ for all }1\le \alpha.
\end{align*}
Finally, using Lemma~\ref{Lem_A}, we have that 
\begin{align*}
f(\alpha)&\le c^2\big(\P\big(G(o,\alpha_{\rm c}\alpha)\big)^2+\phi_1(\alpha_{\rm c}\alpha)+\phi_2(\alpha_{\rm c}\alpha) \big)=f(\alpha/10)^2+g(\alpha)\quad\text{ for all }\alpha \ge 10.
\end{align*}
Hence, since $\lim_{\alpha\uparrow\infty}g(\alpha)=0$, an application of Lemma~\ref{Lem_F} gives the result. 
\end{proof}

\subsection{Proof of Theorem~\ref{thm_InOut} }\label{Proof_thm_InOut}
We prove the theorem in two parts. 
\begin{proof}[Proof of Theorem~\ref{thm_InOut} Part (2) and (3)]
Note that Part (2) is the statement of Theorem~\ref{Thm_Survival} which puts no restrictions in the white-knight intensity. Further, Part (3) follows from \cite{HJCW22}[Theorem 3.3] where  a sub-critical percolation regime is established for sufficiently fast speeds, independent of the device intensities. This in particular implies that global survival is possible only with probability zero. 
\end{proof}
Hence, it only remains to prove Part (1). The main idea is that small velocities make it impossible to transmit the infection through crossings if sufficiently many white knights are available. We say that a street $s=(c_1,c_2)$ is {\em closed} if 
\begin{itemize}
\item[(A)] we have that $|s|>2v{\rho_{\rm W}}$,
\item[(B)] there exist white knights $Y^s\subset Y^{{\lambda_{\rm W}}}$ such that for all $t\ge 0$ and $x\in s$ we have $|x-Y_{i,t}|<r/2$ for some $Y_i\in Y^s$, and
\item[(C)] if at time $t\ge 0$, there exists an infected $X_i\in X^\lambda$ with $X_{i,t}=c_1$ (respectively $X_{i,t}=c_2$), then, $\{X_j\in X^\lambda\cap s\colon |X_{j,u}-c_2|\le v{\rho_{\rm W}},\text{ for some }u\in \tau(t)\}=\emptyset$ (respectively $\{X_j\in X^\lambda\cap s\colon |X_{j,u}-c_1|\le v{\rho_{\rm W}},\text{ for some }u\in \tau(t)\}=\emptyset$). Here, $\tau(t)$ is the time interval in which the infection reaches $s\cap B_{v{\rho_{\rm W}}+r}(c_2)$ (respectively in $s\cap B_{v{\rho_{\rm W}}+r}(c_1)$).
\end{itemize}
A street is called {\em open} if it is not closed.
The key idea is that a closed street cannot be used for the transmission of the infection. Indeed, Condition~(A) guarantees that the street is sufficiently long in order to allow the white knights to patch, and the probability that the condition is satisfied tends to one as $v$ tends to zero. Further, Condition~(B) guarantees the existence of white knights at all times close to every point on the street and in particular close to the crossings. These white knights help to patch any incoming infected device and the condition becomes likely to be satisfied for large white-knight intensities. Now, for Condition~(C), note first that a large device intensity $\lambda$ makes it probable that an infection travels through a street unpatched since we assume that ${\rho_{\rm W}}>{\rho_{\rm I}}$. Also, due to the continuous movement of the devices, it is unavoidable that an infected device enters the street at some time. However, in order for an infected device to leave the street unpatched, it needs to receive the infection at the opposite end of the street, sufficiently close to the crossing, i.e., in $s\cap B_{v{\rho_{\rm W}}+r}(c_2)$ (respectively $s\cap B_{v{\rho_{\rm W}}+r}(c_1)$). Only in this case it can exit the street unpatched from the blocking white knight that is positioned there. Very roughly then, again for small $v$, Condition~(C) becomes likely. 

What makes the probability that Condition~(C) is satisfied challenging to control is the fact that, in principle, a street at any given time could host devices from distant spatial areas. In order to cope with this we again employ a multi-scale argument very similar to the one presented in the previous Section~\ref{Proof_Thm_Extinction}. 
Using the definitions for $\ell_S$ from Section~\ref{Proof_Thm_Extinction}, we now consider the thinned point cloud
$$X^{\lambda,{\rm th}}=\{X_i\in X^\lambda\colon \text{there exists }y\in{\mathrm{supp}}(\kappa^S(X_i,{\mathrm d} y))\text{ and } s\in\ell_S(X_i,y)\text{ such that $s$ is open}\}.$$
Again, the idea is that any device that is not in $X^{\lambda,{\rm th}}$ only finds closed streets on any possible path and hence can not contribute to the spread of the infection. Thus, it suffices to show subcriticality in the associated geostatistical Cox--Boolean model
$$\mathcal{C}=\bigcup_{X_i\in X^{\lambda,{\rm th}}}B_{\ell_S(X_i)}(X_i).$$
However, the situation is more complicated compared to the proof of Theorem~\ref{Thm_Extinction} since the thinning procedure depends on the entire point cloud. Still, we can employ the Lemmas~\ref{Lem_A} and \ref{Lem_F} and only need to replace the Lemma~\ref{Lem_C} by the following statement, which is presented in terms of $\P^*$, the Palm version of $\P$.
\begin{lemma}\label{Lem_C2}
There exists $c>0$ such that 
$$\P\big(G(o,\alpha)\big)\le c \alpha^2\P^*\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^S(o,{\mathrm d} y))\text{ and } s\in\ell_S(o,y)\text{ such that $s$ is open}\big)$$ 
and 
$$\limsup_{{\lambda_{\rm W}}\uparrow\infty}\limsup_{v\downarrow0}\P^*\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^S(o,{\mathrm d} y))\text{ and } s\in\ell_S(o,y)\text{ such that $s$ is open}\big)=0.$$ 
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{Lem_C2}]
First note that we can use the Slivnyak--Mecke formula and translation invariance to estimate 
\begin{align*}
\P\big(G(o,\alpha)\big)&\le \P\big(X^{\lambda,{\rm th}}(B_{10\alpha})>0\big)\le \mathbb E\big[X^{\lambda,{\rm th}}(B_{10\alpha})\big]\\
&\le \lambda 10^2\alpha^2\P^*\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^{S}(o,{\mathrm d} y))\text{ and } s\in\ell_{S}(o,y)\text{ such that $s$ is open}\big).
\end{align*}
We highlight here that, different to the situation of Lemma~\ref{Lem_C}, the probability also extends to the devices $X^\lambda$ since openness is defined not only with respect to white knights and the street system. 

In order to show the second statement of the lemma, we employ stabilization based on the stabilizing random field $\{R_x\}_{x\in\mathbb R^2}$. For all sufficiently large $n$, we have that for $\varepsilon'>0$,
\begin{align*}
\P^*\big(&\text{there exists }y\in{\mathrm{supp}}(\kappa^{S}(o,{\mathrm d} y))\text{ and } s\in\ell_{S}(o,y)\text{ such that $s$ is open}\big)\\
&\le \P^*\big(R(Q_{2n})<n/2, \text{there exists }y\in{\mathrm{supp}}(\kappa^{S}(o,{\mathrm d} y))\text{ and } s\in\ell_{S}(o,y)\text{ such that $s$ is open}\big)+\varepsilon'\\
&\le \P^*\big(\text{there exists an open $s\in Q_{2n}$}\big)+\varepsilon',
\end{align*}
where we used that $\P(R(Q_{2n})\ge n/2)$ tends to zero as $n$ tends to infinity, as well as the fact that under the event $R(Q_{2n})< n/2$ any shortest path starting in $o$ must be contained in $Q_{2n}$. Now, by the Markov inequality, 
\begin{align*}
\P^*\big(\text{there exists an open $s\in Q_{2n}$}\big)\le \mathbb E^*\Big[\sum_{s\in S\cap Q_{2n}}\P^*(s\text{ is open}|S)\Big]
\end{align*}
and, using dominated convergence and the fact that $\mathbb E^*[\#\{s\in S\cap Q_{2n}\}]<\infty$ for PVT and PDT, it suffices to show that, for all $s\in S\cap Q_{2n}$, 
$$\limsup_{{\lambda_{\rm W}}\uparrow\infty}\limsup_{v\downarrow0}\P^*(s\text{ is open}|S)=0,$$
almost-surely with respect to $S$.
In view of the definition of closed streets, let $C(v)$ denote the event that Condition~(A) is satisfied for $s$ and similar $D({\lambda_{\rm W}})$ and $E(v,\lambda,{\lambda_{\rm W}})$ for the events associated to the Conditions~(B) and (C). Then, 
\begin{align*}
\P^*(s\text{ is open}|S)\le \P(C^c(v)|S)+\P( D^c({\lambda_{\rm W}})|S)+\P( E^c(v,\lambda,{\lambda_{\rm W}})|S), 
\end{align*}
where the first and third summands tend to zero as $v$ tends to zero and then also the second summand tends to zero as ${\lambda_{\rm W}}$ tends to infinity. This finishes the proof.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm_InOut} Part 1]
For the proof we can follow the same line of arguments as used for the proof of Theorem~\ref{Thm_Extinction} above, only replacing Lemma~\ref{Lem_C} by Lemma~\ref{Lem_C2} and subsequently adjusting the arguments. We need to prove existence of ${\lambda_{\rm W,c}}<\infty$ such that for all ${\lambda_{\rm W}}>{\lambda_{\rm W,c}}$ there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that $\lim_{\alpha\uparrow\infty}\P(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb R^2))> X^{\lambda,{\rm th}}(B_{8\alpha}))=0$  for all $v<v_{c}(\lambda,{\lambda_{\rm W}})$. Referencing to~\cite{HJCW22}[Lemma III.2, Lemma III.4 and III.5] as well as to Lemma~\ref{Lem_A} and the initial argument presented in Theorem~\ref{Thm_Extinction} it suffices to note that, with the help of Lemma~\ref{Lem_C2}, there exists ${\lambda_{\rm W,c}}<\infty$ such that for all ${\lambda_{\rm W}}>{\lambda_{\rm W,c}}$ there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that for all $v<v_{c}(\lambda,{\lambda_{\rm W}})$ 
$$
\P^*\big(\text{there exists }y\in{\mathrm{supp}}(\kappa^S(o,{\mathrm d} y))\text{ and } s\in\ell_S(o,y)\text{ such that $s$ is open}\big)<\frac 1 2 (100c\alpha_{\rm c})^{-2},
$$
where $\alpha_{\rm c}$ is defined as in the proof of Theorem~\ref{Thm_Extinction}. Then, using the same definitions for $f$ and $g$, the result follows by an application of Lemma~\ref{Lem_F}. 
\end{proof}
\subsection{Proof of Theorem~\ref{Thm_Survival} }\label{ProofThm_Survival}
\begin{proof}[Proof of Theorem~\ref{Thm_Survival}]
For simplicity we only prove the case where $\mu=\delta_v$. The general case can be verified using a thinning argument where we ignore all devices that have a speed greater than $v_{\rm min}+\varepsilon$ for an arbitrarily small $\varepsilon$, only leading to a potentially larger value of $\lambda_{\mathrm c}$.

First note that the assumption that $\kappa$ is translation covariant and $c$-well behaved for some $c>2{\varrho_{\rm I,min}} v$ ensures that it is possible for the kernel to generate a shortest path from one street to an adjacent street while spending a sufficient amount of time on both streets to allow an infection to be past onto another device. Further, the requirement that $r>{\varrho_{\rm I,min}} v$ ensures that the communication radius is large enough such that an infection transmission can happen between devices with initial and target locations in the vicinity of the crossing. Also recall that we require that the streets of length larger than $2v{\varrho_{\rm I,min}}$ percolate with positive probability.

Since we assume ${\lambda_{\rm W}}$ to be arbitrary, in spirit, we think of white knights to be everywhere. We consider open streets as well as open crossings. Let us start with the streets. Let $0<\varepsilon<\min(r/4 ,v({\varrho_{\rm I,min}} -{\varrho_{\rm W,min}})/2)$ and define $n(s)= \lfloor |s|/(2r)\rfloor$, which will represent the number of devices on the street $s$ on which the infection will be propagated. We call a street $s=(c_1,c_2)$ {\em open}, if there are devices $X_0^s, \dots, X_{n(s)}^s$ such that
\begin{itemize}
\item[(S1)]$|s|>v{\varrho_{\rm I,min}}+\varepsilon$,
\item[(S2)] for all $t$, $X_{i,t}^s \in B_\varepsilon(c_1+\frac{ri}{2}\frac{c_2-c_1}{|c_2-c_1|})$ for all $i \in \{1,\dots, n(s)\}$,
\item[(S3)] ${\rho_{\rm I}}(X_i^s,X_{i+1}^s)<{\varrho_{\rm W,min}}$ for all $i \in \{1,\dots, n(s)-1\}$, and
\item[(S4)] ${\rho_{\rm I}}(X_i^s,X_{i-1}^s)<{\varrho_{\rm W,min}}$ for all $i \in \{2,\dots, n(s)\}$.
\end{itemize}
In words a street is open if, by Condition (S1), the street is long enough such that an infection transmission is possible. By Condition~(S2), there is a chain of devices reaching from the crossing $c_1$ to the crossing $c_2$ and such that each pair of neighboring devices has distance less than $r$. In particular, if device $X_0^s$ is infected, Condition (S3) ensures that the infection can not be stopped by white knights as they are too slow and hence the infection propagates from $X_0^s$ to $X^s_{n(s)}$.
Similarly, Condition (S4) ensures that an infection also propagates from $X^s_{n(s)}$ to $X^s_0$.
Let us call $X^s_{c_1}=X^s_1$ and $X^s_{c_2}=X^s_{n(s)}$ the {\em boundary devices} of an open street $s=(c_1,c_2)$.

Next, for a street $s=(c_1,c_2)$, denote by $[a,b]_{s,c_1}$ the segment $\{x \in s \colon x = c_1 + d\frac{c_2-c_1}{|c_2-c_1|}\text{ for some } d\in [a,b]\}$ on $s$ such that $\{0\}_{s,c_1}$ is identified with the crossing $c_1$. Consider again $0<\varepsilon<\min(r/4 ,v({\varrho_{\rm I,min}} -{\varrho_{\rm W,min}})/2)$ to be small enough such that $v_{\mathrm{min}} {\varrho_{\rm I,min}} +\varepsilon< c/2 $. We call a crossing $c$ {\em open at time $t$} if, for any ordered pair of open streets $s_1,s_2$ that share the crossing $c$, there exists a {\em bridging device}  $X^{s_1,s_2}$ such that 
\begin{itemize}
  \item[(C1)] $X_{0}^{s_1,s_2}\in A_{s_1}=[v{\varrho_{\rm I,min}}/2+\varepsilon/2,v{\varrho_{\rm I,min}}/2+\varepsilon]_{s_1,c}$,
  \item[(C2)] $X^{s_1,s_2}$ has its target location in the segment $B_{s_2}=[v{\varrho_{\rm I,min}}/2,v{\varrho_{\rm I,min}}/2+\varepsilon]_{s_2,c}$,
  \item[(C3)] $X_{t}^{s_1,s_2}\in J_{s_1}=[0,\varepsilon/2]_{s_1,c}$ and is on its way back to its starting location,
  \item[(C4)] ${\rho_{\rm I}}(X^{s_1,s_2},X_{c}^{s_1})< {\varrho_{\rm I,min}}+\varepsilon/(2v)$, and
  \item[(C5)] ${\rho_{\rm I}}(X^{s_1,s_2},X_{c}^{s_2})< {\varrho_{\rm I,min}}+\varepsilon/v$.
  \end{itemize}
These conditions roughly guarantee that, for sufficiently long streets, the bridging device $X^{s_1,s_2}$ propagates the infection from $s_1$ to $s_2$. The time the bridging device stays on each of the streets is insufficiently long to allow to be patched by white knights. This is ensured by Condition (C1) and (C2), as the distance the devices travel on each street before entering the crossing is at most $v{\varrho_{\rm I,min}}+2\varepsilon<v{\varrho_{\rm W,min}}$, since $\varepsilon$ is sufficiently small.
Together, since $s_2$ is open, then, due to the configuration of the devices on the open street and the fact that $r>v{\varrho_{\rm I,min}} $, $X^{s_1,s_2}$ will pass the infection on to the boundary device $X^{s_2}_c$ of the chain of transmitting devices on $s_2$. This is ensured by Condition (C5). Finally, Conditions~(C3) and (C4) ensure that $X^{s_1,s_2}$ becomes infected if $X^{s_1}_c$, becomes infected at time $t$, as it is on its way back to its starting location and will spend at least a time of ${\varrho_{\rm W,min}}+\varepsilon/(2v)$  in the vicinity of $X^{s_1}_c$.

The following statement guarantees that there exists a uniform bound for the probability that a given crossing is open for all sufficiently large times. In particular, this bound converges to one as $\lambda$ tends to infinity. Recall that we denoted by $H$ the range of dependence of $\kappa$ and define $L= {H+v{\varrho_{\rm I,min}}/2+\varepsilon}$.
\begin{lemma}\label{lem_mixing}
For almost-all $S$ we have that $(\{c \text{ is open at time } t\})_{c\in S}$ is an independent family of events, conditioned on $S$. Further, there exists $T>0$ such that for almost-all $S$ and all crossings $c\in S$ there exists a constant $C_c=C_c(S\cap B_L(c),T)$ such that
\begin{align*}
    \P(c \text{ is open at time } t\mid S)\ge 1-\exp(-\lambda C_c)\qquad \text{ for all } t\ge T.
\end{align*}
\end{lemma}
We present the proof of the lemma later in this section. 
Consider $\tau_{s_1,c}$, the time that a boundary device associated to the crossing $c$ is infected, then, for $\tau_{s_1,c}>T$ and conditioned on $S$, we can couple the event that $c$ is open at time $\tau_{s_1,c}$ 
to a Bernoulli random variable $\mathcal{B}_c$ with parameter $1-\exp(-\lambda C_c)$, and note that the $\mathcal{B}_c$ are independent conditioned on $S$ and only depend on $S$ in the region $S\cap B_L(c)$.
For $\tau_{s_1,c}= \infty$, i.e., if the boundary device never becomes infected, we assume that the crossing is open nonetheless.
We say that a crossing is {\em open} if $\mathcal{B}_c=1$.
In order to make use of this observation, we need to ensured that the infection survives up until time $T$. 
However, note that survival for long but finite times has a positive probability and can be even constructed locally by an appropriate isolation procedure for the typical device.

Now, we consider the model at time $t\ge T$ and show that the graph of open streets and open crossings percolates with positive probability via stabilization arguments. For this, let us fix an $a$ such that ${\varrho_{\rm I,min}} <a/2 <a_{\rm c}/2$
and recall that we have the random field of stabilizing radii $\{R_x\}_{x\in\mathbb R^2}$ for the street system at our disposal, in addition to the random field $\{R^a_x\}_{x\in\mathbb R^2}$ that acts as a replacement for the asymptotically essentially connectedness of the thinned graph $S^a$.
Then, we say that $z \in \mathbb Z^2$ is \textit{$n$-open} if 
\begin{itemize}
\item[(a)] $R(Q_{3n}(nz))< n$,
\item[(b)] $R^a(Q_{n}(nz))< n/2$,
\item[(c)] every street fully contained in $Q_{3n}(nz)\cap S^a$ is open, and
\item[(d)] every crossing in $Q_{3n}(nz)$ is open.
\end{itemize} 
Condition~(b) limits the size of cells of points in $Q_{n}(nz)$ to a diameter of at most $n/2$ with respect to $S^a$. Further, under the condition, there exists a unique giant component in $Q_{n}(nz)\cap S^a$, build by the boundaries of the cells, that is connected in $Q_{3n}(nz)$. If now adjacent sites $z_1$ and $z_2$ both satisfy Condition~(b), they form a joint connected component, since there exist cells that are simultaneously in $Q_{n}(nz_1)$ and in $Q_{n}(nz_2)$.
Due to Condition~(c) and (d) the infection is able to propagate through open streets and open crossings unpatched.

Now, we can use the domination-by-product-measures theorem~\cite[Theorem 0.0]{domProd} to establish Bernoulli site percolation of $n$-open sites on $\mathbb Z^2$ since (a) guarantees $7$-dependence of $S^a$ and under Condition~(a) also the Conditions~(b) and~(c) can be decided within the $7$-dependent region. By choosing $n>L$, the crossing dependent constant $C_{\rm c}$ from Lemma~\ref{lem_mixing} is also $7$-dependent like the street system.
Hence, using translation invariance, it suffices to show that
\begin{align*}
\liminf_{n\uparrow \infty}\liminf_{\lambda \uparrow \infty} \P( o \text{ is $n$-open} )=1.
\end{align*}
To show this, we denote by $A(n)$, $B(n)$, $C(n,\lambda)$, and $D(n,\lambda)$, the events that the Conditions~(a), (b), (c), and (d) are not satisfied for $z=o$.
Then, $\P( o \text{ is $n$-open})\ge 1-\P(A(n)\cup B(n))-\P(C(n,\lambda))-\P(D(n,\lambda))$.
By assumption, we have that $\limsup_{n\uparrow \infty}\P(A(n)\cup B(n)) = 0$ independently of $\lambda$. For fixed $n$, using Markov inequality and dominated convergence, we now also have $\limsup_{\lambda \uparrow \infty}(\P(C(n,\lambda)) +\P(D(n,\lambda))) = 0$, as $\lambda$ increases, where we used that the expected number of streets and crossings in finite.

Finally, due to the translation invariance of the model, existence of an infinite cluster of open sites implies existence of an infinite cluster of good streets and good crossings and thus to a positive probability that the typical device transmits the infection to infinity. 
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem_mixing}]\label{proof_lem_mixing}
First note that the conditions only dependent on the bridging devices, which have their initial position and their destination close to the crossing and move over the crossing and are therefor independent for distinct crossings. This guarantees the independence of the events. 

In order to bound the probability of openness at late times, we compute the intensity of devices that satisfy the Conditions (1), (2) and (3) at time $t$ and then show that, as $t$ tends to infinity, this intensity converges to a positive crossing-dependent value. In turn, the probability that no such devices exist can be seen as a Poisson void probability.

Let us start by deriving expressions for the intensity of devices at time $t$ in the desired interval, i.e., devices that start on $s_1$ in the interval $A_{s_1}$ and have a target location in $B_{s_2}$, on $s_2$. Recall that the streets $s_1$ and $s_2$ have a joint crossing $c$. 
Note that, by the displacement theorem for Poisson point processes, the number of devices that satisfy the conditions is a Poisson distributed random variable and its parameter is given by $\lambda p_S(s_1,s_2)$, where 
\begin{align*}
p_S(s_1,s_2)=\int_{A_{s_1}}{\mathrm d} x \int_{B_{s_2}}\kappa^S(x,{\mathrm d} z)
\mathbbmss{1}\{0\le r_t(|x,z|_S)-|z,c|_S \le \varepsilon/2\}, 
\end{align*}
where $|z,x|_S$ denotes the shortest distance between $x$ and $z$ on $S$ and $r_t(|z,x|_S)$ is uniquely defined as $vt=|z,x|_S n_t+r_t(|z,x|_S)$ with $n_t$ the largest odd number such that $0\le r_t(|z,x|_S)\le 2|z,x|_S$. In words, $n_t(|z,x|_S)$ denotes the number of times that the device has traveled from $x$ to $z$ and back, and since we want the device to be on the way back, we require $n_t(|z,x|_S)$ to be odd. Then, $r_t(|z,x|_S)$ is the distance from $z$ on the way back. Subtracting the distance from $z$ to the joint crossing $c$, we obtain the traveled distance on $s_1$, which has to lie in the interval $(0,\varepsilon)_{s_1,c}$.

We will show that there is a time $T$ such that for all $t \ge T$ we can bound $p_S(s_1,s_2)$ from below by a constant that does only depend on $S\cap B_{L}(c)$. 

Let us define for $x\in A_{s_1}$ by $0<\eta_x=\eta_x(s_1,s_2,S\cap B_{L}(x),\kappa^S(x,{\mathrm d} y))= \inf_{y \in B_{s_2}}\kappa^S (x,y)$ the minimal density for the kernel, starting in $x$ and having a destination in $B_{s_2}$. This value $\eta_x$ is positive as the kernel is $c$-continuous, $B_{s_2} \subset B_c(x)$, and we use our assumption that $2v {\varrho_{\rm I,min}} +2\varepsilon< c$.

In order to simplify the notation for $A_{s_1}$ and $B_{s_2}$ let us abbreviate $b=v{\varrho_{\rm W,min}}/2$ and associate bijectively the interval $A=[b+\varepsilon/2,b+\varepsilon]$ to $A_{s_1}$ such that $x\mapsto {x}_{s_1,c}$. We associate $B=[b,b+\varepsilon]$ in a similar way to $B_{s_2}$.
 Then, 
\begin{align*}
p_S(s_1,s_2)\ge \int_{A}{\mathrm d} x \eta_x\int_{B}{\mathrm d} z
\mathbbmss{1}\{0\le r_t(x+z)-z \le \varepsilon/2\}
\end{align*}
where we also dropped the singular part in $\kappa^S(x,{\mathrm d} y)$.
Now, let us define by
\begin{align*}
\phi_x(s)=\lim_{a \downarrow 0}\frac{1}{a}\int_{B}{\mathrm d} z\mathbbmss{1}\{0\le r_s(x+z)-z\le a\} 
\end{align*}
the density of points points started at $x$ that are on their way back from their target location, and that are at time $s$ precisely at the origin. Note, that a device satisfies Condition~(3) at time $t$ if and only if it passes the crossing on its way to its starting location in the time interval $[t-\varepsilon(2v)^{-1},t]$. Therefore,  we can rewrite
\begin{align}
\int_{A}{\mathrm d} x \eta_x\int_{B}{\mathrm d} z
\mathbbmss{1}\{0\le r_t(x+z)-z \le \varepsilon/2\}=\int_{t-\varepsilon/(2v)}^t{\mathrm d} s\int_{A}{\mathrm d} x \eta_x\phi_x(s), \label{eq_mixing_lemma_3}
\end{align}
and we will prove that $\P$-almost surely, conditioned on $S$, there exist constants $T$ and $C$ that depend on $v, {\varrho_{\rm I,min}}$, and  $\varepsilon$ but neither on $S$, $x$ nor $\kappa^S$, such that $\inf_{t\ge T}\phi_x(t)>C>0$. 
In order to derive this statement, the main tool is the following identity for $\phi_x(t)$
\begin{align}\label{Eq_phi}
\phi_{x}(t) = \sum_{n=1}^\infty (2\varepsilon n)^{-1} \mathbbmss{1}\{x+2b+(n-1)2(b+x)\le vt<x+2b+2\varepsilon+(n-1)2(x+b+\varepsilon)\}.
\end{align}
Let us explain this formula.
Here, $n$ represents the number of of times that a device, started in $x$ reaches its destination, which is uniformly distributed in $B$.
With every reflection in $B$, the density is distributed to an interval of length $2\varepsilon n$, where $\varepsilon$ is the length of $B$, again uniformly at random. 
Now, $x+2b$, respectively $x+2b+2\varepsilon$ are the shortest, respectively the longest, distance that the device, started in $x$, travels before it re-visits the crossing $c$ for the first time. For each consecutive visit in the origin an additional length of $2(b+x)$, respectively $2(x+b+\varepsilon)$ has to be added as the length of a complete cycle from the origin to its destination and back.

In other words, each indicator becomes non-zero in $x+2b+(n-1)2(x+b)$ and adds a mass of $(2\varepsilon n)^{-1}$ over a length of $2n\varepsilon$. For $n<(x+b)/\varepsilon$ the indicator functions are disjoined, i.e. for $vt<x+2b+((x+b)/\varepsilon-1)2(x+b)$ there is at most one indicator function that is non-zero.
Now, for $n>(x+b)/\varepsilon$, the interval for $vt$ in which the indicator function is non-negative is longer than the minimal cycle length. In other words, it is possible that at least two indicator functions are non-negative.

Now, for $(x+b)/\varepsilon<n<2(x+b)/\varepsilon$ the indicator functions contribute for more than the length of a full cycle $2(x+b)$ and less than twice that length.
Therefore, for each $vt \in [x+2b+(n_1-1)2(b+x),x+2b+(n_2-1)2(b+x)]$ there are either exactly one or two non-negative indicators, where $n_i=\lceil i(x+b)/\varepsilon \rceil$. As each indicator has a weight of at least $(4\epsilon(x+b)/\varepsilon)^{-1}$ we can bound $\phi_x(t)>(4\epsilon(x+b)/\varepsilon)^{-1}$.

More general, in the interval $vt \in [x+2b+(n_m-1)2(b+x),x+2b+(n_{m+1}-1)2(b+x)]$
there are either $m$ or $m+1$ overlapping indicator functions whose contribution can be each be bounded by $(2n_{m+1}\varepsilon)^{-1}$. Therefore, we can bound  
\begin{align}\label{eq_lem_mixing_2}
    \phi_x(t)>m(2(n_{m+1})\varepsilon)^{-1} = \frac{m}{2\varepsilon \lceil (m+1) (x+b)/\varepsilon \rceil} \ge \frac{1}{2\varepsilon \lceil 2(x+b)/\varepsilon \rceil}\ge \frac{1}{4(2b+\varepsilon)}=C>0,
\end{align}
where we used that the bound monotonously increases in $m$ and $x\le b+\varepsilon$.
Therefore, for $T=(x+2b+(n_1-1)2(b+x))v^{-1}$ the desired bound is realized and integration of~\ref{eq_mixing_lemma_3} yields
\begin{align*}
    p_S(s_1,s_2)\ge \varepsilon(2v)^{-1}(8b+4\varepsilon)^{-1} \int_{A}{\mathrm d} x \eta_x=\tilde p_S(s_1,s_2)>0. 
\end{align*}

Noting that the requirements in Conditions~(4) and (5) can be considered independently, respectively with probability $p_4={\varrho_{\rm I}}([{\varrho_{\rm I,min}},{\varrho_{\rm I,min}}+\varepsilon/2])$ and $p_5={\varrho_{\rm I}}([{\varrho_{\rm I,min}},{\varrho_{\rm I,min}}+\varepsilon])$, the number of devices that satisfy the criterion of a bridging device $X^{s_1,s_2}$ can be bounded from below by an Poisson-distributed random variable with expectation $\tilde p_S(s_1,s_2)p_4p_5$.
As $c$ is open if there is at least one bridging device for each combination of open adjacent streets, we can bound
\begin{align*}
    \P(c \text{ is open at time } t\mid S)\ge \prod_{\substack{i\neq j, s_i,s_j \text{ open}\\c \in s_i,c \in s_j}} \big(1-\exp(-\lambda \tilde p(s_i,s_j)p_4p_5)\big)\qquad \text{ for all } t\ge T,
\end{align*}
which proves the lemma.
\end{proof}

\subsection{Global survival}
Let us start with regimes that ensure global survival of the infection even in the presence of white knights. For this, we guarantee the existence of a sequence of streets on which the infection can be transmitted to infinity with positive probability. 

Note that, due to the transmission mechanism, streets need a certain minimal length in order to be useful for transmissions. 
We thus define the thinned system $S^a=\{s\in S \colon |s|\ge a\}$ that only contains streets of length greater or equal than $a\ge 0$. Let us further denote by $R^a_x$ the distance of the furthest point from $x\in \mathbb R^2$ that is reachable without crossing $S^a$, i.e., we define the cell of $x$ by
$$C^a_x=\sup\{y \in \mathbb R^2\colon \text{ there exists a continuous function $f\colon [0,1] \mapsto \mathbb R^2 \setminus S^a$ with }f(0)=x, f(1)=y\}$$
and define $R^a_x=\sup\{|y-x|\colon y \in C^a_x\}$. We say that the thinned graph is {\em $R^a$-connected} if $\lim_{n\uparrow \infty}\P\big(\sup_{x \in Q_n\cap \mathbb Q^2}R^a_x<n\big) =1$ and define 
\begin{align}\label{def_ac}
a_{\rm c}:=\sup\{a>0\colon S^a \text{ is } R^a \text{-connected}\}.
\end{align}
In words, for $a<a_{\rm c}$, the thinned street system satisfies a slightly stronger version of asymptotically essentially connectedness, and we have some control over the cell sizes. Note that  
$a_{\rm c}>0$ for asymptotically connected street systems, see~\cite[Theorem 2.8]{HJCW22}. 

In order to exhibit a regime of global survival, we will furthermore require a slightly stronger version of well behavedness. We say that a kernel $\kappa$ is {\em $c$-continuous} if, for almost-all $S$ and $x\in S$, the uniquely defined absolutely-continuous part of $\kappa^S(x,{\mathrm d} y)$ is non trivial and its density $\kappa^S(x,y)$ with respect to the Lebesgue measure on $S$ satisfies
\begin{align*}
    \inf_{y \in B_c(x)}\kappa^S(x,y)>0.
\end{align*}
Note that $c$-continuous implies $c$-well behaved.

Using these definitions, we can state our first main result on global survival of the infection. 

\begin{theorem}[Global survival]\label{Thm_Survival}
Let $\kappa$ be $c$-continuous for some $c>0$ and ${\varrho_{\rm W,min}}>{\varrho_{\rm I,min}}>0$. Then, for all sufficiently small $v_{\mathrm{min}}$ such that $0<v_{\mathrm{min}} {\varrho_{\rm I,min}} < \min(a_{\rm c}/2,r,c/2) $, there exists $\lambda_{\mathrm c}$ such that for all $\lambda>\lambda_{\mathrm c}$ and all ${\lambda_{\rm W}}\ge 0$,
$$\P(\text{infection survives globally})>0.$$
\end{theorem}
The punchline of the preceding statement is the fact that the critical device intensity is independent of the white-knight intensity. In other words, no matter how dense we can pack white knights into the system, survival is guaranteed as long as we have sufficiently many devices in the system. Roughly, the reason for this phenomenon is that the white knights simply act too slow since ${\varrho_{\rm W,min}}>{\varrho_{\rm I,min}}>0$ when there are sufficiently many devices that can transmit the infection to their neighbors before white knights are able to eliminate it. The idea of the proof is to establish an infinite cluster of good streets that have the property that, if the malware enters the street on one side, it must also exit the street on the other side. We present the details in Section~\ref{ProofThm_Survival}.

\medskip
Next, we present our result on the extinction of the infection in infinite components.
\begin{theorem}[Extinction]\label{Thm_Extinction}
Consider a street system given by a Poisson--Voronoi or Poisson--Delaunay tessellation. Let $\kappa$ be well behaved with bounded support and assume that $r>v_{\mathrm{max}}{\varrho_{\rm W,min}}$ and ${\varrho_{\rm I,min}}>{\varrho_{\rm W,min}}>0$. Then, for all $\lambda\ge 0$, there exists ${\lambda_{\rm W,c}}$ such that for all ${\lambda_{\rm W}}>{\lambda_{\rm W,c}}$ we have that
$$\P(\text{infection survives globally})=0.$$
\end{theorem}
The proof rests on a multi-scale argument and will be given in Section~\ref{Proof_Thm_Extinction}.

\medskip
Finally, we present a result that features the remarkable non-monotone behavior of our model with respect to the speed parameter. More precisely, the following result states that both, too high and too low velocities, are detrimental for the propagation of the infection. We state the result only for fixed velocities and infection and patch times and note that a corresponding result for general $\mu_{\rm v}$, $\varrho_{\rm I}$ and $\varrho_{\rm W}$ also holds.
\begin{theorem}[Speed-dependent survival and extinction]\label{thm_InOut}
Consider a street system given by a Poisson--Voronoi or Poisson--Delaunay tessellation. Let $\kappa$ have bounded support and be $c$-well behaved for some $c>0$. Assume further that $\varrho_{\rm W}=\delta_{\rho_{\rm W}}$ and $\varrho_{\rm I}=\delta_{{\rho_{\rm I}}}$ with $\rho_{\rm W}>{\rho_{\rm I}}$, and let $0<v_o< \min(a_{\rm c}/2,r,c/2)/{\rho_{\rm I}} $. Then, there exists $\lambda_{\rm c}>0$ and ${\lambda_{\rm W,c}}>0$ (independent of $\lambda_{\rm c}$ and $v_o$), such that for all $\lambda >\lambda_{\rm c}$ and all $\lambda_{\rm W}>{\lambda_{\rm W,c}}$ we have that
\begin{enumerate}
\item[(1)] there exists $v_o>v_{\rm c}(\lambda,\lambda_{\rm W})>0$ such that for all $\mu=\delta_{v}$ with $v<v_{\rm c}(\lambda,\lambda_{\rm W})$ we have 
$$\P(\text{infection survives globally})=0,$$
\item[(2)] for $\mu=\delta_{v_o}$ we have $\P(\text{infection survives globally})>0$, and
\item[(3)] there exists $\infty>v^{\rm c}>v_o$ (independent of $\lambda$ and $\lambda_{\rm W}$), such that for all $v>v^{\rm c}$ we have
$$\P(\text{infection survives globally})=0.$$
\end{enumerate}
\end{theorem}
In words, the choice $\lambda >\lambda_{\rm c}$ puts us in a parameter regime in which survival of the infection is possible, as described in Theorem~\ref{Thm_Survival}. However, in the presence of sufficiently many white knights and sufficiently large or small velocity we are again in regimes of extinction. One important aspect here is that sufficiently small velocities and large white-knight intensities lead to absence of global survival also in the case where the infection is faster than the patching. We present the proof in Section~\ref{Proof_thm_InOut}.
