Chapter 3 Characterisation of QSDs for Birth-Death and Epidemic
3.3 Pure Death Processes
3.3.2 General Death Processes
We now extend our process to the case with general death rates. If we return to the finite case, where the process evolves on{1, . . . , L}we obtain existence of a QSD by Theorem 2.1.10. Indeed, one can easily see that the probability measure giving full weight to state 1 is still a QSD in the general case; think of it as the 1-LCD. On the other hand, the claim of uniqueness cannot hold in general. Indeed, consider the following example.
Example 3.3.4. Consider the death process (X(t))t≥0 on S ={1, . . . ,5}with the
death rates γ1 = 3, γ2 = 2, γ3 = 3, γ4 = 1, γ5 = 3. If we consider the initial
condition X(0) = 5, then by solving the x-invariance equation, we obtain the QSD given by u(x) = (1/3,1/3,1/9,2/9,0). For X(0) = 4 we obtain the same result. However, one can also show that (1,0,0,0,0) is still a QSD for this process, and so uniqueness doesn’t hold.
In order to better characterise the QSDs of the process, we instead consider the LCDs starting from different initial conditions. For a fixed starting positionj, we know the j-LCD exists and is unique, but more generally, we would like to know for which pairs (i, j) the i-LCD and j-LCD coincide. We take the known results regarding the existence of the minimal QSDs on a reducible state space in Theorem 2.1.11, and from van Doorn and Pollett [2008] and extend them for the death process to fully characterise the set ofj-LCDs.
Theorem 3.3.5. Let (X(t))t≥0 be a general death process on S∪ {0}={0, . . . , L} with rates {γ1, . . . , γL} (γi > 0 for all i = 1, . . . , L). For j = 1, . . . , L define
γ∗(j) = min{γi : i ≤ j}, and let i∗(j) = min{i ≤ j : γi = γ∗}. Then the j-LCD
gives weight precisely to the states{1, . . . , i∗(j)}.
Proof. For each j = 1, . . . , L, we can consider the process starting from j to be a process which evolves on{0, . . . , j}only without loss of generality. Then we have the decay parameter for the process must beα =γ∗(j). If γi 6=γ∗(j) for all i6=i∗(j),
thenαhas geometric multiplicity one and therefore, by Theorem 2.1.11, there exists a QSD which isγ∗(j)-invariant forQand gives weight to all the states{1, . . . , i∗(j)}. This is because state i∗(j) is minimal for α = γ∗(j). Furthermore, we must have that this is the j-LCD by the same theorem. If γk = γ∗(j) for some k ≤ j with
k6=i∗(j), then straight away it must be the case thatk > i∗(j) by definition ofi∗(j). Moreover, we can see from the x-invariance equations, that under γ∗(j)-invariance we must have
−γ∗(j)ui∗(j)=−γ∗(j)ui∗(j)+γi∗(j)+1ui∗(j)+1
which forcesui∗(j)+1= 0, and henceforth ui = 0 fori > i∗(j), and in particularuk. Therefore, despite the multiplicity, we still have uniqueness of the γ∗(j)-invariant QSD.
gets hung up on during its progression to zero. What this result tells us is that from any initial starting location j, the QSD gives weight up to the tightest bottleneck below the starting location. In the linear case (Section 3.3.1), the only bottleneck is in state 1, so all LCDs are the same. In Example 3.3.4 above, we have bottlenecks at states 2 and 4, so the 3-LCD gives weight to states 1 and 2, and the 5-LCD gives weight to the first four states.
Note that in the above theorem, this can be extended immediately to include other initial distributions with finite support. For initial distributionvwith finite support, ifjv= max supp(v) then thev-LCD is exactly the jv-LCD.
Note that we can use Theorem 3.3.5 to write down the full list of QSDs for the finite death process.
Corollary 3.3.6. For the general death process onLstates with death rates(γi)Li=1,
let A={1≤i≤L:γi < γj for all j < i}. Then the full set of QSDs for the death
process is precisely the set ofi-LCDs for i∈A. Each i-LCD is γ∗(i)-invariant for
Q.
The above theorem and corollary fully characterise the QSDs and LCDs for the general death process on a finite state space. We note that since these exist on the finite state space there are only finitely many such distributions.
Countable State Space
If we now consider the countable case, then we have death rates (γi)∞i=1. One can
apply Theorem 3.3.5 to obtain the finite support LCDs.
Corollary 3.3.7. Let (X(t))t≥0 be a general death process on N∪ {0} with death
rates γi ≥c > 0 for some c >0. Then for each j ≥1 there exists a j-LCD which
gives weight precisely to states {1, . . . , i∗(j)} and which is γ∗(j)-invariant. This
j-LCD is also equal to the v-LCD for anyv withmax(supp(v)) =j.
This holds since for any j > 0 we can consider the process to be evolving just on
{1, . . . , j} and this returns us to the finite case. We can however extend this result
to take advantage of the countable setting. The following result is fairly general, but can be applied in the special case of the linear death process and generate all of the QSDs.
Theorem 3.3.8. Let (X(t))t≥0 be a general death process on N∪ {0} with death
rates c < γi < C strictly bounded away from zero and infinity by c, C independent
of i, and attaining the infimum, γ∗ = infjγj, so γk=γ∗ for somek. Then for each
x∈(0, γ∗) there exists anx-invariant QSD which is a proper probability distribution giving mass to all states in N.
Proof. Note that in this case, since we are considering the countable state space, we need only consider the smallest bottleneck, which we force to exist by construc- tion. In this case, we can iteratively solve thex-invariance equations, which all give positive solutions since they are of the form
uj+1=
γj−x
γj+1
uj
whereγj −x >0 for all x < γ∗. This can then be normalised since
uj+1 = γ1−x γj+1 j Y i=2 γi−x γi u1 < γ1−x γ∗ 1− x C j−1 u1
This follows since 0< γj −x < γj. This boundedness shows boundedness above of
P
iui by the geometric series, and hence we can renormalise the sum to obtain the