Stability and Bifurcation Analysis in A SEIR Epidemic Model with Nonlinear Incidence Rates

Transcription

1 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 Sabiliy and Bifurcaion Analysis in A SEIR Epidemic Model wih Nonlinear Incidence Raes Changjin Xu and Maoxin liao Absrac In his paper, a special SEIR epidemic model wih nonlinear incidence raes is considered. By analyzing he associaed characerisic ranscendenal equaion, i is found ha Hopf bifurcaion occurs when hese delays pass hrough a sequence of criical value. Some explici formulae for deermining he sabiliy and he direcion of he Hopf bifurcaion periodic soluions bifurcaing from Hopf bifurcaions are obained by using he normal form heory and cener manifold heory. Some numerical simulaion for jusifying he heoreical analysis are also presened. Finally, biological explanaions and main conclusions are given. Index Terms SEIR epidemic model; sabiliy; Hopf bifurcaion; periodic soluion I. INTRODUCTION I N recen years, grea aenion has been paid o he dynamics properies (including sable, unsable, persisen and oscillaory behavior ) of he epidemic models which have significan biological bacground. Many excellen and ineresing resuls have been obained [-13]. I is well nown ha epidemic models are invesigaed on he ransmission dynamics of infecious diseases in hos populaion. In his paper, we assume ha disease spreads in a single hos populaion hrough direc conac of hoss and a hos says in a laen period before becoming infecious afer he iniial infecion. An infecious hos may die from disease or recover wih acquired immuniy o he disease a he infecious sage. The hos populaion is pariioned ino four classes: he suscepible, exposed (laen), infecious, and recovered wih sizes denoed by S, E, I, and R, respecively. The hos oal populaion N = S + E + I + R. Then, we consider he following differenial equaions: Ṡ() = µ µs αi p S q, Ė() = αi p S q (ɛ + µ)e, = ɛe (γ + µ)i, Ṙ() = γi µr, p, q, α, µ, ɛ and γ are posiive parameers. For he meaning of he parameers p, q, α, µ, ɛ and γ, one can see Sun e al.[]. Considering he biological meaning of sysem (1), we can easily obain ha he feasible region for sysem (1) is R 4 +. Manuscrip received July 2, 21; revised Sepember 29, 21. This wor is suppored by Naional Naural Science Foundaion of China (No ),he Scienific Research Fund of Hunan Provincial Educaion Deparmen(No.1C6), he Science and echnology Program of Hunan Province (No.21FJ621) and and Docoral Foundaion of Guizhou College of Finance and Economics (21). C. Xu is wih Guizhou Key Laboraory of Economics Sysem Simulaion, School of Mahemaics and Saisics, Guizhou College of Finance and Economics, Guiyang, China. xcj43@126.com. M. Liao is wih School of Mahemaics and Physics, Nanhua Universiy, Hengyang, 4211, P.R.China. (1) Adding he all he equaions of (1), we ge Ṡ + Ė + I + Ṙ = µ(s + E + I + R 1), which has he following implicaion: he hree-dimensional simplex Σ = {(S, E, I, R) R 4 + : S + E + I + R = 1} is posiively invarian. On he simplex Σ, R() = 1. According he above discussion and under he assumpion p = 1, Sun e al.[] obained he following hree-dimensional sysem Ṡ() = µ µs αis q, Ė() = αis q (ɛ + µ)e, = ɛe (γ + µ)i and invesigaed he global sabiliy of (2). In order o reflec he dynamical behaviors of he models depending on he pas informaion, i is more reasonable o incorporae ime delays ino he sysem. Based on his idea and under he assumpion p = q = 1, in his paper, we consider he following delay differenial equaion: Ṡ() = µ µs αis, Ė() = αis (ɛ + µ)e( τ), = ɛe( τ) (γ + µ)i. The dynamics of sysem (3) wih delays could be more complicaed and ineresing. To obain a deep and clear undersanding of dynamics of SEIR epidemic model wih nonlinear incidence raes, in his paper, we sudy he sabiliy, he local Hopf bifurcaion for sysem (3). The remainder of he paper is organized as follows. In Secion 2, we invesigae he sabiliy of he posiive equilibrium and he occurrence of local Hopf bifurcaions. In Secion 3, he direcion and sabiliy of he local Hopf bifurcaion are esablished. In Secion 4, numerical simulaions are carried ou o illusrae he validiy of he main resuls. Biological explanaions and some main conclusions are drawn in Secion. II. STABILITY OF THE POSITIVE EQUILIBRIUM AND LOCAL HOPF BIFURCATIONS In his secion, we shall sudy he sabiliy of he posiive equilibrium and he exisence of local Hopf bifurcaions. One can see ha if he following condiion (H1) αɛµ > µ(ɛ + µ)(γ + µ) (2) (3) (Advance online publicaion: 24 Augus 211)

2 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 holds, hen Eq. (3) has an unique posiive equilibrium E (S, E, I ), q2ω (q 3 q 1 ω 2 ) 2 = p 2 1ω 4 + (ω 3 p 2 ω) 2, S µ = µ + αi, E = γ + µ I, I αɛµ µ(ɛ + µ)(γ + µ) =. namely, ɛ α(ɛ + µ)(γ + µ) (4) ω 6 +(p 2 Le E = (S, E, I 1 2p 2 +2q 1 q 3 q1)ω 2 4 +(p 2 2 q2)ω 2 2 q3 2 =. (1) ) be he arbirary equilibrium poin, and se x() = S, y() = E, z() = I, Le z = ω 2, hen (1) become hen (4) becomes ẋ() = (µ + αi )x() + αs z() αx()z(), ẏ() = αi x() + αs z() (ɛ + µ)y( τ) + αx()z(), ż() = (γ + µ)z() + ɛy( τ). () The linearizaion of Eq. () a (,, ) is ẋ() = (µ + αi )x() + αs z(), ẏ() = αi x() + αs z() (ɛ + µ)y( τ), ż() = (γ + µ)z() + ɛy( τ). whose characerisic equaion is (6) λ 3 + p 1 λ 2 + p 2 λ + (q 1 λ 2 + q 2 λ + q 3 )e λτ =, (7) p 1 = 2µ + γ + αi, p 2 = (µ + αi )(γ + µ), q 1 = γ + µ, q 2 = (2µ + γ + αi )(ɛ + µ) ɛαs, q 3 = (µ + αi )(ɛ + µ)(γ + µ) α 2 ɛs I (µ + αi )ɛαs. In order o invesigae he disribuion of roos of he ranscendenal equaion 7), he following Lemma is useful. Lemma 1 [2] For he ranscendenal equaion P (λ, e λτ1,, e λτm ) = λ n + p () 1 λn + + p () n λ + p() n [ ] + p (1) 1 λn + + p (1) n λ + p(1) n e λτ1 + [ ] + p (m) 1 λ n + + p (m) n λ + p(m) n e λτm =, as (τ 1, τ 2, τ 3,, τ m ) vary, he sum of orders of he zeros of P (λ, e λτ1,, e λτm ) in he open righ half plane can change, and only a zero appears on or crosses he imaginary axis. For τ =, (7) becomes λ 3 + (p 1 + q 1 )λ 2 + (p 2 + q 2 )λ + q 3 =. (8) A se of necessary and sufficien condiions ha all roos of (8) have a negaive real par is given by he well-nown Rouh-Hurwiz crieria in he following form: (H2) (p 1 + q 1 )(p 2 + q 2 ) q 3 >, q 3 >. which leads o z 3 + r 1 z 2 + r 2 z + r 3 =, (11) r 1 = p 2 1 2p 2 + 2q 1 q 3 q 2 1, r 2 = p 2 2 q 2 2, r 3 = q 2 3. Denoe h(z) = z 3 + r 1 z 2 + r 2 z + r 3. (12) Since lim z + h(z) = + and r 3 <, we can conclude ha Eq. (11) has a leas one posiive roo. Wihou loss of generaliy, we assume ha (11) has hree posiive roos, defined by z 1, z 2, z 3, respecively. Then Eq. (1) has hree posiive roos By (9), we have ω 1 = z 1, ω 2 = z 2, ω 3 = z 3. cos ω τ = p 1ω 2 (q 3 q 1 ω 2 ) + (ω 3 p 2 ω)q 2 ω (q 3 q 1 ω 2 ) 2 + (q 2 ω) 2. Thus, if we denoe { τ = 1 [ p1 ω 2 (q 3 q 1 ω 2 ) + (ω 3 ] } p 2 ω)q 2 ω arccos ω (q 3 q 1 ω 2 ) 2 + (q 2 ω) 2 +2jπ, (13) = 1, 2, 3; j =, 1,, hen ±iω is a pair of purely imaginary roos of Eq. (7) wih τ. Define τ = τ () () = min {τ }, ω = ω. (14) {1,2,3} The above analysis leads o he following resul: Lemma 2 If (H1) and (H2) hold, hen all roos of (7) have a negaive real par when τ [, τ ) and (7) admis a pair of purely imaginary roos ±ω when τ = τ ( = 1, 2, 3; j =, 1, 2, ). In he sequel, we assume ha (H3) [(4p 2 + q 2 ) 2p 1 (p 1 + q 1 )] 2 < 12p 2 (p 2 + q 2 ). Le λ(τ) = α(τ) + iω(τ) be a roo of (7) near τ = τ, and ) =, and ω(τ ) = ω. Due o funcional differenial α(τ For ω >, iω is a roo of (7) if and only if Thus, Re ( ) dλ dτ λ=iω = iω 3 p 1 ω 2 +ip 2 ω+( q 1 ω 2 +iq 2 ω+q 3 )(cos ωτ i sin ωτ) =. (p 2 + q 2 3ω 2)ω [(q 3 q 1 ω 2) sin ω τ Separaing he real and imaginary pars, we ge { (q3 q 1 ω 2 ) cos ωτ + q 2 ω sin ωτ = p 1 ω 2, q 2 ω cos ωτ (q 3 q 1 ω 2 ) sin ωτ = ω 3 p 2 ω. (9) M 2 + N 2 + 2(p 1 + q 1 )ω 2[(q 3 q 1 ω 2) cos ω τ equaion heory, for every τ, = 1, 2, 3; j =, 1, 2,, here exiss ε > such ha λ(τ) is coninuously differeniable in τ for τ τ < ε. Subsiuing λ(τ) ino he lef hand of (7) and aing derivaive wih respec o τ, we have ( ) dλ = (3λ2 + 2p 1 λ + p 2 )e λτ dτ λ(q 1 λ 2 + q 2 λ + q 3 ) + q 1 λ + q 2 λ(q 1 λ 2 + q 2 λ + q 3 ) τ λ. q 2 ω cos ω τ ] + q 2 ω sin ω τ ] M 2 + N 2, (Advance online publicaion: 24 Augus 211)

3 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 M = [ω (q 3 q 1 ω 2 ) sin ω τ N = [ω (q 3 q 1 ω 2 ) cos ω τ Togeher wih (9), i follows ha Re ( dλ dτ q 2 ω 2 cos ω τ ]2 + q 2 ω 2 sin ω τ ) = τ=τ ]2. ω 2 {3ω4 [4p 2 + q 2 2p 1 (p 1 + q 1 )]ω 2 + p 2(p 2 + q 2 )} M 2 + N 2 By he assumpion (H3), so we have ( ) dλ signre = signre dτ τ=τ ( dλ dτ ) τ=τ >. According o above analysis and he resuls of Kuang [3] and Hale[4], we have Theorem 1 If (H1), (H2) and (H3) hold, hen he equilibrium E of sysem (3) is asympoically sable for τ [, τ ) and unsable for τ τ, sysem (3) undergoes a Hopf bifurcaion a he equilibrium E when τ = τ, = 1, 2, 3; j =, 1, 2,. Proof The proof of he sabiliy of he equilibrium E can be obained by Lemma 2. When τ = τ, = 1, 2, 3; j =, 1, 2,. (7) has a simple purely imaginary roos λ = ±ω i, and all roos λ j λ, λ saisfy λ j imω for any ineger m, since here is no oher purely imaginary roos excep for λ = ±iω. Furhermore, Re(λ (τ )) >, = 1, 2, 3; j =, 1, 2,. Due o he Hopf bifurcaion heorem [4], we complee he proof. III. DIRECTION AND STABILITY OF THE HOPF BIFURCATION In he previous secion, we obained condiions for Hopf bifurcaion o occur when τ = τ, = 1, 2, 3; j =, 1, 2,. In his secion, we shall derived he explici formulae deermining he direcion, sabiliy, and period of hese periodic soluions bifurcaing from he posiive equilibrium E (S, E, I ) a hese criical value of τ, by using echniques from normal form and cener manifold heory [1]. Throughou his secion, we always assume ha sysem (3) undergoes Hopf bifurcaion a he posiive equilibrium E (S, E, I ) for τ = τ, = 1, 2, 3; j =, 1, 2,., and hen ±iω are corresponding purely imaginary roos of he characerisic equaion a he posiive equilibrium E (S, E, I ). For convenience, le x() = x(τ), ȳ() = y(τ), z() = z(τ) and τ = τ + µ, τ is defined by (2.1) and µ R, drop he bar for he simplificaion of noaions, hen sysem () can be wrien as an FDE in C = C([, ]), R 3 ) as u() = L µ (u ) + F (µ, u ), (1) u() = (x(), y(), z()) T C and u (θ) = u(+θ) = (x( + θ), y( + θ), z( + θ)) T C, and L µ : C R, F : R C R are given by L µ φ = (τ + µ) (µ + α)i αs αi αs (γ + µ) φ 1() φ 2 () φ 3 () and +(τ + µ) (ɛ + µ) ɛ φ 1() φ 2 () φ 3 () f(µ, φ) = (τ + µ) αφ 1()φ 3 () αφ 1 ()φ 3 () (16), (17) respecively, φ(θ) = (φ 1 (θ), φ 2 (θ), φ 3 (θ)) T C. From he discussion in Secion 2, we now ha if µ =, hen sysem (1) undergoes a Hopf bifurcaion a he posiive equilibrium E (S, E, I ) and he associaed characerisic equaion of sysem (1) has a pair of simple imaginary roos ±ω τ. By he represenaion heorem, here is a marix funcion wih bounded variaion componens η(θ, µ), θ [, ] such ha L µ φ = dη(θ, µ)φ(θ), for φ C. (18) In fac, we can choose η(θ, µ) = (τ + µ) (µ + αi αi (ɛ + µ) δ(θ) ɛ (τ + µ) αs αs (γ + µ) δ is he Dirac dela funcion. For φ C([, ], R 3 ), define and A(µ)φ = { dφ(θ) dθ Rφ = δ(θ + 1), (19), θ <, dη(s, µ)φ(s), θ = (2) {, θ <, f(µ, φ), θ =. (21) Then (1) is equivalen o he absrac differenial equaion u = A(µ)u + R(µ)u, (22) u (θ) = u( + θ), θ [, ]. For ψ C([, 1], (R 3 ) ), define { A dψ(s) ψ(s) = ds, s (, 1], dηt (, )ψ( ), s =. For φ C([, ], R 3 ) and ψ C([, 1], (R 3 ) ), define he bilinear form < ψ, φ >= ψ()φ() θ ξ= ψ T (ξ θ)dη(θ)φ(ξ)dξ, η(θ) = η(θ, ), he A = A() and A are adjoin operaors. By he discussions in Secion 2, we now ha ±iω τ are eigenvalues of A(), and hey are also eigenvalues of A corresponding o iω τ and iω τ respecively. By direc compuaion, we can obain q(θ) = (1, a 1, a 2 ) T e iωτ θ, q (s) = D(1, a 1, a 2)e iωτ s, (Advance online publicaion: 24 Augus 211)

4 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 D = 1 B, a 1 = (iω + γ + µ)(iω + µ + αi ), αɛs iωτ e a 2 = iω + µ + αi αs, a 1 = iω + µ + αi αi, a 2 = αs ( iω + 2αI + µ) αi, ( iω + γ + µ) B = 1 + ā 1 a 1 + ā 2 a 2 + a iωτ 1ā 1 (ɛ + µ)e + ā 1 a iωτ 2ɛe. Furhermore, < q (s), q(θ) >= 1 and < q (s), q(θ) >=. Nex, we use he same noaions as hose in Hassard [1] and we firs compue he coordinaes o describe he cener manifold C a µ =. Le u be he soluion of Eq. (1) when µ =. Define z() =< q, u >, W (, θ) = u (θ) 2Re{z()q(θ)}. (23) on he cener manifold C, and we have W (z(), z(), θ) = W 2 z 2 W (, θ) = W (z(), z(), θ), (24) 2 + W 11z z + W 2 z 2 2 +, (2) and z and z are local coordinaes for cener manifold C in he direcion of q and q. Noing ha W is also real if u is real, we consider only real soluions. For soluions u C of (1), ż() = iω τ z + q (θ)f(, W (z, z, θ) + 2Re{zq(θ)} = iω τ z + q ()f. Tha is g(z, z) = g 2 z 2 ż() = iω τ z + g(z, z), 2 + g 11z z + g 2 z g 21 z 2 z 2 +. Hence, we have g(z, z) = q ()f (z, z) = f(, u ) = Dτ Dτ α(1 + ā 1)a 2 z α(1 + ā 1)ā 2 2 z 2 + [ 1 2 W (1) 2 ()ā W (3) 2 z 2 z + h.o.. Then we obain Dτ g 2 = 2 Dτ α(1 + ā 1)a 2, g 11 = 2 Dτ α(1 + ā 1)Re{a 2 }, g 2 = 2 Dτ α(1 + ā 1)ā 2 2, g 21 = 2 Dτ α(1 + ā 1) +W (1) 11 ()a 2 + W (3) 11 () ]. α(1 + ā 1)Re{a 2 }z z Dτ α(1 + ā 1) ] (1) () + W 11 ()a 2 + W (3) 11 () [ 1 2 W (1) 2 ()ā W (3) 2 () For unnown W (1) (3) 2 (), W 2 we sill need o compue hem. (), W (1) 11 (), W (3) 11 () in g 21, Form (22), (23), we have { W AW 2Re{ q = () fq(θ)}, θ <, AW 2Re{ q () fq(θ)} + f, θ = = AW + H(z, z, θ), (26) H(z, z, θ) = H 2 (θ) z2 2 +H 11(θ)z z+h 2 (θ) z2 +. (27) 2 Comparing he coefficiens, we obain (AW 2iτ ω )W 2 = H 2 (θ), (28) We now ha for θ [, ), AW 11 (θ) = H 11 (θ). (29) H(z, z, θ) = q ()f q(θ) q () f q(θ) = g(z, z)q(θ) ḡ(z, z) q(θ). (3) Comparing he coefficiens of (3) wih (27) gives ha H 2 (θ) = g 2 q(θ) ḡ 2 q(θ). (31) H 11 (θ) = g 11 q(θ) ḡ 11 q(θ). (32) From (3.14),(3.17) and he definiion of A, we ge Ẇ 2 (θ) = 2iω τ W 2(θ) + g 2 q(θ) + g 2 q(θ). (33) Noing ha q(θ) = q()e W 2 (θ) = ig 2 iωτ ω τ 2iωτ +E 1 e θ, we have iωτ q()e θ + iḡ 2 3ω τ q()e iωτ θ θ, (34) E 1 = (E (1) 1, E(2) 1, E(3) 1 ) R3 is a consan vecor. Similarly, from (29), (32) and he definiion of A, we have W 11 (θ) = ig 11 ω τ Ẇ 11 (θ) = g 11 q(θ) + g 11 q(θ), (3) iωτ q()e θ + iḡ 11 ω τ iωτ q()e θ +E 2. (36) E 2 = (E (1) 2, E(2) 2, E(3) 2 ) R3 is a consan vecor In wha follows, we shall see appropriae E 1,E 2 in (34), (36), respecively. I follows from he definiion of A and (31), (32) ha dη(θ)w 2 (θ) = 2iω τ W 2() H 2 () (37) and dη(θ)w 11 (θ) = H 11 (), (38) η(θ) = η(, θ). From (28), we have H 2 () = g 2 q() g 2 q() + 2τ H 11 () = g 11 q() g 11 () q()+2τ αa 2 αā 1a 2, (39) αre{a 2} αā 1Re{a 2 }. (4) (Advance online publicaion: 24 Augus 211)

5 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 Noing ha ( iω τ I ( iω τ I e iωτ e iωτ ) θ dη(θ) q() =, ) θ dη(θ) q() = and subsiuing (34) and (39) ino (37), we have (2iω τ I 2iωτ e θ dη(θ))e 1 = 2τ αa 2 αā 1a 2. Tha is l 1 αs αi l 2 E 1 = 2 αa 2 αā 1a 2, 2iωτ ɛe l 3 l 1 = 2iω + µ + αi 2iωτ, l 2 = 2iω + (ɛ + µ)e, l 3 = 2iω + γ + µ. I follows ha E (1) 1 = 11, E (2) 1 = 12, E (3) 1 = 13, (41) l 1 αs 1 = de αi l 2, 2iωτ ɛe l 3 αa 2 αs 11 = 2 de αa 1a 2 l 2, 2iωτ ɛe l 3 l αa αs 12 = 2 de, 1 αi 2 αa 1a 2 l 3 l 1 αa 2 13 = 2 de αi l 2 αa 1a 2. 2iωτ ɛe Similarly, subsiuing (3) and (4) ino (38), we have dη(θ))e 2 = 2τ αre{a 2} αā 1Re{a 2 }. Tha is µ + αi αs αi ɛ + µ αs ɛ γ + µ I follows ha E 2 = 2 αre{a 2 } αā 1Re{a 2 }. E (1) 2 = 21, E (2) 2 = 22, E (3) 2 = 23, (42) = de µ + αi αs αi (ɛ + µ) αs, ɛ (γ + µ) 21 = 2 de αre{a 2} αs αā 1Re{a 2 } ɛ + µ αs, ɛ γ + µ 22 = 2 de µ + αi αrea 2 αs αi αā 1Rea 2 αs, (γ + µ) 23 = 2 de µ + αi αrea 2 αi (ɛ + µ) αā 1Rea 2. ɛ From (34),(36),(41),(42), we can calculae g 21 and derive he following values: i c 1 () = 2ω τ µ 2 = Re{c 1()} Re{λ (τ )}, β 2 = 2Re(c 1 ()), (g 2 g 11 2 g 11 2 g 2 2 T 2 = Im{c 1()} + µ 2 Im{λ (τ )}. ω τ 3 ) + g 21 2, These formulaes give a descripion of he Hopf bifurcaion periodic soluions of (1) a τ = τ, ( = 1, 2, 3; j =, 2, 3, ) on he cener manifold. From he discussion above, we have he following resul: Theorem 2 The periodic soluion is supercriical (subcriical) if µ 2 > (µ 2 < ); he bifurcaing periodic soluions are orbially asympoically sable wih asympoical phase (unsable) if β 2 < (β 2 > ); he periodic of he bifurcaing periodic soluions increase (decrease) if T 2 > (T 2 < ). Remar 1 A τt -periodic soluion of (1) is a T -periodic soluion of (). IV. NUMERICAL EXAMPLES In his secion, we presen some numerical resuls of sysem (3) o verify he analyical predicions obained in he previous secion. From secion 3, we may deermine he direcion of a Hopf bifurcaion and he sabiliy of he bifurcaion periodic soluions. Le us consider he following sysem: Ṡ() = S 2IS, Ė() = 2IS.E( τ), = E( τ).4i, (43) which has a posiive equilibrium E (S, E, I ) 1, 1 ) and saisfies he condiions indicaed in Theorem 1. When τ =, he posiive equilibrium E = 1, 1 ) is asympoically sable. Tae j = for example, by some complicaed compuaion by means of Malab 7., we ge ω.4112, τ.67, λ (τ ) i. Thus we can calculae he following values: c 1 () i, µ , β , T Furhermore, i follows ha µ 2 > and β 2 <. Thus, he posiive equilibrium E = 1, 1 ) is sable when τ < τ as is illusraed by he compuer simulaions ( see Figs.1-7 ). When τ passes hrough he criical value τ, he posiive equilibrium E = 1, 1 ) loses is sabiliy and a Hopf bifurcaion occurs, i.e., a family of periodic soluions bifurcaions from he posiive equilibrium E = 1, 1 ). Since µ 2 > and β 2 <, he direcion of he Hopf bifurcaion is τ > τ, and hese bifurcaing periodic soluions from E = 1, 1 ) a τ are sable, which are depiced in Figs (Advance online publicaion: 24 Augus 211)

6 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3.42 Fig.1 6 Fig Fig. 1. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). Fig.. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). 6 Fig.2 6 Fig Fig. 2. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). Fig. 6. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). 6 Fig.3 Fig Fig. 3. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). Fig. 7. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). 6 Fig.4.42 Fig Fig. 4. Behavior and phase porrai of sysem (43) wih τ =. < τ.67. The posiive equilibrium E = 1, 1 ) is asympoically sable. The iniial value is (.4,,.14). Fig. 8. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14). (Advance online publicaion: 24 Augus 211)

7 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 6 Fig.9 6 Fig Fig. 9. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14). Fig. 13. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14). Fig Fig Fig. 1. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14). Fig. 14. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14). V. BIOLOGICAL EXPLANATIONS AND CONCLUSIONS Fig Fig. 11. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14) Fig Fig. 12. Behavior and phase porrai of sysem (43) wih τ =.8 > τ.67. Hopf bifurcaion occurs from he posiive equilibrium E = 1, 1 ). The iniial value is (.4,,.14). 1 Biological explanaions From he analysis in Secion 2, we now ha if he condiions (H1), (H2) and (H3) hold, hen he posiive equilibrium E (S, E, I ) of sysem (3) is asympoically sable when τ [, τ ), and unsable when τ > τ. This shows ha, in his case, he suscepible, exposed (laen), infecious hos populaions will end o sabilizaion, ha is, he suscepible hos populaions will end o S, he exposed (laen) hos populaions will end o E and he infecious hos populaions will end o I, and his fac is no influenced by he delay τ [, τ ). When τ crosses hrough he criical value τ, he posiive equilibrium E (S, E, I ) of sysem (3) loses sabiliy and a Hopf bifurcaion occurs. If he periodic soluion bifurcaing from he Hopf bifurcaion is sable, hen his shows ha he suscepible, exposed (laen), infecious hos populaions may coexis and eep in an oscillaory mode. From discussion in Secion 2, we now ha he posiive equilibrium E (S, E, I ) is always unsable when τ > τ. Therefore, if he above bifurcaing periodic soluion is unsable, hen i is a leas semi-sable (sable inside and unsable ouside) and hence he suscepible, exposed (laen), infecious hos populaions may eep in an oscillaory mode near he posiive equilibrium E (S, E, I ). 2 Conclusions In his paper, we have invesigaed local sabiliy of he posiive equilibrium E (S, E, I ) and local Hopf bifurcaion in a special SEIR epidemic model wih nonlinear incidence raes. we have showed ha if he condiions (H1), (H2) and (H3) hold, he posiive equilibrium E (S, E, I ) of sysem (3) is asympoically sable for all τ [, τ ) and unsable for τ > τ. We have also showed ha, if he (Advance online publicaion: 24 Augus 211)

8 IAENG Inernaional Journal of Applied Mahemaics, 41:3, IJAM_41_3_3 condiions (H1), (H2) and (H3) hold, as he delay τ increases, he equilibrium loses is sabiliy and a sequence of Hopf bifurcaions occur a he posiive equilibrium E (S, E, I ), i.e., a family of periodic orbis bifurcaes from he he posiive equilibrium E (S, E, I ). A las, he direcion of Hopf bifurcaion and he sabiliy of he bifurcaing periodic orbis are discussed by applying he normal form heory and he cener manifold heorem. A numerical example verifying our heoreical resuls is also correc. ACKNOWLEDGMENT The auhors would lie o han reviewers for heir valuable commens ha led o ruly significan improvemen of he manuscrip. REFERENCES [1] B. Hassard, D. Kazarino and Y. Wan, Theory and applicaions of Hopf bifurcaion, Cambridge, Cambridge Universiy Press,1981. [2] S. G. Ruan and J. J. Wei, On he zero of some ranscendenial funcions wih applicaions o sabiliy of delay differenial equaions wih wo delays, Dynamics of Coninuous, Discree and Impulsive Sysems, Series A: Mahemaical Analysis, vol. 1, no. 1, pp , Jul. 23. [3] Y. Kuang, Delay Differenial Equaions Wih Applicaions in Populaion Dynamics, Academic Press, INC, [4] J. Hale, Theory of Funcional Differenial Equaion, Springer-Verlag, [] C. J. Sun, Y. P. Lin and S. P. Tang, Global sabiliy for an special SEIR epidemic model wih nonlinear incidence raes, Chaos, Solions & Fracals, vol. 33, no. 1, pp , Jul. 27. [6] Y. Naaa and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environmen, Journal of Mahemaical Analysis and Applicaions, vol. 363, no. 1, pp , Mar. 21. [7] G. H. Li and Z. Jin, Global ssbiliy of an SEI epidemic model wih general conac rae, Chaos, Solions & Fracals, vol. 23, no. 3, pp , Jun. 2. [8] M. Y. Li and J. S. Muldowney, Global sabiliy for he SEIR model in epidemiology, Mahemaical Biosciences, vol. 12, no. 2, pp , Feb [9] B. K. Mishra and N. Jha, SEIQRS model for he ransmission of malicious objecs in compuer newor, Applied Mahemaical Modelling, vol. 34, no. 3, pp , Mar. 21. [1] J. Hou and Z. D. Teng, Coninuous and implusive vaccinaion of SEIR epidemic models wih sauraion incidence raes, Mahemaics and Compuers in Simulaion, vol. 79, no. 1, pp , Jun. 29. [11] X. Z. Meng, J. J. Jiao and L. S. Chen, Two profiless delays for an SEIRS epidemic disease model wih verical ransmission and pulse vaccinaion, Chaos, Solions & Fracals, vol. 4: no., pp , Jun. 29. [12] X. Wang, Y. D. Tao and X. Y. Song, Pusle vaccinaion on SEIR epidemic model wih nonlinear incidence rae, Applied Mahemaics and Compuaion, vol. 21, no. 2, pp , Apr. 29. [13] X. Z. Li and L. L. Zhou, Global sabiliy of an SEIR epidemic model wih verical ransmission and sauraing conac rae, Chaos, Solions & Fracals, vol. 4, no. 2, pp , Apr. 29. Changjin Xu is a lecurer in Guizhou Key Laboraory of Economics Sysem Simulaion, School of Mahemaics and Saisics, Guizhou College of Finance and Economics, Guiyang, China. He received his M.S. from Kunming Universiy of Science and Technology, Kunming, in 24 and Ph. D. from Cenral Souh Universiy, Changsha, China in 21. His research ineress focus on he sabiliy and bifurcaion heory of delayed differenial and equaion.. (Advance online publicaion: 24 Augus 211)