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Quasi-stationary distributions of some infection models

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 (N I ) Recovery: I  I1 γ I

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Quasi-stationary distributions of some infection models

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  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). TransitionsRates Transmission: I  I+1(β/N) I (NI) Recovery: I  I1γ I An example of a finite state-space birth-death process. [QSD – Nåsell papers]

  3. Birth-death processes State i = 0,1,…,N, birth ratesλi, 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 <STm2. In terms of re-started processes, m1 = Φ(1,0,0,…,0).Now (1,0,0,…,0) <LRq, so we have m1 <LRΦ(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) <LRρ. 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 <LRm2.

  6. SIS/W model Population of S susceptibles, I infectives, W free-living bacteriaN = S + I constant (herd size) TransitionsRates 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 βin 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 Itinfectives 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.TransitionsRates 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

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