The critical infection rate of the high-dimensional two-stage contact process

Statistics and Probability Letters 140 (2018) 115–125

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The critical infection rate of the high-dimensional two-stage contact process Xiaofeng Xue School of Science, Beijing Jiaotong University, Beijing 100044, China

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In this paper we are concerned with the two-stage contact process on the lattice Zd introduced in Krone (1999). We give a limit theorem of the critical infection rate of the process as the dimension d of the lattice grows to infinity. A linear system and a two-stage SIR model are two main tools for the proof of our main result. © 2018 Elsevier B.V. All rights reserved.

Article history: Received 6 November 2017 Received in revised form 19 April 2018 Accepted 3 May 2018 Available online 9 May 2018 Keywords: Contact process Infection rate SIR model

1. Introduction In this paper we are concerned with the two-stage contact process on the lattice Zd , which is introduced in Krone (1999). For each x ∈ Zd , we use ∥x∥ to denote the l1 -norm of x, i.e.,

∥ x∥ =

d ∑ |xi | i=1

for x = (x1 , . . . , xd ). For any x, y ∈ Zd , we write x ∼ y when and only when ∥x − y∥ = 1. In other words, we use x ∼ y to denote that x and y are neighbors. We use O to denote the origin (0, 0, . . . , 0). d The two-stage contact process {ηt }t ≥0 on Zd is a continuous time Markov process with state space X1 = {0, 1, 2}Z . The transition rates function is given as follows.

ηt (x) → i at rate

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 if ηt (x) = 2 and i = 0, γ if ηt (x) = 1 and i = 2, 1+ δ if ηt (x) = 1 and i = 0, ∑

λ

1{ηt (y)=2} if ηt (x) = 0 and i = 1,

(1.1)

y:y∼x

0 otherwise

for each x ∈ Z d and t ≥ 0, where λ, γ , δ are positive constants and 1A is the indicator function of the event A. The constant λ is called the infection rate of the process. For later use, we denote by ⪯ the partial order on the state space X1 that ξ ⪯ η when and only when ξ (x) ≤ η(x) for each x ∈ Zd . E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2018.05.006 0167-7152/© 2018 Elsevier B.V. All rights reserved.

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X. Xue / Statistics and Probability Letters 140 (2018) 115–125

The process {ηt }t ≥0 intuitively describes the spread of an epidemic on Zd . Each vertex is in one of three states, which are ‘healthy’, ‘semi-infected’ and ‘fully-infected’. A healthy vertex is infected at rate proportional to the number of fullyinfected neighbors to become a ‘semi-infected’ one while a ‘semi-infected’ vertex waits for an exponential time with rate γ to become fully-infected or waits for an exponential time with rate 1 + δ to become healthy. A ‘fully-infected’ vertex waits for an exponential time with rate 1 to become healthy. Our process can be defined equivalently through a graphic representation, the theory of which is introduced in Harris (1978). We consider the set Zd × [0, +∞), i.e., at each vertex there is a time-arrow [0, +∞). For each x ∈ Zd and each y ∼ x, let {wt (x)}t ≥0 be a Poisson process with rate one, {µt (x)}t ≥0 be a process with rate γ , {πt (x)}t ≥0 be a Poisson process with rate δ while {ut (x, y)}t ≥0 be a Poisson process with rate λ and all these Poisson processes be independent. At each event moment t of w· (x), we put a ‘∗’ on (x, t). At each event moment l of µ· (x), we put a ‘⋆’ on (x, l). At each event moment s of π· (x), we put a ‘∆’ on (x, s). At each event moment r of u· (x, y), we put an arrow ‘→’ from (x, r) to (y, r). Then {ηt }t ≥0 evolves as follows. At t = 0, some vertices are in state 2 while others are in state 0 according to the initial configuration of the process. For each vertex x, we observe the time-arrow on x through the positive direction. If we meet an arrow ‘→’ from a neighbor in state 2 to x at some moment r while x is in state 0 at moment r −, the moment just before r, then x becomes in state 1 at moment r. If we meet a ‘∗’ at some moment t, then x is in state 0 at t, no matter what the state x is in at t −. If we meet a ‘∆’ at some moment s while x is in state 1 at s−, then x becomes in state 0 at s. If we meet a ‘⋆’ at some moment l while x is in state 1 at l−, then x becomes in state 2 at l. Otherwise, x stays its state. According to the theory introduced in Harris (1978), the process evolving as above has the transition rates function given in (1.1). The two-stage contact process {ηt }t ≥0 is introduced by Krone in Krone (1999), where a duality relation between this two-stage contact process and an ‘on-off’ process is given. In Foxall (2015), Fox gives a simple proof of the duality relation given in Krone (1999) and answers most of the open questions posed in Krone (1999). If γ = +∞, i.e., a semi-infected vertex becomes a fully-infected one immediately, then equivalently there is only one infected state for the process and hence the model reduces to the classic contact process introduced in Harris (1974). For a survey of the classic contact process, see Chapter 6 of Liggett (1985) and Part I of Liggett (1999). 2. Main result In this section we give our main result. First we introduce some notations and definitions. Throughout this paper we assume that {x : η0 (x) = 1} = ∅, i.e., there is no semi-infected vertex at t = 0. For each A ⊆ Zd , we write ηt as ηtA when {x} {x : η0 (x) = 2} = A. If A = {x} for some x ∈ Zd , then we write ηtA as ηtx instead of ηt . When we omit the superscript A, then d A we mean that A = Z . For any t ≥ 0, we use Ct to denote

{

x : ηtA (x) ̸ = 0

} λ,γ ,δ

as the set of infected vertices at the moment t. We denote by Pd the probability measure of the two-stage contact process ) λ,γ ,δ ( O {ηt }t ≥0 with parameter λ, γ , δ defined as in Eq. (1.1). It is not difficult to check that Pd Ct ̸ = ∅ for all t ≥ 0 is increasing with λ. One way to check this is utilizing the graphic representation given in Section 1. We write ut (x, y) as uλt (x, y) to point out the infection rate λ, then for λ1 < λ2 , according to the property of independent Poisson processes, we can write λ

λ

λ −λ1

ut 2 (x, y) = ut 1 (x, y) + ut 2 λ u· 1 (x, y)

(x, y),

λ −λ u· 2 1 (x, y)

λ

λ

where and are independent. Then {ηt 1 }t ≥0 and {ηt 2 }t ≥0 , our two processes with infection rates λ1 , λ2 respectively and with the same initial state that O is the unique vertex in state 2, are coupled under the same probability λ λ space. Under this coupling, it is easy to check that the condition ηt 1 ⪯ ηt 2 holds after each state-transition, which shows that λ ,γ ,δ ( O

Pd 1

λ ,γ ,δ ( O

)

Ct ̸ = ∅ for all t ≥ 0 ≤ Pd 2

Ct ̸ = ∅ for all t ≥ 0 .

)

Therefore, it is reasonable to define

{ } ) λ,γ ,δ ( O λc (d, γ , δ ) = sup λ : Pd Ct ̸ = ∅ for all t ≥ 0 = 0 .

(2.1)

λc (d, γ , δ ) is called the critical infection rate of the two-stage contact process, the infection rate below which infected vertices die out with probability one conditioned on O being the unique fully-infected vertex at t = 0. Now we give our main result, which is a limit theorem of λc (d, γ , δ ) as the dimension d grows to infinity. Theorem 2.1. For any γ , δ > 0, if λc (d, γ , δ ) are defined as in Eq. (2.1), then lim 2dλc (d, γ , δ ) = 1 +

d→+∞

1+δ

γ

.

Remark 1. To motivate our main result, consider the branching process analogue of our process. That is to say, each fullyinfected vertex has a lifetime which follows the exponential distribution with rate one while generates a semi-infected vertex at each given neighbor at rate λ, in other words, generates a semi-infected vertex at total rate 2dλ. Each semi-infected

X. Xue / Statistics and Probability Letters 140 (2018) 115–125

117

γ

vertex becomes fully-infected with probability 1+δ+γ , otherwise dies. As a result, the number of fully-infected children of γ a fully-infected vertex with lifetime T follows the Poisson distribution with rate 2dλT 1+δ+γ . Since T is an exponential time with rate one, the mean of the number of the fully-infected children is

(

E 2dλT

) γ γ = 2dλ . 1+δ+γ 1+δ+γ

Then, according to the classic theory about the Galton–Watson tree, the counterpart of λc for this branching process analogue 1+δ+γ is exact 2dγ , which makes the above mean equal one. Our main result shows that the critical value of the contact process analogue has similar asymptotic behavior with that of the branching process analogue as the dimension d grows to infinity. □ Remark 2. Let αc (d) be the critical infection rate for the classic contact process on Zd , then it is shown in Griffeath (1983) that lim 2dαc (d) = 1.

(2.2)

d→+∞

Our main result can be considered as an extension of conclusion (2.2) since when γ = +∞ the two-stage contact process reduces to the classic contact process. □ Remark 3. It is shown in Foxall (2015) that λc (d, γ , δ ) = +∞ when γ < conclusion since γ > 4d1−1 for sufficiently large d. □ Remark 4. It is shown in Griffeath (1983) that αc (d) ≤ f (γ , δ ) > 0 such that

λc (d, γ , δ ) ≤

1 2d

(1 +

1+δ

γ

)+

f (γ , δ ) d2

1

+ o(

d2

1 2d

+

1 2d2

1 . 4d−1

Our main result does not contradict this

+ o( d12 ). Hence it is natural to guess that there exists

).

However, according to our current approach we have not managed to obtain such a f yet. We will work on this question as a further study. □ The proof of Theorem 2.1 is divided into Sections 3 and 4. In Section 3, we give the proof of lim infd→+∞ 2dλc (d, γ , δ ) ≥ d 1 + 1+δ . For this purpose, we will introduce a linear system with state space {Z2+ }Z as a main auxiliary model, where γ

. The proof utilizes an approach Z+ = {0, 1, 2, . . .}. In Section 4, we give the proof of lim supd→+∞ 2dλc (d, γ , δ ) ≤ 1 + 1+δ γ given in Xue (2017). The idea of this approach is inspired by the strategy introduced by Kesten in Kesten (1990), which is about the asymptotic behavior of the critical probability of the bond percolation on lattices. We will introduce a two-stage SIR(susceptible–infected–recovered) model, the critical infection rate of which is an upper bound of λc (d, γ , δ ). 3. The proof of lim infd →+∞ 2d λc (d , γ, δ) ≥ 1 +

1+δ

γ

In this section we give the proof of lim infd→+∞ 2dλc (d, γ , δ ) ≥ 1 + 2 Zd

1+δ

γ

. First we introduce an auxiliary model, which is

a linear system with state space {Z+ } . For a survey of the linear system, see Chapter 9 of Liggett (1985). Let {(ζt , θt )}t ≥0 be d

a continuous-time Markov process with state space {Z2+ }Z , where Z+ = {0, 1, 2, . . .}. That is to say, at each vertex x there ( ) is a vector ζ (x), θ (x) . The transition rates function of {(ζt , θt )}t ≥0 is given as follows. For each x ∈ Zd and t ≥ 0,

) ζt (x), θt (x) → (a, b) at rate ⎧ 1 if a = b = 0, ⎪ ⎪ ⎪δ if a = ζt (x) and b = 0, ⎨ γ if a = ζt (x) + θt (x) and b = 0, ⎪ ⎪ ⎪ ⎩λ if y ∼ x, a = ζt (x) and b = θt (x) + ζt (y), 0 otherwise.

(

(3.1)

{(ζt , θt )}t ≥0 can be defined equivalently through a graphic representation. For x ∈ Zd and y ∼ x, let wt (x), µt (x), πt (x), ut (x, y) be defined as in Section one. Then {(ζt , θt )}t ≥0 evolves as follows. For each x, we observe the time-arrow on x through the positive direction. ( If we meet ) a ‘∗’ at some moment t, then the state of x is (0, 0) at t. If we ( meet a ‘∆’ at some ) moment s, then the state of x is ζs− (x), 0 at s. If we meet a ‘⋆’ at some moment l, then the state of x(is ζl− (x) + θl− (x), 0 at ) l. If we meet an arrow ‘→’ from y to x at some moment r for some neighbor y, then the state of x is ζr − (x), θr − (x) + ζr − (y) at r. Otherwise, x stays its state. According to the theory introduced in Harris (1978), the process evolving as above has the transition rates function given in (3.1). According to the above graphic representation, the auxiliary model {(ζt , θt )}t ≥0 and the two-stage contact process have the following coupling relationship.

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X. Xue / Statistics and Probability Letters 140 (2018) 115–125

Lemma 3.1. For x ∈ Zd and t ≥ 0, let 2 if ζt (x) > 0, ˆ ηt (x) = 1 if ζt (x) = 0 and θt (x) > 0, 0 if ζt (x) = θt (x) = 0,

{

then {ˆ ηt }t ≥0 is a two-stage contact process with transition rates function given in Eq. (1.1). Proof of Lemma 3.1. We only need to show that {ˆ ηt }t ≥0 evolves in the same way as that of {ηt }t ≥0 according to the graphic representations of our models. We observe the time-arrow on x through the positive direction. There are five cases to check. Case 1, where we meet a ‘→’ from some neighbor y to x at some moment r that ˆ ηr − (y) = 2 and ˆ ηr − (x) = 0, then ζr − (y) > 0, ζr − (x) = θr − (x) = 0 and

( ) ( ) ( ) ζr (x), θr (x) = ζr − (x), θr − (x) + ζr − (y) = 0, ζr − (y) , i.e., ˆ ηr (x) = 1. Case 2, where we meet a ‘∗’ at some moment t, then ζt (x), θt (x) = (0, 0) and hence ˆ ηt (x) = 0. Case 3, where we meet a ‘⋆’ at some moment l and ˆ ηl− (x) = 1, then ζl− (x) = 0, θl− (x) > 0 and

(

(

)

) ( ) ( ) ζl (x), θl (x) = ζl− (x) + θl− (x), 0 = θl− (x), 0 ,

i.e., ˆ ηl (x) = 2. Case 4, where we meet a ‘∆’ at some moment s while ˆ ηs− (x) = 1, then ζs− (x) = 0, θs− (x) > 0 and

(

) ( ) ( ) ζs (x), θs (x) = ζs− (x), 0 = 0, 0 ,

i.e., ˆ ηs (x) = 0. Case 5, where none of the above four cases occurs, then it is easy to check that ˆ ηt (x) stays its state. In all the five cases, ˆ ηt evolves in the same way as that of ηt and the proof is complete. □ According to Lemma 3.1, from now on we write

ηt (x) = 2 × 1{ζt (x)>0} + 1{ζt (x)=0 and θt (x)>0} for each x ∈ Zd . Conditioned on all the vertices being in state 2 at t = 0, it is shown in Krone (1999) that the distribution of ηt converges d weakly to a probability distribution ν = ν λ,γ ,δ on {0, 1, 2}Z as t grows to infinity. As a result, λ,γ ,δ (

lim Pd

t →+∞

) ( ) ηt (O) = 1 or 2 = ν λ,γ ,δ η(O) = 1 or 2 .

(3.2)

Note that when we omit the superscript of ηt we mean that all the vertices are in state 2 at t = 0. Our proof of lim infd→+∞ 2dλc (d, γ , δ ) ≥ 1 + 1+δ relies on the following proposition, which is Theorem 1.2 of Foxall γ (2015). Proposition 3.2 (Foxall, 2015). P λ,γ ,δ CtO ̸ = ∅ for all t ≥ 0 > 0 if and only if

(

)

( ) ν λ,γ ,δ η(O) = 1 or 2 > 0. For the proof of this proposition, see Section 3.5 of Foxall (2015). Now we give the proof of lim inf 2dλc (d, γ , δ ) ≥ 1 + d→+∞

1+δ

γ

.

Proof of lim infd→+∞ 2dλc (d, γ , δ ) ≥ 1 + Lemma 3.1 and Markov’s inequality,

1+δ

γ

. We assume that ζ0 (x) = 1 and θ0 (x) = 0 for all x ∈ Zd , then according to

λ,γ ,δ (

) ) ) λ,γ ,δ ( λ,γ ,δ ( ηt (O) = 1 or 2 = Pd ζt (O) ≥ 1 + Pd ζt (O) = 0, θt (O) ≥ 1 ) ) λ,γ ,δ ( λ,γ ,δ ( ≤ Pd ζt (O) ≥ 1 + Pd θt (O) ≥ 1

Pd

λ,γ ,δ

≤ Ed

λ,γ ,δ

ζt (O) + Ed

θt (O),

λ,γ ,δ

(3.3) λ,γ ,δ

where Ed is the expectation operator with respect to Pd . According to the transition rates function of (ζt , θt ) and Theorem 9.1.27 of Liggett (1985), which is an extended version of Hille–Yosida Theorem for the linear system,

⎧ d λ,γ ,δ λ,γ ,δ λ,γ ,δ ⎪ ⎪ ⎨ dt Ed ζt (O) = −Ed ζt (O) + γ Ed θt (O), ∑ λ,γ ,δ d λ,γ ,δ λ,γ ,δ ⎪ Ed ζt (y). ⎪ ⎩ dt Ed θt (O) = −(1 + γ + δ )Ed θt (O) + λ y:y∼O

X. Xue / Statistics and Probability Letters 140 (2018) 115–125 λ,γ ,δ

Conditioned on ζ0 (x) = 1, θ0 (x) = 0 for all x ∈ Zd , Ed homogeneity of our process, hence

(

d dt

λ,γ ,δ

Ed ζt (O) λ,γ ,δ Ed θt (O)

−1

γ

2dλ

−(1 + γ + δ )

(

)

)(

ζt (y) does not depend on the choice of y according to the spatial

λ,γ ,δ

Ed ζt (O) . λ,γ ,δ Ed θt (O)

=

119

)

(3.4)

We use G to denote

(

−1

γ

2dλ

−(1 + γ + δ )

) .

Let c1 , c2 be the two eigenvalues of G. When 2dλγ < 1 + γ + δ , c1 and c2 are reals that c1 , c2 < 0 and c1 ̸ = c2 since c1,2 = −

γ +δ+2 2

√ ±

(γ + δ )2 + 8dλγ 2

,

where 0 < (γ + δ )2 + 8dλγ < (γ + δ )2 + 4(1 + γ + δ ) = (γ + δ + 2)2 . By Eq. (3.4) and the classic theory of the linear ODE (See Theorem 9.5.2 of Adkins and Davidson, 2012), when 2dλγ < 1+γ +δ , λ,γ ,δ λ,γ ,δ since c1 ̸ = c2 , Ed ζt (O) = a1 ec1 t + a2 ec2 t and Ed θt (O) = b1 ec1 t + b2 ec2 t for any t ≥ 0, where a1 , a2 , b1 , b2 are four constants. As a result, when 2dλγ < 1 + γ + δ , λ,γ ,δ

lim Ed

t →+∞

λ,γ ,δ

ζt (O) = lim Ed t →+∞

θt (O) = 0,

since c1 , c2 < 0. Therefore, by Eqs. (3.2) and (3.3),

( ) ν λ,γ ,δ η(O) = 1 or 2 = 0 when λ <

1 (1 2d

+

1+δ

λc (d, γ , δ ) ≥

γ

(3.5)

). By Eq. (3.5) and Proposition 3.2,

1 2d

(1 +

1+δ

γ

)

and hence lim infd→+∞ 2dλc (d, γ , δ ) ≥ 1 +

1+δ

γ

.

□

4. The proof of lim supd →+∞ 2d λc (d , γ, δ) ≤ 1 +

1+δ

γ

In this section we give the proof of lim supd→+∞ 2dλc (d, γ , δ ) ≤ 1 + 1+δ . The proof utilizes the approach given in Xue γ (2017), the idea of which is inspired by the strategy introduced by Kesten in Kesten (1990). First we introduce a two-stage SIR(susceptible–infected–recovered) model. The two-stage SIR model {ρt }t ≥0 is a d continuous-time Markov process with state space {−1, 0, 1, 2}Z . The transition rates function of {ρt }t ≥0 is given as follows. d For any x ∈ Z and t ≥ 0,

ρt (x) → i at rate

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 if ρt (x) = 2 and i = −1, 1 + δ if ρt (x) = 1 and i = −1, γ∑ if ρt (x) = 1 and i = 2,

λ

1{ρt (y)=2} if ρt (x) = 0 and i = 1,

(4.1)

y:y∼x

0 otherwise,

where λ, γ , δ is defined as in Eq. (1.1). Intuitively, for the two-stage SIR model, vertices in state −1 are recovered. A recovered vertex can never be infected again and cannot infect others. A fully-infected vertex waits for an exponential time with rate 1 to become recovered while a semi-infected vertex waits for an exponential time with rate 1 + δ to become recovered. {ρt }t ≥0 can be defined equivalently through the graphic representation. Let wt (x), πt (x), ut (x, y) and µt (x) be defined as in Section 1, then {ρt }t ≥0 evolves as follows. For each vertex x, we observe the time-arrow on x through the positive direction. If we meet a ‘→’ from a neighbor in state 2 to x at some moment r while x is in state 0 at moment r −, then x becomes in state 1 at moment r. If we meet a ‘∗’ at some moment t while x is not in state 0 at t −, then x is in state −1 at t. If we meet a ‘∆’ at some moment s while x is in state 1 at s−, then x becomes in state −1 at s. If we meet a ‘⋆’ at some moment l while x is in state 1 at l−, then x becomes in state 2 at l. Otherwise, x stays its state. According to the theory introduced in Harris (1978), the process evolving as above has the transition rates function given in (4.1).

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Throughout this section we assume that there is no vertex in state 1 or −1 at t = 0 for the two-stage SIR model. We write

ρt as ρtO when {x : ρ0 (x) = 2} = {O}. We use DOt to denote {x : ρtO (x) = 1 or ρtO (x) = 2}

λ,γ ,δ

as the set of vertices in state 1 or 2 at the moment t for the two-stage SIR model. We use Pd to also denote the probability measure of the two-stage SIR model with parameters λ, γ , δ . If we utilize the graphic representations of our contact process and SIR model for {ηt }t ≥0 and {ρt }t ≥0 with the same infection rate λ and the same initial configuration that O is in state 2 d while other vertices are in state 0, then {ηt }t ≥0 and {ρt }t ≥0 are coupled under the same probability space. For η ∈ {0, 1, 2}Z Zd d and ρ ∈ {−1, 0, 1, 2} , we also write ρ ⪯ η when ρ (x) ≤ η(x) for each x ∈ Z . Then, according the definitions of the graphic representations of {ηt }t ≥0 and {ρt }t ≥0 , the condition ρt ⪯ ηt holds after each state-transition. As a result, λ,γ ,δ ( O

Pd

)

λ,γ ,δ ( O

Ct ̸ = ∅ for all t ≥ 0 ≥ Pd

Dt ̸ = ∅ for all t ≥ 0

)

(4.2)

for any λ, γ , δ > 0. For later use, we introduce some independent exponential times. For each x ∈ Zd , let W (x) be an exponential time with rate 1, Y (x) be an exponential time with rate 1 + δ while Γ (x) be an exponential time with rate γ . For each pair of neighbors x, y ∈ Zd , let U(x, y) be an exponential time with rate λ. Note that we care about the order of x and y, hence U(x, y) ̸ = U(y, x). We assume that all these exponential times are independent. For each n ≥ 1, we define

{

Ln = ⃗ x = (x0 , x1 , . . . , xn ) ∈ {Zd }n+1 :x0 = O, xi+1 ∼ xi for all 0 ≤ i ≤ n − 1 and xi ̸ = xj for any 0 ≤ i < j ≤ n

}

as the set of self-avoiding paths starting at O with length n. For each n ≥ 1 and each ⃗ x = (x0 , . . . , xn ) ∈ Ln , we use A⃗x to denote the event that U(xi , xi+1 ) < W (xi ) and Γ (xi+1 ) < Y (xi+1 ) for all 0 ≤ i ≤ n − 1. Then, the two-stage SIR model and these exponential times have the following coupling relationship. Lemma 4.1. {ρtO }t ≥0 and {W (x)}x∈Zd , {Y (x)}x∈Zd , {Γ (x)}x∈Zd , {U(x, y)}x∼y can be coupled under a same probability space such that for each n ≥ 1 and any ⃗ x = (x0 , . . . , xn ) ∈ Ln , A⃗x ⊆ {ρt (xn ) = 2 for some t ≥ 0} ⊆ {xn ∈ DOt for some t ≥ 0}. According to Lemma 4.1, in the sense of coupling, the ending vertex xn of the self-avoiding path ⃗ x has ever been fullyinfected on the event A⃗x . The detailed proof of Lemma 4.1 is a little tedious. Here we give an intuitive explanation that should be enough to convince the reader. Explanation of Lemma 4.1. The meanings of the exponential times we introduce are as follows. If a vertex x becomes semi-infected at some moment, then x waits for Y (x) units of time to become recovered or waits for Γ (x) units of time to become fully-infected, depending on whether Y (x) < Γ (x) or Γ (x) < Y (x). If x becomes fully-infected at some moment, then x waits for W (x) units of time to become recovered. For any y ∼ x, the fully-infected vertex x waits for U(x, y) units of time to infect y. This infection, which makes y semi-infected, really occurs when and only when y has not been infected by others at an earlier moment and U(x, y) < W (x). In other words, if we utilize the graphic representation of the SIR model, then we can obtain Γ , Y , W , U as follows. For O, W (O) is the first event moment of w· (O) while U(O, y) is the first event moment of u· (O, y) for any neighbor y of O. Y (O), Γ (O) can be defined as any independent exponential times with rates 1 + δ and γ , which are not utilized in the evolution of ρt . For x ̸ = O and y ∼ x, let t(x) = inf{s : there is an arrow ‘ → ’ from (z , s) to (x, s) for some z in state 2 at s}, t1 (x) = inf{s > t(x) : there is an ‘ ⋆ ’ on (x, s)}, t2 (x) = inf{s > t(x) : there is an ‘ ∗ ’ or ‘∆’ on (x, s)}, t3 (x) = inf{s > t1 (x) : there is an ‘ ∗ ’ on (x, s)}, t4 (x, y) = inf{s > t1 (x) : there is an arrow ‘ → ’ from (x, s) to (y, s)}. Then, if t(x) = +∞, W (x), Y (x), Γ (x), U(x, y) can be defined as any independent exponential times with rate 1, 1 + δ, γ , λ, which is not utilized in the evolution of ρt . If t(x) < +∞, then Γ (x) = t1 (x)−t(x), U(x, y) = t4 (x, y)−t1 (x), W (x) = t3 (x)−t1 (x) and

{ Y (x) =

t2 (x) − t(x) t1 (x) − t(x) + ˜ Y (x)

if t2 (x) ≤ t1 (x), if t1 (x) < t2 (x),

where ˜ Y (x) is an exponential time with rate 1 + δ which is independent of our process and not utilized in the evolution of

ρt .

X. Xue / Statistics and Probability Letters 140 (2018) 115–125

121

On the event A⃗x , we can deduce that x1 , . . . , xn all belong to t ≥0 DOt according to the following analysis. For x1 , there are two cases. The first case is that x1 is in state 0 at the moment before t = U(O, x1 ), then x1 becomes semi-infected at t = U(O, x1 ) since U(O, x1 ) < W (O) and ρ0 (O) = 2. Then, x1 becomes fully-infected at the moment U(O, x1 ) + Γ (x1 ) since Γ (x1 ) < Y (x1 ). The second case is that x1 becomes semi-infected at some moment s < U(O, x1⋃ ), then x becomes O fully-infected at s + Γ (x1 ) since Γ (x1 ) < Y (x1 ). In both cases, x1 has ever been fully-infected, i.e., x1 ∈ t ≥0 Dt . Applying ⋃ O this step repeatedly shows that x2 , x3 , . . . , xn ∈ t ≥0 Dt . □

⋃

Utilizing an approach given in Xue (2017), we consider a special type of self-avoiding paths. For each n ≥ 1, we define

{

Rn = ⃗ x = (x0 , . . . , xn ) ∈ Ln : xi+1 − xi ∈ {±ej : 1 ≤ j ≤ d − ⌊ such that ⌊log d⌋ ∤ (i + 1); xi+1 − xi ∈ {ej : d − ⌊

d log d

d log d

⌋} for any i

⌋ + 1 ≤ j ≤ d}

}

for any i such that ⌊log d⌋ | (i + 1) , where we use a | b to denote that b is divisible by a and {ej }1≤j≤d are the elementary unit vectors on Zd , i.e., ej = (0, . . . , 0, 1jth , 0, . . . , 0) for 1 ≤ j ≤ d. The idea of defining self-avoiding paths with the above form is inspired by the strategy introduced by Kesten in Kesten (1990), which relates the estimation of the upper bound of the critical value of percolation model (or contact process) with the times of collisions of two independent stochastic walks on the lattice. For some technique reason, this strategy requires the stochastic walk is oriented on a minor part of dimensions. Then this walk will not backtrack too much, which is easier for us to control the collision-times of two versions of this walk. ⋂independent ⋃ According to Lemma 4.1, on the event n≥1 ⃗x∈Rn A⃗x , there are vertices with arbitrarily large l1 -norm that have ever been fully-infected, which makes infected vertices survival since each fully-infected vertex waits for an exponential time with rate 1 to become recovered. Therefore, P λ,γ ,δ DOt ̸ = ∅ for all t ≥ 0 ≥ P

(

)

(⋂ ⋃

A⃗x .

)

n ≥1 ⃗ x∈Rn

Note that P

(⋃

⃗x∈Rn A⃗x

(⋂ ⋃

)

⊇

(⋃

⃗x∈Rn+1 A⃗x

)

A⃗x = lim P

)

(⋃

n→+∞

n≥1 ⃗ x∈Rn

for each n, then

A⃗x .

)

⃗x∈Rn

As a result, by Eq. (4.2), λ,γ ,δ ( O

) ( ) Ct ̸ = ∅ for all t ≥ 0 ≥ P λ,γ ,δ DOt ̸ = ∅ for all t ≥ 0 ⋃ ⋂ ⋃ ) ) ( ( A⃗x . A⃗x = lim P ≥P

Pd

n→+∞

n ≥1 ⃗ x∈Rn

To bound P

(⋃

⃗x∈Rn

⃗x∈Rn A⃗x

)

from below, we introduce a self-avoiding random walk {Sn }n≥0 on Zd such that

(S0 , S1 , . . . , Sn ) ∈ Rn for each n ≥ 1. Note that from now on we assume that d is sufficiently large that 2(d − ⌊

d log d

⌋) − ⌊log d⌋ ≥ 1.

We define S0 = O. For i ≥ 1 that ⌊log d⌋ | i,

⏐ ⏐

(

)

P Si = Si−1 + el ⏐Sj , 0 ≤ j ≤ i − 1 =

1

⌊ logd d ⌋

for each d − ⌊ logd d ⌋ + 1 ≤ l ≤ d. For i ≥ 1 that ⌊log d⌋ ∤ i,

(

⏐ ⏐

)

P Si = y⏐Sj , 0 ≤ j ≤ i − 1 =

1

|Hi−1 |

for any y ∈ Hi−1 , where

{

Hi−1 = z : z − Si−1 ∈ {±ej : 1 ≤ j ≤ d − ⌊

d log d

} ⌋} and Sj ̸= z for all 0 ≤ j ≤ i − 1

(4.3)

122

X. Xue / Statistics and Probability Letters 140 (2018) 115–125

while |A| is the cardinality of the set A. Note that Hi−1 is a random set measurable with respect to the σ -field generated by S0 , S1 , . . . , Si−1 . We claim that

|Hi−1 | ≥ 2(d − ⌊

d log d

⌋) − ⌊log d⌋

(4.4)

for each i ≥ 1. This claim holds according to the following analysis. For each x = (x1 , . . . , xd ) ∈ Zd , we define d ∑

u(x) =

|xi |,

d i=d−⌊ log ⌋+1 d

then u(S· ) increases by 1 every ⌊log d⌋ steps and hence

⏐{ }⏐⏐ ⏐ ⏐ 0 ≤ j ≤ i − 1 : u(Sj ) = u(Si−1 ) ⏐ ≤ ⌊log d⌋. As a result,

|{S0 , S1 , . . . , Si−1 }

⋂ d ⌋}| ≤ ⌊log d⌋, {Si−1 ± ej : 1 ≤ j ≤ d − ⌊

(4.5)

log d

since u(z) = u(Si−1 ) for any z ∈ {Si−1 ± ej : 1 ≤ j ≤ d − ⌊ logd d ⌋}. Eq. (4.4) follows from Eq. (4.5) directly.

According to the definition of {Sn }n≥0 , it is easy to check that (S0 , S1 , . . . , Sn ) ∈ Rn for each n ≥ 1. We let {Vn }n≥0 be an independent copy of {Sn }n≥0 with V0 = O. For simplicity, we use S⃗n to denote (S0 , . . . , Sn ) and use V⃗n to denote (V0 , . . . , Vn ) ⃗ = (y0 , . . . , yn ) ∈ Rn , we define for each n ≥ 1, then S⃗n , V⃗n ∈ Rn . For any ⃗ x = (x0 , . . . , xn ), y

⃗) = 0 ≤ i ≤ n : yi = xj for some 0 ≤ j ≤ n F (⃗ x, y

{

}

and

⃗) = 0 ≤ i ≤ n − 1 : yi = xj and yi+1 = xj+1 for some 0 ≤ j ≤ n − 1 . K (⃗ x, y

{

}

We use ˆ P to denote the probability measure of {Sn , Vn }n≥0 and use ˆ E to denote the expectation operator with respect to ˆ P, then the following lemma is crucial for us to prove lim supd→+∞ 2dλc (d, γ , δ ) ≤ 1 + 1+δ . γ Lemma 4.2. For each n ≥ 1, P

(⋃

)

A⃗x ≥

⃗x∈Rn

1

[ ⃗ ⃗ ⃗ ⃗ ( ]. ) ⃗n )|−1 ( λ+1 )|K (S⃗n ,V⃗n )| |F (S ,V )\K (Sn ,Vn )| 1+γ +δ |F (S⃗n ,V ˆ E 2 n n γ

λ

We give the proof of Lemma 4.2 at the end of this section. Now we show that how to utilize Lemma 4.2 to prove lim supd→+∞ 2dλc (d, γ , δ ) ≤ 1 + 1+δ . γ Proof of lim supd→+∞ 2dλc (d, γ , δ ) ≤ 1 +

1+δ

γ

F (S⃗, V⃗ ) = i ≥ 0 : Vi = Sj for some j ≥ 0

{

. Let

}

and K (S⃗, V⃗ ) = i ≥ 0 : Vi = Sj and Vi+1 = Sj+1 for some j ≥ 0 ,

{

}

then lim |K (S⃗n , V⃗n )| = |K (S⃗, V⃗ )| while

n→+∞

lim |F (S⃗n , V⃗n )| = |F (S⃗, V⃗ )|

n→+∞

and hence

( 1 + γ + δ )|F (S⃗n ,V⃗n )|−1 ( λ + 1 )|K (S⃗n ,V⃗n )| ] n→+∞ γ λ [ |F (S⃗,V⃗ )\K (S⃗,V⃗ )| ( 1 + γ + δ ) ⃗ ⃗ ( λ + 1 ) ⃗ ⃗ ] |F (S ,V )|−1 |K (S ,V )| =ˆ E 2 γ λ ( ) [( 2(1 + γ + δ ) )|F (S⃗,V⃗ )\K (S⃗,V⃗ )| ( λ + 1 1 + γ + δ )|K (S⃗,V⃗ )| ] γ ˆ = E 1+γ +δ γ λ γ [

|F (S⃗n ,V⃗n )\K (S⃗n ,V⃗n )|

lim ˆ E 2

according to the monotone convergence theorem. Then by Eq. (4.3) and Lemma 4.2, λ,γ ,δ ( O

Pd

≥ (

)

Ct ̸ = ∅ for all t ≥ 0

1

γ 1+γ +δ

) [( 2(1+γ +δ) )|F (S⃗,V⃗ )\K (S⃗,V⃗ )| ( λ+1 1+γ +δ )|K (S⃗,V⃗ )| ] . ˆ E γ

λ

γ

(4.6)

X. Xue / Statistics and Probability Letters 140 (2018) 115–125

123

Ref. Xue (2017) gives a detailed calculation of the function |F (S⃗,V⃗ )\K (S⃗,V⃗ )| |K (S⃗,V⃗ )|

[

J(C1 , C2 ) = ˆ E C1

C2

] .

According to Lemma 3.4 of Xue (2017), there exists M1 , M2 which do not depend on C1 , C2 and the dimension d of the lattice that J(C1 , C2 ) ≤ M2 C1

n=0

for any C1 , C2 > 0. For given ϑ > 1, let λ = (⌊log d⌋

3 ⌊log d⌋−1

3

+∞ [ ∑

1 ) λ+ λ

(⌊log d⌋ 2(d − ⌊

d log d

ϑ 1+γ +δ , γ

2d

+

2(d − ⌊ logd d ⌋) − ⌊log d⌋

)C2

C2

+

⌋) − ⌊log d⌋

⌊

d log d

3

+

M1 (log d)5 C1 ]n d

⌋⌊log d⌋

then it is easy to check that λ+1 1+γ +δ λ γ

1+γ +δ

γ

⌊log d⌋−1

⌊ logd d ⌋⌊log d⌋3

+

M1 (log d)5

2(1+γ +δ )

γ

d

<1

for sufficiently large d and hence J

( 2(1 + γ + δ ) λ + 1 1 + γ + δ ) , < +∞ γ λ γ

for sufficiently large d. As a result, by Eq. (4.6), λ,γ ,δ ( O

Pd

)

Ct ̸ = ∅ for all t ≥ 0 ≥ (

when λ =

ϑ 1+γ +δ γ

2d

1 γ 1+γ +δ

) ( 2(1+γ +δ) J

γ

, λ+λ 1 1+γγ +δ

) >0

with ϑ > 1 and d is sufficiently large. Therefore,

ϑ 1+γ +δ ϑ 1+δ = (1 + ) γ 2d γ

λc (d, γ , δ ) ≤

2d

for sufficiently large d and hence lim sup 2dλc (d, γ , δ ) ≤ ϑ (1 +

1+δ

γ

d→+∞

)

for ϑ > 1. Since ϑ > 1 is arbitrary, let ϑ ↓ 1 then the proof is complete. □ To finish this section, we need to prove Lemma 4.2. The proof of Lemma 4.2 relies on the following proposition, which is Lemma 3.3 of Xue (2017). Proposition 4.3 (Xue, 2017). If B1 , B2 , . . . , Bn are n arbitrary events∑ defined under the same probability space such that P(Bi ) > 0 n for 1 ≤ i ≤ n and p1 , p2 , . . . , pn are n positive constants such that j=1 pj = 1, then P(

+∞ ⋃ j=1

1

Bj ) ≥ ∑ n

i=1

⋂

P(B

∑n

B)

i j j=1 pi pj P(B )P(B ) i

.

j

For the proof of Proposition 4.3, see Section 3 of Xue (2017). At last we give the proof of Lemma 4.2. Proof of Lemma 4.2. For each ⃗ x ∈ Rn , let p⃗x be the probability that S⃗n = ⃗ x, then by Proposition 4.3, P

(⋃

1

)

A⃗x ≥

(

∑

⃗x∈Rn

⃗x∈Rn

(

P A⃗x

⋂

Ay⃗

P A⃗x

∑

⃗∈Rn y

⋂

Ay⃗

)

.

(4.7)

p⃗x py⃗ ( ) ( ) P A⃗x P Ay⃗

)

Now we bound ( ) ( ) from above. For P A⃗x P Ay⃗

⃗x = (x0 , . . . , xn ), y⃗ = (y0 , . . . , yn ) ∈ Rn , { } if xi ̸ ∈ y0 , . . . , yn for some 0 < i ≤ n, then the factor ( ) ( ) P U(xi , xi+1 ) < W (xi ) P Γ (xi ) < Y (xi ) ( ⋂ ) ( ) ( )

appears once in both P A⃗x then the factor

Ay⃗ and P A⃗x P Ay⃗ , which can be canceled. Similarly, if yj ̸ ∈ x0 , . . . , xn for some 0 < j ≤ n,

P U(yj , yj+1 ) < W (yj ) P Γ (yj ) < Y (yj )

(

) (

)

{

}

124

X. Xue / Statistics and Probability Letters 140 (2018) 115–125

(

appears once in both P A⃗x and the factor

⋂

⃗) \ {0}, then xi = yj for some 0 < i ≤ n Ay⃗ and P A⃗x P Ay⃗ , which can be canceled. If j ∈ F (⃗ x, y

)

( ) ( )

P Γ (yj ) < Y (yj ) = P Γ (xi ) < Y (xi ) =

(

)

(

)

γ 1+γ +δ

( ) ( )

(

appears twice in P A⃗x P Ay⃗ but appears once in P A⃗x

⋂

)

Ay⃗ since

⋂ {Γ (xi ) < Y (xi )} = {Γ (yj ) < Y (yj )},

{Γ (yj ) < Y (yj )}

(

which generates a factor

1+γ +δ

γ

for

P A⃗x

⋂

Ay⃗

)

( ) ( ).

P A⃗x P Ay⃗

⃗), then xi = yj while xi+1 = yj+1 for some 0 ≤ i ≤ n − 1 and the If j ∈ K (⃗ x, y

factor P U(yj , yj+1 ) < W (yj ) = P U(xi , xi+1 ) < W (xi ) =

(

)

(

)

( ) ( )

(

appears twice in P A⃗x P Ay⃗ but appears once in P A⃗x

{U(yj , yj+1 ) < W (yj )}

)

Ay⃗ since

⋂ {U(xi , xi+1 ) < W (xi )} = {U(yj , yj+1 ) < W (yj )}, (

which generates a factor

⋂

λ 1+λ

λ+1 λ

P A⃗x

⋂

Ay⃗

)

⃗) \ K (⃗x, y⃗), then xi = yj while xi+1 ̸= yj+1 for some 0 ≤ i ≤ n − 1 and x, y for ( ) ( ) . If j ∈ F (⃗ P A⃗x P Ay⃗

hence)P (A⃗x Ay⃗ has the factor ) P U(xi , xi+1 ) < W (xi ), U(xi , yj+1 ) < W (xi ) while P A⃗x P Ay⃗ has the factor P U(xi , xi+1 ) < W (xi ) P U(xi , yj+1 ) < W (xi ) , which generates a factor

(

)

⋂

(

P U(xi , xi+1 ) < W (xi ), U(xi , yj+1 ) < W (xi )

(

)

) )=

P U(xi , xi+1 ) < W (xi ) P U(xi , yj+1 ) < W (xi )

(

(

for

P A⃗x

) (

⋂

Ay⃗

2λ + 2 2λ + 1

( ) ( )

(

≤2

)

( ) ( ).

P A⃗x P Ay⃗

In conclusion,

⋂ ) ( 1 + γ + δ )|F (⃗xn ,⃗yn )\{O}| ( 1 + λ )|K (⃗x,⃗y)| ( )|F (⃗x,⃗y)\K (⃗x,⃗y)| Ay⃗ ( ) ( )≤ 2 γ λ P A⃗x P Ay⃗ |F (⃗x,⃗y)\K (⃗x,⃗y)| ( 1 + γ + δ )|F (⃗ x,⃗ y)|−1 ( λ + 1 )|K (⃗ x,⃗ y)| =2 . γ λ (

P A⃗x

(4.8)

By Eq. (4.8) and the definition of p⃗x ,

∑∑

(

p⃗x py⃗

⃗x∈Rn y⃗∈Rn

⋂ ) Ay⃗ ( ) ( )

P A⃗x

P A⃗x P Ay⃗

( 1 + γ + δ )|F (⃗x,⃗y)|−1 ( λ + 1 )|K (⃗x,⃗y)| ] γ λ ⃗x∈Rn y⃗∈Rn [ |F (S⃗ ,V⃗ )\K (S⃗ ,V⃗ )| ( 1 + γ + δ ) ⃗ ⃗ ] |F (Sn ,Vn )|−1 ( λ + 1 )|K (S⃗n ,V⃗n )| n n n n . =ˆ E 2 γ λ

≤

∑∑

[

|F (⃗x,⃗y)\K (⃗x,⃗y)|

p⃗x py⃗ 2

(4.9)

Lemma 4.2 follows from Eqs. (4.7) and (4.9) directly. □ Acknowledgments The author is grateful to the reviewers, their comments were a great help for us to improve this paper. The author is grateful to the financial support from the National Natural Science Foundation of China with grant number 11501542 and the financial support from Beijing Jiaotong University with grant number KSRC16006536. References Adkins, W.A., Davidson, M.G., 2012. Ordinary Differential Equations. Springer, New York. Foxall, E., 2015. New results for the two-stage contact process. J. Appl. Probab. 52, 258–268. Griffeath, D., 1983. The binary contact path process. Ann. Probab. 11, 692–705. Harris, T.E., 1974. Contact interactions on a lattice. Ann. Probab. 2, 969–988. Harris, T.E., 1978. Additive set-valued Markov processes and graphical methods. Ann. Probab. 6, 355–378. Kesten, H., 1990. Asymptotics in high dimensions for percolation. In: Disorder in physical systems, a volume in honor of John Hammersley on the occasion of his 70th birthday, Oxford, pp. 219–240. Krone, S., 1999. The two-stage contact process. Ann. Appl. Probab. 9, 331–351.

X. Xue / Statistics and Probability Letters 140 (2018) 115–125 Liggett, T.M., 1985. Interacting Particle Systems. Springer, New York. Liggett, T.M., 1999. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York. Xue, X.F., 2017. Asymptotic for critical value of the large-dimensional SIR epidemic on clusters. J. Theoret. Probab. published online.

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