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附录
引理4.1证明 首先将接触过程和与分支过程按下述方式进行马氏性耦合,以{(μt(i),ζt(i)),(θt(i),ρt(i))}形式表示接触过程与分支过程的耦合,其中第一个分量(μt(i),ζt(i))表示接触过程半感染个体数与全感染个体数的向量形式,第二个分量(θt(i),ρt(i)) 表示分支过程半感染个体数与全感染个体数的向量形式.
设{(μt(i),ζt(i)),(θt(i),ρt(i))}当前状态为{(i,j),(k,l)},其中i≤ k,j ≤l.则有以下耦合过程: {(0,1),(0,1)}→{(0,0),(0,0)},以速率1; →{(0,1),(1,1)},以速率λ∑iρt(i). {(1,0),(0,0)}→{(0,1),(0,1)},以速率γ; →{(0,0),(0,0)},以速率1+δ; →{(1,0),(2,0)},以速率λ∑iρt(i). {(0,0),(0,1)}→{(0,0),(0,0)},以速率1; →{(0,0),(1,1)},以速率λ∑iρt(i)-λ∑i≠jζt(j); →{(1,0),(1,1)},以速率λ∑i≠jζt(j). {(0,0),(1,0)}→{(0,0),(0,0)},以速率1+δ; →{(0,0),(0,1)},以速率γ; →{(0,0),(2,0)},以速率λ∑iρt(i)-λ∑i≠jζt(j); →{(1,0),(2,0)},以速率λ∑i≠jζt(j). {(0,0),(1,1)}→{(0,0),(0,1)},以速率1+δ; →{(0,0),(0,2)},以速率γ; →{(0,0),(1,0)},以速率1; →{(0,0),(2,1)},以速率λ∑iρt(i)-λ∑i≠jζt(j); →{(1,0),(2,1)},以速率λ∑i≠jζt(j). {(0,1),(1,1)}→{(0,1),(0,1)},以速率1+δ; →{(0,1),(0,2)},以速率γ; →{(0,0),(1,0)},以速率1; →{(0,1),(2,1)},以速率λ∑iρt(i). {(1,0),(1,1)}→{(0,0),(0,1)},以速率1+δ; →{(0,1),(0,2)},以速率γ; →{(1,0),(0,0)},以速率1; →{(1,0),(2,1)},以速率λ∑iρt(i).
{(0,0),(θt(i),ρt(i))}→0,0,θt(i)+1,ρt(i),以速率(1+δ)θt(i); →0,0,θt(i),ρt(i)+1,以速率γθt(i); →1,0,θt(i)+1,ρt(i), 以速率λ∑iρt(i); →0,0,θt(i)+1,ρt(i) ,以速率λ∑iρt(i)-λ∑i≠jζt(j); →0,0,θt(i),ρt(i)-1 ,以速率ρt(i).
0,1,θt(i),ρt(i)→0,1,θt(i)-1,ρt(i),以速率(1+δ)θt(i);
→0,1,θt(i)-1,ρt(i)+1,以速率γθt(i); →0,1,θt(i)+1,ρt(i),以速率λ∑iρt(i); →0,0,θt(i),ρt(i)-1,以速率1; →0,1,θt(i),ρt(i)-1,以速率ρt(i).
1,0,θt(i),ρt(i)→1,0,θt(i)-1,ρt(i),以速率(θt(i)-1)(1+δ); →0,0,θt(i)-1,ρt(i),以速率1+δ; →1,0,θt(i)-1,ρt(i)+1,以速率(θt(i)-1)γ; →0,1,θt(i)-1,ρt(i)+1,以速率γ; →1,0,θt(i)+1,ρt(i),以速率λ∑iρt(i); →1,0,θt(i),ρt(i)-1,以速率ρt(i); 0,0,0,ρt(i)→0,0,0,ρt(i)-1,以速率ρt(i); →0,0,1,ρt(i),以速率λ∑iρt(i)-λ∑i≠jζt(j); →1,0,1,ρt(i),以速率λ∑i≠jζt(j).
0,0,θt(i),0→0,0,θt(i)-1,1,以速率γθt(i); →0,0,θt(i)-1,0,以速率(1+δ)θt(i); →0,0,θt(i)+1,0,以速率λ∑iρt(i)-λ∑i≠jζt(j); →1,0,θt(i)+1,0,以速率λ∑i≠jζt(j).
1,0,θt(i),0→1,0,θt(i)+1,0,以速率λ∑jρt(j); →0,1,θt(i)-1,1,以速率γ; →0,0,θt(i)-1,0,以速率1+δ; →1,0,θt(i)-1,1,以速率(θt(i)-1)γ; →1,0,θt(i)-1,0,以速率(1+δ)(θt(i)-1).
0,1,0,ρt(i)→0,0,0,ρt(i)-1,以速率1; →0,1,1,ρt(i),以速率λ∑jρt(j); →0,1,0,ρt(i)-1,以速率ρt(i). 在此耦合下, μt(i)≤θt(i),ζt(i)≤ρt(i). 即接触过程半感染个体数与全感染个体数均不大于分支过程半感染个体数与全感染个体数.证毕.
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