# Universality of SIS epidemics starting from small initial conditions
111Supported by the ÚNKP-21-1 New National Excellence Program of the Ministry
for Innovation and Technology from the source of the National Research,
Development and Innovation Fund. [width=0.05]logo1.png [width=0.05]logo2.jpg
222 Partially supported by the ERC Synergy under Grant No. 810115 - DYNASNET.

Dániel Keliger
Department of Stochastics, Institute of Mathematics,
Budapest University of Technology and Economics
Műegyetem rkp. 3., H-1111 Budapest, Hungary;
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
e-mail<EMAIL_ADDRESS>
###### Abstract

We are investigating deterministic SIS dynamics on large networks starting
from only a few infected individuals. Under mild assumptions we show that any
two epidemic curves – on the same network and with the same parameters – are
almost identical up to time translation when initial conditions are small
enough regardless of how infections are distributed at the beginning. The
limit of an epidemic starting from the infinite past with infinitesimal
prevalence is identified as the nontrivial eternal solution connecting the
disease free state with the endemic equilibrium. Our framework covers several
benchmark models including the N-Intertwined Mean Field Approximation (NIMFA)
and the Inhomogeneous Mean Field Approximation (IMFA).

##  1 Introduction

As it has been recently shown by the SARS-CoV-2 pandemic, understanding the
propagation of diseases is a crucial task for interconnected societies. Making
predictions about how much capacities hospitals will need, how effective
lockdown measures are, and what is the optimal protocol for distributing
vaccines are just few of the urgent questions faced by policy makers.

Such decisions can be supported by epidemiological models where the
consequences of certain policies can be simulated without the economic and
societal costs of large-scale experimentation. These models usually group
individuals into compartments (such as susceptible, infected, recovered,
exposed) who make random transitions between such compartments with transition
rates depending on the state of individuals being connected to her on a
contact network. Examples are the SIS model with two states: _susceptible_ (S)
and _infected_ (I), where infected individuals are cured at a constant rate
and susceptible individuals are getting infected at a rate proportional to the
number of their infected neighbors. Another example is the SI model, where no
curing events happen, making it more suitable to model e.g. the diffusion of
information.

Although these models provide flexibility, they have certain shortcomings
arising from the high dimensionality of the problem. The exact stochastic
model with a population of $n$ individuals has a state space of size $2^{n}$,
making direct calculations infeasible, invoking the need for Monte Carlo
simulations or certain reductions. Besides computational limitations,
obtaining detailed information about the entire contact network or the
complete initial configuration of infections can pose a challenge as well.

A common way to mitigate the problem of dimensionality is to apply mean field
approximations at several scales (and levels of accuracy). Examples from less
to more detailed methods are: the Homogenous Mean Field Approximation (HMFA),
where it is assumed that the population is well mixed [15], the Inhomogenous
Mean Field Approximation (IMFA), which keeps track of the degree distribution
and assumes individuals are statistically equivalent in the same degree class
[18] and the N-intertwined Mean Field Approximation (NIMFA) which treats the
population at the level of individuals keeping the full contact network and
only neglecting dynamical correlations between individuals [17]. One may also
apply so-called metapopulation models where people are grouped into smaller
communities – cities, regions etc. [4].

These mean-field models yield ODE systems where each equation describes the
evolution of one smaller community or individual. For example, in the HMFA
case, only one equation is needed, while NIMFA works with $n$ equations, which
is still much less than the $2^{n}$ equations needed for the exact stochastic
process. In this work, we utilize a recent graphon-based approach resulting in
a PDE [25, 8] which enables studying continuous populations among with all the
previously mentioned mean-field approximations.

We start from the observation that real world pandemics usually start from
only a few infected individuals, motivating the study of solutions from small
but non-zero initial conditions. Clearly, there are some discrepancies between
such curves as an epidemics starting from infection ratio $1\%$ needs less
time to saturate than an epidemics starting from $0.01\%$, resulting in a
delay. However, after accounting for this time translation, it turns out that
these curves are remarkable close to each other regardless of how the initial
infections are distributed. We dub this phenomenon the Universality of Small
Initial Conditions (USIC).

Figure 1: The ratio of infected individuals for an SI epidemics (parameters:
infection rate $\beta=1$, curing rate $\gamma=0$) on a power-law network with
parameter $p=0.4$ starting from a ratio of infected individuals
$10^{-1},\dots,10^{-5}$ at time 0. (For further details, see Section 2.)

Based on USIC, one can also expect a certain limit object to arise. At time
$t=0$ set the ratio to be an arbitrary value between $0$ and equilibrium
prevalence, say, $10\%$. If the initial infection is small, it should
originate from a far away past time, hence, in the limit there is an epidemic
curve starting from the infinite past from infinitesimal amount of infections.
We call such limit the _nontrivial eternal solution_ as it is defined for all
past future values as well.

Figure 2: Blue: The epidemic curves from Figure 1 after after a time shift.
Orange: The eternal solution given by (18).

Roughly speaking, USIC states that - under the same network and parameters -
there is only one ”relevant” epidemic curve (up to translation), and it is
given by the nontrivial eternal solution, which is robust to perturbations. It
is worth emphasizing though that the nontrivial eternal solution is only
relevant until the underlying parameters: the network, the infection and
curing rates remain constant. In real life epidemice, this can be violated by
imposing lockdown measures, introducing vaccination, by mutations and the
changing awareness of the population. Thus, we only expect the nontrivial
eternal solution to characterize the early phase of the epidemic where
assuming constant parameters is plausible.

We will show (see (17)) that the nontrivial eternal solution at the beginning
increases exponentially in time and infections are distributed according to
the leading eigenvector of the operator corresponding to the linearized PDE
(12) which is nothing but the eigenvector centrality when the graph is finite.
It is worth mentioning that the eigenvalue centrality can be estimated from
samples when the graph is dense enough [1].

In summary, at the early stage of the epidemic one can assume the parameters
to be constant and prevalence to be small. To make predictions, the full
initial configuration is not needed, only the ratio of infected. Furthermore,
there is no need for the full contact network, instead, the eigenvector
centrality is sufficient, which can be estimated from sampling. This results
in a major reduction of dimensionality and will make predictions robust.

The paper is structured in the following manner: The rest of Section 1 lists
the related works. Section 2 introduces the notion of graphon kernels and the
corresponding PDE describing the SIS process on said kernel. Results are
stated in Section 3. Section 4 outlines the major steps in proving USIC, while
the remaining proofs are given in section 5. Finally Section 7 gives a brief
summary along with possible directions for future research.

### Related works

[6] studies the basic properties of a PDE describing the SIS dynamics on
general community structures of which (4) is a special case. The ideas
presented there are heavily utilized in our proofs.

There is a huge body of research for eternal solutions of reaction-diffusion
systems. [20] is an example where the eternal solution connects two stationary
solutions reminiscent of our setup. For the SIS model on $\mathbb{R}$, [27]
constructs certain eternal solutions.

A diagram similar to Figure 1 appears in the work of Volz [26] studying the
stochastic SIR process on graphs generated by the configuration model. His
approach turns out to be exact in the large graph limit [5, 13]. Furthermore,
in Theorem 2.9 of [13], the authors show that starting an epidemic with $o(n)$
infections will converge to an ODE conditioning on large outbreaks. This
result implicitly contains USIC as time being translated to start from
$s_{0}n$ after which all (surviving) processes follow the same deterministic
dynamics given by the ODE. Note that the role of small initial conditions is
relevant as the ODE uses the degree distribution of the susceptible vertices,
which is roughly the same as the global degree distribution when there are
only $o(n)$ infections initially.

[2, 3] consider a stochastic SIR model on configuration model graphs starting
from a single infected individual. A notable difference is that they allow
non-Markovian transitions and construct the stochastic time delay obtained
from the martingale of the corresponding branching process.

[7] studies the stochastic SIS process and approximates it by a birth-death
process. The number of SI links is estimated for a given number of infected
vertices on a “typical” trajectory. This is achieved empirically in two
scenarios: first, the epidemics starts from one initially infected vertex to
get the segment between $0$ and the endemic state (“left side”); in the second
scenario, an epidemic starts with everyone being infected to get the segment
from the endemic state to $n$ (“right side”). We conjecture that the left side
of the curve corresponds to the nontrivial eternal solution in the limit
$n\to\infty$ as it represents a “typical” trajectory starting from small
initial conditions. Further remarks are given in Section 3.4.

##  2 Main concepts and notations

###  2.1 Graphons and kernels

In this section we describe contact patterns between individuals represented
by graphon kernels. For a more detailed overview on graphons see [16].

Individuals are represented by the variable $x\in[0,1]$. For example, in the
finite case, with vertices indexed from $1$ to $n$, individual $i$ can be
represented with the value $x=\frac{i}{n}.$

Connections are given by a symmetric kernel
$W:[0,1]^{2}\mapsto\mathbb{R}^{+}_{0}$ in $L^{2}\left([0,1]^{2}\right).$ One
can interpret $W(x,y)$ as the probability of $x,y\in[0,1]$ being connected, or
the strength of their interaction.

We will make use of the following connectivity property: For all measurable
$A\subseteq[0,1]$ with both $A$ and $A^{c}$ having positive measure one has

$\displaystyle\int_{A}\int_{A^{c}}W(x,y)\mathrm{d}x\mathrm{d}y>0.$ (1)

Due to our assumption, the corresponding integral operator

$\displaystyle\mathbb{W}f(x):=\int_{0}^{1}W(x,y)f(y)\mathrm{d}y$

is a Hilbert-Schmidt operator. If we also assume that $W$ is irreducible –
that is, for some integer $r$, the iterated kernel $W^{(r)}$ is positive a.e.
– then there is a leading eigenvalue $\forall i\ \lambda_{1}>|\lambda_{i}|$
and the corresponding eigenvalue $\varphi_{1}$ is positive a.e. [11]. Also, we
can choose an orthonormal eigenvector basis $(\varphi_{k})_{k=1}^{\infty}$ and
express $W$ as

$\displaystyle
W(x,y)=\sum_{k=1}^{\infty}\lambda_{k}\varphi_{k}(x)\varphi_{k}(y).$

Besides the leading eigenvalue $\lambda_{1}$, we pay special attention to
$\lambda_{2},$ the second largest eigenvalue. $\lambda_{1}-\lambda_{2}>0$ is
referred to as the _spectral gap_.

Note that irreducibility and (1) are equivalent when $W$ is made of constant
blocks, which corresponds to a finite weighted graph. Otherwise, we will
assume $W>0$ a.e. when the two conditions are trivially satisfied.

For all of our examples, $\varphi_{1}$ is uniformly positive, that is

$\displaystyle\exists m>0\ \forall x\in[0,1]\quad\varphi_{1}(x)\geq m.$ (2)

###  2.2 Finite and annealed graphs

We call the kernel $W$ discrete if there is a non-degenerate finite partition
$(I_{i})_{i=1}^{n}$ of $[0,1]$ and non-negative $W_{ij}$ constants such that

$\displaystyle
W(x,y)=\sum_{i=1}^{n}\sum_{j=1}^{n}W_{ij}{\mathds{1}}_{\left\\{I_{i}\right\\}}(x){\mathds{1}}_{\left\\{I_{j}\right\\}}(y)$

meaning, $W$ is piece-wise constant. For functions in the form

$f(x)=\sum_{i=1}^{n}f_{i}{\mathds{1}}_{\left\\{I_{i}\right\\}}(x)$

applying the integral operator leads to

$\displaystyle(x\in I_{i})\
\mathbb{W}f(x)=\sum_{j=1}^{n}W_{ij}\int_{I_{j}}f(y)\mathrm{d}y=\sum_{j=1}^{n}W_{ij}|I_{j}|f_{j}$

motivating the definition of the (asymmetric) weights $w_{ij}:=W_{ij}|I_{j}|$
and the matrix $\mathcal{W}:=(w_{ij})_{i,j=1}^{n}.$

Setting $I_{i}=]\frac{i-1}{n},\frac{i}{n}]$ (with $I_{1}=[0,\frac{1}{n}]$)
gives a finite weighted graph with $n$ vertices and weights given by the (now
symmetric) matrix $\mathcal{W}.$ In particular, setting $\mathcal{W}=A$ to an
adjacency matrix leads to a classical, unweighted graph, with (1) describing
its connectivity.

Another interesting case covered is annealed networks [10]. Let
$k_{1},\dots,k_{n}$ denotes the finite number of possible degrees of a graph.
$p(k_{i})$ is the ratio of vertices having degree $k_{i}$, and
$p(k_{i}|k_{j})$ refers to the probability of a stub of degree $k_{j}$
connecting to a stub of degree $k_{i}$. In the special case of uncorrelated
networks, $p(k_{i}|k_{j})=\frac{k_{i}p(k_{i})}{\langle k\rangle}$ where
$\langle k\rangle$ is the average degree.

The interval corresponding to the degree class $k_{i}$ is set to be
$I_{i}=]p(k_{i-1}),p(k_{i})]$ (with $I_{1}=[0,p(k_{1})]).$ Naturally,
$|I_{i}|=p(k_{i}).$

The weights are $w_{ij}=k_{i}p(k_{j}|k_{i})$.

Finally, we have to check that $W_{ij}=\frac{w_{ij}}{|I_{j}|}$ is symmetric so
that the weight matrix $\mathcal{W}$ indeed corresponds to a discrete $W$
kernel.

$\displaystyle W_{ij}=$
$\displaystyle\frac{k_{i}p(k_{j}|k_{i})}{p(k_{j})}=\langle
k\rangle\frac{p(k_{i},k_{j})}{p(k_{i})p(k_{j})},$

where $(2-\delta_{k_{i},k_{j}})p(k_{i},k_{j})$ is the probability that a
randomly chosen edge connects two vertices with degree $k_{i}$ and $k_{j}$,
hence $W_{ij}$ is indeed symmetric.

In general, $\mathcal{W}$ is symmetric with respect to the scalar product
$[f,g]:=\sum_{i=1}^{n}f_{i}g_{i}\frac{1}{|I_{i}|},$ hence all the eigenvalues
are real. It is easy to see that for any $\phi$ eigenvector of $\mathcal{W},$
$\varphi(x)=\sum_{i=1}^{n}\phi_{i}{\mathds{1}}_{\left\\{I_{i}\right\\}}(x)$ is
an eigenvector of $\mathbb{W}.$ The other way round,
$\phi_{i}:=\frac{1}{|I_{i}|}\int_{I_{i}}\varphi(x)\mathrm{d}x$ also makes
$\phi$ an eigenvector. In particular, the Perron–Frobenius vector of
$\mathcal{W},$ $\phi^{(1)}$ corresponds to
$\varphi_{1}(x)=\sum_{i=1}^{n}\phi_{i}^{(1)}{\mathds{1}}_{\left\\{I_{i}\right\\}}(x),$
making $\varphi_{1}$ uniformly positive with $m:=\min_{1\leq i\leq
n}\phi_{i}^{(1)}$ in the discrete case.

###  2.3 Rank-$1$ kernels

The kernel $W$ has rank $1$ if it has the form
$W(x,y)=\lambda_{1}\varphi_{1}(x)\varphi_{1}(y).$ Note that uncorrelated
annealed graphs are included in this class as
$W_{ij}=\frac{k_{i}k_{j}}{\langle d\rangle}.$ The parameters are

$\displaystyle\phi_{i}^{(1)}=$ $\displaystyle\frac{1}{\sqrt{\langle
k^{2}\rangle}}k_{i}$ $\displaystyle\lambda_{1}=$ $\displaystyle\frac{\langle
k^{2}\rangle}{\langle k\rangle},$

where $\langle k^{2}\rangle$ is the second moment of the degrees.

An important example to keep in mind is the rank-$1$ graphon with
eigenfunction $\varphi_{1}(x)=\sqrt{1-2p}x^{-p}:$

$\displaystyle W(x,y)=\lambda_{1}(1-2p)x^{-p}y^{-p}\ \ 0\leq p<\frac{1}{2}.$
(3)

The condition $0\leq p<\frac{1}{2}$ is needed so that $\varphi_{1}$ is an
$L^{2}\left([0,1]\right)$ function.

The corresponding degree function is

$\displaystyle
d(x):=\int_{0}^{1}W(x,y)\mathrm{d}y=\lambda_{1}\frac{1-2p}{1-p}x^{-p}$

hence the “degree distribution” of the graphon has a power law decay:

$\displaystyle\mathbb{P}\left(d(U)\geq x\right)=\mathbb{P}\left(U\leq
d^{-1}(x)\right)=\left(\lambda_{1}\frac{1-p}{1-2p}\right)^{\frac{1}{p}}x^{-\frac{1}{p}},$

where $U$ is a uniform random variable on $[0,1].$

Note that in this example, $\varphi_{1}(x)$ satisfies the uniform positive
assumption (2) with $m=\sqrt{1-2p}.$

###  2.4 The PDE dynamics

The (deterministic) SIS process on the graphon kernel $W$ is described by the
PDE

$\displaystyle\partial_{t}u=\beta(1-u)\mathbb{W}u-\gamma u.$ (4)

with parameters $\beta>0,\gamma\geq 0$. Here, $u(t,x)$ corresponds to the
probability that individual $x\in[0,1]$ is infected at time $t,$ or, in the
case of metapopulation models, the ratio of infected within population $x$.
Curing happens at a constant rate $\gamma$, and a susceptible individual $x$
becomes infected at rate

$\beta\mathbb{W}u(t,x)=\beta\int_{0}^{1}W(x,y)u(t,y)\mathrm{d}y,$

where the integral corresponds to the number of its infected neighbours.

When $\gamma>0$, one can set $\gamma=1$ via rescaling time. The $\gamma=0$
case corresponds to the SI process, where no curing is allowed. The SI model
is more suitable to describe the diffusion of information rather than viral
infections.

Although (4) describes a deterministic dynamics, it has plenty of connections
to the stochastic SIS process.

When $W$ is discrete, (4) gives back certain well known mean field
approximations. Defining

$\displaystyle z_{i}(t):=\frac{1}{|I_{i}|}\int_{I_{i}}u(t,y)\mathrm{d}y$ (5)

leads to the finite ODE system

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}z_{i}(t)=\beta(1-z_{i}(t))\sum_{j=1}^{n}w_{ij}z_{j}(t)-\gamma
z_{j}(t)$ (6)

referred to as NIMFA or quenched mean field approximation in the literature.
It has been shown that NIMFA gives an upper bound on the exact infection
probabilities [21, 24], and for large average degrees, the error can be
arbitrarily small [22, 23, 12].

When $n=1$, without loss of generality, one can set $w_{11}=1$, deriving HMFA:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}z(t)=\beta(1-z(t))z(t)-\gamma
z(t),$

which is shown to be the exact limit of the prevalence on complete graphs
[15].

To get IMFA [18] one must use annealed networks with weights
$w_{ij}=k_{i}p(k_{j}|k_{i}).$

$\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}z_{i}(t)=&\beta
k_{i}(1-z_{i}(t))\Theta_{i}(t)-\gamma z_{i}(t)\\\
\Theta_{i}(t)=&\sum_{j=1}^{n}p(k_{j}|k_{i})z_{j}(t)\end{split}$ (7)

Smooth $W$ kernels can also arise as the limit of the discrete system (6) when
the underlying graph converges to $W$ in the graphon sense [16, 1]. In certain
special cases with diverging average degree, the stochastic SIS model
converges to (4), making (4) an exact limit, not just an approximation [14].

Not all solutions of (4) are physically relevant as $u(t,x)$ must retain a
probabilistic interpretation. Hence we restrict our attention to the domain

$\displaystyle\Delta:=\\{\left.f\in L^{2}([0,1])\right|0\leq f(x)\leq 1\
a.e.\\}.$ (8)

######  Proposition 1.

Assume $0\leq W\in L^{2}\left([0,1]^{2}\right)$ and $u(0)\in\Delta.$ Then
there is a unique solution to (4) with initial condition $u(0)$ and it
satisfies $\forall t\in\mathbb{R}_{0}^{+}\ u(t)\in\Delta.$

Proposition 1 has already been proven in [6, Proposition 2.9. (ii)] under the
assumption that $W$ is bounded – in which case some further smoothness
properties also follow. Extending it to square-integrable $W$ is relatively
straightforward using the following approximation result:

######  Proposition 2.

Let $0\geq W_{1},W_{2}\in L^{2}\left([0,1]^{2}\right)$ such that
$\|W_{1}\|_{2},\|W_{2}\|_{2}\leq\lambda.$ Let $u_{1},u_{2}$ be two solutions
of (4) with initial conditions $u_{1}(0),u_{2}(0)\in\Delta.$ Then

$\displaystyle\sup_{0\leq t\leq
T}\|u_{1}(t)-u_{2}(t)\|_{2}=O\left(\|u_{1}(0)-u_{2}(0)\|_{2}+\|W_{1}-W_{2}\|_{2}\right)$
(9)

with constants in the $O(.)$ notation depending only on $T$ and $\lambda$.

An easy consequence of Proposition 2 is that we can approximate (4) with the
finite ODE system (6).

Note that Proposition 1 only guarantees $u(t)\in\Delta$ for $t>0$, and it
might not be true for the past. For example, let $A\subseteq[0,1]$ be such
that both $A,A^{c}$ has positive measure and
$u(0,x)={\mathds{1}}_{\left\\{A^{c}\right\\}}(x).$ Then, based on (1),

$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}t}\int_{A}u(t,x)\mathrm{d}x\right|_{t=0}=\beta\int_{A}\mathbb{W}u(0,x)\mathrm{d}x=\beta\int_{A}\int_{A^{c}}W(x,y)\mathrm{d}y\mathrm{d}x>0$

making $\int_{A}u(-t,x)\mathrm{d}x<0$ for some small $t>0.$

We call a solution $u(t)$ _eternal_ if it satisfies

$u(t)\in\Delta\quad\forall t\in\mathbb{R}.$

Obviously, stationary solutions are always eternal. In the supercritical case
(for bounded $W$) there are two such stationary solutions [6, Proposition
4.13.]: the unstable disease free state $u(t)\equiv 0$ and a nonzero endemic
state $\psi$ satisfying

$\displaystyle\beta(1-\psi)\mathbb{W}\psi=\gamma\psi.$ (10)

######  Remark 1.

When $W$ is rank-$1$, the nonzero solution of (10) is

$\displaystyle\psi(x)=$
$\displaystyle\frac{\beta\lambda_{1}c\varphi_{1}(x)}{\gamma+\beta\lambda_{1}c\varphi_{1}(x)}$
$\displaystyle 0<c=$ $\displaystyle\langle\varphi_{1},\psi\rangle\Rightarrow$
$\displaystyle 1=$
$\displaystyle\int_{0}^{1}\frac{\varphi_{1}^{2}(x)}{\frac{\gamma}{\beta\lambda_{1}}+c\varphi_{1}(x)}\mathrm{d}x$

hence the nontrivial solution of (10) is unique for rank-$1$ graphons too with
$\psi\in\Delta$. Later, in Lemmas 11 and 12 we will see that $u(t)\to\psi$
holds in this slightly more general case as well when $u\not\equiv 0.$

We call an eternal solution $u(t)$ _nontrivial_ if it is neither the disease
free, nor the endemic state.

When considering discrete $W$, we have to further restrict the physically
viable solutions, since we cannot distinguish between elements of $I_{i}$ due
to all $x\in I_{i}$ referring to the same vertex $i$. The restricted domain is
defined as

$\displaystyle\Delta_{I}:=\\{\left.f\in\Delta\right|\forall\ 1\leq i\leq n\
f|_{I_{i}}\textit{ is constant.}\\}$ (11)

Note that for discrete $W$, $\Delta_{I}$ is forward invariant, hence (5)
creates a one-to-one map between the solutions of (4) and (12) on the domain
$\Delta_{I}.$ For discrete $W$, we will work with $\Delta_{I}$ instead of
$\Delta$ in this paper.

###  2.5 Linearization of the PDE

The expansion of $u(t)$ by the orthonormal basis
$\left(\varphi_{k}\right)_{k=1}^{\infty}$ is written as

$\displaystyle u(t)$ $\displaystyle=\sum_{k=1}^{\infty}c_{k}(t)\varphi_{k}$
$\displaystyle c_{k}(t)$ $\displaystyle:=\langle u(t),\varphi_{k}\rangle.$

We linearize (4) around $u=0$ to get

$\displaystyle\begin{split}\partial_{t}v&=\mathcal{A}v:=\left(\beta\mathbb{W}-\gamma\mathbb{I}\right)v\\\
v(t_{0})&=u(t_{0}).\end{split}$ (12)

The initial time $t_{0}$ will vary in the theorems and proofs, however, it
will be clear from the context which version of $v$ we are referring to.

The corresponding expansion of $v(t)$ with respect to
$\left(\varphi_{k}\right)_{k=1}^{\infty}$ is

$\displaystyle v(t)$
$\displaystyle=\sum_{k=1}^{\infty}\tilde{c}_{k}(t)\varphi_{k}$
$\displaystyle\tilde{c}_{k}(t)$ $\displaystyle:=\langle
v(t),\varphi_{k}\rangle.$

Since the right hand side of (4) is cooperative [6, Proposition 2.7. (iv)], it
follows that

$\displaystyle\forall t\geq t_{0}\quad 0\leq u(t)\leq v(t).$

######  Remark 2.

Technically, [6, Proposition 2.7. (iv)] only applies for bounded $W$. To
extend it $W\in L^{2}\left([0,1]^{2}\right)$ kernels we have to generalize [6,
Lemma 2.8. (iv)] to include $h\in L^{2}([0,1])$ functions as well. This can be
done by setting $h_{N}(x)=h(x){\mathds{1}}_{\left\\{|h|\leq N\right\\}}(x)$
then taking the limit $h_{N}\to h$ in $L^{2}([0,1]).$

Note that $\varphi_{k}$ are eigenvectors to $\mathcal{A}$ as well with
eigenvalues

$\displaystyle\alpha_{k}=\beta\lambda_{k}-\gamma.$

To solution to the linear system (12) can be written as

$\displaystyle v(t)=e^{\mathcal{A}(t-t_{0})}v(t_{0}).$

Since $\mathcal{A}$ is bounded, the exponential can be understood as a power
series and

$\displaystyle v(t)$
$\displaystyle=\sum_{k=1}^{\infty}c_{k}(t_{0})e^{\alpha_{k}(t-t_{0})}\varphi_{k}.$
(13)

Note that $\tilde{c}(t_{0})=c_{k}(t_{0}).$

It is easy to see that there is a phase transition for $v(t)$ at
$\beta_{c}=\frac{\gamma}{\lambda_{1}}.$ When $\beta<\beta_{c},$ $v(t)\to 0$
for all initial conditions. However, when $\beta>\beta_{c},$ the leading term
$c_{1}(t_{0})e^{\alpha_{1}(t-t_{0})}\varphi_{1}$ survives if
$c_{1}(t_{0})=\int_{0}^{1}u(t_{0},x)\varphi_{1}(x)\mathrm{d}x>0.$
$\varphi_{1}>0$ a.e. makes $\varphi_{1}\mathrm{d}x$ being equivalent to the
Lebesgue measure, therefore, $c_{1}(t_{0})>0$ is equivalent to
$\|u(t_{0})\|_{1}>0,$ meaning, we are not considering the disease free
epidemic.

From now on, it is assumed that we are in the super critical case, that is,
$\beta\lambda_{1}>\gamma.$

##  3 Results

###  3.1 USIC

######  Theorem 1.

(Main)

Assume $W\in L^{2}([0,1]^{2})$ is non-negative along with the dynamics being
supercritical: $\beta\lambda_{1}>\gamma.$

Further assume one of the following set of assumptions:

  * •

$W$ is discrete and connected,

  * •

$0<m_{0}\leq W(x,y)\leq M$ and
$\gamma<\beta\lambda_{1}<\gamma+2\beta(\lambda_{1}-\lambda_{2}),$

  * •

$W$ is rank-$1$, $\varphi_{1}$ is uniformly positive and in
$L^{2+\rho}([0,1])$ for some $\rho>0$.

Then, for all $\varepsilon,\eta>0$ there is a $\delta>0$ such that for all
$u_{1}(0),u_{2}(0)\in\Delta$ (in the discrete $W$ case
$u_{1}(0),u_{2}(0)\in\Delta_{I}$) with
$0<\|u_{1}(0)\|_{2},\|u_{2}(0)\|_{2}\leq\delta$ there are time shifts
$t_{1},t_{2}\geq 0$ such that

$\displaystyle\sup_{t\geq
0}\|u_{1}(t+t_{1})-u_{2}(t+t_{2})\|_{2}\leq\varepsilon,$ (14)

while

$\displaystyle(i=1,2)\ \sup_{0\leq t\leq t_{i}}\|u_{i}(t)\|_{2}\leq\eta.$ (15)

Note that the third set of assumptions are satisfied for power law kernels
(3).

(15) requires some explanations. $\eqref{eq:USIC_main_part}$ without (15) is
meaningless as both $u_{1}(t)$ and $u_{2}(t)$ converge to the endemic state,
hence we could choose $t_{1},t_{2}$ to be large enough so that
$u_{i}(t_{i})\approx\psi$ ($i=1,2$) satisfying the statement in a trivial
manner. (15) mitigates this problem by asserting that the solutions are small
even after the time shifts, hence they can be close to each other even from an
early stage of the epidemic.

(13) suggests the following heuristic outline for the proof of Theorem 1:

  * •

When the initial conditions are small, the solution $u(t)$ can be approximated
by $v(t)$ until it reaches a relatively small $\varepsilon^{\prime}$-level.

  * •

If it has enough time before reaching the $\varepsilon^{\prime}$-level, the
leading term in (13) will dominate, making
$v(t)\approx\varepsilon^{\prime}\varphi_{1}.$

  * •

Different initial conditions can take different amount of time till they reach
the $\varepsilon^{\prime}$-level; typically smaller initial conditions require
more time. However, after reaching the $\varepsilon^{\prime}$-level, both
curves can be approximated by a solution started from
$\varepsilon^{\prime}\varphi_{1}$ (or an appropriately truncated version in
case $\varphi_{1}$ is unbounded).

  * •

Since both solutions are converging to the endemic state $\psi$, there is “not
enough time” for the error to accumulate, making these solutions stay close to
each other.

To summarize, when two initial conditions are small enough – but not
identically $0$ – we can apply an appropriate time shift after which the two
solutions remain close to each other. We call this phenomenon _universality of
small initial conditions_ (USIC).

###  3.2 Basic properties of eternal solutions

Consider the following heuristics.

Take a sequence of initial conditions $u_{n}(-t_{n})\to 0$ with
$-t_{n}\to-\infty$ and at time $t=0$ take some intermediate value between the
disease free and the endemic state, say
$\|u_{n}(0)\|_{1}=\frac{1}{2}\|\psi\|_{1}.$

Since $u_{n}(-t_{n})$ is getting smaller and smaller, USIC suggest the
solutions $u_{n}(t)$ are becoming more and more similar to each other, hence
there should be a limit by the Cauchy-argument. This limit should satisfy
$\lim_{t\to-\infty}u(t)=0,\ \lim_{t\to\infty}u(t)=\psi$ while $u(t)\in\Delta$
for all $t\in\mathbb{R}.$ Since $\|u(0)\|_{1}=\frac{1}{2}\|\psi\|_{1},$ the
limit object is different from the disease free and the endemic state, hence,
it describes a nontrivial eternal solution making them a natural limit object
to study.

######  Theorem 2.

(Existence of nontrivial eternal solutions)

Assume $0\leq W\in L^{2}([0,1]^{2})$ is irreducible along with
supercriticality and $\varphi_{1}\in L^{2+\rho}([0,1])$ for some $\rho>0$.
Then, there is a nontrivial eternal solution such that

$\displaystyle\lim_{t\to-\infty}\frac{\langle\varphi_{1},u(t)\rangle}{\|u(t)\|_{2}}=1.$
(16)

Also, when $W$ is discrete, the nontrivial eternal solution takes values from
$\Delta_{I}.$

Note that in Theorem 1 $\varphi_{1}\in L^{2+\rho}([0,1])$ is either assumed
explicitly or a consequence of $W$ being bounded.

The significance of (16) is that it allows us to describe the shape of
$u(t,x)$ in the early stages of the epidemic, where the eternal solution is
most applicable. Since initially there are only a few infections (see Lemma 1
below), $u(t)$ can be approximated by (13), and while the weights are mostly
concentrated on $\varphi_{1}$ due to (16). Therefore, after appropriate time
translation we get

$\displaystyle u(t,x)\approx\|u(0)\|_{2}e^{\alpha_{1}t}\varphi_{1}(x)$ (17)

when $\|u(0)\|_{2}$ is small.

######  Lemma 1.

Assume $0\leq W\in L^{2}([0,1]^{2})$ with the connectivity property (1) and
supercriticality. Let $u(t)$ be a nontrivial eternal solution. Then
$\lim_{t\to-\infty}u(t)=0$ a.e.

Next we turn to uniqueness of the eternal solution. Of course, any uniqueness
can only be up to time translation, since if $u(t)$ is an eternal solution,
then any time translated version $u(t+\tau)$ will also be an eternal solution.

Without the connectivity assumption of Lemma 1, several fundamentally
different solutions may arise. For example, when the graph is the union of two
disjoint complete graphs, we could treat the solutions on them separately,
hence a mixture of disease free state on one component, endemic state on the
other is possible.

However, under USIC, the only ambiguity that can occur is due to time
translation. Combining it with Lemma 1 _the_ nontrivial eternal solution can
be interpreted as an epidemic started from the infinite past from an
infinitesimally small initially infected population with exponential growth
(17) with ”spatial” distribution described by $\varphi_{1}(x)$ in the
beginning.

Since epidemics usually starts from only a small amount of initial infections,
USIC shows that the nontrivial eternal solution is the limit object, reducing
the problem to a simple curve from the original infinite dimensional problem,
at least for given parameters $\beta,\gamma,W$.

######  Theorem 3.

(Uniqueness of nontrivial eternal solutions)

Assume the conditions of Theorem 1. Let $u_{1},u_{2}$ be two nontrivial
eternal solutions. Then there is a $\tau$ translation time such that
$u_{1}(t+\tau)=u_{2}(t).$

###  3.3 Explicit formulas and approximations

So far we mostly gave implicit descriptions of the nontrivial eternal
solutions, apart from (17). In this section, we aim to give some more explicit
formulas for some cases.

We highlight the work [19] deriving a $\tanh(t)$-formula

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}c(t)=$
$\displaystyle(\beta\lambda_{1}-\gamma)c(t)\left(1-c(t)\right)$ $\displaystyle
u(t,x)\approx$ $\displaystyle(\beta\lambda_{1}-\gamma)c(t)\varphi_{1}(x)$

for epidemics on finite networks when the infection rate $\beta$ is just
slightly above critical. Intuitively, when we are close to criticality, the
endemic state becomes small, hence one can linearize (10) to get
$\beta\mathbb{W}\psi\approx\gamma\psi$ making
$\psi\approx(\beta\lambda_{1}-\gamma)\varphi_{1}.$ This means that the initial
growth phase (17) looks similar to the saturation phase around $\psi$,
resulting in a logistic curve with “spatial” distribution given by
$\varphi_{1}(x).$

We believe these ideas can be generalized for a wider class of $W$ kernels.

On the other end of the spectrum we present an explicit formula for eternal
solutions when the infection rate is much higher than critical; in the limit,
this is the SI process. Without loss of generality, we can set
$\alpha_{1}=\beta\lambda_{1}=1.$ Also, we assume
$W(x,y)=\lambda_{1}\varphi_{1}(x)\varphi_{2}(x)$ is rank-$1$.

Let $u(t)$ be a nontrivial eternal solution. Treating $\mathbb{W}u(t)$ as a
known function, (4) can be solved in the SI case as

$\displaystyle
u(t,x)=1-\exp\left(-\int_{t_{0}}^{t}\beta\mathbb{W}u(s,x)\mathrm{d}s\right)(1-u(t_{0},x)).$

From Lemma 1 $u(t_{0})\to 0$ as $t_{0}\to-\infty$ resulting in

$\displaystyle
u(t,x)=1-\exp\left(-\int_{-\infty}^{t}\beta\mathbb{W}u(s,x)\mathrm{d}s\right).$

Since $W$ is rank-$1$

$\displaystyle\int_{-\infty}^{t}\beta\mathbb{W}u(s,x)\mathrm{d}s=\underbrace{\beta\lambda_{1}}_{=1}\varphi_{1}(x)\int_{-\infty}^{t}c_{1}(s)\mathrm{d}s=:\Omega(t)\varphi_{1}(x),$

which yields

$\displaystyle u(t,x)=1-e^{-\Omega(t)\varphi_{1}(x)}$ (18)

successfully separating the temporal and spatial variables. Our goal now is to
express the temporal part $\Omega(t).$

Define the function

$\displaystyle
F(\omega):=\int_{0}^{1}\varphi_{1}(x)\left(1-e^{-\omega\varphi_{1}(x)}\right)\mathrm{d}x.$
(19)

Note that for $\omega>0$, $F$ is positive and for $\omega\geq 0$, it is
Lipschitz continuous with constant $1$. It is also easy to see that
$\Omega(0)>0$ and $\Omega(t)$ is increasing.

Observe
$\frac{\mathrm{d}}{\mathrm{d}t}\Omega(t)=c_{1}(t)=\langle\varphi_{1},u(t)\rangle,$
resulting in the dynamics

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\Omega(t)=F(\Omega(t)).$ (20)

To solve (20) define

$G(\omega):=\int\frac{1}{F(\omega)}\mathrm{d}\omega.$

Note that $G$ is strictly increasing, hence invertible, leading to

$\displaystyle\Omega(t)=G^{-1}\left(t+G(\Omega(0))\right).$ (21)

###  3.4 Remarks to F. Di Lauro et al.

As it was mentioned in Section 1, [7] by F. Di Lauro et al. approximates the
stochastic SIS process with a birth-death process by identifying birth rates
on a typical trajectory of a given number of infected vertices. The left side
of this curve corresponds to the average obtained from simulations starting
from a single infected individual.

Set $\varepsilon>0$ and wait until the epidemic reaches $\lfloor\varepsilon
n\rfloor$ infections. When $n$ and the average degrees are large enough,
either (4) or (6) is a good approximation of the stochastic SIS process [22,
23, 12, 14], thus, the typical rates correspond to the rates given be the ODE
and PDE approximations. By letting $\varepsilon\to 0^{+}$ we will get a curve
starting from infinitesimal prevalence corresponding to the nontrivial eternal
solution.

Hence, the problem reduces to finding the mapping

$\displaystyle\chi:\int_{0}^{1}u(t,x)\mathrm{d}x\mapsto\int_{0}^{1}(1-u(t,x))\mathbb{W}u(t,x)\mathrm{d}x$
(22)

which maps the number of vertices to the number of SI links along nontrivial
eternal solution under suitable normalization.

Similar considerations work for the right side starting with all individuals
being infected.

Notably, in the SI process, the endemic state coincides with the state of
every individual being infected, hence, only the left side is relevant. If we
further assume $W$ being rank-$1$, we can even give $\chi$ explicitly based on
(18).

For the mapping $\chi$, time plays little role so we apply the more practical
coordinate system $\omega=\Omega(t),$ and set

$\displaystyle U(\omega,x):=1-e^{-\omega\varphi_{1}(x)}.$

Note that $U\left(\Omega(t),x\right)=u(t,x).$

The prevalence at “time” $\omega$ is defined as

$\displaystyle\bar{U}(\omega):=\int_{0}^{1}U(\omega,x)\mathrm{d}x.$

Clearly,

$\displaystyle\chi\left(\bar{U}(\omega)\right)=\int_{0}^{1}\left(1-U(\omega,x)\right)\mathbb{W}U(\omega,x)=F(\omega)\left(\bar{\varphi}_{1}-F(\omega)\right),$

where $\bar{\varphi}_{1}:=\int_{0}^{1}\varphi_{1}(x)\mathrm{d}x.$

Finally, we introduce the notation $H:=F\circ\bar{U}^{-1}$ to conclude

$\displaystyle\chi=H\left(\bar{\varphi}_{1}-H\right).$ (23)

Besides large average degrees one can study annealed networks in which case
(7) will be the exact limit. In the uncorrelated case (23) is still
applicable. Let $G$ denote the momentum generator function of the degree
distribution ${p(k_{i})}_{i=1}^{n}.$

$\displaystyle G(\omega):=\sum_{i=1}^{n}p(k_{i})e^{\omega k_{i}}$

Then $\bar{U}$ and $F$ can be obtained as

$\displaystyle\bar{U}(\omega)=$
$\displaystyle\sum_{i=1}^{n}p(k_{i})\left(1-\exp\left(-\frac{\omega}{\sqrt{\left\langle
k^{2}\right\rangle}}k_{i}\right)\right)=1-G\left(-\frac{\omega}{\sqrt{\left\langle
k^{2}\right\rangle}}\right)$ $\displaystyle F(\omega)=$
$\displaystyle\sum_{i=1}^{n}p(k_{i})\frac{k_{i}}{\sqrt{\left\langle
k^{2}\right\rangle}}\left(1-\exp\left(-\frac{\omega}{\sqrt{\left\langle
k^{2}\right\rangle}}k_{i}\right)\right)$ $\displaystyle=$
$\displaystyle\underbrace{\frac{\langle k\rangle}{\sqrt{\left\langle
k^{2}\right\rangle}}}_{=\bar{\varphi}_{1}}-G^{\prime}\left(-\frac{\omega}{\sqrt{\left\langle
k^{2}\right\rangle}}\right).$

##  4 Main steps of proving USIC

This section contains the proof of Theorem 1. The main structure of the proof
follows the outline in Section 3.1. Several lemmas will be stated throughout
whose proofs will be presented in Section 5.

Throughout the chapter we assume $W(x,y)\geq 0$ is Hilbert-Schmidt,
irreducible and satisfies the connectivity property (1) along with
supercriticality, that is, $\gamma<\beta\lambda_{1}.$

We also mention that when $W$ is uniformly positive, then $\varphi_{1}$ is
also uniformly positive as

$\displaystyle
0<m:=\frac{m_{0}\|\varphi_{1}\|_{1}}{\lambda_{1}}\leq\frac{1}{\lambda_{1}}\int_{0}^{1}W(x,y)\varphi_{1}(y)\mathrm{d}y=\varphi_{1}(x),$

hence $\varphi_{1}(x)\geq m>0$ can be assumed in all three cases of Theorem 1.

###  4.1 Reaching the $\varepsilon^{\prime}$-level

We fix some small $\varepsilon^{\prime}>0$ to which we provide an propitiate
$\delta>0$ controlling the size of the initial conditions. More precisely, we
assume

$\displaystyle 0<\|u(0)\|_{2}\leq\delta.$

Define the time it takes for the leading term in (13) to reach the
$\varepsilon^{\prime}$-level from $c_{1}(t_{0})$ as

$\displaystyle\bar{t}$
$\displaystyle:=\frac{1}{\alpha_{1}}\log\frac{\varepsilon^{\prime}}{c(t_{0})},$
(24)

which is equivalent to $\tilde{c}_{1}(t_{0}+\bar{t})=\varepsilon^{\prime}.$

There are two crucial steps to make our heuristic rigorous until we reach the
$\varepsilon^{\prime}$-level: we should mitigate the error arising from
linearization, and prove that the leading term in (13) is indeed dominant at
time $t_{0}+\bar{t}.$ The following two lemmas work towards these goals.

######  Lemma 2.

$\displaystyle\sup_{t_{0}\leq t\leq
t_{0}+\bar{t}}\|u(t)-v(t)\|_{2}\leq\frac{\beta\lambda_{1}}{\alpha_{1}}\left(\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}\right)^{2}(\varepsilon^{\prime})^{2}$
(25)

######  Lemma 3.

$\displaystyle
v(t_{0}+\bar{t})=\varepsilon^{\prime}\left[\varphi_{1}+O\left(\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}e^{-(\alpha_{1}-\alpha_{2})\bar{t}}\right)\right]\
\ \textit{in }L^{2}([0,1]).$ (26)

The problem with Lemma 2 and 3 is the appearance of the expression
$\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}$ in the error terms which unfortunately
can be arbitrarily large. One such example is the complete graph $W(x,y)\equiv
1$ with

$\displaystyle u(t_{0},x)=\delta{\mathds{1}}_{\left\\{0\leq
x\leq\delta\right\\}};$ (27)

then

$\displaystyle c_{1}(t_{0})$
$\displaystyle=\int_{0}^{1}u(t_{0},x)\underbrace{\varphi_{1}(x)}_{=1}\mathrm{d}x=\delta^{2}$
$\displaystyle\|u(t_{0})\|_{2}^{2}$
$\displaystyle=\int_{0}^{1}u^{2}(t_{0},x)\mathrm{d}x=\delta^{3}$
$\displaystyle\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}$
$\displaystyle=\delta^{-\frac{1}{2}}\to\infty\ \textit{as }\delta\to 0^{+}.$

Note that

$\|u(t_{0})\|_{2}^{2}=\sum_{k=1}^{\infty}c_{k}^{2}(t_{0}),$

hence, $\frac{c_{1}(t_{0})}{\|u(t_{0})\|_{2}}$ measures how much weight the
$\varphi_{1}$ component has initially, resulting in large error terms when
being small.

There are two ways to deal with this problem.

First is setting $t_{0}=0$ to be the beginning and only considering small
initial conditions where $\varphi_{1}$ already has prescribed weight, or, in
other words, assuming $\frac{\|u(0)\|_{2}}{c_{1}(0)}\leq K$ for some constant
$K$. This limits the class of initial conditions we may consider.

This approach works well for the construction of eternal solutions where the
initial condition is chosen to be roughly $c\varphi_{1}$, making the
$\varphi_{1}$ component dominant from the beginning.

Another important case when such an assumption holds naturally is when $W$ is
discrete, as we are only considering piece-wise constant initial conditions.
This excludes counterexamples like (27) where the initially infected are
concentrated in a small region.

######  Lemma 4.

Assume $W$ discrete with $J:=\min_{i}|I_{i}|$ and
$u(0)\in\Delta_{I}\setminus\\{0\\}.$ Then

$\displaystyle\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{2}\leq\frac{1}{m^{2}J}.$

Assuming $\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{2}$ is bounded at
$t_{0}=0$, Lemmas 2 and 3 lead to

$\displaystyle
u(\bar{t})=\varepsilon^{\prime}\left[\varphi_{1}+O\left(\varepsilon^{\prime}+e^{-(\alpha_{1}-\alpha_{2})\bar{t}}\right)\right]\
\ \textit{in }L^{2}([0,1]).$ (28)

It is worth noting that

$\bar{t}=\frac{1}{\alpha_{1}}\log\frac{\varepsilon^{\prime}}{c(0)}\to\infty\ \
\textit{as }\delta\to 0^{+},$

since $c_{1}(0)\leq\|u(0)\|_{2}\leq\delta.$

The other method is to initiate a new period and set $t_{0}=\hat{t},$ where
$\hat{t}$ is the time till $\varphi_{1}$ receives enough weight so that
$\frac{\|u(\hat{t})\|_{2}}{c_{1}(\hat{t})}=O(1).$ The potential danger with
this approach is that this event might happen _later_ than when we reach the
$\varepsilon^{\prime}$-level.

With the choice of

$\displaystyle\hat{t}:=\frac{1}{\alpha_{1}-\alpha_{2}}\log\frac{\|u(0)\|_{2}}{c_{1}(0)}$
(29)

we can guarantee

$\displaystyle\left(\frac{\|v(\hat{t})\|_{2}}{\tilde{c}_{1}(\hat{t})}\right)^{2}$
$\displaystyle=1+\frac{1}{c_{1}^{2}(0)}\sum_{k>1}c_{k}^{2}(0)e^{-2(\alpha_{1}-\alpha_{k})\hat{t}}$
$\displaystyle\leq
1+\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{2}e^{-2(\alpha_{1}-\alpha_{2})\hat{t}}=2$

where $v$ here stands for the solution of (12) with initial condition
$v(0)=u(0).$

However, what we need to bound is $\frac{\|u(\hat{t})\|_{2}}{c_{1}(\hat{t})}$
instead, which can be achieved via the following lemma:

######  Lemma 5.

$\displaystyle\left(\frac{\|u(\hat{t})\|_{2}}{c_{1}(\hat{t})}\right)^{2}\leq\frac{2}{\left(1-\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\right)^{2}}$ (30)

Thus, it remains to bound the denominator of (30).

######  Lemma 6.

Assume $W$ is rank-$1$, $\varphi_{1}(x)\geq m>0,$ and $\varphi_{1}\in
L^{2+\rho}([0,1])$ for some $\rho>0$.

Then there is a $\delta_{0}$ such that for any $0<\delta\leq\delta_{0}$,

$\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\leq\frac{1}{2}.$

######  Lemma 7.

Assume $W(x,y)\leq M,$ $\varphi_{1}(x)\geq m>0$ and
$\beta\lambda_{1}<\gamma+2\beta(\lambda_{1}-\lambda_{2}).$ Then there is a
$\delta_{0}$ such that for any $0<\delta\leq\delta_{0}$

$\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\leq\frac{1}{2}.$

Assuming the denominator of (30) is bounded from below, Lemmas 2, 3 and 5
result in

$\displaystyle
u(\hat{t}+\bar{t})=\varepsilon^{\prime}\left[\varphi_{1}+O\left(\varepsilon^{\prime}+e^{-(\alpha_{1}-\alpha_{2})\bar{t}}\right)\right]\
\ \textit{in }L^{2}([0,1]).$ (31)

Finally, we show that $\bar{t}$ can be arbitrarily large as $\delta$
decreases, that is, $\varphi_{1}$ starts to dominate before we reach the
$\varepsilon^{\prime}$-level.

######  Lemma 8.

Assume $\beta\lambda_{1}<\gamma+2\beta(\lambda_{1}-\lambda_{2}).$ Then
$\bar{t}\to\infty$ as $\delta\to 0^{+}.$

The conditions of Lemma 8 can be interpreted the following way: when the
spectral gap is large, the coefficient of $\varphi_{1}$ can increase more
rapidly than the coefficients of the other components, enabling $\varphi_{1}$
to dominate before reaching the $\varepsilon^{\prime}$-level.

For rank-$1$ graphons, $\lambda_{2}=0,$ hence, the condition of Lemma 8
trivially holds.

As $\bar{t}\to\infty$, we may choose $\delta$ to be small enough so that
$e^{-(\alpha_{1}-\alpha_{2})\bar{t}}\leq\varepsilon^{\prime}$, hence, by (28)
or (31),

$\displaystyle
u_{i}(t_{i})=\varepsilon^{\prime}\varphi_{1}+O\left(\left(\varepsilon^{\prime}\right)^{2}\right)\
\ \textit{in }L^{2}([0,1])\ \quad(i=1,2)$

for some appropriate $t_{1},t_{2}.$

Finally, we have to check (15).

$\displaystyle\sup_{t_{0}\leq t\leq\bar{t}}\|u(t)\|_{2}\leq\sup_{t_{0}\leq
t\leq\bar{t}}\|v(t)\|_{2}\leq\|u(t_{0})\|_{2}e^{\alpha_{1}\bar{t}}=\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}\varepsilon^{\prime}\leq\eta$

when $\varepsilon^{\prime}$ is small enough. If $t_{0}=0$, (15) is already
satisfied. Otherwise, one must check the interval $[0,\hat{t}]$ as well. That
is only needed when $W$ is either bounded or rank-$1$, in which case
$\gamma<\lambda_{1}\beta<\gamma+2\beta(\lambda_{1}-\lambda_{2})$ is assumed.

Let $v$ be the solution of (12) with initial condition $v(0)=u(0).$ Using
(37),

$\displaystyle\|u(t)\|_{2}^{2}\leq$
$\displaystyle\|v(t)\|_{2}^{2}=c_{1}^{2}(0)e^{2\alpha_{1}t}+\sum_{k>1}c_{k}^{2}(0)e^{\alpha_{k}t}\leq
c_{1}(0)e^{2\alpha_{1}\hat{t}}+\|u(0)\|_{2}^{2}e^{2\alpha_{2}\hat{t}}$
$\displaystyle=$ $\displaystyle
c_{1}^{2}(0)e^{2\alpha_{1}\hat{t}}\left(1+\underbrace{\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}e^{-(\alpha_{1}-\alpha_{2})\hat{t}}\right)^{2}}_{=1}\right)$
$\displaystyle\sup_{0\leq t\leq\hat{t}}\|u(t)\|_{2}\leq$
$\displaystyle\sqrt{2}c_{1}(0)e^{\alpha_{1}\hat{t}}=O\left(c_{1}(0)^{1-\frac{\alpha_{1}}{2(\alpha_{1}-\alpha_{2})}}\right)\to
0$ (32)

as $\delta\to 0+,$ thus, it is smaller than $\eta$ for small enough
$\delta>0.$

###  4.2 Reaching a small, but constant level

In this section we want to reach a level $\varepsilon_{0}$ where
$\varepsilon_{0}$ is still relatively small, but does not depend
$\varepsilon^{\prime}$.

Applying an appropriate time translation, we can set

$u_{1}(0),u_{2}(0)=\varepsilon^{\prime}\varphi_{1}+O\left((\varepsilon^{\prime})^{2}\right).$

When $\varepsilon_{0}$ is sufficiently small, $u_{1}$ and $u_{2}$ can still be
approximated by the linearized versions $v_{1},v_{2}$, although, since
$\varepsilon_{0}$ is set at a constant value (depending only on
$W,\beta,\gamma$), a macroscopic error term remains.

Starting from $v(0)=\varepsilon^{\prime}\varphi_{1}$ we need

$\displaystyle
t^{*}=\frac{1}{\alpha_{1}}\log\frac{\varepsilon_{0}}{\varepsilon^{\prime}}$

time to reach level $\varepsilon_{0}$. Also, in the linear system the error
propagation rate is at most $\alpha_{1}$ hence

$\displaystyle\sup_{0\leq t\leq
t^{*}}\|v_{1}(t^{*})-v_{2}(t^{*})\|_{2}\leq\|v_{1}(0)-v_{2}(0)\|_{2}e^{\alpha_{1}t^{*}}=O\left((\varepsilon^{\prime})^{2}\frac{\varepsilon_{0}}{\varepsilon^{\prime}}\right)=O\left(\varepsilon^{\prime}\right)$

and the error could still be arbitrarily small even after reaching level
$\varepsilon_{0}.$

What is left to do is to mitigate the effect of linearization. It turns out
that similar, somewhat weaker but still sufficient conclusions can be drawn to
the nonlinear system as well.

######  Lemma 9.

For $0<\varepsilon^{\prime}\leq\varepsilon_{0}$

$u_{1}(t^{*}),u_{2}(t^{*})=\varepsilon_{0}\varphi+O\left(\varepsilon_{0}^{2}\right)\
\ \textit{in }L^{2}([0,1]),$

hence, for small enough $\varepsilon_{0}>0$

$\|u_{1}(t^{*})\|_{2},\|u_{2}(t^{*})\|_{2}\geq\frac{\varepsilon_{0}}{2}.$

######  Lemma 10.

For small enough $\varepsilon_{0}>0$

$\displaystyle\sup_{0\leq t\leq
t^{*}}\|u_{1}(t)-u_{2}(t)\|_{2}=O\left(\sqrt{\varepsilon^{\prime}}\right).$

We mention that Lemma 10 can easily be modified to get an error bound
$O\left((\varepsilon^{\prime})^{1-\vartheta}\right)$ for arbitrary
$0<\vartheta<1.$

###  4.3 Getting $\varepsilon$-close to the endemic state

We shift the time so that
$\|u_{1}(0)-u_{2}(0)\|_{2}=O\left(\sqrt{\varepsilon^{\prime}}\right)$ and
$u_{1}(0),u_{2}(0)=\varepsilon_{0}\varphi+O\left(\varepsilon_{0}^{2}\right).$

In this section we want to get the two solutions $u_{1},u_{2}$
$\varepsilon$-close to the endemic state $\psi$. The deviations are denoted by
$\tilde{u}_{i}(t):=u_{i}(t)-\psi$ ($i\in\\{1,2\\}$).

Observe that the right hand side of (4) is Lipschitz continuous on $\Delta$
with respect to $\|\cdot\|_{2}$ with constant $L:=2\beta\lambda_{1}+\gamma,$
hence, the error $\|u_{1}(t)-u_{2}(t)\|_{2}$ can only increase by a constant
factor during the finite time interval $[0,t^{**}]$ resulting in

$\displaystyle\sup_{0\leq t\leq
t^{**}}\|u_{1}(t)-u_{2}(t)\|_{2}=O_{t^{**}}\left(\sqrt{\varepsilon^{\prime}}\right).$
(33)

where the subscript $t^{**}$ emphasizes the dependence on $t^{**}$ (but not on
$\delta$). The question is, can we set a $t^{**}=t^{**}(\varepsilon)$ (but
independent of $\delta,\varepsilon^{\prime}$) such that $u_{1},u_{2}$ are
already close to equilibrium?

######  Lemma 11.

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\tilde{u}(t)\|_{2}^{2}\leq-2\beta\int_{0}^{1}\left(\tilde{u}(t,x)\right)^{2}\mathbb{W}u(t,x)\mathrm{d}x$
(34)

Since $u_{1}(0),u_{2}(0)$ are already reached the macroscopic level
$\varepsilon_{0}$ we hope that some $\tilde{\varepsilon}_{0}>0$ number exist
such that $\mathbb{W}u(t,x)\geq\tilde{\varepsilon}_{0},$ or in other words,
every susceptible individual gets infected with a uniformly positive rate.
Thus, by (34) and Grönwall’s lemma

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\tilde{u}(t)\|_{2}^{2}\leq$
$\displaystyle-2\beta\tilde{\varepsilon}_{0}\int_{0}^{1}\left(\tilde{u}(t,x)\right)^{2}\mathrm{d}x=-2\beta\tilde{\varepsilon}_{0}\|\tilde{u}(t)\|_{2}$
$\displaystyle\|\tilde{u}(t^{**})\|_{2}\leq$ $\displaystyle
e^{-\beta\tilde{\varepsilon}_{0}t^{**}}\|\tilde{u}(0)\|_{2}\leq
2e^{-\beta\tilde{\varepsilon}_{0}t^{**}}=\frac{\varepsilon}{2}.$

for $t^{**}=\frac{1}{\beta\tilde{\varepsilon}_{0}}\log\frac{\varepsilon}{4}.$

Note that, for this choice of $t^{**}$ the right hand side of (33) becomes
dependent on $\varepsilon.$

The following lemma provides the existence of such $\tilde{\varepsilon}_{0}.$

######  Lemma 12.

Assume one of the following:

  * •

$W$ is discrete,

  * •

$0<m_{0}\leq W(x,y)\leq M,$

  * •

$W$ is rank-$1$ and $0<m\leq\varphi_{1}(x)$.

Also, assume
$u(0)=\varepsilon_{0}\varphi_{1}+O\left(\varepsilon_{0}^{2}\right)$ for some
small enough $\varepsilon_{0}$. Then there exists an
$\tilde{\varepsilon}_{0}>0$ such that for all $t\geq 0,\ x\in[0,1]$,

$\mathbb{W}u(t,x)\geq\tilde{\varepsilon}_{0}.$

Observe (34) ensures that the deviation from $\psi$ can only decrease. Hence,
for all $t\geq t^{**}$

$\displaystyle\|u_{1}(t)-u_{2}(t)\|_{2}=$
$\displaystyle\|\tilde{u}_{1}(t)-\tilde{u}_{2}(t)\|_{2}\leq\|\tilde{u}_{1}(t)\|_{2}+\|\tilde{u}_{2}(t)\|_{2}$
$\displaystyle\leq$
$\displaystyle\|\tilde{u}_{1}(t^{**})\|_{2}+\|\tilde{u}_{2}(t^{**})\|_{2}\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.$

To sum up, we can derive USIC the following way: we choose the small, but
constant $\varepsilon_{0}$ and $\tilde{\varepsilon}_{0}$ values depending on
$W$. Then we set the arbitrarily small $\varepsilon,\eta>0$ values for which
we can choose a small enough $\varepsilon^{\prime}$ (depending on
$\varepsilon$ and $\eta$) such that
$O\left(\sqrt{\varepsilon^{\prime}}\right)$ and any other error term goes
under $\varepsilon$ in addition to $O(\varepsilon^{\prime})\leq\eta$. Lastly,
we choose an appropriate $\delta>0$ for such $\varepsilon^{\prime}$ and for
(32).

This way, at every step of the argument, the error is at most $\varepsilon$
and initially both $u_{1}$ and $u_{2}$ are under level $\eta$.

##  5 Proofs

###  5.1 Smaller statements

Here we gathered some lemmas that are used in proofs but not vital to
understanding the main ideas.

######  Lemma 13.

Denote $g(t):=\beta u(t)\mathbb{W}u(t).$

$\displaystyle u(t)=v(t)-\int_{t_{0}}^{t}e^{\mathcal{A}(t-s)}g(s)\mathrm{d}s$
(35)

###### Proof.

(Lemma 13)

Note that

$\displaystyle\partial_{t}u$
$\displaystyle=\beta(1-u)\mathbb{W}u-u=(\beta\mathbb{W}-\mathbb{I})u-\beta
u\mathbb{W}u=\mathcal{A}u-g.$

Treating $g$ as a known function makes (4) an inhomogeneous linear problem
with solution

$\displaystyle u(t)=$ $\displaystyle
e^{A(t-t_{0})}\left(u(t_{0})-\int_{t_{0}}^{t}e^{-\mathcal{A}(s-t_{0})}g(s)\mathrm{d}s\right)=$
$\displaystyle\underbrace{e^{A(t-t_{0})}u(t_{0})}_{=v(t)}-\int_{t_{0}}^{t}e^{\mathcal{A}(t-s)}g(s)\mathrm{d}s.$

∎

######  Lemma 14.

Assume $\varphi_{1}$ is uniformly positive with constant $m>0.$ Then

$\displaystyle\|u(t)\|_{2}\leq\frac{1}{\sqrt{m}}\sqrt{c_{1}(t)}.$

###### Proof.

(Lemma 14)

$\displaystyle\|u(t)\|_{2}^{2}$
$\displaystyle=\int_{0}^{1}u^{2}(t,x)\mathrm{d}x\leq\int_{0}^{1}u(t,x)\mathrm{d}x=\frac{1}{m}\int_{0}^{1}mu(t,x)\mathrm{d}x$
$\displaystyle\leq\frac{1}{m}\int_{0}^{1}\varphi_{1}(x)u(t,x)\mathrm{d}x=\frac{1}{m}c_{1}(t)$

∎

###  5.2 Existence, uniqueness and approximation of the PDE

Here we give the proofs of Propositions 1 and 2.

###### Proof.

(Proposition 2) Let $f_{i}\in\Delta$ and $0\leq W_{i}\in
L^{2}\left([0,1]^{2}\right),\ \|W_{i}\|_{2}\leq\lambda\ (i=1,2).$ Define

$\displaystyle\mathcal{G}[f_{i},W_{i}](x):=\beta(1-f_{i})\mathbb{W}_{i}f_{i}-\gamma
f_{i}.$

$\displaystyle\|\mathcal{G}[f_{1},W_{1}]-\mathcal{G}[f_{2},W_{2}]\|_{2}\leq$
$\displaystyle\beta\|(1-f_{1})\mathbb{W}_{1}f_{1}-(1-f_{2})\mathbb{W}_{2}f_{2}\|_{2}+\gamma\|f_{1}-f_{2}\|_{2}$
$\displaystyle\leq$
$\displaystyle\beta\|\mathbb{W}_{1}-\mathbb{W}_{2}\|+\beta\|(1-f_{1})\mathbb{W}_{2}f_{1}-(1-f_{2})\mathbb{W}_{2}f_{2}\|_{2}$
$\displaystyle+\gamma\|f_{1}-f_{2}\|_{2}$ $\displaystyle\leq$
$\displaystyle\beta\|\mathbb{W}_{1}-\mathbb{W}_{2}\|+\beta\|\mathbb{W}_{2}(f_{1}-f_{2})\|_{2}$
$\displaystyle+\beta\|\mathbb{W}_{2}f_{2}\|_{2}\|f_{1}-f_{2}\|_{2}+\gamma\|f_{1}-f_{2}\|_{2}$
$\displaystyle\leq$
$\displaystyle\beta\|\mathbb{W}_{1}-\mathbb{W}_{2}\|_{2}+(2\beta\lambda+\gamma)\|f_{1}-f_{2}\|_{2}$

Note that

$\displaystyle
u_{i}(t)=u_{i}(0)+\int_{0}^{t}\mathcal{G}[u_{i}(s),W_{i}]\mathrm{d}s,$

hence, applying triangle inequality and Grönwall’s lemma yields

$\displaystyle\sup_{0\leq t\leq
T}\|u_{1}(t)-u_{2}(t)\|_{2}\leq\left(\|u_{1}(0)-u_{2}(0)\|_{2}+\beta
T\|\mathbb{W}_{1}-\mathbb{W}_{2}\|_{2}\right)e^{(2\beta\lambda_{1}+\gamma)T}.$

To finish, we observe
$\|\mathbb{W}_{1}-\mathbb{W}_{2}\|_{2}\leq\|W_{1}-W_{2}\|_{2}.$ ∎

###### Proof.

(Proposition 1) The uniqueness simply follows from the right hand side of (4)
being Lipschitz on $\Delta.$

Let $u_{N}$ be a sequence of solutions with kernel

$W_{N}(x,y):=\max\\{W(x,y),N\\}$

and initial condition $u_{N}(0)=u(0).$ [6, Proposition 2.9. (ii)] guarantees
that such solutions exist on $\mathbb{R}_{0}^{+}$ and $u_{N}(t)\in\Delta.$

From (9) we conclude

$\displaystyle\sup_{0\leq t\leq
T}\|u_{N}(t)-u_{M}(t)\|_{2}=O(\|W_{N}-W_{M}\|_{2})\to 0$

as $N,M\to\infty$, making $\left(u_{N}\right)_{N=1}^{\infty}$ a Cauchy-
sequence on $[0,T].$ Since $T$ is arbitrary, the domain of $u$ can be extended
to $\mathbb{R}_{0}^{+}.$

It is straightforward to check that $u(t)\in\Delta$ and satisfies (4). ∎

###  5.3 Proofs of Subsection 4.1

###### Proof.

(Lemma 2)

Recall Lemma 13.

$\displaystyle\|g(s)\|_{2}$
$\displaystyle=\beta\|u(s)\mathbb{W}u(s)\|_{2}\leq\beta\lambda_{1}\|u(s)\|_{2}^{2}\leq\beta\lambda_{1}\|v(s)\|_{2}^{2}$
$\displaystyle\leq\beta\lambda_{1}\|u(t_{0})\|_{2}^{2}e^{2\alpha_{1}(s-t_{0})}$
$\displaystyle\|u(t)-v(t)\|_{2}$
$\displaystyle\leq\int_{t_{0}}^{t}\left\|e^{\mathcal{A}(t-s)}g(s)\right\|_{2}\mathrm{d}s\leq\int_{t_{0}}^{t}e^{\alpha_{1}(t-s)}\left\|g(s)\right\|_{2}\mathrm{d}s$
$\displaystyle\leq\beta\lambda_{1}\|u(t_{0})\|_{2}^{2}e^{\alpha_{1}(t-t_{0})}\int_{t_{0}}^{t}e^{\alpha_{1}(s-t_{0})}\mathrm{d}s$
$\displaystyle\leq\frac{\beta\lambda_{1}}{\alpha_{1}}\|u(t_{0})\|_{2}^{2}e^{2\alpha_{1}(t-t_{0})}$

Thus,

$\displaystyle\sup_{t_{0}\leq t\leq
t_{0}+\bar{t}}\|u(t)-v(t)\|_{2}\leq\frac{\beta\lambda_{1}}{\alpha_{1}}\|u(t_{0})\|_{2}^{2}e^{2\alpha_{1}\bar{t}}=\frac{\beta\lambda_{1}}{\alpha_{1}}\left(\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}\right)^{2}(\varepsilon^{\prime})^{2}.$
(36)

∎

###### Proof.

(Lemma 3)

$\displaystyle v(t_{0}+\bar{t})$
$\displaystyle=c_{1}(t_{0})e^{\alpha_{1}\bar{t}}\varphi_{1}+\sum_{k>1}c_{k}(t_{0})e^{\alpha_{k}\bar{t}}\varphi_{k}$
$\displaystyle=\underbrace{c_{1}(t_{0})e^{\alpha_{1}\bar{t}}}_{=\varepsilon^{\prime}}\left(\varphi_{1}+\frac{1}{c_{1}(t_{0})}\sum_{k>1}c_{k}(t_{0})e^{-(\alpha_{1}-\alpha_{k})\bar{t}}\varphi_{k}\right)$

$\displaystyle\left\|\frac{1}{c_{1}(t_{0})}\sum_{k>1}c_{k}(t_{0})e^{-(\alpha_{1}-\alpha_{k})\bar{t}}\varphi_{k}\right\|_{2}^{2}=\frac{1}{c_{1}^{2}(t_{0})}\sum_{k>1}c_{k}^{2}(t_{0})e^{-2(\alpha_{1}-\alpha_{k})\bar{t}}\leq$
$\displaystyle\frac{e^{-2(\alpha_{1}-\alpha_{2})\bar{t}}}{c_{1}^{2}(t_{0})}\sum_{k>1}c_{k}^{2}(t_{0})\leq\left(\frac{\|u(t_{0})\|_{2}}{c_{1}(t_{0})}\right)^{2}e^{-2(\alpha_{1}-\alpha_{2})\bar{t}}$

∎

###### Proof.

(Lemma 4)

$\displaystyle\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{2}$
$\displaystyle=\frac{\int_{0}^{1}u^{2}(0,x)\mathrm{d}x}{\left(\int_{0}^{1}\varphi_{1}(x)u(0,x)\mathrm{d}x\right)^{2}}\leq\frac{1}{m^{2}}\int_{0}^{1}\left(\frac{u(0,x)}{\|u(0)\|_{1}}\right)^{2}\mathrm{d}x$
$\displaystyle=:\frac{1}{m^{2}}\int_{0}^{1}\rho^{2}(x)\mathrm{d}x\leq\frac{\|\rho\|_{\infty}}{m^{2}}$

as $\rho(x):=\frac{u(0,x)}{\|u(t_{0})\|_{1}}$ is a density function.

It remains to give an upper bound on $\rho(x).$

$\displaystyle(x\in I_{i})\ \
\rho(x)=\frac{z_{i}(0)}{\sum_{j=1}^{n}z_{j}(0)|I_{j}|}\leq\frac{1}{J}\frac{z_{i}(0)}{\sum_{j=1}^{n}z_{j}(0)}\leq\frac{1}{J}.$

∎

###### Proof.

(Lemma 5)

Recall Lemma 13. Let $v$ be the solution of (12) with $v(0)=u(0).$

$\displaystyle u(t)$
$\displaystyle=v(t)-\beta\int_{0}^{t}e^{\mathcal{A}(t-s)}u(s)\mathbb{W}u(s)\mathrm{d}s$
$\displaystyle=v(t)-\beta\int_{0}^{t}\sum_{k=1}^{\infty}\langle
u(s)\mathbb{W}u(s),\varphi_{k}\rangle\underbrace{e^{\mathcal{A}(t-s)}\varphi_{k}}_{e^{\alpha_{k}(t-s)}\varphi_{k}}\mathrm{d}s$
$\displaystyle c_{1}(t)$
$\displaystyle=c_{1}(0)e^{\alpha_{1}t}-\beta\int_{0}^{t}e^{\alpha_{1}(t-s)}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s$
$\displaystyle=c_{1}(0)e^{\alpha_{1}t}\left(1-\frac{\beta}{c_{1}(0)}\int_{0}^{t}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\right)$

$\|u(t)\|_{2}$ will be bounded in the usual way.

$\displaystyle\|u(t)\|_{2}^{2}$ $\displaystyle\leq\|v(t)\|_{2}^{2}\leq
c_{1}^{2}(0)e^{2\alpha_{1}t}+\|u(0)\|_{2}^{2}e^{2\alpha_{2}t}$

This results in

$\displaystyle\left(\frac{\|u(\hat{t})\|_{2}}{c_{1}(\hat{t})}\right)^{2}$
$\displaystyle\leq\frac{1+\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{2}e^{-2(\alpha_{1}-\alpha_{2})\hat{t}}}{\left(1-\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\right)^{2}}$
$\displaystyle=\frac{2}{\left(1-\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\right)^{2}}.$

∎

###### Proof.

(Lemma 6)

Note that
$\mathbb{W}u(s)=\lambda_{1}\varphi_{1}c_{1}(s)\leq\lambda_{1}\varphi_{1}c_{1}(0)e^{\alpha_{1}s}.$
We have

$\displaystyle\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\leq\beta\lambda_{1}\int_{0}^{\hat{t}}\left\langle
u(s),\varphi_{1}^{2}\right\rangle\mathrm{d}s$

Use Hölder’s inequality with $p=1+\frac{2}{\rho},\ q=1+\frac{\rho}{2}.$

$\displaystyle\left\langle
u(s),\varphi_{1}^{2}\right\rangle\leq\|u(s)\|_{p}\left\|\varphi_{1}^{2}\right\|_{q}.$

Since $\varphi_{1}\in L^{2+\rho}([0,1])$

$\displaystyle\left\|\varphi_{1}^{2}\right\|_{q}^{q}=\int_{0}^{1}\varphi_{1}^{2q}(x)\mathrm{d}x=\int_{0}^{1}\varphi_{1}^{2+\rho}(x)\mathrm{d}x<\infty.$

As for the first term, since $0\leq u(s,x)\leq 1$ and $1<p<\infty$

$\displaystyle\|u(s)\|_{p}^{p}=$
$\displaystyle\int_{0}^{1}\left(u(s,x)\right)^{p}\mathrm{d}x\leq\int_{0}^{1}u(s,x)\mathrm{d}x\leq\frac{1}{m}\int_{0}^{1}\varphi_{1}(x)u(s,x)\mathrm{d}x$
$\displaystyle=$
$\displaystyle\frac{1}{m}c_{1}(s)\leq\frac{1}{m}c_{1}(0)e^{\alpha_{1}s}$

leading to

$\displaystyle\int_{0}^{\hat{t}}\left\langle
u(s),\varphi_{1}^{2}\right\rangle\mathrm{d}s\leq\left(\frac{c_{1}(0)}{m}\right)^{\frac{1}{p}}\left\|\varphi_{1}^{2}\right\|_{q}\int_{0}^{\hat{t}}e^{\frac{\alpha_{1}}{p}s}\mathrm{d}s\leq\frac{p}{\alpha_{1}}m^{-\frac{1}{p}}\left\|\varphi_{1}^{2}\right\|_{q}\left(c_{1}(0)e^{\alpha_{1}\hat{t}}\right)^{\frac{1}{p}}.$

Based on Lemma 14,

$\displaystyle\begin{split}e^{\alpha_{1}\hat{t}}=&\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{\frac{\alpha_{1}}{\alpha_{1}-\alpha_{2}}}=O\left(c_{1}(0)^{-\frac{\alpha_{1}}{2(\alpha_{1}-\alpha_{2})}}\right)\\\
c_{1}(0)e^{\alpha_{1}\hat{t}}=&O\left(c_{1}(0)^{1-\frac{\alpha_{1}}{2(\alpha_{1}-\alpha_{2})}}\right)\to
0\end{split}$ (37)

as $\delta\to 0^{+}$ since

$\displaystyle\frac{\alpha_{1}}{2(\alpha_{1}-\alpha_{2})}=\frac{\beta\lambda_{1}-\gamma}{2\beta\lambda_{1}}=\frac{1}{2}\left(1-\frac{\gamma}{\beta\lambda_{1}}\right)<1.$

Hence, we can find a small enough $\delta_{0}$ such that
$0<\delta\leq\delta_{0}$ implies

$\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\leq\frac{1}{2}.$

∎

###### Proof.

(Lemma 7) Since $W$ is bounded and $\varphi_{1}(x)\geq m$,

$\displaystyle\begin{split}\mathbb{W}u(t,x)=&\int_{0}^{1}W(x,y)u(t,y)\mathrm{d}y\leq\frac{M}{m}\int_{0}^{1}mu(t,y)\mathrm{d}y\leq\\\
&\frac{M}{m}\int_{0}^{1}\varphi_{1}(y)u(t,y)\mathrm{d}y=\frac{M}{m}\langle
u(t),\varphi_{1}\rangle=\frac{M}{m}c_{1}(t)\end{split}$ (38)

uniformly in $x\in[0,1].$

$\displaystyle\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\leq\frac{\beta
M}{mc_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}c_{1}(s)\langle
u(s),\varphi_{1}\rangle\mathrm{d}s=$ $\displaystyle\frac{\beta
M}{mc_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}c_{1}^{2}(s)\mathrm{d}s\leq\frac{\beta
M}{mc_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}c_{1}^{2}(0)e^{2\alpha_{1}s}\mathrm{d}s=$
$\displaystyle\frac{\beta
Mc_{1}(0)}{m}\int_{0}^{\hat{t}}e^{\alpha_{1}s}\mathrm{d}s\leq\frac{\beta
Mc_{1}(0)}{\alpha_{1}m}e^{\alpha_{1}\hat{t}}$

Using (37)

$\displaystyle
c_{1}(0)e^{\alpha_{1}\hat{t}}=O\left(c_{1}(0)^{1-\frac{\alpha_{1}}{2(\alpha_{1}-\alpha_{2})}}\right)\to
0$

as $\delta\to 0^{+}$ since $\frac{\alpha_{1}}{2(\alpha_{1}-\alpha_{2})}<1$ due
to our assumption. Hence, we can find a small enough $\delta_{0}$ such that
$0<\delta\leq\delta_{0}$ implies

$\frac{\beta}{c_{1}(0)}\int_{0}^{\hat{t}}e^{-\alpha_{1}s}\langle
u(s)\mathbb{W}u(s),\varphi_{1}\rangle\mathrm{d}s\leq\frac{1}{2}.$

∎

###### Proof.

(Lemma 8)

$\displaystyle\bar{t}=\frac{1}{\alpha_{1}}\log\frac{\varepsilon^{\prime}}{c_{1}(\hat{t})}=\frac{1}{\alpha_{1}}\log\varepsilon^{\prime}+\frac{1}{\alpha_{1}}\log\frac{1}{c_{1}(\hat{t})}$

Since $\varepsilon^{\prime}$ is fixed,
$\frac{1}{\alpha_{1}}\log\varepsilon^{\prime}$ is just a constant and we can
neglect it.

$\displaystyle\frac{1}{\alpha_{1}}\log\frac{1}{c_{1}(\hat{t})}\geq\frac{1}{\alpha_{1}}\log\frac{1}{c_{1}(0)e^{\alpha_{1}\hat{t}}}=\frac{1}{\alpha_{1}}\log\frac{1}{c_{1}(0)}-\hat{t}=$
$\displaystyle-\frac{1}{\alpha_{1}}\log
c_{1}(0)-\frac{1}{\alpha_{1}-\alpha_{2}}\log\frac{\|u(0)\|_{2}}{c_{1}(0)}=-\log\left[c_{1}(0)^{\frac{1}{\alpha_{1}}}\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{\frac{1}{\alpha_{1}-\alpha_{2}}}\right]$

Clearly, it is enough to show that the argument of the logarithm is small.
Using Lemma 14 with
$\nu:=\frac{1}{\alpha_{1}}-\frac{1}{2(\alpha_{1}-\alpha_{2})}$ shows

$\displaystyle
c_{1}(0)^{\frac{1}{\alpha_{1}}}\left(\frac{\|u(0)\|_{2}}{c_{1}(0)}\right)^{\frac{1}{\alpha_{1}-\alpha_{2}}}=O\left(c_{1}^{\nu}(0)\right).$

Observe

$\displaystyle\nu>$ $\displaystyle 0$ $\displaystyle\frac{1}{\alpha_{1}}>$
$\displaystyle\frac{1}{2(\alpha_{1}-\alpha_{2})}$ $\displaystyle\alpha_{1}<$
$\displaystyle 2(\alpha_{1}-\alpha_{2})$
$\displaystyle\beta\lambda_{1}-\gamma<$ $\displaystyle
2\beta(\lambda_{1}-\lambda_{2})$

making $\nu$ positive by the assumption.

Therefore $\bar{t}\geq-\nu\log c_{1}(0)+O(1)\to\infty$ as $\delta\to 0^{+}.$ ∎

###  5.4 Proofs of Subsection 4.2

###### Proof.

(Lemma 9)

Let $u(t)$ be either $u_{1}(t)$ or $u_{2}(t).$ From the proof of Lemma 2,

$\displaystyle\left\|u(t^{*})-v(t^{*})\right\|_{2}\leq\frac{\beta\lambda_{1}}{\alpha_{1}}\|u(0)\|_{2}^{2}e^{2\alpha_{1}t^{*}}=O\left(\left(\varepsilon^{\prime}\frac{\varepsilon_{0}}{\varepsilon^{\prime}}\right)^{2}\right)=O\left(\varepsilon_{0}^{2}\right),$

making $u(t^{*})=v(t^{*})+O\left(\varepsilon_{0}^{2}\right).$

$\displaystyle v(t^{*})=$ $\displaystyle
e^{\mathcal{A}t^{*}}u(0)=e^{\mathcal{A}t^{*}}(\varepsilon^{\prime}\varphi_{1}+O\left((\varepsilon^{\prime})^{2}\right))=\varepsilon_{0}\varphi_{1}+e^{\mathcal{A}t^{*}}O\left((\varepsilon^{\prime})^{2}\right))$
$\displaystyle=$
$\displaystyle\varepsilon_{0}\varphi_{1}+O\left(e^{\alpha_{1}t^{*}}(\varepsilon^{\prime})^{2}\right))=O\left(\frac{\varepsilon_{0}}{\varepsilon^{\prime}}(\varepsilon^{\prime})^{2}\right)=\varepsilon_{0}\varphi_{1}+O\left(\varepsilon_{0}\varepsilon^{\prime}\right)$
$\displaystyle\overset{\varepsilon^{\prime}\leq\varepsilon_{0}}{=}$
$\displaystyle\varepsilon_{0}\varphi_{1}+O\left(\varepsilon_{0}^{2}\right)$

∎

###### Proof.

(Lemma 10),

Using Lemma 13,

$\displaystyle u_{i}(t)=$ $\displaystyle
e^{\mathcal{A}t}u_{i}(0)-\beta\int_{0}^{t}e^{\mathcal{A}(t-s)}u_{i}(s)\mathbb{W}u_{i}(s)\mathrm{d}s$
$\displaystyle u_{1}(t)-u_{2}(t)=$ $\displaystyle
e^{\mathcal{A}t}(u_{1}(0)-u_{2}(0))$
$\displaystyle-\int_{0}^{t}e^{\mathcal{A}(t-s)}\left(u_{1}(s)\mathbb{W}u_{1}(s)-u_{2}(s)\mathbb{W}u_{2}(s)\right)\mathrm{d}s.$

Notice

$\displaystyle\left\|u_{1}(s)\mathbb{W}u_{1}(s)-u_{2}(s)\mathbb{W}u_{2}(s)\right\|_{2}\leq$
$\displaystyle\|\mathbb{W}u_{1}(s)\|_{2}\|u_{1}(s)-u_{2}(s)\|_{2}+\|u_{2}(s)\|_{2}\|\mathbb{W}\left(u_{1}(s)-u_{2}(s)\right)\|_{2}\leq$
$\displaystyle\lambda_{1}\left(\|u_{1}(s)\|_{2}+\|u_{2}(s)\|_{2}\right)\|u_{1}(s)-u_{2}(s)\|_{2}\leq$
$\displaystyle\lambda_{1}\left(\|u_{1}(0)\|_{2}+\|u_{2}(0)\|_{2}\right)e^{\alpha_{1}s}\|u_{1}(s)-u_{2}(s)\|_{2}.$

Set $0\leq t\leq t^{*}.$

$\displaystyle\|u_{1}(t)-u_{2}(t)\|_{2}\leq$ $\displaystyle
e^{\alpha_{1}t}\|u_{1}(0)-u_{2}(0)\|_{2}+$
$\displaystyle\beta\lambda_{1}\left(\|u_{1}(0)\|_{2}+\|u_{2}(0)\|_{2}\right)\int_{0}^{t}e^{\alpha_{1}(t-s)}e^{\alpha_{1}s}\|u_{1}(s)-u_{2}(s)\|_{2}\mathrm{d}s\leq$
$\displaystyle\underbrace{e^{\alpha_{1}t^{*}}\|u_{1}(0)-u_{2}(0)\|_{2}}_{=O\left(\varepsilon^{\prime}\right)}+$
$\displaystyle\underbrace{\beta\lambda_{1}\left(\|u_{1}(0)\|_{2}+\|u_{2}(0)\|_{2}\right)e^{\alpha_{1}t^{*}}}_{=O\left(\varepsilon_{0}\right)}\int_{0}^{t}\|u_{1}(s)-u_{2}(s)\|_{2}\mathrm{d}s$

Let $K$ be the constant corresponding to $O(\varepsilon_{0})$. Choose
$\varepsilon_{0}$ to be small enough to satisfy
$\frac{K\varepsilon_{0}}{\alpha_{1}}\leq\frac{1}{2}.$ Then applying Grönwall’s
lemma yields

$\displaystyle\sup_{0\leq t\leq t^{*}}\|u_{1}(t)-u_{2}(t)\|_{2}=$
$\displaystyle
O\left(\varepsilon^{\prime}e^{K\varepsilon_{0}t^{*}}\right)=O\left(\varepsilon^{\prime}\left(\frac{\varepsilon_{0}}{\varepsilon^{\prime}}\right)^{\frac{K\varepsilon_{0}}{\alpha_{1}}}\right)=O\left(\varepsilon^{\prime}\sqrt{\frac{\varepsilon_{0}}{\varepsilon^{\prime}}}\right)$
$\displaystyle=$ $\displaystyle O\left(\sqrt{\varepsilon^{\prime}}\right).$

∎

###  5.5 Proofs of Subsection 4.3

###### Proof.

(Lemma 11)

From [6, Proposition 4.13.] we know that the operator
$\mathcal{B}=\beta(1-\psi)\mathbb{W}$ has spectral radius $\gamma$ with $\psi$
as the corresponding Perron-Frobenius eigenfunction. This makes
$\Lambda:=\mathcal{B}-\gamma\mathbb{I}$ a negative operator as

$\displaystyle\left\langle\Lambda
u,u\right\rangle=\left\langle\mathcal{B}u,u\right\rangle-\gamma\|u\|_{2}^{2}\leq\gamma\|u\|_{2}^{2}-\gamma\|u\|_{2}^{2}=0.$

Also, $\Lambda\psi=0$.

Next, we calculate the time derivative of $\tilde{u}.$

$\displaystyle\partial_{t}\tilde{u}=$
$\displaystyle\partial_{t}(u-\psi)=\partial_{t}u=\beta(1-u)\mathbb{W}u-\gamma
u$ $\displaystyle=$ $\displaystyle\beta(1-\psi-\tilde{u})\mathbb{W}u-\gamma
u=\Lambda u-\beta\left(\mathbb{W}u\right)\tilde{u}$

Since $\Lambda u=\Lambda(\tilde{u}+\psi)=\Lambda\tilde{u}$ we end up with the
expression

$\displaystyle\partial_{t}\tilde{u}=\Lambda\tilde{u}-\beta\left(\mathbb{W}u\right)\tilde{u}.$

Finally, we give an upper bound on the derivative of
$\|\tilde{u}(t)\|_{2}^{2}.$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\tilde{u}(t)\|_{2}^{2}=$
$\displaystyle
2\left\langle\partial_{t}\tilde{u}(t),\tilde{u}(t)\right\rangle=2\left\langle\Lambda\tilde{u}(t),\tilde{u}(t)\right\rangle-2\beta\left\langle(\mathbb{W}u(t))\tilde{u}(t),\tilde{u}(t)\right\rangle$
$\displaystyle\leq$
$\displaystyle-2\beta\left\langle(\mathbb{W}u(t))\tilde{u}(t),\tilde{u}(t)\right\rangle=-2\beta\int_{0}^{1}\left(\tilde{u}(t,x)\right)^{2}\mathbb{W}u(t,x)\mathrm{d}x$

∎

The proof of Lemma 11 will be handled in three parts based on the set of
assumptions we have.

For Case 1, first we show that $u(t)$ can be bounded from below by a curve
which is increasing and uniformly positive. The following lemma is a
reformulation of [6, Proposition 4.8.].

######  Lemma 15.

Assume $\lambda_{1}\beta>\gamma$ and $\varphi_{1}$ is bounded. Define
$\varphi_{\theta}:=\theta\varphi_{1}$ and let $u_{\theta}$ be a solution to
(4) with initial condition $u_{\theta}(0)=\varphi_{\theta}.$

Then there is a small $\theta_{0}>0$ such that for all
$0<\theta\leq\theta_{0}$ $\varphi_{\theta}\in\Delta$ and $u_{\theta}(t,x)$ is
monotone increasing in $t\geq 0.$

###### Proof.

(Lemma 15)

Since we are in the supercritical regime we can set a small $\epsilon>0$ such
that $\mu:=\beta(1-\epsilon)\lambda_{1}-\gamma\geq 0.$ For such $\epsilon$ we
can set a $\theta_{0}$ such that for all $0<\theta\leq\theta_{0}$
$0\leq\varphi_{\theta}(x)\leq\epsilon$ making $\varphi_{\theta}\in\Delta.$

Since $\varphi_{\theta}$ is an eigenvector of $\mathbb{W}$ with eigenvalue
$\lambda_{1}$

$\displaystyle\beta(1-\epsilon)\mathbb{W}\varphi_{\theta}=\beta(1-\epsilon)\lambda_{1}\varphi_{\theta}=(\gamma+\mu)\varphi_{\theta},$

implying

$\displaystyle
0\leq\mu\varphi_{\theta}=\beta(1-\epsilon)\mathbb{W}\varphi_{\theta}-\gamma\varphi_{\theta}\leq\beta(1-\varphi_{\theta})\mathbb{W}\varphi_{\theta}-\gamma\varphi_{\theta}.$

[6, Proposition 2.12.] implies $u_{\theta}(t)$ is increasing in $t\geq 0.$ ∎

###### Proof.

(Case 1 of Lemma 12)

For $x\in I_{i}$

$\displaystyle
u(0,x)=\frac{1}{|I_{i}|}\int_{I_{i}}u(0,y)\mathrm{d}y=\varepsilon_{0}\phi_{i}^{(1)}+O\left(\varepsilon_{0}^{2}\right)\geq\frac{\varepsilon_{0}}{2}\phi_{i}^{(1)}$

for small enough $\varepsilon_{0}$ uniformly in $i$. Therefore,
$u(0,x)\geq\frac{\varepsilon_{0}}{2}\varphi_{1}(x).$

In the discrete case,

$\|\varphi_{1}\|_{\infty}=\max_{1\leq i\leq n}\phi_{i}^{(1)}<\infty,$

making Lemma 15 applicable. Set
$\theta:=\min\\{\frac{\varepsilon_{0}}{2},\theta_{0}\\}.$ This makes $u(0)\geq
u_{\theta}(0),$ hence $u(t)\geq u_{\theta}(t)$ for $t\geq 0$ as (4) is
cooperative, and

$\displaystyle\mathbb{W}u(t,x)\geq\mathbb{W}u_{\theta}(t,x)\geq\mathbb{W}u_{\theta}(0,x)=\mathbb{W}\varphi_{\theta}=\lambda_{1}\varphi_{\theta}\geq\lambda_{1}\theta
m=:\tilde{\varepsilon}_{0}>0$

∎

###### Proof.

(Case 2 of Lemma 12)

Since $W$ is uniformly positive,

$\displaystyle\mathbb{W}u(t,x)=$
$\displaystyle\int_{0}^{1}W(x,y)u(t,y)\mathrm{d}y\geq
m_{0}\int_{0}^{1}u(t,y)\mathrm{d}y\geq m_{0}\int_{0}^{1}u^{2}(t,y)\mathrm{d}y$
$\displaystyle=$ $\displaystyle
m_{0}\|u(t)\|_{2}^{2}=m_{0}\left(\|\varphi_{1}\|_{2}\|u(t)\|_{2}\right)^{2}\geq
m_{0}\langle\varphi_{1},u(t)\rangle^{2}=m_{0}c_{1}^{2}(t),$

thus, it is enough to have a lower bound for $c_{1}(t).$

Since $u(0)=\varepsilon_{0}\varphi_{1}+O\left(\epsilon_{0}^{2}\right)$ one has
$c_{1}(0)\geq\frac{\varepsilon_{0}}{2}$ for small enough $\varepsilon_{0}.$

We will show that $c_{1}(t)$ can not go below a certain small positive value
after reaching such level.

Recall Lemma 13. From (35), scalar product of both sides by $\varphi_{1}$
gives

$\displaystyle
c_{1}(t)=c_{1}(0)e^{\alpha_{1}t}-\beta\int_{0}^{t}e^{\alpha_{1}(t-s)}\langle
u(s)\mathbb{W}u(s),\varphi\rangle\mathrm{d}s.$

The derivative is

$\displaystyle\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}c_{1}(t)=&\alpha_{1}c_{1}(0)e^{\alpha_{1}t}-\beta\langle
u(t)\mathbb{W}u(t),\varphi_{1}\rangle-\alpha_{1}\int_{0}^{t}e^{\alpha_{1}(t-s)}\langle
u(s)\mathbb{W}u(s),\varphi\rangle\mathrm{d}s\\\
=&\alpha_{1}c_{1}(t)-\beta\langle
u(t)\mathbb{W}u(t),\varphi_{1}\rangle.\end{split}$ (39)

Using (38)

$\displaystyle\langle
u(t)\mathbb{W}u(t),\varphi_{1}\rangle\leq\frac{M}{m}c_{1}(t)\langle
u(t),\varphi_{1}\rangle=\frac{M}{m}c_{1}^{2}(t),$

which concludes

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}c_{1}(t)\geq
c_{1}(t)\left(\alpha_{1}-\frac{\beta M}{m_{0}}c_{1}(t)\right).$

Assuming $\frac{\varepsilon_{0}}{2}\leq\frac{m_{0}\alpha_{1}}{\beta M}$
implies $c_{1}(t)\geq\frac{\varepsilon_{0}}{2}$ for all $t\geq 0$. ∎

###### Proof.

(Case 3 of Lemma 12)

Since $W$ is rank-$1$,

$\displaystyle\mathbb{W}u(t,x)=\lambda_{1}c_{1}(t)\varphi_{1}(x)\geq\lambda_{1}mc_{1}(t),$

hence, it is enough to get a lower bound on $c_{1}(t)$.For this, we will use
(39) once again. As $Wu(t,x)=\lambda_{1}c_{1}(t)\varphi_{1}(x)$ (39) takes the
form

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}c_{1}(t)=c_{1}(t)\left(\alpha_{1}-\beta\lambda_{1}\left\langle
u(t),\varphi_{1}^{2}\right\rangle\right).$

Choose $K$ large enough so that $\int_{\varphi_{1}(x)\geq
K}\varphi_{1}^{2}(x)\mathrm{d}x\leq\frac{\alpha_{1}}{2\beta\lambda_{1}}.$

$\displaystyle\beta\lambda_{1}\left\langle
u(t),\varphi_{1}^{2}\right\rangle\leq\beta\lambda_{1}\left\langle{\mathds{1}}_{\left\\{\varphi_{1}\geq
K\right\\}},\varphi_{1}^{2}\right\rangle+\beta\lambda_{1}K\langle
u(t),\varphi_{1}\rangle\leq\frac{\alpha_{1}}{2}+\beta\lambda_{1}Kc_{1}(t),$

resulting in

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}c_{1}(t)\geq
c_{1}(t)\left(\frac{\alpha_{1}}{2}-\beta\lambda_{1}Kc_{1}(t)\right).$

The rest is similar to Case 2. ∎

##  6 Proofs regarding eternal solutions

###### Proof.

(Theorem 2)

Set a small enough $\varepsilon_{0}>0$ whose value is chosen later. Also,
define $\varepsilon_{n}:=\varepsilon_{0}e^{-\alpha_{1}n}.$ We will often use
the identity

$e^{\alpha n}=\frac{\varepsilon_{0}}{\varepsilon_{n}}.$

Let $(u_{n})_{n=1}^{\infty}$ be a set of solutions with initial conditions

$\displaystyle
u_{n}(-n)=\max\\{\varepsilon_{n}\varphi_{1},1\\}=\varepsilon_{n}\left(\varphi_{1}-\left(\varphi_{1}-\frac{1}{\varepsilon_{n}}\right){\mathds{1}}_{\left\\{\varphi_{1}\geq\frac{1}{\varepsilon_{n}}\right\\}}\right)=:\varepsilon_{n}\left(\varphi_{1}-\eta_{n}\right).$

Clearly,

$\displaystyle\|\eta_{n}\|_{2}^{2}=\int_{\varphi_{1}\geq\frac{1}{\varepsilon_{n}}}\left(\varphi_{1}(x)-\frac{1}{\varepsilon_{n}}\right)^{2}\mathrm{d}x\searrow
0$

as $\|\eta_{n}\|_{2}^{2}$ is the area under $\varphi_{1}^{2}$ and above the
line $\frac{1}{\varepsilon_{n}}.$

To get a rate of convergence as $n\to\infty.$, define the random variable
$\xi$ on $[0,1]$ with density function $\varphi_{1}^{2}(x).$ Due to Markov’s
inequality,

$\displaystyle\|\eta_{n}\|_{2}^{2}\leq$
$\displaystyle\int_{\varphi_{1}\geq\frac{1}{\varepsilon_{n}}}\varphi_{1}^{2}(x)\mathrm{d}x$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(\varphi_{1}(\xi)\geq\frac{1}{\varepsilon_{n}}\right)=\mathbb{P}\left(\varphi_{1}^{\rho}(\xi)\geq\varepsilon_{n}^{-\rho}\right)\leq\varepsilon_{n}^{\rho}\mathbb{E}\left(\varphi_{1}^{\rho}(\xi)\right)$
$\displaystyle=$
$\displaystyle\varepsilon_{n}^{\rho}\int_{0}^{1}\varphi_{1}^{2+\rho}(x)\mathrm{d}x=O\left(\varepsilon_{n}^{\rho}\right)$
$\displaystyle\|\eta_{n}\|_{2}^{2}=$ $\displaystyle
O\left(\varepsilon_{n}^{\frac{\rho}{2}}\right).$

Define $c_{l}^{(n)}(t):=\langle\varphi_{l},u_{n}(t)\rangle.$

Set $k_{0}$ be large enough and let $n,k$ be $n\geq k\geq k_{0}.$

With a slight modification of (36),

$\displaystyle\sup_{0\leq t\leq n-k_{0}}\|u_{n}(t-n)-v_{n}(t-n)\|_{2}\leq$
$\displaystyle\frac{\beta\lambda_{1}}{\alpha_{1}}\|u_{n}(-n)\|_{2}^{2}e^{2\alpha_{1}(n-k_{0})}$
$\displaystyle\leq$
$\displaystyle\frac{\beta\lambda_{1}}{\alpha_{1}}\varepsilon_{n}^{2}e^{2\alpha_{1}(n-k_{0})}=\frac{\beta\lambda_{1}}{\alpha_{1}}\varepsilon_{k_{0}}^{2}$

To approximate $v_{n}(-k_{0})$ first notice

$\displaystyle
c_{l}^{(n)}(-n)=\varepsilon_{n}\left(\delta_{1,l}-\langle\varphi_{l},\eta_{n}\rangle\right),$

hence (13) shows

$\displaystyle v_{n}(-k)=$
$\displaystyle\sum_{l}c_{l}^{(n)}(-n)e^{\alpha_{l}(n-k_{0})}=\varepsilon_{n}e^{\alpha_{1}(n-k_{0})}\left(\varphi_{1}+O\left(\|\eta_{n}\|_{2}\right)\right)$
$\displaystyle=$
$\displaystyle\varepsilon_{k_{0}}\left(\varphi_{1}+O\left(\|\eta_{k_{0}}\|\right)\right).$

Define $\nu_{k_{0}}:=\varepsilon_{k_{0}}+\|\eta_{k_{0}}\|_{2}$. Without loss
of generality, one can assume $0<\rho\leq 2,$ hence
$\nu_{k_{0}}=O\left(\varepsilon_{n}^{\frac{\rho}{2}}\right).$ Combining the
two bounds yields

$\displaystyle\|u_{n}(-k_{0})-\varepsilon_{k_{0}}\varphi_{1}\|_{2}=O\left(\varepsilon_{k_{0}}\nu_{k_{0}}\right).$

The same argument can be used for index $k$, resulting in

$\displaystyle
u_{n}(-k_{0}),u_{k}(-k_{0})=\varepsilon_{k_{0}}\varphi_{1}+O\left(\varepsilon_{k_{0}}\nu_{k_{0}}\right)\
\textit{in}\ L^{2}([0,1])$ (40)

which resembles the setup of Subsection 4.2. Indeed, with some modification of
the proof of Lemma 10 for $0\leq t\leq k_{0}$

$\displaystyle\|u_{n}(t-k_{0})-u_{k}(t-k_{0})\|_{2}\leq$
$\displaystyle\underbrace{e^{\alpha_{1}k_{0}}\|u_{n}(-k_{0})-u_{k}(-k_{0})\|_{2}}_{=O\left(\nu_{k_{0}}\right)}+$
$\displaystyle\underbrace{\beta\lambda_{1}\left(\|u_{n}(-k_{0})\|_{2}+\|u_{k}(-k_{0})\|_{2}\right)e^{\alpha_{1}k_{0}}}_{=O\left(\varepsilon_{0}\right)}\int_{0}^{t}\|u_{n}(s-k_{0})-u_{2}(s-k_{0})\|_{2}\mathrm{d}s.$

Choose $\varepsilon_{0}$ small enough such that
$O(\varepsilon_{0})\leq\frac{\alpha_{1}\rho}{4}.$ Then by Grönwall’s lemma

$\displaystyle\sup_{0\leq t\leq
k_{0}}\|u_{n}(t-k_{0})-u_{k}(t-k_{0})\|_{2}=O\left(\nu_{k_{0}}\exp\left(\frac{\alpha_{1}\rho}{4}k_{0}\right)\right)=O\left(\varepsilon_{k_{0}}^{\frac{\rho}{4}}\right).$
(41)

Now, fix an arbitrarily large $\tau>0.$ Define the norm

$\displaystyle\|u_{n}-u_{k}\|_{[-\tau,0]}:=\sup_{-\tau\leq t\leq
0}\|u_{n}(t)-u_{k}(0)\|_{2}.$

$\tau\leq k_{0}$ for large enough $k_{0}$, hence (41) yields

$\displaystyle\|u_{n}-u_{k}\|_{[-\tau,0]}=O\left(\varepsilon_{k_{0}}^{\frac{\rho}{4}}\right)\to
0$ (42)

as $k_{0}\to\infty$, making $(u_{n})_{n=1}^{\infty}$ a Cauchy sequence for
this norm, with limit $u$.

Note that (40) is also applicable for $k_{0}=0$ making
$u_{n}(0)=\varepsilon_{0}\varphi_{1}+O\left(\varepsilon_{0}\nu_{0}\right)$ in
$L^{2}([0,1]).$ Choosing $\varepsilon_{0}$ small enough shows that $u$ cannot
be neither the disease-free nor the endemic state.

$u(t)$ inherits being in $\Delta$ and satisfies (4), so it can be extended to
$[-\tau,\infty[.$ To extend backwards in time, we consider
$\tau^{\prime}>\tau>0$ with corresponding limit $u^{\prime}.$ Clearly
$\left.u^{\prime}\right|_{[-\tau,0]}=u$, otherwise $u_{n}(t)\to u(t)$ would
not hold for $-\tau\leq t\leq 0.$ In conclusion, $u(t)$ can be extended to
$\mathbb{R}$ as well making it a nontrivial eternal solution.

When $W$ is discrete, $\varepsilon_{0}$ can be chosen small enough so that
$u_{n}(-n)=\varepsilon_{n}\varphi_{1}\in\Delta_{I}.$ $u(t)\in\Delta_{I}$ also
holds in this case.

Finally, we have to show (16). We refine the sequence $u_{n}$ into

$\displaystyle u_{n,m}\left(-\frac{n}{m}\right)=$
$\displaystyle\max\\{\varepsilon_{\frac{n}{m}}\varphi_{1},1\\}=:\varepsilon_{\frac{n}{m}}\left(\varphi_{1}-\eta_{n,m}\right).$

$(u_{n,m})_{n=1}^{\infty}$ is also Cauchy and since it includes
$(u_{n})_{n=1}^{\infty}$ as a subsequence,

$\lim_{n\to\infty}\|u_{n,m}(t)-u(t)\|_{2}=0$

holds as well.

Similarly to (40),

$u_{n,m}\left(-\frac{k_{0}}{m}\right)=\varepsilon_{\frac{k_{0}}{m}}\varphi_{1}+O\left(\varepsilon_{\frac{k_{0}}{m}}\nu_{k_{0},m}\right)$

where

$\nu_{k_{0},m}:=\varepsilon_{\frac{k_{0}}{m}}+\|\eta_{k_{0},m}\|_{2}.$

Fix $\varepsilon>0$. Then we are going to select a large enough $m$ depending
on $\varepsilon,$ then $k_{0}$ depending on $m$, and finally $n$ depending on
$m$.

$\displaystyle
1\geq\frac{\left\langle\varphi_{1},u_{n,k}\left(-\frac{k_{0}}{m}\right)\right\rangle}{\|u_{n,k}\left(-\frac{k_{0}}{m}\right)\|_{2}}=\frac{\varepsilon_{\frac{k_{0}}{m}}+O\left(\varepsilon_{\frac{k_{0}}{m}}\nu_{k_{0},m}\right)}{\varepsilon_{\frac{k_{0}}{m}}+O\left(\varepsilon_{\frac{k_{0}}{m}}\nu_{k_{0},m}\right)}=1+O(\nu_{k_{0},m})\geq
1-\frac{\varepsilon}{4}$

for large enough $\frac{k_{0}}{m}$.

Since $u_{n,m}\left(-\frac{k_{0}}{m}\right)\to u\left(-\frac{k_{0}}{m}\right)$
as $n\to\infty$ in $L^{2}([0,1])$, we can choose $n$ large enough so that

$\displaystyle
1\geq\frac{\left\langle\varphi_{1},u\left(-\frac{k_{0}}{m}\right)\right\rangle}{\|u\left(-\frac{k_{0}}{m}\right)\|_{2}}\geq
1-\frac{\varepsilon}{2}$

We have to extend this result to
$t\in\left[-\frac{k_{0}}{m},-\frac{k_{0}-1}{m}\right[.$ Note that

$\displaystyle\sup_{-\frac{k_{0}}{m}\leq
t\leq-\frac{k_{0}-1}{m}}\left\|u\left(t\right)-u\left(-\frac{k_{0}}{m}\right)\right\|_{2}\leq\left\|u\left(-\frac{k_{0}}{m}\right)\right\|_{2}e^{\frac{L}{m}}$

where $L$ is the Lipschitz-constant of (4) on $\Delta$.

$\displaystyle\left|\frac{\langle\varphi_{1},u(-t)\rangle}{\|u(-t)\|_{2}}-\frac{\left\langle\varphi_{1},u\left(-\frac{k_{0}}{m}\right)\right\rangle}{\|u\left(-\frac{k_{0}}{m}\right)\|_{2}}\right|\leq$
$\displaystyle\left|\frac{\|u\left(-\frac{k_{0}}{m}\right)\|_{2}}{\|u(-t)\|_{2}}-1\right|+\frac{\left|\left\langle\varphi_{1},u(-t)-u\left(-\frac{k_{0}}{m}\right)\right\rangle\right|}{\|u\left(-\frac{k_{0}}{m}\right)\|_{2}}=O\left(e^{\frac{L}{m}}\right)$

For $m$ large enough,

$\displaystyle 1\geq\frac{\langle\varphi_{1},u(-t)\rangle}{\|u(-t)\|_{2}}\geq
1-\varepsilon,$

concluding the proof. ∎

###### Proof.

(Lemma 1)

Let $u(t)$ be a nontrivial eternal solution and $\tilde{u}(t):=u(t)-\psi.$
Using Lemma 11 for $-t\leq 0$,

$\displaystyle 4\geq\|\tilde{u}(-t)\|_{2}^{2}=$
$\displaystyle\|\tilde{u}(0)\|_{2}^{2}+\int_{-t}^{0}\left(-\frac{\mathrm{d}}{\mathrm{d}t}\|\tilde{u}(s)\|_{2}^{2}\right)\mathrm{d}s$
$\displaystyle\geq$
$\displaystyle\|\tilde{u}(0)\|_{2}^{2}+2\beta\int_{-t}^{0}\int_{0}^{1}\left(\tilde{u}(s,x)\right)^{2}\mathbb{W}u(s,x)\mathrm{d}x\mathrm{d}s.$

Since $\left(\tilde{u}(s,x)\right)^{2}\mathbb{W}u(s,x)\geq 0$ the right hand
side is monotone increasing in $t$ and bounded, hence

$\displaystyle\int_{-\infty}^{0}\int_{0}^{1}\left(\tilde{u}(s,x)\right)^{2}\mathbb{W}u(s,x)\mathrm{d}x\mathrm{d}s<\infty,$

implying

$\displaystyle\lim_{t\to-\infty}\int_{0}^{1}\left(\tilde{u}(t,x)\right)^{2}\mathbb{W}u(t,x)\mathrm{d}x=0.$

Due to

$\displaystyle\int_{0}^{1}\left(\tilde{u}(s,x)\right)^{2}\mathbb{W}u(s,x)\mathrm{d}x\leq
4\int_{0}^{1}\int_{0}^{1}W(x,y)\mathrm{d}x\mathrm{d}y<\infty,$

we can take the limit inside:

$\displaystyle\lim_{t\to-\infty}\left(\tilde{u}(t,x)\right)^{2}\mathbb{W}u(t,x)=0.$

Define

$\displaystyle u(-\infty,x):=\limsup_{t\to-\infty}u(t,x),$ $\displaystyle
A:=\\{x\in[0,1]\left|\mathbb{W}u(-\infty,x)=0\right.\\}.$

Clearly $u(-\infty,x)=\psi(x)$ for almost every $x\in A^{c}.$

Not that, $A^{c}$ can not have measure $1$ as

$\displaystyle\|\tilde{u}(-t)\|_{2}^{2}\geq\|\tilde{u}(0)\|_{2}^{2}>0,$

since $u(0)\neq\psi.$

Assume $A^{c}$ has measure strictly between $0$ and $1$. From the connectivity
assumption

$D:=\int_{A}\int_{A^{c}}W(x,y)\mathrm{d}y\mathrm{d}x>0.$

When $W(x,y)\geq m_{0}>0$, we have

$\displaystyle\psi(x)=\frac{\beta\mathbb{W}\psi(x)}{\gamma+\beta\mathbb{W}\psi(x)}\geq\frac{\beta
m_{0}\|\psi\|_{1}}{\gamma+\beta m_{0}\|\psi\|_{1}}>0.$

When $W$ is discrete,

$\displaystyle\min_{x\in[0,1]}\psi(x)>0.$

based on [9], hence, in all cases we may assume $\psi$ is uniformly positive
with lower bound $\psi_{\min}>0.$

Now if

$\displaystyle 0=$
$\displaystyle\int_{A}\mathbb{W}u(-\infty,x)\mathrm{d}x=\int_{A}\int_{0}^{1}W(x,y)u(-\infty,y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq$
$\displaystyle\int_{A}\int_{A^{c}}W(x,y)u(-\infty,y)\mathrm{d}y\mathrm{d}x=\int_{A}\int_{A^{c}}W(x,y)\psi(y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq$
$\displaystyle\psi_{\min}\int_{A}\int_{A^{c}}W(x,y)\mathrm{d}y\mathrm{d}x=\psi_{\min}D>0,$

leading to a contradiction.

What remains is to show $\mathbb{W}u(-\infty)=0$ a.e. implies $u(-\infty)=0$
a.e. Define

$B_{\epsilon}:=\\{x\in[0,1]\left|u(-\infty,x)\geq\epsilon\right.\\}.$

It is enough to show $B_{\epsilon}$ has zero measure as $B_{\frac{1}{n}}$ are
increasing sets. Note that $B_{\epsilon}$ can not have measure $1$ as that
would lead to

$\displaystyle
0=\int_{0}^{1}\mathbb{W}u(-\infty,x)\mathrm{d}x\geq\epsilon\int_{0}^{1}\int_{0}^{1}W(x,y)\mathrm{d}y\mathrm{d}x>0.$

Assume $B_{\epsilon}$ has measure between $0$ and $1$. Then

$\displaystyle 0=$
$\displaystyle\int_{B_{\epsilon}^{c}}\mathbb{W}u(-\infty,x)\mathrm{d}x\geq\int_{B_{\epsilon}^{c}}\int_{B_{\epsilon}}W(x,y)u(-\infty,y)\mathrm{d}y\mathrm{d}x$
$\displaystyle\geq$
$\displaystyle\epsilon\int_{B_{\epsilon}^{c}}\int_{B_{\epsilon}}W(x,y)\mathrm{d}y\mathrm{d}x>0$

concluding the proof. ∎

###### Proof.

(Theorem 3)

Indirectly assume

$\sup_{t\in\mathbb{R}}\inf_{\tau\in\mathbb{R}}\|u_{1}(t+\tau)-u_{2}(t)\|_{2}>0,$

that is, there exists a $t$ such that

$\inf_{\tau\in\mathbb{R}}\|u_{1}(t+\tau)-u_{2}(t)\|_{2}>0.$

With appropriate time translation we get

$\displaystyle\inf_{\tau\in\mathbb{R}}\|u_{1}(\tau)-u_{2}(0)\|_{2}=:2\varepsilon>0.$
(43)

Set $\eta=\frac{\varepsilon}{2}$ and choose a corresponding $\delta>0$
according to Proposition 1.

Due to Lemma 1, $\lim_{t\to-\infty}u_{i}(t)=0$ a.e. ($i=1,2$), hence, for
$T\geq 0$ large enough,

$0<\|u_{1}(-T)\|_{2},\|u_{2}(-T)\|_{2}\leq\delta,$

meaning there must be times $t_{1},t_{2}\geq 0$ such that

$\displaystyle\sup_{t\geq 0}\|u_{1}(t+t_{1}-T)-u_{2}(t+t_{2}-T)\|_{2}\leq$
$\displaystyle\varepsilon,$ $\displaystyle\sup_{0\leq t\leq
t_{2}}\|u_{2}(t-T)\|_{2}\leq$ $\displaystyle\frac{\varepsilon}{2}.$

$T>t_{2}$ would give $\|u_{1}(t_{1}-t_{2})-u_{2}(0)\|_{2}\leq\varepsilon,$
violating (43) with $\tau=t_{1}-t_{2}.$

So $T\in[0,t_{2}]$, and then
$\|u_{2}(0)\|_{2}=\|u_{2}(T-T)\|\leq\frac{\varepsilon}{2}.$

Set $\tau$ to be a large negative number such that
$\|u_{1}(\tau)\|_{2}\leq\varepsilon.$ However, that implies

$\displaystyle 2\varepsilon\leq$
$\displaystyle\|u_{1}(\tau)-u_{2}(0)\|_{2}\leq\|u_{1}(\tau)\|_{2}+\|u_{2}(0)\|_{2}\leq\varepsilon+\|u_{2}(0)\|_{2},$
$\displaystyle\varepsilon\leq$ $\displaystyle\|u_{2}(0)\|_{2},$

resulting in a contradiction. ∎

## Acknowledgment

The author is thankful to Illés Horváth, Péter L. Simon and István Z. Kiss for
insightful discussions.

##  7 Outlook

In this paper we investigated the deterministic SIS process in general
communities described by graphons, starting from small initial conditions. We
have shown that after appropriate time translation, the solutions will be
close to each other and identified their limit as the nontrivial eternal
solution. This results in a huge reduction of complexity as one can neglect
the exact initial configuration of infections as it can be well-approximated
by the scaled version of the eigenvector centrality.

Future work may include extending these results to other compartmental models,
like SIR and SEIR. Another direction would be finding the stochastic analogue
to USIC on sparse network in the $n\to\infty$ local limit sense.

## References

  * [1] M. Avella-Medina, F. Parise, M. T. Schaub, and S. Segarra.  Centrality measures for graphons: Accounting for uncertainty in networks.  IEEE Transactions on Network Science and Engineering, 7(1):520–537, Jan 2020.
  * [2] A. D. Barbour and G. Reinert.  Approximating the epidemic curve.  Electronic Journal of Probability, 18:1–30, 2013.
  * [3] S. BHAMIDI, R. V. D. HOFSTAD, and J. KOMJÁTHY.  The front of the epidemic spread and first passage percolation.  Journal of Applied Probability, 51A:101–121, 2014.
  * [4] S. Bonaccorsi, S. Ottaviano, D. Mugnolo, and F. D. Pellegrini.  Epidemic outbreaks in networks with equitable or almost-equitable partitions.  SIAM Journal on Applied Mathematics, 75(6):2421–2443, 2015.
  * [5] L. Decreusefond, J.-S. Dhersin, P. Moyal, and V. C. Tran.  Large graph limit for an SIR process in random network with heterogeneous connectivity.  Annals of Applied Probability, 22, 04 2012.
  * [6] J.-F. Delmas, D. Dronnier, and P.-A. Zitt.  An infinite-dimensional SIS model, 2020.
  * [7] F. Di Lauro, J.-C. Croix, L. Berthouze, and I. Z. Kiss.  PDE limits of stochastic SIS epidemics on networks.  Journal of Complex Networks, 8(4), 12 2020.  cnaa043.
  * [8] S. Gao and P. E. Caines.  Spectral representations of graphons in very large network systems control.  2019 IEEE 58th Conference on Decision and Control (CDC), Dec 2019\.
  * [9] Ágnes Bodó and P. L. Simon.  Transcritical bifurcation yielding global stability for network processes.  Nonlinear Analysis, 196:111808, 2020.
  * [10] B. Guerra and J. Gómez-Gardeñes.  Annealed and mean-field formulations of disease dynamics on static and adaptive networks.  Physical review. E, Statistical, nonlinear, and soft matter physics, 82:035101, 09 2010.
  * [11] T. Horiguchi and Y. Fukui.  A variation of the Jentzsch theorem for a symmetric integral kernel and its application.  Interdisciplinary Information Sciences, 2, 01 1996.
  * [12] I. Horváth and D. Keliger.  Accuracy criterion for mean field approximations of markov processes on hypergraphs, 2022.
  * [13] S. Janson, M. Luczak, and P. Windridge.  Law of large numbers for the sir epidemic on a random graph with given degrees.  Random Structures & Algorithms, 45(4):726–763, 2014.
  * [14] D. Keliger, I. Horváth, and B. Takács.  Local-density dependent Markov processes on graphons with epidemiological applications.  Stochastic Processes and their Applications, 2022.
  * [15] T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes.  Journal of Applied Probability, 7:49–58, 04 1970.
  * [16] L. Lovász.  Large Networks and Graph Limits, volume 60 of Colloquium Publications.  American Mathematical Society, 2012.
  * [17] P. Mieghem.  The N-intertwined SIS epidemic network model.  Computing, 93:147–169, 12 2011.
  * [18] R. Pastor-Satorras and A. Vespignani.  Epidemic spreading in scale-free networks.  Phys. Rev. Lett., 86:3200–3203, Apr 2001.
  * [19] B. Prasse and P. Mieghem.  Time-dependent solution of the NIMFA equations around the epidemic threshold.  Journal of mathematical biology, 81, 12 2020.
  * [20] M. Robinson.  Construction of eternal solutions for a semilinear parabolic equation, 2008.
  * [21] P. L. Simon and I. Z. Kiss.  On bounding exact models of epidemic spread on networks, 2017.  https://arxiv.org/abs/1704.01726.
  * [22] A. Sridhar and S. Kar.  Mean-field approximation for stochastic population processes in networks under imperfect information, 2021.
  * [23] A. Sridhar and S. Kar.  On the accuracy of deterministic models for viral spread on networks, 2021\.
  * [24] P. Van Mieghem, F. D. Sahneh, and C. Scoglio.  An upper bound for the epidemic threshold in exact Markovian SIR and SIS epidemics on networks.  In 53rd IEEE Conference on Decision and Control, pages 6228–6233, 2014.
  * [25] R. Vizuete, P. Frasca, and F. Garin.  Graphon-based sensitivity analysis of SIS epidemics, 12 2019.
  * [26] E. Volz.  SIR dynamics in random networks with heterogeneous connectivity.  Journal of mathematical biology, 56:293–310, 04 2008.
  * [27] S.-L. Wu and P. Weng.  Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z.  Journal of Mathematical Analysis and Applications, 394(2):603–615, 2012.