1. Introduction
In this paper we are concerned with the susceptible–infected–removed (SIR) model with vertex-dependent recovery and infection rates on the complete graph. We first introduce the model. For a given integer
$N\geq 1$
, our SIR model
$\{\xi_t\}_{t\geq 0}$
is a continuous-time Markov process with state space
\begin{align*}\{-1, 0, 1\}^N=\left\{\xi=\left(\xi(1), \xi(2),\ldots, \xi(N)\right)\;:\;\xi(i)\in\{-1, 0, 1\}\text{ for }1\leq i\leq N\right\},\end{align*}
i.e. each vertex on the complete graph
$\{1,2,\ldots,N\}$
is in one of three states namely
$-1$
, 0 or 1. For any configuration
$\xi\in \{-1,0,1\}^N$
and
$i\in \{1,2,\ldots,N\}$
,
$\xi(i)$
is the state of vertex i.
Throughout this paper, we use
$\mathbb{T}$
to denote the one-dimensional torus (0, 1]. There are two functions
$\lambda\;:\; \mathbb{T}^2\rightarrow (0, +\infty)$
and
$\psi\;:\; \mathbb{T}\rightarrow (0, +\infty)$
such that for all
$1\leq i\leq N$
:
-
(1) if
$\xi_t(i)=1$
then
$\xi_t(i)$
jumps to
$-1$
at rate
$\psi\!\left(\frac{i}{N}\right)$
; -
(2) if
$\xi_t(i)=0$
then
$\xi_t(i)$
jumps to 1 at rate where
\begin{align*}\frac{1}{N}\sum_{j=1}^N\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)1_{\{\xi_t(j)=1\}},\end{align*}
$1_A$
is the indicator function of the event A.
We further assume that a vertex in state
$-1$
is frozen in this state forever. The process
$\{\xi_t\}_{t\geq 0}$
can be defined equivalently via its generator. According to the transitions rates given above, the generator
$\mathcal{L}$
of
$\{\xi_t\}_{t\geq 0}$
is given by
\begin{align}\mathcal{L}f(\xi)&=\sum_{i=1}^N\psi\!\left(\frac{i}{N}\right)1_{\{\xi(i)=1\}}\big(f\big(\xi^{i, -1}\big)-f(\xi)\big) \nonumber\\[5pt] &\quad+\frac{1}{N}\sum_{i=1}^N\sum_{j=1}^N\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)1_{\{\xi(i)=0, \xi(j)=1\}}\big(f\big(\xi^{i, 1}\big)-f(\xi)\big) \end{align}
for any
$\xi\in \{-1, 0, 1\}^N$
and
$f\;:\; \{-1, 0, 1\}^N\rightarrow \mathbb{R}$
, where
\begin{align*}\xi^{i, k}(j)=\left\{\begin{array}{l@{\quad}l}\xi(j) & \text{ if }j\neq i,\\[5pt] k & \text{ if }j=i,\end{array}\right.\end{align*}
for all
$1\leq i\leq N$
and
$k\in\{1, -1\}$
. From now on, to distinguish among different N, we write
$\xi_t$
and
$\mathcal{L}$
as
$\xi_t^N$
and
$\mathcal{L}^N$
, respectively.
Intuitively,
$\{\xi_t^N\}_{t\geq 0}$
describes the spread of a SIR epidemic between N individuals. States
$-1$
, 0, and 1 represent ‘removed’, ‘susceptible’, and ‘infected’, respectively. If the ith individual is susceptible and the jth is infected, then j infects i at rate
$N^{-1}\lambda\!\left(i/N, j/N\right)$
and j becomes removed at rate
$\psi\!\left({j}/{N}\right)$
. A removed individual will never be infected again.
We further assume that
$\lambda$
and
$\psi$
are piecewise smooth. In detail, we assume that there exist integer
$m\geq 1$
and
$0=b_0 \lt b_1 \lt b_2 \lt \cdots < b_m=1$
such that
$\lambda(u, v)=\lambda_{i,j}(u, v)$
for any
$0\leq i, j\leq m-1, (u, v)\in (b_i, b_{i+1}]\times (b_j, b_{j+1}]$
and
$\psi(u)=\psi_i(u)$
for any
$0\leq i\leq m-1$
,
$u\in (b_i, b_{i+1}]$
, where
$\lambda_{i,j}\in C^1([b_i, b_{i+1}]\times[b_j, b_{j+1}])$
for all
$0\leq i, j\leq m-1$
and
$\psi_i\in C^1[b_i, b_{i+1}]$
for all
$0\leq i\leq m-1$
.
When
$\psi\equiv 1$
and
$\lambda\equiv \lambda_0\in \mathbb{R}$
, our model reduces to the classic spatial-homogeneous SIR model (see [Reference Anderson and May1]) such that
$(\mathcal{S}_t^N, \mathcal{I}_t^N)$
jumps to
$(\mathcal{S}_t^N, \mathcal{I}_t^N-1)$
at rate
$\mathcal{I}_t^N$
or jumps to
$(\mathcal{S}_t^N-1, \mathcal{I}_t^N+1)$
at rate
$(\lambda_0/N)\mathcal{S}_t^N\mathcal{I}_t^N$
, where
\begin{align*}\mathcal{S}_t^N=\sum_{i=1}^N1_{\left\{\xi_t^N(i)=0\right\}} \quad\text{and} \quad \mathcal{I}_t^N=\sum_{i=1}^N1_{\left\{\xi_t^N(i)=1\right\}}\end{align*}
are total numbers of susceptible individuals and infected individuals, respectively. For the classic SIR model, the law of large numbers (LLN) and the central limit theorem (CLT) of
$\{(\mathcal{S}_t^N, \mathcal{I}_t^N)\}_{t\geq 0}$
can be derived directly as an application of the theory of density-dependent Markov chains (DDMCs) given in [Reference Kurtz7] and Chapter 11 of [Reference Ethier and Kurtz3]. For a detailed review, see Subsection 2.2.
When
$\lambda(u, v)=\vartheta(u)\vartheta(v)$
for all
$u, v\in \mathbb{T}$
and some piecewise smooth
$\vartheta$
on
$\mathbb{T}$
, our model reduces to the SIR model where all individuals have different activities. In detail, the ith vertex has activity
$\vartheta(i/N)$
and the infection between two vertices occurs at a rate proportional to the product of the activities of these two vertices. Furthermore, if there exist
$\psi_1, \ldots, \psi_m, \vartheta_1, \ldots, \vartheta_m\in \mathbb{R}$
such that
$\psi(u)=\psi_i, \vartheta(u)=\vartheta_i$
for all
$1\leq i\leq m$
and
$u\in (b_{i-1}, b_i]$
, then our model further reduces to the SIR model with m types. In detail, the individual in the ith type has recovery rate
$\psi_i$
and activity
$\vartheta_i$
. The proportion of the ith-type individuals is
$b_i-b_{i-1}$
.
Our model also includes the case where the infection rates are distance-dependent. In detail, let
$D(\cdot, \cdot)$
be the metric function on
$\mathbb{T}$
, i.e.
\begin{align*}D(u, v)=\min\{v-u, 1+u-v\}\end{align*}
for any
$0 < u\leq v\leq 1$
. If
$\lambda(u, v)=g(D(u, v))$
, where g is decreasing on
$[0, 1/2]$
, then our model reduces to the SIR epidemic where an infectious individual infects nearer susceptible individuals at larger rates.
In this paper, we investigate the hydrodynamic limit and fluctuation of our process
$\{\xi_t^N\}_{t\geq 0}$
, i.e. LLN and CLT of
\begin{align*}\frac{1}{N}\sum_{i=1}^N1_{\left\{\xi_t^N(i)=0\right\}}f\!\left(\frac{i}{N}\right) \quad \text{and} \quad \frac{1}{N}\sum_{i=1}^N1_{\left\{\xi_t^N(i)=1\right\}}f\!\left(\frac{i}{N}\right)\end{align*}
for any piecewise smooth f on
$\mathbb{T}$
as
$N\rightarrow+\infty$
. This investigation is motivated by the aforesaid LLN and CLT given in [Reference Kurtz7] of
$\mathcal{S}_t^N$
and
$\mathcal{I}_t^N$
in the spatial-homogenous case, where
$f\equiv 1$
and recovery and infection rates do not rely on coordinates of individuals. To reveal the contribution of each vertex to the spread of the epidemic, it is natural to discuss general cases where recovery and infection rates are vertex-dependent and f can be any piecewise smooth function on
$\mathbb{T}$
. Hence, we consider empirical density fields
\begin{align*}\frac{1}{N}\sum_{i=1}^N1_{\{\xi_t(i)=0\}}\delta_{\frac{i}{N}}({\mathrm{d}}u) \quad\text{and} \quad \frac{1}{N}\sum_{i=1}^N1_{\{\xi_t(i)=1\}}\delta_{\frac{i}{N}}({\mathrm{d}}u)\end{align*}
as linear operators from
$\hat{C}^1(\mathbb{T})$
to
$\mathbb{R}$
, where
$\hat{C}^1(\mathbb{T})$
is the set of piecewise smooth functions on
$\mathbb{T}$
. We directly give LLN and CLT of these
$(\hat{C}^1(\mathbb{T}))^\prime$
-valued random elements. For mathematical details, see Section 2.
We are also inspired by previous investigations about hydrodynamics of different types of interacting particle systems on complete graphs such as Ehrenfest models, voter models, symmetric exclusion processes and binary contact path processes (see [Reference Xue16, Reference Xue17]). For those models, hydrodynamic limits are driven by
$(C^\infty(\mathbb{T}))^\prime$
-valued linear ordinary differential equations (ODEs) and fluctuations are driven by
$(C^\infty(\mathbb{T}))^\prime$
-valued Ornstein–Uhlenbeck (O-U) processes. It is natural to ask whether analogue results hold for the SIR model. We show that the hydrodynamic limit of our SIR model is driven by a deterministic
$(\hat{C}^1(\mathbb{T}))^\prime$
-valued process with density which is the solution to a nonlinear
$\hat{C}^1(\mathbb{T})$
-valued ODE and the fluctuation of our SIR model is driven by a
$(\hat{C}^1(\mathbb{T}))^\prime$
-valued O-U process. For mathematical details, see Section 2.
According to the definition of the SIR model,
$\mathcal{S}_t^N$
is decreasing in t and any infectious individual will become removed. Hence, there exists a random variable
$\mathcal{S}_{\infty}^N$
such that
\begin{align*}\lim_{t\rightarrow+\infty}(\mathcal{S}_t^N, \mathcal{I}_t^N)=\left(\mathcal{S}_\infty^N, 0\right)\!.\end{align*}
Since
$\mathcal{S}_\infty^N$
is the final size of the susceptible individuals, it is natural to investigate the limit theorems of
$\mathcal{S}_\infty^N$
as
$N\rightarrow +\infty$
. By utilizing a time-change argument, Watson [Reference Watson13] gives the LLN and the CLT of
$\mathcal{S}_\infty^N$
for the classic SIR model. In this paper, we obtain a LLN of
$\mathcal{S}_\infty^N$
for our SIR model where transition rates are vertex-dependent. For some technical reasons, a CLT for
$\mathcal{S}_\infty^N$
of our model is still open. For mathematical details, see Section 2.
Another important question for the SIR model is whether an outbreak of the epidemic occurs conditioned on only one individual being infectious initially. For the classic SIR, a well-known result is that:
-
(1) when
$\lambda_0<1$
, for any
\begin{align*}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(N-\mathcal{S}_\infty^N\geq \epsilon N\big|\mathcal{I}_0^N=1, \mathcal{S}_0^N=N-1\right)=0\end{align*}
$\epsilon>0$
;
-
(2) when
$\lambda_0>1$
, there exists
$c>0$
independent of N such that
\begin{align*}\inf_{N\geq 1}\mathrm{P}\!\left(N-\mathcal{S}_\infty^N\geq cN\big|\mathcal{I}_0^N=1, \mathcal{S}_0^N=N-1\right)>0.\end{align*}
Xue [Reference Xue15] extends the above results to the SIR model with independent and identically distributed (i.i.d.) edge-dependent infection weights. In this paper, we give a similar phase transition for the SIR model with vertex-dependent activities and recovery rates mentioned above. For mathematical details, see Section 2.
2. Main results
In this section, we give our main results and apply them in several examples, including the spatial-homogeneous case to revisit the LLN and CLT given in [Reference Kurtz7].
2.1. Main results
In this subsection, we give our main result. For later use, we first introduce some notation and definitions. As mentioned in Section 1, we denote by
$\hat{C}^1(\mathbb{T})$
the set of piecewise smooth functions on
$\mathbb{T}$
. We let
$\hat{C}^1(\mathbb{T})$
be endowed with the norm
$\|\cdot\|_{\hat{C}^1(\mathbb{T})}$
defined as
\begin{align*}\|f\|{}_{\hat{C}^1(\mathbb{T})}=\sup_{u\in \mathbb{T}}\!\left(|f(u)|+|f^\prime(u)|\right)\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T})$
.
For any
$t\geq 0$
and
$1\leq i\leq N$
, we define
\begin{align*}S_t^N(i)=1_{\left\{\xi_t^N(i)=0\right\}} \quad \text{and} \quad I_t^N(i)=1_{\left\{\xi_t^N(i)=1\right\}}.\end{align*}
According to the evolution of our SIR model,
$S_t^N(i)$
is decreasing in t. Hence, it is reasonable to define
\begin{equation}S_\infty^N(i)=\lim_{t\rightarrow+\infty}S_t^N(i).\end{equation}
Actually,
$S_\infty^N(i)=1$
if and only if i has never been infected. Consequently,
$\sum_{i=1}^NS_\infty^N(i)$
is the final size of susceptible individuals.
For any
$a\in[0, 1]$
, let
$\delta_a({\mathrm{d}}u)$
be the Dirac measure concentrated on a, then we define
\begin{align*}\mu_t^N({\mathrm{d}}u)=\frac{1}{N}\sum_{i=1}^NI_t^N(i)\delta_{\frac{i}{N}}({\mathrm{d}}u)\end{align*}
and
\begin{align*}\theta_t^N({\mathrm{d}}u)=\frac{1}{N}\sum_{i=1}^NS_t^N(i)\delta_{\frac{i}{N}}({\mathrm{d}}u)\end{align*}
for all
$t\geq 0$
and
$N\geq 1$
. Consequently,
$\mu_t^N$
and
$\theta_t^N$
can be considered as random elements in
$(\hat{C}^1(\mathbb{T}))^\prime$
such that
\begin{align*}\mu_t^N(\,f)=\frac{1}{N}\sum_{i=1}^NI_t^N(i)\,f\!\left(\frac{i}{N}\right) \quad \text{and} \quad \theta_t^N(\,f)=\frac{1}{N}\sum_{i=1}^NS_t^N(i)\,f\!\left(\frac{i}{N}\right)\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T})$
. We denote by P the probability measure of our process
$\{\xi_t\}_{t\geq 0}$
. Throughout this paper, we adopt the following initial assumption.
Assumption (A):
$\{\xi_0^N(i)\}_{i=1}^N$
are independent and there exists
$\phi\in \hat{C}^1(\mathbb{T})$
such that
\begin{align*}\mathrm{P}\!\left(\xi_0^N(i)=1\right)=\phi\!\left(\frac{i}{N}\right)=1-\mathrm{P}\!\left(\xi_0^N(i)=0\right)\end{align*}
for all
$1\leq i\leq N$
.
The following theorem is our first main result, which gives LLNs of
$\mu_t^N$
and
$\theta_t^N$
as
$N\rightarrow+\infty$
, i.e., the hydrodynamic limit of
$\xi_t^N$
.
Theorem 2.1. Under Assumption (A),
\begin{align*}\lim_{N\rightarrow+\infty}\mu_t^N(\,f)=\int_0^1 \rho_1(t,u)\,f(u)\,{\mathrm{d}}u \quad \text{and} \quad \lim_{N\rightarrow+\infty}\theta_t^N(\,f)=\int_0^1\rho_0(t,u)\,f(u)\,{\mathrm{d}}u\end{align*}
in probability for any
$f\in \hat{C}^1(\mathbb{T})$
, where
$\{(\rho_1(t, \cdot), \rho_0(t, \cdot))\}_{t\geq 0}$
is the unique solution to the
$(\hat{C}^1(\mathbb{T}))^2$
-valued ODE
\begin{equation}\left\{\begin{array}{l}\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_1(t,u)=-\psi(u)\rho_1(t,u)+\rho_0(t,u)\int_0^1\lambda(u,v)\rho_1(t,v)\,{\mathrm{d}} v,\quad 0\leq u\leq 1,\\[9pt]\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_0(t,u)=-\rho_0(t,u)\int_0^1\lambda(u, v)\rho_1(t,v)\,{\mathrm{d}}v, \quad 0\leq u\leq 1,\\[9pt]\rho_1(0, u)=\phi(u), \quad 0\leq u\leq 1,\\[9pt]\rho_0(0, u)=1-\phi(u), \quad 0\leq u\leq 1.\end{array}\right.\end{equation}
Remark 2.1. Note that Equation (2.2) satisfies local Lipschitz’s condition under the norm
$\|(\cdot, \cdot)\|_{(\hat{C}^1(\mathbb{T}))^2}$
on
$(\hat{C}^1(\mathbb{T}))^2$
given by
\begin{align*}\|(\,f,g)\|_{(\hat{C}^1(\mathbb{T}))^2}=\|f\|_{\hat{C}^1(\mathbb{T})}+\|g\|_{\hat{C}^1(\mathbb{T})}\end{align*}
for all
$(\,f,g)\in (\hat{C}^1(\mathbb{T}))^2$
. Hence, by Theorem 1.2 in Chapter 19 of [Reference Lang8], there exists
$T_0>0$
such that the solution to Equation (2.2) on
$[0, T_0]\times(\hat{C}^1(\mathbb{T}))^2$
exists and is unique. Furthermore, it is easy to check that the solution
$(\rho_1(t, \cdot), \rho_0(t, \cdot))$
satisfies that
$\rho_1(t, u), \rho_0(t,u)\geq 0$
and
$\rho_1(t,u)+\rho_0(t,u)\leq 1$
for any
$u\in \mathbb{T}$
and
$t\leq T_0$
. Therefore, there exists
$\epsilon_0>0$
such that the unique solution to Equation (2.2) on
$[0, T_0]\times(\hat{C}^1(\mathbb{T}))^2$
can be extended to the unique solution on
$[0, T_0+n\epsilon_0]\times (\hat{C}^1(\mathbb{T}))^2$
for any
$n\geq 1$
. Hence, the solution to Equation (2.2) on
$[0, +\infty)\times (\hat{C}^1(\mathbb{T}))^2$
exists and is unique.
Remark 2.2. A mean field analysis gives an intuitive explanation of Theorem 2.1. The Kolmogorov–Chapman (KC) equations of
$\{\xi_t\}_{t\geq 0}$
are driven by
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathrm{P}(I_t(i)=1)=&-\psi\!\left(\frac{i}{N}\right)\mathrm{P}(I_t(i)=1)\\[5pt] &+\frac{1}{N}\sum_{j=1}^N\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{P}\!\left(S_t(i)=1, I_t(j)=1\right)\end{align*}
and
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\mathrm{P}(S_t(i)=1)=-\frac{1}{N}\sum_{j=1}^N\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{P}\!\left(S_t(i)=1, I_t(j)=1\right)\!.\end{align*}
For large N, since the infection rate between two different vertices is
$O({1}/{N})$
, it is natural to think that states of different vertices are approximately independent and hence the term
$\mathrm{P}(S_t(i)=1, I_t(j)=1)$
in the KC equation can be replaced by
\begin{align*}\mathrm{P}\!\left(S_t(i)=1\right)\mathrm{P}\!\left(I_t(j)=1\right)\end{align*}
with small error. For large N,
$\mathrm{P}(S_t(i)=1)$
(respectively,
$\mathrm{P}(I_t(j)=1)$
) can be approximately considered as the density of
$\theta_t^N$
(respectively,
$\mu^N_t$
) on
${i}/{N}$
and then the aforesaid KC equation is approximately a discrete version of Equation (2.2). In fact, the idea of our proof of Theorem 2.1 is based on the above mean field analysis. We will rigorously prove that
$\textrm{Cov}(S_t^N(i), I_t^N(j))$
converges to 0 as
$N\rightarrow+\infty$
when
$i\neq j$
. For mathematical details, see Section 3.
Now we discuss the fluctuation of our process
$\{\xi_t^N\}_{t\geq 0}$
, i.e. the CLT from the LLN given in Theorem 2.1. To give our main result, we introduce some notation and definitions. For
$N\geq 1$
and
$t\geq 0$
, we define
\begin{align*}\eta_t^N({\mathrm{d}}u)=\frac{1}{\sqrt{N}}\sum_{i=1}^N\!\left(I_t^N(i)-\mathrm{E}I_t^N(i)\right)\delta_{\frac{i}{N}}({\mathrm{d}} u)\end{align*}
and
\begin{align*}\beta_t^N({\mathrm{d}} u)=\frac{1}{\sqrt{N}}\sum_{i=1}^N\!\left(S_t^N(i)-\mathrm{E}S_t^N(i)\right)\delta_{\frac{i}{N}}({\mathrm{d}}u),\end{align*}
where E is the expectation with respect to P. Let
$(\hat{C}^1(\mathbb{T}))^\prime$
be equipped with the weak topology, i.e.
$\nu_n$
converges to
$\nu$
in
$(\hat{C}^1(\mathbb{T}))^\prime$
if and only if
\begin{align*}\lim_{n\rightarrow+\infty}\nu_n(\,f)=\nu(\,f)\end{align*}
for all
$f\in \hat{C}^1(\mathbb{T})$
, then
$\{(\eta_t^N, \beta_t^N)\}_{0\leq t\leq T}$
can be considered as random elements in
$\mathcal{D}([0, T], ((\hat{C}^1(\mathbb{T}))^\prime)^2)$
for given
$T>0$
, where
$\mathcal{D}([0, T], ((\hat{C}^1(\mathbb{T}))^\prime)^2)$
is the set of càdlàg functions from [0, T] to
$((\hat{C}^1(\mathbb{T}))^\prime)^2$
endowed with the Skorohod topology.
We denote by
$\{\mathcal{W}_t\}_{t\geq 0}$
the standard Brownian motion on
$L^2(\mathbb{T})$
, which can be considered as a
$(\hat{C}^1(\mathbb{T}))^\prime$
-valued process such that
$\{\mathcal{W}_t(\,f)\}_{t\geq 0}$
is a real-valued Brownian motion with
\begin{align*}\textrm{Cov}\!\left(\mathcal{W}_t(\,f), \mathcal{W}_t(\,f)\right)=t\int_0^1f^2(u)\,{\mathrm{d}}u\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T})$
and
$t\geq 0$
.
Remark 2.3. As a
$(\hat{C}^1(\mathbb{T}))^\prime$
-valued process,
$\{\mathcal{W}_t\}_{t\geq 0}$
can be defined formally as follows. Let
$e_n(u)=\sqrt{2}\cos(2n\pi u)$
,
$e_{-n}(u)=\sqrt{2}\sin\!(2n\pi u)$
for all
$n\geq 1$
and
$e_0(u)\equiv 1$
, then for any
$g\in C^1(\mathbb{T})$
,
\begin{align*}\sum_{-\infty < n < +\infty}n^2\!\left(\int_0^1g(u)e_n(u)\,{\mathrm{d}}u\right)^2=\int_0^1(g^\prime(u))^2\,{\mathrm{d}} u < +\infty.\end{align*}
Let
$\{B_t^n\;:\;t\geq 0\}_{-\infty < n < +\infty}$
be independent copies of standard
$\mathbb{R}$
-valued Brownian motion, then for any
$t\geq 0$
and
$g\in C^1(\mathbb{T})$
,
\begin{align*}\mathcal{W}_t(g)=\left\langle \sum_{-\infty < n < +\infty}\frac{e_nB_t^n}{n^2}, g\right\rangle_1\!,\end{align*}
where
\begin{align*}\langle h_1, h_2\rangle_1=\sum_{-\infty < n < +\infty}n^2\!\left(\int_0^1h_1(u)e_n(u)\,{\mathrm{d}} u\right)\left(\int_0^1h_2(u)e_n(u)\,{\mathrm{d}}u\right)\end{align*}
for any
\begin{align*}h_1, h_2\in \left\{h\in L^2(\mathbb{T})\;:\;\sum_{-\infty < n < +\infty}n^2\!\left(\int_0^1h(u)e_n(u)\,{\mathrm{d}} u\right)^2<+\infty\right\}.\end{align*}
For any
$f\in \hat{C}^1(\mathbb{T})$
, let
$\{f_n\}_{n\geq 1}$
be a sequence in
$C^1(\mathbb{T})$
such that
$f_n\rightarrow f$
in
$L^2(\mathbb{T})$
, then, since
\begin{align*}\textrm{Var}(\mathcal{W}_t(\,f_n))=t\int_0^1f_n^2(u)\,{\mathrm{d}}u,\end{align*}
$\{\mathcal{W}_t(\,f_n)\}_{n\geq 1}$
is a
$L^2$
-Cauchy sequence and we define
$\mathcal{W}_t(\,f)$
as the
$L^2$
-limit of
$\{\mathcal{W}_t(\,f_n)\}_{n\geq 1}$
.
For any
$t\geq 0$
, we denote by
$\mathcal{A}_{1,t}$
the linear operator from
$\hat{C}^1(\mathbb{T})$
to
$\hat{C}^1(\mathbb{T})$
such that
\begin{align*}\mathcal{A}_{1,t\;}f(u)=\,f(u)\int_0^1\lambda(u,v)\rho_1(t,v)\,{\mathrm{d}}v\end{align*}
and by
$\mathcal{A}_{0, t}$
the linear operator from
$\hat{C}^1(\mathbb{T})$
to
$\hat{C}^1(\mathbb{T})$
such that
\begin{align*}\mathcal{A}_{0, t\;}f(u)=\int_0^1\lambda(v,u)\rho_0(t,v)\,f(v)\,{\mathrm{d}}v\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T})$
and
$u\in \mathbb{T}$
, where
$(\rho_1(t,\cdot), \rho_0(t,\cdot))$
is the solution to Equation (2.2). We denote by
$\mathcal{A}_\psi$
the linear operator from
$\hat{C}^1(\mathbb{T})$
to
$\hat{C}^1(\mathbb{T})$
such that
\begin{align*}\mathcal{A}_\psi f(u)=\psi(u)\,f(u)\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T}), u\in \mathbb{T}$
. For any
$t\geq 0$
, we denote by
$b_t$
the linear operator from
$\hat{C}^1(\mathbb{T})$
to
$\hat{C}^1(\mathbb{T})$
such that
\begin{align*}b_{t\;}f(u)=\sqrt{\psi(u)\rho_1(t,u)}\,f(u)\end{align*}
and by
$\alpha_t$
the linear operator from
$\hat{C}^1(\mathbb{T})$
to
$\hat{C}^1(\mathbb{T})$
such that
\begin{align*}\alpha_{t\;}f(u)=\sqrt{\int_0^1\lambda(u,v)\rho_0(t, u)\rho_1(t,v)\,{\mathrm{d}}v}f(u)\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T})$
and
$u\in \mathbb{T}$
. For any linear operator
$\mathcal{P}$
from
$\hat{C}^1(\mathbb{T})$
to
$\hat{C}^1(\mathbb{T})$
, we denote by
$\mathcal{P}^{*}$
the adjoint operator of
$\mathcal{P}$
, i.e.
$\mathcal{P}^{*}$
is the linear operator from
$(\hat{C}^1(\mathbb{T}))^\prime$
to
$(\hat{C}^1(\mathbb{T}))^\prime$
such that
\begin{align*}\mathcal{P}^{*}\nu(\,f)=\nu(\mathcal{P}f)\end{align*}
for any
$f\in \hat{C}^1(\mathbb{T})$
and
$\nu\in (\hat{C}^1(\mathbb{T}))^\prime$
.
Let
$\{\mathcal{W}_t^1\}_{t\geq 0}$
and
$\{\mathcal{W}_t^2\}_{t\geq 0}$
be two independent copies of
$\{\mathcal{W}_t\}_{t\geq 0}$
, then we introduce the definition of the solution to the following
$((\hat{C}^1(\mathbb{T}))^\prime)^2$
-valued stochastic differential equation (SDE):
\begin{equation}\left\{\begin{array}{l} {\mathrm{d}}\eta_t=\mathcal{A}_{1,t}^{*}\beta_t\,{\mathrm{d}}t+\mathcal{A}_{0, t}^{*}\eta_t\,{\mathrm{d}} t-\mathcal{A}_\psi^{*}\eta_t\,{\mathrm{d}}t+b_t^*\,{\mathrm{d}}\mathcal{W}_t^1+\alpha_t^*\,{\mathrm{d}} \mathcal{W}_t^2,\\[5pt]{\mathrm{d}}\beta_t=-\mathcal{A}_{1,t}^{*}\beta_t\,{\mathrm{d}}t-\mathcal{A}_{0, t}^{*}\eta_t\,{\mathrm{d}} t-\alpha_t^*\,{\mathrm{d}}\mathcal{W}_t^2,\\[5pt](\eta_0, \beta_0)\text{ is independent of }\{\mathcal{W}_t^1\}_{t\geq 0}\text{ and }\{\mathcal{W}_t^2\}_{t\geq 0}.\end{array}\right.\end{equation}
According to an analysis similar with that leading to Theorem 1.4 of [Reference Holley and Stroock5], with given initial distribution of
$(\eta_0, \beta_0)$
, there exists an unique random element
$\{(\eta_t, \beta_t)\}_{t\geq 0}$
such that:
-
(1)
$\{(\eta_t(\,f), \beta_t(\,f))\}_{t\geq 0}$
is a
$\mathbb{R}^2$
-valued continuous process for any
$f\in \hat{C}^1(\mathbb{T})$
; -
(2) for any
$G\in C_c^\infty(\mathbb{R}^2)$
and
$f, g\in \hat{C}^1(\mathbb{T})$
, is a martingale, where u is the first coordinate and v is the second coordinate.
\begin{align*}\Bigg\{&G(\eta_t(\,f), \beta_t(g))-G(\eta_0(\,f), \beta_0(g))\\[5pt] &\text{ }-\int_0^t\frac{\partial}{\partial u}G(\eta_s(\,f), \beta_s(g))\left(\beta_s(\mathcal{A}_{1,s}\;f)+\eta_s(\mathcal{A}_{0,s}\;f-\mathcal{A}_\psi f)\right){\mathrm{d}}s\\[5pt] &\text{ }+\int_0^t\frac{\partial}{\partial v}G(\eta_s(\,f), \beta_s(g))\left(\beta_s(\mathcal{A}_{1,s}g)+\eta_s(\mathcal{A}_{0, s}g)\right){\mathrm{d}}s\\[5pt] &\text{ }-\frac{1}{2}\int_0^t\frac{\partial^2}{\partial u^2}G(\eta_s(\,f), \beta_s(g))\\[5pt] &\quad\quad\quad\times\left(\int_0^1\rho_1(s,u)\psi(u)\,f^2(u)\,{\mathrm{d}} u+\int_0^1\int_0^1\lambda(u,v)\rho_0(s,u)\rho_1(s,v)\,f^2(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\right){\mathrm{d}}s\\[5pt] &\text{ }-\frac{1}{2}\int_0^t\int_0^1\int_0^1\frac{\partial^2}{\partial v^2}G(\eta_s(\,f), \beta_s(g))\lambda(u,v)\rho_0(s,u)\rho_1(s,v)g^2(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\,{\mathrm{d}}s\\[5pt] &\text{ }+\int_0^t\int_0^1\int_0^1\frac{\partial^2}{\partial uv}G(\eta_s(\,f), \beta_s(g))\lambda(u,v)\rho_0(s,u)\rho_1(s,v)\,f(u)g(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\,{\mathrm{d}}s\Bigg\}_{t\geq 0}\end{align*}
According to Itô’s formula, it is nature to define the above
$\{(\eta_t, \beta_t)\}_{t\geq 0}$
as the solution to Equation (2.3), which is called a generalized O-U process as introduced in [Reference Holley and Stroock5].
Remark 2.4. As introduced in [Reference Holley and Stroock5], the formal definition of the solution to Equation (2.3) is the unique solution to the corresponding martingale problem. However, the analysis leading to Theorem 1.4 of [Reference Holley and Stroock5] implies that we can define the solution to Equation (2.3) equivalently through solving this equation directly. In detail, Equation (2.3) can be written as
\begin{align*}{\mathrm{d}}\begin{pmatrix} \eta_t\\[5pt] \beta_t\end{pmatrix}=\Sigma_{1,t}^*\begin{pmatrix} \eta_t\\[5pt] \beta_t\end{pmatrix}\,{\mathrm{d}}t+\Sigma_{2,t}^*\,{\mathrm{d}}\begin{pmatrix} \mathcal{W}_t^1\\[5pt] \mathcal{W}_t^2\end{pmatrix},\end{align*}
where
\begin{align*}\Sigma_{1,t}^*=\begin{pmatrix} \mathcal{A}_{1,t}^*\;\;\;\;\;&\mathcal{A}_{0,t}^*-\mathcal{A}_\psi^*\\[5pt] -\mathcal{A}_{1,t}^*\;\;\;\;\;&-\mathcal{A}_{0,t}^*\end{pmatrix}\quad \text{and}\quad \Sigma_{2,t}^*=\begin{pmatrix}b_t^*\;\;\;\;\;&\alpha_t^*\\[5pt] 0\;\;\;\;\;&-\alpha_t^*\end{pmatrix}.\end{align*}
Let
$\{\Upsilon^*(t,s)\}_{0\leq s\leq t<+\infty}$
be the solution to the matrix-valued ODE
\begin{equation}\left\{\begin{array}{l} \dfrac{\partial}{\partial t}\Upsilon^*(t, s)=\Sigma_{1,t}^*\Upsilon^*(t,s),\\[5pt]\Upsilon^*(s,s)=\begin{pmatrix}I_d^*\;\;\;\;\;&0 \\[5pt] 0\;\;\;\;\;& I_d^*\end{pmatrix},\quad s\geq 0,\end{array}\right.\end{equation}
where
$I_d^*$
is the identity function on
$(\hat{C}^1(\mathbb{T}))^\prime$
. Note that the existence and uniqueness of
$\Upsilon^*$
follows from Theorem 1.2 in Chapter 19 of [Reference Lang8]. Then, according to Itô’s formula,
\begin{equation}\begin{pmatrix} \eta_t\\[5pt] \beta_t\end{pmatrix}=\Upsilon^*(t,0)\begin{pmatrix} \eta_0\\[5pt] \beta_0\end{pmatrix}+\int_0^t\Upsilon^*(t,s)\Sigma_{2,s}^*\,{\mathrm{d}}\begin{pmatrix} \mathcal{W}_s^1\\[5pt] \mathcal{W}_s^2\end{pmatrix}.\end{equation}
Now we can state our second main result, which gives the fluctuation of our SIR model.
Theorem 2.2. Under Assumption (A), for a given
$T>0$
,
$\{(\eta_t^N, \beta_t^N)\}_{0\leq t\leq T}$
converges weakly, with respect to the Skorohod topology of
$\mathcal{D}([0, T], ((\hat{C}^1(\mathbb{T}))^\prime)^2)$
, to
$\{(\eta_t, \beta_t)\}_{0\leq t\leq T}$
as
$Narrow+\infty$
, where
$\{(\eta_t, \beta_t)\}_{t\geq 0}$
is the solution to Equation (2.3) with the following initial distribution:
-
(1)
$\beta_0=-\eta_0$
; -
(2) for any
$f\in \hat{C}^1(\mathbb{T})$
,
$\eta_0(\,f)$
follows
$\mathbb{N}(0, \int_0^1f^2(u)\phi(u)(1-\phi(u))\,{\mathrm{d}}u)$
, where
$\mathbb{N}(a, \sigma^2)$
is the normal distribution with mean a and variance
$\sigma^2$
.
Theorem 2.2 shows that the fluctuation of
$\{\xi_t^N\}_{t\geq 0}$
is driven by a generalized O-U process. According to Remark 2.4, for any
$f,g\in \hat{C}^1(\mathbb{T})$
and
$t, s\geq 0$
,
$(\eta_t^N(\,f), \beta_s^N(g))$
converges weakly to the normal distribution of
$(\eta_t(\,f), \beta_s(g))$
given by (2.5).
Remark 2.5. As far as we understand, Theorem 2.2 relies heavily on the assumption that the recovery and infection periods in our model have the exponential distributions. Roughly speaking, when the recovery and infection periods are exponential, the generator of the corresponding model has a difference form. By the Taylor’s expansion formula up to the second order, the aforesaid difference form converges to a gradient operator plus a Laplace operator under proper time–space scaling, which is the generator of a diffusion process driven by a Brownian motion. As far as we know, when the recovery and infection periods have general distributions other than exponential, the corresponding process has no generator with a difference form and the related fluctuation limit is not driven by a diffusion process as given by (2.3).
Pang and Pardoux [Reference Pang and Pardoux11] investigated functional LLN and fluctuation limits for the non-Markovian epidemic models where the recovery periods and infection periods have general distributions other than exponential. It was shown in [Reference Pang and Pardoux11] that the functional LLN of the non-Markovian epidemic model is driven by a system of deterministic integral equations and the fluctuation limit is a system of stochastic Volterra integral equations driven by Gaussian processes. The model in [Reference Pang and Pardoux11] and our model are two different extensions of the classic SIR model. The model in [Reference Pang and Pardoux11] is non-Makrovian and with vertex-independent transition rates while our model is Markovian and with vertex-dependent transition rates. Roughly speaking, for Makovian models, since KC equation holds, the LLN and CLT of the model are usually driven by ODEs and SDEs, respectively. For non-Markovian models, because of the absence of the memoryless property, the LLN and the CLT of the model are driven by integral equations and stochastic integral equations, respectively. For the model with vertex-independent transition rates, according to the spatial homogeneity, limit theorems are discussed for the numbers of infectious and susceptible individuals. For the model with vertex-dependent transition rates, since the contributions of each vertex to the spread of the epidemic is different, limit theorems should be discussed for the empirical density fields of the model. This is why the LLN and CLT in [Reference Pang and Pardoux11] are driven by real-valued integral equations and stochastic integral equations, respectively, whereas the LLN and CLT in this paper are driven by linear-operator-valued differential equations and SDEs, respectively. It is natural to study a non-Markovian SIR model with vertex-dependent transition rates, which includes the models in [Reference Pang and Pardoux11] and this paper as two special examples. According to aforesaid explanations, it is natural to believe that the LLN and CLT of the non-Markovian SIR model with vertex-dependent transition rates should be driven by linear-operator-valued integral equations and stochastic integral equations, respectively. We will work on this topic as a further investigation.
Our next main result is concerned with the LLN of the final size of the susceptible individuals. To give this LLN, we require the following lemma.
Lemma 2.1. Let
$\{(\rho_1(t, \cdot), \rho_0(t, \cdot))\}_{t\geq 0}$
be the solution to (2.2), then:
-
(1)
for any
\begin{align*}\lim_{t\rightarrow+\infty}\rho_1(t, u)=0\end{align*}
$u\in \mathbb{T}$
;
-
(2) there exists
$\hat{\rho}_0\;:\; \mathbb{T}\rightarrow [0, 1]$
such that for any
\begin{align*}\lim_{t\rightarrow+\infty}\rho_0(t, u)=\hat{\rho}_0(u)\end{align*}
$u\in \mathbb{T}$
.
The intuitive meaning of Lemma 2.1 is clear since the number of susceptible individuals is decreasing in t and all infectious individuals will become removed finally. A rigorous proof of Lemma 2.1 is given in Section 6.
Now we give our LLN of the final size of the susceptible individuals.
Theorem 2.3. Let
$S_\infty^N(i)$
be defined as in (2.1) and
$\hat{\rho}_0$
be defined as in Lemma 2.1. If
\begin{equation}\max_{u,v\in \mathbb{T}}\lambda(u, v)\int_0^1(1-\phi(u))\,{\mathrm{d}}u<\min_{u\in \mathbb{T}}\psi(u),\end{equation}
then, under Assumption (A),
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_\infty^N(i)=\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}}u\end{align*}
in probability.
We believe that Theorem 2.3 holds without Condition (2.6). However, our current proof of Theorem 2.3 requires Condition (2.6) to stochastically dominate the number of infectious individuals from above by a subcritical birth–death process. For mathematical details, see Section 6. We will work on how to cancel Condition (2.6) as a further investigation.
As we have mentioned in Section 1, for the classic SIR model, Watson [Reference Watson13] gives the CLT of the final size of the susceptible individuals as
$N\rightarrow+\infty$
via a time-change argument. It is natural to ask whether an analogue CLT holds for our model. In detail, is there a Gaussian distribution
$\mathbb{N}(0, \sigma^2)$
such that
$\sqrt{N}(N^{-1}\sum_{i=1}^NS_\infty^N(i)-\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}} u)$
converges weakly to
$\mathbb{N}(0, \sigma^2)$
as
$N\rightarrow+\infty$
? We guess that the answer to this question is positive and
$\sigma^2$
should be given by
\begin{align*}\sigma^2=\lim_{t\rightarrow+\infty}\sigma_t^2,\end{align*}
where
$\sigma_t^2=\textrm{Var}(\beta_t(1))$
and
$\beta_t$
is defined as in Theorem 2.2. However, we have not yet managed to prove this guess. The time-change argument given in [Reference Watson13] is not applicable to our model since this argument relies heavily on the fact that
$\{(\mathcal{S}_t^N, \mathcal{I}_t^N)\}_{t\geq 0}$
is a Markov process for the classic SIR model but
$\{(\sum_{i=1}^NS_t^N(i), \sum_{i=1}^NI_t^N(i))\}_{t\geq 0}$
is not Markovian for our model with vertex-dependent transition rates. We will work on the CLT of the final size of the susceptible individuals as a further investigation.
As we have mentioned in Section 1, another important problem for the SIR model is whether an outbreak of the disease occurs conditioned on only one individual being infectious at
$t=0$
. We obtain a phase transition conclusion for this problem in the case where all individuals have different activities, i.e.
$\lambda(u, v)=\vartheta(u)\vartheta(v)$
for some piecewise smooth
$\vartheta$
on
$\mathbb{T}$
. To give our result, for
$1\leq i\leq N$
, we denote by
$\hat{A}_i^N$
the event that
$\xi_0^N(i)=1$
and
$\xi_0^N(j)=0$
for all
$j\neq i$
. We further denote by
$\mathcal{R}_\infty^N$
the final size of the removed individuals for the process
$\{\xi_t^N\}_{t\geq 0}$
. Note that
$\mathcal{R}_\infty^N$
is the number of individuals which have been infectious. We have the following theorem.
Theorem 2.4. Assuming that
$\lambda(u, v)=\vartheta(u)\vartheta(v)$
for some piecewise smooth
$\vartheta$
on
$\mathbb{T}$
and any
$u, v\in \mathbb{T}$
. If
\begin{align*}\int_0^1\frac{\vartheta^2(u)}{\psi(u)}\,{\mathrm{d}}u<1,\end{align*}
then
\begin{align*}\lim_{N\rightarrow+\infty}\!\left(\sup_{1\leq i\leq N}\mathrm{P}\!\left(\mathcal{R}_\infty^N\geq \epsilon N\Big|\hat{A}_i^N\right)\right)=0\end{align*}
for any
$\epsilon>0$
. If
\begin{align*}\int_0^1\frac{\vartheta^2(u)}{\psi(u)}\,{\mathrm{d}}u>1,\end{align*}
then there exists
$c>0$
independent of N such that
\begin{align*}\inf_{N\geq 1, 1\leq i\leq N}\mathrm{P}\!\left(\mathcal{R}_\infty^N\geq cN\Big|\hat{A}_i^N\right)>0.\end{align*}
We omit the proof of Theorem 2.4 in this paper since it utilizes an argument nearly the same as that introduced in the proof of the main theorem in [Reference Xue15], where a strategy given in [Reference Kesten6] to bound the probability of the existence of an infection path from below plays the key role. In Section 7, we gives an intuitive explanation of why phase transition occurs when
$\int_0^1\frac{\vartheta^2(u)}{\psi(u)}\,{\mathrm{d}}u$
increases from less than 1 to greater than 1.
Remark 2.6. By Theorem 2.4, it is natural to ask whether a hydrodynamic limit theorem as Theorem 2.1 or a fluctuation theorem as Theorem 2.2 holds for our model starting from the initial condition where only one individual is infected. Currently, we only have a vague guess of this question. As far as we guess, when
$\lambda(\cdot, \cdot)$
is sufficiently large (e.g.
$\lambda(\cdot, \cdot)$
makes
$\int_0^1(\vartheta^2(u)/\psi(u))\,{\mathrm{d}}u>1$
for the model where individuals have different activities), with probability
$p_N$
such that
$\inf_Np_N>0$
,
$\mathcal{I}_t^N$
reaches the scale O(1)N at some moment
$\tau_N=O(1)\log N$
. An intuitive explanation of this guess is that
$\mathcal{I}_t^N$
grows exponentially as a supercritical branching process for small t when
$\lambda(\cdot, \cdot)$
is large. Then, the process
$\{\xi_{t+\tau_N}^N\}_{t\geq 0}$
evolves similarly to a version of our model starting from an initial condition under Assumption (A) and analogues of Theorems 2.1 and 2.2 should hold for the process
$\{\xi_{t+\tau_N}^N\}_{t\geq 0}$
. However, a precise statement of the above guess and a corresponding rigorous proof require much more detailed analysis than which given in this paper. We will work on this problem as a further investigation.
Remaining sections of this paper are organized as follows. We prove a replacement lemma in Section 3 as a preliminary, which shows that states of different vertices are approximately independent as
$N\rightarrow+\infty$
. We prove Theorem 2.1 in Section 4. With our replacement lemma, Theorem 2.1 follows from a classic mean-variance analysis. We prove Theorem 2.2 in Section 5. According to the Dynkin’s martingale formula and our replacement lemma, we can show that each weak limit of subsequence of
$\{(\eta_t^N, \beta_t^N)\;:\;0\leq t\leq T\}_{N\geq 1}$
is the solution to the martingale problem with respect to Equation (2.3) and, hence, Theorem 2.2 holds after we check that
$\{(\eta_t^N, \beta_t^N)\;:\;0\leq t\leq T\}_{N\geq 1}$
are tight. We prove Theorem 2.3 in Section 6. The main strategy in the proof is to bound the number of infectious individuals from above by a subcritical birth–death process. We give an heuristic explanation of Theorem 2.4 in Section 7, which shows that, conditioned on only one individual being infectious at
$t=0$
, the spread of the disease in the initial few steps is similar to a branching process where the mean of the offspring of a given individual is
$\int_0^1(\vartheta^2(u)/\psi(u))\,{\mathrm{d}}u$
.
2.2. A revisit of the spatial-homogeneous case
As an application of our main result, in this section we revisit LLN and CLT of the spatial-homogeneous SIR model given in [Reference Kurtz7], i.e. throughout this subsection we assume that
$\psi\equiv 1$
,
$\lambda\equiv \lambda_0\in \mathbb{R}$
and
$\phi\equiv \phi_0\in [0, 1]$
. As we have introduced in Section 1,
$\{(\mathcal{I}_t^N, \mathcal{S}_t^N)^{\textsf{T}}\}_{t\geq 0}$
is a
$\mathbb{R}^2$
-valued continuous-time Markov process such that
\begin{align*}\left(\mathcal{I}_t^N, \mathcal{S}_t^N\right)^{\textsf{T}} \text{ jumps to } \left(\mathcal{I}_t^N, \mathcal{S}_t^N\right)^{\textsf{T}}+l_k \text{ at rate }NF_k\!\left(\frac{\mathcal{I}_t^N}{N}, \frac{\mathcal{S}_t^N}{N}\right)\end{align*}
for
$k=1, 2$
, where
$\textsf{T}$
is the transposition operator,
$l_1=(\!-\!1, 0)^{\textsf{T}}, l_2=(1, -1)^{\textsf{T}}$
,
$F_1(u,v)=u$
and
$F_2(u,v)=\lambda_0uv$
. As a result,
$\{(\mathcal{I}_t^N, \mathcal{S}_t^N)^{\textsf{T}}\}_{t\geq 0}$
is an example of the DDMC introduced in [Reference Kurtz7]. Then, as direct applications of main results given in [Reference Kurtz7], we have the following two theorems.
Theorem 2.5. (Kurtz [Reference Kurtz7]). As
$N\rightarrow+\infty$
,
$(\mathcal{I}_t^N/N, \mathcal{S}_t^N/N)^{\textsf{T}}$
converges in probability to
$x_t$
, where
\begin{equation}\left\{\begin{array}{l} \dfrac{{\mathrm{d}}}{{\mathrm{d}}t}x_t=\sum_{k=1}^2l_kF_k(x_t),\\[9pt] x_0=(\phi_0, 1-\phi_0)^{\textsf{T}}.\end{array}\right.\end{equation}
Theorem 2.6. (Kurtz [Reference Kurtz7]). As
$N\rightarrow+\infty$
,
$$\left(\frac{\mathcal{I}_t^N-\mathrm{E}\mathcal{I}_t^N}{\sqrt{N}}, \frac{\mathcal{S}_t^N-\mathrm{E}\mathcal{S}_t^N}{\sqrt{N}}\right)^{\textsf{T}}$$
converges in distribution to
$V_t$
, where
\begin{equation}{\mathrm{d}}V_t=\sum_{k=1}^2l_k\nabla^\textsf{T}F_k(x_t)V_t,{\mathrm{d}} t+\left(\sum_{k=1}^2l_kF_k(x_t)l_k^\textsf{T}\right)^{\frac{1}{2}}\,{\mathrm{d}}\mathcal{B}_t,\end{equation}
where
$\mathcal{B}_t$
is a two-dimensional standard Brownian motion and
$\nabla^\textsf{T}=({\partial}/{\partial u}, {\partial}/{\partial v})$
.
In [Reference Kurtz7], LLN and CLT of DDMC are derived through writing the process as the solution to a SDE driven by Poisson processes. Here we prove Theorems 2.5 and 2.6 via our main results Theorems 2.1 and 2.2.
Proof. Alternative proof of Theorem 2.5 Since
$N^{-1}\mathcal{I}_t^N=\mu_t^N(1)$
and
$N^{-1}\mathcal{S}_t^N=\theta_t^N(1)$
, by Theorem 2.1,
$N^{-1}\mathcal{I}_t^N, N^{-1}\mathcal{S}_t^N$
converges in probability to
\begin{align*}i_t=\int_0^1\rho_1(t,u)\,{\mathrm{d}}u \quad\text{and} \quad s_t=\int_0^1\rho_0(t,u)\,{\mathrm{d}}u,\end{align*}
respectively, where
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_1(t,u)=-\rho_1(t,u)+\lambda_0\rho_0(t,u)\int_0^1\rho_1(t,v)\,{\mathrm{d}}v,\end{align*}
and
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_0(t,u)=-\lambda_0\rho_0(t,u)\int_0^1\rho_1(t,v)\,{\mathrm{d}}v.\end{align*}
As a result,
\begin{align*} \left\{\begin{array}{l} \dfrac{{\mathrm{d}}}{{\mathrm{d}}t}i_t=-i_t+\lambda_0i_ts_t,\\[9pt] \dfrac{{\mathrm{d}}}{{\mathrm{d}}t}s_t=-\lambda_0i_ts_t,\\[9pt] (i_0, s_0)=(\phi_0, 1-\phi_0),\end{array}\right. \end{align*}
which is Equation (2.7).
Proof. Alternative proof of Theorem 2.6 Since
\begin{align*}\frac{\mathcal{I}_t^N-\mathrm{E}\mathcal{I}_t^N}{\sqrt{N}}=\eta_t^N(1) \quad \text{and} \quad \frac{\mathcal{S}_t^N-\mathrm{E}\mathcal{S}_t^N}{\sqrt{N}}=\beta_t^N(1),\end{align*}
by Theorem 2.2,
$$\left(\frac{\mathcal{I}_t^N-\mathrm{E}\mathcal{I}_t^N}{\sqrt{N}}, \frac{\mathcal{S}_t^N-\mathrm{E}\mathcal{S}_t^N}{\sqrt{N}}\right)^{\textsf{T}}$$
converges in distribution to
$(\eta_t(1), \beta_t(1))^\textsf{T}$
, where
\begin{align*} \left\{\begin{array}{l}{\mathrm{d}}\eta_t(1)=\beta_t(\mathcal{A}_{1,t}1)\,{\mathrm{d}}t+\eta_t(\mathcal{A}_{0, t}1)\,{\mathrm{d}} t-\eta_t(\mathcal{A}_{\psi}1)\,{\mathrm{d}}t+\|b_t1\|_2\,{\mathrm{d}}\hat{B}_t^1+\|\alpha_t1\|_2\,{\mathrm{d}} \hat{B}_t^2,\\[5pt]{\mathrm{d}}\beta_t(1)=-\beta_t(\mathcal{A}_{1,t}1)\,{\mathrm{d}}t-\eta_t(\mathcal{A}_{0,t}1)\,{\mathrm{d}} t-\|\alpha_t1\|_2\,{\mathrm{d}}\hat{B}_t^2,\end{array}\right.\end{align*}
where
$\|f\|_2=\sqrt{\int_0^1f^2(u)\,{\mathrm{d}}u}$
for any
$f\in C[0, 1]$
and
$\{(\hat{B}_t^1, \hat{B}_t^2)\}_{t\geq 0}$
is a
$\mathbb{R}^2$
-valued standard Brownian motion. By direct calculation,
\begin{align*}\beta_t(\mathcal{A}_{1,t}1)=\lambda_0i_t\beta_t(1), \quad \eta_t(\mathcal{A}_{0, t}1)=\lambda_0s_t\eta_t(1) \quad\text{and} \quad \mathcal{A}_\psi1=1.\end{align*}
Therefore, let
$V_t=(\eta_t(1), \beta_t(1))^\textsf{T}$
, then
\begin{align*}{\mathrm{d}}V_t=\begin{pmatrix} \lambda_0s_t-1\;\;\;\;\;& \lambda_0i_t\\[5pt] -\lambda_0s_t\;\;\;\;\;&-\lambda_0i_t\end{pmatrix}V_t\,{\mathrm{d}} t+\begin{pmatrix} \|b_t1\|_2\;\;\;\;\; & \|\alpha_t1\|_2\\[5pt] 0\;\;\;\;\; &-\|\alpha_t1\|_2\end{pmatrix}\,{\mathrm{d}}\begin{pmatrix} \hat{B}_t^1\\[5pt] \hat{B}_t^2\end{pmatrix}.\end{align*}
By direct calculation, it is easy to check that
\begin{align*}\sum_{k=1}^2l_k\nabla^\textsf{T}F_k(x_t)=\begin{pmatrix} \lambda_0s_t-1\;\;\;\;\;& \lambda_0i_t\\[5pt] -\lambda_0s_t\;\;\;\;\;&-\lambda_0i_t\end{pmatrix}\end{align*}
and
\begin{align*}\begin{pmatrix} \|b_t1\|_2\;\;\;\;\;& \|\alpha_t1\|_2\\[5pt] 0\;\;\;\;\;&-\|\alpha_t1\|_2\end{pmatrix}\begin{pmatrix} \|b_t1\|_2\;\;\;\;\;& \|\alpha_t1\|_2\\[5pt] 0\;\;\;\;\;&-\|\alpha_t1\|_2\end{pmatrix}^\textsf{T}=\sum_{k=1}^2l_kF_k(x_t)l_k^\textsf{T}\end{align*}
and, hence,
$\{V_t\}_{t\geq 0}$
is the solution to Equation (2.8).
2.3. Applications of main results to other examples
In this subsection, we apply our main results to other examples than the classic SIR model. We first give the LLN of the SIR model with vertex-dependent activities.
Example 2.1. The SIR model with vertex-dependent activities. Assuming that
$\psi\equiv \psi_0\in (0, +\infty)$
and
$\lambda(u, v)=\vartheta(u)\vartheta(v)$
for some piecewise smooth
$\vartheta$
on
$\mathbb{T}$
and all
$u, v\in \mathbb{T}$
, then our model reduces to the SIR model with vertex-dependent activities as we have explained in Section 1.
To give our result, we define
\begin{align*}\mathcal{U}_1(y)=\int_0^1(1-\phi(u))y^{\vartheta(u)}\,{\mathrm{d}}u\end{align*}
and
\begin{align*}\mathcal{U}_2(y)=\int_0^1\vartheta(u)\,{\mathrm{d}}u+\psi_0\log y-\int_0^1\theta(u)(1-\phi(u))y^{\vartheta(u)}\,{\mathrm{d}}u\end{align*}
for all
$y>0$
. We have the following corollary of Theorem 2.1, which gives the LLNs of the number of susceptible individuals and the total activities of infectious individuals.
Corollary 2.1. Under the assumption of Example 2.1 and Assumption (A), for any
$t\geq 0$
,
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_t^N(i)=\mathcal{U}_1(k(t)) \quad \text{and} \quad\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NI_t^N(i)\vartheta\!\left(\frac{i}{N}\right)=\mathcal{U}_2(k(t))\end{align*}
in probability, where k(t) is the solution to the ODE
\begin{equation}\left\{\begin{array}{l}\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}k(t)=-k(t)\mathcal{U}_2(k(t)),\\[9pt] k(0)=1.\end{array}\right.\end{equation}
Proof. By Theorem 2.1,
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_t^N(i)=\int_0^1\rho_0(t, u)\,{\mathrm{d}}u \quad \text{and} \quad\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NI_t^N(i)\vartheta\!\left(\frac{i}{N}\right)=\int_0^1\rho_1(t, u)\vartheta(u)\,{\mathrm{d}}u\end{align*}
in probability. Therefore, we only need to show that
\begin{align*}\int_0^1\rho_0(t, u)\,{\mathrm{d}}u=\mathcal{U}_1(k(t)) \quad \text{and} \quad \int_0^1\rho_1(t, u)\vartheta(u)\,{\mathrm{d}}u=\mathcal{U}_2(k(t)).\end{align*}
For simplicity, we denote by
$g_1(t)$
the term
$\int_0^1\rho_0(t, u)\,{\mathrm{d}}u$
and by
$g_2(t)$
the term
$\int_0^1\rho_1(t, u)\vartheta(u)\,{\mathrm{d}}u$
. By Theorem 2.1, under the assumption of Example 2.1,
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_0(t, u)=-\rho_0(t, u)\vartheta(u)g_2(t),\end{align*}
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_1(t, u)=-\psi_0\rho_1(t, u)+\rho_0(t, u)\vartheta(u)g_2(t)\end{align*}
and, hence,
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}g_2(t)=-\psi_0h_2(t)+\int_0^1\rho_0(t, u)(\vartheta(u))^2\,{\mathrm{d}}ug_2(t).\end{align*}
Let r(t) satisfy that
\begin{align*}\int_0^{r(t)}g_2(s)\,{\mathrm{d}}s=t,\end{align*}
then
\begin{equation}\frac{{\mathrm{d}}}{{\mathrm{d}}t}r(t)=\frac{1}{g_2(r(t))}.\end{equation}
Let
$\tilde{\rho}_l(t, u)=\rho_l(r(t), u)$
for
$l=0, 1$
and
$\tilde{g}_j(t)=g_j(r(t))$
for
$j=1, 2$
, then by (2.10),
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{\rho}_0(t, u)=-\tilde{\rho}_0(t, u)\vartheta(u)\end{align*}
and
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{g}_2(t)=-\psi_0+\int_0^1\tilde{\rho}_0(t, u)(\vartheta(u))^2\,{\mathrm{d}}u.\end{align*}
Then, by
$\rho_0(0, u)=1-\phi(u)$
,
\begin{align*}\tilde{\rho}_0(t, u)=(1-\phi(u)){\mathrm{e}}^{-\vartheta(u)t}\end{align*}
and, hence,
\begin{align*}\tilde{g}_1(t)=\int_0^1 (1-\phi(u)){\mathrm{e}}^{-\vartheta(u)t}\,{\mathrm{d}}u,\end{align*}
\begin{align*}\tilde{g}_2(t)=\int_0^1\vartheta(u)\,{\mathrm{d}}u-\psi_0t-\int_0^1\vartheta(u)(1-\phi(u)){\mathrm{e}}^{-\vartheta(u)t}\,{\mathrm{d}} u.\end{align*}
Therefore,
\begin{align*}g_1(t)=\int_0^1 (1-\phi(u)){\mathrm{e}}^{-\vartheta(u)z(t)}\,{\mathrm{d}}u=\mathcal{U}_1({\mathrm{e}}^{-z(t)})\end{align*}
and
\begin{align*}g_2(t)=\int_0^1\vartheta(u)\,{\mathrm{d}}u-\psi_0z(t)-\int_0^1\vartheta(u)(1-\phi(u)){\mathrm{e}}^{-\vartheta(u)z(t)}\,{\mathrm{d}} u=\mathcal{U}_2({\mathrm{e}}^{-z(t)}),\end{align*}
where
\begin{align*}r(z(t))=z(r(t))=t.\end{align*}
At last, we only need to show that
${\mathrm{e}}^{-z(t)}$
is the solution to Equation (2.9). By (2.10),
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}{\mathrm{e}}^{-z(t)}&=-{\mathrm{e}}^{-z(t)}\frac{{\mathrm{d}}}{{\mathrm{d}}t}z(t)\\[5pt] &=-{\mathrm{e}}^{-z(t)}\frac{1}{r^\prime(z(t))}=-{\mathrm{e}}^{-z(t)}g_2(r(z(t)))=-{\mathrm{e}}^{-z(t)}g_2(t)\\[5pt] &=-{\mathrm{e}}^{-z(t)}\mathcal{U}_2({\mathrm{e}}^{-z(t)}).\end{align*}
Then, since
${\mathrm{e}}^{-z(0)}={\mathrm{e}}^{-0}=1$
,
${\mathrm{e}}^{-z(t)}$
is the solution to (2.9) and the proof is complete.
We have the following corollary of Theorem 2.3, which gives the LLN of the final size of the susceptible individuals of the SIR model with vertex-dependent activities.
Corollary 2.2. Under the assumption of Example 2.1 and Assumption (A), if
\begin{align*}\left(\max_{u\in \mathbb{T}}\vartheta(u)\right)^2\int_0^1(1-\phi(u))\,{\mathrm{d}}u<\psi_0,\end{align*}
then
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_{\infty}^N(i)=\mathcal{U}_1\big({\mathrm{e}}^{-z(\infty)}\big)\end{align*}
in probability, where
\begin{align*}z(\infty)=\int_0^{+\infty}\mathcal{U}_2(k(s))\,{\mathrm{d}}s\end{align*}
and
$\{k(s)\}_{s\geq 0}$
is the solution to (2.9).
Proof. Since
$(\!\max_{u\in \mathbb{T}}\vartheta(u))^2\int_0^1(1-\phi(u))\,{\mathrm{d}}u<\psi_0$
, Assumption (2.6) holds. Therefore, by Theorem 2.3 and Corollary 2.1,
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_{\infty}^N(i)=\mathcal{U}_1\big(\lim_{t\rightarrow+\infty}k(t)\big)\end{align*}
in probability. According to the proof of Corollary 2.1,
$k(t)={\mathrm{e}}^{-z(t)}$
, where z(t) satisfies
\begin{align*}\int_0^{t}g_2(s)\,{\mathrm{d}}s=z(t)\end{align*}
and
$g_2(s)=\mathcal{U}_2(k(s))$
. Hence,
\begin{align*}\lim_{t\rightarrow+\infty}k(t)=\exp\!\left\{-\int_0^{+\infty}\mathcal{U}_2(k(s))\right\}\end{align*}
and the proof is complete.
By applying Theorem 2.2 under the assumption of Example 2.1, we can further obtain the CLTs of
$\sum_{i=1}^NS_t^N(i)$
and
$\sum_{i=1}^NI_t^N(i)\vartheta(i/N)$
. In detail,
\begin{align*}\frac{1}{\sqrt{N}}\!\left(\sum_{i=1}^N\!\left(S_t^N(i)-\mathrm{E}S_t^N(i)\right)\!, \sum_{i=1}^N\!\left(I_t^N(i)-\mathrm{E}I_t^N(i)\right)\vartheta(i/N)\right)\end{align*}
converges weakly to
$(\beta_t(1), \eta_t(\vartheta))$
as
$N\rightarrow+\infty$
, where
$\{(\beta_t, \eta_t)\}_{t\geq 0}$
is the solution to (2.3). However, under the assumption of Example 2.1,
$(\beta_t, \eta_t)$
does not have a briefer expression than that given in Subsection 2.1, so we do not list the CLT of Example 2.1 as a main result of this paper.
As the second application, we give the LLN and the CLT of the SIR model with distance-dependent infection rates.
Example 2.2. The SIR model with distance-dependent infection rates. Assuming that
$\psi\equiv \psi_0\in (0, +\infty)$
and
\begin{align*}\lambda(u, v)=g(D(u, v))\end{align*}
for some
$g\;:\;\mathbb{T}\rightarrow [0, 1/2]$
and all
$u, v\in \mathbb{T}$
, where
$D(\cdot, \cdot)$
is the metric function on
$\mathbb{T}$
defined as in Section 1, then our model reduces to the SIR model with distance-dependent infection rates. We further assume that
$\phi$
in Assumption (A) is a constant, i.e.
$\phi\equiv \phi_0\in (0, 1)$
.
To give our result, we denote by
$\tilde{c}$
the term
\begin{align*}\int_0^{1/2}g(v)\,{\mathrm{d}}v+\int_{1/2}^1g(1-v)\,{\mathrm{d}}v.\end{align*}
We further define
\begin{align*}\mathcal{U}_3(y)=(1-\phi_0)y^{\tilde{c}}\end{align*}
and
\begin{align*}\mathcal{U}_4(y)=1+\psi_0\log y-y^{\tilde{c}}(1-\phi_0)\end{align*}
for all
$y>0$
. We have the following corollary of Theorem 2.1, which gives the LLNs of the numbers of the susceptible and infectious individuals of the SIR model with distance-dependent infection rates.
Corollary 2.3. Under the assumption of Example 2.2 and Assumption (A), for any
$t\geq 0$
,
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_t^N(i)=\mathcal{U}_3(\tilde{k}(t))\text{ and }\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NI_t^N(i)=\mathcal{U}_4(\tilde{k}(t))\end{align*}
in probability, where
$\{\tilde{k}(t)\}_{t\geq 0}$
is the solution to
\begin{equation}\left\{\begin{array}{l}\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}\tilde{k}(t)=-\tilde{k}(t)\mathcal{U}_4(\tilde{k}(t)),\\[9pt] \tilde{k}(0)=1.\end{array}\right.\end{equation}
Under the assumption of Example 2.2,
$\rho_0(t, u)$
and
$\rho_1(t, u)$
are independent of u, which can be briefly written as
$\rho_0(t)$
and
$\rho_1(t)$
. By Theorem 2.2,
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_t^N(i)=\rho_0(t)\quad\text{and}\quad\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NI_t^N(i)=\rho_1(t)\end{align*}
in probability and
\begin{align*} \left\{\begin{array}{l}\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_0(t)=-\tilde{c}\rho_0(t)\rho_1(t),\\[9pt] \dfrac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_1(t)=-\psi_0\rho_1(t)+\tilde{c}\rho_0(t)\rho_1(t).\end{array}\right.\end{align*}
By introducing
$\tilde{r}(t)$
and
$\tilde{z}(t)$
such that
\begin{align*}\int_0^{\tilde{r}(t)}\rho_1(s)\,{\mathrm{d}}s=t\end{align*}
and
\begin{align*}\tilde{r}(\tilde{z}(t))=t,\end{align*}
Corollary 2.3 follows from the same argument as that in the proof of Corollary 2.1, the detail of which we omit here.
We further have the following analogue of Corollary 2.2.
Corollary 2.4. Under the assumption of Example 2.2 and Assumption (A), if
\begin{align*}\max_{u\in [0, 1/2]}g(u)(1-\phi_0)<\psi_0,\end{align*}
then
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\sum_{i=1}^NS_{\infty}^N(i)=\mathcal{U}_3\big({\mathrm{e}}^{-\tilde{z}(\infty)}\big)\end{align*}
in probability, where
\begin{align*}\tilde{z}(\infty)=\int_0^{+\infty}\mathcal{U}_4(\tilde{k}(s))\,{\mathrm{d}}s\end{align*}
and
$\{\tilde{k}(s)\}_{s\geq 0}$
is the solution to (2.11).
The proof of Corollary 2.4 utilizes the same argument as that leading to Corollary 2.2, the detail of which we omit here.
By applying Theorem 2.2 under the assumption of Example 2.2, we have the following CLT of the numbers of susceptible and infectious individuals of the SIR model with distance-dependent infection rates.
Corollary 2.5. Under the assumption of Example 2.2 and Assumption (A), as
$N\rightarrow+\infty$
,
\begin{align*}\frac{1}{\sqrt{N}}\!\left(\sum_{i=1}^N\!\left(S_t^N(i)-\mathrm{E}S_t^N(i)\right)\!, \sum_{i=1}^N\!\left(I_t^N(i)-\mathrm{E}I_t^N(i)\right)\right)\end{align*}
converges weakly to
$\tilde{V}_t$
, where
\begin{equation*}{\mathrm{d}}\tilde{V}_t=\Sigma_t\tilde{V}_t\,{\mathrm{d}}t+\begin{pmatrix}\sqrt{\psi_0\mathcal{U}_4(\tilde{k}(t))}\;\;\;\;\; & \sqrt{\tilde{c}\mathcal{U}_3(\tilde{k}(t))\mathcal{U}_4(\tilde{k}(t))}\\[5pt] 0\;\;\;\;\; & -\sqrt{\tilde{c}\mathcal{U}_3(\tilde{k}(t))\mathcal{U}_4(\tilde{k}(t))}\end{pmatrix}\,{\mathrm{d}}\mathcal{B}_t,\end{equation*}
where
\begin{align*}\Sigma_t=\begin{pmatrix}\mathcal{U}_3(\tilde{k}(t))\tilde{c}-\psi_0\;\;\;\;\; & \mathcal{U}_4(\tilde{k}(t))\tilde{c}\\[5pt] -\mathcal{U}_3(\tilde{k}(t))\tilde{c}\;\;\;\;\; & -\mathcal{U}_4(\tilde{k}(t))\tilde{c}\end{pmatrix},\end{align*}
$\mathcal{B}_t$
is a two-dimensional standard Brownian motion and
$\{\tilde{k}(t)\}_{t\geq 0}$
is the solution to (2.11).
Note that, although Corollary 2.5 has a similar statement to that of Theorem 2.6, Example 2.2 is not a special case of the DDMCs and, hence, Corollary 2.5 cannot be obtained by applying the CLT given in [Reference Kurtz7] directly.
According to the proof of Corollary 2.3, under the assumption of Example 2.2,
\begin{align*}\rho_0(t, u)=\rho_0(t)=\mathcal{U}_3(\tilde{k}(t))\end{align*}
and
\begin{align*}\rho_1(t, u)=\rho_1(t)=\mathcal{U}_4(\tilde{k}(t)).\end{align*}
Hence,
\begin{align*}\beta_t(\mathcal{A}_{1,t}1)=\mathcal{U}_4(\tilde{k}(t))\tilde{c}\beta_t(1), \quad\eta_t(\mathcal{A}_{0, t}1)=\mathcal{U}_3(\tilde{k}(t))\tilde{c}\eta_t(1)\end{align*}
and
\begin{align*}\mathcal{A}_\psi1=\psi_01, \text{ }b_t1=\sqrt{\psi_0\mathcal{U}_4(\tilde{k}(t))}1, \quad \alpha_t1=\sqrt{\tilde{c}\mathcal{U}_3(\tilde{k}(t))\mathcal{U}_4(\tilde{k}(t))}1.\end{align*}
Consequently, Corollary 2.5 follows from the same argument as that given in our alternative proof of Theorem 2.6, the detail of which we omit here.
3. Replacement lemma
In this section we prove a replacement lemma which shows that states of different vertices are approximately independent as
$N\rightarrow+\infty$
. For simplicity, we define
\begin{align*}H_{t, 0}^N(i)=S_t^N(i) \quad \text{and} \quad H_{t, 1}^N(i)=I_t^N(i)\end{align*}
for
$1\leq i\leq N$
, then our replacement lemma is as follows.
Lemma 3.1. There exist
$C_1, C_2, C_3<+\infty$
independent of N such that for all
$0\leq t\leq T$
,
$N\geq 4$
,
$(r_1, r_2, r_3, r_4)\in \{0, 1\}^4$
and any four different
$i,j,k,l\in \{1,2,\ldots,N\}$
,
\begin{align}\left|\mathrm{E}\!\left(H_{t, r_1}^N(i)H_{t, r_2}^N(j)\right)-\mathrm{E}H_{t, r_1}^N(i)\mathrm{E}H_{t, r_2}^N(j)\right|\leq \frac{C_1}{N},\end{align}
\begin{align}\left|\mathrm{E}\!\left(H_{t, r_1}^N(i)H_{t, r_2}^N(j)H_{t, r_3}^N(k)\right)-\mathrm{E}H_{t, r_1}^N(i)\mathrm{E}H_{t, r_2}^N(j)\mathrm{E}H_{t, r_3}^N(k)\right|\leq \frac{C_2}{N}\end{align}
and
\begin{align}\left|\mathrm{E}\!\left(H_{t, r_1}^N(i)H_{t, r_2}^N(j)H_{t, r_3}^N(k)H_{t, r_4}^N(l)\right)-\mathrm{E}H_{t, r_1}^N(i)\mathrm{E}H_{t, r_2}^N(j)\mathrm{E}H_{t, r_3}^N(k)\mathrm{E}H_{t, r_4}^N(l)\right|\leq \frac{C_3}{N}.\end{align}
For different i, j and given
$t>0$
, conditioned on
$I_t(j)=1$
, it is easy to check that i is infected by j before the moment t with probability
$O(N^{-1})$
. However, this is not enough for the proof of Lemma 3.1, since j may infect some other individuals during [0, t] and these individuals can further infect i. Thus, roughly speaking, we need to show that j is not in the set of individuals which can transmit the disease to i before moment t via an infection path ending at i with high probability.
The idea of the proof of Lemma 3.1 is as follows. We introduce some independent exponential times describing the recovery times of infected vertices and the infection times between infected and susceptible vertices. Then,
$\{\xi_t\}_{t>0}$
can be defined as some random variables being measurable with respect to the sigma-algebra generated by
$\xi_0$
and above exponential times. By further introducing some independent copies of above exponential times, we can construct
$\hat{\xi}_t(j), \hat{\xi}_t(k), \hat{\xi}_t(l)$
such that:
-
(1)
$(\xi_t(i), \hat{\xi}_t(j), \hat{\xi}_t(k), \hat{\xi}_t(l))$
are independent; -
(2)
$\hat{\xi}_t(m)$
and
$\xi_t(m)$
have the same distribution for
$m=j, k, l$
; -
(3) conditioned on an event with probability
$1-O(1/N)$
,
\begin{align*}(\hat{\xi}_t(j), \hat{\xi}_t(k), \hat{\xi}_t(l))=\left(\xi_t(j), \xi_t(k), \xi_t(l)\right)\!.\end{align*}
In detail, for each
$1\leq i\leq N$
, let
$K_i^N$
be an exponential time with rate
$\psi({i}/{N})$
. For any
$1\leq i\neq j\leq N$
, let
$U_{(i, j)}^N$
be an exponential time with rate
$N^{-1}\lambda({i}/{N}, {j}/{N})$
. We further assume that all these exponential times are independent. The meanings of these exponential times are as follows. For any
$1\leq i\neq j\leq N$
, if j is infected at some moment, then j waits for
$K_j^N$
to become removed and waits for
$U_{(i, j)}^N$
to infect i. This infection really occurs when
$U_{(i, j)}^N<K_j^N$
and i is susceptible at the moment just before being infected by j. According to the above explanation, let
$\xi_0^N$
be distributed as in Assumption (A), then
$\xi_t^N(i)$
is determined by
$\xi_0^N$
and above exponential times. In detail, for an integer
$n\geq 0$
and a self-avoiding path
$m_0, m_1, m_2,\ldots, m_n$
on
$\{1,2,\ldots,N\}$
, we call
$m_0, m_1, m_2,\ldots, m_n$
an infection path before moment t if following three conditions hold:
-
(1)
$\xi_0^N(m_0)=1$
; -
(2)
$U_{(m_{q+1}, m_q)}^N<K_{m_q}^N$
for all
$0\leq q\leq n-1$
; -
(3)
$\sum_{q=0}^{n-1}U^N_{(m_{q+1}, m_q)}\leq t$
;
For later use, if a self-avoiding path
$m_0, m_1,\ldots, m_n$
satisfies condition (3), then we call this path a semi-infection path before moment t. Of course, an infection path before moment t is a semi-infection path before moment t.
For any
$i\in \{1,2,\ldots, N\}$
,
$\xi_s^N(i)=1$
for some
$s\leq t$
when and only when there exists an infection path before moment t ending at i. For readers not familiar with this property of the SIR model, we give an explanation here. We only show the sufficiency since the necessity follows from a similar analysis. Assume that
$m_0, m_1,\ldots, m_n$
is an infection path before moment t with
$m_n=i$
. Now we investigate
$m_1$
. If
$\xi_s^N(m_1)=0$
for all
$s<U^N_{(m_1, m_0)}$
, then
$m_1$
is infected by
$m_0$
at the moment
$U^N_{(m_1, m_0)}$
and consequently
$\xi^N_{U^N_{(m_1, m_0)}}(m_1)=1$
since
$\xi_0^N(m_0)=1$
and
$U^N_{(m_1, m_0)}<K_{m_0}^N$
. In conclusion,
$\xi^N(m_1)$
becomes 1 at some moment
$a_1\leq U^N_{(m_1, m_0)}$
. Similarly, if
$\xi_s^N(m_2)=0$
for all
$s<a_1+U^N_{(m_2, m_1)}$
, then
$m_2$
is infected by
$m_1$
at the moment
$a_1+U^N_{(m_2, m_1)}$
since
$\xi^N(m_1)$
becomes infected at the moment
$a_1$
and
$U^N_{(m_2, m_1)}<K^N_{m_1}$
. In conclusion,
$\xi^N(m_2)$
becomes 1 at some moment
\begin{align*}a_2\leq a_1+U^N_{(m_2, m_1)}\leq \sum_{q=0}^1U^N_{(m_{q+1}, m_q)}.\end{align*}
By induction, there exist
$a_1\leq a_2\leq \ldots\leq a_n$
such that
$\xi^N(m_r)$
becomes 1 at some moment
\begin{align*}a_r\leq a_{r-1}+U^N_{(m_r, m_{r-1})}\leq \sum_{q=0}^{r-1}U^N_{(m_{q+1}, m_q)}\end{align*}
for all
$1\leq r\leq n$
. Therefore,
$\xi^N(i)=\xi^N(m_n)$
becomes 1 at some moment
\begin{align*}a_n\leq \sum_{q=0}^{n-1}U^N_{(m_{q+1}, m_q)}\leq t.\end{align*}
According to the above property of the SIR model. If there is no infection path before moment t ending at i, then
$\xi_t^N(i)=0$
. If there exists some infection path before moment t ending at i, then we choose the one making
$\sum_{q=0}^{n-1}U^N_{(m_{q+1}, m_q)}$
minimum. Let c be the aforesaid minimum, i.e. c is the moment at which i is infected. Consequently,
$\xi_t^N(i)=1$
when
$c+K_{i}^N>t$
and
$\xi_t^N(i)=-1$
when
$c+K_i^N\leq t$
. The above description of the SIR model can be considered as an application of the graphical method introduced in [Reference Harris4].
Britton et al. [Reference Britton, Leung and Trapman2] introduces a family of epidemic models constructed by independent waiting and infection times. The main difference between the models in [Reference Britton, Leung and Trapman2] and this paper is that in our model all N individuals may have different recovery rates and all infection rates between every pair of two individuals may be different while in the model of [Reference Britton, Leung and Trapman2] individuals are divided into K types such that the individuals in the same type are not distinguished, where K is independent of the population N. Thus, in our model we need introduce
$N(N-1)$
different infection times and N different recovery times while
$K^2$
different distributions of the point processes indicating infection moments between two individuals in different types are required for the model in [Reference Britton, Leung and Trapman2]. Both the models in [Reference Britton, Leung and Trapman2] and this paper include the Markovian SIR model with K types as special cases.
For given four different
$i,j,k,l\in \{1,2,\ldots, N\}$
, to define copies of
$\xi_t^N(j), \xi_t^N(k), \xi_t^N(l)$
, we introduce following notation. For any
$1\leq m\leq N$
, let
$\xi_{0, 2}^N(m), \xi_{0, 3}^N(m), \xi_{0, 4}^N(m)$
be three independent copies of
$\xi_0^N(m)$
. Similarly, for any
$1\leq m\neq q\leq N$
, we let
$K_{m,2}^N, K_{m,3}^N, K_{m,4}^N$
be three independent copies of
$K_m^N$
and
$U_{(m,q),2}^N, U_{(m,q),3}^N, U_{(m,q),4}^N$
be three independent copies of
$U_{(m,q)}^N$
. We assume all these random variables are independent. For
$t>0$
,
$1\leq m\leq N$
and
$0\leq r\leq N-1$
, we define random set
$\Xi^{N, t}_{m,r}\subseteq \{1,2,\ldots,N\}$
by induction as follows.
-
(1)
$\Xi^{N, t}_{m, 0}=\{m\}$
; -
(2) after
$\Xi^{N,t}_{m, r}$
are defined for
$0\leq r\leq q < N-1$
,
$\Xi^{N, t}_{m, q+1}$
is defined as the set of
$\varphi\in \{1,2,\ldots, N\}\setminus(\bigcup_{r=0}^q\Xi^{N, t}_{m, r})$
such that there exists a self-avoiding path such that
\begin{align*}\varphi(\!=m_{q+1}), m_q, \ldots, m_1, m(\!=m_0)\end{align*}
$m_r\in \bigcup_{l=1}^r\Xi^{N, t}_{m,l}$
for
$1\leq r\leq q$
and
\begin{align*}\sum_{r=0}^{q}U^N_{(m_r, m_{r+1})}\leq t.\end{align*}
In conclusion,
$\Xi^{N, t}_{m, q}$
is the set of vertices
$\varphi$
such that:
-
(1) there is a semi-infection path before moment t from
$\varphi$
to m with length q; -
(2) there is no semi-infection path before moment t from
$\varphi$
to m with length smaller than q.
Note that if
$\Xi^{N, t}_{m, r}=\emptyset$
for some
$r\geq 1$
, then
$\Xi^{N,t}_{m, q}=\emptyset$
for all
$q\geq r$
. Let
\begin{align*}\Gamma_{m}^{N, t}=\bigcup_{r=0}^{N-1}\Xi^{N, t}_{m , r}.\end{align*}
According to definitions of
$\{\Xi^{N, t}_{m , r}\}_{1\leq r\leq N-1}$
and
$\Gamma_{m}^{N, t}$
,
$\bigcup_{r=0}^q\Xi^{N, t}_{m , r}$
is the set of vertices
$\varphi$
such that there is a semi-infection path before moment t from
$\varphi$
to m with length at most q and
$\Gamma_{m}^{N, t}$
is the set of vertices
$\varphi$
such that there is a semi-infection path before moment t from
$\varphi$
to m. Therefore,
$\xi_t^N(m)$
is determined by values of
$\{\xi_0^N(\varphi)\}_{\varphi\in \Gamma_{m}^{N, t}}$
,
$\{K_\varphi^N\}_{\varphi\in \Gamma_{m}^{N, t}}$
and
$\{U^N_{(\varphi, \varrho)}\;:\;\varphi,\varrho\in \Gamma_{m}^{N, t}, \varrho\neq \varphi\}$
since according to these values we can deduce whether infection path before moment t ending at m exists and which infection path before moment t ending at m makes
$\sum_{q=0}^{n-1}U^N_{(m_{q+1}, m_q)}$
minimum if such path exists. Consequently, we can write
\begin{align*}\xi_t^N(m)=Q_m\big(\Gamma_{m}^{N, t}, \big\{\xi_0^N(\varphi)\big\}_{\varphi\in \Gamma_{m}^{N, t}}, \big\{K_\varphi^N\big\}_{\varphi\in \Gamma_{m}^{N, t}}, \big\{U^N_{(\varphi, \varrho)}\;:\;\varphi,\varrho\in \Gamma_{m}^{N, t}, \varrho\neq \varphi\big\}\big),\end{align*}
where the precise expression of the function
$Q_m$
is clear but a little complex so we omit here.
For given four different
$i,j,k,l\in \{1,2,\ldots, N\}$
, let
$\{\Gamma_{m}^{N, t}\;:\;m=i,j,k,l\}$
be defined as above. Now we define the copy
$\hat{\Gamma}_m^{N,t}$
of
$\Gamma_m^{N, t}$
for
$m=j,k,l$
. For
$m=j,k,l$
, let
\begin{align*}\hat{\Gamma}_{m}^{N, t}=\bigcup_{r=0}^{N-1}\hat{\Xi}^{N, t}_{m , r},\end{align*}
where
$\{\hat{\Xi}^{N, t}_{m, r}\}_{r=0}^{N-1}$
is defined by induction as follows:
-
(1)
$\hat{\Xi}^{N, t}_{m, 0}=\{m\}$
; -
(2) after
$\hat{\Xi}^{N,t}_{m, r}$
are defined for
$0\leq r\leq q < N-1$
,
$\hat{\Xi}^{N, t}_{m, q+1}$
is defined as the set of
$\varphi\in \{1,2,\ldots, N\}\setminus(\bigcup_{r=0}^q\hat{\Xi}^{N,t}_{m, r})$
such that there exists a self-avoiding path such that
\begin{align*}\varphi(\!=m_{q+1}), m_q, \ldots, m_1, m(\!=m_0)\end{align*}
$m_r\in \bigcup_{l=1}^r\hat{\Xi}^{N, t}_{m,l}$
for
$1\leq r\leq q$
and where for
\begin{align*}\sum_{r=0}^{q}\tilde{U}^{N, m}_{(m_r, m_{r+1})}\leq t,\end{align*}
$m=j,k,l$
and all
$1\leq \varphi\neq \varrho\leq N$
,
\begin{align*} \tilde{U}^{N, m}_{(\varphi, \varrho)}=\left\{\begin{array}{l@{\quad}l} U^N_{(\varphi, \varrho)} & \text{ if } m=j\text{ and }\varphi\not\in \Gamma_{i}^{N, t}, \\[5pt] U^N_{(\varphi, \varrho), 2} & \text{ if } m=j\text{ and }\varphi \in \Gamma_{i}^{N, t}, \\[5pt] U^N_{(\varphi, \varrho)} & \text{ if } m=k\text{ and }\varphi \not\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}, \\[5pt] U^N_{(\varphi, \varrho), 3} & \text{ if } m=k\text{ and }\varphi \in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t},\\[5pt] U^N_{(\varphi, \varrho)} & \text{ if } m=l\text{ and }\varphi\not\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}\bigcup \hat{\Gamma}_{k}^{N, t}, \\[5pt] U^N_{(\varphi, \varrho), 4} & \text{ if } m=l\text{ and }\varphi\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}\bigcup \hat{\Gamma}_{k}^{N, t}.\end{array}\right.\end{align*}
The random sets
$\Gamma_i^{N,t}, \hat{\Gamma}_j^{N, t}, \hat{\Gamma}_k^{N, t}, \hat{\Gamma}_l^{N, t}$
can be obtained from a breadth-first search (BFS). In detail, to obtain
$\Xi^{N, t}_{i, 1}$
, we check the values of
$U^N_{(i, \varphi)}$
for all
$\varphi\neq i$
and then
$\Xi^{N, t}_{i, 1}$
is the set of
$\varphi$
satisfying
$U^N_{(i, \varphi)}\leq t$
. Then to obtain
$\Xi_{i, 2}^{N, t}$
, we check the values of
$U^N_{(\varphi, \varrho)}$
for all
$\varphi\in \Xi^{N, t}_{i, 1}, \varrho\neq \varphi$
and then
$\Xi^{N, t}_{i, 2}$
is the set of
$\varrho\not\in \{i\}\bigcup\Xi^{N, t}_{i, 1}$
such that there exists
$\varphi\in \Xi^{N, t}_{i, 1}$
such that
$U^N_{(i, \varphi)}+U^N_{(\varphi, \varrho)}\leq t$
. In general, in the rth step of the BFS, we check the values of
$U^N_{(\varphi, \varrho)}$
for all
$\varphi\in \bigcup_{q=0}^{r-1}\Xi^{N, t}_{i, q}, \varrho\neq \varphi$
and then
$\Xi^{N, t}_{i, r}$
is the set of
$\varrho\in \{1,2,\ldots, N\}\setminus(\bigcup_{q=0}^{r-1}\Xi^{N, t}_{i, q})$
such that there exists a self-avoiding path
\begin{align*}\varrho(\!=m_{r}), m_{r-1}, \ldots, m_1, m(\!=m_0)\end{align*}
such that
$m_q\in \bigcup_{n=1}^q\Xi^{N, t}_{i,n}$
for
$1\leq q\leq r-1$
and
\begin{align*}\sum_{q=0}^{r-1}U^N_{(m_q, m_{q+1})}\leq t.\end{align*}
After
$\Gamma_i^{N, t}$
is obtained, we can obtain
$\hat{\Gamma}_j^{N, t}$
according to the same BFS procedure. However, in the second BFS, if some
$\varphi\in \Gamma_i^{N, t}$
is included in
$\bigcup_{q=1}^r\hat{\Xi}^{N, t}_{j, q}$
, then in the
$r+1$
th step, we check the value of
$U^N_{(\varphi, \varrho), 2}$
for
$\varrho\neq \varphi$
instead of
$U^N_{(\varphi, \varrho)}$
. Similarly, in the third (respectively, fourth) BFS to obtain
$\hat{\Gamma}_k^{N, t}$
(respectively,
$\hat{\Gamma}_l^{N, t}$
), if some
$\varphi\in \Gamma_i^{N, t}\bigcup\hat{\Gamma}_j^{N, t}$
(respectively,
$\varphi\in \Gamma_i^{N, t}\bigcup\hat{\Gamma}_j^{N, t}\bigcup \hat{\Gamma}_k^{N, t}$
) is included in
$\bigcup_{q=1}^r\hat{\Xi}^{N, t}_{k, q}$
(respectively,
$\bigcup_{q=1}^r\hat{\Xi}^{N, t}_{l, q}$
), then in the
$r+1$
th step, we check the value of
$U^N_{(\varphi, \varrho), 3}$
(respectively,
$U^N_{(\varphi, \varrho), 4}$
) for
$\varrho\neq \varphi$
.
For
$m=j,k,l$
, let
$\hat{\xi}_t^N(m)$
be determined by values of
$\{\tilde{\xi}_0^{N, m}(\varphi)\}_{\varphi\in \hat{\Gamma}_{m}^{N, t}}$
,
$\{\tilde{K}_\varphi^{N, m}\}_{\varphi\in \hat{\Gamma}_{m}^{N, t}}$
and
$\{\tilde{U}^{N,m}_{(\varphi, \varrho)}\;:\;\varphi\in \hat{\Gamma}_{m}^{N, t}, \varrho\neq \varphi\}$
in the same way as
$\xi_t^N(m)$
is determined by values of
$\{\xi_0^N(\varphi)\}_{\varphi\in \Gamma_{m}^{N, t}}$
,
$\{K_\varphi^N\}_{\varphi\in \Gamma_{m}^{N, t}}$
and
$\{U^N_{(\varphi, \varrho)}\;:\;\varphi\in \Gamma_{m}^{N, t}, \varrho\neq \varphi\}$
, where
\begin{align*} \tilde{\xi}_0^{N, m}(\varphi)=\left\{\begin{array}{l@{\quad}l} \xi_0^N(\varphi) & \text{ if } m=j\text{ and }\varphi\not\in \Gamma_{i}^{N, t}, \\[5pt] \xi_{0,2}^N(\varphi) & \text{ if } m=j\text{ and }\varphi \in \Gamma_{i}^{N, t}, \\[5pt] \xi_0^N(\varphi) & \text{ if } m=k\text{ and }\varphi \not\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}, \\[5pt] \xi_{0,3}^N(\varphi) & \text{ if } m=k\text{ and }\varphi \in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t},\\[5pt] \xi_0^N(\varphi) & \text{ if } m=l\text{ and }\varphi\not\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}\bigcup \hat{\Gamma}_{k}^{N, t}, \\[5pt] \xi_{0,4}^N(\varphi) & \text{ if } m=l\text{ and }\varphi\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}\bigcup \hat{\Gamma}_{k}^{N, t}\end{array}\right. \end{align*}
and
\begin{align*} \tilde{K}_\varphi^{N, m}=\left\{\begin{array}{l@{\quad}l} K_\varphi^N & \text{ if } m=j\text{ and }\varphi\not\in \Gamma_{i}^{N, t}, \\[5pt] K_{\varphi,2}^N & \text{ if } m=j\text{ and }\varphi \in \Gamma_{i}^{N, t}, \\[5pt] K_\varphi^N & \text{ if } m=k\text{ and }\varphi \not\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}, \\[5pt] K_{\varphi,3}^N & \text{ if } m=k\text{ and }\varphi \in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t},\\[5pt] K_\varphi^N & \text{ if } m=l\text{ and }\varphi\not\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}\bigcup \hat{\Gamma}_{k}^{N, t}, \\[5pt] K_{\varphi,4}^N & \text{ if } m=l\text{ and }\varphi\in \Gamma_{i}^{N, t}\bigcup \hat{\Gamma}_{j}^{N, t}\bigcup \hat{\Gamma}_{k}^{N, t},\end{array}\right.\end{align*}
i.e.
\begin{align*}\hat{\xi}_t^N(m)=Q_m\big(\hat{\Gamma}_{m}^{N, t}, \big\{\tilde{\xi}_0^{N, m}(\varphi)\big\}_{\varphi\in \hat{\Gamma}_{m}^{N, t}}, \big\{\tilde{K}_\varphi^{N, m}\big\}_{\varphi\in \hat{\Gamma}_{m}^{N, t}}, \big\{\tilde{U}^{N,m}_{(\varphi, \varrho)}\;:\;\varphi,\varrho\in \hat{\Gamma}_{m}^{N, t}, \varrho\neq \varphi\big\}\big)\end{align*}
for
$m=j,k,l$
. The following lemma is crucial for us to prove Lemma 3.1.
Lemma 3.2. Let i, j, k, l be four different elements in
$\{1,2,\ldots,N\}$
, then
$\hat{\xi}_t^N(m)$
and
$\xi_t^N(m)$
have the same distribution for
$m=j,k,l$
. Furthermore,
\begin{align*}\xi_t^N(i), \hat{\xi}_t^N(j), \hat{\xi}_t^N(k), \hat{\xi}_t^N(l)\end{align*}
are independent.
Proof of Lemma
3.2. We only show that
$\hat{\xi}_t^N(j),\xi_t^N(j)$
have the same distribution and
$\xi_t^N(i), \hat{\xi}_t^N(j)$
are independent since other parts of Lemma 3.2 follow from similar analyses.
For any
$B\subseteq\{1,2,\ldots, N\}$
such that
$B\ni i$
, if after the first BFS we obtain
$\Gamma_i^{N,t}=B$
, then in this BFS we only utilize the information of
$\{U^N_{(\varphi, \varrho)}\;:\;\varphi\in B, \varrho\neq \varphi\}$
, i.e. the event
$\{\Gamma_i^{N,t}=B\}$
is measurable with respect to the sigma-algebra generated by
$\{U^N_{(\varphi, \varrho)}\;:\;\varphi\in B, \varrho\neq \varphi\}$
. When
$\Gamma_i^{N,t}=B$
,
$\xi_t^N(i)$
is determined by
\begin{align*}\big\{\xi_0^N(\varphi)\big\}_{\varphi\in B}, \big\{K_\varphi^N\big\}_{\varphi\in B}, \big\{U^N_{(\varphi, \varrho)}\;:\;\varphi,\varrho\in B, \varrho\neq \varphi\big\}.\end{align*}
Conditioned on
$\Gamma_i^{N,t}=B$
,
$\hat{\xi}_t^N(j)$
is determined by
\begin{align*}\big\{\xi_{0,2}^N(\varphi)\big\}_{\varphi\in B}, \big\{\xi_0^N(\varphi)\big\}_{\varphi\not\in B}, \big\{K_{\varphi,2}^N\big\}_{\varphi\in B}, \big\{K_\varphi^N\big\}_{\varphi\not \in B},\big\{U^N_{(\varphi, \varrho),2}\;:\;\varphi\in B, \varrho\neq \varphi\big\}\end{align*}
and
$\{U^N_{(\varphi, \varrho)}\;:\;\varphi\not\in B, \varrho\neq \varphi\}$
, all information of which are not utilized in the first BFS to obtain
$\Gamma_i^{N,t}=B$
and the value of
$\xi_t^N(i)$
. Hence,
\begin{align}&\mathrm{P}\Big(\hat{\xi}_t^N(j)=b, \xi_t^N(i)=c\Big|\Gamma_i^{N,t}=B\Big)=\mathrm{P}\Big(\hat{\xi}_t^N(j)=b\Big|\Gamma_i^{N,t}=B\Big)\mathrm{P}\Big(\xi_t^N(i)=c\Big|\Gamma_i^{N,t}=B\Big) \end{align}
for any
$b,c\in \{-1, 0, 1\}$
. Furthermore,
\begin{align*}\left\{\xi_{0,2}^N(\varphi)\right\}_{\varphi\in B}\bigcup\left\{\xi_0^N(\varphi)\right\}_{\varphi\not\in B} \quad \text{and} \quad \left\{\xi_0^N(\varphi)\right\}_{\varphi\in \{1,2\ldots, N\}}\end{align*}
have the same distribution. Similarly,
\begin{align*}\big\{K_{\varphi,2}^N\big\}_{\varphi\in B}\bigcup\left\{K_\varphi^N\right\}_{\varphi\not \in B} \quad \text{and} \quad \left\{K_\varphi^N\right\}_{\varphi\in \{1,2,\ldots,N\}}\end{align*}
have the same distribution and
\begin{align*}\big\{U^N_{(\varphi, \varrho),2}\;:\;\varphi\in B, \varrho\neq \varphi\big\}\bigcup \big\{U^N_{(\varphi, \varrho)}\;:\;\varphi\not\in B, \varrho\neq \varphi\big\} \text{ and }\big\{U^N_{(\varphi, \varrho)}\;:\;\varrho\neq \varphi\big\}\end{align*}
have the same distribution. Therefore, utilizing the fact that
$\Gamma_i^{N,t}=B$
is measurable with respect to
$\{U^N_{(\varphi, \varrho)}\;:\;\varphi\in B, \varphi\neq \varrho\}$
, we have
\begin{equation}\mathrm{P}\big(\hat{\xi}_t^N(j)=b\big|\Gamma_i^{N,t}=B\big)=\mathrm{P}\big(\xi_t^N(j)=b\big)\end{equation}
for any
$b\in \{-1,0,1\}$
. By Equations (3.4) and (3.5), an application of the total probability formula shows that
$\xi_t^N(i), \hat{\xi}_t^N(j)$
are independent and
$\hat{\xi}_t^N(j),\xi_t^N(j)$
have the same distribution.
Now we can give our proof of Lemma 3.1.
Proof of Lemma
3.1. Throughout this proof we assume that i, j, k, l are four given different vertices in
$\{1,2,\ldots, N\}$
and
$0\leq t\leq T$
. We only give the proof of Equation (3.3) since the other two follow from similar analysis.
For any
$B\subset \{1,2,\ldots, N\}$
and
$m\not\in B$
, we denote by
$\Xi^{N,t,B}_{m, r}$
the result of the rth step of the BFS as that giving
$\Gamma_m^{N,t}$
except that vertices in B are not investigated, i.e. we have:
-
(1)
$\Xi^{N, t, B}_{m, 0}=\{m\}$
; -
(2) after
$\Xi^{N,t, B}_{m, r}$
are defined for
$0\leq r\leq q < N-1$
,
$\Xi^{N, t, B}_{m, q+1}$
is defined as the set of
$\varphi\in (\{1,2,\ldots, N\}\setminus B)\setminus(\bigcup_{r=0}^q\Xi^{N, t, B}_{m, r})$
such that there exists a self-avoiding path such that
\begin{align*}\varphi(\!=m_{q+1}), m_q, \ldots, m_1, m(\!=m_0)\end{align*}
$m_r\in \bigcup_{l=1}^r\Xi^{N, t, B}_{m,l}$
for
$1\leq r\leq q$
and
\begin{align*}\sum_{r=0}^{q}U^N_{(m_r, m_{r+1})}\leq t.\end{align*}
We further define
$\Gamma_m^{N, t, B}=\bigcup_{r=0}^{N-1}\Xi^{N,t,B}_{m, r}$
, i.e.,
$\Gamma_m^{N, t, B}$
is the set of vertices
$\varphi$
such that there is a semi-infection path before moment t from
$\varphi$
to m avoiding B. For simplicity, we denote
\begin{align*}B_1^N=\Gamma_i^{N, t, \{j, k, l\}}, B_2^N=\Gamma_j^{N, t, B_1^N\bigcup\{k,l\}}, B_3^N=\Gamma_k^{N, t, \bigcup_{m=1}^2B_m^N\bigcup \{l\}}\end{align*}
and
\begin{align*}B_4^N=\Gamma_l^{N, t, \bigcup_{m=1}^3B_m^N}.\end{align*}
We denote by
$\Omega_{N, 1}$
the event that
$U^N_{(\varphi, \varrho)}>T$
for any
$(\varphi, \varrho)\in B_{c_1}^N\times B_{c_2}^N$
for all
$1\leq c_2 < c_1\leq 4$
, by
$\Omega_{N, 2}$
the event that
$U^N_{(\varphi, j)}>T$
for any
$\varphi\in B_1^N$
, by
$\Omega_{N, 3}$
the event that
$U^N_{(\varphi, k)}>T$
for any
$\varphi\in B_1^N\bigcup B_2^N$
, by
$\Omega_{N, 4}$
the event that
$U^N_{(\varphi, l)}>T$
for any
$\varphi\in B_1^N\bigcup B_2^N\bigcup B_3^N$
and by
$\Omega_N$
the intersection of
$\Omega_{N, m}$
for
$m=1,2,3,4$
. On the event
$\Omega_N$
,
\begin{align*}B_1^N=\Gamma_i^{N, t}, \quad B_2^N=\Gamma_j^{N, t}=\hat{\Gamma}_j^{N, t}, \quad B_3^N=\Gamma_k^{N, t}=\hat{\Gamma}_k^{N, t},\end{align*}
and
\begin{align*}B_4^N=\Gamma_l^{N, t}=\hat{\Gamma}_l^{N, t}.\end{align*}
Consequently, on the event
$\Omega_N$
,
$\Gamma_{d_1}^{N,t}\bigcap \Gamma_{d_2}^{N, t}=\emptyset$
for any different
$d_1, d_2\in \{i,j,k,l\}$
and
$\hat{\xi}_t^N(m)=\xi_t^N(m)$
for
$m=j,k,l$
. Then by Lemma 3.2, for
$(r_1, r_2, r_3, r_4)=(1,1,1,1)$
,
\begin{align}&\left|\mathrm{E}\!\left(H_{t, r_1}^N(i)H_{t, r_2}^N(j)H_{t, r_3}^N(k)H_{t, r_4}^N(l)\right)-\mathrm{E}H_{t, r_1}^N(i)\mathrm{E}H_{t, r_2}^N(j)\mathrm{E}H_{t, r_3}^N(k)\mathrm{E}H_{t, r_4}^N(l)\right| \notag\\[5pt] &=\Bigg|P\!\left(\xi_t^N(m)=1\text{ for }m=i,j,k,l\right)-\prod_{m=i,j,k,l}P\!\left(\xi_t^N(m)=1\right)\Bigg| \notag\\[5pt] &=\Bigg|P\!\left(\xi_t^N(m)=1\text{ for }m=i,j,k,l\right)-P(\xi_t^N(i)=1)\prod_{m=j,k,l}P\big(\hat{\xi}_t^N(m)=1\big)\Bigg|\notag\\[5pt] &=\big|P\big(\xi_t^N(m)=1\text{ for }m=i,j,k,l\big)-P\big(\xi_t(i)=1, \hat{\xi}_t^N(m)=1\text{ for }m=j,k,l\big)\big|\notag\\[5pt] &\leq \big|P\big(\xi_t^N(m)=1\text{ for }m=i,j,k,l, \Omega_N^c\big)\notag\\[5pt] &\quad\quad\quad-P\big(\xi_t(i)=1, \hat{\xi}_t^N(m)=1\text{ for }m=j,k,l, \Omega_N^c\big)\big|\notag\\[5pt] &\leq 2P(\Omega_N^c).\end{align}
Equation (3.6) holds for other
$(r_1, r_2, r_3, r_4)\in \{1, 0\}^4$
according to a similar analysis. For any
$\varrho\neq\varphi$
,
\begin{align*}P\big(U^N_{(\varrho, \varphi)}\leq T\big)=1-e^{-\frac{1}{N}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)T}\leq \frac{\|\lambda\|_\infty T}{N},\end{align*}
where
$\|\lambda\|_\infty=\sup_{0\leq u,v\leq 1}|\lambda(u, v)|$
. Note that information of
\begin{align*}\big\{U^N_{(\varphi, \varrho)}\;:\;(\varphi, \varrho)\in B_{c_1}^N\times B_{c_2}^N\text{ for some }1\leq c_2 < c_1\leq 4\big\}, \big\{U^N_{(\varphi, j)}\big\}_{\varphi\in B_1^N},\end{align*}
$\{U^N_{(\varphi, k)}\}_{\varphi\in B_1^N\bigcup B_2^N}$
and
$\{U^N_{(\varphi, l)}\}_{\varphi\in B_1^N\bigcup B_2^N\bigcup B_3^N}$
are not utilized in the definition of
$B_1^N, B_2^N, B_3^N, B_4^N$
, as a result,
\begin{align*}P(\Omega_N^c)\leq \frac{\|\lambda\|_\infty T}{N}\mathrm{E}\!\left(3|B_1^N|+2|B_2^N|+|B_3^N|+\sum_{1\leq c_2 < c_1\leq 4}|B_{c_2}^N||B_{c_1}^N|\right)\!,\end{align*}
where
$|A|$
is the cardinality of the set A. For any self-avoiding path
$m_0, m_1,\ldots, m_q$
on
$\{1,2,\ldots,N\}$
with length q and
$\tau>0$
,
\begin{align*}P\!\left(\sum_{r=0}^{q-1}U^N_{(m_r, m_{r+1})}\leq t\right)\leq {\mathrm{e}}^{\tau t}\prod_{r=0}^{q-1}\mathrm{E} {\mathrm{e}}^{-\tau U^N_{(m_r, m_{r+1})}}\leq {\mathrm{e}}^{\tau t}\!\left(\frac{\|\lambda\|_\infty}{N\tau+\|\lambda\|_\infty}\right)^q.\end{align*}
Let
$\tau=2\|\lambda\|_\infty$
, then for any
$m\in \{1,2,\ldots, N\}$
,
\begin{align*}\mathrm{E}|\Gamma_m^{N, t}|\leq \sum_{q=0}^{+\infty}N^q {\mathrm{e}}^{2\|\lambda\|_\infty T}\!\left(\frac{1}{2N}\right)^q=2 {\mathrm{e}}^{2\|\lambda\|_\infty T}.\end{align*}
For any
$A\subseteq\{1,2,\ldots,N\}\setminus\{j,k,l\}$
such that
$A\ni i$
,
$\{B_1^N=A\}$
is measurable with respect to the sigma-algebra generated by
$\{U^N_{\varphi, \varrho}\;:\;\varphi\in A, \varrho\neq \varphi\}$
. Therefore,
\begin{align*}\mathrm{E}\big(\big|B_2^N\big|\big|B_1^N=A\big)&=\mathrm{E}\Big(|\Gamma_j^{N,t,A\bigcup\{k,l\}}|\Big|B_1^N=A\Big)\\[5pt] &=\mathrm{E}|\Gamma_j^{N,t,A\bigcup\{k,l\}}|\leq \mathrm{E}|\Gamma_j^{N,t}|\end{align*}
and, hence,
\begin{align*}\mathrm{E}\!\left(\big|B_1^N\big|\big|B_2^N\big|\right)&\leq \big(\mathrm{E}\big|\Gamma_i^{N,t}\big|\big)\big(\mathrm{E}\big|\Gamma^{N, t}_j\big|\big)\leq 4{\mathrm{e}}^{4\|\lambda\|_\infty T}.\end{align*}
The above inequality holds for other
$m_2<m_1$
according to similar analysis. As a result,
\begin{equation}P(\Omega_N^c)\leq \frac{\|\lambda\|_\infty T}{N}\big(12{\mathrm{e}}^{2\|\lambda\|_\infty T}+24{\mathrm{e}}^{4\|\lambda\|_\infty T}\big).\end{equation}
4. Proof of Theorem 2.1
In this section we prove Theorem 2.1. It is easy to check that Theorem 2.1 follows from the following two lemmas.
Lemma 4.1. For any
$f\in \hat{C}^1(\mathbb{T})$
,
\begin{align*}\lim_{N\rightarrow+\infty}\textrm{Var}\!\left(\mu_t^N(\,f)\right)=\lim_{N\rightarrow+\infty}\textrm{ Var}\!\left(\theta_t^N(\,f)\right)=0.\end{align*}
Lemma 4.2. For any
$f\in \hat{C}^1(\mathbb{T})$
,
\begin{align*}\lim_{N\rightarrow+\infty}\mathrm{E}\!\left(\mu_t^N(\,f)\right)=\int_0^1\rho_1(t,u)\,f(u)\,{\mathrm{d}}u\end{align*}
and
\begin{align*}\lim_{N\rightarrow+\infty}\mathrm{E}\!\left(\theta_t^N(\,f)\right)=\int_0^1\rho_0(t, u)\,f(u)\,{\mathrm{d}}u.\end{align*}
Proof of Theorem
2.1. According to the fact that
$(c+d)^2\leq 2c^2+2d^2$
for
$c,d\in \mathbb{R}$
,
\begin{align*}&\left(\mu_t^N(\,f)-\int_0^1\rho_1(t,u)\,f(u)\,{\mathrm{d}}u\right)^2\\[5pt] &\leq 2\left(\mu_t^N(\,f)-\mathrm{E}\!\left(\mu_t^N(\,f)\right)\right)^2+2\!\left(\mathrm{E}\!\left(\mu_t^N(\,f)\right)-\int_0^1\rho_1(t,u)\,f(u)\,{\mathrm{d}} u\right)^2.\end{align*}
Then, by Lemmas 4.1 and 4.2,
$\mu_t^N(\,f)$
converges to
$\int_0^1\rho_1(t,u)\,f(u)\,{\mathrm{d}}u$
in
$L^2$
and, hence, in probability as
$N\rightarrow+\infty$
. Similar analysis holds for
$\theta_t^N(\,f)$
.
The remainder of this section is devoted to proofs of Lemmas 4.1 and 4.2.
Proof of Lemma
4.1. Since T can be chosen arbitrarily, we assume that
$0 < t\leq T$
without loss of generality. By Lemma 3.1,
\begin{align*}&\textrm{Var}\!\left(\mu_t^N(\,f)\right)=\frac{1}{N^2}\sum_{i=1}^N\sum_{j=1}^N\textrm{Cov}\!\left(I_t^N(i), I_t^N(j)\right)\,f\!\left(\frac{i}{N}\right)\,f\!\left(\frac{j}{N}\right)\\[5pt] &\leq \frac{\|f\|_\infty^2}{N^2}\sum_{i=1}^N\sum_{j=1}^N\!\left|\mathrm{E}\!\left(H_{t, 1}^N(i)H_{t, 1}^N(j)\right)-\mathrm{E}H_{t, 1}^N(i)\mathrm{E}H_{t, 1}^N(j)\right|\\[5pt] &=\frac{\|f\|_\infty^2}{N^2}\sum_{i=1}^N\sum_{j\neq i}\!\left|\mathrm{E}\!\left(H_{t, 1}^N(i)H_{t, 1}^N(j)\right)-\mathrm{E}H_{t, 1}^N(i)\mathrm{E}H_{t, 1}^N(j)\right|\\[5pt] &\quad\quad+\frac{\|f\|_\infty^2}{N^2}\sum_{i=1}^N\!\left|\mathrm{E}\!\left(H_{t, 1}^N(i)H_{t, 1}^N(i)\right)-\mathrm{E}H_{t, 1}^N(i)\mathrm{E}H_{t, 1}^N(i)\right|\\[5pt] &\leq \frac{2N\|f\|_\infty^2}{N^2}+\frac{\|f\|_\infty^2N(N-1)C_1}{N^3}=\|f\|_\infty^2\left(\frac{(N-1)C_1}{N^2}+\frac{2}{N}\right)\!,\end{align*}
where
$\|f\|_\infty=\sup_{0\leq u\leq 1}|f(u)|$
. As a result,
\begin{align*}\lim_{N\rightarrow+\infty}\textrm{Var}\!\left(\mu_t^N(\,f)\right)=0.\end{align*}
The second part of Lemma 4.1 that
$\lim_{N\rightarrow+\infty}\textrm{Var}(\theta_t^N(\,f))=0$
follows from a similar analysis.
Proof of Lemma
4.2. We still assume that
$0 < t\leq T$
without loss of generality. For
$N\geq 1$
,
$1\leq i\leq N$
and any
$0\leq t\leq T$
, let
\begin{align*}\rho_1^N(t, u)=P(I_t^N(i)=1)\text{ and }\rho_0^N(t, u)=P(S_t^N(i)=1)\end{align*}
when
$({i-1})/{N}<u\leq {i}/{N}$
. Then, to prove Lemma 4.2 we only need to show that
\begin{equation}\lim_{N\rightarrow+\infty}\sup_{0\leq t\leq T}\sup_{0 < u\leq 1}\!\left|\rho_m^N(t, u)-\rho_m(t, u)\right|=0\end{equation}
for
$m=0, 1$
. According to the generator
$\mathcal{L}^N$
of
$\{\xi_t^N\}_{t\geq 0}$
given in Section 1 and Chapman–Kolmogorov equation, for any
$1\leq i\leq N$
and
$({i-1})/{N}<u\leq {i}/{N}$
,
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_1^N(t, u)&=\frac{{\mathrm{d}}}{{\mathrm{d}}t}P\big(I_t^N(i)=1\big) \\[5pt] &=-\psi\!\left(\frac{i}{N}\right)P\big(I_t^N(i)=1\big)+\frac{1}{N}\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)P\big(S_t^N(i)=1, I_t^N(j)=1\big)\end{align*}
and
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_0^N(t, u)&=\frac{{\mathrm{d}}}{{\mathrm{d}}t}P\big(S_t^N(i)=1\big) \\[5pt] &=-\frac{1}{N}\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)P\big(S_t^N(i)=1, I_t^N(j)=1\big).\end{align*}
For all
$N\geq 1$
, let
\begin{align*}\Delta_\lambda^N=\sup\big\{|\lambda(u_1, v_1)-\lambda(u_2, v_2)|:\;|u_1-u_2|, |v_1-v_2|\leq N^{-1}\big\},\end{align*}
\begin{align*}\Delta_\psi^N=\sup\big\{|\psi(u_1)-\psi(u_2)|:\;|u_1-u_2|\leq N^{-1}\big\}\end{align*}
and
\begin{align*}\Delta_\phi^N=\sup\big\{|\phi(u_1)-\phi(u_2)|:\;|u_1-u_2|\leq N^{-1}\big\}.\end{align*}
By Lemma 3.1,
$|P(S_t^N(i)=1, I_t^N(j)=1)-\rho_0^N(t, {i}/{N})\rho_t^N(t, {j}/{N})|\leq {C_1}/{N}$
for
$i\neq j$
and hence
\begin{align*}&\left|\frac{1}{N}\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)P\big(S_t^N(i)=1, I_t^N(j)=1\big)-\rho_0^N(t,u)\int_0^1\lambda(u, v)\rho_1^N(t,v)dv\right|\\[5pt] &\leq \Delta_\lambda^N+\frac{C_1\|\lambda_\infty\|}{N}+\frac{\|\lambda\|_\infty}{N}\end{align*}
and
$|\psi({i}/{N})P(I_t^N(i)=1)-\psi(u)\rho_1^N(t,u)|\leq \Delta_\psi^N$
for
$({i-1})/{N}<u\leq {i}/{N}$
. For all
$N\geq 1$
, let
\begin{align*}M_t^N=\sup_{0 < u\leq 1}\!\left\{|\rho_0^N(t,u)-\rho_0(t,u)|+|\rho_1^N(t,u)-\rho_1(t, u)|\right\},\end{align*}
then
\begin{align*}|\rho_0^N(t,u)\lambda(u, v)\rho_1^N(t,v)-\rho_0(t,u)\lambda(u, v)\rho_1(t,v)|\leq 2\|\lambda\|_\infty M_t^N\end{align*}
for any
$0 < u,v\leq 1$
. According to Assumption (A),
$M_0^N\leq 2\Delta_\phi^N$
. In conclusion, by writing Equation (2.2) and above two Chapman–Kolmogorov Equations as their integration forms, we have
\begin{align*}M_t^N\leq \!\left(2\Delta_\phi^N+T\!\left(\Delta_\psi^N+2\Delta_\lambda^N+\frac{2C_1\|\lambda_\infty\|}{N}+\frac{2\|\lambda\|_\infty}{N}\right)\right)+4\|\lambda\|_\infty\int_0^tM_s^N\,{\mathrm{d}}s\end{align*}
for
$0\leq t\leq T$
. As a result, according to Gronwall’s inequality,
\begin{align*}M_t^N\leq \!\left(2\Delta_\phi^N+T\!\left(\Delta_\psi^N+2\Delta_\lambda^N+\frac{2C_1\|\lambda_\infty\|}{N}+\frac{2\|\lambda\|_\infty}{N}\right)\right)e^{4\|\lambda\|_\infty T}\end{align*}
for
$0\leq t\leq T$
. Consequently,
$\lim_{N\rightarrow+\infty}\sup_{0\leq t\leq T}M_t^N=0$
and, hence, Equation (4.1) holds.
5. Proof of Theorem 2.2
In this section we prove Theorem 2.2. As a preliminary, we first need the tightness of
$\{(\eta_t^N, \beta_t^N)\;:\;0\leq t\leq T\}_{N\geq 1}$
.
Lemma 5.1. Under Assumption (A),
$\{(\eta_t^N, \beta_t^N)\;:\;0\leq t\leq T\}_{N\geq 1}$
are tight with respect to the Skorohod topology of
$\mathcal{D}([0, T], ((\hat{C}^1(\mathbb{T}))^\prime)^2)$
.
Proof. Since
$\hat{C}^1(\mathbb{T})$
is a nuclear Fréchet space, according to Theorem 4.1 of [Reference Mitoma10] and Aldous’ criteria, Lemma 5.1 follows from the following four equations.
-
(1) For all
$0\leq t\leq T$
and
$f\in \hat{C}^1(\mathbb{T})$
, (5.1)
\begin{equation}\lim_{M\rightarrow+\infty}\limsup_{N\rightarrow+\infty}P\!\left(\left|\beta_t^N(\,f)\right|\geq M\right)=0.\end{equation}
-
(2) For all
$\epsilon>0$
and
$f\in C^{\infty}(\mathbb{T})$
, (5.2)where
\begin{equation}\lim_{\delta\rightarrow0}\limsup_{N\rightarrow+\infty}\sup_{\varsigma\in \mathcal{T}, s<\delta}P\!\left(\left|\beta^N_{\varsigma+s}(\,f)-\beta_\varsigma^N(\,f)\right|>\epsilon\right)=0,\end{equation}
$\mathcal{T}$
is the set of stopping times of
$\{\xi_t^N\}_{t\geq 0}$
bounded by T.
-
(3) For all
$0\leq t\leq T$
and
$f\in \hat{C}^1(\mathbb{T})$
,
\begin{equation*}\lim_{M\rightarrow+\infty}\limsup_{N\rightarrow+\infty}P\!\left(\left|\eta_t^N(\,f)\right|\geq M\right)=0.\end{equation*}
-
(4) For all
$\epsilon>0$
and
$f\in \hat{C}^1(\mathbb{T})$
,
\begin{equation*}\lim_{\delta\rightarrow0}\limsup_{N\rightarrow+\infty}\sup_{\varsigma\in \mathcal{T}, s<\delta}P\!\left(\left|\eta^N_{\varsigma+s}(\,f)-\eta_\varsigma^N(\,f)\right|>\epsilon\right)=0.\end{equation*}
Here we only check Equations (5.1) and (5.2) since the other two follows from a similar analysis. According to Dynkin’s martingale formula,
\begin{align*}\beta_t^N(\,f)=\beta_0^N(\,f)+\int_0^t\!\left(\partial_s+\mathcal{L}^N\right)\beta_s^N(\,f)\,{\mathrm{d}} s+\mathcal{M}_{t,\beta}^N(\,f),\end{align*}
where
$\{\mathcal{M}_{t,\beta}^N(\,f)\}_{t\geq 0}$
is a martingale with mean zero and quadratic variation process
$\langle\mathcal{M}^N_\beta(\,f)\rangle_t$
given by
\begin{align*}\big\langle\mathcal{M}^N_\beta(\,f)\big\rangle_t=\int_0^t\mathcal{L}^N\big(\big(\beta_s^N(\,f)\big)^2\big)-2\beta_s^N(\,f)\mathcal{L}^N\beta_s^N(\,f)\,{\mathrm{d}}s.\end{align*}
As a result, to prove Equations (5.1) and (5.2), we only need to check the following four equations.
-
(1) For all
$0\leq t\leq T$
and
$f\in \hat{C}^1(\mathbb{T})$
, (5.3)
\begin{equation}\lim_{M\rightarrow+\infty}\limsup_{N\rightarrow+\infty}P\!\left(\left|\int_0^t\!\left(\partial_s+\mathcal{L}^N\right)\beta_s^N(\,f)\,{\mathrm{d}} s\right|\geq M\right)=0.\end{equation}
-
(2) For all
$\epsilon>0$
and
$f\in \hat{C}^1(\mathbb{T})$
, (5.4)
\begin{equation}\lim_{\delta\rightarrow0}\limsup_{N\rightarrow+\infty}\sup_{\varsigma\in \mathcal{T}, s<\delta}P\!\left(\left|\int_\varsigma^{\varsigma+s} \!\left(\partial_u+\mathcal{L}^N\right)\beta_u^N(\,f)\,{\mathrm{d}} u\right|>\epsilon\right)=0.\end{equation}
-
(3) For all
$0\leq t\leq T$
and
$f\in \hat{C}^1(\mathbb{T})$
, (5.5)
\begin{equation}\lim_{M\rightarrow+\infty}\limsup_{N\rightarrow+\infty}P\big(\big|\mathcal{M}_{t, \beta}^N(\,f)\big|\geq M\big)=0.\end{equation}
-
(4) For all
$\epsilon>0$
and
$f\in \hat{C}^1(\mathbb{T})$
, (5.6)
\begin{equation}\lim_{\delta\rightarrow0}\limsup_{N\rightarrow+\infty}\sup_{\varsigma\in \mathcal{T}, s<\delta}P\big(\big|\mathcal{M}_{\varsigma+s, \beta}^N(\,f)-\mathcal{M}_{\varsigma, \beta}^N(\,f)\big|>\epsilon\big)=0.\end{equation}
For Equation (5.3), by direct calculation,
\begin{align*}&\int_0^t\!\left(\partial_s+\mathcal{L}^N\right)\beta_s^N(\,f)\,{\mathrm{d}}s \\[5pt] &=\int_0^t\!\left(\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\left(S_t^N(i)I_t^N(j)-\mathrm{E}\!\left(S_t^N(i)I_t^N(j)\right)\right)\,f\!\left(\frac{i}{N}\right)\right)\,{\mathrm{d}} s.\end{align*}
According to Lemma 3.1, for any different i, j, k, l,
\begin{equation}\left|\textrm{Cov}\!\left(S_t^N(i)I_t^N(j), S_t^N(k)I_t^N(l)\right)\right|\leq \frac{C_3+2C_1}{N}+\frac{C_1^2}{N^2}\leq \frac{C_5}{N},\end{equation}
where
$C_5=C_3+2C_1+C_1^2$
. As a result, by Cauchy–Schwarz inequality,
\begin{align}&\mathrm{E}\!\left(\!\left(\int_0^t\!\left(\partial_s+\mathcal{L}^N\right)\beta_s^N(\,f)ds\right)^2\right)\notag\\[5pt] &\leq T^2\|\lambda\|_\infty^2\|f\|_\infty^2\!\left(\frac{C_5}{N}\frac{N(N-1)(N-2)(N-3)}{N^3}+\frac{6N^3}{N^3}\right) \notag\\[5pt] &\leq T^2\|\lambda\|_\infty^2\|f\|_\infty^2\!\left(C_5+6\right)\!.\end{align}
Equation (5.3) follows from Equation (5.8) and Markov’s inequality.
For Equation (5.4), by Cauchy–Schwarz inequality and Equation (5.7),
\begin{align}{\mathrm{E}}\!\left(\left|\int_\varsigma^{\varsigma+s} \!\left(\partial_u+\mathcal{L}^N\right)\beta_u^N(\,f)\,{\mathrm{d}} u\right|^2\right)&\leq \delta\int_0^{T+\delta}\mathrm{E}\!\left(\!\left(\!\left(\partial_u+\mathcal{L}^N\right)\beta_u^N(\,f)\right)^2\right)\,{\mathrm{d}} u \notag\\[5pt] &\leq \delta(T+\delta)\|\lambda\|_\infty^2\|f\|_\infty^2(C_5+6).\end{align}
Equation (5.4) follows from Equation (5.9) and Markov’s inequality.
For Equation (5.5), by direct calculation,
\begin{align}&\mathcal{L}^N\!\left(\!\left(\beta_s^N(\,f)\right)^2\right)-2\beta_s^N(\,f)\mathcal{L}^N\beta_s^N(\,f) \notag\\[5pt] &=\frac{1}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)S_s^N(i)I_s^N(j)\,f^2\!\left(\frac{i}{N}\right)\leq \|\lambda\|_\infty\|f\|_\infty^2.\end{align}
Hence,
\begin{equation}\mathrm{E}\!\left(\big|\mathcal{M}_{t, \beta}^N(\,f)\big|^2\right)=\mathrm{E}\langle\mathcal{M}^N_\beta(\,f)\rangle_t\leq T\|\lambda\|_\infty\|f\|^2_\infty.\end{equation}
Equation (5.5) follows from Equation (5.11) and Markov’s inequality.
For Equation (5.6), by Equation (5.10),
\begin{align}\mathrm{E}\Big(\big|\mathcal{M}_{\varsigma+s, \beta}^N(\,f)-\mathcal{M}_{\varsigma, \beta}^N(\,f)\big|^2\Big)&=\mathrm{E}\big(\big\langle\mathcal{M}^N_\beta(\,f)\big\rangle_{\varsigma+s}-\big\langle\mathcal{M}^N_\beta(\,f)\big\rangle_{\varsigma}\big) \notag\\[5pt] &\leq \delta \|\lambda\|_\infty\|f\|^2_\infty.\end{align}
Equation (5.6) follows from Equation (5.12) and Markov’s inequality.
In conclusion, Equations (5.3)–(5.6) all hold and, hence, Equations (5.1) and (5.2) hold.
For later use, we need the following lemma.
Lemma 5.2. For any
$h\in C([0, 1]\times[0, 1])$
and
$0\leq t\leq T$
,
\begin{align*}\lim_{N\rightarrow+\infty}\int_0^t\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j\neq i}\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right){\mathrm{d}}s=0\end{align*}
in probability.
Proof. Since
$S_t^N(i), I_t^N(i)\leq 1$
, we only need to show that
\begin{equation}\lim_{N\rightarrow+\infty}\int_0^t\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right){\mathrm{d}}s=0\end{equation}
in probability. Let
$\mathcal{H}$
be the set of
$h\in C([0, 1]\times[0, 1])$
that
$h(u,v)=h_1(u)h_2(v)$
for all
$u,v\in [0, 1]$
for some
$h_1, h_2\in C[0, 1]$
. We first show that Equation (5.13) holds for all
$h\in\mathcal{H}$
and, hence, for all
$h\in\textrm{span}(\mathcal{H})$
. For any
$h\in \mathcal{H}$
,
\begin{align*}&\mathrm{E}\!\left|\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right)\right|\\[5pt] &\quad =\mathrm{E}\!\left|\left(\frac{1}{N}\sum_{i=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)h_1\!\left(\frac{i}{N}\right)\right)\left(\frac{1}{N^{0.5}}\sum_{j=1}^N\!\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h_2\!\left(\frac{j}{N}\right)\right)\right|.\end{align*}
According to Lemma 3.1 and the analysis leading to Lemma 4.1,
\begin{align*}\sup_{0\leq s\leq t}\mathrm{E}\!\left(\!\left(\frac{1}{N}\sum_{i=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)h_1\!\left(\frac{i}{N}\right)\right)^2\right)=O(N^{-1})\end{align*}
and
\begin{align*}\sup_{0\leq s\leq t}\mathrm{E}\!\left(\!\left(\frac{1}{N^{0.5}}\sum_{j=1}^N\!\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h_2\!\left(\frac{j}{N}\right)\right)^2\right)=O(1).\end{align*}
By Cauchy–Schwarz inequality, for
$h\in \mathcal{H}$
,
\begin{equation}\sup_{0\leq s\leq t}\mathrm{E}\!\left|\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right)\right|=O\big(N^{-1/2}\big).\end{equation}
As a result, for
$h\in \mathcal{H}$
, Equation (5.13) follows from Markov’s inequality.
For general h, since
$\textrm{span}(\mathcal{H})$
is dense in
$C([0, 1]\times[0, 1])$
, for any
$\varepsilon>0$
there exists
$h_\epsilon\in \textrm{span}(\mathcal{H})$
such that
$\|h_\epsilon-h\|_\infty\leq \epsilon$
. According to Lemma 3.1, there exists
$C_8<+\infty$
independent of
$N, h, h_\epsilon$
such that
\begin{align*}&\mathrm{E}\!\left(\!\left(\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)(h-h_\epsilon)\left(\frac{i}{N}, \frac{j}{N}\right)\right)^2\right)\\[5pt] &\leq C_8\|h-h_\epsilon\|^2\leq C_8\varepsilon^2\end{align*}
for all
$0\leq s\leq T$
. Therefore, by Equation (5.14) and Hölder’s inequality,
\begin{align*}&\sup_{0\leq s\leq t}\mathrm{E}\!\left|\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right)\right|\\[5pt] &\leq \sqrt{C_8}\varepsilon+O\!\left(N^{-\frac{1}{2}}\right)\!.\end{align*}
Let
$N\rightarrow+\infty$
, then
\begin{align*}&\lim_{N\rightarrow+\infty}\sup_{0\leq s\leq t}\mathrm{E}\!\left|\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right)\right|\\[5pt] &\leq \sqrt{C_8}\varepsilon\end{align*}
and, hence,
\begin{align*}\lim_{N\rightarrow+\infty}\sup_{0\leq s\leq t}\mathrm{E}\!\left|\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j=1}^N\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)h\!\left(\frac{i}{N}, \frac{j}{N}\right)\right|=0\end{align*}
since
$\varepsilon$
is arbitrary. Consequently, for general
$h\in \mathcal{H}$
, Equation (5.13) follows from Markov’s inequality.
At last, we prove Theorem 2.2.
Proof of Theorem 2.2. By Lemma 5.1, any subsequence
\begin{align*}\left\{\left(\eta_t^{N_j}, \beta_t^{N_j}\right)\;:\;0\leq t\leq T\right\}_{j\geq 1}\end{align*}
of
$\{(\eta_t^N, \beta_t^N)\;:\;0\leq t\leq T\}_{N\geq 1}$
has a weakly convergent subsequence
\begin{align*}\left\{\left(\eta_t^{N_{j_k}}, \beta_t^{N_{j_k}}\right)\;:\;0\leq t\leq T\right\}_{k\geq 1}\end{align*}
with respect to the Skorohod topology of
$\mathcal{D}([0, T],((\hat{C}^1(\mathbb{T}))^\prime)^2)$
. Let
$(\hat{\eta}_t, \hat{\beta}_t)_{0\leq t\leq T}$
be the weak limit of
$\{(\eta_t^{N_{j_k}}, \beta_t^{N_{j_k}})\;:\;0\leq t\leq T\}_{k\geq 1}$
, then we only need to show that
$(\hat{\eta}_t, \hat{\beta}_t)_{0\leq t\leq T}$
is the solution to Equation (2.3) with the required initial distribution given in Theorem 2.2. For simplicity, from now on we write
\begin{align*}\left\{\left(\eta_t^{N_{j_k}}, \beta_t^{N_{j_k}}\right)\;:\;0\leq t\leq T\right\}_{k\geq 1}\end{align*}
as
$\{(\eta_t^N, \beta_t^N)\;:\;0\leq t\leq T\}_{N\geq 1} $
.
The fact that
$\hat{\beta}_0=-\hat{\eta}_0$
and
$\hat{\eta}_0(\,f)$
follows
$\mathbb{N}(0, \int_0^1f^2(u)\phi(u)(1-\phi(u))\,{\mathrm{d}}u)$
for all
$f\in \hat{C}^1(\mathbb{T})$
follows from Assumption (A). The fact that
$(\hat{\eta}_t(\,f), \hat{\beta}_t(\,f))_{0\leq t\leq T}$
is continuous for any
$f\in \hat{C}^1(\mathbb{T})$
follows from the fact that
\begin{align*}\left|\eta_t^N(\,f)-\eta_{t-}^N(\,f)\right|, \left|\beta_t^N(\,f)-\beta^N_{t-}(\,f)\right|\leq \frac{\|f\|_\infty}{\sqrt{N}},\end{align*}
where
$t-$
is the moment just before t. Therefore, we only need to show that
$\{\mathcal{M}_{t,\hat{\eta}, \hat{\beta}}(G, f, g)\}_{0\leq t\leq T}$
is a martingale for any
$G\in C_c^\infty(\mathbb{R}^2)$
and
$f,g\in \hat{C}^1(\mathbb{T})$
, where
$\mathcal{M}_{t,\hat{\eta}, \hat{\beta}}(G, f, g)$
is defined as
\begin{align*}&G(\hat{\eta}_t(\,f), \hat{\beta}_t(g))-G(\hat{\eta}_0(\,f), \hat{\beta}_0(g))\\[5pt] &\text{ }-\int_0^t\frac{\partial}{\partial u}G(\hat{\eta}_s(\,f), \hat{\beta}_s(g))\big(\hat{\beta}_s(\mathcal{A}_{1,s}\;f)+\hat{\eta}_s(\mathcal{A}_{0,s}\;f-\mathcal{A}_\psi f)\big){\mathrm{d}}s\\[5pt] &\text{ }+\int_0^t\frac{\partial}{\partial v}G(\hat{\eta}_s(\,f), \hat{\beta}_s(g))\big(\hat{\beta}_s(\mathcal{A}_{1,s}g)+\hat{\eta}_s(\mathcal{A}_{0, s}g)\big){\mathrm{d}}s\\[5pt] &\text{ }-\frac{1}{2}\int_0^t\frac{\partial^2}{\partial u^2}G(\hat{\eta}_s(\,f), \hat{\beta}_s(g))\\[5pt] &\quad\quad\quad\times\left(\int_0^1\rho_1(s,u)\psi(u)\,f^2(u)\,{\mathrm{d}} u+\int_0^1\int_0^1\lambda(u,v)\rho_0(s,u)\rho_1(s,v)\,f^2(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\right){\mathrm{d}}s\\[5pt] &\text{ }-\frac{1}{2}\int_0^t\int_0^1\int_0^1\frac{\partial^2}{\partial v^2}G(\hat{\eta}_s(\,f), \hat{\beta}_s(g))\lambda(u,v)\rho_0(s,u)\rho_1(s,v)g^2(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\,{\mathrm{d}}s\\[5pt] &\text{ }+\int_0^t\int_0^1\int_0^1\frac{\partial^2}{\partial uv}G(\hat{\eta}_s(\,f), \hat{\beta}_s(g))\lambda(u,v)\rho_0(s,u)\rho_1(s,v)\,f(u)g(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\,{\mathrm{d}}s.\end{align*}
For all
$N\geq 1$
, let
\begin{align*}&\mathcal{M}^N_{t,\eta, \beta}(G, f, g) \\[5pt] &=G\!\left(\eta_t^N(\,f), \beta_t^N(g)\right)-G\!\left(\eta_0^N(\,f), \beta_0^N(g)\right)-\int_0^t\!\left(\partial_s+\mathcal{L}^N\right)G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\,{\mathrm{d}} s,\end{align*}
then
$\{\mathcal{M}_{t,\eta, \beta}^N(G, f, g)\}_{0\leq t\leq T}$
is a martingale according to Dynkin’s martingale formula. According to the definition of
$\mathcal{L}^N$
and Chapman–Kolmogorov equation, we have
\begin{align*}&\mathcal{L}^NG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\\[5pt] &=\sum_{i=1}^N\psi\!\left(\frac{i}{N}\right)I_s^N(i)\!\left(G\!\left(\eta_s^N(\,f)-\frac{f\!\left(\frac{i}{N}\right)}{\sqrt{N}}, \beta_s^N(g)\right)-G(\eta_s^N(\,f), \beta_s^N(g))\right)\\[5pt] &\quad+\frac{1}{N}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)S_s^N(i)I_s^N(j)\\[5pt] &\quad\quad\quad\quad\times\left(G\!\left(\eta_s^N(\,f)+\frac{f\!\left(\frac{i}{N}\right)}{\sqrt{N}}, \beta_s^N(g)-\frac{g\!\left(\frac{i}{N}\right)}{\sqrt{N}}\right)-G(\eta_s^N(\,f), \beta_s^N(g))\right)\end{align*}
and
\begin{align*}&\partial_sG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\\[5pt] &=\frac{1}{\sqrt{N}}\partial_uG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\sum_{i=1}^N\psi\!\left(\frac{i}{N}\right)\mathrm{E}I_s^N(i)\,f\!\left(\frac{i}{N}\right)\\[5pt] &\quad-\frac{1}{N^{1.5}}\partial_uG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\,f\!\left(\frac{i}{N}\right)\\[5pt] &\quad+\frac{1}{N^{1.5}}\partial_vG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)g\!\left(\frac{i}{N}\right)\!.\end{align*}
Then, by Taylor’s expansion up to the second order,
\begin{align*}&\left(\partial_s+\mathcal{L}^N\right)G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right) \\[5pt] &=\partial_uG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\Bigg(-\eta_s^N(\mathcal{A}_\psi f)\\[5pt] &\quad\quad\quad+\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\left(S_s^N(i)I_s^N(j)-\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\right)\,f\!\left(\frac{i}{N}\right)\Bigg)\\[5pt] &\quad-\partial_vG(\eta_s^N(\,f), \beta_s^N(g))\\[5pt] &\quad\quad\quad\times\left(\frac{1}{N^{1.5}}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\left(S_s^N(i)I_s^N(j)-\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\right)g\!\left(\frac{i}{N}\right)\right)\\[5pt] &\quad+\frac{1}{2}\partial_{uu}^2G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\Bigg(\frac{1}{N}\sum_{i=1}^N\psi\!\left(\frac{i}{N}\right)I_s^N(i)\,f^2\!\left(\frac{i}{N}\right)\\[5pt] &\quad\quad\quad+\frac{1}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)S_s^N(i)I_s^N(j)\,f^2\!\left(\frac{i}{N}\right)\Bigg)\\[5pt] &\quad-\frac{\partial_{uv}^2G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)S_s^N(i)I_s^N(j)\,f\!\left(\frac{i}{N}\right)g\!\left(\frac{i}{N}\right)\\[5pt] &\quad+\frac{1}{2}\frac{\partial_{vv}^2G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)S_s^N(i)I_s^N(j)g^2\!\left(\frac{i}{N}\right)+\varepsilon_{s,1}^N,\end{align*}
where
$\varepsilon_{s,1}^N$
is the Lagrange remainder such that
\begin{align*}\|\varepsilon_{s, 1}^N\|\leq \frac{3\|G^{\prime\prime\prime}\|_\infty\left(\|f\|_\infty^3+\|g\|_\infty^3\right)\left(\|\psi\|_\infty+\|\lambda\|_\infty\right)}{2\sqrt{N}}.\end{align*}
According to the above equation, Lemma 3.1 and Cauchy–Schwarz inequality, there exists
$C_6<\infty$
independent of N such that
\begin{align*}\sup_{N\geq 4, 0\leq t\leq T}\mathrm{E}\big(\big(\mathcal{M}^N_{t,\eta, \beta}(G, f, g)\big)^2\big)\leq C_6.\end{align*}
Hence, for any
$0\leq t\leq T$
,
$\{\mathcal{M}^N_{t, \eta, \beta}(G, f, g)\}_{N\geq 4}$
are uniformly integrable. Therefore, according to Theorem 5.3 of [Reference Whitt14], we only need to show that
$\mathcal{M}^N_{t, \eta, \beta}(G, f, g)$
converges weakly to
$\mathcal{M}_{t,\hat{\eta}, \hat{\beta}}(G, f, g)$
as
$N\rightarrow+\infty$
to complete this proof.
According to Lemma 3.1 and Cauchy–Schwarz inequality, random variables
$I_s^N(i)$
and
$S_s^N(i)I_s^N(j)$
in the last three terms of the expression of
$(\partial_s+\mathcal{L}^N)G(\eta_s^N(\,f), \beta_s^N(g))$
can be replaced by their means with small errors. Similarly, according to Lemmas 3.1 and 5.2, the random variable
\begin{align*}S_s^N(i)I_s^N(j)-\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\end{align*}
in the first two terms of the expression of
$(\partial_s+\mathcal{L}^N)G(\eta_s^N(\,f), \beta_s^N(g)) $
can be replaced by
\begin{align*}(\mathrm{E}S_s^N(i))\!\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)+(\mathrm{E}I_s^N(j))\!\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\end{align*}
with a small error since
\begin{align*}&S_s^N(i)I_s^N(j)-\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\\[5pt] &\quad-\left((\mathrm{E}S_s^N(i))\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)+(\mathrm{E}I_s^N(j))\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\right)\\[5pt] &=\left(S_s^N(i)-\mathrm{E}S_s^N(i)\right)\left(I_s^N(j)-\mathrm{E}I_s^N(j)\right)-\left(\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)-\mathrm{E}S_s^N(i)\mathrm{E}I_s^N(j)\right)\!.\end{align*}
In detail,
\begin{align*}&(\partial_s+\mathcal{L}^N)G(\eta_s^N(\,f), \beta_s^N(g)) \\[5pt] &=\varepsilon_{s,1}^N+\varepsilon_{s,2}^N+\partial_uG(\eta_s^N(\,f), \beta_s^N(g))\Big(\!-\eta_s^N(\mathcal{A}_\psi f)+\eta_s^N(\mathcal{A}_{0,s\;}^Nf)+\beta_s^N(\mathcal{A}_{1,s\;}^Nf)\Big)\\[5pt] &\quad-\partial_vG(\eta_s^N(\,f), \beta_s^N(g))\left(\beta_s^N(\mathcal{A}_{1,s}^Ng)+\eta_s^N(\mathcal{A}_{0,s}^Ng)\right)\\[5pt] &\quad+\frac{1}{2}\partial_{uu}^2G(\eta_s^N(\,f), \beta_s^N(g))\Bigg(\frac{1}{N}\sum_{i=1}^N\psi\!\left(\frac{i}{N}\right)\mathrm{E}I_s^N(i)\,f^2\!\left(\frac{i}{N}\right)\\[5pt] &\quad\quad\quad+\frac{1}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\,f^2\!\left(\frac{i}{N}\right)\Bigg)\\[5pt] &\quad-\frac{\partial_{uv}^2G(\eta_s^N(\,f), \beta_s^N(g))}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)\,f\!\left(\frac{i}{N}\right)g\!\left(\frac{i}{N}\right)\\[5pt] &\quad+\frac{1}{2}\frac{\partial_{vv}^2G(\eta_s^N(\,f), \beta_s^N(g))}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{E}\!\left(S_s^N(i)I_s^N(j)\right)g^2\!\left(\frac{i}{N}\right)\!,\end{align*}
where
\begin{align*}\lim_{N\rightarrow+\infty}\int_0^t\varepsilon_{s,1}^N\,{\mathrm{d}}s=0\end{align*}
in probability,
\begin{align*}\mathrm{E}\!\left(\!\left(\int_0^t \varepsilon_{2,s}^N\,{\mathrm{d}}s\right)^2\right)\leq \frac{C_7}{N}\end{align*}
for some
$C_7<+\infty$
independent of N,
\begin{align*}\mathcal{A}_{0,s\;}^Nf(u)=\frac{1}{N}\sum_{j=1}^N\lambda\!\left(\frac{j}{N}, u\right)\mathrm{E}S_s^N(j)\,f\!\left(\frac{j}{N}\right)\end{align*}
and
\begin{align*}\mathcal{A}_{1,s\;}^Nf(u)=f(u)\frac{1}{N}\sum_{j=1}^N\lambda\!\left(u, \frac{j}{N}\right)\mathrm{E}I_s^N(j).\end{align*}
According to Equation (4.1), Lemma 3.1 and Chebyshev’s inequality,
\begin{align*}\lim_{N\rightarrow+\infty}\int_0^t\eta_s^N\!\left(\mathcal{A}_{0,s}(\,f)-\mathcal{A}_{0,s}^N(\,f)\right){\mathrm{d}}s=0\end{align*}
and
\begin{align*}\lim_{N\rightarrow+\infty}\int_0^t\beta_s^N\!\left(\mathcal{A}_{1,s}(\,f)-\mathcal{A}_{1,s}^N(\,f)\right){\mathrm{d}}s=0\end{align*}
in probability. By Lemma 3.1,
$\mathrm{E}(S_s^N(i)I_s^N(j))$
in the last three terms of the above expression of
$(\partial_s+\mathcal{L}^N)G(\eta_s^N(\,f), \beta_s^N(g))$
can be replaced by
$\mathrm{E}S_s^N(i)\mathrm{E}I_s^N(j)$
with small error. Furthermore, by Equation (4.1), in the new expression of
$(\partial_s+\mathcal{L}^N)\,G\,(\eta_s^N(\,f), \beta_s^N(g))$
we can replace
$\frac{1}{N}\sum_{i=1}^N\psi({i}/{N})\mathrm{E}I_s^N(i)\,f^2({i}/{N})$
by
\begin{align*}\int_0^1\psi(u)\rho_1(s,u)\,f^2(u)\,{\mathrm{d}}u\end{align*}
and replace any term with form
$$\frac{1}{N^2}\sum_{i=1}^N\sum_{j\neq i}\lambda\!\left(\frac{i}{N}, \frac{j}{N}\right)\mathrm{E}S_s^N(i)\mathrm{E}I_s^N(j)h\!\left(\frac{i}{N}, \frac{j}{N}\right)$$
by
\begin{align*}\int_0^1\int_0^1\lambda(u,v)\rho_0(s,u)\rho_1(s,v)h(u,v)\,{\mathrm{d}}u\,{\mathrm{d}}v\end{align*}
with small errors. In conclusion,
\begin{align*}&\left(\partial_s+\mathcal{L}^N\right)G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right) \\[5pt] &=\varepsilon_{s,4}^N+\partial_uG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\Big(\!-\eta_s^N(\mathcal{A}_\psi f)+\eta_s^N(\mathcal{A}_{0,s\;}f)+\beta_s^N(\mathcal{A}_{1,s\;}f)\Big)\\[5pt] &\quad-\partial_vG\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\left(\beta_s^N(\mathcal{A}_{1,s}g)+\eta_s^N(\mathcal{A}_{0,s}g)\right)\\[5pt] &\quad+\frac{1}{2}\partial_{uu}^2G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\Bigg(\int_0^1\psi(u)\rho_1(s, u)\,f^2(u)\,{\mathrm{d}}u\\[5pt] &\quad\quad\quad+\int_0^1\int_0^1\lambda(u,v)\rho_0(s,u)\rho_1(s,v)\,f^2(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\Bigg)\\[5pt] &\quad-\partial_{uv}^2G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\int_0^1\int_0^1\lambda(u, v)\rho_0(s,u)\rho_1(s,v)\,f(u)g(u)\,{\mathrm{d}}u\,{\mathrm{d}}v\\[5pt] &\quad+\frac{1}{2}\partial_{vv}^2G\!\left(\eta_s^N(\,f), \beta_s^N(g)\right)\int_0^1\int_0^1\lambda(u,v)\rho_0(s,u)\rho_1(s,v)g^2(u)\,{\mathrm{d}}u\,{\mathrm{d}}v,\end{align*}
where
\begin{align*}\lim_{N\rightarrow+\infty}\int_0^t \varepsilon_{s, 4}^N\,{\mathrm{d}}s=0\end{align*}
in probability. Since
$\{(\hat{\eta}_t, \hat{\beta}_t)\;:\;0\leq t\leq T\}$
is a weak limit of
\begin{align*}\left\{\left(\eta_t^N, \beta_t^N\right)\;:\;0\leq t\leq T\right\}_{N\geq 1},\end{align*}
let
$N\rightarrow+\infty$
in the above expression of
$(\partial_s+\mathcal{L}^N)G(\eta_s^N(\,f), \beta_s^N(g)) $
, then we have
$\mathcal{M}^N_{t, \eta, \beta}(G, f, g)$
converges weakly to
$\mathcal{M}_{t,\hat{\eta}, \hat{\beta}}(G, f, g)$
as
$N\rightarrow+\infty$
and the proof is complete.
6. Proof of Theorem 2.3
In this section, we prove Theorem 2.3. As a preliminary, we first prove Lemma 2.1.
Proof of Lemma
2.1. As we have mentioned in Remark 2.1,
$\rho_1(t, u), \rho_0(t, u)\geq 0$
. Therefore,
$\rho_0(t, u), \rho_0(t, u)+\rho_1(t, u)$
are both decreasing in t since
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\rho_0(t,u)=-\rho_0(t,u)\int_0^1\lambda(u, v)\rho_1(t,v)\,{\mathrm{d}}v\leq 0\end{align*}
and
\begin{align*}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\!\left(\rho_0(t, u)+\rho_1(t, u)\right)=-\psi(u)\rho_1(t,u)\leq 0.\end{align*}
Consequently, there exist
$\hat{\rho}_1(u), \hat{\rho}_0(u)$
such that
\begin{align*}\lim_{t\rightarrow+\infty}\!\left(\rho_1(t, u)+\rho_0(t, u)\right)=\hat{\rho}_1(u)+\hat{\rho}_0(u), \quad\lim_{t\rightarrow+\infty}\rho_0(t, u)=\hat{\rho}_0(u)\end{align*}
and, hence,
$\lim_{t\rightarrow+\infty}\rho_1(t, u)=\hat{\rho}_1(u)$
. If
$\hat{\rho}_1(u)>0$
then
\begin{align*}\lim_{t\rightarrow+\infty}\!\left(\rho_0(t, u)+\rho_1(t, u)\right)=-\infty\end{align*}
according to
$\frac{d}{dt}(\rho_0(t, u)+\rho_1(t, u))=-\psi(u)\rho_1(t,u)$
, which is contradictory. Hence,
\begin{align*}\hat{\rho}_1(u)=0\end{align*}
and the proof is complete.
Now we prove Theorem 2.3.
Proof of Theorem
2.3. We only need to show that, for any
$\epsilon>0$
,
\begin{equation}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\sum_{i=1}^NS_\infty^N(i)>\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}} u+\epsilon\right)=0\end{equation}
and
\begin{equation}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\sum_{i=1}^NS_\infty^N(i)>\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}} u-\epsilon\right)=1.\end{equation}
We first check (6.1). According to the definition of
$\hat{\rho}_0$
, there exists
$T_\epsilon>0$
such that
\begin{align*}\int_0^1\rho_0(T_\epsilon, u)\,{\mathrm{d}}u-\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}}u<\frac{\epsilon}{3}.\end{align*}
Since
$N^{-1}\sum_{i=1}^NS_t^N(i)=\theta_t^N(1)$
, by Theorem 2.1,
\begin{align*}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\sum_{i=1}^NS_{T_\epsilon}^N(i)> \int_0^1\rho_0(T_\epsilon, u)\,{\mathrm{d}}u+\frac{\epsilon}{3}\right)=0\end{align*}
and, hence,
\begin{equation}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\sum_{i=1}^NS_{T_\epsilon}^N(i)> \int_0^1\hat{\rho}_0(u)\,{\mathrm{d}}u+\epsilon\right)=0.\end{equation}
Since
$S_t^N(i)$
is decreasing in t,
$\sum_{i=1}^NS_\infty^N(i)\leq \sum_{i=1}^NS_{T_\epsilon}^N(i)$
and then (6.1) follows from (6.3).
Now we check (6.2). Under Assumption (2.6), we can let
$\epsilon$
be sufficiently small such that
\begin{align*}&\max_{u,v\in \mathbb{T}}\lambda(u, v)\!\left(\int_0^1(1-\phi(u))\,{\mathrm{d}}u+2\epsilon\right)\\[5pt] &<\frac{\max_{u,v\in \mathbb{T}}\lambda(u, v)\int_0^1(1-\phi(u))\,{\mathrm{d}}u+\min_{u\in \mathbb{T}}\psi(u)}{2}<\min_{u\in \mathbb{T}}\psi(u).\end{align*}
By Lemma 2.1, there exists
$\hat{T}_\epsilon>0$
such that
\begin{align*}\int_0^1\rho_1(\hat{T}_\epsilon, u)\,{\mathrm{d}}u<\frac{\epsilon}{3}\end{align*}
and
\begin{align*}\left|\int_0^1\rho_0(\hat{T}_\epsilon, u)\,{\mathrm{d}}u-\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}} u\right|<\frac{\epsilon}{3}.\end{align*}
Let
$D_{N, \epsilon}$
be the event that
\begin{align*}\left|\frac{1}{N}\sum_{i=1}^NI_{\hat{T}_\epsilon}^N(i)-\int_0^1\rho_1(\hat{T}_\epsilon, u)\,{\mathrm{d}} u\right|<\frac{\epsilon}{3}\end{align*}
and
\begin{align*}\left|\frac{1}{N}\sum_{i=1}^NS_{\hat{T}_\epsilon}^N(i)-\int_0^1\rho_0(\hat{T}_\epsilon, u)\,{\mathrm{d}} u\right|<\frac{\epsilon}{3}.\end{align*}
By Theorem 2.1,
\begin{equation}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(D_{N, \epsilon}\right)=1.\end{equation}
Conditioned on
$D_{N, \epsilon}$
, for
$t\geq \hat{T}_\epsilon$
,
$\sum_{i=1}^NI_{t}^N(i)\rightarrow \sum_{i=1}^NI_{t}^N(i)-1$
at rate
\begin{align*}\sum_{i=1}^NI_t^N(i)\psi(i/N)\geq \min_{u\in \mathbb{T}}\psi(u)\sum_{i=1}^NI_{t}^N(i)\end{align*}
and
$\sum_{i=1}^NI_{t}^N(i)\rightarrow \sum_{i=1}^NI_{t}^N(i)+1$
at rate
\begin{align*}&\frac{1}{N}\sum_{i=1}^N\sum_{j=1}S_t^N(i)I_t^N(j)\lambda\!\left(i/N, j/N\right)\\[5pt] &\leq \max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\frac{1}{N}\sum_{i=1}S_t^N(i)\right)\left(\sum_{j=1}^NI_t^N(j)\right)\\[5pt] &\leq \max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\frac{1}{N}\sum_{i=1}S_{\hat{T}_\epsilon}^N(i)\right)\left(\sum_{i=1}^NI_t^N(i)\right)\\[5pt] &\leq \max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\int_0^1\rho_0(\hat{T}_\epsilon, u)\,{\mathrm{d}} u+\frac{\epsilon}{3}\right)\left(\sum_{i=1}^NI_t^N(i)\right)\\[5pt] &\leq \max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}} u+\frac{2\epsilon}{3}\right)\left(\sum_{i=1}^NI_t^N(i)\right)\\[5pt] &\leq \max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\int_0^1\rho_0(0, u)\,{\mathrm{d}} u+\frac{2\epsilon}{3}\right)\left(\sum_{i=1}^NI_t^N(i)\right)\\[5pt] &=\max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\int_0^1(1-\phi(u))\,{\mathrm{d}} u+\frac{2\epsilon}{3}\right)\left(\sum_{i=1}^NI_t^N(i)\right)\!.\end{align*}
Let
$\{\hat{I}_t^N\}_{t\geq 0}$
be the birth–death process with initial condition
\begin{align*}\hat{I}_0^N=N\!\left(\int_0^1\rho_1(\hat{T}_\epsilon, u)\,{\mathrm{d}}u+\frac{\epsilon}{3}\right)\!,\end{align*}
and transition rates given by
\begin{align*} \hat{I}_t^N\rightarrow\left\{\begin{array}{l@{\quad}l} \hat{I}_t^N-1 & \text{ at rate } \hat{B}\hat{I}_t^N,\\[5pt] \hat{I}_t^N+1 & \text{ at rate } \hat{A}\hat{I}_t^N,\end{array}\right. \end{align*}
where
$\hat{B}=\min_{u\in \mathbb{T}}\psi(u)$
and
\begin{align*}\hat{A}=\frac{\max_{u,v\in \mathbb{T}}\lambda(u, v)\int_0^1(1-\phi(u))\,{\mathrm{d}}u+\min_{u\in \mathbb{T}}\psi(u)}{2}.\end{align*}
Since
\begin{align*}\max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\int_0^1(1-\phi(u))\,{\mathrm{d}}u+\frac{2\epsilon}{3}\right)\leq \max_{u, v\in \mathbb{T}}\lambda(u, v)\left(\int_0^1(1-\phi(u))\,{\mathrm{d}}u+2\epsilon\right)\leq \hat{A},\end{align*}
the process
$\{\sum_{i=1}^NI_{t+\hat{T}_\epsilon}^N(i)\}_{t\geq 0}$
, conditioned on
$D_{N, \epsilon}$
, is stochastically dominated from above by
$\{\hat{I}_t^N\}_{t\geq 0}$
. In the birth–death process
$\{\hat{I}_t^N\}_{t\geq 0}$
, each individual dies at rate
$\hat{B}$
and gives birth to a new individual at rate
$\hat{A}$
. Since
$\hat{A}<\hat{B}$
, the process is subcritical and will be extinct finally. Let
$\mathcal{V}_N$
be the number of all descendants of the initial
$\hat{I}_0^N$
individuals, then, conditioned on
$D_{N, \epsilon}$
,
\begin{align*}\sum_{i=1}^NS_{\hat{T}_\epsilon}^N(i)-\sum_{i=1}^NS_{\infty}^N(i)\end{align*}
is stochastically dominated from above by
$\mathcal{V}_N$
. According to the strong Markov property of the birth–death process, it is easy to check that the mean
$\hat{C}$
of the number of all descendants of a given individual in
$\{\hat{I}_t^N\}_{t\geq 0}$
satisfies
\begin{align*}\hat{C}=\frac{\hat{B}}{\hat{A}+\hat{B}}\times 0+\frac{\hat{A}}{\hat{A}+\hat{B}}\times (1+2\hat{C})\end{align*}
and, hence,
\begin{align*}\hat{C}=\frac{\hat{A}}{\hat{B}-\hat{A}}.\end{align*}
Then, according to the LLN of the sum of i.i.d. random variables,
\begin{align*}\lim_{N\rightarrow+\infty}\frac{1}{N}\mathcal{V}_N=\hat{C}\!\left(\int_0^1\rho_1(\hat{T}_\epsilon, u)\,{\mathrm{d}} u+\frac{\epsilon}{3}\right)\end{align*}
in probability. Since
\begin{align*}\int_0^1\rho_1(\hat{T}_\epsilon, u)\,{\mathrm{d}}u+\frac{\epsilon}{3}\leq \frac{2\epsilon}{3}<\epsilon,\end{align*}
we have
\begin{align*}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\mathcal{V}_N\geq 2\epsilon \hat{C}\right)=0\end{align*}
and, hence,
\begin{align*}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\sum_{i=1}^NS_{\hat{T}_\epsilon}^N(i)-\frac{1}{N}\sum_{i=1}^NS_{\infty}^N(i)\geq 2\epsilon \hat{C}\Bigg|D_{N, \epsilon}\right)=0.\end{align*}
On
$D_{N, \epsilon}$
,
$N^{-1}\sum_{i=1}^NS_{\hat{T}_\epsilon}^N(i)>\int_0^1\hat{\rho}_0(u)\,{\mathrm{d}} u-{\epsilon}/{3}$
. Therefore,
\begin{equation}\lim_{N\rightarrow+\infty}\mathrm{P}\!\left(\frac{1}{N}\sum_{i=1}^NS_{\infty}^N(i)\geq \int_0^1\hat{\rho}_0(u)\,{\mathrm{d}}u-\epsilon\!\left(2\hat{C}+\frac{1}{3}\right)\Bigg|D_{N, \epsilon}\right)=1.\end{equation}
Since
$\epsilon$
is arbitrary, Equation (6.2) follows from (6.4) and (6.5).
7. An intuitive explanation of Theorem 2.4
In this section, we give an intuitive explanation of Theorem 2.4. Throughout this section, we assume that
$\lambda(u, v)=\vartheta(u)\vartheta(v)$
for some piecewise smooth
$\vartheta$
on
$\mathbb{T}$
. If i is the unique infectious individual at
$t=0$
and other individuals are susceptible, then, for integer
$k<N$
and given different
$x_1, x_2, \ldots, x_k \in \{1,2,\ldots, N\}\setminus\{i\}$
, the path
$i(\!=x_0), x_1, x_2, \ldots, x_k$
is an infection path with probability
\begin{align*}\prod_{j=0}^{k-1}\mathrm{P}\!\left(U^N_{(x_{j+1}, x_j)} < K_{x_j}^N\right)&=\prod_{j=0}^{k-1}\frac{\frac{1}{N}\vartheta(x_j/N)\vartheta(x_{j+1}/N)}{\frac{1}{N}\vartheta(x_j/N)\vartheta(x_{j+1}/N)+\psi(x_j/N)}\\[5pt] &\approx \prod_{j=0}^{k-1}\frac{\frac{1}{N}\vartheta(x_j/N)\vartheta(x_{j+1}/N)}{\psi(x_j/N)}\\[5pt] &=\frac{1}{N^k}\frac{\vartheta(i/N)\vartheta(x_k/N)}{\psi(i/N)}\left(\prod_{j=1}^{k-1}\frac{\left(\vartheta(x_j/N)\right)^2}{\psi(x_j/N)}\right)\end{align*}
for sufficiently large N. Hence, the mean of the number of the infection paths starting from i with length k is approximate to
\begin{equation}\frac{1}{N^k}\sum_{x_1, x_2, \ldots, x_k}\frac{\vartheta(i/N)\vartheta(x_k/N)}{\psi(i/N)}\left(\prod_{j=1}^{k-1}\frac{\left(\vartheta(x_j/N)\right)^2}{\psi(x_j/N)}\right)\!,\end{equation}
where the sum is over all self-avoiding paths
$x_1, \ldots, x_k$
on
$\{1, 2, \ldots, N\}\setminus\{i\}$
. When N is much larger than k, the simple random walk on
$\{1, 2, \ldots, N\}$
is self-avoiding and avoids i in the initial k steps with high probability. Hence, in (7.1), we can further let the sum be over all
$(x_1, \ldots, x_k)\in \{1, 2,\ldots, N\}^k$
with small error generated. Therefore, the mean of the number of the infection paths starting from i with length k is approximate to
\begin{align*}&\frac{1}{N^k}\sum_{(x_1, x_2, \ldots, x_k)\in \{1, 2, \ldots, N\}^k}\frac{\vartheta(i/N)\vartheta(x_k/N)}{\psi(i/N)}\!\left(\prod_{j=1}^{k-1}\frac{\left(\vartheta(x_j/N)\right)^2}{\psi(x_j/N)}\right)\\[5pt] &=\frac{\vartheta(i/N)}{\psi(i/N)}\left(\frac{1}{N}\sum_{x\in \{1,2,\ldots, N\}}\frac{\left(\vartheta(x/N)\right)^2}{\psi(x/N)}\right)^{k-1}\!\left(\frac{1}{N}\sum_{y\in \{1, 2, \ldots, N\}}\vartheta(y/N)\right)\\[5pt] &=O(1)\!\left(\int_0^1\frac{\vartheta^2(u)}{\psi(u)}\,{\mathrm{d}} u+o(1)\right)^{k-1}=O(1)\!\left(\int_0^1\frac{\vartheta^2(u)}{\psi(u)}\,{\mathrm{d}}u\right)^{k}\end{align*}
for sufficiently large N. As a result, the initial several steps of the spread of the disease are similar to a branching process with offspring distribution with mean
$\int_0^1({\vartheta^2(u)}/{\psi(u)})\,{\mathrm{d}}u$
. Hence, intuitively, the disease outbreaks when
$\int_0^1({\vartheta^2(u)}/{\psi(u)})\,{\mathrm{d}}u>1$
or dies out quickly when
$\int_0^1({\vartheta^2(u)}/{\psi(u)})\,{\mathrm{d}}u<1$
.
Acknowledgements
The author is grateful to the referees and editors for their careful reading of the paper as well as their comments and suggestions.
Funding information
The author is grateful to the financial support from the National Natural Science Foundation of China with grant number 12371142 and the Fundamental Research Funds for the Central Universities with grant number 2022JBMC039.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
