Quasi-stationary 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 state-space birth-death process.

[QSD – Nåsell papers]

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,

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 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.

SIS/W modelPopulation of S susceptibles, I infectives, W free-living 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 (N-I)

Depletion:

(I, W) (I, W1) μW + (ν/N) (1-p)W (N-I) + (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 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.

‘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 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,

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*/(1-i*))]

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 (N-I)

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).

(Piecewise-deterministic 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 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.

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, I1) 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.

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

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