The SIS Epidemic Model with Markovian Switching

Transcription

1 The SIS Epidemic with Markovian Switching Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH (Joint work with A. Gray, D. Greenhalgh and J. Pan)

2 Outline Motivation 1 Motivation Prey-predator model Prey-predator model with Markovian switching SIS 2 3

3 Outline Motivation Prey-predator model Prey-predator model with Markovian switching SIS 1 Motivation Prey-predator model Prey-predator model with Markovian switching SIS 2 3

4 Prey-predator model Prey-predator model with Markovian switching SIS The prey-predator model is described by ẋ 1 (t) = x 1 (t)(a 1 b 1 x 2 (t)), ẋ 2 (t) = x 2 (t)( c 1 + d 1 x 1 (t)), (1.1) on t 0, where x 1 (t) and x 2 (t) are the numbers of preys and predators.

5 Prey-predator model Prey-predator model with Markovian switching SIS Noting we have dx 1 (t) dx 2 (t) = x 1(t)(a 1 b 1 x 2 (t)) x 2 (t)( c 1 + d 1 x 1 (t)), c 1 + d 1 x 1 (t) x 1 (t) dx 1 (t) = a 1 b 1 x 2 (t) dx 2 (t). x 2 (t) Integrating yields c 1 log(x 1 (t)) + d 1 x 1 (t) a 1 log(x 2 (t)) + b 1 x 2 (t) = const. This implies the solution is periodic.

6 Prey-predator model Prey-predator model with Markovian switching SIS Example in mode 1: ẋ 1 (t) = x 1 (t)(1 2x 2 (t)), ẋ 2 (t) = x 2 (t)( 1 + x 1 (t)) (1.2) with x 1 (0) = 1 and x 2 (0) = 2. The solution is shown in Figure 1.

7 Prey-predator model Prey-predator model with Markovian switching SIS States x1 x2 x Time x1 Figure 1

8 Prey-predator model Prey-predator model with Markovian switching SIS Example in mode 2: ẋ 1 (t) = x 1 (t)(0.5 5x 2 (t)), ẋ 2 (t) = x 2 (t)( 6 + 2x 1 (t)) (1.3) with x 1 (0) = 1 and x 2 (0) = 2. The solution is shown in Figure 2.

9 Prey-predator model Prey-predator model with Markovian switching SIS States x1 x2 x Time x1 Figure 2

11 Prey-predator model Prey-predator model with Markovian switching SIS The prey-predator model is now switching from mode 1 or 2 to the other according to the continuous-time Markov chain r(t) on the state space S = {1, 2} with the with the generator ( ) 3 3 Γ = 1 1 starting from r(0) = 1. So the model becomes where ẋ 1 (t) = x 1 (t)(a r(t) b r(t) x 2 (t)), ẋ 2 (t) = x 2 (t)( c r(t) + d r(t) x 1 (t)), (1.4) a 1 = 1, b 1 = 2, c 1 = 1, d 1 = 1; a 2 = 0.5, b 2 = 5, c 2 = 6, d 2 = 2.

12 Prey-predator model Prey-predator model with Markovian switching SIS Takeuchi et al. (2006) revealed a very interesting and surprising result: all positive trajectories of equation (1.4) always exit from any compact set of R 2 + with probability 1; that is equation (1.4) is neither permanent nor dissipative. Figure 3 supports this result clearly.

13 Prey-predator model Prey-predator model with Markovian switching SIS r(t) t States x1 x t Figure 3

15 Prey-predator model Prey-predator model with Markovian switching SIS The classical SIS epidemic model is described by { ds(t) dt di(t) dt = µn βs(t)i(t) + γi(t) µs(t), = βs(t)i(t) (µ + γ)i(t), subject to S(t) + I(t) = N, along with the initial values S(0) = S 0 > 0 and I(0) = I 0 > 0, where I(t) and S(t) are respectively the number of infectious and susceptible individuals at time t in a population of size N, and µ and γ 1 are the average death rate and the average infectious period respectively. β is the disease transmission coefficient, so that β = λ/n, where λ is the disease contact rate of an infective individual. (1.5)

16 Prey-predator model Prey-predator model with Markovian switching SIS It is easy to see that I(t) obeys the scalar Lotka Volterra model di(t) dt = I(t)[βN µ γ βi(t)], (1.6) which has the explicit solution [ ( ) ] e (βn µ γ)t 1 β β 1, I I(t) = 0 βn µ γ + βn µ γ [ ] 1 1, (1.7) + βt I0 if βn µ γ 0 and = 0, respectively. Defining the basic reproduction number for the deterministic SIS model we can conclude: R D 0 = βn µ + γ, (1.8)

17 Prey-predator model Prey-predator model with Markovian switching SIS If R D 0 1, lim t I(t) = 0. If R0 D > 1, lim t I(t) = βn µ γ β. In this case, I(t) will monotonically decrease or increase to βn µ γ β if I(0) > βn µ γ β I(t) βn µ γ β or < βn µ γ β if I(0) = βn µ γ β., respectively, while

18 Prey-predator model Prey-predator model with Markovian switching SIS If R D 0 1, lim t I(t) = 0. If R0 D > 1, lim t I(t) = βn µ γ β. In this case, I(t) will monotonically decrease or increase to βn µ γ β if I(0) > βn µ γ β I(t) βn µ γ β or < βn µ γ β if I(0) = βn µ γ β., respectively, while

20 The stochastic has the form { ds(t) dt di(t) dt = µ r(t) N β r(t) S(t)I(t) + γ r(t) I(t) µ r(t) S(t), = β r(t) S(t)I(t) (µ r(t) + γ r(t) )I(t), (2.1) where r(t) is a Markov chain on the state space S = {1, 2} with the generator ( ) ν12 ν Γ = 12. ν 21 ν 21

21 There is a sequence {τ k } k 0 of finite-valued F t -stopping times such that 0 = τ 0 < τ 1 < < τ k almost surely and r(t) = r(τ k )I [τk,τ k+1 )(t). (2.2) k=0 Moreover, given that r(τ k ) = 1, the random variable τ k+1 τ k follows the exponential distribution with parameter ν 12, while given that r(τ k ) = 2, τ k+1 τ k follows the exponential distribution with parameter ν 21. Furthermore, this Markov chain has a unique stationary distribution Π = (π 1, π 2 ) given by π 1 = ν 21 ν 12 + ν 21, π 2 = ν 12 ν 12 + ν 21. (2.3)

22 We assume that the system parameters β i, µ i, γ i (i S) are all positive numbers. Given that I(t) + S(t) = N, we see that I(t), the number of infectious individuals, obeys the stochastic Lotka Volterra model with Markovian switching given by where di(t) dt = I(t)[α r(t) β r(t) I(t)], (2.4) α i := β i N µ i γ i, i S. (2.5)

23 Theorem For any given initial value I(0) = I 0 (0, N), there is a unique solution I(t) on t R + to equation (2.4) such that P(I(t) (0, N) for all t 0) = 1. Moreover, the solution has the explicit form ( ) t exp 0 α r(s)ds I(t) = 1 I 0 + ( t ). (2.6) 0 exp s 0 α r(u)du β r(s) ds

25 Recall that for the deterministic SIS epidemic model (1.6), the basic reproduction number R0 D was also the threshold between disease extinction and persistence, with extinction for R0 D 1 and persistence for R0 D > 1. In the stochastic model, there are different types of extinction and persistence, for example almost sure extinction, extinction in mean square and extinction in probability. In the rest of the paper we examine a threshold T S 0 = π 1 β 1 N + π 2 β 2 N π 1 (µ 1 + γ 1 ) + π 2 (µ 2 + γ 2 ) (2.7) for almost sure extinction or persistence of our stochastic epidemic model.

26 Proposition We have the following alternative condition on the value of T0 S: T0 S < 1 if and only if π 1α 1 + π 2 α 2 < 0; T0 S = 1 if and only if π 1α 1 + π 2 α 2 = 0; T0 S > 1 if and only if π 1α 1 + π 2 α 2 > 0.

30 Theorem If T0 S < 1, then, for any given initial value I 0 (0, N), the solution of the stochastic SIS epidemic model (2.4) obeys lim sup t 1 t log(i(t)) α 1π 1 + α 2 π 2 a.s. (2.8) By Proposition 2, we hence conclude that I(t) tends to zero exponentially almost surely. In other words, the disease dies out with probability one.

32 Theorem If T0 S > 1, then, for any given initial value I 0 (0, N), the solution of the stochastic SIS model (2.4) has the properties that lim inf I(t) π 1α 1 + π 2 α 2 a.s. (2.9) t π 1 β 1 + π 2 β 2 and lim sup I(t) π 1α 1 + π 2 α 2 a.s. (2.10) t π 1 β 1 + π 2 β 2 In other words, the disease will reach the neighbourhood of the level π 1α 1 +π 2 α 2 π 1 β 1 +π 2 β 2 infinitely many times with probability one.

33 To reveal more properties of the stochastic SIS model, we observe from Proposition 2 that T0 S > 1 is equivalent to the condition that π 1 α 1 + π 2 α 2 > 0. This may be divided into two cases: (a) both α 1 and α 2 are positive; (b) only one of α 1 and α 2 is positive. Without loss of generality, we may assume that 0 < α 1 /β 1 = α 2 /β 2 or 0 < α 1 /β 1 < α 2 /β 2 in Case (a), while α 1 /β 1 0 < α 2 /β 2 in Case (b). So there are three different cases to be considered under condition T S 0 > 1.

34 Lemma The following statements hold with probability one: (i) If 0 < α 1 /β 1 = α 2 /β 2, then I(t) = α 1 /β 1 for all t > 0 when I 0 = α 1 /β 1. (ii) If 0 < α 1 /β 1 < α 2 /β 2, then I(t) (α 1 /β 1, α 2 /β 2 ) for all t > 0 whenever I 0 (α 1 /β 1, α 2 /β 2 ). (iii) If α 1 /β 1 0 < α 2 /β 2, then I(t) (0, α 2 /β 2 ) for all t > 0 whenever I 0 (0, α 2 /β 2 ).

38 Theorem Assume that T0 S > 1 and let I 0 (0, N) be arbitrary. The following statements hold with probability one: (i) If 0 < α 1 /β 1 = α 2 /β 2, then lim t I(t) = α 1 /β 1. (ii) If 0 < α 1 /β 1 < α 2 /β 2, then α 1 lim inf β 1 t (iii) If α 1 /β 1 0 < α 2 /β 2, then 0 lim inf t I(t) lim sup I(t) α 2. t β 2 I(t) lim sup I(t) α 2. t β 2

42 Theorem Assume that T0 S > 1 and 0 < α 1 β 1 < α 2 β 2, and let I 0 (0, N) be arbitrary. Then for any ε > 0, sufficiently small for α 1 β 1 + ε < π 1α 1 + π 2 α 2 π 1 β 1 + π 2 β 2 < α 2 β 2 ε, the solution of the stochastic SIS epidemic model (2.4) has the properties that ( P lim inf I(t) < α ) 1 + ε e ν 12T 1 (ε), (2.11) t β 1 and ( P lim sup I(t) > α ) 2 ε e ν 21T 2 (ε), (2.12) t β 2 where T 1 (ε) > 0 and T 2 (ε) > 0 are defined by

43 T 1 (ε) = 1 ( ( β1 log β ) ( 2 α1 ) ( εβ1 )) +log +ε log α 1 α 1 α 2 β 1 α 1 (2.13) and T 2 (ε) = 1 ( ( β1 log β ) ( 2 α2 ) ( εβ2 )) +log ε log. (2.14) α 2 α 1 α 2 β 2 α 2

44 Theorem Assume that T0 S > 1 (namely π 1α 1 + π 2 α 2 > 0) and α 1 β 1 0 < α 2 β 2. Let I 0 (0, N) be arbitrary. Then for any ε > 0, sufficiently small for ε < π 1α 1 + π 2 α 2 π 1 β 1 + π 2 β 2 < α 2 β 2 ε, the solution of the stochastic SIS model (2.4) has the properties that ( ) P lim inf I(t) < ε e ν 12T 3 (ε), (2.15) t and ( P lim sup I(t) > α ) 2 ε e ν 21T 4 (ε), (2.16) t β 2 where T 3 (ε) > 0 and T 4 (ε) > 0 are defined by

45 T 3 (ε) = 1 ( ( β2 log β ) ( 1 +log ε α ) ( 1 α1 )) log ε α 1 α 2 α 1 β 1 β 1 (2.17) and T 4 (ε) = 1 ( ( 2 log α 2 ε β ) ( 2 α2 ) ( + log ε log ε β )) 2. (2.18) α 2 β 2 α 2

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