The SIS and SIR stochastic epidemic models revisited

Transcription

1 The SIS and SIR stochastic epidemic models revisited Jesús Artalejo Faculty of Mathematics, University Complutense of Madrid Madrid, Spain jesus_artalejomat.ucm.es BCAM Talk, June 16, 2011

2 Talk Schedule Part I. Quasi-stationary and ratio of expectations distributions 1. Introduction 2. Quasi-stationarity 3. RE-distribution 4. Application to the SIS stochastic model 5. Application to the SIR stochastic model

3 Part II. Extinction time and time to infection 1. Introduction 2. The SIS model: Time to infection 3. The SIR model: Extinction time

4 Part I. Quasi-stationary and ratio of expectations distributions This first part of the talk is based on the paper: Artalejo, J.R. and Lopez-Herrero, M.J. (2010) Quasi-stationary and ratio of expectations distributions: A comparative study Journal of Theoretical Biology 266 (2010),

5 1. Introduction We deal with continuous time Markov chains {X(t); t 0} (CTMC) with state space S = S A S T, where S T is a set of transient states and S A is a set of absorbing states (e.g. S A = {0}). The stationary distribution is degenerate so the fundamental problem is: There exist a need for probabilistic measures of the system behavior before absorption. Analogues of the stationary distribution of the irreducible case should be considered and compared.

6 In the spirit of the seminal work by Darroch and Seneta (1967), we will consider two possibilities: Quasi-stationary distribution. Ratio of expectations (RE) distribution. A comparison between both distributions is justified only if the convergence to quasi-stationary regime is relatively fast.

7 2. Quasi-stationarity The starting point is the conditional probability u i (t) =P {X(t) =i T>t}, i S T, where T =sup{t 0 X(t) S T } denotes the absorption time. Definition 1. Suppose that the chain starts with the initial distribution a i = P {X(0) = i}, i S T. If there exists a starting distribution a i = u i such that P {X(t) =i T>t} = u i, i S T, for all t 0, thenu = {u i ; i S T } is called a quasi-stationary distribution.

8 There also exists a limiting interpretation which states that lim t P {X(t) =i T>t} = u i, i S T, independently of the initial distribution {a i ; i S T }. The above limiting result shows that the quasi-stationary distribution is a good measure of the system dynamics before absorption, but restricting only to those realizations in which the time to absorption is sufficiently large.

9 The existence and computation of the quasi-stationary distribution becomes difficult when S T is infinite. A. S T is finite and irreducible There exists a unique quasi-stationary distribution, but an analytical (explicit form) solution only exists in a few special cases. Example 1. If {X(t); t 0} is a birth and death process with S A = {0}, then (1 δ in )μ i+1 u i+1 (λ i + μ i )u i + λ i 1 u i 1 = μ 1 u 1 u i, 1 i N, where S = {0,...,N} and λ 0 =0. The above non-linear equation also holds when S = N.

10 Particularizations for the case S = N 1. The case of homogeneous rates λ 0 =0,λ i = λ, i 1, μ i = μ, i 1. We have S A = {0}, S T = N {0} and u i = ³ 1 R 2 i ³ R i 1,i 1, for R = λ μ < The linear growth model λ i = iλ, i 0, μ i = iμ, i 1. Once more, we have S A = {0}. The quasi-stationary distribution is a geometric law u i =(1 R)R i 1,i 1, for R = λ μ < 1.

11 Computation of the quasi-stationary distribution Q ST : sub-generator associated to the transient states S T. α : eigenvalue with maximal real part of Q ST (α is real and positive). α 0 : real part of the eigenvalue with the next smallest real part. The vector u = {u i ; i S T } is a left eigenvector corresponding to α. α is a rate of convergence to absorption, while α 0 α is a rate of convergence to the quasi-stationary regime. The quasi-stationary distribution should be used only if R u =2α/α 0 < 1 α<α 0 α.

12 Approximations of the quasi-stationary distribution {X(t); t 0} is a birth and death process with S = {0,...,N} and S A = {0}. A first approximation is based on a birth and death process, X (0) (t), withthe same rates except μ (0) 1 =0. The resulting limiting probabilities are given by p (0) i = p (0) 1 π i, 1 i N, p (0) 1 = ³ P Ni=1 π i 1, where π 1 =1and π i = Q i 1 k=1 λ k μ k+1, 2 i N.

13 A second approximation uses a birth and death process, X (1) (t), withthesame birth rates but μ (1) i = μ i 1, 1 i N. The new chain is positive recurrent and the limiting probabilities are as follows p (1) i = p (1) 1 τ i, 1 i N, p (1) 1 = ³ P Ni=1 τ i 1, where τ 1 =1and τ i = Q i 1 k=1 λ k μ k, 2 i N.

14 Recursive computation of the quasi-stationary distribution {X(t); t 0} is a birth and death process with S = {0,...,N} and S A = {0}. Then, we have u i = u 1 π i ix k=1 1 (1 δ k1 ) P k 1 j=1 u j τ k, 2 i N, NX i=1 u i =1.

15 Algorithm 1. Step 1. Start with u (0) i = p (0) i (or p (1) i ), 1 i N. Step 2. For i =2,...,N, compute u (1) i = u (0) 1 π i ix k=1 1 (1 δ k1 ) µ u (0) 1 +(1 δ k2 ) P k 1 j=2 u(1) j. τ k Step 3. Compute the normalization u (1) 1 = u (0) 1 u (0) 1 + P N i=2 u (1) i, u (1) i = u (1) i u (0) 1 + P N i=2 u (1) i, 2 i N. Step 4. Increase n and repeat the same steps until for any given >0. max 1 i N u(n) i u (n 1) <, i

16 B. S T is finite but reducible (van Doorn and Pollett, 2008) Suppose that S T consists of L communicating classes S k,for1 k L. A partial order on {S k ;1 k L} is defined by writing S i S j when class S i is accessible from S j. Let α k be the (negative) eigenvalue with maximal real part of the sub-generator Q k corresponding to the states in S k. Then, the eigenvalue of Q ST with maximal real part is obtained as α, where α =min 1 k L α k. We also define I(α) ={k : α k = α} and a(α) =mini(α). Theorem 1. If α has a geometric multiplicity one, then the Markov chain has a unique quasi-stationary distribution {u i ; i S T } from which S a(α) is accessible. The jth component of {u i ; i S T } is positive if and only if state j is accessible from S a(α). A simple necessary and sufficient condition for establishing that α has geometric multiplicity one is that {S k ; k I(α)} is linearly ordered, that is, S i S j i j, foralli, j I(α).

17 Example 2. The quasi-stationary distribution of the pure death process on S = {0, 1, 2} with absorbing state 0 and μ i > 0, fori S T = {1, 2}, is given by (u 1,u 2 )= ( (1, 0), if μ1 μ 2, ³ μ2 μ 1, 1 μ 2 μ 1, if μ2 <μ 1. In particular, if μ 1 = μ 2 = μ then all the quasi-stationary mass is concentrated at the state i =1.

18 3. RE-distribution Let T j be the time that the CTMC spends in state j S T before absorption. Definition 2. Given that X(0) = i S T, we define the RE-distribution, P i =(P i (j)), as follows P i (j) = E i[t j ] E i [T ],i,j S T. The ratio of means distribution (Darroch and Seneta, 1967) corresponds with the unconditional version: P (j) = P P i S T a i E i [T j ] i S T a i E i [T ],j S T.

19 Remarks and computation The RE-distribution always exists provided that the expected time to absorption E i [T ] <. The RE-distribution assigns positive probability to all state j accessible from the initial state i. Let us construct the ideal replicated model obtained by assuming that at each extinction the biological model restarts in the same initial state i S T. The stationary distribution of this replicated (regenerative) model amounts to the RE-distribution P i. For a fixed j S T,afirst-step argument yields E i h Tj i = δ ij q i + P k S T k6=i q ik q i E k h Tj i,i ST.

20 4. Application to the SIS stochastic model Closed population model of N individuals. Classified either as susceptible or infective individual. Susceptible can be infected, then they recover and return to the susceptible pool. Evolution of the epidemic: Birth and death process {I(t); t 0}. I(t) :number of infective individuals at time t. S = {0, 1,...,N} (0 is an absorbing state).

21 Classical SIS rates Infection rate λ i = β N i(n i). Recovery rate μ i = γi. R 0 = β γ denotes the transmission factor. λ 1 ¾ μ 2 - μ N 1 - λ N 1¾ μ N ¾μ 1 - λ2 ¾ N 1 N Figure 1. States and transitions of the birth and death model

22 In the case of a birth and death process, we have E 0 h Tj i =0, (λ i + μ i )E i h Tj i = μi E i 1 h Tj i + λi E i+1 h Tj i, i 6= j, 1 i N, (λ j + μ j )E j h Tj i = μj E j 1 h Tj i + λj E j+1 h Tj i +1. Theorem 2. For the SIS epidemic model, the RE-distribution reduces to. P i (j) = NP j=1 min(i,j) 1 P j k=1 min(i,j) 1 P j k=1 ³ R0 j k (N k)! N (N j)! ³ R0 N j k (N k)! (N j)!, 1 j N.

23 Stochastic ordering relationships For a birth and death process with S = {0,...,N} and S A = {0}: P 1 = p (0), p (0) st u, P 1 st u st P N, P i st P i 0, 1 i i 0 N, where X st Y F X (x) F Y (x), for all x R.

24 For the SIS model: u st p (1), P N st p (1), for N fixed and R 0 sufficiently large, p (1) st P N, for N fixed and R 0 sufficiently small.

25 Numerical example N =100,γ=1and several choices of R 0 = β. R u < 1 to use u is meaningful. bp and P e are mixtures of the RE-distributions. p (1) u = max 1 j N p(1) j u j (maximum pointwise distance). R p (1) u P 1 u P N u bp u ep u Table 1. Distributions distances with respect to u

26 5. Application to the SIR stochastic model Closed population model of N individuals. Classified as susceptible, infective or removed individuals. Susceptible can be infected, then they recover and become immune. Evolution of the epidemic: Bidimensional CTMC {(I(t),S(t)); t 0}. I(t) :number of infective individuals at time t (I(0) = m 1). S(t) :number of susceptibles at time t (S(0) = n). S = {(i, j); 0 j n, 0 i m+n j} (S T = {(0,j); 0 j n}).

27 Classical SIR rates Infection rate λ ij = β N ij. Recovery rate μ i = γi. S(t) μ 1 ¾ ¾ ¾ μ 1 ¾ ¾ ¾ ¾ μ 1 ¾ ¾ ¾ ¾ ¾ μ 1 R R R μ 2 λ 13 μ 2 λ 12 μ 2 λ 11 μ 2 R R R μ 3 λ 23 μ 3 λ 22 μ 3 λ 21 μ 3 R R R λ 33 ¾ ¾ ¾ ¾ ¾ ¾ I(t) Figure 2. States and transitions of the SIR epidemic model μ 4 λ 32 μ 4 λ 31 μ 4 R R λ 42 μ 5 λ 41 μ 5 R λ 51 μ 6

28 The quasi-stationary probabilities are u (i,j) = δ (1,0)(i,j), 0 j n, 1 i m + n j, because min (i,j) ST (λ ij +μ i )=λ 10 +μ 1 = γ, for μ i = γi,and1 i m+n. The RE-distribution of the SIR epidemic model is given by P (m,n) (i, j) = np j=0 A ij λ ij +μ i m+n j P i=1, (i, j) S A T, ij λ ij +μ i where A ij is the probability of reaching the state (i, j) S starting from (m, n) before the extinction occurs.

29 Algorithm 2. Step 1. Set A ij = δ (i,j)(m,n). Step 2. For i = m 1,m 2,..., 0 calculate A in = A i+1,n μ i+1 λ i+1,n + μ i+1. Step 3. For k =1, calculate 3.a. λ m+k 1,n k+1 A m+k,n k = A m+k 1,n k+1. λ m+k 1,n k+1 + μ m+k 1

30 3.b. For i = m + k 1,m+ k 2,...,2 compute λ i 1,n k+1 μ A i,n k = A i+1 i+1,n k + A i 1,n k+1. λ i+1,n k + μ i+1 λ i 1,n k+1 + μ i 1 3.c For i =0, 1 calculate A i,n k = A i+1,n k μ i+1 λ i+1,n k + μ i+1. Step 4. Set k = k +1. If k n go to Step 3.a. This scheme also gives a method for computing the final size of the epidemic: A 0j : the probability that j susceptible individuals survive.

31 Remarks The RE-distribution gives positive probability to all transient states S T. Once a state has been visited, the system leaves it forever. It supports the idea that in a relatively rapid period the extinction time is reached. We observe that R u 1. More concretely, we have R u = ( 1, if γ β/n, if γ>β/n. 2γ β/n+γ,

32 Part II. Extinction time and time to infection This second part of the talk is based on the paper: Artalejo, J.R., Economou, A. and Lopez-Herrero, M.J. (2011) Stochastic epidemic models revisited: Analysis of some continuous performance measures Journal of Biological Dynamics (forthcoming).

33 1. Introduction Let {X(t); t 0} be a CTMC with finite state space S = S A S T. L : the unconditional first passage time to the absorbing set S A, given that the initial distribution is α. The general theory says that L follows a PH distribution, so we have P {L x} =1 α exp{q ST x}e, x 0. If M = Q ST is invertible, then P {L < } =1, ϕ(s) =E[e sl ]= α(si M) 1 Me, E[L k ]=k!α( M 1 ) k e, for k 1.

34 Computation of P {L x} (or f L (x)) and E[L k ] The extinction time is the time to reach the absorbing set S A. The time to infection can be viewed as the first passage time to an auxiliary state a. Method 1. To compute a Padé approximation to the matrix exponential exp{mx} (MATLAB). Method 2. To employ numerical inversion of Laplace transforms (Abate and Whitt, 1995). Method 3. To exploit the specific matrix structure (birth and death, triangular) to avoid subtractions, matrices with positive and negative entries, etc.

35 2. The SIS model: Time to infection At time t, population consists of i infected individuals and N i non-infected individuals. Select at random one of those N i non-infected individuals. Define S i as the time until the selected individuals gets infected. Trivial cases: S 0 =, S N has no sense.

36 For a given i, 0 i N 1, we define v i = P {S i < }, ψ i (s) = E[e ss i 1 {Si < } ],Re(s) 0, Mi k = E[Si k 1 {S i < } ], k 0, where 1 A is the indicator random variable on the set A. The probabilities v i are strictly between 0 and 1 because P {S i < } λ i λ i +μ i... λ N 1 λ N 1 +μ N 1 > 0, P {S i = } μ i λ i +μ i... μ 1 λ 1 +μ 1 > 0.

37 Three possibilities for the next event: recovery. infection of a non-tagged individual. infection of the selected individual. A first step argument leads to the tri-diagonal system ψ 0 (s) = 0, ψ i (s) = μ i N i 1 ψ s + λ i + μ i 1 (s)+ ψ i s + λ i + μ i N i i+1 (s) λ + i 1 s + λ i + μ i N i, 1 i N 1. λ i

38 Theorem 3. The Laplace transforms ψ i (s) are computed by ψ N 1 (s) = ψ i (s) = 2 ³ λ N 1 (s + g N 2 + λ N 2 )+μ N 1 D N 2 2(s + λ N 1 )(s + g N 2 + λ N 2 )+μ N 1 (2(s + g N 2 )+λ N 2 ), N 2 X k=i D k λ k +ψ N 1 (s) N k N k 1 N 2 Y k=i ky n=i λ n N n 1 s + g n + λ n N n λ k N k 1 s + g k + λ k N k, 1 i N 2, where the coefficients are given by the recursive scheme λ i 1 N i+1 s + g i 1 + g i = μ i,d i = s + g i 1 + λ i 1 g 1 = μ 1, D 1 = λ 1 N 1, λ i N i + μ i D i 1 s + g i 1 + λ i 1, 2 i N 2.

39 The Tauberian result f Si (0) = lim s sψ i (s) gives f Si (0) = λ i N i, for 1 i N 1. Observe that v i = M 0 i = P {S i < } = ψ i (0), 0 i N 1. By differentiating equations for ψ i (s) k 1 times, and setting s =0,weget M0 k =0, (λ i + μ i ) Mi k = μ imi 1 k + λ N i 1 i Mi+1 k N i + kmk 1 i, 1 i N 1. Recursive modifications: (i) replace ψ i (s) by Mi k, (ii) set s = 0, (iii) use D 1 = km f 1 k 1 and D i = km f i k 1 + μ i D i 1 (g i 1 + λ i 1 ) 1.

40 Numerical example Head lice infection (Stone et al. (2008)) N =100,ξ=0.01 (external rate of infection), i =1, 40, 90 P {S i < } β =0.05 β =0.5 β =1.0 β =5.0 β = γ = γ = γ = γ = Table 2. Probability of becoming infected

41 (i, β, γ) = (1, 1.5, 1.0) (i, β, γ) = (1, 1.5, 2.0) (i, β, γ) = (2, 1.5, 1.0) x Figure 3. Density function of S i when N =100

42 3. The SIR model: Extinction time At time t, population consists of i infected individuals and j susceptibles. Let L ij be the extinction time (i.e. absorption by S T ). We notice that u ij = P {L ij < } =1, (i, j) S T. We define ϕ ij (s) = E[e sl ij], (i, j) S, Re(s) 0, Mij k = E[L k ij ], (i, j) S, k 0.

43 Using a firststepargumentwefind that ϕ 0j (s) = 1, 0 j n, ϕ ij (s) = μ i s + λ ij + μ i ϕ i 1,j (s)+ λ ij s + λ ij + μ i ϕ i+1,j 1 (s), 0 j n, 1 i m + n j. Solving the second equation for j =0,wehave ϕ i0 (s) = μ i s + μ i ϕ i 1,0 (s) = = iy k=1 μ k s + μ k, 1 i m + n. Computation in the natural order 1 j n and 1 i m + n j. Numerical inversion leads to f Lij (x) with f Lij (0) = δ i1 μ 1.

44 For determining the moments, we first notice that M0j 0 =1, Mk 0j =0, 0 j n, k 1, and for the transient states M 0 ij =1, 0 j n, 1 i m + n j. Moments of order k 1 can be computed in the natural order from M k i0 = k ix M k ij = l=1 M k 1 l0 μ l, 1 i m + n, k 1, λ ij μ i Mi 1,j k λ ij + μ + Mi+1,j 1 k i λ ij + μ + k M k 1 i λ ij + μ i 1 j n, 1 i m + n j, k 1. ij,

45 (β, γ) = (0.05, 5.0) (β, γ) = (10.0, 5.0) x Figure 4. Density function of L 1,29 when γ =5.0