A periodic epidemic model in a patchy environment

Transcription

1 J Math Anal Appl 325 (2007) wwwelseviercom/locate/jmaa A periodic epidemic model in a patchy environment Fang Zhang, Xiao-Qiang Zhao Department of Mathematics and Statistics, Memorial University of Newfoundland, St John s, NL A1C 5S7, Canada Received 19 April 2005 Available online 2 March 2006 Submitted by CE Wayne Abstract An epidemic model in a patchy environment with periodic coefficients is investigated in this paper By employing the persistence theory, we establish a threshold between the extinction and the uniform persistence of the disease Further, we obtain the conditions under which the positive periodic solution is globally asymptotically stable At last, we present two examples and numerical simulations 2006 Elsevier Inc All rights reserved Keywords: Epidemic model; Population dispersal; Persistence and extinction of disease; Periodic solution 1 Introduction It has been observed that population dispersal affects the spread of many infectious diseases In 1976, Hethcote [5] put forth an epidemic model with population dispersal between two patches After him, Brauer and van den Driessche [2] proposed a model with immigration of infectives In [10], Wang and Zhao presented a disease transmission model with population dispersal among n patches S i = B i(n i )N i μ i S i β i S i I i + γ i I i + a ij S j, 1 i n, (11) I i = β is i I i (μ i + γ i )I i + b ij I j, 1 i n, Supported in part by the NSERC of Canada and the MITACS of Canada * Corresponding author addresses: fzhang@mathmunca (F Zhang), xzhao@mathmunca (X-Q Zhao) X/$ see front matter 2006 Elsevier Inc All rights reserved doi:101016/jjmaa

2 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) with the properties a ji = 0, b ji = 0, 1 i n, (12) and established a threshold between the extinction and the uniform persistence of the disease for this model They also considered the global attractivity of the disease free equilibrium under the condition that the dispersal rates of susceptible and infective individuals are the same in each patch Recently, the uniqueness and the global attractivity of the endemic equilibrium of this model has been studied by Jin and Wang [8] However, these authors only considered the constant coefficients in model (11) Since the periodicity has been observed in the incidence of many infectious diseases, such as measles, chickenpox, mumps, rubella, poliomyelitis, diphtheria, pertussis and influenza(see, eg, [6]), it is more realistic to assume that all the coefficients depend on time periodically As mentioned in [4], the seasonality is an important factor for the spread of infectious diseases, such as the marked change of the contact rate caused by the school system or the weather changes (eg, measles), the emergence of the insects caused by the seasonal variation (eg, temperature, humidity, etc) We will assume that these coefficients are periodic with a common period due to the seasonal effects In this paper, we consider the following periodic system: S i = B i(t, N i )N i μ i (t)s i β i (t)s i I i + γ i (t)i i + a ij (t)s j, 1 i n, I i = β i(t)s i I i ( μ i (t) + γ i (t) ) (13) I i + b ij (t)i j, 1 i n, with all functions being continuous, ω-periodic in t HereS i,i i are the numbers of susceptible and infectious individuals in patch i, respectively N i = S i + I i is the number of the population in patch i, B i (t, N i ) is the birth rate of the population in the ith patch, μ i (t) is the death rate of the population in the ith patch, and γ i (t) is the recovery rate of infectious individuals in the ith patch a ii (t), b ii (t) 0 represent the emigration rates of susceptible and infectious individuals in the ith patch, respectively a ij (t), b ij (t), j i, represent the immigration rates of susceptible and infectious individuals from jth patch to ith patch Since the death rates and birth rates of the individuals during the dispersal process are ignored in this model, we have a ji (t) = 0, b ji (t) = 0, 1 i n, t [0,ω] (14) We further assume that (A1) a ij (t) 0, b ij (t) 0, a ii (t) 0, b ii (t) 0, 1 i j n, t [0,ω], and two n n matrices (a ij (t)) and (b ij (t)) are irreducible (A2) B i (t, N i )>0, (t, N i ) R + (0, ), 1 i n (A3) B i (t, N i ) is continuously differentiable with B i(t,n i ) N i < 0, (t, N i ) R + (0, ), 1 i n (A4) Bi u( ) := lim N i Bi u(n i)<μ l i,1 i n, where Bu i (N i) := max t [0,ω] B i (t, N i ), μ l i := min t [0,ω] μ i (t)

3 498 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Biologically, (A1) implies that these n patches cannot be separated into two groups such that there is no immigration of susceptible and infective individuals from first group to second group (see the definition of irreducibility in Section 2); (A2) and (A3) mean that each birth rate function is positive and decreasing; and (A4) represents the case where each birth rate cannot exceed the death rate when the population number is sufficiently large This paper is organized as follows In Section 2, a threshold between the extinction and persistence of the disease is established In Section 3, we prove the uniqueness and the global asymptotic stability of the positive periodic solution when susceptible and infectious individuals have the same dispersal rates, and the global attractivity of the positive periodic solution when the dispersal rates of susceptible and infectious individuals are very close Finally, we present the numerical simulations for the model with two patches 2 Threshold dynamics Let (R k, R k + ) be the standard ordered k-dimensional Euclidean space with a norm For u, v R k, we write u v provided u v R k +, u>vprovided u v Rk + \{0}, and u v provided u v Int(R k + ) Recall that a k k matrix (a ij ) is said to be cooperative if all of its off-diagonal entries are nonnegative; irreducible if its index set {1, 2,,k} cannot be split into two complementary sets (without common indices) {m 1,m 2,,m μ } and {n 1,n 2,,n ν } (μ + ν = k) such that a mp n q = 0, 1 p μ, 1 q ν Let A(t) be a continuous, cooperative, irreducible, and ω-periodic k k matrix function, Φ A( ) (t) be the fundamental solution matrix of the linear ordinary differential system x = A(t)x, and r(φ A( ) (ω)) be the spectral radius of Φ A( ) (ω) It then follows from [1, Lemma 2] (see also [7, Theorem 11]) that Φ A( ) (t) is a matrix with all entries positive for each t>0 By the Perron Frobenius theorem, r(φ A( ) (ω)) is the principal eigenvalue of Φ A( ) (ω) in the sense that it is simple and admits an eigenvector v 0 The following result is useful for our subsequent comparison arguments Lemma 21 Let μ = 1 ω ln r(φ A( )(ω)) Then there exists a positive, ω-periodic function v(t) such that e μt v(t) is a solution of x = A(t)x Proof Let v 0 be an eigenvector associated with the principal eigenvalue r(φ A( ) (ω)) By the change of variable x(t) = e μt v(t), we reduce the linear system x = A(t)x to v = A(t)v μv = ( A(t) μi ) v (21) Thus, v(t) := Φ (A( ) μi) (t)v is a positive solution of (21) It is easy to see that e μt Φ (A( ) μi) (t) = Φ A( ) (t) Moreover, v(ω) = Φ (A( ) μi) (ω)v = e μω Φ A( ) (ω)v = e μω r ( Φ A( ) (ω) ) v = v = v(0) Thus, v(t) is a positive ω-periodic solution of (21), and hence, x(t) = e μt v(t) is a solution of x = A(t)x Let P : R 2n + R2n + be the Poincaré map associated with (13), that is, P ( x 0) = u ( ω,x 0), x 0 R 2n +,

4 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) where u(t, x 0 ) is the unique solution of (13) with u(0,x 0 ) = x 0 In order to find the disease free periodic solutions of (13), we consider S i = B i(t, S i )S i μ i (t)s i + a ij (t)s j, 1 i n (22) Let P 1 : R n + Rn + be the Poincaré map associated with (22), that is, P 1 ( S 0 ) = u 1 ( ω,s 0 ), S 0 R n +, where u 1 (t, S 0 ) is the solution of (22) with u 1 (0,S 0 ) = S 0 If z is a nonnegative constant, we define an auxiliary matrix B 1 (t, z) μ 1 (t) + a 11 (t) a 1n (t) a M(t,z) = 21 (t) a 2n (t) a n1 (t) B n (t, z) μ n (t) + a nn (t) This matrix will be used to prove the existence and the uniqueness of a positive fixed point of P 1 and is different from the standard Jacobian matrix Let F : R 1 + Rn + Rn be defined by the right-hand side of (22) It is easy to see that F has the following properties: (B1) F i (t, S) 0 for every S 0 with S i = 0, t R 1 +,1 i n; (B2) F i S j 0, i j, (t, S) R 1 + Rn +, and D SF(t,0) is irreducible for each t R 1 +,S Rn + ; (B3) for each t 0, F(t, ) is strictly subhomogeneous on R n + in the sense that F(t,αS) > αf(t,s), S 0, α (0, 1); (B4) F(t,0) 0, and F(t,S)<D S F(t,0)S, t 0,S 0 Note that the nonlinear system (22) is dominated by the linear system S = D S F(t,0)S It then follows that for any S 0 R n +, the unique solution u 1(t, S 0 ) of (22) satisfying u 1 (0,S 0 ) = S 0 exists globally on [0, ) and u 1 (t, S 0 ) 0, t 0 We claim that (22) admits a bounded positive solution Indeed, in view of (A4), we can choose a sufficient large real number K such that ω 0 (μ i(t) B i (t, K)) dt > 0, i = 1,,n Then by Lemma 21, there is a positive, ω-periodic function v(t) = (v 1 (t), v 2 (t),,v n (t)) such that V(t)= e μt v(t) is a solution of V = M(t,K)V, where μ = ω 1 ln r(φ M(,K)(ω)) LetΣ(t) = n i=1 V i (t) = e μt n i=1 v i (t) By the first equation in (14), it easily follows that Σ (t) a(t)σ(t), t 0, where a(t) = max{b i (t, K) μ i (t): 1 i n} Thus, lim t Σ(t)= 0, and hence μ<0, ie, r(φ M(,K) (ω)) < 1 Choose l>0 large enough such that lv i (t) > K, 1 i n, t [0,ω] Set H(t) lv(t)ifwerewrite(22)ass = F(t,S), it is easy to see that F ( t,h(t) ) < M(t, K)H (t), t 0, (23) where (A3) is used By the standard comparison theorem (see, eg, [9, Theorem B1]), it follows that ( ) 0 <u 1 mω, lv(0) ΦM(,K) (mω)lv(0) = r ( Φ M(,K) (mω) ) lv(0) = r ( Φ M(,K) (ω) ) m lv(0)<lv(0), m 0,

5 500 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) that is, P m 1 (lv(0)) < lv(0), m 0 Consequently, P m 1 (lv(0)) is bounded In order for P 1 to admit a positive fixed point, we need to assume that (A5) r(φ M(,0) (ω)) > 1 By [11, Theorem 212], it then follows that the Poincaré map P 1 has a unique positive fixed point S (0) = (S1 (0), S 2 (0),,S n (0)) which is globally attractive for S0 R n + \{0} Thus, E 0 = (S1 (0), S 2 (0),, S n (0), 0,,0) is the unique disease free fixed point of the Poincaré map P To investigate the global dynamics of (13), we first show that (13) admits a family of compact, positively invariant sets For convenience, we denote the positive solution (S 1 (t),, S n (t), I 1 (t),, I n (t)) of (13) by (S(t), I (t)) Lemma 22 Let (A1) (A5) hold Then there is N > 0 such that every forward solution in R 2n + of (13) eventually enters into G N := {(S, I) R 2n + : n i=1 (S i + I i ) N }, and for each N N, G N is positively invariant for (13) Proof Let N = n i=1 N i, N i = S i + I i By (13) and (14), we have N = ( Bi (t, N i ) μ i (t) ) N i i=1 ( B u i (N i ) μ l ) i Ni (24) i=1 If B u i (0+) := lim N i 0+ B u i (N i)<μ l i, i = 1, 2,,n, then there exists α>0 such that N (t) αn(t), t 0, and hence, Lemma 22 holds for any positive number N Otherwise, we partition {1, 2,,n} into two sets P 1 and P 2 such that B u i (0+)>μl i, i P 1, B u i (0+) μl i, i P 2 Without loss of generality, we suppose that P 1 ={1,,m} and P 2 ={m + 1,,n}Fori P 1, since B u i (0+) >μl i and B u i ( ) <μl i, (A3) implies that there is a unique k i > 0 such that B u i (k i) μ l i = 0 It follows from (A4) that there is N 0 > 0 such that ( B u i (N) μ l i) m N< k j Bj u (0+) 1, N N 0, 1 i n Let N = nn 0 By the definition of N, it is easy to see that N N implies N i0 N 0 for some 1 i 0 n It then follows from (24) that N (t) m Bj u (0+)k j + ( Bi u 0 (N i0 ) μ l ) i Ni0 0 < 1, if N(t) N, which implies that G N, N N, is positively invariant and every forward orbit enters into G N eventually

6 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Define b 11 (t) b 1n (t) b M 1 (t) = 21 (t) b 2n (t), b n1 (t) b nn (t) where b ii (t) = β i (t)s i (t) μ i(t) γ i (t) + b ii (t), 1 i n Clearly, M 1 (t) is irreducible and has nonnegative off-diagonal elements In the case where the susceptible and infective individuals in each patch have the same dispersal rate, we have the following result on the global attractivity of the ω-periodic solution (S (t), 0) Theorem 21 Let (A1) (A5) hold and r(φ M1 ( )(ω)) < 1 Ifa ij (t) = b ij (t) for 1 i, j n, t [0,ω], then lim t (S(t) S (t)) = 0, lim t I(t)= 0 for all (S 0,I 0 ) (R n + \{0}) Rn + Proof Let us consider a nonnegative solution (S(t), I (t)) of (13) We want to show that lim I(t)= 0 (25) t By (13), we have N i = B i(t, N i )N i μ i (t)n i + a ij (t)n j, 1 i n (26) By the afore-mentioned conclusion for (22), the Poincaré map associated with (26) has a unique positive fixed point S (0) which is globally attractive in R n + \{0} It then follows that for any ɛ>0, there holds N(t) = S(t) + I(t)<S (t) + ɛ, where ɛ = (ɛ,,ɛ) Int(R n + ), when t is sufficiently large Thus, if t is sufficiently large, we have I i <β i(t) ( Si (t) + ɛ) I i ( μ i (t) + γ i (t) ) I i + b ij (t)i j, 1 i n (27) It then suffices to show that positive solutions of the following auxiliary system I ˇ i = β i(t) ( Si (t) + ɛ) Iˇ i ( μ i (t) + γ i (t) ) Iˇ i + b ij (t) Iˇ j, 1 i n, (28) tend to zero as t goes to infinity Let M 2 (t) be the matrix defined by M 2 (t) = diag ( β 1 (t), β 2 (t),,β n (t) ) Since r(φ M1 ( )(ω)) < 1 and r(φ M1 ( )+ɛm 2 ( )(ω)) is continuous for small ɛ, we can fix ɛ>0 small enough such that r(φ M1 ( )+ɛm 2 ( )(ω)) < 1 By Lemma 21, there is a positive, ω-periodic function v(t) = ( v 1 (t), v 2 (t),, v n (t)) such that ρe μt v(t) is a solution of (28) for any constant ρ, where μ = ω 1 ln r(φ M 1 ( )+ɛm 2 ( )(ω)) I 0 R n +, we can choose a real number ρ0 > 0 such that I 0 ρ 0 v(0) By the standard comparison theorem (see, eg, [9, Theorem A4]), we then get (25) For any (S 0,I 0 ) (R n + \{0}) Rn +,wehaven0 = S 0 + I 0 R n + \{0} By the global attractivity of S (0) for P 1, it then follows that ( S(t) S (t) ) ( = lim N(t) I(t) S (t) ) = 0 lim t t

7 502 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) If the susceptible and infective individuals in each patch have different dispersal rate, and the initial value I 0 is small, we still have the result on the attractivity of the ω-periodic solution (S (t), 0) Theorem 22 Let (A1) (A5) hold and r(φ M1 ( )(ω)) < 1 Then there exists δ>0 such that for every (S 0,I 0 ) G N with S 0 0 and Ii 0 <δ, 1 i n, the solution (S(t), I (t)) of (13) satisfies lim t (S(t) S (t)) = 0, lim t I(t)= 0 Proof Let us consider an auxiliary system S i = B i(t, S i ) S i μ i (t) S i + ( B i (t, 0+) + γ i (t) ) ɛ + a ij (t) S j, 1 i n, (29) where ɛ>0 is a small constant to be determined By (A5) and the previous analysis of system (22), we can restrict ɛ small enough such that (29) admits a globally attractive and positive ω- periodic solution S (t, ɛ) LetS ɛ (t, N ) be the solution of (29) through (N,,N ) at t = 0 We choose an integer n 1 > 0 such that S ɛ (t, N )<S (t, ɛ) + ɛ, t n 1 ω Define a matrix M 1 (t, ɛ) by β 1 (t)s1 (t, ɛ) μ 1(t) γ 1 (t) + b 11 (t) b 1n (t) b 21 (t) b 2n (t) b n1 (t) β n (t)sn (t, ɛ) μ n(t) γ n (t) + b nn (t) Since M 1 (t, 0) = M 1 (t) and r(φ M1 (,ɛ)+ɛm 2 (t)(ω)) is continuous for small ɛ, we can now restrict ɛ small enough such that r(φ M1 (,ɛ)+ɛm 2 ( )(ω)) < 1 By Lemma 21, there is a positive ω-periodic function v(t) = (v 1 (t),,v n (t)) such that I(t) ˇ = e μ3t v(t) is a solution of I ˇ = (M 1 (t, ɛ) + ɛm 2 (t)) I ˇ, where μ 3 = ω 1 ln r(φ M 1 (,ɛ)+ɛm 2 ( )(ω)) Choose k>0small enough such that kv(t) < ɛ for all t [0,ω] Now we define another auxiliary system I ˆ i = β i(t)n Iˆ i ( μ i (t) + γ i (t) ) Iˆ i + b ij (t) Iˆ j, 1 i n (210) Let I(t,δ) ˆ be the solution of (210) through (δ,,δ) R n at t = 0 We restrict δ>0small enough such that I(t,δ)<ke ˆ μ3t v(t) kv(t) < ɛ, t [0,n 1 ω] (211) Let (S(t), I (t)) be a nonnegative solution of (13) with (S 0,I 0 ) G N, S 0 0 and Ii 0 <δ, 1 i n It then follows that S(t) 0, t >0 We further claim that I(t) ke μ3t v(t), t 0 Suppose not By the comparison principle and (211), there exist q, 1 q n, and T 1 >n 1 ω such that I(t) ke μ3t v(t), for 0 t T 1, I q (T 1 ) = k ( e μ 3T 1 v(t 1 ) ) q, I q (t) > k ( e μ 3T 1 v(t 1 ) ) q, for 0 <t T 1 1 (212)

8 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Note that for 0 t T 1,wehave S i <B i(t, S i )S i μ i (t)s i + ( B i (t, 0+) + γ i (t) ) ɛ + a ij (t)s j, 1 i n (213) It follows from the comparison principle that S(T 1 )<S (T 1,ɛ)+ ɛ Then, for 0 t T 1 1, we have S(t) < S (t, ɛ) + ɛ, and hence I i <β i(t) ( S i (t, ɛ) + ɛ) I i ( μ i (t) + γ i (t) ) I i + b ij (t)i j, 1 i n Since I(T 1 ) ke μ 3T 1 v(t 1 ), the comparison principle implies that and hence, I(t)<ke μ 3t v(t), for 0 t T 1 1, I q (t) < k ( e μ 3t v(t) ) q, for 0 <t T 1 1, which contradicts to (212) This proves the claim By the claim above, (213) holds for all t 0 Thus, the comparison principle implies that S(t) < S (t, ɛ) + ɛ, t n 1 ω By a similar argument as above, it then follows that I(t)<ke μ 3t v(t), t >T 1 Consequently, I(t) 0ast Since P m (x 0 ) = u(mω, x 0 ), x 0 R 2n +,wehave P m( S 0,I 0) = u ( mω, ( S 0,I 0)) = ( S(mω), I (mω) ), ( S 0,I 0) R n + Rn + Given (S 0,I 0 ) G N with S 0 0 and Ii 0 <δ,1 i n, it easily follows that S(t) Int(R n + ), t >0 Let ω = ω ( S 0,I 0) { := (S,I ): {m k } such that lim P m ( k S 0,I 0) } = (S,I ) k be the omega limit set of (S 0,I 0 ) for P Since lim t I(t) = 0, there holds ω = ω {0} We claim that ω {0} Assume that, by contradiction, ω ={0} lim n P n (S 0,I 0 ) = (0, 0), then lim t S(t) = 0 By assumption (A5), we can choose a small η>0 such that r(φ M(,0) ηe (ω)) > 1, where E = diag(1,,1) It follows that there exists t >0 such that B i ( t,ni (t) ) β i (t)i i (t) B i (t, 0+) η, t t, 1 i n Then S(t) = (S 1 (t),,s n (t)) satisfies S i (t) > ( B i (t, 0+) η ) S i μ i (t)s i + a ij (t)s j, t t, 1 i n (214) Let p(t) = (p 1 (t),, p n (t)) be the positive ω-periodic function such that e μ4t p(t) is a solution of the linear system Ŝ i = ( B i (t, 0+) η ) Ŝ i μ i (t)ŝ i + a ij (t)ŝ j, 1 i n, (215)

9 504 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) where μ 4 = ω 1 ln(r(φ M(,0) ηe(ω))) > 0 Since S( t) Int(R n + ), we can choose a small number α>0 such that S( t) αp(0) Then the comparison theorem implies that S(t) αe μ4(t t) p(t t) αe μ4(t t) min p(t t), t t 0 t t, and hence lim t S i (t) =,1 i n, a contradiction Note that for any S 0 R n +,wehave u(t, (S 0, 0)) = (u 1 (t, S 0 ), 0), t 0 It then follows that P m( S 0, 0 ) = ( P1 m ( S 0 ), 0 ), S 0 R n +,m 0 Since ω is an internal chain transitive set for P, and hence, ω is an internal chain transitive set for P 1 Let W s( S (0) ) := { S 0 : P1 m ( S 0 ) S (0): m } Since ω {0} and S (0) is globally attractive for P 1 in R n + \{0},wehave ω W s (S (0)) By [11, Theorem 121], we then get ω ={S (0)}, and hence ω ={(S (0), 0)} Thus, lim t (S(t) S (t)) = 0 and lim t I(t)= 0 The following result shows that actually r(φ M1 ( )(ω)) is a threshold parameter for the extinction and the uniform persistence of the disease When r(φ M1 ( )(ω)) > 1, the model (13) admits at least one positive periodic solution, and the disease is uniformly persistent Theorem 23 Let (A1) (A5) hold and r(φ M1 ( )(ω)) > 1 Then system (13) admits at least one positive periodic solution, and there is a positive constant ɛ such that for all (S 0,I 0 ) R n + Int(R n + ), the solution (S(t), I (t)) of (13) satisfies lim inf I i(t) ɛ, 1 i n t Proof Define X = R 2n +, X 0 = R n + Int( R n +), X0 = X \ X 0 We first prove that P is uniformly persistent with respect to (X 0, X 0 ) By the form of (13), it is easy to see that both X and X 0 are positively invariant Clearly, X 0 is relatively closed in X Furthermore, system (13) is point dissipative (see Lemma 22) Set M = {( S 0,I 0) X 0 : P m( S 0,I 0) X 0, m 0 } We now show that M = { (S, 0): S 0 } (216) Obviously, {(S, 0): S 0} M For any (S 0,I 0 ) X 0 \{(S, 0): S 0}, we partition {1, 2,,n} into two sets Q 1 and Q 2 such that Ij 0 = 0, j Q 1, Ii 0 > 0, i Q 2, where Q 1 and Q 2 are nonempty j Q 1, i Q 2,wehave I j (0) b jii i (0)>0

10 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) It follows that (S(t), I (t)) / X 0 for 0 <t 1 Thus, the positive invariance of X 0 implies (216) It is clear that there are two fixed points of P in M, which are M 0 = (0, 0) and M 1 = (S (0), 0) Choose η>0 small enough such that r(φ M1 ( ) ηm 2 ( )(ω)) > 1 Let us consider a perturbed system of (22) Ŝ i = B i(t, Ŝ i + δ)ŝ i ( μ i (t) + β i (t)δ ) Ŝ i + a ij (t)ŝ j, 1 i n (217) As in our previous analysis of system (22), we can choose δ>0 small enough such that the Poincaré map associated with (217) admits a unique positive fixed point S (0,δ) which is globally attractive in R n + \{0} By the implicit function theorem, it follows that S (0,δ) is continuous in δ Thus, we can fix a small number δ>0 such that S (t, δ) > S (t) η, t 0, where η = (η,,η) By the continuity of solutions with respect to the initial values, there exists δ0 > 0 such that for all (S0,I 0 ) X 0 with (S 0,I 0 ) M i δ0,wehave u(t, (S 0,I 0 )) u(t, M i ) <δ, t [0,ω], i = 0, 1 We now claim that lim sup d ( P m( S 0,I 0) ),M i δ 0, i = 0, 1 m Suppose, by contradiction, that lim sup n d(p m (S 0,I 0 ), M i )<δ0 for some (S0,I 0 ) X 0 and i Without loss of generality, we can assume that d(p m (S 0,I 0 ), M i )<δ0, m 0 Then, we have u(t, P m (S 0,I 0 )) u(t, M i ) <δ, m 0, t [0,ω] For any t 0, let t = mω + t, where t [0,ω)and m =[ ω t ] is the greatest integer less than or equal to ω t Thus, we get u ( t, ( S 0,I 0)) u(t, M i ) = u ( t,p m( S 0,I 0)) u(t,m i ) <δ, t 0 Let (S(t), I (t)) = u(t, (S 0,I 0 )) It then follows that 0 I i (t) δ, t 0, 1 i n Then for t 0, we have S i B i(t, S i + δ)s i ( μ i (t) + β i (t)δ ) S i + a ij (t)s j, 1 i n (218) Since the fixed point S (0,δ) of the Poincaré map associated with (217) is globally attractive and S (t, δ) > S (t) η, there is T>0such that S(t) S (t) η for t T As a consequence, for t T, there holds I i β i(t) ( S i (t) η) I i ( μ i (t) + γ i (t) ) I i + b ij (t)i j, 1 i n (219) Since r(φ M1 ( ) ηm 2 ( )(ω)) > 1, it is easy to see that lim t I i (t) =, i = 1, 2,,n, which leads to a contradiction Note that S (0) is globally attractive in R n + \{0} for P 1 By the afore-mentioned claim, it then follows that M 0 and M 1 are isolated invariant sets in X, W s (M 0 ) X 0 =, and W s (M 1 ) X 0 = Clearly, every orbit in M converges to either M 0 or M 1, and M 0 and M 1 are acyclic in M By [11, Theorem 131] for a stronger repelling property of X 0, we conclude that P is uniformly persistent with respect to (X 0, X 0 ) Thus, [11, Theorem 311] implies the uniform persistence of the solutions of system (13) with respect to (X 0, X 0 ) By [11, Theorem 136], P has a fixed point ( S(0), I(0)) X 0 Then S(0) R n + and I(0) Int(R n + )We further claim that S(0) R n + \{0} Suppose that S(0) = 0 By the second equation in (14), we

11 506 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) then get 0 = n i=1 (μ i (t) + γ i (t)) I i (0), and hence I i (0) = 0, i= 1, 2,,n, a contradiction By the first equation in (13) and the irreducibility of the cooperative matrix (a ij (t)), it follows that u(t, ( S(0), I(0))) Int(R n + ), t >0 Then ( S(0), I(0)) is a componentwise positive fixed point of P Thus, ( S(t), I(t)) is a positive ω-periodic solution of (13) 3 The positive periodic solutions In the case where the dispersal rate of susceptible individuals and infective individuals are equal, we are able to prove the uniqueness and global asymptotic stability of the positive ω- periodic solution under the condition that r(φ M1 ( )(ω)) > 1 Theorem 31 Let (A1) (A5) hold and r(φ M1 ( )(ω)) > 1 Ifa ij (t) = b ij (t) for 1 i, j n, t [0,ω], then system (13) admits a unique positive ω-periodic solution which is globally asymptotically stable in R n + (Rn + \{0}) Proof By (13), when a ij (t) = b ij (t), wehave S i = B i(t, N i )N i μ i (t)s i β i (t)s i I i + γ i (t)i i + Then we obtain I i = β i(t)s i I i ( μ i (t) + γ i (t) ) I i + N i = B i(t, N i )N i μ i (t)n i + a ij (t)s j, 1 i n, a ij (t)i j, 1 i n (31) a ij (t)n j, 1 i n (32) By the afore-mentioned conclusion for (22), the Poincaré map associated with (32) has a unique positive fixed point S (0) which is globally attractive for N R n + \{0} Then (31) is equivalent to the following system: N i = B i(t, N i )N i μ i (t)n i + a ij (t)n j, 1 i n, I i = β i(t)(n i I i )I i ( μ i (t) + γ i (t) ) (33) I i + a ij (t)i j, 1 i n Since lim t (N(t) S (t)) = 0, the second equation of (33) has the following limiting system: I i = β i(t) ( Si (t) ) I i I i ( μ i (t) + γ i (t) ) I i + a ij (t) I j, 1 i n (34) Let F 1 : R 1 + Rn + Rn be defined by the right-hand side of (34) Clearly, F 1 satisfies (B1) (B4) Let P 2 : R n + Rn + be the Poincaré map associated with (34), that is, ( P 2 I 0 ) ( = u 2 ω,i 0 ), I 0 R n +, where u 2 (t, I 0 ) is the solution of (34) with u 2 (0,I 0 ) = I 0 We claim that (34) admits a bounded positive solution

12 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Define ā 11 (t) a 1n (t) a M 2 (t, Z) = 21 (t) a 2n (t), a n1 (t) ā nn (t) where ā ii (t) = β i (t)(si (t) Z) μ i(t) γ i (t) + a ii (t),1 i n We can choose a sufficiently large real number Z>0 such that ω 0 (β i(t)(si (t) Z) μ i(t) γ i (t)) dt < 0, 1 i n By Lemma 21, there is a positive, ω-periodic function v(t) = (v 1 (t), v 2 (t),, v n (t)) such that V(t)= e μ5t v(t) is a solution of V = M 2 (t, Z)V, where μ 5 = 1 ω ln r(φ M 2 (,Z)(ω)) Let Σ(t) = n i=1 V i (t) = e μ 5t n i=1 v i (t) By the first equation in (14), it easily follows that Σ (t) b(t)σ(t), t 0, where b(t) = max{β i (t)(si (t) Z) μ i(t) γ i (t): 1 i n} Thus, lim t Σ(t) = 0, and hence μ 5 < 0, ie, r(φ M2 (,Z)(ω)) < 1 Choose l>0 large enough such that lv i (t) > Z, t [0,ω], i = 1, 2,,n Set H(t) lv(t)ifwerewrite(34)asi = F 1 (t, I), it is easy to see that ( ) F 1 t,h(t) <M2 (t, Z)H (t), t 0 (35) By the standard comparison theorem (see, eg, [9, Theorem B1]), it follows that ( ) 0 <u 2 mω, lv(0) ΦM2 (,Z)(mω)lv(0) = r ( Φ M2 (,Z)(mω) ) lv(0) = r ( Φ M2 (,Z)(ω) ) m lv(0)<lv(0), m 0 That is, P2 m(lv(0)) < lv(0), m 0 Consequently, P 2 m (lv(0)) is bounded Since r(φ M2 (,0)(ω)) = r(φ M1 ( )(ω)) > 1 By [11, Theorem 212], it then follows that the Poincaré map P 2 has a unique positive fixed point I(0) which is globally attractive for I 0 R n + \{0} Thus, the Poincaré map P associated with (31) admits a unique fixed point (S (0) I(0), I(0)) It then follows from Theorem 23 that the unique fixed point is positive We denote it by ( S(0), I(0)) Let P 3 : X := R 2n + R2n + be the Poincaré map associated with (33), that is, ( P 3 x 0 ) ( = u 3 ω,x 0 ), x 0 R 2n +, where u 3 (t, x 0 ) is the solution of (33) with u 3 (0,x 0 ) = x 0 Thus, we have ( N 0,I 0) ( ( = u 3 mω, N 0,I 0)), ( N 0,I 0) R n + Rn + P m 3 Let (N 0,I 0 ) (R n + \{0}) (Rn + \{0}) be fixed It then follows that ( ) ( ( N(t),I(t) = u3 t, N 0,I 0)) Int ( R n ) ( ) + Int R n +, t >0 Let ω = ω(n 0,I 0 ) be the omega limit set of (N 0,I 0 ) for P 3 Since lim t (N(t) S (t)) = 0, there holds ω ={S (0)} ω We claim that ω {0} Assume that, by contradiction, ω ={0} Then lim m P3 m(n 0,I 0 ) = (S (0), 0), that is, lim t (N(t) S (t)) = 0, lim t I(t)= 0 Since r(φ M1 ( )(ω)) > 1, we can choose a small η>0 such that r(φ M1 ( ) ηe(ω)) > 1, where E = diag(1,,1) It follows that there exists t >0 such that β i (t) ( N i (t) I i (t) ) β i (t)si (t) η, t t, 1 i n Then I(t)= (I 1 (t),,i n (t)) satisfies I i β i (t)si (t) η) I i ( μ i (t) + γ i (t) ) I i + a ij (t)i j, t t, 1 i n (36)

13 508 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Let q(t) = (q 1 (t),, q n (t)) be the positive ω-periodic function such that e μ 6t q(t) is a solution of the linear system I ˆ i = ( β i (t)si (t) η) Iˆ i ( μ i (t) + γ i (t) ) Iˆ i + a ij (t) ˆ I j, 1 i n, (37) where μ 6 = ω 1 ln(r(φ M 1 ( ) ηe(ω))) > 0 Since I( t) Int(R + n ), we can choose a small number α>0 such that I( t) αq(0) Then the comparison theorem implies that I(t) αe μ6(t t) q(t t) αe μ6(t t) min q(t t), t t 0 t t, and hence lim t I i (t) =,1 i n, a contradiction Note that for any I 0 R n +,wehave u 3 (t, (S (0), I 0 )) = (S (t), u 2 (t, I 0 )), t 0 It then follows that P m 3 ( S (0), I 0) = ( S (0), P m 2 ( I 0 )), I 0 R n +,m 0 Since ω is an internal chain transitive set for P 3, ω is an internal chain transitive set for P 2 Let W s( I(0) ) { := I 0 ( ( : P m 2 I 0 )) } = I(0) lim m Since ω {0} and I(0) is globally attractive for P 2 in R n + \{0}, wehave ω W s ( I(0)) By [11, Theorem 121], we then get ω ={ I(0)}, and hence ω ={(S (0), I(0))}, which implies that the positive fixed point ( S(0), I(0)) of P is globally attractive in R n + (Rn + \{0}) It follows that system (13) admits a unique positive ω-periodic solution ( S(t), I(t)) such that lim t (S(t) S(t)) = 0 and lim t (I (t) I(t))= 0, (S 0,I 0 ) R n + (Rn + \{0}) It remains to prove the stability of ( S(t), I(t)) for (13), which is equivalent to the stability of ( N(t), I(t)):= ( S(t) + I(t), I(t)) for (33) The associated Jacobian matrix is [ ] A1 (t) 0 A(t) =, A 2 (t) A 3 (t) where a11 (t) a 12(t) a 1n (t) a 21 (t) a22 A 1 (t) = (t) b 2n(t) a n1 (t) a n2 (t) ann (t) with aii (t) = B i(t,n i ) N i Ni = N i N i + B i (t, N i ) μ i (t) + a ii (t), A 2 (t) = diag ( ) β 1 (t) I 1,β 2 (t) I 2,,β n (t) I n, and b11 (t) a 12(t) a 1n (t) a 21 (t) b22 A 3 (t) = (t) b 2n(t) a n1 (t) a n2 (t) bnn (t) with b ii (t) = β i(t) N i 2β i (t) I i μ i (t) γ i (t) + a ii (t)

14 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Obviously, ā11 (t) a 12(t) a 1n (t) a 21 (t) ā22 A 1 (t) < (t) b 2n(t) := C 1(t) a n1 (t) a n2 (t) ānn (t) with āii (t) = B i(t, N i ) μ i (t) + a ii (t), and b 11 (t) a 12(t) a 1n (t) a 21 (t) b 22 A 3 (t) < (t) b 2n(t) := C 3(t) a n1 (t) a n2 (t) b nn (t) with b ii (t) = β i(t) N i β i (t) I i μ i (t) γ i (t) + a ii (t) The comparison principle implies that Φ A1 ( )(t) Φ C1 ( )(t), Φ A3 ( )(t) Φ C3 ( )(t), and hence Φ A1 ( )(ω) Φ C1 ( )(ω), Φ A3 ( )(ω) Φ C3 ( )(ω) By [9, Theorem A4], we have μ(φ A1 ( )(ω)) < μ(φ C1 ( )(ω)) and μ(φ A3 ( )(ω)) < μ(φ C3 ( )(ω)) Notice that ( N 1 (t),, N n (t)) is a positive ω-periodic solution of the system N = C 1 (t)n Then we have Φ C1 ( )(t) N 1 (0) N n (0) = N 1 (t) N n (t) It follows that N 1 (0) N 1 (ω) N 1 (0) Φ C1 ( )(ω) N n (0) = N n (ω) = N n (0), and hence μ(φ C1 ( )(ω)) = 1 On the other hand, ( I 1 (t),, I n (t)) is a positive ω-periodic solution of the system I = C 3 (t)i Thus, we obtain I 1 (0) I 1 (t) Φ C3 ( )(t) I n (0) = I n (t) It follows that I 1 (0) I 1 (ω) I 1 (0) Φ C3 ( )(ω) I n (0) = I n (ω) = I n (0), and hence μ(φ C3 ( )(ω)) = 1 Consequently, we have μ ( Φ A ( )(ω) ) = max { μ ( Φ A1 ( )(ω) ),μ ( Φ A3 ( )(ω) )} < 1, which implies the stability of ( N(t), I(t)) At last, we prove the global attractivity of positive periodic solution in the case where {b ij (t)} is very close to {a ij (t)}letλ 0 be the set of all continuous and ω-periodic n n matrix functions satisfying a ij (t) > 0, i j, a ii (t) < 0 and n a ji (t) = 0

15 510 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Theorem 32 Assume (A1) (A5) hold Let λ 0 ={a ij (t): 1 i, j n} Λ 0 be fixed, λ = {b ij (t): 1 i, j n} Λ 0, M 1λ (t) be the matrix M 1 (t) with parameter λ, and M 1λ0 (t) be the matrix M 1λ (t) with λ = λ 0 If r(φ M1λ0 ( )(ω)) > 1, then there exists ɛ>0 such that for any λ with λ λ 0 ɛ, system(13) admits a unique positive ω-periodic solution ( S λ1 (t), S λ2 (t),, S λn (t), I λ1 (t),, I λn (t)) such that lim t (S i (t) S λi (t)) = 0 and lim t (I i (t) I λi (t)) = 0 for every (S 0,I 0 ) R n + Int(Rn + ) Proof There exist ɛ 0 > 0,η>0, such that r(φ M1λ ( )(ω)) > 1 and r(φ M1λ ηm2 ( )(ω)) > 1 whenever λ λ 0 ɛ 0 We fix a sufficiently small δ>0 such that the Poincaré map associated with (217) admits a unique positive fixed point S (0,δ), which is globally attractive in R n + \{0} and S (t, δ) > S (t) η Letu(t, (S 0,I 0 ), λ) be the solution of (13) with parameter λ and initial value (S 0,I 0 ) X By the continuity of solutions with respect to initial values and parameter λ, there exist positive numbers δ0 and ɛ 0 such that u(t, (S0,I 0 ), λ) u(t, M i,λ 0 ) <δ, t [0,ω], (S 0,I 0 ) M i δ0 and λ λ 0 ɛ0, i = 0, 1 Let ɛ = min{ɛ 0,ɛ0 }Bythe argument similar to that of the claim in the proof of Theorem 23, it follows that for any λ with λ λ 0 ɛ, and all (S 0,I 0 ) R n + Int(Rn + ), there holds lim sup d ( P m ( λ S 0,I 0) ),M i δ 0, i = 0, 1, m where P λ is the Poincaré map associated with (13) with parameter λ Moreover, Lemma 22 implies that solutions of (13) in X are uniformly bounded and ultimately bounded for each λ Λ 0 It follows that P has a global attractor A λ X 0 for each λ Λ 0 LetΛ 1 = Λ 0 {λ: λ λ 0 ɛ } Then there exists a bounded and closed set G in R n + Rn +, such that λ Λ1 A λ G Hence, by [11, Theorem 142], there exists a δ 0 > 0 such that for any λ Λ 1, lim inf d( P m ( m λ S 0,I 0) ), X 0 δ0 Since λ Λ 1 P(A λ ) = λ Λ 1 A λ G = G X 0, λ Λ 1 P(A λ ) is compact By applying [11, Theorem 141] on the perturbation of a globally stable fixed point, we complete the proof 4 Numerical simulations In order to simulate the periodic solutions, we consider the case that the patch number is 2 For simplicity, we assume that the contact rate β i (t), i = 1, 2, is ω-periodic with the expression β 1 (t) = β 2 (t) = m sin(pt) + q, and other parameters are independent of time t Then ω = 2π p, and assumption (14) is equivalent to that a 12 = a 22, a 21 = a 11, b 12 = b 22, b 21 = b 11 Thus, (13) reduces to S 1 = B 1(N 1 )N 1 (μ 1 a 11 )S 1 β(t)s 1 I 1 + γ 1 I 1 a 22 S 2, S 2 = B 2(N 2 )N 2 (μ 2 a 22 )S 2 β(t)s 2 I 2 + γ 2 I 2 a 11 S 1, I 1 = β(t)s (41) 1I 1 (μ 1 + γ 1 b 11 )I 1 b 22 I 2, I 2 = β(t)s 2I 2 (μ 2 + γ 2 b 22 )I 2 b 11 I 1

16 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) As mentioned in [3], we choose B i (N i ) = r i N i + c i, where c i <μ i, i = 1, 2 Suppose that r 1 = r 2 = r, c 1 = c 2 = c, μ 1 = μ 2 = μ, γ 1 = γ 2 = γ, a 11 = a 22 = b 11 = b 22 = θ<0 Then (41) reduces to S 1 = r (μ + θ c)s 1 ( m sin(pt) + q ) S 1 I 1 + (c + γ)i 1 + θs 2, S 2 = r (μ + θ c)s 2 ( m sin(pt) + q ) S 2 I 2 + (c + γ)i 2 + θs 1, I 1 = ( m sin(pt) + q ) (42) S 1 I 1 (μ + γ + θ)i 1 + θi 2, I 2 = ( m sin(pt) + q ) S 2 I 2 (μ + γ + θ)i 2 + θi 1 It is easy to verify that conditions (A1) (A5) are satisfied In this case, (S1 (0), S 2 (0)) can be obtained explicitly as S1 (0) = S 2 (0) = r μ c Under all assumptions above, we get [( β(t) r μ c M 1 (t) = μ γ θ θ [( μ γ θ θ = θ μ γ θ θ )] β(t) μ c r μ γ θ )] + [( β(t) r μ c 0 0 β(t) r μ c Let A be a 2 2 constant matrix, and α(t) be a continuous ω-periodic function Note that if x(t) is a solution of x = (A + α(t)i)x, then y(t) = e t 0 α(s)ds x(t) satisfies )] y (t) = e t 0 α(s)ds( x α(t)x ) = e t 0 α(s)ds Ax(t) = Ae t 0 α(s)ds x(t) = Ay(t) Thus, we have φ A+α( )I (t) = e t 0 α(s)ds e At By the above observation, it follows that r ( Φ M1 ( )(ω) ) = e μ c r ω 0 β(t)dt e (μ+γ)ω Fix μ = 02, c = 01, θ = 1, γ = 4, m = 1, p = 2π, q = 01, r = 1 Since ω = 1, we have r ( Φ M1 ( )(1) ) < 1 By Theorem 21, system (13) has a positive ω-periodic solution such that lim t (S(t) S (t)) = 0 and lim t I(t)= 0 for all (S 0,I 0 ) R n + Rn + Our numerical simulations in Fig 1 confirm this result Fix μ = 2, c = 1, θ = 1, γ = 01, m = 1, p = 2π, q = 1, r = 10 We then have ω = 1 and r(φ M1 ( )(1)) > 1 By Theorem 31, system (13) has a unique positive ω-periodic solution such that lim t (S(t) S(t)) = 0 and lim t (I (t) I(t))= 0 for all (S 0,I 0 ) R n + (Rn + \{0}) Our numerical simulations in Fig 2 confirm this result Acknowledgment We are very grateful to the anonymous referee for careful reading and helpful suggestions which led to an improvement of our original manuscript

17 512 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Fig 1 The extinction of the disease

18 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Fig 1 (continued)

19 514 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Fig 2 The uniform persistence of the disease

20 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) Fig 2 (continued)

21 516 F Zhang, X-Q Zhao / J Math Anal Appl 325 (2007) References [1] G Aronsson, RB Kellogg, On a differential equation arising from compartmental analysis, Math Biosci 38 (1973) [2] F Brauer, P van den Driessche, Models for transmission of disease with immigration of infectives, Math Biosci 171 (2001) [3] K Cooke, P van den Driessche, X Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J Math Biol 39 (1999) [4] O Diekmann, JAP Heesterbeek, Mathematical Epidemiology of Infectious diseases, Model Building, Analysis and Interpretation, Wiley, 2000 [5] HW Hethcote, Qualitative analysis of communicable disease models, Math Biosci 28 (1976) [6] HW Hethcote, SA Levin, Periodicity in epidemiological models, in: Applied Mathematical Ecology, Springer- Verlag, Berlin, 1989 [7] MW Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J Math Anal 16 (1985) [8] Y Jin, W Wang, The effect of population dispersal on the spread of a disease, J Math Anal Appl 308 (2005) [9] HL Smith, P Waltman, The Theory of the Chemostat, Cambridge Univ Press, 1995 [10] W Wang, X-Q Zhao, An epidemic model in a patchy environment, Math Biosci 190 (2004) [11] X-Q Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003