1. Quasi-stationary distributions of some infection models
Damian Clancy
University of Liverpool, UK

2. SIS model
Population of S susceptibles, I infectives. N = S + I constant (population size).
Transitions Rates
Transmission: I  I+1 (β/N) I (NI)
Recovery: I  I1 γ I
An example of a finite state-space birth-death process.
[QSD – Nåsell papers]

3. Birth-death processes
State i = 0,1,…,N, birth rates λi, death rates μi, absorbing state i=0.
Suppose every time state i=0 is reached, the process is re-started, from state i = 1,2,…,N with probability νi. Re-started process has stationary distribution Φ(ν).
The QSD q of the original process satisfies Φ(q) = q.
(Ferrari, Kesten, Martinez and Picco, Ann Prob, 1995)
For birth-death processes,

4. Likelihood ratio ordering
Now straightforward to show that the map Φ preserves LR ordering of distributions.
For two distributions ν and ρ,
[Stochastic ordering: ]

5. Back to the SIS infection model
Kryscio and Lefevre, JAP 1989, write down two distributions m1 and m2, show that m1 <ST q and conjecture that q <ST m2.
In terms of re-started processes, m1 = Φ(1,0,0,…,0). Now (1,0,0,…,0) <LR q, so we have m1 <LR Φ(q) = q.
On the other hand, a little algebra shows that m2 = Φ(ρ) for a distribution ρ which can be shown to satisfy Φ(ρ) <LR ρ.
Repeated iteration of the map Φ, starting from the distribution ρ, gives a sequence of distributions decreasing in the LR-ordering sense and converging to the fixed point q, hence we have q <LR m2.

6. SIS/W model
Population of S susceptibles, I infectives, W free-living bacteria N = S + I constant (herd size)
Transitions Rates
Direct transmission: (I, W)  (I+1, W) (β/N) I (NI)
Recovery: (I, W)  (I1, W) γ I
Shedding: (I, W)  (I, W+1) λ I
Indirect transmission: (I, W)  (I+1, W1) (ν/N) pW (N-I)
Depletion:
(I, W)  (I, W1) μW + (ν/N) (1-p)W (N-I) + (ν/N) WI
[Total W  W1 rate of μW + νW]

7. Deterministic model
Deterministic SIS/W model (with N = 1) has equilibrium points at (0,0) and
where

8. Limiting Conditional Distributions
State space (I, W) {0, 1, …, N}×{0, 1, …}
Absorbing state (I, W) = (0,0)
LCDs of interest:

9. Existence of a QSD
Ferrari, Kesten, Martinez and Picco (Ann Prob, 1995): For a Markov process on {0,1,…} with absorbing state 0, states {1,2,…} a single communicating class, then denoting by T the hitting time of state 0, if
(AR)
then a QSD exists if and only if
SIS/W model: Take X = I + (N+1)W, then conditions of FKMP are satisfied with absorbing state (I, W) = (0,0).

10. Asymptotic remoteness:
Each bacterium has survival time Exp(μ + ν), so
Second condition:
FKMP point out this is equivalent to existence of a non-negative “uniform supermartingale” f(Xt)
We can take f(Xt) = Wt
Outside a finite part of the state space (the part where W is small), f(Xt) is expected to decrease at a rate bounded away from zero.

11. ‘Outbreak’ LCD
There exists a QSD conditional upon not hitting (I, W) = (0, 0).
During one outbreak of infection in the animals, would be interested in the LCD conditional upon not hitting the set of states I = 0.
Does not seem possible to establish QSD existence using FKMP, failure of asymptotic remoteness since all states with I = 1 (and W = 0, 1, 2, 3, …) are adjacent to states with I = 0.

12. Normal approximation
Provided R0 > 1, the process converges towards the non-zero equilibrium. Deviations from the equilibrium can be approximated by a 2-dimensional Ornstein-Uhlenbeck process, whose equilibrium distribution is bivariate Normal, mean zero, variance matrix expressible in terms of the parameters of the process. In particular,
where

13. Effect of indirect transmission
Consider varying any one of the parameters (p, λ, μ, ν) at the same time as varying β in such a way that R0 remains unchanged. That is, keep the total amount of transmission per animal constant, so i* (mean proportion of infected animals) remains constant, but vary the proportion of transmission which is direct/indirect. Can show that as the proportion of transmission which is indirect increases, the variance of the number of infected animals in (quasi-)equilibrium decreases. [Provided ν <  (2 + γ i*/(1-i*))]

14. Approximating PDMP
Closed population of size N consisting of It infectives and NIt susceptibles, together with Wt bacteria. Infection: I  I+1 rate (β/N) I (NI) + (ν/N) p WD (N-I) Recovery: I  I1 rate γ I Bacterial shedding / depletion: The process (I, W) is Markov, with uncountable state space {0,1,…,N}×[0,), and absorbing state (0, 0). (Piecewise-deterministic Markov process.)

15. A sample path
(N=70) I W

16. Computed (marginal) LCDs

17. Persistence times
Have seen that indirect transmission leads to lower variance, compared to direct transmission.
Hence indirect transmission can lead to much smaller quasi-stationary probability of being in states with I = 1.
This means that indirect transmission can result in much longer persistence times, compared to direct transmission, even though the average amount of transmission is the same.

18. An open population SIR model
Population of S susceptibles, I infectives. Transitions Rates
Transmission: (S, I)  (S-1, I+1) β I S
Recovery: (S, I)  (S, I1) γ I
Birth: (S, I)  (S+1, I) μ S
State-space (S, I) D = {0, 1, …}×{0, 1, …}
Irreducible class (S, I)  C = {1, 2, …}×{1, 2, …}, from which exit occurs almost surely within finite time.

19. Empirical LCD S I

20. Approximation of LCD
The marginal LCDs of S and I both look very much like Geometric distributions, and seem to be approximately uncorrelated.
Further, the deterministic version of this model has a (neutral) eqiulibrium at (s*, i*) = (γ/ β, μ /β), and it seems (numerically) that these values approximate the means of the marginal LCDs.
So we can approximate the LCD as the product of independent Geometric distributions, with means given by the deterministic equilibrium values of S, I.
(Although this approximation does not satisfy the equations for a QSD.)

21. Duration
Starting from the QSD, the time until exit from the irreducible class C is Exponentially distributed. Using our bivariate Geometric approximation, can approximate the mean exit time as