- Email: [email protected]
Global stability of a SIR epidemic model with nonlinear incidence rate and time delay
Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Global stability of a SIR epidemic model with nonlinear incidence rate and time delay Rui Xu a,b,∗ , Zhien Ma b a
Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, PR China
b
Department of Applied Mathematics, Xi’an Jiaotong University, Xi’an, 710049, PR China
article
info
Article history: Received 25 January 2008 Accepted 8 October 2008 Keywords: SIR epidemic model Nonlinear incidence Time delay Stability
a b s t r a c t In this paper, a SIR epidemic model with nonlinear incidence rate and time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease free equilibrium is discussed. It is proved that if the basic reproductive number R0 > 1, the system is permanent. By comparison arguments, it is shown that if R0 < 1, the disease free equilibrium is globally asymptotically stable. If R0 > 1, by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Let S (t ) denote the number of members of a population susceptible to the disease, I (t ) the number of infective members and R(t ) the number of members who have been removed from the possibility of infection through full immunity. In [7], Cooke formulated a SIR model with time delay effect by assuming that the force of infection at time t is given by β S (t )I (t −τ ), where β is the average number of contacts per infective per day and τ > 0 is a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible human. Cooke considered the following model S˙ (t ) = B − µ1 S (t ) − β S (t )I (t − τ ), ˙I (t ) = β S (t )I (t − τ ) − (µ2 + γ )I (t ), R˙ (t ) = γ I (t ) − µ3 R(t ),
(1.1)
where parameters µ1 , µ2 , µ3 are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that µ1 ≤ min{µ2 , µ3 }. The parameters B and γ are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. Much attention has been paid to the analysis of the stability of the disease free equilibrium and the endemic equilibrium of system (1.1) (see, for example, [1–5,7,10,12–15]). In [3], Beretta et al. considered the global stability of the disease free equilibrium and the endemic equilibrium of system (1.1). They showed that the disease free equilibrium is globally stable for any delay τ while the endemic equilibrium is not feasible. By constructing a suitable Lyapunov functional, sufficient conditions were derived to guarantee that if the endemic equilibrium is feasible, it is also globally stable for the delay being sufficiently
∗ Corresponding address: Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang 050003, Hebei Province, PR China. E-mail address: [email protected] (R. Xu). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.013
3176
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
small. In [12], Ma et al. derived an explicit expression of the lower bound of the component I (t ) of solution of system (1.1) which was proposed as an open problem. They therefore gave an estimation of the length of the time delay ensuring the global asymptotic stability of the endemic equilibrium. Incidence rate plays an important role in the modelling of epidemic dynamics. It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate. In many epidemic models, the bilinear incidence rate β SI and the standard incidence rate β SI /N are frequently used. The bilinear incidence rate is based on the law of mass action. This contact law is more appropriate for communicable diseases such as influenza etc., but not for sexually transmitted diseases. It has been pointed out that for standard incidence rate, it may be a good approximation if the number of available partners is large enough and everybody could not make more contacts than is practically feasible. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [6] introduced a saturated incidence rate g (I )S into epidemic models, where g (I ) tends to a saturation level when I gets large, i.e., g (I ) =
βI 1 + αI
,
where β I measures the infection force of the disease and 1/(1 + α I ) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. This incidence rate seems more reasonable than the bilinear incidence rate β IS, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters. Motivated by the work of Beretta et al. [3], Capasso and Serio [6] and Ma et al. [12], in this paper, we are concerned with the effect of time delay and nonlinear incidence rate on the dynamics of a SIR epidemic model. To this end, we consider the following delay differential equations S˙ (t ) = B − µ1 S (t ) −
β S (t )I (t − τ ) 1 + α I (t − τ )
,
˙I (t ) = β S (t )I (t − τ ) − (µ2 + γ )I (t ), 1 + α I (t − τ ) R˙ (t ) = γ I (t ) − µ3 R(t ),
(1.2)
where µ1 ≤ min{µ2 , µ3 }. The initial conditions for system (1.2) take the form S (θ ) = φ1 (θ ),
φi (θ ) ≥ 0,
I (θ ) = φ2 (θ ),
R(θ ) = φ3 (θ),
θ ∈ [−τ , 0], φi (0) > 0,
i = 1, 2, 3,
(1.3)
where (φ1 (θ ), φ2 (θ ), φ3 (θ )) ∈ C ([−τ , 0], R3+0 ), the Banach space of continuous functions mapping the interval [−τ , 0] into R3+0 , where R3+0 = {(x1 , x2 , x3 ) : xi ≥ 0, i = 1, 2, 3}. It is well known by the fundamental theory of functional differential equations [9], system (1.2) has a unique solution (S (t ), I (t ), R(t )) satisfying the initial conditions (1.3). It is easy to show that all solutions of system (1.2) with initial conditions (1.3) are defined on [0, +∞) and remain positive for all t ≥ 0. The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of a disease free equilibrium and an endemic equilibrium of system (1.2) is discussed. In Section 3, it is proved that system (1.2) is permanent if the basic reproductive number R0 > 1. In Section 4, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Alternatively, by constructing a suitable Lyapunov functional, it is shown that if R0 > 1, the endemic equilibrium of system Eq. (1.2) is globally stable provided that the parameter α and the time delay τ are sufficiently small. By comparison arguments, it is proved that if R0 < 1, the disease free equilibrium is globally stable. A brief discussion is given in Section 5 to conclude this work. 2. Local stability In this section, we discuss the local stability of an endemic equilibrium and a disease-free equilibrium of system (1.2) by analyzing the corresponding characteristic equations, respectively. System (1.2) always has a disease-free equilibrium E1 (B/µ1 , 0, 0). Further, if Bβ > µ1 (µ2 + γ ), then system (1.2) has a unique endemic equilibrium E ∗ (S ∗ , I ∗ , R∗ ), where Bα + µ2 + γ
, β + αµ1 Bβ − µ1 (µ2 + γ ) I∗ = , (µ2 + γ )(β + αµ1 ) γ [Bβ − µ1 (µ2 + γ )] R∗ = . µ3 (µ2 + γ )(β + αµ1 ) S∗ =
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3177
Let R0 =
Bβ
µ1 (µ2 + γ )
.
R0 is called the basic reproductive number of system (1.2). It is easy to show that if R0 > 1, the endemic equilibrium E ∗ exists; if R0 < 1, E ∗ is not feasible. The characteristic equation of system (1.2) at the endemic equilibrium E ∗ is of the form
(λ + µ3 )[λ2 + p1 λ + p0 + (q1 λ + q0 )e−λτ ] = 0,
(2.1)
where
βI∗ , p0 = (µ2 + γ ) µ1 + 1 + αI ∗ βI∗ p 1 = µ1 + µ2 + γ + , 1 + αI ∗ βµ1 S ∗ βS∗ q0 = − , q1 = − . ∗ 2 (1 + α I ) (1 + α I ∗ )2
Clearly, λ = −µ3 is always a root of Eq. (2.1). All other roots of (2.1) are determined by the following equation
λ2 + p1 λ + p0 + (q1 λ + q0 )e−λτ = 0.
(2.2)
When τ = 0, Eq. (2.2) becomes
λ2 + (p1 + q1 )λ + p0 + q0 = 0.
(2.3)
It is readily seen that if R0 > 1, then
βI∗ µ1 + > 0, p0 + q0 = (µ2 + γ ) µ1 − 1 + αI ∗ 1 + αI ∗ βI∗ µ2 + γ p1 + q1 = µ1 + + µ2 + γ − > 0. 1 + αI ∗ 1 + αI ∗ Hence, if R0 > 1, the endemic equilibrium E ∗ of system (1.2) is locally stable when τ = 0. If iω(ω > 0) is a solution of (2.1), separating real and imaginary parts, we derive that
p1 ω = q0 sin ωτ − q1 ω cos ωτ ,
(2.4)
ω2 − p0 = q0 cos ωτ + q1 ω sin ωτ . Squaring and adding the two equations of (2.4), it follows that
ω4 + (p21 − 2p0 − q21 )ω2 + p20 − q20 = 0.
(2.5)
Letting z = ω2 , then Eq. (2.5) becomes z 2 + (p21 − 2p0 − q21 )z + p20 − q20 = 0.
(2.6)
It is easy to show that
2 βI∗ (µ2 + γ )2 µ1 + + (µ2 + γ )2 − > 0, ∗ 1 + αI (1 + α I ∗ )2 βI∗ βµ1 S ∗ µ1 βI∗ + µ − + > 0. p20 − q20 = (µ2 + γ ) (µ2 + γ ) µ1 + 1 1 + αI ∗ (1 + α I ∗ )2 1 + αI ∗ 1 + αI ∗ p21 − 2p0 − q21 =
Hence, if R0 > 1, Eq. (2.6) has no positive roots. Accordingly, if R0 > 1, the endemic equilibrium E ∗ of system Eq. (1.2) is locally stable for all τ ≥ 0. The characteristic equation of system (1.2) at the disease-free equilibrium E1 (B/µ1 , 0, 0) takes the form
Bβ 2 −λτ (λ + µ3 ) λ + (µ1 + µ2 + γ )λ + µ1 (µ2 + γ ) − (λ + µ1 )e = 0. µ1
(2.7)
Clearly, Eq. (2.7) always has a negative real root λ = −µ3 . All other roots are given by the roots of equation
λ2 + (µ1 + µ2 + γ )λ + µ1 (µ2 + γ ) −
Bβ
µ1
(λ + µ1 )e−λτ = 0.
(2.8)
3178
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Let f1 (λ) = λ2 + (µ1 + µ2 + γ )λ + µ1 (µ2 + γ ) −
Bβ
µ1
(λ + µ1 )e−λτ .
If R0 > 1, then it is readily seen that for λ real, f1 (0) = µ1 (µ2 + γ ) − Bβ < 0,
lim f1 (λ) = +∞.
t →+∞
Hence, f1 (λ) = 0 has a positive real root. Therefore, if R0 > 1, the disease-free equilibrium E1 (B/µ1 , 0, 0) is unstable. If R0 < 1, it is easy to show that the disease-free equilibrium E1 (B/µ1 , 0, 0) of system (1.2) is stable when τ = 0. Using a similar argument one can show that if R0 < 1, the equilibrium E1 of system (1.2) is locally asymptotically stable. Theorem 2.1. For system (1.2), if the basic reproductive number R0 < 1, the disease-free equilibrium E1 (B/µ1 , 0, 0) is locally asymptotically stable while the endemic equilibrium is not feasible. If R0 > 1, the endemic equilibrium E ∗ exists and is locally asymptotically stable while the disease-free equilibrium E1 is unstable. 3. Permanence In this section, we are concerned with the permanence of system (1.2). Lemma 3.1. For any solution (S (t ), I (t ), R(t )) of system (1.2), we have that lim sup(S (t ) + I (t ) + R(t )) ≤ B/µ1 . t →+∞
Proof. Let N (t ) = S (t ) + I (t ) + R(t ). Noting that µ1 ≤ min{µ2 , µ3 }, it follows from system (1.2) that N˙ (t ) = B − µ1 S (t ) − µ2 I (t ) − µ3 R(t ) ≤ B − µ1 N (t ). A standard comparison argument shows that lim supt →+∞ N (t ) ≤ B/µ1 . This completes the proof.
Lemma 3.2. If R0 > 1, then for any solution (S (t ), I (t ), R(t )) of system (1.2) with initial conditions (1.3), we have that lim inf S (t ) ≥ t →+∞
B(µ1 + Bα) B(αµ1 + β) + µ21
:= v1 ,
lim inf I (t ) ≥ I ∗ e−(µ2 +γ )τ := v2 ,
(3.1)
t →+∞
lim inf R(t ) ≥ γ v2 /µ3 := v3 . t →+∞
Proof. Let (S (t ), I (t ), R(t )) be any solution of system (1.2) with initial conditions (1.3). By Lemma 3.1, it follows that lim sup I (t ) ≤ B/µ1 . t →+∞
Hence, for ε > 0 sufficiently small, there is a T1 > 0 such that if t > T1 , I (t ) < B/µ1 + ε . We therefore derive from the first equation of system (1.2) that, for t > T1 + τ , S˙ (t ) ≥ B −
µ1 +
β(B/µ1 + ε) S (t ), 1 + α(B/µ1 + ε)
which yields lim inf S (t ) ≥ t →+∞
B[1 + α(B/µ1 + ε)] . µ1 + (αµ1 + β)(B/µ1 + ε)
Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that lim inf S (t ) ≥ t →+∞
B(µ1 + Bα) B(αµ1 + β) + µ21
:= v1 .
We now show that lim inft →+∞ I (t ) ≥ v2 . For t ≥ 0, define a differentiable function V (t ) = I (t ) + β S ∗
Z
t t −τ
I ( u) 1 + α I (u)
du.
(3.2)
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3179
Calculating the derivative of V (t ) along solutions of system (1.2) we derive that V˙ (t ) =
β I (t − τ ) (S (t ) − S ∗ ) + 1 + α I (t − τ )
βS∗ − µ2 − γ 1 + α I (t )
I (t ).
(3.3)
β qI ∗ For any 0 < q < 1, we have that S ∗ < B/[µ1 + 1+α qI ∗ ] := k. There is a constant ρ ≥ 1 sufficiently large such that S ∗ < k(1 − e−kρτ ) := S ∆ . We now claim that it is impossible that I (t ) ≤ qI ∗ for all t ≥ ρτ . Otherwise, if I (t ) ≤ qI ∗ for all t ≥ ρτ , we derive from the first equation of system (1.2) that, for t ≥ ρτ + τ ,
S˙ (t ) ≥ B −
µ1 +
β qI ∗ 1 + α qI ∗
S (t ).
(3.4)
It then follows from (3.4) that, for t > ρτ + τ ,
S (t ) ≥ e−k(t −ρτ −τ ) S (ρτ + τ ) + B
>
B k
Z
t
ρτ +τ
ek(θ−ρτ −τ ) dθ
(1 − e−k(t −ρτ −τ ) ).
(3.5)
Hence, for t > 2ρτ + τ , we derive from (3.5) that S (t ) >
B k
(1 − e−kρτ ) = S ∆ > S ∗ .
(3.6)
Noting that I (t ) ≤ qI ∗ < I ∗ , it follows from (3.3) and (3.6) that, for t > 2ρτ + τ ,
β I (t − τ ) (S (t ) − S ∗ ) + 1 + α I (t − τ ) β I (t − τ ) (S (t ) − S ∗ ) = 1 + α I (t − τ ) β I (t − τ ) (S ∆ − S ∗ ). > 1 + α I (t − τ )
V˙ (t ) ≥
βS∗ − µ2 − γ 1 + αI ∗
I (t )
(3.7)
Setting i=
min I (θ + 2ρτ + 2τ ),
θ∈[−τ ,0]
we will claim that I (t ) ≥ i for all t ≥ 2ρτ + τ . Otherwise, if there is a T ≥ 0 such that I (t ) ≥ i for 2ρτ + τ ≤ t ≤ 2ρτ + 2τ + T , I (2ρτ + 2τ + T ) = i and ˙I (2ρτ + 2τ + T ) ≤ 0, it follows from (3.6) and the second equation of system (1.2) that, for t1 = 2ρτ + 2τ + T ,
˙I (t1 ) = β S (t1 )I (t1 − τ ) − (µ2 + γ )I (t1 ) 1 + α I (t1 − τ ) β S (t1 ) ≥ − (µ2 + γ ) i 1 + α I (t1 − τ ) βS∆ > − (µ + γ ) i 2 1 + αI ∗ > 0.
(3.8)
This is a contradiction. Hence, I (t ) ≥ i for all t ≥ 2ρτ + τ . Accordingly, for t ≥ 2ρτ + 2τ , it follows from (3.7) that V˙ (t ) >
βi 1 + αi
(S ∆ − S ∗ ),
which yields V (t ) → +∞ as t → +∞. It follows from (3.2) that there is a T2 > 0 such that if t > T2 , V (t ) ≤
B
µ1
+
Bβ S ∗
µ1 + Bα
+ 1.
A contradiction occurs. Hence, the claim is proved.
3180
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
By the claim, we are left to consider two possibilities. First, I (t ) ≥ qI ∗ for all t sufficiently large. Second, I (t ) oscillates about qI ∗ for all t sufficiently large. We now show that I (t ) ≥ qv2 for all t sufficiently large. The conclusion is obvious for the first case. For the second case, let t1 < t2 be sufficiently large such that I (t1 ) = I (t2 ) = qI ∗ ,
I (t ) < qI ∗ ,
t1 < t < t2 .
If t2 − t1 ≤ τ , it follows from the second equation of system (1.2) that
˙I (t ) > −(µ2 + γ )I (t ), which yields, for t1 < t < t2 , I (t ) > I (t1 )e−(µ2 +γ )(t −t1 ) > qv2 . If t2 − t1 > τ , we obtain from the second equation of system (1.2) that I (t ) ≥ qv2 for t ∈ [t1 , t1 + τ ]. We now claim that I (t ) ≥ qv2 for all t ∈ [t1 + τ , t2 ]. Otherwise, there is a T ∗ > 0 such that I (t ) ≥ qv2 for t1 ≤ t ≤ t1 + τ + T ∗ , I (t1 + τ + T ∗ ) = qv2 and ˙I (t1 + τ + T ∗ ) ≤ 0. On the other hand, it follows from the second equation of system (1.2) that, for t0 = t1 + τ + T ∗ ,
˙I (t0 ) = β S (t0 )I (t0 − τ ) − (µ2 + γ )I (t0 ) 1 + α I (t0 − τ ) β S ∆ qv2 − (µ2 + γ )qv2 ≥ 1 + α qv2 βS∆ + γ ) − (µ ≥ qv2 2 1 + αI ∗ > 0,
(3.9)
a contradiction. Hence, I (t ) ≥ qv2 for t ∈ [t1 , t2 ]. Since the interval [t1 , t2 ] is chosen arbitrarily, we conclude that I (t ) ≥ qv2 for all t sufficiently large for the second case. Since q(0 < q < 1) is chosen arbitrarily, we have that lim inft →+∞ I (t ) ≥ v2 . The proof is complete. From Lemmas 3.1 and 3.2, we therefore have the following result. Theorem 3.1. If R0 > 1, then system (1.2) is permanent. 4. Global stability In this section, we discuss the global stability of the endemic equilibrium E ∗ and the disease-free equilibrium E1 of system (1.2), respectively. We first introduce a result which is useful in discussing the global stability of the equilibria E1 and E ∗ . We now consider the following equation with time delay u˙ (t ) =
au(t − τ ) 1 + α u(t − τ )
− cu(t ),
u(θ ) = φ(θ ) ≥ 0,
θ ∈ [−τ , 0), φ(0) > 0,
(4.1)
where a, c and α are positive constants, τ ≥ 0. Eq. (4.1) always has a trivial equilibrium u = 0. If a > c, then Eq. (4.1) has a unique positive equilibrium u∗ = (a − c )/(α c ). For Eq. (4.1), we have the following result. Lemma 4.1. If a > c, then the positive equilibrium u∗ = (a − c )/(α c ) of (4.1) is globally asymptotically stable; if a < c, then the trivial equilibrium (0, 0) of (4.1) is globally asymptotically stable. Proof. Let f (u) = au/(1 + α u), g (u) = u. Clearly, f (u) and g (u) are strictly increasing. By Theorem 9.1 in Kuang [11], we see that if a > c, the positive equilibrium u∗ of system (4.1) is globally asymptotically stable. If a < c, we derive from (4.1) that u˙ (t ) ≤ au(t − τ ) − cu(t ). Consider the following auxiliary equation
v˙ (t ) = av(t − τ ) − c v(t ). Define V (t ) =
1 2
1
v 2 (t ) + a 2
Z
t
v 2 (s)ds. t −τ
(4.2)
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3181
Calculating the derivative of V (t ) along solution of Eq. (4.2), it follows that d dt
V (t ) = av(t )v(t − τ ) − c v 2 (t ) +
1 2
av 2 ( t ) −
1 2
av 2 (t − τ )
1 1 1 a(v 2 (t ) + v 2 (t − τ )) − c v 2 (t ) + av 2 (t ) − av 2 (t − τ ) 2 2 2 = (a − c )v 2 (t ).
≤
Hence, if a < c, limt →+∞ v(t ) = 0. By comparison it follows that, if a < c, limt →+∞ u(t ) = 0. This completes the proof. Theorem 4.1. If R0 > 1, Bα > µ2 + γ , then the endemic equilibrium E ∗ (S ∗ , I ∗ , R∗ ) of system (1.2) is globally asymptotically stable. Proof. Let (S (t ), I (t ), R(t )) be any positive solution of system (1.2) with initial conditions (1.3). Let U1 = lim sup S (t ), t →+∞
V1 = lim inf S (t ), t →+∞
U2 = lim sup I (t ),
U3 = lim sup R(t ),
t →+∞
V2 = lim inf I (t ), t →+∞
t →+∞
V3 = lim inf R(t ). t →+∞
In the following we claim that U1 = V1 = S , U2 = V2 = I ∗ , U3 = V3 = R∗ . It follows from the first equation of system (1.2) that ∗
S˙ (t ) ≤ B − µ1 S (t ). By comparison we derive that lim sup S (t ) ≤ B/µ1 := M1S . t →+∞
Hence, for ε > 0 sufficiently small there is a T1 > 0 such that if t > T1 , S (t ) ≤ M1S + ε. We obtain from the second equation of system (1.2) that for t > T1 + τ S ˙I (t ) ≤ β(M1 + ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
Consider the following auxiliary equation u˙ (t ) =
β(M1S + ε)u(t − τ ) − (µ2 + γ )u(t ). 1 + α u(t − τ )
Noting that R0 > 1, by Lemma 4.1 it follows that lim u(t ) =
t →+∞
β(M1S + ε) − (µ2 + γ ) . α(µ2 + γ )
By comparison we derive that U2 = lim sup I (t ) ≤ t →+∞
β(M1S + ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U2 ≤ M1I , where M1I =
β M1S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T2 > T1 + τ such that if t > T2 , I (t ) ≤ M1I + ε. It follows from the third equation of system (1.2) that for t > T2 , R˙ (t ) ≤ γ (M1I + ε) − µ3 R(t ). A standard comparison argument shows that U3 = lim sup R(t ) ≤ γ (M1I + ε)/µ3 . t →+∞
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U3 ≤ M1R , where M1R = γ M1I /µ3 .
3182
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
We derive from the first equation of system (1.2) that for t > T2 + τ S˙ (t ) ≥ B − µ1 S (t ) −
β(M1I + ε) S (t ). 1 + α(M1I + ε)
By comparison it follows that V1 = lim inf S (t ) ≥ t →+∞
B[1 + α(M1I + ε)]
µ1 + (β + αµ1 )(M1I + ε)
.
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V1 ≥ N1S , where N1S =
B(1 + α M1I )
µ1 + (β + αµ1 )M1I
.
Hence, for ε > 0 sufficiently small there is a T3 > T2 + τ such that if t > T3 , S (t ) ≥ N1S − ε. We derive from the second equation of system (1.2) that for t > T3 + τ S ˙I (t ) ≥ β(N1 − ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
Consider the following auxiliary equation
v˙ (t ) =
β(N1S − ε)v(t − τ ) − (µ2 + γ )v(t ). 1 + αv(t − τ )
Provided that β(N1S − ε) > µ2 + γ , by Lemma 4.1 we derive that lim v(t ) =
t →+∞
β(N1S − ε) − (µ2 + γ ) . α(µ2 + γ )
By comparison it follows that V2 = lim inf I (t ) ≥ t →+∞
β(N1S − ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V2 ≥ N1I , where N1I =
β N1S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T4 > T3 + τ such that if t > T4 , I (t ) ≥ N1I − ε. We derive from the third equation of system (1.2) that, for t > T4 , R˙ (t ) ≥ γ (N1I − ε) − µ3 R(t ). By comparison it follows that V3 = lim inf R(t ) ≥ t →+∞
γ (N1I − ε) . µ3
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V3 ≥ N1R , where N1R = γ N1I /µ3 . Again, it follows from the first equation of system (1.2) that, for t > T4 , S˙ (t ) ≤ B − µ1 S (t ) −
β(N1I − ε) S (t ). 1 + α(N1I − ε)
By comparison we derive that U1 = lim sup S (t ) ≤ t →+∞
B[1 + α(N1I − ε)]
µ1 + (β + αµ1 )(N1I − ε)
.
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U1 ≤ M2S , where M2S =
B(1 + α N1I )
µ1 + (β + αµ1 )N1I
.
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Hence, for ε > 0 sufficiently small there is a T5 > T4 such that if t > T5 , S (t ) ≤ M2S + ε. It follows from the second equation of system (1.2) that, for t > T5 + τ , S ˙I (t ) ≤ β(M2 + ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
By Lemma 4.1 and a comparison argument we derive that U2 = lim sup I (t ) ≤ t →+∞
β(M2S + ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U2 ≤ M2I , where M2I =
β M2S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T6 > T5 + τ such that if t > T6 , I (t ) ≤ M2I + ε. It follows from the third equation of system (1.2) that, for t > T6 , R˙ (t ) ≤ γ (M2I + ε) − µ3 R(t ). By comparison we obtain that U3 = lim sup R(t ) ≤ γ (M2I + ε)/µ3 . t →+∞
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U3 ≤ M1R , where M2R = γ M2I /µ3 . We derive from the first equation of system (1.2) that, for t > T6 + τ , S˙ (t ) ≥ B − µ1 S (t ) −
β(M2I + ε) S (t ). 1 + α(M2I + ε)
By comparison it follows that V1 = lim inf S (t ) ≥ t →+∞
B[1 + α(M2I + ε)]
µ1 + (β + αµ1 )(M2I + ε)
.
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V1 ≥ N2S , where N2S =
B(1 + α M2I )
µ1 + (β + αµ1 )M2I
.
Hence, for ε > 0 sufficiently small there is a T7 > T6 + τ such that if t > T7 , S (t ) ≥ N2S − ε. It follows from the second equation of system (1.2) that, for t > T7 + τ , S ˙I (t ) ≥ β(N2 − ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
By Lemma 4.1 and a comparison argument we derive that V2 = lim inf I (t ) ≥ t →+∞
β(N2S − ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V2 ≥ N2I , where N2I =
β N2S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T8 > T7 + τ such that if t > T8 , I (t ) ≥ N2I − ε. It follows from the third equation of system (1.2) that, for t > T8 , R˙ (t ) ≥ γ (N2I − ε) − µ3 R(t ). By comparison we derive that V3 = lim inf R(t ) ≥ t →+∞
γ (N2I − ε) . µ3
3183
3184
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V3 ≥ N2R , where N2R = γ N2I /µ3 . Continuing this process, we obtain six sequences MnS , MnI , MnR , NnS , NnI , NnR (n = 1, 2, . . .) such that for n ≥ 2, MnS =
B(1 + α NnI −1 )
µ1 + (β + αµ1 )NnI −1
,
β MnS − (µ2 + γ ) , α(µ2 + γ ) B(1 + α MnI ) NnS = , µ1 + (β + αµ1 )MnI MnI =
(4.3)
β NnS − (µ2 + γ ) , α(µ2 + γ ) γ MnI γ NnI MnR = , NnR = . µ3 µ3 NnI =
It is readily seen that NnS ≤ V1 ≤ U1 ≤ MnS ,
NnI ≤ V2 ≤ U2 ≤ MnI ,
NnR ≤ V3 ≤ U3 ≤ MnR .
(4.4)
.
(4.5)
We derive from (4.3) that MnS+1 =
B2 α 2 MnS
(β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2
Noting that MnS ≥ S ∗ and Bα > µ2 + γ , it follows from (4.5) that MnS+1 − MnS =
=
B2 α 2 MnS
(β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2
− MnS
(Bα − µ2 − γ )MnS [Bα + µ2 + γ − (β + αµ1 )MnS ] (β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2
(Bα − µ2 − γ )MnS [Bα + µ2 + γ − (β + αµ1 )S ∗ ] (β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2 = 0. ≤
Thus, the sequence MnS is monotonically non-increasing. Hence, limn→+∞ MnS exists. Taking n → +∞, it follows from (4.5) that lim
n→+∞
MnS+1
B2 α 2 lim MnS
=
n→+∞
(β + αµ1 )(Bα − µ2 − γ ) lim MnS + (µ2 + γ )2
.
(4.6)
n→+∞
Noting that lim MnS+1 = lim MnS ,
n→+∞
n→+∞
we therefore derive from (4.6) that lim MnS =
n→+∞
Bα + µ2 + γ
β + αµ1
= S∗.
(4.7)
It follows from (4.7) and the second equation of (4.3) that lim
n→+∞
MnI
β lim MnS − (µ2 + γ ) =
n→+∞
α(µ2 + γ )
=
Bβ − µ1 (µ2 + γ )
(µ2 + γ )(β + αµ1 )
= I ∗.
Similarly, one can derive from (4.3) and (4.7) that lim NnS = S ∗ ,
n→+∞
lim NnI = I ∗ ,
n→+∞
lim MnR = R∗ ,
n→+∞
lim NnR = R∗ .
n→+∞
(4.8)
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3185
It follows from Eqs. (4.4), (4.7) and (4.8) that U1 = V1 = S ∗ ,
U2 = V2 = I ∗ ,
U3 = V3 = R∗ .
We therefore have that lim S (t ) = S ∗ ,
lim I (t ) = I ∗ ,
t →+∞
lim R(t ) = R∗ .
t →+∞
t →+∞
Noting that if R0 > 1, the endemic equilibrium E ∗ is locally asymptotically stable, we therefore conclude that E ∗ is globally asymptotically stable. The proof is complete. We note that Theorem 4.1 is not valid when α is sufficiently small. Alternatively, we now give a different result on the global stability of the endemic equilibrium E ∗ of system (1.2) by means of Lyapunov functional technique used by Beretta et al. in [3]. For ε > 0 sufficiently small, we denote M (ε) = B/µ1 + ε, v2 = I ∗ e−(µ2 +γ )τ , m(ε) = v2 − ε > 0, A1 (ε) =
µ1 +
βI∗ 1 + αI ∗
1 M (ε)
,
β 2I ∗ (2I ∗ + 2S ∗ + M (ε)), 2m(1 + α I ∗ )2 β2 (4.9) B2 (ε) = [2S ∗ (I ∗ + S ∗ + M (ε)) + M (ε)I ∗ ], 2m(ε)(1 + α I ∗ )2 αβ αβ a0 (ε) = (µ2 + γ ) A1 (ε) − A1 (ε)(µ2 + γ ) − µ1 , m(ε) m(ε) αβ a1 (ε) = [B1 (ε)(µ2 + γ ) + B2 (ε)µ1 ] 2A1 (ε)(µ2 + γ ) − (µ1 + µ2 + γ ) + A1 (ε)B2 (ε)(µ2 + γ − µ1 )2 , m(ε) B1 (ε) =
a2 (ε) = [B1 (ε)(µ2 + γ ) + B2 (ε)µ1 ]2 + B1 (ε)B2 (ε)(µ2 + γ − µ1 )2 . Theorem 4.2. If R0 > 1, the endemic equilibrium E ∗ (S ∗ , I ∗ , R∗ ) of system (1.2) is globally asymptotically stable provided that A1 (0) > αβ/v2 and τ < τ0 , where
τ0 =
a1 (0) −
p
a1 (0)2 − 4a0 (0)a2 (0) 2a2 (0)
.
Proof. Let (S (t ), I (t ), R(t )) be any positive solution of system (1.2) with initial conditions (1.3). On substituting x(t ) = S (t ) − S ∗ , y(t ) = I (t ) − I ∗ into system (1.2), it follows that
βI∗ β S (t ) x(t ) − y(t − τ ), 1 + αI ∗ (1 + α I (t − τ ))(1 + α I ∗ ) βI∗ βS∗ β S (t ) y˙ (t ) = x(t ) − y(t ) + y(t − τ ). ∗ ∗ 1 + αI 1 + αI (1 + α I (t − τ ))(1 + α I ∗ )
x˙ (t ) = −µ1 x(t ) −
(4.10)
Define a Lyapunov function
x y V1 (x, y) = x − S ∗ ln 1 + ∗ + y − I ∗ ln 1 + ∗ . S I Calculating the derivative of V1 along solutions of system (4.10), we derive that
βI∗ αβ S ∗ S (t )I (t − τ ) 2 = I ( t ) x ( t ) − y2 (t ) dt S (t )I (t ) 1 + αI ∗ (1 + α I (t − τ ))(1 + α I ∗ ) αβ I ∗ S (t )I (t − τ ) β S (t ) ∗ ∗ + x ( t ) y ( t ) + ( S y ( t ) − I x ( t ))( y ( t − τ ) − y ( t )) (1 + α I (t − τ ))(1 + α I ∗ ) (1 + α I (t − τ ))(1 + α I ∗ ) βI∗ αβ I ∗ I (t − τ ) 1 2 ≤ − µ1 + x (t ) + x(t )y(t ) ∗ 1 + αI S (t ) I (t )(1 + α I (t − τ ))(1 + α I ∗ )
dV1
1
+
− µ1 +
β I (t )(1 + α I ∗ )
|(S ∗ y(t ) − I ∗ x(t ))(y(t − τ ) − y(t ))|.
(4.11)
3186
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
For ε > 0 sufficiently small, by Lemmas 3.1 and 3.2, there is a T > 0 such that if t > T , S (t ) ≤ M (ε),
I (t ) ≥ m(ε) > 0.
(4.12)
It follows from (4.11) and (4.12) that, for t > T + τ ,
βI∗ ≤ − µ1 + dt 1 + αI ∗
dV1
+
β m(ε)(1 + α I ∗ )
1 M (ε)
x2 (t ) +
αβ m(ε)
|x(t )y(t )|
|(S ∗ y(t ) − I ∗ x(t ))(y(t − τ ) − y(t ))|.
(4.13)
Note that, for t > T + 2τ , y(t − τ ) − y(t ) = −
β 1 + αI ∗
t
Z
[I ∗ x(θ ) − S ∗ y(θ )]dθ − t −τ
β 1 + αI ∗
Z
S (θ )
t t −τ
1 + α I (θ − τ )
y(θ − τ )dθ .
(4.14)
Hence, using the inequality a2 + b2 ≥ 2ab, we derive from (4.12) and (4.14) that, for t > T + 2τ ,
|(S ∗ y(t ) − I ∗ x(t ))(y(t − τ ) − y(t ))| ≤
βS∗ 2 1 + αI ∗ 1
Z
t
[I ∗ (x2 (θ ) + y2 (t )) + S ∗ (y2 (θ ) + y2 (t ))]dθ t −τ
Z t βI∗ [I ∗ (x2 (θ ) + x2 (t )) + S ∗ (y2 (θ ) + x2 (t ))]dθ 2 1 + α I ∗ t −τ Z t Z t 1 β M (ε)S ∗ 1 β M (ε)I ∗ 2 2 + (y (θ − τ ) + y (t ))dθ + (y2 (θ − τ ) + x2 (t ))dθ 2 1 + α I ∗ t −τ 2 1 + α I ∗ t −τ 1 β S∗τ 1 β I ∗τ = (I ∗ + S ∗ + M (ε))y2 (t ) + (I ∗ + S ∗ + M (ε))x2 (t ) ∗ 2 1 + αI 2 1 + αI ∗ Z Z 1 β(S ∗ + I ∗ ) t ∗ 2 1 β M (ε)(S ∗ + I ∗ ) t 2 ∗ 2 + [ I x (θ ) + S y (θ )] d θ + y (θ − τ )dθ . 2 1 + αI ∗ 2 1 + αI ∗ t −τ t −τ +
1
(4.15)
It therefore follows from (4.13) and (4.15) that, for t > T + 2τ , dV1 dt
βI∗ 1 αβ x2 (t ) + |x(t )y(t )| 1 + α I ∗ M (ε) m(ε) β 1 β S∗τ 1 β I ∗τ + (I ∗ + S ∗ + M (ε))y2 (t ) + (I ∗ + S ∗ + M (ε))x2 (t ) ∗ ∗ m(ε)(1 + α I ) 2 1 + α I 2 1 + αI ∗ Z Z 1 β M (ε)(S ∗ + I ∗ ) t 2 1 β(S ∗ + I ∗ ) t ∗ 2 ∗ 2 [ I x (θ ) + S y (θ )] d θ + y (θ − τ ) d θ . + 2 1 + αI ∗ 2 1 + αI ∗ t −τ t −τ
≤ − µ1 +
(4.16)
Define V2 =
β 2 (S ∗ + I ∗ ) 2 m(ε)(1 + α I ∗ )2 1
+
t
Z
t −τ
1 β 2 M (ε)(S ∗ + I ∗ )
t
Z Z
2 m(ε)(1 + α I ∗ )2
v t t −τ
[I ∗ x2 (θ ) + S ∗ y2 (θ )]dθ dv t
Z v
y2 (θ − τ )dθ dv +
1 β 2 M (ε)τ (S ∗ + I ∗ ) 2 m(ε)(1 + α I ∗ )2
Z
t
y2 (u)du.
(4.17)
t −τ
Then, we derive from (4.16) and (4.17) that, for t > T + 2τ , d dt
(V1 + V2 ) ≤ −(A1 (ε) − B1 (ε)τ )x2 (t ) + B2 (ε)τ y2 (t ) +
αβ m(ε)
|x(t )y(t )|,
(4.18)
where A1 (ε), B1 (ε) and B2 (ε) are defined in (4.9). We now define V3 ( x , y ) =
1 2
c (x(t ) + y(t ))2 ,
(4.19)
where c is a positive constant to be determined later. Hence, it follows from (4.18) and (4.19) that, for t > T + 2τ , d dt
(V1 + V2 + V3 ) ≤ −(A1 (ε) + c µ1 − B1 (ε)τ )x2 (t ) − c (µ1 + µ2 + γ )x(t )y(t ) − [c (µ2 + γ ) − B2 (ε)τ ]y2 (t ) +
αβ m(ε)
|x(t )y(t )|
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3187
≤ −(A1 (ε) + c µ1 − B1 (ε)τ )x2 (t ) − [c (µ2 + γ ) − B2 (ε)τ ]y2 (t ) αβ + c (µ1 + µ2 + γ ) + |x(t )y(t )| m(ε) := −(|x(t )|, |y(t )|)Q (|x(t )|, |y(t )|)T ,
(4.20)
where
Q =
A1 (ε) + c µ1 − B1 (ε)τ
αβ − c (µ1 + µ2 + γ ) + 2 m(ε) 1
−
αβ 2 m(ε) . c (µ2 + γ ) − B2 (ε)τ 1
c (µ1 + µ2 + γ ) +
The matrix Q is positively definite if and only if A1 (ε) + c µ1 − B1 (ε)τ > 0,
(4.21)
and
4(A1 (ε) − B1 (ε)τ + c µ1 )[c (µ2 + γ ) − B2 (ε)τ ] − c (µ1 + µ2 + γ ) +
αβ m(ε)
2
> 0.
(4.22)
Let F (c ) = (µ2 + γ − µ1 )2 c 2 + 4B2 (ε)τ (A1 (ε) − B1 (ε)τ ) +
αβ m(ε)
2
αβ (µ1 + µ2 + γ ) c . − 4(µ2 + γ )(A1 (ε) − B1 (ε)τ ) − 4µ1 B2 (ε)τ − 2 m(ε)
(4.23)
If τ = 0, then
αβ αβ 2 F (c ) = (µ2 + γ − µ1 ) c − 4A1 (ε)(µ2 + γ ) − 2 (µ1 + µ2 + γ ) c + . m(ε) m(ε) 2 2
It is easy to show that if A1 (0) > αβ/v2 , for ε > 0 sufficiently small, F (c ) = 0 has two positive real roots c01 < c02 . We may choose c01 < c < c02 such that F (c ) < 0. Let 0 < τ < A1 (ε)/B1 (ε) and A1 (0) > αβ/v2 hold. Clearly, F (0) = 4B2 (ε)τ (A1 (ε) − B1 (ε)τ ) + (αβ/m(ε))2 > 0. If A1 (0) > αβ/v2 , then A1 (ε) > αβ/m(ε) for ε > 0 sufficiently small. The equation F (c ) = 0 has two positive real roots c1 and c2 (c1 ≤ c2 ) if and only if
τ<
2A1 (ε)(µ2 + γ ) −
αβ
m(ε)
(µ1 + µ2 + γ )
2[B1 (ε)(µ2 + γ ) + B2 (ε)µ1 ]
:= τ1 (ε),
(4.24)
and
∆(τ ) = a2 (ε)τ 2 − a1 (ε)τ + a0 (ε) ≥ 0,
(4.25)
(ε)− 4a0 (ε)a2 (ε) > where ai (ε) > 0(i = 1, 2, 3) are defined in (4.9). It is not difficult to show that if A1 (0) > αβ/v 0 for ε > 0 sufficiently small. Hence, the equation ∆(τ ) = 0 has two positive real roots τ2 (ε) and τ3 (ε) (τ2 (ε) < τ3 (ε)). It is not difficult to show that τ2 (ε) < τ1 (ε) < τ3 (ε) since µ2 + γ > µ1 . We choose τ0 = τ2 (0). Noting that the functions A1 (ε), B1 (ε) and B2 (ε) are continuous with respect to ε > 0 sufficiently small, we conclude that τ2 (ε) is also continuous with respect to ε > 0 sufficiently small. Hence, for any 0 ≤ τ < τ0 , we have that τ < τ2 (ε) for ε > 0 sufficiently small. It therefore follows from ∆(τ ) > 0 that there is a positive constant c = c (τ ) such that F (c ) < 0. Accordingly, the matrix Q is positively definite. Hence, we derive from (4.20) that, for t sufficiently large, 2 2 , then a1
d dt
(V1 + V2 + V3 ) ≤ −δ(x2 (t ) + y2 (t ))
(4.26)
for some positive constant δ = δ(τ ) for 0 ≤ τ ≤ τ2 (ε). Hence, x2 (t ) and y2 (t ) are integrable on [0, +∞). By Lemmas 3.1 and 3.2 and (4.10), we see that x2 (t ) and y2 (t ) are uniformly continuous on [0, +∞). Therefore, applying Barbalat’s lemmas (Lemmas 1.2.2 and 1.2.3, Gopalsamy [8]), we conclude that lim x2 (t ) = lim y2 (t ) = 0,
t →+∞
t →+∞
which yields lim S (t ) = S ∗ ,
t →+∞
lim I (t ) = I ∗ .
t →+∞
(4.27)
3188
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
It therefore follows from (4.27) and the third equation of system (1.2) that lim R(t ) = R∗ .
t →+∞
Hence, the endemic equilibrium E ∗ of system (1.2) is globally attractive. Noting that E ∗ is locally asymptotically stable if R0 > 1, it is globally asymptotically stable. The proof is complete. Remark. By Theorem 4.2, we see that if the basic reproductive number R0 > 1, the endemic equilibrium E ∗ of system (1.2) is globally asymptotically stable provided that the parameter α and the time delay τ are sufficiently small. Theorem 4.3. If R0 < 1, then the disease-free equilibrium E1 (B/µ1 , 0, 0) of system (1.2) is globally asymptotically stable. Proof. Let (S (t ), I (t ), R(t )) be any positive solution of system (1.2) with initial conditions (1.3). If R0 < 1, we may choose ε > 0 sufficiently small satisfying
β(B/µ1 + ε) < µ2 + γ .
(4.28)
We derive from the first equation of system (1.2) that S˙ (t ) ≤ B − µ1 S (t ). By comparison it follows that lim sup S (t ) ≤ B/µ1 .
(4.29)
t →+∞
Hence, for ε > 0 sufficiently small satisfying (4.28), there is a T1 > 0 such that if t > T1 , S (t ) ≤ B/µ1 + ε. For ε > 0 sufficiently small satisfying (4.28), we derive from the second equation of system (1.2) that, for t > T1 + τ ,
˙I (t ) ≤ β(B/µ1 + ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ ) Consider the following auxiliary equation u˙ (t ) =
β(B/µ1 + ε)u(t − τ ) − (µ2 + γ )u(t ). 1 + α u(t − τ )
By Lemma 4.1 and (4.28), it follows that lim u(t ) = 0.
t →+∞
By comparison we derive that lim sup I (t ) = 0. t →+∞
Hence, for ε > 0 sufficiently small satisfying (4.28), there is a T2 > T1 + τ such that if t > T2 , I (t ) < ε. For ε > 0 sufficiently small satisfying (4.28), we obtain from the third equation of system (1.2) that for t > T2 , R˙ (t ) ≤ γ ε − µ3 R(t ). A standard comparison argument shows that lim sup R(t ) = 0. t →+∞
We derive from the first equation of system (1.2) that, for t > T2 + τ , S˙ (t ) ≥ B − µ1 S (t ) −
βε 1 + αε
S (t ).
By comparison it follows that lim inf S (t ) ≥ t →+∞
B(1 + αε)
µ1 + (β + αµ1 )ε
.
Letting ε → 0, we derive that lim inf S (t ) ≥ B/µ1 . t →+∞
This, together with (4.29), yields lim S (t ) = B/µ1 .
t →+∞
Noting that if R0 < 1, the disease-free equilibrium E1 of system (1.2) is locally asymptotically stable, we conclude that E1 is globally asymptotically stable. This completes the proof.
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3189
5. Discussion In this paper, we have formulated a delayed SIR epidemic model with nonlinear incidence rate introduced by Capasso and Serio [6]. We have shown that if the basic reproductive number R0 > 1, system (1.2) is permanent. The global stability of the disease free equilibrium and the endemic equilibrium of system (1.2) was also addressed, respectively. By comparison arguments, we have shown that if the basic reproductive number R0 < 1, the disease free equilibrium is globally asymptotically stable while the endemic equilibrium is not feasible. By the iteration technique, we have shown that if R0 > 1, the endemic equilibrium of system (1.2) is globally asymptotically stable provided Bα > µ2 + γ . Noting that this additional condition is not valid when the parameter α is sufficiently small, alternatively, by constructing a suitable Lyapunov functional, a different result was given to show that if R0 > 1, the endemic equilibrium of system (1.2) is globally asymptotically stable provided that the parameter α and the time delay τ are sufficiently small. Acknowledgments The authors wish to thank the reviewers for their valuable comments and suggestions that greatly improved the presentation of this work. This work was supported by the National Natural Science Foundation of China (Nos. 10671209, 10531030), China Postdoctoral Science Foundation (No. 20060391010), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] R.M. Anderson, R.M. May, Population biology of infectious diseases: Part I, Nature 280 (1979) 361–367. [2] E. Beretta, V. Capasso, F. Rinaldi, Global stability results for a generalized Lotka–Volterra system with distributed delays: Applications to predator–prey and epidemic systems, J. Math. Biol. 26 (1988) 661–668. [3] E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal. 47 (2001) 4107–4115. [4] E. Beretta, Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol. 33 (1995) 250–260. [5] E. Beretta, Y. Takeuchi, Convergence results in SIR epidemic model with varying population sizes, Nonlinear Anal. 28 (1997) 1909–1921. [6] V. Capasso, G. Serio, A generalization of the Kermack–Mckendrick deterministic epidemic model, Math. Biosci. 42 (1978) 41–61. [7] K.L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math. 9 (1979) 31–42. [8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, Norwell, MA, 1992. [9] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, Heidelberg, 1977. [10] H.W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 7 (1976) 335–356. [11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [12] W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004) 1141–1145. [13] W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (2002) 581–591. [14] Y. Takeuchi, W. Ma, Stability analysis on a delayed SIR epidemic model with density dependent birth process, Dynam. Cont. Discrete Impul. Syst. 5 (1999) 171–184. [15] Y. Takeuchi, W. Ma, E. Beretta, Global asymptotic properties of a SIR epidemic model with night incubation time, Nonlinear Anal. 42 (2000) 931–947.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Global stability of a SIR epidemic model with nonlinear incidence rate and time delay Rui Xu a,b,∗ , Zhien Ma b a
Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, PR China
b
Department of Applied Mathematics, Xi’an Jiaotong University, Xi’an, 710049, PR China
article
info
Article history: Received 25 January 2008 Accepted 8 October 2008 Keywords: SIR epidemic model Nonlinear incidence Time delay Stability
a b s t r a c t In this paper, a SIR epidemic model with nonlinear incidence rate and time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease free equilibrium is discussed. It is proved that if the basic reproductive number R0 > 1, the system is permanent. By comparison arguments, it is shown that if R0 < 1, the disease free equilibrium is globally asymptotically stable. If R0 > 1, by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Let S (t ) denote the number of members of a population susceptible to the disease, I (t ) the number of infective members and R(t ) the number of members who have been removed from the possibility of infection through full immunity. In [7], Cooke formulated a SIR model with time delay effect by assuming that the force of infection at time t is given by β S (t )I (t −τ ), where β is the average number of contacts per infective per day and τ > 0 is a fixed time during which the infectious agents develop in the vector, and it is only after that time that the infected vector can infect a susceptible human. Cooke considered the following model S˙ (t ) = B − µ1 S (t ) − β S (t )I (t − τ ), ˙I (t ) = β S (t )I (t − τ ) − (µ2 + γ )I (t ), R˙ (t ) = γ I (t ) − µ3 R(t ),
(1.1)
where parameters µ1 , µ2 , µ3 are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. It is natural biologically to assume that µ1 ≤ min{µ2 , µ3 }. The parameters B and γ are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. Much attention has been paid to the analysis of the stability of the disease free equilibrium and the endemic equilibrium of system (1.1) (see, for example, [1–5,7,10,12–15]). In [3], Beretta et al. considered the global stability of the disease free equilibrium and the endemic equilibrium of system (1.1). They showed that the disease free equilibrium is globally stable for any delay τ while the endemic equilibrium is not feasible. By constructing a suitable Lyapunov functional, sufficient conditions were derived to guarantee that if the endemic equilibrium is feasible, it is also globally stable for the delay being sufficiently
∗ Corresponding address: Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang 050003, Hebei Province, PR China. E-mail address: [email protected] (R. Xu). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.013
3176
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
small. In [12], Ma et al. derived an explicit expression of the lower bound of the component I (t ) of solution of system (1.1) which was proposed as an open problem. They therefore gave an estimation of the length of the time delay ensuring the global asymptotic stability of the endemic equilibrium. Incidence rate plays an important role in the modelling of epidemic dynamics. It has been suggested by several authors that the disease transmission process may have a nonlinear incidence rate. In many epidemic models, the bilinear incidence rate β SI and the standard incidence rate β SI /N are frequently used. The bilinear incidence rate is based on the law of mass action. This contact law is more appropriate for communicable diseases such as influenza etc., but not for sexually transmitted diseases. It has been pointed out that for standard incidence rate, it may be a good approximation if the number of available partners is large enough and everybody could not make more contacts than is practically feasible. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [6] introduced a saturated incidence rate g (I )S into epidemic models, where g (I ) tends to a saturation level when I gets large, i.e., g (I ) =
βI 1 + αI
,
where β I measures the infection force of the disease and 1/(1 + α I ) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. This incidence rate seems more reasonable than the bilinear incidence rate β IS, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters. Motivated by the work of Beretta et al. [3], Capasso and Serio [6] and Ma et al. [12], in this paper, we are concerned with the effect of time delay and nonlinear incidence rate on the dynamics of a SIR epidemic model. To this end, we consider the following delay differential equations S˙ (t ) = B − µ1 S (t ) −
β S (t )I (t − τ ) 1 + α I (t − τ )
,
˙I (t ) = β S (t )I (t − τ ) − (µ2 + γ )I (t ), 1 + α I (t − τ ) R˙ (t ) = γ I (t ) − µ3 R(t ),
(1.2)
where µ1 ≤ min{µ2 , µ3 }. The initial conditions for system (1.2) take the form S (θ ) = φ1 (θ ),
φi (θ ) ≥ 0,
I (θ ) = φ2 (θ ),
R(θ ) = φ3 (θ),
θ ∈ [−τ , 0], φi (0) > 0,
i = 1, 2, 3,
(1.3)
where (φ1 (θ ), φ2 (θ ), φ3 (θ )) ∈ C ([−τ , 0], R3+0 ), the Banach space of continuous functions mapping the interval [−τ , 0] into R3+0 , where R3+0 = {(x1 , x2 , x3 ) : xi ≥ 0, i = 1, 2, 3}. It is well known by the fundamental theory of functional differential equations [9], system (1.2) has a unique solution (S (t ), I (t ), R(t )) satisfying the initial conditions (1.3). It is easy to show that all solutions of system (1.2) with initial conditions (1.3) are defined on [0, +∞) and remain positive for all t ≥ 0. The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of a disease free equilibrium and an endemic equilibrium of system (1.2) is discussed. In Section 3, it is proved that system (1.2) is permanent if the basic reproductive number R0 > 1. In Section 4, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Alternatively, by constructing a suitable Lyapunov functional, it is shown that if R0 > 1, the endemic equilibrium of system Eq. (1.2) is globally stable provided that the parameter α and the time delay τ are sufficiently small. By comparison arguments, it is proved that if R0 < 1, the disease free equilibrium is globally stable. A brief discussion is given in Section 5 to conclude this work. 2. Local stability In this section, we discuss the local stability of an endemic equilibrium and a disease-free equilibrium of system (1.2) by analyzing the corresponding characteristic equations, respectively. System (1.2) always has a disease-free equilibrium E1 (B/µ1 , 0, 0). Further, if Bβ > µ1 (µ2 + γ ), then system (1.2) has a unique endemic equilibrium E ∗ (S ∗ , I ∗ , R∗ ), where Bα + µ2 + γ
, β + αµ1 Bβ − µ1 (µ2 + γ ) I∗ = , (µ2 + γ )(β + αµ1 ) γ [Bβ − µ1 (µ2 + γ )] R∗ = . µ3 (µ2 + γ )(β + αµ1 ) S∗ =
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3177
Let R0 =
Bβ
µ1 (µ2 + γ )
.
R0 is called the basic reproductive number of system (1.2). It is easy to show that if R0 > 1, the endemic equilibrium E ∗ exists; if R0 < 1, E ∗ is not feasible. The characteristic equation of system (1.2) at the endemic equilibrium E ∗ is of the form
(λ + µ3 )[λ2 + p1 λ + p0 + (q1 λ + q0 )e−λτ ] = 0,
(2.1)
where
βI∗ , p0 = (µ2 + γ ) µ1 + 1 + αI ∗ βI∗ p 1 = µ1 + µ2 + γ + , 1 + αI ∗ βµ1 S ∗ βS∗ q0 = − , q1 = − . ∗ 2 (1 + α I ) (1 + α I ∗ )2
Clearly, λ = −µ3 is always a root of Eq. (2.1). All other roots of (2.1) are determined by the following equation
λ2 + p1 λ + p0 + (q1 λ + q0 )e−λτ = 0.
(2.2)
When τ = 0, Eq. (2.2) becomes
λ2 + (p1 + q1 )λ + p0 + q0 = 0.
(2.3)
It is readily seen that if R0 > 1, then
βI∗ µ1 + > 0, p0 + q0 = (µ2 + γ ) µ1 − 1 + αI ∗ 1 + αI ∗ βI∗ µ2 + γ p1 + q1 = µ1 + + µ2 + γ − > 0. 1 + αI ∗ 1 + αI ∗ Hence, if R0 > 1, the endemic equilibrium E ∗ of system (1.2) is locally stable when τ = 0. If iω(ω > 0) is a solution of (2.1), separating real and imaginary parts, we derive that
p1 ω = q0 sin ωτ − q1 ω cos ωτ ,
(2.4)
ω2 − p0 = q0 cos ωτ + q1 ω sin ωτ . Squaring and adding the two equations of (2.4), it follows that
ω4 + (p21 − 2p0 − q21 )ω2 + p20 − q20 = 0.
(2.5)
Letting z = ω2 , then Eq. (2.5) becomes z 2 + (p21 − 2p0 − q21 )z + p20 − q20 = 0.
(2.6)
It is easy to show that
2 βI∗ (µ2 + γ )2 µ1 + + (µ2 + γ )2 − > 0, ∗ 1 + αI (1 + α I ∗ )2 βI∗ βµ1 S ∗ µ1 βI∗ + µ − + > 0. p20 − q20 = (µ2 + γ ) (µ2 + γ ) µ1 + 1 1 + αI ∗ (1 + α I ∗ )2 1 + αI ∗ 1 + αI ∗ p21 − 2p0 − q21 =
Hence, if R0 > 1, Eq. (2.6) has no positive roots. Accordingly, if R0 > 1, the endemic equilibrium E ∗ of system Eq. (1.2) is locally stable for all τ ≥ 0. The characteristic equation of system (1.2) at the disease-free equilibrium E1 (B/µ1 , 0, 0) takes the form
Bβ 2 −λτ (λ + µ3 ) λ + (µ1 + µ2 + γ )λ + µ1 (µ2 + γ ) − (λ + µ1 )e = 0. µ1
(2.7)
Clearly, Eq. (2.7) always has a negative real root λ = −µ3 . All other roots are given by the roots of equation
λ2 + (µ1 + µ2 + γ )λ + µ1 (µ2 + γ ) −
Bβ
µ1
(λ + µ1 )e−λτ = 0.
(2.8)
3178
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Let f1 (λ) = λ2 + (µ1 + µ2 + γ )λ + µ1 (µ2 + γ ) −
Bβ
µ1
(λ + µ1 )e−λτ .
If R0 > 1, then it is readily seen that for λ real, f1 (0) = µ1 (µ2 + γ ) − Bβ < 0,
lim f1 (λ) = +∞.
t →+∞
Hence, f1 (λ) = 0 has a positive real root. Therefore, if R0 > 1, the disease-free equilibrium E1 (B/µ1 , 0, 0) is unstable. If R0 < 1, it is easy to show that the disease-free equilibrium E1 (B/µ1 , 0, 0) of system (1.2) is stable when τ = 0. Using a similar argument one can show that if R0 < 1, the equilibrium E1 of system (1.2) is locally asymptotically stable. Theorem 2.1. For system (1.2), if the basic reproductive number R0 < 1, the disease-free equilibrium E1 (B/µ1 , 0, 0) is locally asymptotically stable while the endemic equilibrium is not feasible. If R0 > 1, the endemic equilibrium E ∗ exists and is locally asymptotically stable while the disease-free equilibrium E1 is unstable. 3. Permanence In this section, we are concerned with the permanence of system (1.2). Lemma 3.1. For any solution (S (t ), I (t ), R(t )) of system (1.2), we have that lim sup(S (t ) + I (t ) + R(t )) ≤ B/µ1 . t →+∞
Proof. Let N (t ) = S (t ) + I (t ) + R(t ). Noting that µ1 ≤ min{µ2 , µ3 }, it follows from system (1.2) that N˙ (t ) = B − µ1 S (t ) − µ2 I (t ) − µ3 R(t ) ≤ B − µ1 N (t ). A standard comparison argument shows that lim supt →+∞ N (t ) ≤ B/µ1 . This completes the proof.
Lemma 3.2. If R0 > 1, then for any solution (S (t ), I (t ), R(t )) of system (1.2) with initial conditions (1.3), we have that lim inf S (t ) ≥ t →+∞
B(µ1 + Bα) B(αµ1 + β) + µ21
:= v1 ,
lim inf I (t ) ≥ I ∗ e−(µ2 +γ )τ := v2 ,
(3.1)
t →+∞
lim inf R(t ) ≥ γ v2 /µ3 := v3 . t →+∞
Proof. Let (S (t ), I (t ), R(t )) be any solution of system (1.2) with initial conditions (1.3). By Lemma 3.1, it follows that lim sup I (t ) ≤ B/µ1 . t →+∞
Hence, for ε > 0 sufficiently small, there is a T1 > 0 such that if t > T1 , I (t ) < B/µ1 + ε . We therefore derive from the first equation of system (1.2) that, for t > T1 + τ , S˙ (t ) ≥ B −
µ1 +
β(B/µ1 + ε) S (t ), 1 + α(B/µ1 + ε)
which yields lim inf S (t ) ≥ t →+∞
B[1 + α(B/µ1 + ε)] . µ1 + (αµ1 + β)(B/µ1 + ε)
Since this inequality holds for arbitrary ε > 0 sufficiently small, it follows that lim inf S (t ) ≥ t →+∞
B(µ1 + Bα) B(αµ1 + β) + µ21
:= v1 .
We now show that lim inft →+∞ I (t ) ≥ v2 . For t ≥ 0, define a differentiable function V (t ) = I (t ) + β S ∗
Z
t t −τ
I ( u) 1 + α I (u)
du.
(3.2)
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3179
Calculating the derivative of V (t ) along solutions of system (1.2) we derive that V˙ (t ) =
β I (t − τ ) (S (t ) − S ∗ ) + 1 + α I (t − τ )
βS∗ − µ2 − γ 1 + α I (t )
I (t ).
(3.3)
β qI ∗ For any 0 < q < 1, we have that S ∗ < B/[µ1 + 1+α qI ∗ ] := k. There is a constant ρ ≥ 1 sufficiently large such that S ∗ < k(1 − e−kρτ ) := S ∆ . We now claim that it is impossible that I (t ) ≤ qI ∗ for all t ≥ ρτ . Otherwise, if I (t ) ≤ qI ∗ for all t ≥ ρτ , we derive from the first equation of system (1.2) that, for t ≥ ρτ + τ ,
S˙ (t ) ≥ B −
µ1 +
β qI ∗ 1 + α qI ∗
S (t ).
(3.4)
It then follows from (3.4) that, for t > ρτ + τ ,
S (t ) ≥ e−k(t −ρτ −τ ) S (ρτ + τ ) + B
>
B k
Z
t
ρτ +τ
ek(θ−ρτ −τ ) dθ
(1 − e−k(t −ρτ −τ ) ).
(3.5)
Hence, for t > 2ρτ + τ , we derive from (3.5) that S (t ) >
B k
(1 − e−kρτ ) = S ∆ > S ∗ .
(3.6)
Noting that I (t ) ≤ qI ∗ < I ∗ , it follows from (3.3) and (3.6) that, for t > 2ρτ + τ ,
β I (t − τ ) (S (t ) − S ∗ ) + 1 + α I (t − τ ) β I (t − τ ) (S (t ) − S ∗ ) = 1 + α I (t − τ ) β I (t − τ ) (S ∆ − S ∗ ). > 1 + α I (t − τ )
V˙ (t ) ≥
βS∗ − µ2 − γ 1 + αI ∗
I (t )
(3.7)
Setting i=
min I (θ + 2ρτ + 2τ ),
θ∈[−τ ,0]
we will claim that I (t ) ≥ i for all t ≥ 2ρτ + τ . Otherwise, if there is a T ≥ 0 such that I (t ) ≥ i for 2ρτ + τ ≤ t ≤ 2ρτ + 2τ + T , I (2ρτ + 2τ + T ) = i and ˙I (2ρτ + 2τ + T ) ≤ 0, it follows from (3.6) and the second equation of system (1.2) that, for t1 = 2ρτ + 2τ + T ,
˙I (t1 ) = β S (t1 )I (t1 − τ ) − (µ2 + γ )I (t1 ) 1 + α I (t1 − τ ) β S (t1 ) ≥ − (µ2 + γ ) i 1 + α I (t1 − τ ) βS∆ > − (µ + γ ) i 2 1 + αI ∗ > 0.
(3.8)
This is a contradiction. Hence, I (t ) ≥ i for all t ≥ 2ρτ + τ . Accordingly, for t ≥ 2ρτ + 2τ , it follows from (3.7) that V˙ (t ) >
βi 1 + αi
(S ∆ − S ∗ ),
which yields V (t ) → +∞ as t → +∞. It follows from (3.2) that there is a T2 > 0 such that if t > T2 , V (t ) ≤
B
µ1
+
Bβ S ∗
µ1 + Bα
+ 1.
A contradiction occurs. Hence, the claim is proved.
3180
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
By the claim, we are left to consider two possibilities. First, I (t ) ≥ qI ∗ for all t sufficiently large. Second, I (t ) oscillates about qI ∗ for all t sufficiently large. We now show that I (t ) ≥ qv2 for all t sufficiently large. The conclusion is obvious for the first case. For the second case, let t1 < t2 be sufficiently large such that I (t1 ) = I (t2 ) = qI ∗ ,
I (t ) < qI ∗ ,
t1 < t < t2 .
If t2 − t1 ≤ τ , it follows from the second equation of system (1.2) that
˙I (t ) > −(µ2 + γ )I (t ), which yields, for t1 < t < t2 , I (t ) > I (t1 )e−(µ2 +γ )(t −t1 ) > qv2 . If t2 − t1 > τ , we obtain from the second equation of system (1.2) that I (t ) ≥ qv2 for t ∈ [t1 , t1 + τ ]. We now claim that I (t ) ≥ qv2 for all t ∈ [t1 + τ , t2 ]. Otherwise, there is a T ∗ > 0 such that I (t ) ≥ qv2 for t1 ≤ t ≤ t1 + τ + T ∗ , I (t1 + τ + T ∗ ) = qv2 and ˙I (t1 + τ + T ∗ ) ≤ 0. On the other hand, it follows from the second equation of system (1.2) that, for t0 = t1 + τ + T ∗ ,
˙I (t0 ) = β S (t0 )I (t0 − τ ) − (µ2 + γ )I (t0 ) 1 + α I (t0 − τ ) β S ∆ qv2 − (µ2 + γ )qv2 ≥ 1 + α qv2 βS∆ + γ ) − (µ ≥ qv2 2 1 + αI ∗ > 0,
(3.9)
a contradiction. Hence, I (t ) ≥ qv2 for t ∈ [t1 , t2 ]. Since the interval [t1 , t2 ] is chosen arbitrarily, we conclude that I (t ) ≥ qv2 for all t sufficiently large for the second case. Since q(0 < q < 1) is chosen arbitrarily, we have that lim inft →+∞ I (t ) ≥ v2 . The proof is complete. From Lemmas 3.1 and 3.2, we therefore have the following result. Theorem 3.1. If R0 > 1, then system (1.2) is permanent. 4. Global stability In this section, we discuss the global stability of the endemic equilibrium E ∗ and the disease-free equilibrium E1 of system (1.2), respectively. We first introduce a result which is useful in discussing the global stability of the equilibria E1 and E ∗ . We now consider the following equation with time delay u˙ (t ) =
au(t − τ ) 1 + α u(t − τ )
− cu(t ),
u(θ ) = φ(θ ) ≥ 0,
θ ∈ [−τ , 0), φ(0) > 0,
(4.1)
where a, c and α are positive constants, τ ≥ 0. Eq. (4.1) always has a trivial equilibrium u = 0. If a > c, then Eq. (4.1) has a unique positive equilibrium u∗ = (a − c )/(α c ). For Eq. (4.1), we have the following result. Lemma 4.1. If a > c, then the positive equilibrium u∗ = (a − c )/(α c ) of (4.1) is globally asymptotically stable; if a < c, then the trivial equilibrium (0, 0) of (4.1) is globally asymptotically stable. Proof. Let f (u) = au/(1 + α u), g (u) = u. Clearly, f (u) and g (u) are strictly increasing. By Theorem 9.1 in Kuang [11], we see that if a > c, the positive equilibrium u∗ of system (4.1) is globally asymptotically stable. If a < c, we derive from (4.1) that u˙ (t ) ≤ au(t − τ ) − cu(t ). Consider the following auxiliary equation
v˙ (t ) = av(t − τ ) − c v(t ). Define V (t ) =
1 2
1
v 2 (t ) + a 2
Z
t
v 2 (s)ds. t −τ
(4.2)
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3181
Calculating the derivative of V (t ) along solution of Eq. (4.2), it follows that d dt
V (t ) = av(t )v(t − τ ) − c v 2 (t ) +
1 2
av 2 ( t ) −
1 2
av 2 (t − τ )
1 1 1 a(v 2 (t ) + v 2 (t − τ )) − c v 2 (t ) + av 2 (t ) − av 2 (t − τ ) 2 2 2 = (a − c )v 2 (t ).
≤
Hence, if a < c, limt →+∞ v(t ) = 0. By comparison it follows that, if a < c, limt →+∞ u(t ) = 0. This completes the proof. Theorem 4.1. If R0 > 1, Bα > µ2 + γ , then the endemic equilibrium E ∗ (S ∗ , I ∗ , R∗ ) of system (1.2) is globally asymptotically stable. Proof. Let (S (t ), I (t ), R(t )) be any positive solution of system (1.2) with initial conditions (1.3). Let U1 = lim sup S (t ), t →+∞
V1 = lim inf S (t ), t →+∞
U2 = lim sup I (t ),
U3 = lim sup R(t ),
t →+∞
V2 = lim inf I (t ), t →+∞
t →+∞
V3 = lim inf R(t ). t →+∞
In the following we claim that U1 = V1 = S , U2 = V2 = I ∗ , U3 = V3 = R∗ . It follows from the first equation of system (1.2) that ∗
S˙ (t ) ≤ B − µ1 S (t ). By comparison we derive that lim sup S (t ) ≤ B/µ1 := M1S . t →+∞
Hence, for ε > 0 sufficiently small there is a T1 > 0 such that if t > T1 , S (t ) ≤ M1S + ε. We obtain from the second equation of system (1.2) that for t > T1 + τ S ˙I (t ) ≤ β(M1 + ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
Consider the following auxiliary equation u˙ (t ) =
β(M1S + ε)u(t − τ ) − (µ2 + γ )u(t ). 1 + α u(t − τ )
Noting that R0 > 1, by Lemma 4.1 it follows that lim u(t ) =
t →+∞
β(M1S + ε) − (µ2 + γ ) . α(µ2 + γ )
By comparison we derive that U2 = lim sup I (t ) ≤ t →+∞
β(M1S + ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U2 ≤ M1I , where M1I =
β M1S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T2 > T1 + τ such that if t > T2 , I (t ) ≤ M1I + ε. It follows from the third equation of system (1.2) that for t > T2 , R˙ (t ) ≤ γ (M1I + ε) − µ3 R(t ). A standard comparison argument shows that U3 = lim sup R(t ) ≤ γ (M1I + ε)/µ3 . t →+∞
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U3 ≤ M1R , where M1R = γ M1I /µ3 .
3182
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
We derive from the first equation of system (1.2) that for t > T2 + τ S˙ (t ) ≥ B − µ1 S (t ) −
β(M1I + ε) S (t ). 1 + α(M1I + ε)
By comparison it follows that V1 = lim inf S (t ) ≥ t →+∞
B[1 + α(M1I + ε)]
µ1 + (β + αµ1 )(M1I + ε)
.
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V1 ≥ N1S , where N1S =
B(1 + α M1I )
µ1 + (β + αµ1 )M1I
.
Hence, for ε > 0 sufficiently small there is a T3 > T2 + τ such that if t > T3 , S (t ) ≥ N1S − ε. We derive from the second equation of system (1.2) that for t > T3 + τ S ˙I (t ) ≥ β(N1 − ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
Consider the following auxiliary equation
v˙ (t ) =
β(N1S − ε)v(t − τ ) − (µ2 + γ )v(t ). 1 + αv(t − τ )
Provided that β(N1S − ε) > µ2 + γ , by Lemma 4.1 we derive that lim v(t ) =
t →+∞
β(N1S − ε) − (µ2 + γ ) . α(µ2 + γ )
By comparison it follows that V2 = lim inf I (t ) ≥ t →+∞
β(N1S − ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V2 ≥ N1I , where N1I =
β N1S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T4 > T3 + τ such that if t > T4 , I (t ) ≥ N1I − ε. We derive from the third equation of system (1.2) that, for t > T4 , R˙ (t ) ≥ γ (N1I − ε) − µ3 R(t ). By comparison it follows that V3 = lim inf R(t ) ≥ t →+∞
γ (N1I − ε) . µ3
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V3 ≥ N1R , where N1R = γ N1I /µ3 . Again, it follows from the first equation of system (1.2) that, for t > T4 , S˙ (t ) ≤ B − µ1 S (t ) −
β(N1I − ε) S (t ). 1 + α(N1I − ε)
By comparison we derive that U1 = lim sup S (t ) ≤ t →+∞
B[1 + α(N1I − ε)]
µ1 + (β + αµ1 )(N1I − ε)
.
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U1 ≤ M2S , where M2S =
B(1 + α N1I )
µ1 + (β + αµ1 )N1I
.
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Hence, for ε > 0 sufficiently small there is a T5 > T4 such that if t > T5 , S (t ) ≤ M2S + ε. It follows from the second equation of system (1.2) that, for t > T5 + τ , S ˙I (t ) ≤ β(M2 + ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
By Lemma 4.1 and a comparison argument we derive that U2 = lim sup I (t ) ≤ t →+∞
β(M2S + ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U2 ≤ M2I , where M2I =
β M2S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T6 > T5 + τ such that if t > T6 , I (t ) ≤ M2I + ε. It follows from the third equation of system (1.2) that, for t > T6 , R˙ (t ) ≤ γ (M2I + ε) − µ3 R(t ). By comparison we obtain that U3 = lim sup R(t ) ≤ γ (M2I + ε)/µ3 . t →+∞
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U3 ≤ M1R , where M2R = γ M2I /µ3 . We derive from the first equation of system (1.2) that, for t > T6 + τ , S˙ (t ) ≥ B − µ1 S (t ) −
β(M2I + ε) S (t ). 1 + α(M2I + ε)
By comparison it follows that V1 = lim inf S (t ) ≥ t →+∞
B[1 + α(M2I + ε)]
µ1 + (β + αµ1 )(M2I + ε)
.
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V1 ≥ N2S , where N2S =
B(1 + α M2I )
µ1 + (β + αµ1 )M2I
.
Hence, for ε > 0 sufficiently small there is a T7 > T6 + τ such that if t > T7 , S (t ) ≥ N2S − ε. It follows from the second equation of system (1.2) that, for t > T7 + τ , S ˙I (t ) ≥ β(N2 − ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ )
By Lemma 4.1 and a comparison argument we derive that V2 = lim inf I (t ) ≥ t →+∞
β(N2S − ε) − (µ2 + γ ) . α(µ2 + γ )
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V2 ≥ N2I , where N2I =
β N2S − (µ2 + γ ) . α(µ2 + γ )
Hence, for ε > 0 sufficiently small there is a T8 > T7 + τ such that if t > T8 , I (t ) ≥ N2I − ε. It follows from the third equation of system (1.2) that, for t > T8 , R˙ (t ) ≥ γ (N2I − ε) − µ3 R(t ). By comparison we derive that V3 = lim inf R(t ) ≥ t →+∞
γ (N2I − ε) . µ3
3183
3184
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V3 ≥ N2R , where N2R = γ N2I /µ3 . Continuing this process, we obtain six sequences MnS , MnI , MnR , NnS , NnI , NnR (n = 1, 2, . . .) such that for n ≥ 2, MnS =
B(1 + α NnI −1 )
µ1 + (β + αµ1 )NnI −1
,
β MnS − (µ2 + γ ) , α(µ2 + γ ) B(1 + α MnI ) NnS = , µ1 + (β + αµ1 )MnI MnI =
(4.3)
β NnS − (µ2 + γ ) , α(µ2 + γ ) γ MnI γ NnI MnR = , NnR = . µ3 µ3 NnI =
It is readily seen that NnS ≤ V1 ≤ U1 ≤ MnS ,
NnI ≤ V2 ≤ U2 ≤ MnI ,
NnR ≤ V3 ≤ U3 ≤ MnR .
(4.4)
.
(4.5)
We derive from (4.3) that MnS+1 =
B2 α 2 MnS
(β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2
Noting that MnS ≥ S ∗ and Bα > µ2 + γ , it follows from (4.5) that MnS+1 − MnS =
=
B2 α 2 MnS
(β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2
− MnS
(Bα − µ2 − γ )MnS [Bα + µ2 + γ − (β + αµ1 )MnS ] (β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2
(Bα − µ2 − γ )MnS [Bα + µ2 + γ − (β + αµ1 )S ∗ ] (β + αµ1 )(Bα − µ2 − γ )MnS + (µ2 + γ )2 = 0. ≤
Thus, the sequence MnS is monotonically non-increasing. Hence, limn→+∞ MnS exists. Taking n → +∞, it follows from (4.5) that lim
n→+∞
MnS+1
B2 α 2 lim MnS
=
n→+∞
(β + αµ1 )(Bα − µ2 − γ ) lim MnS + (µ2 + γ )2
.
(4.6)
n→+∞
Noting that lim MnS+1 = lim MnS ,
n→+∞
n→+∞
we therefore derive from (4.6) that lim MnS =
n→+∞
Bα + µ2 + γ
β + αµ1
= S∗.
(4.7)
It follows from (4.7) and the second equation of (4.3) that lim
n→+∞
MnI
β lim MnS − (µ2 + γ ) =
n→+∞
α(µ2 + γ )
=
Bβ − µ1 (µ2 + γ )
(µ2 + γ )(β + αµ1 )
= I ∗.
Similarly, one can derive from (4.3) and (4.7) that lim NnS = S ∗ ,
n→+∞
lim NnI = I ∗ ,
n→+∞
lim MnR = R∗ ,
n→+∞
lim NnR = R∗ .
n→+∞
(4.8)
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3185
It follows from Eqs. (4.4), (4.7) and (4.8) that U1 = V1 = S ∗ ,
U2 = V2 = I ∗ ,
U3 = V3 = R∗ .
We therefore have that lim S (t ) = S ∗ ,
lim I (t ) = I ∗ ,
t →+∞
lim R(t ) = R∗ .
t →+∞
t →+∞
Noting that if R0 > 1, the endemic equilibrium E ∗ is locally asymptotically stable, we therefore conclude that E ∗ is globally asymptotically stable. The proof is complete. We note that Theorem 4.1 is not valid when α is sufficiently small. Alternatively, we now give a different result on the global stability of the endemic equilibrium E ∗ of system (1.2) by means of Lyapunov functional technique used by Beretta et al. in [3]. For ε > 0 sufficiently small, we denote M (ε) = B/µ1 + ε, v2 = I ∗ e−(µ2 +γ )τ , m(ε) = v2 − ε > 0, A1 (ε) =
µ1 +
βI∗ 1 + αI ∗
1 M (ε)
,
β 2I ∗ (2I ∗ + 2S ∗ + M (ε)), 2m(1 + α I ∗ )2 β2 (4.9) B2 (ε) = [2S ∗ (I ∗ + S ∗ + M (ε)) + M (ε)I ∗ ], 2m(ε)(1 + α I ∗ )2 αβ αβ a0 (ε) = (µ2 + γ ) A1 (ε) − A1 (ε)(µ2 + γ ) − µ1 , m(ε) m(ε) αβ a1 (ε) = [B1 (ε)(µ2 + γ ) + B2 (ε)µ1 ] 2A1 (ε)(µ2 + γ ) − (µ1 + µ2 + γ ) + A1 (ε)B2 (ε)(µ2 + γ − µ1 )2 , m(ε) B1 (ε) =
a2 (ε) = [B1 (ε)(µ2 + γ ) + B2 (ε)µ1 ]2 + B1 (ε)B2 (ε)(µ2 + γ − µ1 )2 . Theorem 4.2. If R0 > 1, the endemic equilibrium E ∗ (S ∗ , I ∗ , R∗ ) of system (1.2) is globally asymptotically stable provided that A1 (0) > αβ/v2 and τ < τ0 , where
τ0 =
a1 (0) −
p
a1 (0)2 − 4a0 (0)a2 (0) 2a2 (0)
.
Proof. Let (S (t ), I (t ), R(t )) be any positive solution of system (1.2) with initial conditions (1.3). On substituting x(t ) = S (t ) − S ∗ , y(t ) = I (t ) − I ∗ into system (1.2), it follows that
βI∗ β S (t ) x(t ) − y(t − τ ), 1 + αI ∗ (1 + α I (t − τ ))(1 + α I ∗ ) βI∗ βS∗ β S (t ) y˙ (t ) = x(t ) − y(t ) + y(t − τ ). ∗ ∗ 1 + αI 1 + αI (1 + α I (t − τ ))(1 + α I ∗ )
x˙ (t ) = −µ1 x(t ) −
(4.10)
Define a Lyapunov function
x y V1 (x, y) = x − S ∗ ln 1 + ∗ + y − I ∗ ln 1 + ∗ . S I Calculating the derivative of V1 along solutions of system (4.10), we derive that
βI∗ αβ S ∗ S (t )I (t − τ ) 2 = I ( t ) x ( t ) − y2 (t ) dt S (t )I (t ) 1 + αI ∗ (1 + α I (t − τ ))(1 + α I ∗ ) αβ I ∗ S (t )I (t − τ ) β S (t ) ∗ ∗ + x ( t ) y ( t ) + ( S y ( t ) − I x ( t ))( y ( t − τ ) − y ( t )) (1 + α I (t − τ ))(1 + α I ∗ ) (1 + α I (t − τ ))(1 + α I ∗ ) βI∗ αβ I ∗ I (t − τ ) 1 2 ≤ − µ1 + x (t ) + x(t )y(t ) ∗ 1 + αI S (t ) I (t )(1 + α I (t − τ ))(1 + α I ∗ )
dV1
1
+
− µ1 +
β I (t )(1 + α I ∗ )
|(S ∗ y(t ) − I ∗ x(t ))(y(t − τ ) − y(t ))|.
(4.11)
3186
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
For ε > 0 sufficiently small, by Lemmas 3.1 and 3.2, there is a T > 0 such that if t > T , S (t ) ≤ M (ε),
I (t ) ≥ m(ε) > 0.
(4.12)
It follows from (4.11) and (4.12) that, for t > T + τ ,
βI∗ ≤ − µ1 + dt 1 + αI ∗
dV1
+
β m(ε)(1 + α I ∗ )
1 M (ε)
x2 (t ) +
αβ m(ε)
|x(t )y(t )|
|(S ∗ y(t ) − I ∗ x(t ))(y(t − τ ) − y(t ))|.
(4.13)
Note that, for t > T + 2τ , y(t − τ ) − y(t ) = −
β 1 + αI ∗
t
Z
[I ∗ x(θ ) − S ∗ y(θ )]dθ − t −τ
β 1 + αI ∗
Z
S (θ )
t t −τ
1 + α I (θ − τ )
y(θ − τ )dθ .
(4.14)
Hence, using the inequality a2 + b2 ≥ 2ab, we derive from (4.12) and (4.14) that, for t > T + 2τ ,
|(S ∗ y(t ) − I ∗ x(t ))(y(t − τ ) − y(t ))| ≤
βS∗ 2 1 + αI ∗ 1
Z
t
[I ∗ (x2 (θ ) + y2 (t )) + S ∗ (y2 (θ ) + y2 (t ))]dθ t −τ
Z t βI∗ [I ∗ (x2 (θ ) + x2 (t )) + S ∗ (y2 (θ ) + x2 (t ))]dθ 2 1 + α I ∗ t −τ Z t Z t 1 β M (ε)S ∗ 1 β M (ε)I ∗ 2 2 + (y (θ − τ ) + y (t ))dθ + (y2 (θ − τ ) + x2 (t ))dθ 2 1 + α I ∗ t −τ 2 1 + α I ∗ t −τ 1 β S∗τ 1 β I ∗τ = (I ∗ + S ∗ + M (ε))y2 (t ) + (I ∗ + S ∗ + M (ε))x2 (t ) ∗ 2 1 + αI 2 1 + αI ∗ Z Z 1 β(S ∗ + I ∗ ) t ∗ 2 1 β M (ε)(S ∗ + I ∗ ) t 2 ∗ 2 + [ I x (θ ) + S y (θ )] d θ + y (θ − τ )dθ . 2 1 + αI ∗ 2 1 + αI ∗ t −τ t −τ +
1
(4.15)
It therefore follows from (4.13) and (4.15) that, for t > T + 2τ , dV1 dt
βI∗ 1 αβ x2 (t ) + |x(t )y(t )| 1 + α I ∗ M (ε) m(ε) β 1 β S∗τ 1 β I ∗τ + (I ∗ + S ∗ + M (ε))y2 (t ) + (I ∗ + S ∗ + M (ε))x2 (t ) ∗ ∗ m(ε)(1 + α I ) 2 1 + α I 2 1 + αI ∗ Z Z 1 β M (ε)(S ∗ + I ∗ ) t 2 1 β(S ∗ + I ∗ ) t ∗ 2 ∗ 2 [ I x (θ ) + S y (θ )] d θ + y (θ − τ ) d θ . + 2 1 + αI ∗ 2 1 + αI ∗ t −τ t −τ
≤ − µ1 +
(4.16)
Define V2 =
β 2 (S ∗ + I ∗ ) 2 m(ε)(1 + α I ∗ )2 1
+
t
Z
t −τ
1 β 2 M (ε)(S ∗ + I ∗ )
t
Z Z
2 m(ε)(1 + α I ∗ )2
v t t −τ
[I ∗ x2 (θ ) + S ∗ y2 (θ )]dθ dv t
Z v
y2 (θ − τ )dθ dv +
1 β 2 M (ε)τ (S ∗ + I ∗ ) 2 m(ε)(1 + α I ∗ )2
Z
t
y2 (u)du.
(4.17)
t −τ
Then, we derive from (4.16) and (4.17) that, for t > T + 2τ , d dt
(V1 + V2 ) ≤ −(A1 (ε) − B1 (ε)τ )x2 (t ) + B2 (ε)τ y2 (t ) +
αβ m(ε)
|x(t )y(t )|,
(4.18)
where A1 (ε), B1 (ε) and B2 (ε) are defined in (4.9). We now define V3 ( x , y ) =
1 2
c (x(t ) + y(t ))2 ,
(4.19)
where c is a positive constant to be determined later. Hence, it follows from (4.18) and (4.19) that, for t > T + 2τ , d dt
(V1 + V2 + V3 ) ≤ −(A1 (ε) + c µ1 − B1 (ε)τ )x2 (t ) − c (µ1 + µ2 + γ )x(t )y(t ) − [c (µ2 + γ ) − B2 (ε)τ ]y2 (t ) +
αβ m(ε)
|x(t )y(t )|
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3187
≤ −(A1 (ε) + c µ1 − B1 (ε)τ )x2 (t ) − [c (µ2 + γ ) − B2 (ε)τ ]y2 (t ) αβ + c (µ1 + µ2 + γ ) + |x(t )y(t )| m(ε) := −(|x(t )|, |y(t )|)Q (|x(t )|, |y(t )|)T ,
(4.20)
where
Q =
A1 (ε) + c µ1 − B1 (ε)τ
αβ − c (µ1 + µ2 + γ ) + 2 m(ε) 1
−
αβ 2 m(ε) . c (µ2 + γ ) − B2 (ε)τ 1
c (µ1 + µ2 + γ ) +
The matrix Q is positively definite if and only if A1 (ε) + c µ1 − B1 (ε)τ > 0,
(4.21)
and
4(A1 (ε) − B1 (ε)τ + c µ1 )[c (µ2 + γ ) − B2 (ε)τ ] − c (µ1 + µ2 + γ ) +
αβ m(ε)
2
> 0.
(4.22)
Let F (c ) = (µ2 + γ − µ1 )2 c 2 + 4B2 (ε)τ (A1 (ε) − B1 (ε)τ ) +
αβ m(ε)
2
αβ (µ1 + µ2 + γ ) c . − 4(µ2 + γ )(A1 (ε) − B1 (ε)τ ) − 4µ1 B2 (ε)τ − 2 m(ε)
(4.23)
If τ = 0, then
αβ αβ 2 F (c ) = (µ2 + γ − µ1 ) c − 4A1 (ε)(µ2 + γ ) − 2 (µ1 + µ2 + γ ) c + . m(ε) m(ε) 2 2
It is easy to show that if A1 (0) > αβ/v2 , for ε > 0 sufficiently small, F (c ) = 0 has two positive real roots c01 < c02 . We may choose c01 < c < c02 such that F (c ) < 0. Let 0 < τ < A1 (ε)/B1 (ε) and A1 (0) > αβ/v2 hold. Clearly, F (0) = 4B2 (ε)τ (A1 (ε) − B1 (ε)τ ) + (αβ/m(ε))2 > 0. If A1 (0) > αβ/v2 , then A1 (ε) > αβ/m(ε) for ε > 0 sufficiently small. The equation F (c ) = 0 has two positive real roots c1 and c2 (c1 ≤ c2 ) if and only if
τ<
2A1 (ε)(µ2 + γ ) −
αβ
m(ε)
(µ1 + µ2 + γ )
2[B1 (ε)(µ2 + γ ) + B2 (ε)µ1 ]
:= τ1 (ε),
(4.24)
and
∆(τ ) = a2 (ε)τ 2 − a1 (ε)τ + a0 (ε) ≥ 0,
(4.25)
(ε)− 4a0 (ε)a2 (ε) > where ai (ε) > 0(i = 1, 2, 3) are defined in (4.9). It is not difficult to show that if A1 (0) > αβ/v 0 for ε > 0 sufficiently small. Hence, the equation ∆(τ ) = 0 has two positive real roots τ2 (ε) and τ3 (ε) (τ2 (ε) < τ3 (ε)). It is not difficult to show that τ2 (ε) < τ1 (ε) < τ3 (ε) since µ2 + γ > µ1 . We choose τ0 = τ2 (0). Noting that the functions A1 (ε), B1 (ε) and B2 (ε) are continuous with respect to ε > 0 sufficiently small, we conclude that τ2 (ε) is also continuous with respect to ε > 0 sufficiently small. Hence, for any 0 ≤ τ < τ0 , we have that τ < τ2 (ε) for ε > 0 sufficiently small. It therefore follows from ∆(τ ) > 0 that there is a positive constant c = c (τ ) such that F (c ) < 0. Accordingly, the matrix Q is positively definite. Hence, we derive from (4.20) that, for t sufficiently large, 2 2 , then a1
d dt
(V1 + V2 + V3 ) ≤ −δ(x2 (t ) + y2 (t ))
(4.26)
for some positive constant δ = δ(τ ) for 0 ≤ τ ≤ τ2 (ε). Hence, x2 (t ) and y2 (t ) are integrable on [0, +∞). By Lemmas 3.1 and 3.2 and (4.10), we see that x2 (t ) and y2 (t ) are uniformly continuous on [0, +∞). Therefore, applying Barbalat’s lemmas (Lemmas 1.2.2 and 1.2.3, Gopalsamy [8]), we conclude that lim x2 (t ) = lim y2 (t ) = 0,
t →+∞
t →+∞
which yields lim S (t ) = S ∗ ,
t →+∞
lim I (t ) = I ∗ .
t →+∞
(4.27)
3188
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
It therefore follows from (4.27) and the third equation of system (1.2) that lim R(t ) = R∗ .
t →+∞
Hence, the endemic equilibrium E ∗ of system (1.2) is globally attractive. Noting that E ∗ is locally asymptotically stable if R0 > 1, it is globally asymptotically stable. The proof is complete. Remark. By Theorem 4.2, we see that if the basic reproductive number R0 > 1, the endemic equilibrium E ∗ of system (1.2) is globally asymptotically stable provided that the parameter α and the time delay τ are sufficiently small. Theorem 4.3. If R0 < 1, then the disease-free equilibrium E1 (B/µ1 , 0, 0) of system (1.2) is globally asymptotically stable. Proof. Let (S (t ), I (t ), R(t )) be any positive solution of system (1.2) with initial conditions (1.3). If R0 < 1, we may choose ε > 0 sufficiently small satisfying
β(B/µ1 + ε) < µ2 + γ .
(4.28)
We derive from the first equation of system (1.2) that S˙ (t ) ≤ B − µ1 S (t ). By comparison it follows that lim sup S (t ) ≤ B/µ1 .
(4.29)
t →+∞
Hence, for ε > 0 sufficiently small satisfying (4.28), there is a T1 > 0 such that if t > T1 , S (t ) ≤ B/µ1 + ε. For ε > 0 sufficiently small satisfying (4.28), we derive from the second equation of system (1.2) that, for t > T1 + τ ,
˙I (t ) ≤ β(B/µ1 + ε)I (t − τ ) − (µ2 + γ )I (t ). 1 + α I (t − τ ) Consider the following auxiliary equation u˙ (t ) =
β(B/µ1 + ε)u(t − τ ) − (µ2 + γ )u(t ). 1 + α u(t − τ )
By Lemma 4.1 and (4.28), it follows that lim u(t ) = 0.
t →+∞
By comparison we derive that lim sup I (t ) = 0. t →+∞
Hence, for ε > 0 sufficiently small satisfying (4.28), there is a T2 > T1 + τ such that if t > T2 , I (t ) < ε. For ε > 0 sufficiently small satisfying (4.28), we obtain from the third equation of system (1.2) that for t > T2 , R˙ (t ) ≤ γ ε − µ3 R(t ). A standard comparison argument shows that lim sup R(t ) = 0. t →+∞
We derive from the first equation of system (1.2) that, for t > T2 + τ , S˙ (t ) ≥ B − µ1 S (t ) −
βε 1 + αε
S (t ).
By comparison it follows that lim inf S (t ) ≥ t →+∞
B(1 + αε)
µ1 + (β + αµ1 )ε
.
Letting ε → 0, we derive that lim inf S (t ) ≥ B/µ1 . t →+∞
This, together with (4.29), yields lim S (t ) = B/µ1 .
t →+∞
Noting that if R0 < 1, the disease-free equilibrium E1 of system (1.2) is locally asymptotically stable, we conclude that E1 is globally asymptotically stable. This completes the proof.
R. Xu, Z. Ma / Nonlinear Analysis: Real World Applications 10 (2009) 3175–3189
3189
5. Discussion In this paper, we have formulated a delayed SIR epidemic model with nonlinear incidence rate introduced by Capasso and Serio [6]. We have shown that if the basic reproductive number R0 > 1, system (1.2) is permanent. The global stability of the disease free equilibrium and the endemic equilibrium of system (1.2) was also addressed, respectively. By comparison arguments, we have shown that if the basic reproductive number R0 < 1, the disease free equilibrium is globally asymptotically stable while the endemic equilibrium is not feasible. By the iteration technique, we have shown that if R0 > 1, the endemic equilibrium of system (1.2) is globally asymptotically stable provided Bα > µ2 + γ . Noting that this additional condition is not valid when the parameter α is sufficiently small, alternatively, by constructing a suitable Lyapunov functional, a different result was given to show that if R0 > 1, the endemic equilibrium of system (1.2) is globally asymptotically stable provided that the parameter α and the time delay τ are sufficiently small. Acknowledgments The authors wish to thank the reviewers for their valuable comments and suggestions that greatly improved the presentation of this work. This work was supported by the National Natural Science Foundation of China (Nos. 10671209, 10531030), China Postdoctoral Science Foundation (No. 20060391010), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] R.M. Anderson, R.M. May, Population biology of infectious diseases: Part I, Nature 280 (1979) 361–367. [2] E. Beretta, V. Capasso, F. Rinaldi, Global stability results for a generalized Lotka–Volterra system with distributed delays: Applications to predator–prey and epidemic systems, J. Math. Biol. 26 (1988) 661–668. [3] E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal. 47 (2001) 4107–4115. [4] E. Beretta, Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol. 33 (1995) 250–260. [5] E. Beretta, Y. Takeuchi, Convergence results in SIR epidemic model with varying population sizes, Nonlinear Anal. 28 (1997) 1909–1921. [6] V. Capasso, G. Serio, A generalization of the Kermack–Mckendrick deterministic epidemic model, Math. Biosci. 42 (1978) 41–61. [7] K.L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math. 9 (1979) 31–42. [8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, Norwell, MA, 1992. [9] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, Heidelberg, 1977. [10] H.W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 7 (1976) 335–356. [11] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. [12] W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004) 1141–1145. [13] W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (2002) 581–591. [14] Y. Takeuchi, W. Ma, Stability analysis on a delayed SIR epidemic model with density dependent birth process, Dynam. Cont. Discrete Impul. Syst. 5 (1999) 171–184. [15] Y. Takeuchi, W. Ma, E. Beretta, Global asymptotic properties of a SIR epidemic model with night incubation time, Nonlinear Anal. 42 (2000) 931–947.