Quasistationary distributions of some infection models Damian Clancy University of Liverpool, UK.

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Transcript of Quasistationary distributions of some infection models Damian Clancy University of Liverpool, UK.
Quasistationary distributions of some infection models
Damian Clancy
University of Liverpool, UK
SIS model
Population of S susceptibles, I infectives. N = S + I constant (population size).
Transitions Rates
Transmission: I I+1 (β/N) I (NI)Recovery: I I1 γ I
An example of a finite statespace birthdeath process.
[QSD – Nåsell papers]
Birthdeath 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 restarted, from state i = 1,2,…,N with probability νi. Restarted 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 birthdeath processes,
i
j
i
jk
N
jr rk
k
ii
1
11
1
Likelihood ratio ordering
Now straightforward to show that the map Φ preserves LR ordering of distributions.
For two distributions ν and ρ,
jij
i
j
iLR whenever
[Stochastic ordering:
jj
ii
j
ii
ST each for 11
]
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 restarted 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 LRordering sense and converging to the fixed point q, hence we have q <LR m2.
SIS/W modelPopulation of S susceptibles, I infectives, W freeliving bacteriaN = S + I constant (herd size)
Transitions Rates
Direct transmission: (I, W) (I+1, W) (β/N) I (NI)Recovery: (I, W) (I1, W) γ I
Shedding: (I, W) (I, W+1) λ I
Indirect transmission: (I, W) (I+1, W1) (ν/N) pW (NI)
Depletion:
(I, W) (I, W1) μW + (ν/N) (1p)W (NI) + (ν/N) WI
[Total W W1 rate of μW + νW]
Deterministic model
Deterministic SIS/W model (with N = 1) has equilibrium points at (0,0) and
**0
* /11 iwRi
where
pR0
Limiting Conditional Distributions
0,0, ,, lim t
, ttttwi WIwiWIPm
0min lim
0t u
tuti IiIPr 
State space (I, W) {0, 1, …, N}×{0, 1, …}
Absorbing state (I, W) = (0,0)
LCDs of interest:
Existence of a QSDFerrari, 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
0any for 0
ttTPlim xx
then a QSD exists if and only if
.0 somefor Tx eE
SIS/W model: Take X = I + (N+1)W, then conditions of FKMP are satisfied with absorbing state (I, W) = (0,0).
(AR)
Asymptotic remoteness:
Each bacterium has survival time Exp(μ + ν), so
wetTPwt
x as 01
Second condition:
FKMP point out this is equivalent to existence of a nonnegative “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.
‘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.
Normal approximationProvided R0 > 1, the process converges towards the nonzero equilibrium. Deviations from the equilibrium can be approximated by a 2dimensional OrnsteinUhlenbeck process, whose equilibrium distribution is bivariate Normal, mean zero, variance matrix expressible in terms of the parameters of the process. In particular,
D
spsI
22 1
where
221 spssD
*1 is
Effect of indirect transmissionConsider 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*/(1i*))]
Approximating PDMPClosed population of size N consisting of It infectives and NIt
susceptibles, together with Wt bacteria.
Infection: I I+1 rate (β/N) I (NI) + (ν/N) p WD (NI)
Recovery:I I1 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).
(Piecewisedeterministic Markov process.)
DD WI
dt
dW
A sample path
I
W
(N=70)
Computed (marginal) LCDs
Persistence times
Have seen that indirect transmission leads to lower variance, compared to direct transmission.
Hence indirect transmission can lead to much smaller quasistationary 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.
An open population SIR model
Population of S susceptibles, I infectives.
Transitions Rates
Transmission: (S, I) (S1, I+1) β I S
Recovery: (S, I) (S, I1) γ I
Birth: (S, I) (S+1, I) μ S
Statespace (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.
Empirical LCD
S
I
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.)
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
22