Consider the contact process with parameter $ \lambda \in (0,1)$ on the triangle. Let $X(t)$ denote the total number of infections at time $t$. Compute the transition rates of this process.
Our teacher gave us the solutions: The transition rates $r(i, j)$ from $i$ to $j$ are: $r(3; 2) = 3$, $r(2; 1) = 2$, $r(1; 0) = 1$, $r(1; 2) = 2 \lambda$, $r(2; 3) = 2 \lambda$, and all other rates are zero.
The definition of the rates is the following:
$$c(x,n) = 1 \text { if } n(x)=1$$ $$c(x,n) = \lambda \sum_{y~x} n(y) \text{ if } n(x)=0$$
By calculating the rates with the definition I get for $r(2,1) = 1$, because it goes from a state were two people are injected to a state were just one is.. so $n(x)=1$ and therefore the rate is equal to 1?
And so on... $r(1,0) =1$, $r(1,2) = \lambda$, $r(2,3)= 2 \lambda$.
What am I doing wrong? How am I supposed to compute the rates?