GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS. Yoichi Enatsu and Yukihiko Nakata. Yoshiaki Muroya. (Communicated by Yasuhiro Takeuchi)

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1 MATHEMATICAL BIOSCIENCES doi: /me AND ENGINEERING Volume 7, Numer 2, April 2010 pp GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS Yoichi Enatsu and Yukihiko Nakata Department of Pure and Applied Mathematics Waseda University Ohkuo, Shinjuku-ku, Tokyo, , Japan Yoshiaki Muroya Department of Mathematics Waseda University Ohkuo, Shinjuku-ku, Tokyo, , Japan (Communicated y Yasuhiro Takeuchi) Astract. In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distriuted delays y using a variation of the ackward Euler method. Applying a Lyapunov functional technique, it is shown that the gloal dynamics of each discrete SIR epidemic model are fully determined y a single threshold parameter and the effect of discrete time delays are harmless for the gloal staility of the endemic equilirium of the model. 1. Introduction. The application of theories of functional differential/difference equations in mathematical iology has een developed rapidly. Various mathematical models have een proposed in the literature of population dynamics, ecology and epidemiology. Many authors have studied the epidemic models, which displays the dynamical ehavior of the transmission of infectious disease (see also [1]-[18] and references therein). Cooke [5] formulated an SIR epidemic model with ilinear incidence rate and a discrete time delay which takes the form βsi to investigate the spread of an infectious disease transmitted y a vector (e.g. mosquitoes, rats, etc.) after an incuation time denoting the time during which the infectious agents develop in the vector. This is called the phenomena of time delay effect which now has important iological meanings in epidemic models (see [1, 5]). Based on their ideas, Ma et al. [13] considered a continuous SIR model with a discrete delay and investigated the gloal asymptotic staility of the equilirium, while a continuous SIR model with distriuted delays has een also studied in [1, 2, 12, 14, 17]. The distriuted delays is more appropriate form than the discrete one ecause it is considered more realistic to assume that the time delay is not a fixed time ut a distriuted parameter which is upper ounded y some positive 2000 Mathematics Suject Classification. Primary: 34K20, 34K25; Secondary: 92D30. Key words and phrases. Backward Euler method, discrete SIR epidemic model, distriuted delays, gloally asymptotic staility, Lyapunov functional. The corresponding author is Yoichi Enatsu. The third author is supported y Scientific Research (c), No of Japan Society for the Promotion of Science. 347

2 348 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA finite time. Beretta and Takeuchi [1] have studied the following continuous SIR model with distriuted delays: ds(t) h = βs(t) f(τ)i(t τ)dτ s(t), dt 0 di(t) h = βs(t) f(τ)i(t τ)dτ (µ 2 + λ)i(t), (1.1) dt 0 dr(t) = λi(t) µ 3 r(t), dt where s(t), i(t) and r(t) denote the proportions of the population susceptile to the disease, of infective memers and of memers who have een removed from the possiility of infection at time t, respectively. It is assumed that all neworns are susceptile and all recruitment is into the susceptile class at a constant rate > 0. The positive constants, µ 2 and µ 3 represent the death rates of susceptile, infectious and recovered classes, respectively. Infectiousness is assumed to vary over time from the initial time of infection until a duration h has passed and the function f(τ) denotes the fraction of vector population in which the time taken to ecome infectious is τ. The mass action coefficient is β and f(τ) is chosen so that it is nonnegative and continuous on [0, h] and assume that, for simplicity, h f(τ)dτ = 1. 0 System (1.1) always has a disease-free equilirium E 0 = (/, 0, 0). Furthermore, if R 0 > 1, then system (1.1) has a unique endemic equilirium E = (s, i, r ), where 0 < s = µ 2 + λ <, 0 < i = s β βs <, 0 < r = λi <, (1.2) µ 3 and β R 0 = (µ 2 + λ). (1.3) For the case R 0 1, it is known that disease dies out for (1.1) (see also [1, 2, 17]). For the case R 0 > 1, it is shown that system (1.1) is permanent y Ma et al. [12]. Later, y using a Lyapunov functional for R 0 > 1, complete analysis of system (1.1) has een estalished y McCluskey [14]. Thereafter, using the similar techniques, McCluskey [15] also estalished a complete analysis of an SIR epidemic models with a saturated incidence rate and a discrete delay. Summarizing the aove discussion, the following result holds (see [1, 2, 12, 17] and [14, Theorem 4.1]). Theorem A For system (1.1), if R 0 1, then the disease-free equilirium E 0 is gloally attractive, and locally asymptotically stale if the inequality is strict, and if R 0 > 1, then the endemic equilirium E is gloally asymptotically stale. On the other hand, there occur situations such that constructing discrete epidemic models is more appropriate approach to understand disease transmission dynamics and to evaluate eradication policies ecause they permit aritrary timestep units, preserving the asic features of corresponding continuous-time models. Furthermore, this allows etter use of statistical data for numerical simulations due to the reason that the infection data are compiled at discrete given time intervals. For a discrete epidemic model with immigration of infectives, Jang and Elaydi [10] showed the gloal asymptotic staility of the disease-free equilirium, the local asymptotic staility of the endemic equilirium and the strong persistence

3 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 349 of susceptile class y means of the nonstandard discretization method. Recently, using a discretization called mixed type formula in Izzo and Vecchio [8] and Izzo et al. [9], Sekiguchi [16] otained the permanence of a class of SIR discrete epidemic models with one delay and SEIRS discrete epidemic models with two delays if an endemic equilirium of each model exists. For the detailed property for a class of discrete epidemic models, we refer to [3, 4, 8, 9, 10, 11, 16, 18]. However, in those cases, how to choose the discrete schemes which guarantee the gloal asymptotic staility for the endemic equilirium of the models, was still an open prolem. In this paper, motivated y the aove facts, we propose the following discrete SIR epidemic model which is derived from system (1.1) y applying a variation of ackward Euler method. s(p) = β f(j)i(p j), (1.4) i(p) = β f(j)i(p j) (µ 2 + λ), r(p + 1) r(p) = λ µ 3 r(p + 1), where, β, µ i (i = 1, 2, 3), λ and m are positive constants and f(j) 0, j = 0, 1,, m. For simplicity, we may assume that m f(j) = 1. Similar to the case of continuous system (1.1), system (1.4) always has a diseasefree equilirium E 0 = (/, 0, 0). Furthermore, if R 0 > 1, then system (1.4) has a unique endemic equilirium E = (s, i, r ) defined y (1.2). The initial condition of system (1.4) is s(p) = φ(p) 0, i(p) = ψ(p) 0, r(p) = σ(p) 0, p = m, (m 1),, 1, and s(0) > 0, i(0) > 0, r(0) > 0. (1.5) Remark 1.1. To prove the positivity of s(p), i(p) and r(p) for any p 0, we need to use the ackward Euler discretization instead of the forward Euler discretization (see, e.g., [8, 9]). Moreover, to apply a discrete time analogue of the Lyapunov function proposed y McCluskey [14, 15], we adopt a variation of the ackward Euler method which is different from that of Sekiguchi [16]. β (µ 2+γ) Using the same threshold R 0 = and applying the oth techniques in Izzo and Vecchio [8] and Izzo et al. [9] for the case R 0 1 and McCluskey [14, Theorem 4.1] for the case R 0 > 1 to system (1.4), we estalish a complete analysis of the gloal asymptotic staility for system (1.4). In particular, y applying Lemma 4.1, we offer a simplified proof for the permanence of system (1.4) for R 0 > 1 (cf. Sekiguchi [16]). Our main result in this paper is as follows. Theorem 1.1. For system (1.4), there exists a unique disease-free equilirium E 0 = (/, 0, 0) which is gloally asymptotically stale, if and only if, R 0 1, and there exists a unique endemic equilirium E = (s, i, r ) which is gloally asymptotically stale, if and only if, R 0 > 1. Remark 1.2. Theorem 1.1 for system (1.4) is just a discrete analogue of Theorem A for system (1.1).

4 350 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA The organization of this paper is as follows. In Section 2, we offer some asic results for system (1.4) and some asymptotic properties are investigated. The first part of Theorem 1.1 concerning the gloal asymptotic staility for R 0 1 is given in Section 3, and the second part of Theorem 1.1 concerning the permanence and the gloal staility for R 0 > 1 is given in Section 4. Finally, we offer concluding remarks in Section Basic properties. For system (1.4), since the variale r does not appear in the first and the second equations, it is sufficient to consider the following 2-dimensional system. s(p) = β f(j)i(p j) µ 1, (2.1) i(p) = β f(j)i(p j) (µ 2 + λ), with the initial condition { s(p) = φ(p) 0, i(p) = ψ(p) 0, p = m, (m 1),, 1, (2.2) and s(0) > 0, i(0) > 0. The following results in Sections 2 and 3 are otained y applying techniques in Izzo and Vecchio [8] and Izzo et al. [9] to system (2.1). Lemma 2.1. Let (s(p), i(p)) e the solutions of system (2.1) with the initial condition (2.2). Then s(p) > 0, i(p) > 0 for any p > 0. Proof. Assume that s(p j), i(p j) > 0, j = 0, 1,, m. Then, system (2.1) ecomes {1 + µ 1 + β f(j)i(p j) = + s(p) > 0, (2.3) {1 + (µ 2 + λ) = i(p) + β f(j)i(p j) > 0. Then, y the first equation of (2.3), we have s(p+1) > 0, and y the second equation of (2.3), we have > 0. Hence, y induction, we prove this lemma. Lemma 2.2. Any solution (s(p), i(p)) of system (2.1) with the initial condition (2.2) satisfies limsup (s(p) + i(p)) /µ, where µ = min{, µ 2 + λ. Proof. Let V (p) = s(p) + i(p). From system (2.1), we have that from which we have that V (p + 1) V (p) = (µ 2 + λ) Hence, the proof is complete. µv (p + 1), lim sup V (p) µ.

5 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 351 Now, put s = liminf s(p), i = liminf i(p), and s = limsup s(p), and i = limsup i(p). (2.4) Then y m f(j) = 1, similar to the Izzo et al. [9, Proof of Lemma 3.3], we otain the following asic lemma. Lemma 2.3. For any solution (s(p), i(p)) of system (2.1) with the initial condition (2.2), we have that 0 < + βi s s + βi, 0 i i, βs 1, if i > 0, (2.5) µ 2 + λ βs 1, if i > 0. µ 2 + λ Further, we put P(s) βs µ 2 + λ, (2.6) then we easily otain the following lemma. Lemma 2.4. P(s) is a strictly increasing positive continuous function of s on [0, + ) and ( ) P(0) = 0, and R 0 = P, (2.7) and { P(s) 1, if i > 0, P(s) 1, if i > 0. If R 0 > 1, then there exists a unique solution s = s of P(s) = 1 such that 0 < s = µ 2 + λ β (2.8) <. (2.9) Proof. By the definition of P(s), (1.3) and (2.5), we see that P(s) is a strictly increasing positive continuous function of s on (0, + ), and (2.7) and (2.8) hold. If R 0 > 1, then P(0) = 0 < 1 < R 0 = P( ). Since P(s) is a strictly increasing positive continuous function of s, there exists a unique solution 0 < s = s = µ 2+λ β < of P(s) = Gloal staility of the disease-free equilirium E 0 = (/, 0) for R 0 1. In this section, we prove the first part of Theorem 1.1 for system (1.4). By applying techniques in Izzo et al. [9], we otain the gloal asymptotic staility of the diseasefree equilirium E 0 of system (2.1) for R 0 1. Lemma 3.1. If R 0 1, then lim s(p) =, and lim i(p) = 0, (3.1) and E 0 = (, 0) is gloally asymptotically stale.

6 352 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA Proof. From (2.5) in Lemma 2.3, for any ǫ > 0, there is an integer p 0 0 such that + ǫ for p p 0. Consider the following sequence {w(p) + p=p 0 defined y where Since u(p) = β w(p) = i(p) + u(p), p p 0, (3.2) f(j) u(p + 1) u(p) { p+1 = β f(j) = β we have that k=p+1 j p k=p j s(j + k + 1)i(k), p p 0. s(j + k + 1)i(k) p k=p j { f(j) s(p + j + 2) i(p j), w(p + 1) w(p) = = β = { β +β s(j + k + 1)i(k) f(j)i(p j) (µ 2 + γ) { f(j) s(p + j + 2) i(p j) f(j)s(p + j + 2) (µ 2 + γ) ( ) β f(j) + ǫ (µ 2 + γ) µ 1 { β (µ 2 + γ) + βǫ. Since ǫ > 0 is aritrary, we otain that if R 0 1, then { β w(p + 1) w(p) (µ 2 + γ) 0, (3.3) and the nonnegative sequence {w(p) + p=p 0 is monotone decreasing. Therefore, there exists a nonnegative constant ŵ such that lim w(p) = ŵ. We will prove that ŵ = 0 and (3.1) hold for R 0 1. If R 0 < 1, then β (µ 2 + γ) < 0 and y (3.3), we conclude that lim i(p) = 0. Then, ŵ = 0 and y Lemma 2.3, we otain (3.1). Suppose that R 0 = 1. From (1.4), we have that = + cs(p) β 1 f(j)i(p j), = di(p) + β (3.4) 2 f(j)i(p j),

7 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 353 where = d =, c = 1, β1 = β, µ 2 + γ, β2 = β 1 + µ 2 + γ. Then, y applying the similar techniques in Izzo et al. [9, Proofs of Lemmas ] to (3.4) with Lemmas 2.3 and 2.4, we prove the claim that there is a sequence {p k + k=0 such that each of i(p k j), j = 0, 1,, m converges to 0 as k +. If i = 0, then the claim is evident. Now, suppose that i > 0. By (2.8) in Lemma 2.4, we have that P(s) 1 = R 0 = P( ), which implies s = from (2.5) in Lemma 2.3. Therefore, there exists a sequence {p k + k=1 such that lim k + s(p k + 1) = s. Then, 1 c = = s and s(p k + 1) = c{s(pk + 1) s(p k ) β 1 s(p k + 1) m f(j)i(p k j) 1 c s, as k +. Then, we easily otain that [ c{s(p k + 1) s(p k ) + β 1 s(p k + 1) lim k + Since limsup k + {s(p k + 1) s(p k ) 0, we otain that lim sup k + Thus, it holds that {s(p k + 1) s(p k ) = 0, and limsup lim {s(p k + 1) s(p k ) = 0, and k + k + lim k + ] f(j)i(p k j) = 0. f(j)i(p k j) = 0. f(j)i(p k j) = 0. Hence, it follows that lim k + s(p k ) = lim k + s(p k + 1) = s and lim k + In the same way, we can prove that lim k + s(p k l) = s, lim k + f(j)i(p k j) = 0. f(j)i(p k l j) = 0, for l = 0, 1,, m. Since y m f(j) = 1, there exists at least one 0 j 0 m such that f(j 0 ) > 0, we can otain that i(p k l j 0 ) = 0, l = 0, 1,, m. Hence, the claim is proved. Then, y the definition of w(p), we otain lim k + w(p k + 1) = 0, which implies that ŵ = 0. Hence, y lim w(p) = 0, we can conclude that (3.1) still holds for R 0 = 1. Finally, we will prove that if R 0 1, then E 0 = (, 0) is uniformly stale. First, consider the case that there exists a nonnegative integer p 1 such that >

8 354 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA s = for any p p 1. From the first equation of (2.1), we have that for any p > 0, s(p) = ( ) β f(j)i(p j) 0. (3.5) Then, we otain that s(p) s(p 1 ) for any p p 1. Next, consider the case that there exists a nonnegative integer p 2 such that s(p 2 ) s =. Then, y the first equation of (2.1), we have that s(p 2 + 1) + s(p 2) / 1 + =. Thus, we otain that s(p) s =, for any p p 2. Then, y the second equation of (2.1) and β µ 2 + λ, we have that for any p p 2, i(p) + β(/) m f(j)i(p j) 1 + (µ 2 + λ) 1 + (µ 2 + λ) m f(j) max i(p j) 1 + (µ 2 + λ) 0jm = max i(p j). (3.6) 0jm Moreover, y the first equation of (2.1), we have that = 1 ( s(p) ) β and hence, for any p p 2, we otain that = ( ( ) p p2+1 s(p2 ) ) p p2+1 s(p2 ) + β µ 2 1 f(j)i(p j), β(/ ) max 0jm i(p 2 j) /(1 + ) max i(p 2 j). 0jm Thus, y the aove discussion, we can conclude that E 0 = (, 0) is uniformly stale. Hence, if R 0 1, then E 0 = (, 0) is gloally asymptotically stale. Lemma 3.2. (3.1) implies R 0 1. Proof. Suppose that R 0 = P( ) > 1. Then, y Lemma 2.4, there exists a positive constant solution (s(p), i(p)) = (s, i ) of system (2.1) defined y (1.2). This contradicts the fact that (3.1) holds. Hence, the proof is complete. Proof of the first part of Theorem 1.1. By Lemmas 3.1 and 3.2, we immediately otain the conclusion of this theorem. 4. Gloal staility of the endemic equilirium E = (s, i ) for R 0 > 1. In this section, we assume that R 0 > 1 and similar to the result of McCluskey [14, Theorem 4.1] for the continuous SIR model, we prove the second part of Theorem 1.1 for system (1.4). By Lemmas , we otain the permanence of system (2.1) for R 0 > 1.

9 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 355 Lemma 4.1. If i(p+1) < min i(p j), then s(p+1) < 0jm s. Inversely, if s(p+1) s, then i(p j). min 0jm Proof. By the second equation of (2.1), we have that i(p) = + µ 2 + λ s f(j)i(p j). Therefore, if < min 0jm i(p j), then y i(p) > 0 and m f(j)i(p j) >, we have that > s, from which we otain < s. Then, the remained part of this lemma is evident. Lemma 4.2. If R 0 > 1, then any solution of system (2.1) with initial condition (1.5), lim inf s(p) v 1 := lim inf i(p) v 2 := β > 0, (4.1) µ ( ) m+l0 1 i > 0, (4.2) 1 + µ 2 + λ where for k = + βqi, the constant l 0 1 is sufficiently large such that s < s := k {1 ( 1 1+k )l0. Proof. By the first equation of (2.3), it is straightforward to otain (4.1). We now show that (4.2) holds. For any 0 < q < 1, y (1.2), one can see that s = +βi < +βqi. We first prove the claim that any solution (s(p), i(p)) of system (2.1) does not have the following property: there exists a nonnegative integer p 0 such that i(p) qi for all p p 0. Suppose on the contrary that there exist a solution (s(p), i(p)) of system (2.1) and a nonnegative integer p 0 such that i(p) qi for all p p 0. From system (2.1), one can otain that which yields that s(p) 1 + k k, for p p 0 + m, ( ) p+1 (p0+m) 1 s(p 0 + m) k 1 + k 1 + k k { 1 1 ( 1 1+k )p+1 (p0+m) 1 1 ( k 1+k ) p+1 (p0+m) p (p 0+m) l=0 ( k, for any p p 0 + m. Therefore, we have that { ( ) l0 1 1 = s > s, k 1 + k for any p p 0 + m + l 0 1. (4.3) ) l

10 356 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA Then, y the second part of Lemma 4.1, we otain that min i(p j), for any p p 0 + m + l 0 1. (4.4) 0jm Thus, we otain that there exists a positive constant î such that i(p) î for any p p 0 + m + l 0 1. Moreover, for the sequence {w(p) + p=p 0 defined y (3.2), we have that w(p + 1) w(p) = β f(j)s(p + j + 2) (µ 2 + γ) > {βs (µ 2 + γ) > {βs (µ 2 + γ)î, for any p p 0 + m + l 0 1, which implies y βs (µ 2 + γ) = β(s s ) > 0, that lim w(p) = +. However, y (3.2) and Lemma 2.2, it holds that there is a positive constant p 4 p 0 +m+l 0 1 and w such that w(p) w for any p p 4, which leads to contradiction. Hence, the claim is proved. By the claim, we are left to consider two possiilities. First, i(p) qi for all p sufficiently large. Second, we consider the case that i(p) oscillates aout qi for all p sufficiently large. We now show that i(p) qv 2 for all p sufficiently large. If the first condition that i(p) qi holds for all p sufficiently large, then we get the conclusion of the proof. If the second condition holds, let p 1 < p 2 e sufficiently large such that i(p 1 ), i(p 2 ) > qi, and i(p) qi for any p 1 < p < p 2. Then, y the second equation of system (2.1), we have that that is i(p) (µ 2 + λ), µ 2 + λ i(p), for any p p 1. This implies that ( ) p+1 p1 ( ) p+1 p1 1 1 i(p 1 ) qi, for any p p µ 2 + λ 1 + µ 2 + λ Therefore, we otain that ( ) m+l0 1 qi = qv 2, for any p 1 p p 1 + m + l 0 1. (4.5) 1 + µ 2 + λ If p 2 p 1 + m + l 0, then y applying the similar discussion to (4.3) and (4.4) in place of p 0 y p 1 + 1, we otain that v 2 for p 1 + m + l 0 p p 2 1. Hence, we prove that qv 2 for p 1 p p 2. Since the interval p 1 p p 2 is aritrarily chosen, we conclude that i(p+1) qv 2 for all sufficiently large for the second case. Since q (0 < q < 1) is also aritrarily chosen, we conclude that This completes the proof. lim inf i(p) v 2.

11 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 357 Lemma 4.3. There exists a unique endemic equilirium E = (s, i ) of system (2.1) and if and only if, R 0 > 1. 0 < + i s s s <, and 0 < i i i < µ, (4.6) Proof. By (3.1)-(3.2) and Lemma 4.2, we have that R 0 > 1, if and only if, there exists a unique endemic equilirium E = (s, i ) of system (2.1) and liminf i(p) > 0. Then, y (2.5) and Lemma 2.4, we otain (4.6). This completes the proof. Proof of the second part of Theorem 1.1. Consider the following Lyapunov function (see McCluskey [14, 15]). U(p) = 1 βi U s(p) + 1 βs U i(p) + U + (p), (4.7) where ( ) ( s(p) i(p) U s (p) = g s, U i (p) = g i p ( ) i(k) U + (p) = f(j) g, k=p j i ), and g(x) = x 1 lnx, x > 0. We now show that U(p + 1) U(p) 0. First, we calculate U s (p + 1) U s (p). = U s (p + 1) U s (p) s(p) s ln s(p) s(p) s(p) s = 1 s s ( s(p)) = 1 s s { β ecause ln(1 x) x holds for any x < 1, one can otain that ln s(p) ( = ln ( = 1 f(j)i(p j), (4.8) ( 1 s(p) ) 1 s(p) s(p). ))

12 358 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA Sustituting = βs i + s into (4.8), we see that U s (p + 1) U s (p) 1 s { s βs i + s β f(j)i(p j) = ( s ) 2 s ( +βi m f(j) 1 s )( 1 s Second, similarly, we calculate U i (p + 1) U i (p). = U i (p + 1) U i (p) i(p) i ln i(p) i(p) i(p) i = 1 i i ( i(p)) = 1 i i { β Since µ 2 + λ = βs holds, we otain that U i (p + 1) U i (p) 1 i { i β m ( = βs i f(j) 1 Finally, calculating U + (p + 1) U + (p), we get that ) i(p j) i. f(j)i(p j) (µ 2 + λ). f(j)i(p j) βs )( i(p j) s i ) i. = = = U + (p + 1) U + (p) { p+1 { ( ) i(k) f(j) g k=p+1 j i i p k=p j { ( ) ( ) i(p j) f(j) g g ( ) f(j)g i i ( ) i(k) g i ( i(p j) f(j)g i ). Defining x p+1 = s, y p+1 = i, z p,j = i(p j) i, j = 0, 1,, m,

13 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 359 similar to McCluskey [14, Proof of Theorem 4.1], we otain that U(p + 1) U(p) ( s ) 2 βs i ( f(j) x ) p+1z p,j + lny p+1 lnz p,j x p+1 y p+1 = ( s ) 2 βs i [ ( ) ( )] 1 xp+1 z p,j f(j) g + g 0. x p+1 y p+1 Hence, U(p + 1) U(p) 0 for any p 0. Since U(p) 0 is monotone decreasing sequence, there is a limit lim U(p) 0. Then, lim (U(p+1) U(p)) = 0, from which we otain that lim = s, and i(p j) lim = lim z p,j = 1, y p+1 if f(j) > 0, j = 0, 1,, m. Then, y the first equation of (2.1), we have that for p 0, which implies Using the relations s(p) = ( ) β = ( ) β f(j)i(p j) m = (1 + ) + s(p) m. β f(j)i(p j) i(p+1) f(j)i(p j), lim ( (1 + ) + s(p)) = s > 0, and m f(j)i(p j) lim β = βs > 0, we otain that lim = i, that is, lim (s(p), i(p)) = (s, i ). Since U(p) U(0) for all p 0 and g(x) 0 with equality if and only if x = 1, E is uniformly stale. Hence, the proof is complete.

14 360 YOICHI ENATSU, YUKIHIKO NAKATA AND YOSHIAKI MUROYA 5. Discussions. Recently, in order to investigate the transmission dynamics of infectious diseases, there have een many papers focusing on the analysis of the gloal staility of the disease-free equilirium and the endemic equilirium of a class of discrete and continuous SIR epidemic models. Motivated y the fact that the discrete models are more appropriate forms than the continuous ones in order to directly fit the statistical data concerning infectious diseases with a latent period such as malaria, in this paper, we show that the disease-free equilirium E 0 is gloally asymptotically stale if R 0 1, and the unique endemic equilirium E exists and is gloally asymptotically stale if R 0 > 1 for the discrete SIR epidemic model (1.4) y applying a variation of the ackward Euler discretization (cf. [8, 9]) and Lyapunov functional techniques in McCluskey [14, 15]. We note that its staility conditions no longer need any restriction of the size of time delays for the model (1.4) which is discretized y a variation of the ackward Euler method. From a iological viewpoint, it is noteworthy that the gloal dynamics is determined y a single threshold parameter R 0 without imposing any restriction on the length of an incuation period of the diseases not only for the continuous SIR model in [1, 2, 12, 14, 17] ut also for this discrete SIR model. Applying these techniques to the other types of discrete epidemic models is our future work. Acknowledgments. The work of this paper was partially prepared during the authors visit as memers of International Research Training Group 1529 in Technische Universität Darmstadt on Feruary, 2010, where the authors have had interesting discussions with Professor Matthias Hieer. The authors are grateful to anonymous referee for his/her careful readings and valuale comments which improved the paper in the present style. REFERENCES [1] E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population size, Nonlinear Anal., 28 (1997), [2] E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Gloal asymptotic staility of an SIR epidemic model with distriuted time delay, Nonlinear Anal., 47 (2001), [3] C. Castillo-Chavez and A-A. Yakuu, Dispersal, disease and life-history evolution, Math. Biosci., 173 (2001), [4] C. Castillo-Chavez and A-A. Yakuu, Discrete-time S-I-S models with complex dynamics, Nonlinear Anal., 47 (2001), [5] K. L. Cooke, Staility analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), [6] O. Diekmann, J. A. P. Heestereek and J. A. J. Metz, On the definition and the computation of the asic reproduction R 0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), [7] M. C. M. De Jong, O. Diekmann and J. A. P. Heestereek, The computation of R 0 for discrete-time epidemic models with dynamic heterogeneity, Math. Biosci., 119 (1994), [8] G. Izzo and A. Vecchio, A discrete time version for models of population dynamics in the presence of an infection, J. Comput. Appl. Math., 210 (2007), [9] G. Izzo, Y. Muroya and A. Vecchio, A general discrete time model of population dynamics in the presence of an infection, Discrete Dyn. Nat. Soc., 2009, Art. ID , 15 pages doi: /2009/ [10] S. Jang and S. N. Elaydi, Difference equations from discretization of a continuous epidemic model with immigration of infectives, Can. Appl. Math. Q., 11 (2003), [11] J. Li, Z. Ma and F. Brauer, Gloal analysis of discrete-time SI and SIS epidemic models, Math. Biol. Engi., 4 (2007), [12] W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distriuted time delays, Tohoku Math. J., 54 (2002),

15 GLOBAL STABILITY FOR A CLASS OF DISCRETE SIR EPIDEMIC MODELS 361 [13] W. Ma, M. Song and Y. Takeuchi, Gloal staility of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), [14] C. C. McCluskey, Complete gloal staility for an SIR epidemic model with delay-distriuted or discrete, Nonlinear Anal. RWA., 11 (2010), [15] C. C. McCluskey, Gloal staility for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA., (2010), doi: /j.nonrwa [16] M. Sekiguchi, Permanence of some discrete epidemic models, Int. J. Biomath., 2 (2009), [17] Y. Takeuchi, W. Ma and E. Beretta, Gloal asymptotic properties of a delay SIR epidemic model with finite incuation times, Nonlinear Anal., 42 (2000), [18] Y. Zhou, Z. Ma and F. Brauer, A discrete epidemic model for SARS transmission and control in China, Mathematical and Computer Modeling., 40 (2004), Received July 8, 2009; Accepted Feruary 13, address: yo1.gc-rw.docomo@akane.waseda.jp address: yunayuna.na@gmail.com address: ymuroya@waseda.jp