contradiction. Suppose that there exist $\gamma>0,\delta>0$ such that
${\lambda(\beta,\theta,\gamma)=\lambda(\beta,\theta,\gamma+\delta)=0}$. Let
$\widetilde{\mathcal{L}}$ and $(\widetilde{M}_{t})_{t\geq 0}$ be the generator
and semigroup associated to $(\beta,\theta,\gamma+\delta)$, while we denote by
$\mathcal{L}$ the generator and $h$ the eigenvector for Perron’s root
associated to $(\beta,\theta,\gamma)$. We also define the function $G$

$\displaystyle
G(n):=\sum_{j=1}^{n-1}\frac{n}{j(j+1)}\left(h(j)+h(n-j)-h(n)\right).$

Since $h$ is subadditive (Lemma 6.2) and non-negative (Proposition 4.1), we
have

$\displaystyle
G(n)\leq\sum_{j=1}^{n-1}\frac{n}{j(j+1)}\left(jh(1)+(n-j)h(1)\right)\leq
n^{2}h(1).$

Moreover by Remark 4, $G$ is a non-zero positive function. Then by a similar
argument like (63) and using $\mathcal{L}h=\lambda h=0$, we have

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\widetilde{M}_{t}h=\widetilde{M}_{t}(\widetilde{\mathcal{L}}h)=\widetilde{M}_{t}(\mathcal{L}h+\delta
G)=\widetilde{M}_{t}(\mathcal{L}h)+\delta\widetilde{M}_{t}G=\delta\widetilde{M}_{t}G.$
(71)

By Proposition 4.1, there exist Perron’s eigenelements
$\widetilde{h},\widetilde{\pi}$, and also a constant $\widetilde{\omega}>0$,
corresponding to $\widetilde{\mathcal{L}}$, and the estimate (30) holds with
the corresponding terms. Then we apply (30) and $G(n)\leq n^{2}h(1)$ to obtain

$\displaystyle\widetilde{M}_{t}G\geq\left\langle{\widetilde{\pi},G}\right\rangle\widetilde{h}-Ce^{-\omega
t}.$

Here $C>0$ is a constant that does not depend on $t$. We plug in the above
display to (71) and obtain

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\widetilde{M}_{t}h\geq\delta\left(\left\langle{\widetilde{\pi},G}\right\rangle\widetilde{h}-Ce^{-\omega
t}\right),$

If $\lambda(\beta,\theta,\gamma+\delta)=0$, then by (30),
$\widetilde{M}_{t}h$, as a vector, converges to
$\left\langle{\widetilde{\pi},h}\right\rangle\widetilde{h}$ as $t\to\infty$
for every element. However since $G$ is a non-zero positive function (which
implies $\left\langle{\widetilde{\pi},G}\right\rangle>0$), the above display
shows that $\widetilde{M}_{t}h$ increases at least linearly to infinity as
$t\to\infty$ for every element. This is a contradiction, so by the
monotonicity of $\lambda$ in $\gamma$, we have
$\lambda(\beta,\theta,\gamma+\delta)>0$. We conclude that for fixed
$0<\theta<\beta$, there exists only one $\gamma$ which makes
$\lambda(\beta,\theta,\gamma)=0.$ We denote it by
$\gamma_{c}=\gamma_{c}(\beta,\theta)$.

Step 2: Strictly increasing and continuous function. Using Lemma 6.1 (first
inequality), we have

$\displaystyle\lambda(\beta,\theta+\delta,\gamma_{c}(\beta,\theta))\leq\lambda(\beta,\theta,\gamma_{c}(\beta,\theta))-\delta=-\delta.$

Since $\lambda(\beta,\theta,\gamma)$ is increasing with respect to $\gamma$,
we obtain that ${\gamma_{c}(\beta,\theta+\delta)>\gamma_{c}(\beta,\theta)}$.

Next we prove the continuity of ${\theta\mapsto\gamma_{c}(\beta,\theta)}$. By
the monotonicity of $\lambda$ in $\gamma$, we have for any $\varepsilon>0$,
$\lambda(\beta,\theta,\gamma_{c}(\beta,\theta)+\varepsilon)>0$. Then by the
continuity of $\lambda$ on $\theta$, there exists some $\delta_{0}>0$ such
that for all $\delta\in(0,\delta_{0})$,
$\lambda(\beta,\theta+\delta,\gamma_{c}(\beta,\theta)+\varepsilon)>0$. This
implies, using the uniqueness of $\gamma_{c}$ and monotonicity of $\lambda$ on
$\gamma$, that for any $\delta\in(0,\delta_{0})$

$\displaystyle\gamma_{c}(\beta,\theta+\delta)<\gamma_{c}(\beta,\theta)+\varepsilon,$

which proves the right continuity of $\gamma_{c}(\beta,\theta)$ on $\theta$.
The left continuity can be proved exactly in the same way.

Step 3: Asymptotic behavior. By the monotonicity property of $\lambda$ on
$\gamma$ and regimes for subcriticality and supercriticality given in
Proposition 6.1, we have:

$\displaystyle\theta\left(2^{1-\frac{\theta}{\beta}}-1\right)^{-1}\leq\gamma_{c}(\beta,\theta)\leq\max\left\\{2\theta\left(\frac{1+\frac{\theta}{\beta}}{1-\frac{\theta}{\beta}}\right),\frac{3}{2}\theta\left(1+\frac{\theta}{\beta}\right)\right\\}.$
(72)

Then it suffices to let $\theta\searrow 0$ and $\theta\nearrow\beta$ to prove
the asymptotic limits.

Step 4: Regime. Once we have proved that the critical phase is the single
point for $\beta,\theta$ fixed in an admissible domain, we apply the
monotonicity about $\gamma$ in Proposition 6.1 and justify the regime for
other phases. ∎

###  6.3 Monotonicity of $h$ and $\lambda$

The dependence of $\lambda$ in function of $\theta,\gamma$ has been discussed
in the previous subsections, while the dependence on $\beta$ is more
complicated. Indeed, in terms of number of active clusters, isolation and
fragmentation have opposite effect and increasing the cluster size accelerate
both. This problem is closely related to the monotonicity of the eigenfunction
$h$. The main result in this part is the following one.

######  Proposition 6.3.

The monotonicity of the mappings $h(.,\theta,\gamma)$ and
$\lambda(.,\theta,\gamma)$ depends on the values of $\gamma,\theta$:

  * •

if $\gamma>\theta$, the two functions are increasing;

  * •

if $\gamma=\theta$, the two functions are constants;

  * •

if $\gamma<\theta$, the two functions are decreasing.

The case $\gamma=\theta$ is special: using (14), we find that $h\equiv 1$ is
an eigenvector and ${\lambda=-\theta}$. The other cases are more delicate.
Roughly, increasing $\beta$ makes the infection progress faster and
accelerates the process. If $\gamma>\theta$, this benefits the fragmentation
more and thus makes the number of clusters and $\lambda$ increase. Otherwise,
isolation is more impacted and the converse happens.

This effect of acceleration due to the increase of $\beta$ can be understood
via the following simplified problem. Let us consider a linear birth and death
process $(S_{t})_{t\geq 0}$ starting from a single particle: i.e. each
particle (representing a cluster) lives during an exponential time of
parameter $1/\rho$ and is replaced by two particles (resp. zero) with
probability $p_{2}=\gamma/(\theta+\gamma)$ (resp.
$p_{0}=\theta/(\theta+\gamma)$), corresponding respectively to fragmentation
and isolation. This birth and death process $(S_{t})_{t\geq 0}$ satisfies
$\mathbb{E}[S_{t}]=\exp(\lambda_{BD}t)$ with

$\displaystyle\lambda_{BD}=\rho\left(p_{2}-p_{0}\right)=\rho\left(\frac{\gamma-\theta}{\gamma+\theta}\right).$

The monotonicity of $\lambda_{BD}$ in function of $\rho$ thus depends on the
sign of $(\gamma-\theta)$, i.e. on the fact that the process is subcritical or
supercritical.

Such a dichotomy also exists in our model, and will be clear in a modified
process, which is close to our original GFI process and the Malthusian
coefficients are explicitly linked. This modification yields a simpler
process, since the probability for a cluster to be isolated or fragmented does
not depend on its size any longer and $\gamma=\theta$ provides the critical
case for this latter. This is the key property, which will allow us to make a
coupling for the modified processes with different $\beta$ and then verify the
monotonicities.

We divide the proof into three steps. Firstly, we introduce a modified GFI
process, by slightly changing the isolation term. Secondly, we obtain coupling
results for times and sizes of our modified GFI process. Finally, we use this
coupling to prove Proposition 6.3. We skip some technical details in the
second and the third step, which can be found in the Appendix B of the long
version [bansaye2021growth] of this paper on arXiv.

####  6.3.1 A modified GFI process

In this part, we introduce our modified GFI process. Its dynamic of growth and
fragmentation is the same as the original one. For the dynamic of isolation,
we let each edge (instead of each vertex) have rate $\theta$ to be detected
independently and then the cluster is isolated. This minor modification of the
isolation rate does not change the splitting property, thus the study of the
modified GFI process can also be reduced to its size process
$(\overline{X}_{t},\overline{Y}_{t})_{t\geq 0}$. For the parameters
$(\beta,\theta,\gamma)$, the associated generator $\overline{\mathcal{L}}$ of
the first moment is defined as

$\overline{\mathcal{L}}f(n)=\beta n(f(n+1)-f(n))-\theta(n-1)f(n)\\\
+\sum_{j=1}^{n-1}\frac{\gamma n}{j(j+1)}\left(f(j)+f(n-j)-f(n)\right).$ (73)

Compared to the generator $\mathcal{L}$ defined in (14) with the same
parameters, the isolation rate is $\theta(n-1)$ instead of $\theta n$ for a
cluster of size $n$, and it is clear that we have

$\displaystyle\overline{\mathcal{L}}=\mathcal{L}-\theta Id.$ (74)

Therefore, Lemma 4.2 also applies to $\overline{\mathcal{L}}$. The couple
$(\pi,h)$ associated to $\mathcal{L}$ is also the eigenelement of
$\overline{\mathcal{L}}$, with the Perron’s root / Malthusian exponent

$\displaystyle\overline{\lambda}h=\overline{\mathcal{L}}h=(\lambda-\theta)h\Longrightarrow\overline{\lambda}=\lambda-\theta.$
(75)

For fixed $\gamma,\theta>0$, the mapping
$\beta\mapsto\lambda(\beta,\theta,\gamma)$ shares the same monotonicity with
${\beta\mapsto\overline{\lambda}(\beta,\theta,\gamma)}$, so it suffices to
study the the latter one.

The modified GFI process provides some advantages. We observe that for a
cluster of size $n$, the fragmentation rate and the isolation rate are
respectively $\gamma(n-1)$ and $\theta(n-1)$. So the probability that
isolation happens before fragmentation (and conversely) does not depend on its
size and is equal to $\theta/(\gamma+\theta)$ (resp.
$\gamma/(\gamma+\theta))$. This nice property has inspired the idea of
coupling in the next step.

We finish this part by proving the following lemma.

######  Lemma 6.3.

The modified GFI process has the following criteria of phases: it is
supercritical ($\overline{\lambda}>0$) when $\gamma>\theta$, critical
($\overline{\lambda}=0$) when $\gamma=\theta$ , and subcritical
$\overline{\lambda}<0$ when $\gamma<\theta$.

###### Proof.

For the original GFI process, we know that the case $\gamma=\theta$ gives us
$\lambda=-\theta$. Moreover, the monotonicity of the function
$\gamma\mapsto\lambda(\beta,\theta,\gamma)$ in Lemma 6.1 tells us

$\displaystyle\gamma>\theta\Longrightarrow\lambda>-\theta,\qquad\gamma<\theta\Longrightarrow\lambda<-\theta.$

Combing this with (74) proves Lemma 6.3. ∎

####  6.3.2 Monotone coupling for the modified GFI processes

Let us fix the rates $\theta$ and $\gamma$, and consider an initial cluster of
size $n$ in the modified GFI process with the infection rate $\beta$. This
cluster may grow by infection for a random time $\tau_{n}^{\beta}$, called
lifetime, when (independently of this time) it makes fragmentation with
probability ${\gamma/(\gamma+\theta)}$ or isolation with probability
${\theta/(\gamma+\theta)}$. When it splits, it leaves two child clusters whose
sizes are distributed as the random variable
$(Z^{\beta}_{n,1},Z^{\beta}_{n,2})$ following Proposition 3.1. That is,
$Z^{\beta}_{n,1}$ is the size of the first child, and $Z^{\beta}_{n,2}$ is for
the second child. Here we write $(Z^{\beta}_{n,1},Z^{\beta}_{n,2})=(0,0)$ by
convention if this cluster is isolated.

We say there is a coupling for two random variables $A$ and $B$ if we can find
$A^{\prime}$ distributed as $A$, $B^{\prime}$ distributed as $B$, and
$A^{\prime},B^{\prime}$ are defined in a common probability space. We
sometimes skip the superscripts for these coupled random variables when the
context of coupling is clear. Then we have the following lemma for
$(Z^{\beta}_{n,1},Z^{\beta}_{n,2})$ and $\tau_{n}^{\beta}$ defined above.

######  Lemma 6.4.

For any $1\leq n\leq n^{\prime}$ and $0<\beta\leq\beta^{\prime}$, there exists
a coupling for $\left(\tau_{n}^{\beta},Z^{\beta}_{n,1},Z^{\beta}_{n,2}\right)$
and
$\left(\tau_{n^{\prime}}^{\beta^{\prime}},Z^{\beta^{\prime}}_{n^{\prime},1},Z^{\beta^{\prime}}_{n^{\prime},2}\right)$
such that

$\displaystyle\tau_{n}^{\beta}\geq\tau_{n^{\prime}}^{\beta^{\prime}},\qquad
Z^{\beta}_{n,1}\leq Z^{\beta^{\prime}}_{n^{\prime},1},\qquad
Z^{\beta}_{n,2}\leq Z^{\beta^{\prime}}_{n^{\prime},2}\qquad a.s..$

and in this coupling the two clusters either both reach fragmentation or both
reach isolation.

This lemma implies that the larger cluster size or infection rate will result
in shorter lifetime and larger sizes for child clusters. Concerning the second
statement, since the probability of either event is independent of the cluster
size, we can use a common Bernoulli random variable to determine whether they
split or are isolated in both processes. For (6.4), it is also natural as
larger cluster size or infection rate speed up the growth events.

We now consider two modified GFI processes with common $\theta,\gamma$ but
different infection rates $\beta\leq\beta^{\prime}$ and initial cluster sizes
$n_{0}\leq n^{\prime}_{0}$. We denote the active cluster processes by
$(\overline{\mathscr{X}}^{\beta}_{n_{0},t})_{t\geq
0},(\overline{\mathscr{X}}^{\beta^{\prime}}_{n^{\prime}_{0},t})_{t\geq 0}$ and
their size processes by $(\overline{X}^{\beta}_{n_{0},t})_{t\geq 0}$ and
$(\overline{X}^{\beta^{\prime}}_{n^{\prime}_{0},t})_{t\geq 0}$.

We can construct a coupling of the two modified GFI processes in a common
probability space as follows. Notice that Lemma 6.4 allows us to couple the
lifetimes and cluster sizes of any two clusters. In particular, the lemma
tells us that either fragmentation or isolation occurs to both of them. We can
continue to apply the coupling to their first child clusters and second child
clusters if the fragmentation takes place. If we start from the initial
clusters of both process, we can couple the whole processes in a common
probability space denoted by
$\left(\Omega,\mathcal{G},\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\right)$.
In particular, we can use the UHN labelling to describe the one-to-one mapping
or coupling between clusters from the two processes. For instance, the label
$12$, if it exists, maps the second child of the first child of the initial
cluster from one process to another.

In this coupling, due to Lemma 6.4, any active cluster in
$(\overline{\mathscr{X}}^{\beta}_{n_{0},t})_{t\geq 0}$ has a longer lifetime
and smaller child cluster sizes (if they exist) compared to the corresponding
cluster in
$(\overline{\mathscr{X}}^{\beta^{\prime}}_{n^{\prime}_{0},t})_{t\geq 0}$,
almost surely.

####  6.3.3 Proof of Proposition 6.3

Let us prove the monotonicity result on the modified GFI, which will
immediately give the result on the original process using the link (75)
between the maximal eigenvalues.

To compare the Malthusian exponent and the eigenfunction, it suffices to
compare $\mathbb{E}[\langle\overline{X}^{\beta}_{n_{0},T},1\rangle]$ and
$\mathbb{E}[\langle\overline{X}^{\beta^{\prime}}_{n^{\prime}_{0},T},1\rangle]$
for a large $T$ thanks to the Malthusian behavior in (29).

We compare them in the common probability space
$\left(\Omega,\mathcal{G},\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\right)$
constructed previously. It can be done using a stopping line as follows. Let
us write $\overline{\mathcal{U}}^{\beta}_{T}$ the set of UHN labels of the
active clusters in $\overline{\mathscr{X}}^{\beta}_{n,T}$. The corresponding
clusters in $(\overline{\mathscr{X}}^{\beta’}_{n^{\prime}_{0},t})_{t\geq 0}$
labeled by $\overline{\mathcal{U}}^{\beta}_{T}$ are active or have been active
before time $T$ due to the coupling. Conversely, the active clusters in
$\overline{\mathscr{X}}^{\beta’}_{n^{\prime}_{0},T}$ are issued from active
clusters whose labels are in $\overline{\mathcal{U}}^{\beta}_{T}$, which
allows us to use the branching property (see Figure 7):

$\displaystyle\langle\overline{X}^{\beta^{\prime}}_{n^{\prime}_{0},T},1\rangle=\sum_{u\in\overline{\mathcal{U}}^{\beta}_{T}}N_{T}^{\beta’}(u).$

Here $N_{T}^{\beta’}(u)$ is the number of active clusters in
$\overline{\mathscr{X}}^{\beta’}_{n^{\prime}_{0},T}$ that are issued from the
cluster labeled by $u$ (either its descendants, or itself if it is still alive
up to $T$) in $(\overline{\mathscr{X}}^{\beta’}_{n^{\prime}_{0},t})_{0<t\leq
T}$, and we also notice that
$\langle\overline{X}^{\beta}_{n_{0},T},1\rangle=\sum_{u\in\overline{\mathcal{U}}^{\beta}_{T}}1$.

The comparison of
$\langle\overline{X}^{\beta^{\prime}}_{n^{\prime}_{0},T},1\rangle$ and
$\langle\overline{X}^{\beta}_{n_{0},T},1\rangle$ is reduced to that between
the mean of $N_{T}^{\beta’}(u)$ and $1$. For any active cluster labeled by $u$
that has evolved to time $T$ in
$(\overline{\mathscr{X}}^{\beta}_{n_{0},t})_{t\geq 0}$, the coupled evolution
of the corresponding active cluster in
$(\overline{\mathscr{X}}^{\beta’}_{n^{\prime}_{0},t})_{t\geq 0}$ is at a
random time $\tau_{T}(u)<T$. Thus $N_{T}^{\beta’}(u)$ is the number of
clusters issued from the cluster labeled $u$ in
$(\overline{\mathscr{X}}^{\beta’}_{n^{\prime}_{0},t})_{t\geq 0}$ that evolves
for the extra time $(T-\tau_{T}(u))$. We recall that each active cluster
either fragments or be isolated with fixed probabilities whose difference is
$(\gamma-\theta)/(\gamma+\theta)$. Then, if $\gamma>\theta$ (resp.
$\gamma<\theta$), $N_{T}^{\beta’}(u)$ as the mean number of clusters issued
from the cluster labeled by $u$ during the extra time $(T-\tau_{T}(u))$ is
larger than $1$ (resp. smaller than $1$). This finishes the proof of
Proposition 6.3.

Figure 7: An illustration of the monotone coupling argument and the proof of
Proposition 6.3. The blue crosses represent isolation. The figure on the left
is for $(\overline{\mathscr{X}}^{\beta}_{n_{0},t})_{0\leq t\leq T}$, and the
one on the right is for
$(\overline{\mathscr{X}}^{\beta^{\prime}}_{n^{\prime}_{0},t})_{0\leq t\leq
T}$. With a monotone coupling, the evolution on the right is quicker than the
one on the left, so each cluster at time $T$ on the left can find its coupled
one on the right at a random moment before $T$. The rest of evolution on the
right (the green cones) is not coupled with the left up to time $T$, thus the
monotonicity depends on whether the extra time will make the population grow
or diminish. That is, the monotonicity depends on the phase (supercricial,
critical and subcritical) that the population belongs to.

####  6.3.4 Proof of Theorem 2.4

We combine Proposition 6.1, Proposition 6.2 and Proposition 6.3 to obtain
Theorem 2.4.

##  7 Examples and simulations

In this part, we give some examples about how our model and theorems can be
applied to perform the numerical simulation of epidemics. The key quantity of
our model is the Malthusian exponent $\lambda$. It dictates whether the
infection will grow or vanish, and moreover serves as the growth or decay
rate. We described how $\lambda$ depends on $(\beta,\theta,\gamma)$ in
Proposition 6.1 and 6.3. However, the function $\lambda(\beta,\theta,\gamma)$
is not explicit, so it is useful to investigate its values numerically. We use
Finite Difference Method applied to (13) to approximate $M_{t}$ and then use
(29) to approximate $\lambda$.

We simulated $\lambda$ for different values of the three parameters in Figure
8. Figure 8(a) gives a surface of the function $\lambda$ with $\beta=1$, which
confirms the monotonicity of $\lambda$ on $\theta$ and $\gamma$ (see also
Figure 8(b) and Figure 8(c)). Thus to reduce infection, we should conduct more
contact tracing (increase $\theta$) and avoid loss of contact information
(decrease $\gamma$). These two approaches are natural and intuitive.

Figure 8(d) confirms the monotonicity of $\lambda$ on $\beta$, see Proposition
6.3. When $\theta=0.03,0.06,0.09$ (thus $\theta<\gamma$), $\lambda$ is
increasing on $\beta$; when $\theta=0.12,0.15$ (thus $\theta>\gamma$),
$\lambda$ is a decreasing function of $\beta$.

(a) The values of $\lambda$ with $\beta=1$.

(b) Each curve describes the dependence of $\lambda$ on $\theta$, with
$\gamma,\beta$ fixed.

(c) Each curve describes the dependence of $\lambda$ on $\gamma$, with
$\theta,\beta$ fixed.

(d) Each curve describes the dependence of $\lambda$ on $\beta$, with
$\theta,\gamma$ fixed.

Figure 8: The dependence of $\lambda$ on $(\beta,\theta,\gamma)$.

##  8 Complementary results and further discussions

In this part, we discuss some other properties and generalizations of our
model.

###  8.1 Fragmentation by removing vertices

One can also consider a similar GFI process where the fragmentation is
generated by removing vertices. We call the new process v-GFI process for
short. More precisely, this process has the same dynamics of growth and
isolation, but every vertex rings independently with an exponential clock of
parameter $\gamma$, and once it rings, the associated vertex is removed from
the cluster to generate several subclusters.

For the model of fragmentation by removing vertices on RRT (without
isolation), some interesting properties about the size distribution have been
discussed in [24]. Especially, a generalized splitting property should hold
for our v-GFI process, so we can also follow the same strategy to study its
size process $(X_{t},Y_{t})_{t\geq 0}$. Moreover, by [24, eq.(7)] we deduce
that the generator for the first moment semigroup of $(X_{t})_{t\geq 0}$ is

$\begin{split}\mathcal{L}_{\bullet}f(n)&=\beta n(f(n+1)-f(n))-\theta nf(n)\\\
&\qquad\qquad\qquad-\gamma nf(n)+\gamma
n\sum_{j=1}^{n-1}\left(\frac{1}{j(j+1)}+\frac{1}{(n-j)(n-j+1)}\right)f(j).\end{split}$
(76)

Compared to (14), we have $\mathcal{L}_{\bullet}f=\mathcal{L}f-\gamma f$,
which implies that ${e^{\mathcal{L}_{\bullet}t}=e^{-\gamma
t}e^{\mathcal{L}t}=e^{-\gamma t}M_{t}}$. Therefore we conclude that, quite
surprisingly, $\mathcal{L}$ and $\mathcal{L}_{\bullet}$ have exactly the same
eigenelements $(\pi,h)$ given in Proposition 4.1, with a modification of
Perron’s root $\lambda\mapsto\lambda-\gamma$. We can expect to recover other
similar properties for the v-GFI process.

###  8.2 Role of the initial condition

Although the main results are proved with the initial condition
$G_{0}=\\{0\\}$ (a patient zero), the proofs and results will not change much
if the initial cluster is a RRT of any size. Indeed the reduction of the study
to a growth fragmentation isolation process for sizes works similarly.

Besides, extension of the results to an initial condition given by a
collection of RRTs can be achieved using that each initial cluster evolves
independently by branching property.

Finally, one may wish to start from one (or a collection of) deterministic
finite tree(s).We believe that our work can be adapted or used in that
purpose. For instance using the stopping line when the clusters have size $1$
to exploit the results of this paper.

###  8.3 Model generalizations

We can use the same method to deal with generalizations such as allowing a
cluster to lose several edges at the same time, or infect several individuals
at the same time. More delicate generalizations could be interesting too:

  * •

Adding recovering;

  * •

Breaking the Markov property (adding an age would register the time for
infection and would affect the individual rates in the epidemics:
contamination, loss of contact information, and recovery);

  * •

Improving and enriching the tracing procedure;

  * •

Adding heterogeneity within the population (for tracing and infection).

The reduction we have exploited (from collections of trees to collection of
sizes, due to the nice splitting property of RRT) should fail in general. We
hope there is still some nice probabilistic constructions (even if more
complex) to find but it is still speculative. We are confident on the
extension of asymptotic results on the Malthusian growth and empirical
measures. It could be achieved without going through a reduction of the state
space and without describing the distribution of clusters at fixed times, but
rather exploiting the general statement on semigroups that we have used here
on a more complex state space (space of trees). In particular, the Lyapunov
functions that we have exhibited here may still be relevant to deal with
processes in more general state spaces.

###  8.4 Finite-distance contact tracing

In our model, the contact tracing is implemented for the whole cluster despite
of its size, which is very rare in reality. It is natural to ask whether the
properties observed in our model also hold for the finite-distance contact
tracing. The answer is probably true, but the justification requires some new
mathematical techniques.

##  Appendix A Conditional expectation and variance

We prove here Lemma 5.1 for the conditional expectation and variance estimates
of

$Z_{k,p,K}:=\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]_{>K}}\right\rangle-\left\langle{\widetilde{X}_{k\Delta},[x^{p}]_{>K}}\right\rangle,$

used in (50). Here $(\widetilde{X}_{t})_{t\in[k\Delta,(k+1)\Delta)}$,
containing only growth and no fragmentation nor isolation, is a coupling of
$(X_{t})_{t\in[k\Delta,(k+1)\Delta)}$.

###### Proof of Lemma 5.1.

For the conditional expectation, we decompose it using the genealogy of
clusters

$\mathbb{E}[Z_{k,p,K}|\mathcal{F}_{k\Delta}]=\sum_{u\in\mathcal{U}_{k\Delta}}\mathbb{E}\left[[x^{p}]_{>K}(\widetilde{X}_{(k+1)\Delta}^{u})-[x^{p}]_{>K}(\widetilde{X}_{k\Delta}^{u})\,\,\Big{|}\,\,\mathcal{F}_{k\Delta}\right].$
(77)

We observe that for any $0<a\leq b$,

$\displaystyle[x^{p}]_{>K}(b)-[x^{p}]_{>K}(a)$
$\displaystyle=(b^{p}-a^{p})\mathbf{1}_{\left\\{b>K\right\\}}+a^{p}\mathbf{1}_{\left\\{a\leq
K<b\right\\}}\mathbf{1}_{\left\\{a\leq
K\right\\}}+a^{p}\mathbf{1}_{\left\\{a\leq
K<b\right\\}}\mathbf{1}_{\left\\{a>K\right\\}}.$

On the right hand side, the second term can be bounded by
${K^{p}\mathbf{1}_{\left\\{a\leq K<b\right\\}}}$ and the third term is zero,
so we have

$\displaystyle[x^{p}]_{>K}(b)-[x^{p}]_{>K}(a)\leq(b^{p}-a^{p})+K^{p}\mathbf{1}_{\left\\{a\leq
K<b\right\\}}.$ (78)

Since there is only growth in the process $\widetilde{X}_{t}$ on
$[k\Delta,(k+1)\Delta)$,
$\widetilde{X}_{(k+1)\Delta-}^{u}\geq\widetilde{X}_{k\Delta}^{u}$ for any
$u\in\mathcal{U}_{k\Delta}$. So we can apply (78) with
$a=\widetilde{X}_{k\Delta}^{u}$ and $b=\widetilde{X}_{(k+1)\Delta-}^{u}$ to
obtain that

$\mathbb{E}[Z_{k,p,K}|\mathcal{F}_{k\Delta}]\leq\mathbb{E}\left[\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]}\right\rangle-\left\langle{\widetilde{X}_{k\Delta},[x^{p}]}\right\rangle\,\,|\,\,\mathcal{F}_{k\Delta}\right]\\\
+{K^{p}\sum_{u\in\mathcal{U}_{k\Delta}}\mathbb{E}\left[\mathbf{1}_{\left\\{\widetilde{X}_{k\Delta}^{u}\leq
K<\widetilde{X}_{(k+1)\Delta}^{u}\right\\}}|\mathcal{F}_{k\Delta}\right]}.$
(79)

For the first term on the right hand side in (79), we follow the proof of
Lemma 4.1-(i) to get

$\begin{split}\mathbb{E}\left[\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{p}]}\right\rangle-\left\langle{\widetilde{X}_{k\Delta},[x^{p}]}\right\rangle\,\,|\,\,\mathcal{F}_{k\Delta}\right]&\leq\left(e^{2^{p-1}p\beta\Delta}-1\right)\left\langle{\widetilde{X}_{k\Delta},[x^{p}]}\right\rangle.\end{split}$

For the second term in (79), we can control it by the total number of active
clusters of size smaller than $K$ at $k\Delta$ that grow within
$[k\Delta,(k+1)\Delta)$:

$\displaystyle
K^{p}\sum_{u\in\mathcal{U}_{k\Delta}}\mathbb{E}\left[\mathbf{1}_{\left\\{\widetilde{X}_{(k+1)\Delta}^{u}>K\right\\}}-\mathbf{1}_{\left\\{{\widetilde{X}}_{k\Delta}^{u}\leq
K\right\\}}\,\,\Big{|}\,\,\mathcal{F}_{k\Delta}\right]$ $\displaystyle\leq
K^{p}\sum_{u\in\mathcal{U}_{k\Delta}}\mathbb{E}\left[\mathbf{1}_{\left\\{{\widetilde{X}}_{k\Delta}^{u}\leq
K,\text{ the cluster labeled by }u\text{ grows in }\widetilde{X}_{t}\text{
within
}[k\Delta,(k+1)\Delta)\right\\}}\,\,\Big{|}\,\,\mathcal{F}_{k\Delta}\right]$
$\displaystyle\leq(1-e^{-\beta\Delta
K})K^{p}\left\langle{\widetilde{X}_{k\Delta},1}\right\rangle.$

Plugging in the two inequalities to (79) yields (51).

For the conditional variance, we have

$\displaystyle\operatorname{var}\left[Z_{k,p,K}\,\,|\,\,\mathcal{F}_{k\Delta}\right]=\operatorname{var}\left[\sum_{u\in\mathcal{U}_{k\Delta}}\left([x^{p}]_{>K}(\widetilde{X}_{(k+1)\Delta-}^{u})-[x^{p}]_{>K}(\widetilde{X}_{k\Delta}^{u})\right)\,\,|\,\,\mathcal{F}_{k\Delta}\right].$

By branching property,

$\displaystyle\operatorname{var}\left[Z_{k,p,K}\,\,|\,\,\mathcal{F}_{k\Delta}\right]$
$\displaystyle=\sum_{u\in\mathcal{U}_{k\Delta}}\operatorname{var}\left[\left([x^{p}]_{>K}(\widetilde{X}_{(k+1)\Delta-}^{u})-[x^{p}]_{>K}(\widetilde{X}_{k\Delta}^{u})\right)\,\,|\,\,\mathcal{F}_{k\Delta}\right]$
$\displaystyle\leq\sum_{u\in\mathcal{U}_{k\Delta}}\mathbb{E}\left[\left([x^{p}]_{>K}(\widetilde{X}_{(k+1)\Delta-}^{u})-[x^{p}]_{>K}(\widetilde{X}_{k\Delta}^{u})\right)^{2}\,\,|\,\,\mathcal{F}_{k\Delta}\right]$
$\displaystyle\leq\sum_{u\in\mathcal{U}_{k\Delta}}\mathbb{E}\left[[x^{p}]^{2}_{>K}(\widetilde{X}_{(k+1)\Delta}^{u}-)-[x^{p}]^{2}_{>K}(\widetilde{X}_{k\Delta}^{u})\,\,|\,\,\mathcal{F}_{k\Delta}\right]$
$\displaystyle=\mathbb{E}\left[\left\langle{\widetilde{X}_{(k+1)\Delta-},[x^{2p}]_{>K}}\right\rangle-\left\langle{\widetilde{X}_{k\Delta},[x^{2p}]_{>K}}\right\rangle\,\,|\,\,\mathcal{F}_{k\Delta}\right].$

From the second line to the third line we use $(a-b)^{2}\leq a^{2}-b^{2}$ for
any $a>b>0$. The rest is the same as in the computation of conditional
expectation and we obtain (52). ∎

##  Appendix B Coupling for modified GFI processes

We present here the details of the proof of Proposition 6.3. We will prove the
monotone property Lemma 6.4 in Section B.1, which is divided into the part for
size in Section B.1.1 and the part for waiting time in Section B.1.2. Then we
apply this coupling along one cluster to the whole process in $[0,T]$ and
prove Proposition 6.3 in Section B.2

Throughout this section, we fix $\theta,\gamma,T>0$,
$\beta^{\prime}\geq\beta>0$ and two positive integers $n^{\prime}_{0}\geq
n_{0}$. Our object is a probability space
$\left(\Omega,\mathcal{G},\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\right)$
as a coupling such that two modified GFI processes live in it: one is of
parameters $(\beta,\theta,\gamma)$ starting from an initial RRT of size
$n_{0}$, and the other is of parameters $(\beta^{\prime},\theta,\gamma)$
starting from an initial RRT of size $n^{\prime}_{0}$. We denote by
$(\overline{X}_{t})_{0\leq t\leq T}$ the size process of the former, and
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$ the size process of the latter.
We will usually abuse these two notations to indicate the two modified GFI
processes.

Recall the Ulam-Harris-Neveu notation defined in Section 3.1. For any cluster
$u\in\mathcal{U}$, its birth time is the moment when his parent makes the
fragmentation, and its end of lifetime is the moment for its fragmentation or
isolation event. During its lifetime, it has several growth events but the
label is unchanged.

###  B.1 Coupling for one cluster

####  B.1.1 Coupling for the size distribution

As we said, the construction consists of the coupling of the event and the
size distribution. We focus at first the later one, and study the passage of
size domination between the generation.

######  Algorithm 1 (Coupling for the size distribution).

For the two modified processes $(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$
and $(\overline{X}_{t})_{0\leq t\leq T}$ defined above and each cluster
labeled $u\in\mathcal{U}$ appearing in $(\overline{X}_{t})_{0\leq t\leq T}$,
we use two independent random variables

$\displaystyle\widehat{U}(u)\sim\operatorname{Uniform}[0,1],\qquad\widehat{V}(u)\sim\operatorname{Bernoulli}\left(\frac{\gamma}{\gamma+\theta}\right),$

to sample the fragmentation/isolation event when it comes to the end of
lifetime. We denote by $N^{\prime},N$ respectively for the size of $u$ in
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$ and $(\overline{X}_{t})_{0\leq
t\leq T}$ at the end of its lifetime.

  1. 1.

If $\widehat{V}(u)=1$, the event is fragmentation in both processes. We then
define

$\displaystyle
j:=\left\lfloor\frac{N}{(N-1)\widehat{U}(u)+1}\right\rfloor,\qquad
j^{\prime}:=\left\lfloor\frac{N^{\prime}}{(N^{\prime}-1)\widehat{U}(u)+1}\right\rfloor,$
(80)

and we set $(N-j,j)$ (resp. $(N^{\prime}-j^{\prime},j^{\prime})$) for the size
of the first child cluster and the second child cluster in
$(\overline{X}_{t})_{0\leq t\leq T}$ (resp.
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$).

  2. 2.

If $\widehat{V}(u)=0$, the event is isolation in both processes

We justify that this coupling of size gives us the desired passage of the size
domination between the generation.

######  Proposition B.1.

Algorithm 1 is consistent with the fragmentation and isolation in the modified
GFI process. Moreover, supposing $N^{\prime}\geq N$, we also have the
domination for the size of child clusters

$\displaystyle j^{\prime}\geq j,\qquad N^{\prime}-j^{\prime}\geq N-j.$ (81)

###### Proof.

As defined at the beginning of Section 6.3.1, the fragmentation and isolation
are all attached to the edge in the modified GFI process. Then conditioned
that an event is fragmentation or isolation, the probability to be
fragmentation is $\frac{\gamma}{\gamma+\theta}$, which is independent of its
size. Moreover, following the splitting property 3.1, the event that a cluster
of size $N$ has a second child of size $j$ can be sampled by

$\displaystyle\left(\frac{N}{N-1}\right)\frac{1}{j+1}<\widehat{U}(u)+\frac{1}{N-1}\leq\left(\frac{N}{N-1}\right)\frac{1}{j},$

which is equivalent to (80). This proves the consistence of the coupling of
size.

We then prove the size domination. Notice that the event type is same for the
two processes, because it is determined by the common random variables
$\widehat{U}(u),\widehat{V}(u)$. The domination thus (81) can be reduced to
prove the following inequality: for every $x\in(0,1]$ and two positive
integers $N<N^{\prime}$, we have

$\displaystyle
0\leq\left\lfloor\frac{N^{\prime}}{N^{\prime}x+(1-x)}\right\rfloor-\left\lfloor\frac{N}{Nx+(1-x)}\right\rfloor\leq
N^{\prime}-N.$ (82)

For the first inequality, we notice that

$\displaystyle\frac{N}{Nx+(1-x)}=\frac{1}{x+(1-x)/N},$

so it is increasing with respect to $N$. For the second inequality, we have

$\displaystyle\left\lfloor\frac{N^{\prime}}{N^{\prime}x+(1-x)}\right\rfloor-\left\lfloor\frac{N}{Nx+(1-x)}\right\rfloor$
$\displaystyle\leq\frac{N^{\prime}}{N^{\prime}x+(1-x)}-\left(\frac{N}{Nx+(1-x)}-1\right)$
$\displaystyle=\frac{(N^{\prime}-N)(1-x)}{(1+(N^{\prime}-1)x)(1+(N-1)x)}+1$
$\displaystyle<(N^{\prime}-N)+1.$

From the second line to the third line, because $x>0$, we have $1-x<1$,
${1+(N^{\prime}-1)x>1}$, ${1+(N-1)x>1}$ and “=” cannot be attained. Since the
difference is an integer, this implies the second inequality in (82) and we
finish the proof. ∎

####  B.1.2 Coupling for the waiting time

We now turn to the coupling of the waiting time, and aim to let larger cluster
produce more growths within shorter lifetime. The main idea is a backward
time-change sampling: we sample at first a pre-waiting time for the
fragmentation/isolation event, which is generated by the current cluster size
by supposing no growth events during its lifetime. Then we need to fill in the
lifetime with the growth events, and make necessary time change with respect
to the latest cluster size. After this time change, the pre-waiting time
becomes the real waiting time of the fragmentation/isolation event.

Now, we give the detailed description of the backward time-change sampling in
a modified GFI process of parameter $(\beta,\theta,\gamma)$.

######  Algorithm 2 (Backward time-change sampling).

For $u\in\mathcal{U}$, we sample the events in its lifetime as following. Here
all the random variables
$W(u),\widetilde{W}(u),(Z_{i}(u))_{i\in\mathbb{N}_{+}}$ are independent. We
denote by $n$ the cluster size of $u$ at birth.

  1. (i)

If $n\geq 2$, we apply the following steps:

    1. (i.1)

Set $\tau_{0}(u)=0$.

    2. (i.2)

Sample $W(u)\sim\operatorname{Exp}((\gamma+\theta)(n-1))$ as the pre-waiting
time from $\tau_{0}(u)$ for the fragmentation/isolation event for $u$. To
unify the notation, we set $\kappa_{0}(u):=W(u)$ as the pre-waiting time from
$\tau_{0}(u)$.

    3. (i.3)

For any $i\in\mathbb{N}$, suppose that we have already obtained $\tau_{i}(u)$
for the $i$-th growth event and $\kappa_{i}(u)$ as the pre-waiting time of the
fragmentation/isolation from $\tau_{i}(u)$. We sample the ${(i+1)}$-th growth
event, whose waiting time from $\tau_{i}(u)$ is sampled from a random variable
${Z_{i+1}(u)\sim\operatorname{Exp}(\beta(n+i))}$. Then there are two cases.

       * •

If $Z_{i+1}(u)<\kappa_{i}(u)$, then the $(i+1)$-th growth event happens before
the fragmentation/isolation event. We set

$\displaystyle\tau_{i+1}(u):=\tau_{i}(u)+Z_{i+1}(u),$ (83)

as the moment for the $(i+1)$-th growth event. We also refresh the pre-waiting
time for the fragmentation/isolation event from $\tau_{i+1}(u)$ by

$\displaystyle\kappa_{i+1}(u):=\left(\frac{n+i-1}{n+i}\right)\left(\kappa_{i}(u)-Z_{i+1}(u)\right).$
(84)

       * •

If $Z_{i+1}(u)\geq\kappa_{i}(u)$, then there are only $i$ growth events before
the fragmentation/isolation event. Therefore, we stop the recurrence of (i.3)
and set

$\displaystyle\tau_{\dagger}(u):=\tau_{i}(u)+\kappa_{i}(u),$ (85)

as the length of lifetime of $u$.

  2. (ii)

If $n=1$ at birth, we apply the following steps:

    1. (ii.1)

Sample $\widetilde{W}(u)\sim\operatorname{Exp}(\beta)$ for the waiting time of
the first growth event of $u$. Set $\tau_{0}:=\widetilde{W}(u)$.

    2. (ii.2)

Repeat step (i.2)-(i.3) for $u$ as a cluster of size $2$.

This algorithm is also consistent with the modified GFI process thanks to the
memory-less property of the exponential random variables. More precisely, we
make use of Lemma B.1, which ensures that the time change in the step (i.3)
gives us an new independent exponential random variable after the growth.

######  Lemma B.1.

For three independent random variables $Z_{1},Z_{2},Z_{3}$ such that
${Z_{i}\sim\operatorname{Exp}(\alpha_{i})}$, with ${\alpha>0}$ and
${i=1,2,3}$, we have

$\displaystyle\left(Z_{1},\frac{\alpha_{2}}{\alpha_{3}}(Z_{2}-Z_{1})\right)\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}\stackrel{{\scriptstyle\text{(d)}}}{{=}}(Z_{1},Z_{3})\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}.$

###### Proof.

It suffices to prove that, for any $f\in C_{b}(\mathbb{R}^{2})$, we have

$\displaystyle\mathbb{E}\left[f\left(Z_{1},\frac{\alpha_{2}}{\alpha_{3}}(Z_{2}-Z_{1})\right)\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}\right]=\mathbb{E}[f(Z_{1},Z_{3})\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}].$ (86)

For the left-hand side, we have

$\displaystyle\mathbb{E}\left[f\left(Z_{1},\frac{\alpha_{2}}{\alpha_{3}}(Z_{2}-Z_{1})\right)\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}\right]$
$\displaystyle=\int_{0}^{\infty}\alpha_{1}e^{-\alpha_{1}z_{1}}\int_{z_{1}}^{\infty}\alpha_{2}e^{-\alpha_{2}z_{2}}f\left(z_{1},\frac{\alpha_{2}}{\alpha_{3}}(z_{2}-z_{1})\right)\,{\mathrm{d}}z_{2}{\mathrm{d}}z_{1}$
$\displaystyle=\int_{0}^{\infty}\alpha_{1}e^{-(\alpha_{1}+\alpha_{2})z_{1}}\int_{z_{1}}^{\infty}\alpha_{2}e^{-\alpha_{2}(z_{2}-z_{1})}f\left(z_{1},\frac{\alpha_{2}}{\alpha_{3}}(z_{2}-z_{1})\right)\,{\mathrm{d}}z_{2}{\mathrm{d}}z_{1}.$

By setting $z_{3}=\frac{\alpha_{2}}{\alpha_{3}}(z_{2}-z_{1})$, we have

$\mathbb{E}\left[f\left(Z_{1},\frac{\alpha_{2}}{\alpha_{3}}(Z_{2}-Z_{1})\right)\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}\right]\\\
=\int_{0}^{\infty}\alpha_{1}e^{-(\alpha_{1}+\alpha_{2})z_{1}}\int_{0}^{\infty}\alpha_{3}e^{-\alpha_{3}z_{3}}f(z_{1},z_{3})\,{\mathrm{d}}z_{3}{\mathrm{d}}z_{1}.$
(87)

For the right-hand side of (86), we have

$\begin{split}&\mathbb{E}[f(Z_{1},Z_{3})\mathbf{1}_{\left\\{Z_{1}\leq
Z_{2}\right\\}}]\\\
&=\int_{0}^{\infty}\alpha_{1}e^{-\alpha_{1}z_{1}}\int_{z_{1}}^{\infty}\alpha_{2}e^{-\alpha_{2}z_{2}}\int_{0}^{\infty}\alpha_{3}e^{-\alpha_{3}z_{3}}f\left(z_{1},z_{3}\right)\,{\mathrm{d}}z_{3}{\mathrm{d}}z_{2}{\mathrm{d}}z_{1}\\\
&=\int_{0}^{\infty}\alpha_{1}e^{-(\alpha_{1}+\alpha_{2})z_{1}}\int_{0}^{\infty}\alpha_{3}e^{-\alpha_{3}z_{3}}f(z_{1},z_{3})\,{\mathrm{d}}z_{3}{\mathrm{d}}z_{1}.\end{split}$
(88)

We combine (87) and (88), which proves Lemma B.1. ∎

The advantage of Algorithm 2 is that the waiting time of
fragmentation/isolation event largely depends on one random variable $W(u)$.
This helps us construct a coupling to ensure the domination of lifetime
length.

######  Algorithm 3 (Coupling of waiting time).

For any $u\in\mathcal{U}$ appearing in $(\overline{X}_{t})_{0\leq t\leq T}$,
let $n,n^{\prime}$ be respectively its size in $(\overline{X}_{t})_{0\leq
t\leq T}$ and $(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$ at birth. If
$n\leq n^{\prime}$, we have the following coupling for its evolution in two
processes using Algorithm 2. Here for every quantity
($W(u),Z_{i}(u),\tau_{i}(u),\kappa_{i}(u)\cdots$) in Algorithm 2, we keep its
original notation for that in $(\overline{X}_{t})_{0\leq t\leq T}$, and the
notation with superscript
($W^{\prime}(u),Z^{\prime}_{i}(u),\tau^{\prime}_{i}(u),\kappa^{\prime}_{i}(u)\cdots$)
for that in $(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$.

  1. (i)

Sample two sets of random variables $\Lambda(u),\widehat{\Lambda}(u)$ that

$\displaystyle\Lambda(u):=\\{\chi(u),\eta(u),(\xi_{i}(u))_{i\in\mathbb{N}_{+}}\\},\qquad\widehat{\Lambda}(u):=\\{\widehat{\chi}(u),\widehat{\eta}(u),(\widehat{\xi}_{i}(u))_{i\in\mathbb{N}_{+}}\\}.$

All the random variables are independent and they satisfy

$\displaystyle\chi(u),\eta(u),\xi_{i}(u),\widehat{\chi}(u),\widehat{\eta}(u),\widehat{\xi}_{i}(u)\sim\operatorname{Exp}(1),\quad\forall
i\in\mathbb{N}_{+}.$

The random variables in $\widehat{\Lambda}(u)$ are used in both processes,
while that in $\Lambda(u)$ are only used in $(\overline{X}_{t})_{0\leq t\leq
T}$.

  2. (ii)

We use $\widehat{\eta}(u)$ to generate the common randomness concerning the
waiting time of fragmentation/isolation

$\begin{split}W(u):=\frac{\widehat{\eta}(u)}{(\gamma+\theta)(n-1+\mathbf{1}_{\left\\{n=1\right\\}})},\qquad
W^{\prime}(u):=\frac{\widehat{\eta}(u)}{(\gamma+\theta)(n^{\prime}-1+\mathbf{1}_{\left\\{n^{\prime}=1\right\\}})}.\end{split}$
(89)

Here the indicator function comes from the fact that there is a supplementary
growth event before sampling the fragmentation/isolation event in (ii) of
Algorithm 2; we will also see them in the following paragraphs.

  3. (iii)

For the coupling of the growth events, we distinguish two cases.

    1. (iii.1)

If $n^{\prime}=n$, then we use the random variable $\widehat{\Lambda}(u)$ to
generate the common randomness of growths for the two processes. That is, we
set

$\begin{split}&\widetilde{W}(u)=\frac{\widehat{\chi}(u)}{\beta},\qquad
Z_{i}(u)=\frac{\widehat{\xi}_{i}(u)}{\beta(n+i-1+\mathbf{1}_{\left\\{n=1\right\\}})},\quad\forall
i\in\mathbb{N}_{+},\\\
&\widetilde{W}^{\prime}(u)=\frac{\widehat{\chi}(u)}{\beta^{\prime}},\qquad
Z^{\prime}_{i}(u)=\frac{\widehat{\xi}_{i}(u)}{\beta^{\prime}(n+i-1+\mathbf{1}_{\left\\{n=1\right\\}})},\quad\forall
i\in\mathbb{N}_{+}.\end{split}$ (90)

    2. (iii.2)

If $n^{\prime}>n$, then our strategy is to let the growth in
$(\overline{X}_{t})_{0\leq t\leq T}$ evolve independently until size
$n^{\prime}$, and then make coupling as (iii.1). That is, we let the first
$(n^{\prime}-n)$ growth events in $(\overline{X}_{t})_{0\leq t\leq T}$ be
sampled independently from that of $(\overline{X}^{\prime}_{t})_{0\leq t\leq
T}$ using the random variables in $\Lambda(u)$

$\displaystyle\widetilde{W}(u)=\frac{\chi(u)}{\beta},\qquad
Z_{i}(u)=\frac{\xi_{i}(u)}{\beta(n+i-1+\mathbf{1}_{\left\\{n=1\right\\}})},\quad\forall
1\leq i\leq(n^{\prime}-n-\mathbf{1}_{\left\\{n=1\right\\}}).$

If all the growth events above are before the fragmentation/isolation event in
$(\overline{X}_{t})_{0\leq t\leq T}$, then the current cluster size of $u$ is
also $n^{\prime}$, and we let the rest of the the evolution of $u$ in
$(\overline{X}_{t})_{0\leq t\leq T}$ be coupled with that in
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$

$\begin{split}Z^{\prime}_{i}(u)=\frac{\widehat{\xi}_{i}(u)}{\beta^{\prime}(n^{\prime}+i-1)},\quad
Z_{i+n^{\prime}-n-\mathbf{1}_{\left\\{n=1\right\\}}}(u)=\frac{\widehat{\xi}_{i}(u)}{\beta(n^{\prime}+i-1)},\quad\forall
i\in\mathbb{N}_{+}.\end{split}$ (91)

Otherwise, the number of growth is less than $(n^{\prime}-n)$ in
$(\overline{X}_{t})_{0\leq t\leq T}$ and we only need the firth equation in
(91) for the growth events in $(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$.

It is not hard to verify that Algorithm 3 gives a coupling of Algorithm 2. One
only needs to check that all the random variables in Algorithm 3 have the
proper parameters as in Algorithm 2. Meanwhile, some common random variables
create correlations between the processes and this is the object of the
coupling. As we said, thanks to the backward time-change sampling, we only
need to sample one exponential random variable for the waiting time of
fragmentation/isolation, and it is quite natural to use a common random
variable $\widehat{\eta}(u)$ to create monotonicity. It is also the idea for
the growth events, but the case $n<n^{\prime}$ is a little more delicate.

######  Proposition B.2.

Let $u\in\mathcal{U}$ appearing in $(\overline{X}_{t})_{0\leq t\leq T}$, and
suppose two positive integers $n^{\prime}\geq n$ be respectively its size in
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$ and $(\overline{X}_{t})_{0\leq
t\leq T}$ at birth. Then in Algorithm 3, the length of lifetime is shorter in
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$ than that in
$(\overline{X}_{t})_{0\leq t\leq T}$, and its size at the end of lifetime is
also larger in the former process.

Before proving Proposition B.2, one intermediate step is to define a predicted
length of lifetime after $i$-th growth, as a generalization of (85) in
Algorithm 2

$\displaystyle\tau_{\dagger,i}(u):=\tau_{i}(u)+\kappa_{i}(u).$ (92)

The interpretation is that we have explored the information of events until
the moment $\tau_{i}(u)$, and suppose no more growth event after it. Thus we
make a sum of $\tau_{i}(u)$ and $\kappa_{i}(u)$. Heuristically, the more
growth events we explore, the closer to $\tau_{\dagger}(u)$ this quantity is.
We justify this observation.

######  Lemma B.2.

Let $n$ be the cluster size of $u$ at birth and $k$ be the number of growth
events after $\tau_{0}(u)$ in Algorithm 2, then we have an expression that for
every $0\leq i\leq k$

$\displaystyle\tau_{\dagger,i}(u)=\widetilde{W}(u)\mathbf{1}_{\left\\{n=1\right\\}}+\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}W(u)+\sum_{j=1}^{i}\frac{i+1-j}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}Z_{j}(u).$
(93)

Moreover, the sequence $(\tau_{\dagger,i}(u))_{0\leq i\leq k}$ is decreasing.

###### Proof.

The indicator $\mathbf{1}_{\left\\{n=1\right\\}}$ here is used to distinguish
the special case $n=1$ in Algorithm 2. We prove (93) for the case $n\geq 2$ at
first by recurrence. The case $i=0$ is trivial by the expression that

$\displaystyle\tau_{\dagger,0}(u)=\tau_{0}(u)+\kappa_{0}(u)=0+W(u)=W(u).$

Suppose that we have proved the case for $(i-1)$, then for the case $i$ we
apply the formula (83) and (84)

$\begin{split}\tau_{\dagger,i}(u)&=\tau_{i}(u)+\kappa_{i}(u)\\\
&=\tau_{i-1}(u)+Z_{i}(u)+\left(\frac{n+i-2}{n+i-1}\right)\left(\kappa_{i-1}(u)-Z_{i}(u)\right)\\\
&=\frac{1}{n+i-1}Z_{i}(u)+\tau_{i-1}(u)+\left(\frac{n+i-2}{n+i-1}\right)\kappa_{i-1}(u).\end{split}$
(94)

We then apply the recurrence of (93) at $(i-1)$ and obtain

$\displaystyle\tau_{\dagger,i}(u)$
$\displaystyle=\frac{1}{n+i-1}(Z_{i}(u)+\tau_{i-1}(u))+\left(\frac{n+i-2}{n+i-1}\right)(\kappa_{i-1}(u)+\tau_{i-1}(u))$
$\displaystyle=\frac{1}{n+i-1}\sum_{j=1}^{i}Z_{j}(u)+\left(\frac{n+i-2}{n+i-1}\right)\left(\frac{n-1}{n+i-2}W(u)+\sum_{j=1}^{i-1}\frac{i-j}{n+i-2}Z_{j}(u)\right)$
$\displaystyle=\frac{n-1}{n+i-1}W(u)+\sum_{j=1}^{i}\frac{i+1-j}{n+i-1}Z_{j}(u).$

This is the desired result of (93). For the monotonicity, we apply the
recurrence (94).

$\displaystyle\tau_{\dagger,i}(u)-\tau_{\dagger,i-1}(u)$
$\displaystyle=\frac{1}{n+i-1}Z_{i}(u)+\tau_{i-1}(u)+\left(\frac{n+i-2}{n+i-1}\right)\kappa_{i-1}(u)-(\tau_{i-1}(u)+\kappa_{i-1}(u))$
$\displaystyle=\frac{1}{n+i-1}(Z_{i}(u)-\kappa_{i-1}(u)).$

By (i.3) of Algorithm 2, the existence of $i$-th growth implies
$Z_{i}(u)<\kappa_{i-1}(u)$. Therefore, the difference above is negative and we
finish the proof.

Then for the case $n=1$, it suffice to add the first growth event at
$\tau_{0}(u)$, and treat the rest term as the evolution of a cluster of size
$2$. ∎

###### Proof of Proposition B.2.

We treat the two cases in the step (ii) of Algorithm 3.

  1. 1.

For the case $n^{\prime}=n$, the coupling in (89) and (90) gives us that

$\displaystyle
W^{\prime}(u)=W(u),\qquad\widetilde{W}^{\prime}(u)\leq\widetilde{W}(u),\qquad
Z^{\prime}_{i}(u)\leq Z(u),\forall i\in\mathbb{N}_{+},$ (95)

thanks to the fact $\beta^{\prime}\geq\beta$ and they use the same random
variables in $\widehat{\Lambda}(u)$. This implies in Algorithm 2 that the
waiting time for each growth event is always shorter in
$(\overline{X}^{\prime}_{t})_{t\geq 0}$, while the pre-waiting time
$\kappa_{0}(u),\kappa^{\prime}_{0}(u)$ for the fragmentation/isolation event
are same. By an induction in (84), we also establish
$\kappa^{\prime}_{i}(u)\geq\kappa_{i}(u)$, which ensures that we can explore
more growth events in $(\overline{X}^{\prime}_{t})_{t\geq 0}$. This finishes
the statement about the size in Proposition B.2 for the case $n^{\prime}=n$.

Concerning the length of lifetime, we use the formula (93) of the $i$-th
predicted length of lifetime $\tau_{\dagger,i}(u)$ that

$\displaystyle\tau_{\dagger,i}(u)=\widetilde{W}(u)\mathbf{1}_{\left\\{n=1\right\\}}+\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}W(u)+\sum_{j=1}^{i}\frac{i+1-j}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}Z_{j}(u).$

Compare it with its version in $(\overline{X}^{\prime}_{t})_{t\geq 0}$, we
have

$\widetilde{W}(u)\mathbf{1}_{\left\\{n=1\right\\}}+\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}W(u)+\sum_{j=1}^{i}\frac{i+1-j}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}Z_{j}(u)\\\
\geq\widetilde{W}^{\prime}(u)\mathbf{1}_{\left\\{n=1\right\\}}+\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}W^{\prime}(u)+\sum_{j=1}^{i}\frac{i+1-j}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}Z^{\prime}_{j}(u),$

because of (95). This implies that
$\tau_{\dagger,i}(u)\geq\tau^{\prime}_{\dagger,i}(u)$. As we know, there
exists a $k\in\mathbb{N}_{+}$ such that
$\tau_{\dagger}(u)=\tau_{\dagger,k}(u)$, so we have

$\displaystyle\tau_{\dagger}(u)=\tau_{\dagger,k}(u)\geq\tau^{\prime}_{\dagger,k}(u)\geq\tau^{\prime}_{\dagger}(u).$

Here the last inequality comes from the decreasing property in Lemma B.2 and
the fact there are more growth events in $(\overline{X}^{\prime}_{t})_{t\geq
0}$. This finishes the statement about the length of lifetime in Proposition
B.2 for the case $n^{\prime}=n$.

  2. 2.

For the case $n^{\prime}>n$, there are also two cases.

    1. (a)

There are less than $(n^{\prime}-n)$ growth events for $u$ in
$(\overline{X}_{t})_{t\geq 0}$. For this case, the statement about the size is
obvious as the cluster size is less than $n^{\prime}$ in
$(\overline{X}_{t})_{t\geq 0}$ of $u$ at the end of lifetime. We consider the
length of lifetime in $(\overline{X}_{t})_{t\geq 0}$ that

$\displaystyle\tau_{\dagger,i}(u)$
$\displaystyle=\widetilde{W}(u)\mathbf{1}_{\left\\{n=1\right\\}}+\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}W(u)+\sum_{j=1}^{i}\frac{i+1-j}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}Z_{j}(u)$
$\displaystyle\geq\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}W(u)$
$\displaystyle=\frac{\widehat{\eta}(u)}{(\gamma+\theta)(n+i-1+\mathbf{1}_{\left\\{n=1\right\\}})}.$

In the last line, we make use of (89). Notice that
$n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}$ is the cluster size of $u$ after
several growth events in its lifetime, and it should be less than $n^{\prime}$
by assumption. Thus, we obtain

$\displaystyle\tau_{\dagger,i}(u)\geq\frac{\widehat{\eta}(u)}{(\gamma+\theta)n^{\prime}}=W^{\prime}(u)=\tau^{\prime}_{\dagger,0}(u)\geq\tau^{\prime}_{\dagger}(u),$

where the last inequality comes from the decreasing property in Lemma B.2. We
choose $i$ as the index of the last growth event of $u$ in
$(\overline{X}_{t})_{t\geq 0}$, and this concludes the statement about the
length of lifetime.

    2. (b)

There are at least $(n^{\prime}-n)$ growth events for $u$ in
$(\overline{X}_{t})_{t\geq 0}$. From (ii).b of Algorithm 3, after
$(n^{\prime}-n)$ growth events, $u$ has the same cluster size in two processes
and the growth events are totally coupled. Equation (91) gives us

$\displaystyle Z^{\prime}_{i}(u)\leq
Z_{i+n^{\prime}-n-\mathbf{1}_{\left\\{n=1\right\\}}}(u),\qquad\forall
i\in\mathbb{N}_{+},$ (96)

thus the domination of waiting time for the growth events is also established.
For the pre-waiting time of the fragmentation/isolation event, (84) gives us

$\displaystyle\kappa_{i+1}(u)=\left(\frac{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i+\mathbf{1}_{\left\\{n=1\right\\}}}\right)\left(\kappa_{i}(u)-Z_{i+1}(u)\right)\leq\left(\frac{n+i-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+i+\mathbf{1}_{\left\\{n=1\right\\}}}\right)\kappa_{i}(u).$

So a telescope formula entails

$\begin{split}\kappa_{n^{\prime}-n-\mathbf{1}_{\left\\{n=1\right\\}}}(u)&\leq\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n+(n^{\prime}-n-\mathbf{1}_{\left\\{n=1\right\\}})-1+\mathbf{1}_{\left\\{n=1\right\\}}}\kappa_{0}(u)\\\
&=\frac{n-1+\mathbf{1}_{\left\\{n=1\right\\}}}{n^{\prime}-1}\frac{\widehat{\eta}(u)}{(\gamma+\theta)(n-1+\mathbf{1}_{\left\\{n=1\right\\}})}\\\
&=\kappa^{\prime}_{0}(u).\end{split}$ (97)

Here we use the the fact that $\widetilde{W}^{\prime}(u),\widetilde{W}(u)$ are
sampled by the same random variable $\widehat{\eta}(u)$ in (89). Therefore,
(97) also gives us a domination of the pre-waiting time that
${\kappa_{n^{\prime}-n-\mathbf{1}_{\left\\{n=1\right\\}}}(u)\leq\kappa^{\prime}_{0}(u)}$.
Combining this with (96), we can make use of a similar argument like the case
$n^{\prime}=n$ for the part of $u$ in $(\overline{X}_{t})_{t\geq 0}$ after
$(n^{\prime}-n)$ growth events, together with $u$ in
$(\overline{X}^{\prime}_{t})_{t\geq 0}$ from its birth. This concludes our
proof.

∎

###### Proof of Lemma 6.4.

A combination the coupling in Algorithm 1 and Algorithm 3 gives us desired
result; see Proposition B.1 and Proposition B.2 for proof. ∎

###  B.2 Complete proof of Proposition 6.3

We can now apply the coupling in previous sections to the whole process
$(\overline{X}_{t})_{0\leq t\leq T}$ and $(\overline{X}^{\prime}_{t})_{0\leq
t\leq T}$, which gives us desired property in the previous discussion in
Section 6.3.

######  Algorithm 4 (Coupling for modified GFI processes).

We define a set of random variables

$\displaystyle\Gamma(u):=\\{\widehat{U}(u),\widehat{V}(u)\\}\cup\widehat{\Lambda}(u)\cup\Lambda(u),$
(98)

where $\widehat{U}(u),\widehat{V}(u)$ are defined in Algorithm 1 and
$\widehat{\Lambda}(u),\Lambda(u)$ are defined in Algorithm 3. We construct our
probability space
$\left(\Omega,\mathcal{G},\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\right)$
and processes $(\overline{X}_{t},\overline{X}^{\prime}_{t})_{0\leq t\leq T}$
in it.

  1. (i)

Let $\Omega:=(\Gamma(u))_{u\in\mathcal{U}}$ and define
$\mathcal{G}:=\sigma\left(\cup_{u\in\mathcal{U}}\Gamma(u)\right)$, then sample
i.i.d. random variable sets $(\Gamma(u))_{u\in\mathcal{U}}$ which gives the
probability $\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}$.

  2. (ii)

Starting from the cluster of label $\emptyset$, follow the breadth-first
search order, i.e. by the order $\emptyset,1,2,11,12,21,22,111\cdots$ (see
[16, Chapter 22.2] for reference) and apply Algorithm 3 and Algorithm 1 with
respect to its birth time to each branch.

Recall the Ulam-Harris-Neveu notation defined in Section 3.1, where we use
$\mathcal{U}_{t}$ for the labels of the alive clusters at moment $t$,
$\mathcal{U}_{t}(u)$ for the one issued from $u$ and $\overline{X}^{u}_{t}$
for the cluster size of label $u$. We keep the original version for the
process $(\overline{X}_{t})_{0\leq t\leq T}$ and
$\mathcal{U}_{t},\mathcal{U}^{\prime}_{t}(u),\overline{X}^{\prime u}_{t}$ for
its version in $(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$. The following
proposition captures the intuition that “$(\overline{X}^{\prime}_{t})_{0\leq
t\leq T}$ evolves quicker than $(\overline{X}_{t})_{0\leq t\leq T}$, and for
every cluster the processes are no longer coupled from a certain random time”.

######  Proposition B.3.

The probability space
$\left(\Omega,\mathcal{G},\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\right)$
defined in Algorithm 4 satisfies the following properties.

  * •

$(\overline{X}_{t})_{0\leq t\leq T}$ and $(\overline{X}^{\prime}_{t})_{0\leq
t\leq T}$ live in it as a modified GFI process with respect to their natural
filtration.

  * •

Under the filtration $(\mathcal{G}_{t})_{0\leq t\leq T}$ defined as

$\displaystyle\mathcal{G}_{t}:=\sigma\left((\overline{X}^{\prime
u}_{s})_{0\leq s\leq t},(\overline{X}^{u}_{s})_{0\leq s\leq
T},u\in\mathcal{U}\right),$ (99)

for every $u\in\mathcal{U}_{T}$, there exists a stopping time
$\tau_{T}(u)\in[0,T]$ with respect to $(\mathcal{G}_{t})_{0\leq t\leq T}$ such
that $\overline{X}^{\prime u}_{\tau_{T}(u)}>0$ and
$\left(\sum_{v\in\mathcal{U}^{\prime}_{t}(u)}\delta_{\overline{X}^{\prime
v}_{t}}\right)_{\tau_{T}(u)\leq t\leq T}$ is Markov process with respect to
the filtration $(\mathcal{G}_{t})_{0\leq t\leq T}$.

###### Proof of Proposition B.3.

The first statement is the result of the recurrence of Algorithm 1 and 3. In
this recurrence, Proposition B.2 ensures the monotonicity of the length of
lifetime and cluster size when fragmentation, while Proposition B.1 pass this
monotonicity to the child clusters. Therefore,
$(\overline{X}^{\prime}_{t})_{0\leq t\leq T}$ evolves quicker than
$(\overline{X}_{t})_{0\leq t\leq T}$.

Concerning the second statement, when a cluster of label $u$ stops its
evolution at $T$ in $(\overline{X}_{t})_{0\leq t\leq T}$, its trajectory
translates to the information about $\Gamma(v)_{v\in\mathcal{U}}$, which then
translates to its coupled one in $(\overline{X}^{\prime}_{t})_{0\leq t\leq
T}$: it is still alive at a moment $\tau_{T}(u)$. This dependence implies that
$\tau_{T}(u)$ is a stopping time with respect to $(\mathcal{G}_{t})_{0\leq
t\leq T}$. Naturally, for the rest of process $(\overline{X}^{\prime
u})_{0\leq t\leq T}$ after $\tau_{T}(u)$, because $(\overline{X}_{t})_{0\leq
t\leq T}$ does not provide more information and then we obtain the Markov
property. Thus we finish the proof of the second statement. ∎

Proposition B.3 gives a rigorous proof of Proposition 6.3.

###### Proof of Proposition 6.3.

We use
$\mathbb{E}^{\beta}_{\delta_{n_{0}}},\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}$
respectively for the expectation on the probability space of
$(\overline{X}_{t})_{t\geq 0}$ and $(\overline{X}^{\prime}_{t})_{t\geq 0}$.
Our main object is to compare
$\mathbb{E}^{\beta}_{\delta_{n_{0}}}\left[\left\langle{\overline{X}_{T},1}\right\rangle\right]$
and
$\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}\left[\left\langle{\overline{X}^{\prime}_{T},1}\right\rangle\right]$,
which will illustrate Malthusian behavior for large $T$.

We use the common probability space
$\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}$ constructed in
Algorithm 4, and we write
$\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}$ for the associated
expectation. By Proposition B.3, under
$\mathbb{P}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}$, every cluster in
$\mathcal{U}_{T}$ has its coupled one in $(\overline{X}^{\prime}_{t})_{0\leq
t\leq T}$, which is still alive at the moment $\tau_{T}(u)$ and produces the
clusters in $\mathcal{U}^{\prime}_{T}$. Then we have

$\displaystyle\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}\left[\left\langle{\overline{X}^{\prime}_{T},1}\right\rangle\right]=\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\left[\sum_{v\in\mathcal{U}^{\prime}_{T}}1\right]=\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\left[\sum_{u\in\mathcal{U}_{T}}\sum_{v\in\mathcal{U}^{\prime}_{T}(u)}1\right].$

Since the modified GFI process is càdlàg, we apply the strong Markov property
that

$\displaystyle\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}\left[\left\langle{\overline{X}^{\prime}_{T},1}\right\rangle\right]$
$\displaystyle=\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\left[\sum_{u\in\mathcal{U}_{T}}\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\left[\sum_{v\in\mathcal{U}^{\prime}_{T}(u)}1\Bigg{|}\mathcal{G}_{\tau_{T}(u)}\right]\right]$
$\displaystyle=\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\left[\sum_{u\in\mathcal{U}_{T}}\mathbb{E}^{\beta^{\prime}}_{\delta_{\overline{X}^{\prime
u}_{\tau_{T}(u)}}}\left[\sum_{v\in\mathcal{U}^{\prime}_{T}(u)}1\Bigg{|}\mathcal{G}_{\tau_{T}(u)}\right]\right],$

and the quantity ${\mathbb{E}^{\beta^{\prime}}_{\delta_{\overline{X}^{\prime
u}_{\tau_{T}(u)}}}\left[\sum_{v\in\mathcal{U}^{\prime}_{T}(u)}1\Bigg{|}\mathcal{G}_{\tau_{T}(u)}\right]}$
can be seen as the mean of offspring for a modified GFI process issued from a
RRT of size $\overline{X}^{\prime u}_{\tau_{T}(u)}$.

Lemma 6.3 gives us some indication about the phases of the modified GFI
process, but here we need a more precise result. In fact, the mapping
$t\mapsto\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}[\langle\overline{X}^{\prime}_{t},1\rangle]$
is increasing (resp. decreasing) when $\gamma>\theta$ (resp. $\gamma<\theta$).
The idea is to use the first moment generator (73) and a constant test
function $\mathbf{1}$ that

$\displaystyle\overline{\mathcal{L}}\mathbf{1}(n)=(\gamma-\theta)(n-1).$

Therefore, the sign depends on $(\gamma-\theta)$. For the case
$\gamma>\theta$, we obtain

$\displaystyle\forall
u\in\mathcal{U}_{T},\qquad\mathbb{E}^{\beta^{\prime}}_{\delta_{\overline{X}^{\prime
u}_{\tau_{T}(u)}}}\left[\sum_{v\in\mathcal{U}^{\prime}_{T}(u)}1\Bigg{|}\mathcal{G}_{\tau_{T}(u)}\right]\geq
1,$

and from this we deduce

$\displaystyle\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}\left[\left\langle{\overline{X}^{\prime}_{T},1}\right\rangle\right]\geq\mathbb{E}^{\beta,\beta^{\prime}}_{n_{0},n^{\prime}_{0}}\left[\sum_{u\in\mathcal{U}_{T}}1\right]=\mathbb{E}^{\beta}_{\delta_{n_{0}}}\left[\left\langle{\overline{X}_{T},1}\right\rangle\right].$
(100)

Because of the Malthusian behavior (29), for a large $T$ we have

$\displaystyle\mathbb{E}^{\beta^{\prime}}_{\delta_{n^{\prime}_{0}}}\left[\left\langle{\overline{X}^{\prime}_{T},1}\right\rangle\right]\sim
h^{\prime}(n^{\prime}_{0})e^{\overline{\lambda}^{\prime}T},\qquad\mathbb{E}^{\beta}_{\delta_{n_{0}}}\left[\left\langle{\overline{X}_{T},1}\right\rangle\right]\sim
h(n_{0})e^{\overline{\lambda}T},$

where $h^{\prime},h$ are the associated eigenvectors. The monotonicity (100)
thus implies for the case $\gamma>\theta$:

  * •

when $\beta^{\prime}>\beta$, we have
$\overline{\lambda}^{\prime}\geq\overline{\lambda}$ ;

  * •

when $\beta^{\prime}=\beta,n_{0}^{\prime}>n_{0}$, we have
$h(n^{\prime}_{0})\geq h(n_{0})$.

The case $\gamma<\theta$ is similar. Finally, recall
${\lambda(\beta,\theta,\gamma)=\overline{\lambda}(\beta,\theta,\gamma)+\theta}$
as discussed in Section 6.3.1, we thus obtain the desired result. ∎

[Acknowledgments] This work was partially funded by the Chair “Modélisation
Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MNHN-F.X and ANR
ABIM 16-CE40-0001 and ANR NOLO 20-CE40-0015. L.Y. acknowledges the support of
the National Natural Science Foundation of China (Youth Programme, Grant:
11801458).

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