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Dynamics of a delayed epidemic model with non-monotonic incidence rate
Commun Nonlinear Sci Numer Simulat 15 (2010) 459–468
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Dynamics of a delayed epidemic model with non-monotonic incidence rate q Hai-Feng Huo *, Zhan-Ping Ma Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 23 January 2009 Accepted 11 April 2009 Available online 3 May 2009 PACS: 02.30Ks
a b s t r a c t A delayed epidemic model with non-monotonic incidence rate which describes the psychological effect of certain serious on the community when the number of infectives is getting larger is studied. The disease-free equilibrium is globally asymptotically stable when R0 < 1 and is globally attractive when R0 ¼ 1 are derived. On the other hand, The disease is permanent when R0 > 1 is also obtained. Numerical simulation results are given to support the theoretical predictions. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Epidemic model Time delay Nonlinear incidence Global stability Lyapunov functional Permanence
1. Introduction Epidemiology is the study of the spread of disease in space and time, aiming at tracing factors that give rise to their occurrence. Biologically, it is necessary to consider time delay in epidemic models. In recent years, the dynamics of the epidemic models have received considerable attention [7–12]. The basic and important research subjects for these systems are the existence of the threshold value which distinguishes whether the infections disease will die out, the local and global stability of the disease-free equilibrium and the endemic equilibrium, the persistence and extinction of the disease, etc. In modelling of communicable diseases, the incidence rate (the rate of new infections) is considered to play a key role in ensuring that the model does indeed give a reasonable qualitative description of the disease dynamics [13,14]. In most classical disease transmission models, the incidence rate is assumed to be mass action incidence with bilinear interactions given by kIS, where k is the probability of transmission per contact, and S and I represent the susceptible and infected populations, respectively. However, there are several reasons for using non-linear incidence rates such as saturating and nearly bilinear. For instance, Yorke and London [18] showed that the incidence rate kð1 cIÞIS with positive c and time dependent k is consistent with the results of the simulations for measles outbreaks. To prevent the unboundedness of contact rate, Capasso and Serio [15] used a saturated incidence rate of the form kIS=ð1 þ kdIÞ; d > 0. To incorporate the effect of behavioral changes, Liu p and coworkers [16,17] used a non-linear incidence rate given by kI S=ð1 þ aIq Þ with k; p; q; a > 0. q This work was partially supported by the NNSF of China, the Key Project of Chinese Ministry of Education (209131), the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the NSF of Gansu Province of China (3ZS042-B25-013), the NSF of Bureau of Education of Gansu Province of China for Postgraduate Tutors (0803-01), the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (Q200703) and the Doctor’s Foundation of Lanzhou University of Technology. * Corresponding author. E-mail address: [email protected] (H.-F. Huo).
1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.04.018
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Xiao and Ruan [1] first proposed a incidence rate kSI=ð1 þ aI2 Þ, where kI measure the infection force of the disease and 1=ð1 þ aI2 Þ describes the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals is very large. Based on the above incidence rate, they introduced the following epidemic model with non-monotonic incidence rate
8 0 kSðtÞIðtÞ S ðtÞ ¼ b dSðtÞ 1þ þ cRðtÞ; > > aI2 ðtÞ < 0 kSðtÞIðtÞ > I ðtÞ ¼ 1þaI2 ðtÞ ðd þ lÞIðtÞ; > : 0 R ðtÞ ¼ lIðtÞ ðd þ cÞRðtÞ;
ð1:1Þ
where SðtÞ; IðtÞ and RðtÞ denote the numbers of susceptible, infective, and recovered individuals at time t, respectively. b is the recruitment rate of the population, d is the natural death rate of the population, k is the proportionality constant, l is the natural recovery rate of the infective individuals, c is the rate at which recovered individuals lose immunity and return to the susceptible class, a is the parameter measures the psychological or inhibitory effect. ds sÞ [2], since those infected at time t s become If we assume that the force of infection at any time t is given by ke1þaISIðt 2 ðtsÞ ds 6 1 represents the survival of vector population in which the time taken to beinfectious at time s later. The term 0 < e come infectious is s, which relates the population to its previous life stage [2]. Note that this probability of being alive after time s; eds is independent of the age of the individual. It is the only function for which the model is translation invariant, i.e. a semi-flow [4]. Then the model (1.1) takes the following form
8 0 keds SðtÞIðtsÞ > > > S ðtÞ ¼ b dSðtÞ 1þaI2 ðtsÞ þ cRðtÞ; < ds
ð1:2Þ
sÞ ðd þ lÞIðtÞ; I0 ðtÞ ¼ ke1þaSðtÞIðt > I2 ðtsÞ > > : 0 R ðtÞ ¼ lIðtÞ ðd þ cÞRðtÞ:
The paper is organized as follows. Section 2 contains some preliminaries include initial conditions, existence of equilibria of system (1.2), persistence theory. Section 3 deals with the global stability and the attractiveness of the disease-free equilibrium of system (1.2). Local stability and permanence of the endemic equilibrium of system (1.2) are settled in Section 4. Some numerical simulations and a brief discussion are given in Section 5 and 6, respectively. 2. Preliminaries We denote by C the Banach space of continuous functions u : ½s; 0 ! R3 with norm
jjujj ¼ sup fju1 ðhÞj; ju2 ðhÞj; ju3 ðhÞjg; s6h60
where u ¼ ðu1 ; u2 ; u3 Þ. Further, let
C þ ¼ fu ¼ ðu1 ; u2 ; u3 Þ 2 C : ui ðhÞ P 0 for all h 2 ½s; 0; i ¼ 1; 2; 3g: The initial condition for system (1.2) is given as
SðhÞ ¼ u1 ðhÞ;
IðhÞ ¼ u2 ðhÞ;
RðhÞ ¼ u3 ðhÞ;
h 2 ½s; 0;
ð2:1Þ
where u ¼ ðu1 ; u2 ; u3 Þ 2 C þ . Lemma 2.1. Suppose that ðSðtÞ; IðtÞ; RðtÞÞ is a solution of system (1.2) with initial condition (2.1), then SðtÞ P 0; IðtÞ P 0; RðtÞ P 0 for all t P 0. Denote
X ¼ fðS; I; RÞ : S P 0; I P 0; R P 0; S þ I þ R 6 S0 g; where S0 ¼ bd. Using the fact that SðtÞ þ IðtÞ þ RðtÞ S0 , it is easy to show that X is positively invariant with respect to system (1.2). System (1.2) always has a disease-free equilibrium E0 ¼ ðS0 ; 0; 0Þ. To find the positive equilibria, set ds
b dSðtÞ ds
ke
ke
SðtÞIðt sÞ
1 þ aI2 ðt sÞ
SðtÞIðt sÞ
1 þ aI2 ðt sÞ
þ cRðtÞ ¼ 0;
ðd þ lÞIðtÞ ¼ 0;
lIðtÞ ðd þ cÞRðtÞ ¼ 0: This yields
adðd þ lÞI2 ðtÞ þ keds d þ l
cl
dþc
ds
IðtÞ þ dðd þ lÞ ke
b ¼ 0;
ð2:2Þ
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solving above equation, we obtain ds
I ¼
ke
pffiffiffiffi cl d þ l dþ c þ D 2adðd þ lÞ
;
S ¼
1 cl b dþl I ; d dþc
R ¼
l dþc
I ;
where 2 2ds
D¼k e
! ds cl 2 ke b 2 : dþl 4ad ðd þ lÞ 1 dðd þ lÞ dþc
Define the basic reproduction number as follows ds
R0 ¼
ke S0 : dþl
We denote
1 d
s ¼ log
kS0 : dþl
It is easy to prove the following theorem. Theorem 2.1. When R0 6 1(i.e. s P s ), then system (1.2) only has the disease-free equilibrium E0 and when R0 > 1 (i.e. 0 6 s < s ), then system (1.2) has the endemic equilibrium E ¼ ðS ; I ; R Þ and E is unique. Consider an autonomous system of delay differential equations
_ xðtÞ ¼ Fðxt Þ
ð2:3Þ n
n
such that Fð0Þ ¼ 0 and F : Cð½s; 0; R Þ ! R is Lipschitzian. We will directly give the following Lemma without proof. Lemma 2.2. (See Hale [[3], Corollary 3.1, Chapter 5]). Let x1 ðÞ and x2 ðÞ be nonnegative continuous scalar functions such that xi ðrÞ ¼ 0 ði ¼ 1; 2Þ if and only if r ¼ 0; x1 ðrÞ ! þ1 as r ! þ1, and V : Cð½s; 0; Rn Þ ! Rn is a continuously differentiable scalar functional that for a special set X of solution of (2.3) satisfied
Vð/Þ P x1 ð/ð0ÞÞ;
_ Vð/Þ 6 x2 ðj/ð0ÞjÞ:
ð2:4Þ
Then x ¼ 0 is asymptotically stable in the set X. In order to prove Theorem 4.2 of Section 4, we engage the persistence theory by Hale and Waltman [6] for infinite dimensional systems. Now, we present the persistence theory [6] as follows. Consider a metric space X with metric d. T is a continuous semi-flow on X, i.e. a continuous mapping T : ½0; 1 X ! X with the following properties
T t T s ¼ T tþs ; t; s P 0;
T 0 ðxÞ ¼ x; x 2 X:
Here T t denotes the mapping from X to X given by T t ðxÞ ¼ Tðt; xÞ. The distance dðx; yÞ of a point x 2 X from a subset Y of X is defined by
dðx; yÞ ¼ inf dðx; yÞ: y2Y
Recall that the positive orbit cþ ðxÞ through x is defined as cþ ðxÞ ¼ [tP0 fTðtÞxg, and its x-limit set is xðxÞ ¼ \sP0 CL[tPs fTðtÞxg, where CL means closure. Define W s ðAÞ the stable set of a compact invariant set A as
W s ðAÞ ¼ fx : x 2 X; xðxÞ–/; xðxÞ Ag; e @ the particular invariant sets of interest as define A
e @ ¼ [x2A xðxÞ: A @ (H1) Assume X is the closure of open set X 0 ; @X 0 is nonempty and is the boundary of X 0 . Moreover the C 0 -semigroup TðxÞ on X satisfies
TðtÞ : X 0 ! X 0
TðtÞ : @X 0 ! @X 0 :
Lemma 2.3. (See [6, Theorem 4.1, p. 392]). Suppose TðtÞ satisfies (H1) and (i) There is a t0 P 0 such that TðtÞ is compact for t > t0 . (ii) TðtÞ is point dissipative in X. e @ is isolated and has an acyclic covering M. Then TðtÞ is uniformly persistent if for each M i 2 M; W s ðMi Þ \ X 0 ¼ /. (iii) A
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3. Disease-free equilibrium In this section, we discuss the stability and the attractiveness of the disease-free equilibrium E0 of system (1.2). We have the following main results. Theorem 3.1. If R0 < 1 (i.e. s > s ), then the disease-free equilibrium E0 of system (1.2) is globally asymptotically stable in X. If R0 > 1 (i.e. 0 6 s < s ), E0 becomes unstable. Proof. We consider the following Lyapunov functional
Vðxt Þ ¼ IðtÞ þ x1 RðtÞ þ x2 eds
Z
t
ts
IðuÞ 1 þ aI2 ðuÞ
du þ
x3 2
ðSðtÞ S0 Þ2 ;
where xi > 0; ði ¼ 1; 2; 3Þ. Then Vðxt Þ P min 1; x1 ; x23 ðIðtÞ þ RðtÞ þ ðSðtÞ S0 Þ2 . The time derivative of Vðxt Þ along the solution of system (1.2) becomes ds
V 0 ðxt Þjð1:2Þ ¼
SðtÞIðt sÞ ðd þ lÞIðtÞ þ x1 lIðtÞ x1 ðd þ cÞRðtÞ 1 þ aI2 ðt sÞ IðtÞ Iðt sÞ x2 eds þ x2 eds 1 þ aI2 ðtÞ 1 þ aI2 ðt sÞ ! ds ke SIðt sÞ þ cRðtÞ þ x3 ðSðtÞ S0 Þ dðSðtÞ S0 Þ 1 þ aI2 ðt sÞ
ke
6 dx3 ðSðtÞ S0 Þ2 ðx1 ðd þ cÞ x3 ðSðtÞ S0 ÞcÞRðtÞ
þ x1 l þ x2 eds ðd þ lÞ IðtÞ þ ðkS x2 kSx3 ðS S0 ÞÞ
eds Iðt sÞ 1 þ aI2 ðt sÞ
ð3:1Þ
:
It is easy to know that x1 ðd þ cÞ x3 ðSðtÞ S0 Þc > 0 for any x1 ; x3 > 0; SðtÞ 2 X. Let us choose xi > 0; ði ¼ 1; 2; 3Þ satisfying
kð1 þ x3 S0 Þ2 < 4x2 x3 ;
x1 l þ x2 e
ds
ð3:2Þ
< ðd þ lÞ:
ð3:3Þ
The choice of (3.2) is possible if x2 > kS0 . In fact, (3.2) is equivalent to 2
kS0 x23 þ 2ðkS0 2x2 Þx3 þ k < 0
ð3:4Þ 2
2
which is true given that x3 ¼ S10 if kS0 2x2 < 0 and ðkS0 2x2 Þ > ðkS0 Þ . By (3.3), R0 < 1 and x2 > kS0 , then
d þ l > x1 l þ x2 eds > x1 l þ kS0 eds ; dþl
ð3:5Þ 0
we can choose x1 < l ð1 R0 Þ to satisfy (3.5), i.e. (3.3) holds true. Hence from (3.1), V ðxt Þjð1:2Þ is negative definite and is equal to zero if and only if ðSðtÞ; IðtÞ; RðtÞÞ ¼ E0 . By Lemma 2.2, the conclusion is valid. When R0 > 1, we take a transform xðtÞ ¼ SðtÞ S ; yðtÞ ¼ IðtÞ I ; zðtÞ ¼ RðtÞ R , then system (1.2) will become
8 0 x ðtÞ ¼ b dðxðtÞ þ S Þ þ cðzðtÞ þ R Þ > > > > ds > > ke ðxðtÞ þ S Þ FðI Þ þ F 0 ðI Þyðt sÞ þ oðyðt sÞÞ ; > < ds y0 ðtÞ ¼ ke ðxðtÞ þ S Þ FðI Þ þ F 0 ðI Þyðt sÞ þ oðyðt sÞÞ > > > > ðd þ lÞðyðtÞ þ I Þ; > > > : 0 z ðtÞ ¼ lyðtÞ ðd þ cÞzðtÞ;
ð3:6Þ
2
1aI where FðI Þ ¼ 1þIaI2 ; F 0 ðI Þ ¼ ð1þ . aI2 Þ2 The linear part of (3.6) is
8 ds ds 0 0 > < x ðtÞ ¼ ðd þ ke FðI ÞÞxðtÞ ke S F ðI Þyðt sÞ þ czðtÞ; ds ds 0 0 y ðtÞ ¼ ke FðI ÞxðtÞ þ ke S F ðI Þyðt sÞ ðd þ lÞyðtÞ; > : 0 z ðtÞ ¼ lyðtÞ ðd þ cÞzðtÞ:
ð3:7Þ
If ðS ; I ; R Þ ¼ E0 ¼ ðS0 ; 0; 0Þ, then we obtain the following characteristic equation about the equilibrium E0
ds ðk þ dÞðk þ d þ cÞ k þ d þ l ke S0 ¼ 0:
ð3:8Þ
H.-F. Huo, Z.-P. Ma / Commun Nonlinear Sci Numer Simulat 15 (2010) 459–468 ds
463
ds
Set f ðkÞ ¼ k þ d þ l ke S0 . Obviously, f ð0Þ ¼ d þ l ke S0 < 0 by R0 > 1; f ðkÞ ! þ1 as k ! þ1, then there must exist a k0 > 0 such that f ðk0 Þ ¼ 0, i.e. the characteristic Eq. (3.8) has at least one root with positive real part. Therefore, E0 is unstable. This completes the proof of Theorem 3.1. h Remark 3.1. Theorem 3.1 shows that time delay sðs > s Þ does not affect global asymptotic stability of E0 . Theorem 3.2. If R0 ¼ 1 (i.e.
s ¼ s ), then the disease-free equilibrium E0 ¼ ðS0 ; 0; 0Þ of system (1.2) is globally attractive in X.
Proof. Let uðtÞ ¼ ðSðtÞ; IðtÞ; RðtÞÞ be any solution of system (1.2) with the initial function ðu1 ; u2 ; u3 Þ. Denote ut ðhÞ ¼ ðSðt þ hÞ; Iðt þ hÞ; Rðt þ hÞÞ; h 2 ½s; 0. Consider the following Lyapunov functional ds
Vðut Þ ¼ IðtÞ þ ke
SðtÞ
Z
t
ts
IðuÞ 2
1 þ aI ðuÞ
du þ ðd þ lÞ
Calculating derivative of Vðut Þ, we obtain
"
0
SðtÞ
ds
V ðut Þ ¼ ke
1 þ aI2 ðtÞ
ðd þ lÞ
1 1 þ aI2 ðtÞ
aI3 ðtÞ : 1 þ aI2 ðtÞ
# IðtÞ:
Let G ¼ fu 2 C þ : V 0 ðuÞ ¼ 0g and M be the largest set in G which is invariant with respect to system (1.2). Clearly, M is not empty, since ðS0 ; 0; 0Þ 2 M. If R0 ¼ 1, we have
"
0
V ðut Þ ¼ ðd þ lÞ
SðtÞ S0
#
S0 ð1 þ aI2 ðtÞÞ
IðtÞ 6 0:
Hence, G ¼ fu ¼ ðu1 ; u2 ; u3 Þ 2 C þ : u1 ¼ S0 or u2 ¼ 0g. When SðtÞ ¼ S0 , from SðtÞ þ IðtÞ þ RðtÞ S0 we always have IðtÞ ¼ 0. Therefore, we obtain that M ¼ fðS0 ; 0; 0Þg. Thus, it follows from the Lyapunov–LaSalle invariance principle (See Theorem 3.1 in [Chapter 5, [3]]) that limt!þ1 SðtÞ ¼ S0 and limt!þ1 IðtÞ ¼ limt!þ1 RðtÞ ¼ 0. This completes the proof of Theorem 3.2. h 4. Endemic equilibrium In this section, we discuss the local asymptotic stability and the permanence of the endemic equilibrium E of system (1.2). Theorem 4.1. If R0 > 1 (i.e. 0 6 s < s ), then the endemic equilibrium E of system (1.2) is locally asymptotically stable. Proof. We will discuss the local asymptotic stability of the endemic equilibrium E of system (1.2) through discussing the global stability of the equilibrium (0,0,0) of system (3.6). Consider the Lyapunov functional Vðut Þ ¼ V 1 ðut Þ þ V 2 ðut Þ, where ut ¼ ðxt ; yt ; zt Þ,
1 1 1 V 1 ðut Þ ¼ x1 ðxðtÞ þ yðtÞ þ zðtÞÞ2 þ x2 y2 ðtÞ þ x3 z2 ðtÞ; 2 2 2 Z t 1 ds 0 2 y ðsÞds; V 2 ðut Þ ¼ x2 ke S F ðI Þ 2 ts here, xi > 0 ði ¼ 1; 2; 3Þ. Note that
Vðut Þ P
1 1 1 x1 ðxðtÞ þ yðtÞ þ zðtÞÞ2 þ x2 y2 ðtÞ þ x3 z2 ðtÞ; 2 2 2
the time derivative of Vðut Þ along the solution of system (3.6) is ds
V 0 ðut Þ ¼x1 ðxðtÞ þ yðtÞ þ zðtÞÞ½dðxðtÞ þ yðtÞ þ zðtÞÞ þ x2 ke
FðI ÞxðtÞyðtÞ
ds
x2 ðd þ lÞy2 ðtÞ þ x2 ke S F 0 ðI ÞyðtÞyðt sÞ þ x3 zðtÞz0 ðtÞ
1 ds þ x2 ke S F 0 ðI Þ y2 ðtÞ y2 ðt sÞ : 2 It is easy to show that
1 2
x2 keds S F 0 ðI ÞyðtÞyðt sÞ 6 x2 keds S F 0 ðI Þ y2 ðtÞ þ y2 ðt sÞ ; we have ds
V 0 ðut Þ 6 x1 dðx2 ðtÞ þ y2 ðtÞ þ z2 ðtÞÞ ð2x1 d x2 ke
FðI ÞÞxðtÞyðtÞ
ð2x1 d x3 lÞyðtÞzðtÞ 2x1 dxðtÞzðtÞ x3 ðd þ cÞz2 ðtÞ ds 0
þ x2 ½ke
S F ðI Þ ðd þ lÞy2 ðtÞ:
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By using Eq. (2.2), it is easy to obtain that
h
i
x2 keds S F 0 ðI Þ ðd þ lÞ ¼
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
:
Then ds
V 0 ðut Þ 6 x1 dðx2 ðtÞ þ y2 ðtÞ þ z2 ðtÞÞ ð2x1 d x2 ke
FðI ÞÞxðtÞyðtÞ
ð2x1 d x3 lÞyðtÞzðtÞ 2x1 dxðtÞzðtÞ x3 ðd þ cÞz2 ðtÞ
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
y2 ðtÞ:
ð4:1Þ ds
Choose xi > 0 ði ¼ 1; 2; 3Þ satisfying 2x1 d ¼ x2 ke Then (4.1) becomes 0
2
V ðut Þ 6 x1 dx ðtÞ x1 d þ
FðI Þ and 2x1 d ¼ x3 l.
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
! y2 ðtÞ
ðx1 d þ x3 ðd þ cÞÞz2 ðtÞ 2x1 dxðtÞzðtÞ ¼ gðxðtÞ; zðtÞÞ x1 d þ
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
! y2 ðtÞ;
ð4:2Þ
2
where gðxðtÞ; zðtÞÞ ¼ x1 dx ðtÞ þ 2x1 dxðtÞzðtÞ þ ðx1 d þ x3 ðd þ cÞÞz2 ðtÞ. Since
4ðx1 dÞ2 4x1 d½x1 d þ x3 ðd þ cÞ ¼ 4x1 dx3 ðd þ cÞ < 0; hence gðxðtÞ; zðtÞÞ is negative. From (4.2) we obtain that V 0 ðut Þ is negative definite which implies the equilibrium (0,0,0) of system (3.6) is globally asymptotically stable by Lemma 2.1. Thus, the equilibrium E of system (1.2) is locally asymptotically stable. This completes the proof of Theorem 4.1. h Finally, we study the permanence of the disease of system (1.2), It is clear that the limit set of system (1.2) is on the plane S þ I þ R ¼ S0 . Thus, we focus on the reduced system
8 ds sÞ > < S0 ðtÞ ¼ b dSðtÞ ke SðtÞIðt þ cðS0 SðtÞ IðtÞÞ; 2 1þaI ðtsÞ
ds > sÞ : I0 ðtÞ ¼ ke SðtÞIðt ðd þ lÞIðtÞ: 2
ð4:3Þ
1þaI ðtsÞ
Consequently, the permanence of the disease of system (4.3) is equivalent to the permanence of the disease of system (1.2). we define
X1 ¼ fðS; IÞ : S P 0; I P 0; S þ I 6 S0 g: Clearly, X1 is positively invariant with respect to system (4.3). We have the following Theorem. Theorem 4.2. If R0 > 1 (i.e. 0 6 s < s ), then the disease of system (4.3) is permanent in X1 (i.e. the disease of system (1.2) is permanent in X). Proof. As the first step, we verify that the boundary planes of R2þ ¼ fðS; IÞ : S P 0; I P 0g repel the positive solution to sys2 2 tem (4.3) uniformly. Let C þ 1 ð½s; 0; Rþ Þ denote the space of continuous functions mapping ½s; 0 into Rþ . We choose
C 1 ¼ fðu1 ; u2 Þ 2 C þ1 ð½s; 0; R2þ Þ : u1 ðhÞ > 0; u2 ðhÞ 0; h 2 ½s; 0g; þ 2 0 2 0 Denote X ¼ C þ 1 ð½s; 0; Rþ Þ, and X ¼ IntC 1 ð½s; 0; Rþ Þ, then C 1 ¼ @X . It is easy to see that system (4.3) possesses constant 0 e solution in C 1 ¼ @X : E 2 C 1 with
e E ¼ fðu1 ; u2 Þ 2 C þ1 ð½s; 0; R2þ Þ : u1 ðhÞ S0 ;
u2 ðhÞ 0; h 2 ½s; 0g:
We verify below that the conditions of Lemma 2.3 are satisfied. By the definition of X 0 and @X 0 and system (4.3), it is easy to see that conditions (i) and (ii) of Lemma 2.3 are satisfied and that X 0 and @X 0 are invariant. Hence (H1) is also satisfied (See Fig. 1. Consider condition (iii) of Lemma 2.3. We have
S0 ðtÞjðu1 ;u2 Þ2C 1 0; thus SðtÞjðu1 ;u2 Þ2C 1 0 for all t P 0. Hence we have
I0 ðtÞjðu1 ;u2 Þ2C1 ¼ ðd þ lÞIðtÞ 6 0;
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tau=0.6
4
4
3.5
3.5
3
3
2.5
2
1.5
1.5
1
1
0.5
0.5
0
5
10
15
20
time t
25
30
35
5
0
40
3.5
3.5
3
3
solution
4
2.5
2
1.5
1.5
1
1
0.5
0.5
5
10
15
20
25
30
35
time t
10
15
20
time t
25
30
35
40
40
S(t) I(t) R(t)
tau=600
2.5
2
0
5
4.5
4
0
0
5
S(t) I(t) R(t)
tau=60
4.5
solution
2.5
2
0
S(t) I(t) R(t)
tau=6
4.5
solution
solution
4.5
5
S(t) I(t) R(t)
0
0
5
10
15
20
time t
25
30
35
40
Fig. 1. When b ¼ 4; d ¼ 1; k ¼ 0:8; a ¼ c ¼ l ¼ 1; s ¼ 0:47; s > s (i.e. R0 < 1), E0 is globally asymptotically stable, the disease dies out. We can see from figure that increase of time delay s ðs > s Þ does not affect global asymptotic stability of E0 .
e@ ¼ e from which follows that all points in C 1 approach e E, i.e. C 1 ¼ W s ð e EÞ. Hence A E and clearly it is isolated. Then the flow in e @ is acyclic, satisfying condition (iii) of Lemma 2.3. A EÞ \ X 0 ¼ /, assume the contrary, i.e. W s ð e EÞ \ X 0 –/. Then there exists a positive solution ðSðtÞ; IðtÞÞ Now we show that W s ð e in X1 to system (4.3) with limt!þ1 ðSðtÞ; IðtÞÞ ¼ ðS0 ; 0Þ, and for sufficiently small positive constant e with (See Fig. 2 ds
e<
ke
S0 ðd þ lÞ
ds
þ aðd þ lÞÞ
2ðke
;
there exists a positive constant T ¼ TðeÞ such that
pffiffiffi
pffiffiffi
e > 0; IðtÞ < e for all t P T:
SðtÞ > S0
By the second equation of (4.3) we have ds
I0 ðtÞ >
ke
ðS0 eÞIðt sÞ
1 þ aI2 ðt sÞ
ðd þ lÞIðtÞ; t P T þ s:
ð4:4Þ
Consider the equation
(
ds
u0 ðtÞ ¼ ke
ðS0 eÞuðtsÞ 1þau2 ðtsÞ
ðd þ lÞuðtÞ;
uðtÞ ¼ IðtÞ; t 2 ½T; T þ s:
ð4:5Þ
By (4.4) and the comparison theorem, we have IðtÞ P uðtÞ for all t P T. On the other hand, using Theorem 4.9.1 of [5], [ p. ffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds eÞðdþlÞ 159], we have limt!þ1 uðtÞ ¼ u for all solutions to system (4.5), where u ¼ ke ðSa0ðdþ > e is the unique positive equilÞ
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H.-F. Huo, Z.-P. Ma / Commun Nonlinear Sci Numer Simulat 15 (2010) 459–468 5
tau=0.47
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
5
10
15
20
25
30
35
0
40
S(t) I(t) R(t)
tau=0.47
4.5
solution
solution
4.5
5
S(t) I(t) R(t)
0
50
100
150
time t, t=0 to 40
200
250
300
350
400
time t, t=0 to 400
Fig. 2. When b ¼ 4; d ¼ 1; k ¼ 0:8; a ¼ c ¼ l ¼ 1; s ¼ s ¼ 0:47 (i.e. R0 ¼ 1), SðtÞ approaches to its steady state value while IðtÞ and RðtÞ approach zero as time goes to infinity, which indicate that E0 is globally attractive, the disease dies out.
5
tau=0.1
4
4
3.5
3.5
3 2.5
3 2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
10
20
30
40
50
60
70
80
90
0
100
0
10
20
30
40
5
tau=0.4
3.5
3
3
solution
solution
3.5
2.5
2
1.5
1.5
1
1
0.5
0.5
20
30
40
50
time t
80
90
100
60
70
80
90
100
S(t) I(t) R(t)
2.5
2
10
70
tau=0.45
4.5 4
0
60
5
S(t) I(t) R(t)
4
0
50
time t
time t
4.5
S(t) I(t) R(t)
tau=0.25
4.5
solution
solution
4.5
5
S(t) I(t) R(t)
0
0
10
20
30
40
50
time t
60
70
80
90
100
Fig. 3. When b ¼ 4; d ¼ 1; k ¼ 0:8; a ¼ c ¼ l ¼ 1; s ¼ 0:47; 0 6 s < s (i.e. R0 > 1), the disease is permanent. We can see from figure that SðtÞ approaches to S0 ¼ 4 while IðtÞ and RðtÞ approach zero as time delay s ð0 6 s < s Þ tends to the finite value s , which means that the disease will always exist until s P s (i.e. R0 6 1).
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1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
alpha=0.5
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
5
10
15
20
25
30
35
40
0
alpha=10
0
5
10
time t
15
20
25
30
35
40
time t Fig. 4. The dependence of I on the parameter a.
pffiffiffi pffiffiffi librium of system (4.5). Hence we get limt!þ1 IðtÞ P u > e, contradicting IðtÞ < e as t P T. Thus we have W s ð e EÞ \ X 0 ¼ /. Now we get that system (4.3) satisfies all conditions of Lemma 2.3, thus the disease of system (4.3) is permanent in X1 . This completes the proof of Theorem 4.2. h Remark 4.2. Theorem 4.2 shows that time delay (1.2).
s ð0 6 s < s Þ does not affect the permanence of the disease of system
Remark 4.3. Computer observations (See Fig. 3 in Section 5) suggest that the endemic equilibrium is also globally stable. So, we give an interesting open problem: whether we can also obtain that the endemic equilibrium E is globally stable when R0 > 1. 5. Numerical simulation Throughout this paper, we considered a delayed epidemic model with non-monotonic incidence rate and obtained some results. From Theorems 3.1, 3.2 and 4.2 we have the following cases: 1. When R0 < 1 (i.e. s > s ), the disease-free equilibrium of system (1.2) is globally asymptotically stable, the disease will die out. 2. When R0 ¼ 1 (i.e. s ¼ s ), the disease-free equilibrium of system (1.2) is globally attractive, the disease will die out. 3. When R0 > 1 (i.e. 0 6 s < s ), the disease of system (1.2) is permanent, the disease will always exist. In the following, we present some numerical simulation of examples which validate these theoretical results obtained in this paper. 6. Discussions Recall that the parameter a describes the psychological effect of the general public toward the infectives. Though the basic reproduction number R0 does not depend on a explicitly, numerical simulations shown that when the disease is endemic, the steady value I of the infectives decreases as a increases (see Fig. 4). From the expression of I we can see that I approaches zero as a goes to infinity. In [1], Xiao and Ruan also mentioned the above properties of a. References [1] [2] [3] [4] [5] [6] [7]
Xiao D, Ruan S. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences 2007;208:419–29. Capasso V. In: Mathematical Structure of Epidemic Systems. Lecture notes in biomathematics, vol. 97. Berlin: Springer; 1993. Hale JK. Theory of functional differential equations. New York: Springer; 1977. Hethcote HW, Tudor DW. Integral equation models for endemic infectious diseases. J Math Biol 1980;9:37–47. Kuang Y. Delay differential equations with applications in population dynamics. New York: Academic Press; 1993. Hale JK, Waltman P. Persistence in infinite-dimensional systems. SIAM J Math Anal 1989;20:388–95. Zhang T, Teng Z. Global behavior and permanence of SIRS epidemic model with time delay. Nonlin Anal Real World Appl 2007. doi:10.1016/ j.nonrwa.2007.03.010.
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[8] Faina B, Georgy K, Song B, Castillo-Chavez C. A simple epidemic model with surprising dynamics. Math Biosci Eng 2005;2:133–52. [9] Song B, Castillo-Chavez C, Aparicio JP. Tuberculosis models with fast and slow dynamics: the role of close and casual contacts. Math Biosci 2002;180:187–205. [10] Wang H, Li J, Kuang Y. Mathematical modeling and qualitative analysis of insulin therapies. Math Biosci 2007;210:17–33. [11] Zhen J, Ma Z, Han M. Global stability of an SIRS epidemic model with delays. Acta Mathematica Scientia 2006;26B(2):291–306. [12] Zhen J, Ma Z. The stability of an SIR epidemic model with time delays. Math Biosci 2006;3:101–9. [13] Capasso V. In: Mathematical structure of epidemic systems. Lecture notes in biomathematics, vol. 97. Berlin: Springer; 1993. [14] Levin SA, Hallam TG, Gross LJ. Applied mathematical ecology. New York: Springer; 1989. [15] Capasso V, Serio G. A generalization of the Kermack–McKendrick deterministic epidemic model. Math Biosci 1978;42:43. [16] Lin J, Andreasen V, Levin SA. Dynamics of influenza A drift: the linear three-strain model. Math Biosci 1999;162:33. [17] Liu WM, Hethcote HW, Levin SA. Dynamical behavior of epidemiological models with nonlinear incidence rates. J Math Biol 1987;25:359. [18] Yorke JA, London WP. Recurrent outbreaks of measles,chickenpox and mumps II. Am J Epidemiol 1973;98:469.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Dynamics of a delayed epidemic model with non-monotonic incidence rate q Hai-Feng Huo *, Zhan-Ping Ma Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 23 January 2009 Accepted 11 April 2009 Available online 3 May 2009 PACS: 02.30Ks
a b s t r a c t A delayed epidemic model with non-monotonic incidence rate which describes the psychological effect of certain serious on the community when the number of infectives is getting larger is studied. The disease-free equilibrium is globally asymptotically stable when R0 < 1 and is globally attractive when R0 ¼ 1 are derived. On the other hand, The disease is permanent when R0 > 1 is also obtained. Numerical simulation results are given to support the theoretical predictions. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Epidemic model Time delay Nonlinear incidence Global stability Lyapunov functional Permanence
1. Introduction Epidemiology is the study of the spread of disease in space and time, aiming at tracing factors that give rise to their occurrence. Biologically, it is necessary to consider time delay in epidemic models. In recent years, the dynamics of the epidemic models have received considerable attention [7–12]. The basic and important research subjects for these systems are the existence of the threshold value which distinguishes whether the infections disease will die out, the local and global stability of the disease-free equilibrium and the endemic equilibrium, the persistence and extinction of the disease, etc. In modelling of communicable diseases, the incidence rate (the rate of new infections) is considered to play a key role in ensuring that the model does indeed give a reasonable qualitative description of the disease dynamics [13,14]. In most classical disease transmission models, the incidence rate is assumed to be mass action incidence with bilinear interactions given by kIS, where k is the probability of transmission per contact, and S and I represent the susceptible and infected populations, respectively. However, there are several reasons for using non-linear incidence rates such as saturating and nearly bilinear. For instance, Yorke and London [18] showed that the incidence rate kð1 cIÞIS with positive c and time dependent k is consistent with the results of the simulations for measles outbreaks. To prevent the unboundedness of contact rate, Capasso and Serio [15] used a saturated incidence rate of the form kIS=ð1 þ kdIÞ; d > 0. To incorporate the effect of behavioral changes, Liu p and coworkers [16,17] used a non-linear incidence rate given by kI S=ð1 þ aIq Þ with k; p; q; a > 0. q This work was partially supported by the NNSF of China, the Key Project of Chinese Ministry of Education (209131), the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the NSF of Gansu Province of China (3ZS042-B25-013), the NSF of Bureau of Education of Gansu Province of China for Postgraduate Tutors (0803-01), the Development Program for Outstanding Young Teachers in Lanzhou University of Technology (Q200703) and the Doctor’s Foundation of Lanzhou University of Technology. * Corresponding author. E-mail address: [email protected] (H.-F. Huo).
1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.04.018
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Xiao and Ruan [1] first proposed a incidence rate kSI=ð1 þ aI2 Þ, where kI measure the infection force of the disease and 1=ð1 þ aI2 Þ describes the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals is very large. Based on the above incidence rate, they introduced the following epidemic model with non-monotonic incidence rate
8 0 kSðtÞIðtÞ S ðtÞ ¼ b dSðtÞ 1þ þ cRðtÞ; > > aI2 ðtÞ < 0 kSðtÞIðtÞ > I ðtÞ ¼ 1þaI2 ðtÞ ðd þ lÞIðtÞ; > : 0 R ðtÞ ¼ lIðtÞ ðd þ cÞRðtÞ;
ð1:1Þ
where SðtÞ; IðtÞ and RðtÞ denote the numbers of susceptible, infective, and recovered individuals at time t, respectively. b is the recruitment rate of the population, d is the natural death rate of the population, k is the proportionality constant, l is the natural recovery rate of the infective individuals, c is the rate at which recovered individuals lose immunity and return to the susceptible class, a is the parameter measures the psychological or inhibitory effect. ds sÞ [2], since those infected at time t s become If we assume that the force of infection at any time t is given by ke1þaISIðt 2 ðtsÞ ds 6 1 represents the survival of vector population in which the time taken to beinfectious at time s later. The term 0 < e come infectious is s, which relates the population to its previous life stage [2]. Note that this probability of being alive after time s; eds is independent of the age of the individual. It is the only function for which the model is translation invariant, i.e. a semi-flow [4]. Then the model (1.1) takes the following form
8 0 keds SðtÞIðtsÞ > > > S ðtÞ ¼ b dSðtÞ 1þaI2 ðtsÞ þ cRðtÞ; < ds
ð1:2Þ
sÞ ðd þ lÞIðtÞ; I0 ðtÞ ¼ ke1þaSðtÞIðt > I2 ðtsÞ > > : 0 R ðtÞ ¼ lIðtÞ ðd þ cÞRðtÞ:
The paper is organized as follows. Section 2 contains some preliminaries include initial conditions, existence of equilibria of system (1.2), persistence theory. Section 3 deals with the global stability and the attractiveness of the disease-free equilibrium of system (1.2). Local stability and permanence of the endemic equilibrium of system (1.2) are settled in Section 4. Some numerical simulations and a brief discussion are given in Section 5 and 6, respectively. 2. Preliminaries We denote by C the Banach space of continuous functions u : ½s; 0 ! R3 with norm
jjujj ¼ sup fju1 ðhÞj; ju2 ðhÞj; ju3 ðhÞjg; s6h60
where u ¼ ðu1 ; u2 ; u3 Þ. Further, let
C þ ¼ fu ¼ ðu1 ; u2 ; u3 Þ 2 C : ui ðhÞ P 0 for all h 2 ½s; 0; i ¼ 1; 2; 3g: The initial condition for system (1.2) is given as
SðhÞ ¼ u1 ðhÞ;
IðhÞ ¼ u2 ðhÞ;
RðhÞ ¼ u3 ðhÞ;
h 2 ½s; 0;
ð2:1Þ
where u ¼ ðu1 ; u2 ; u3 Þ 2 C þ . Lemma 2.1. Suppose that ðSðtÞ; IðtÞ; RðtÞÞ is a solution of system (1.2) with initial condition (2.1), then SðtÞ P 0; IðtÞ P 0; RðtÞ P 0 for all t P 0. Denote
X ¼ fðS; I; RÞ : S P 0; I P 0; R P 0; S þ I þ R 6 S0 g; where S0 ¼ bd. Using the fact that SðtÞ þ IðtÞ þ RðtÞ S0 , it is easy to show that X is positively invariant with respect to system (1.2). System (1.2) always has a disease-free equilibrium E0 ¼ ðS0 ; 0; 0Þ. To find the positive equilibria, set ds
b dSðtÞ ds
ke
ke
SðtÞIðt sÞ
1 þ aI2 ðt sÞ
SðtÞIðt sÞ
1 þ aI2 ðt sÞ
þ cRðtÞ ¼ 0;
ðd þ lÞIðtÞ ¼ 0;
lIðtÞ ðd þ cÞRðtÞ ¼ 0: This yields
adðd þ lÞI2 ðtÞ þ keds d þ l
cl
dþc
ds
IðtÞ þ dðd þ lÞ ke
b ¼ 0;
ð2:2Þ
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solving above equation, we obtain ds
I ¼
ke
pffiffiffiffi cl d þ l dþ c þ D 2adðd þ lÞ
;
S ¼
1 cl b dþl I ; d dþc
R ¼
l dþc
I ;
where 2 2ds
D¼k e
! ds cl 2 ke b 2 : dþl 4ad ðd þ lÞ 1 dðd þ lÞ dþc
Define the basic reproduction number as follows ds
R0 ¼
ke S0 : dþl
We denote
1 d
s ¼ log
kS0 : dþl
It is easy to prove the following theorem. Theorem 2.1. When R0 6 1(i.e. s P s ), then system (1.2) only has the disease-free equilibrium E0 and when R0 > 1 (i.e. 0 6 s < s ), then system (1.2) has the endemic equilibrium E ¼ ðS ; I ; R Þ and E is unique. Consider an autonomous system of delay differential equations
_ xðtÞ ¼ Fðxt Þ
ð2:3Þ n
n
such that Fð0Þ ¼ 0 and F : Cð½s; 0; R Þ ! R is Lipschitzian. We will directly give the following Lemma without proof. Lemma 2.2. (See Hale [[3], Corollary 3.1, Chapter 5]). Let x1 ðÞ and x2 ðÞ be nonnegative continuous scalar functions such that xi ðrÞ ¼ 0 ði ¼ 1; 2Þ if and only if r ¼ 0; x1 ðrÞ ! þ1 as r ! þ1, and V : Cð½s; 0; Rn Þ ! Rn is a continuously differentiable scalar functional that for a special set X of solution of (2.3) satisfied
Vð/Þ P x1 ð/ð0ÞÞ;
_ Vð/Þ 6 x2 ðj/ð0ÞjÞ:
ð2:4Þ
Then x ¼ 0 is asymptotically stable in the set X. In order to prove Theorem 4.2 of Section 4, we engage the persistence theory by Hale and Waltman [6] for infinite dimensional systems. Now, we present the persistence theory [6] as follows. Consider a metric space X with metric d. T is a continuous semi-flow on X, i.e. a continuous mapping T : ½0; 1 X ! X with the following properties
T t T s ¼ T tþs ; t; s P 0;
T 0 ðxÞ ¼ x; x 2 X:
Here T t denotes the mapping from X to X given by T t ðxÞ ¼ Tðt; xÞ. The distance dðx; yÞ of a point x 2 X from a subset Y of X is defined by
dðx; yÞ ¼ inf dðx; yÞ: y2Y
Recall that the positive orbit cþ ðxÞ through x is defined as cþ ðxÞ ¼ [tP0 fTðtÞxg, and its x-limit set is xðxÞ ¼ \sP0 CL[tPs fTðtÞxg, where CL means closure. Define W s ðAÞ the stable set of a compact invariant set A as
W s ðAÞ ¼ fx : x 2 X; xðxÞ–/; xðxÞ Ag; e @ the particular invariant sets of interest as define A
e @ ¼ [x2A xðxÞ: A @ (H1) Assume X is the closure of open set X 0 ; @X 0 is nonempty and is the boundary of X 0 . Moreover the C 0 -semigroup TðxÞ on X satisfies
TðtÞ : X 0 ! X 0
TðtÞ : @X 0 ! @X 0 :
Lemma 2.3. (See [6, Theorem 4.1, p. 392]). Suppose TðtÞ satisfies (H1) and (i) There is a t0 P 0 such that TðtÞ is compact for t > t0 . (ii) TðtÞ is point dissipative in X. e @ is isolated and has an acyclic covering M. Then TðtÞ is uniformly persistent if for each M i 2 M; W s ðMi Þ \ X 0 ¼ /. (iii) A
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3. Disease-free equilibrium In this section, we discuss the stability and the attractiveness of the disease-free equilibrium E0 of system (1.2). We have the following main results. Theorem 3.1. If R0 < 1 (i.e. s > s ), then the disease-free equilibrium E0 of system (1.2) is globally asymptotically stable in X. If R0 > 1 (i.e. 0 6 s < s ), E0 becomes unstable. Proof. We consider the following Lyapunov functional
Vðxt Þ ¼ IðtÞ þ x1 RðtÞ þ x2 eds
Z
t
ts
IðuÞ 1 þ aI2 ðuÞ
du þ
x3 2
ðSðtÞ S0 Þ2 ;
where xi > 0; ði ¼ 1; 2; 3Þ. Then Vðxt Þ P min 1; x1 ; x23 ðIðtÞ þ RðtÞ þ ðSðtÞ S0 Þ2 . The time derivative of Vðxt Þ along the solution of system (1.2) becomes ds
V 0 ðxt Þjð1:2Þ ¼
SðtÞIðt sÞ ðd þ lÞIðtÞ þ x1 lIðtÞ x1 ðd þ cÞRðtÞ 1 þ aI2 ðt sÞ IðtÞ Iðt sÞ x2 eds þ x2 eds 1 þ aI2 ðtÞ 1 þ aI2 ðt sÞ ! ds ke SIðt sÞ þ cRðtÞ þ x3 ðSðtÞ S0 Þ dðSðtÞ S0 Þ 1 þ aI2 ðt sÞ
ke
6 dx3 ðSðtÞ S0 Þ2 ðx1 ðd þ cÞ x3 ðSðtÞ S0 ÞcÞRðtÞ
þ x1 l þ x2 eds ðd þ lÞ IðtÞ þ ðkS x2 kSx3 ðS S0 ÞÞ
eds Iðt sÞ 1 þ aI2 ðt sÞ
ð3:1Þ
:
It is easy to know that x1 ðd þ cÞ x3 ðSðtÞ S0 Þc > 0 for any x1 ; x3 > 0; SðtÞ 2 X. Let us choose xi > 0; ði ¼ 1; 2; 3Þ satisfying
kð1 þ x3 S0 Þ2 < 4x2 x3 ;
x1 l þ x2 e
ds
ð3:2Þ
< ðd þ lÞ:
ð3:3Þ
The choice of (3.2) is possible if x2 > kS0 . In fact, (3.2) is equivalent to 2
kS0 x23 þ 2ðkS0 2x2 Þx3 þ k < 0
ð3:4Þ 2
2
which is true given that x3 ¼ S10 if kS0 2x2 < 0 and ðkS0 2x2 Þ > ðkS0 Þ . By (3.3), R0 < 1 and x2 > kS0 , then
d þ l > x1 l þ x2 eds > x1 l þ kS0 eds ; dþl
ð3:5Þ 0
we can choose x1 < l ð1 R0 Þ to satisfy (3.5), i.e. (3.3) holds true. Hence from (3.1), V ðxt Þjð1:2Þ is negative definite and is equal to zero if and only if ðSðtÞ; IðtÞ; RðtÞÞ ¼ E0 . By Lemma 2.2, the conclusion is valid. When R0 > 1, we take a transform xðtÞ ¼ SðtÞ S ; yðtÞ ¼ IðtÞ I ; zðtÞ ¼ RðtÞ R , then system (1.2) will become
8 0 x ðtÞ ¼ b dðxðtÞ þ S Þ þ cðzðtÞ þ R Þ > > > > ds > > ke ðxðtÞ þ S Þ FðI Þ þ F 0 ðI Þyðt sÞ þ oðyðt sÞÞ ; > < ds y0 ðtÞ ¼ ke ðxðtÞ þ S Þ FðI Þ þ F 0 ðI Þyðt sÞ þ oðyðt sÞÞ > > > > ðd þ lÞðyðtÞ þ I Þ; > > > : 0 z ðtÞ ¼ lyðtÞ ðd þ cÞzðtÞ;
ð3:6Þ
2
1aI where FðI Þ ¼ 1þIaI2 ; F 0 ðI Þ ¼ ð1þ . aI2 Þ2 The linear part of (3.6) is
8 ds ds 0 0 > < x ðtÞ ¼ ðd þ ke FðI ÞÞxðtÞ ke S F ðI Þyðt sÞ þ czðtÞ; ds ds 0 0 y ðtÞ ¼ ke FðI ÞxðtÞ þ ke S F ðI Þyðt sÞ ðd þ lÞyðtÞ; > : 0 z ðtÞ ¼ lyðtÞ ðd þ cÞzðtÞ:
ð3:7Þ
If ðS ; I ; R Þ ¼ E0 ¼ ðS0 ; 0; 0Þ, then we obtain the following characteristic equation about the equilibrium E0
ds ðk þ dÞðk þ d þ cÞ k þ d þ l ke S0 ¼ 0:
ð3:8Þ
H.-F. Huo, Z.-P. Ma / Commun Nonlinear Sci Numer Simulat 15 (2010) 459–468 ds
463
ds
Set f ðkÞ ¼ k þ d þ l ke S0 . Obviously, f ð0Þ ¼ d þ l ke S0 < 0 by R0 > 1; f ðkÞ ! þ1 as k ! þ1, then there must exist a k0 > 0 such that f ðk0 Þ ¼ 0, i.e. the characteristic Eq. (3.8) has at least one root with positive real part. Therefore, E0 is unstable. This completes the proof of Theorem 3.1. h Remark 3.1. Theorem 3.1 shows that time delay sðs > s Þ does not affect global asymptotic stability of E0 . Theorem 3.2. If R0 ¼ 1 (i.e.
s ¼ s ), then the disease-free equilibrium E0 ¼ ðS0 ; 0; 0Þ of system (1.2) is globally attractive in X.
Proof. Let uðtÞ ¼ ðSðtÞ; IðtÞ; RðtÞÞ be any solution of system (1.2) with the initial function ðu1 ; u2 ; u3 Þ. Denote ut ðhÞ ¼ ðSðt þ hÞ; Iðt þ hÞ; Rðt þ hÞÞ; h 2 ½s; 0. Consider the following Lyapunov functional ds
Vðut Þ ¼ IðtÞ þ ke
SðtÞ
Z
t
ts
IðuÞ 2
1 þ aI ðuÞ
du þ ðd þ lÞ
Calculating derivative of Vðut Þ, we obtain
"
0
SðtÞ
ds
V ðut Þ ¼ ke
1 þ aI2 ðtÞ
ðd þ lÞ
1 1 þ aI2 ðtÞ
aI3 ðtÞ : 1 þ aI2 ðtÞ
# IðtÞ:
Let G ¼ fu 2 C þ : V 0 ðuÞ ¼ 0g and M be the largest set in G which is invariant with respect to system (1.2). Clearly, M is not empty, since ðS0 ; 0; 0Þ 2 M. If R0 ¼ 1, we have
"
0
V ðut Þ ¼ ðd þ lÞ
SðtÞ S0
#
S0 ð1 þ aI2 ðtÞÞ
IðtÞ 6 0:
Hence, G ¼ fu ¼ ðu1 ; u2 ; u3 Þ 2 C þ : u1 ¼ S0 or u2 ¼ 0g. When SðtÞ ¼ S0 , from SðtÞ þ IðtÞ þ RðtÞ S0 we always have IðtÞ ¼ 0. Therefore, we obtain that M ¼ fðS0 ; 0; 0Þg. Thus, it follows from the Lyapunov–LaSalle invariance principle (See Theorem 3.1 in [Chapter 5, [3]]) that limt!þ1 SðtÞ ¼ S0 and limt!þ1 IðtÞ ¼ limt!þ1 RðtÞ ¼ 0. This completes the proof of Theorem 3.2. h 4. Endemic equilibrium In this section, we discuss the local asymptotic stability and the permanence of the endemic equilibrium E of system (1.2). Theorem 4.1. If R0 > 1 (i.e. 0 6 s < s ), then the endemic equilibrium E of system (1.2) is locally asymptotically stable. Proof. We will discuss the local asymptotic stability of the endemic equilibrium E of system (1.2) through discussing the global stability of the equilibrium (0,0,0) of system (3.6). Consider the Lyapunov functional Vðut Þ ¼ V 1 ðut Þ þ V 2 ðut Þ, where ut ¼ ðxt ; yt ; zt Þ,
1 1 1 V 1 ðut Þ ¼ x1 ðxðtÞ þ yðtÞ þ zðtÞÞ2 þ x2 y2 ðtÞ þ x3 z2 ðtÞ; 2 2 2 Z t 1 ds 0 2 y ðsÞds; V 2 ðut Þ ¼ x2 ke S F ðI Þ 2 ts here, xi > 0 ði ¼ 1; 2; 3Þ. Note that
Vðut Þ P
1 1 1 x1 ðxðtÞ þ yðtÞ þ zðtÞÞ2 þ x2 y2 ðtÞ þ x3 z2 ðtÞ; 2 2 2
the time derivative of Vðut Þ along the solution of system (3.6) is ds
V 0 ðut Þ ¼x1 ðxðtÞ þ yðtÞ þ zðtÞÞ½dðxðtÞ þ yðtÞ þ zðtÞÞ þ x2 ke
FðI ÞxðtÞyðtÞ
ds
x2 ðd þ lÞy2 ðtÞ þ x2 ke S F 0 ðI ÞyðtÞyðt sÞ þ x3 zðtÞz0 ðtÞ
1 ds þ x2 ke S F 0 ðI Þ y2 ðtÞ y2 ðt sÞ : 2 It is easy to show that
1 2
x2 keds S F 0 ðI ÞyðtÞyðt sÞ 6 x2 keds S F 0 ðI Þ y2 ðtÞ þ y2 ðt sÞ ; we have ds
V 0 ðut Þ 6 x1 dðx2 ðtÞ þ y2 ðtÞ þ z2 ðtÞÞ ð2x1 d x2 ke
FðI ÞÞxðtÞyðtÞ
ð2x1 d x3 lÞyðtÞzðtÞ 2x1 dxðtÞzðtÞ x3 ðd þ cÞz2 ðtÞ ds 0
þ x2 ½ke
S F ðI Þ ðd þ lÞy2 ðtÞ:
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By using Eq. (2.2), it is easy to obtain that
h
i
x2 keds S F 0 ðI Þ ðd þ lÞ ¼
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
:
Then ds
V 0 ðut Þ 6 x1 dðx2 ðtÞ þ y2 ðtÞ þ z2 ðtÞÞ ð2x1 d x2 ke
FðI ÞÞxðtÞyðtÞ
ð2x1 d x3 lÞyðtÞzðtÞ 2x1 dxðtÞzðtÞ x3 ðd þ cÞz2 ðtÞ
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
y2 ðtÞ:
ð4:1Þ ds
Choose xi > 0 ði ¼ 1; 2; 3Þ satisfying 2x1 d ¼ x2 ke Then (4.1) becomes 0
2
V ðut Þ 6 x1 dx ðtÞ x1 d þ
FðI Þ and 2x1 d ¼ x3 l.
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
! y2 ðtÞ
ðx1 d þ x3 ðd þ cÞÞz2 ðtÞ 2x1 dxðtÞzðtÞ ¼ gðxðtÞ; zðtÞÞ x1 d þ
2x2 dðd þ lÞaI2 ð1 þ aI2 Þ2
! y2 ðtÞ;
ð4:2Þ
2
where gðxðtÞ; zðtÞÞ ¼ x1 dx ðtÞ þ 2x1 dxðtÞzðtÞ þ ðx1 d þ x3 ðd þ cÞÞz2 ðtÞ. Since
4ðx1 dÞ2 4x1 d½x1 d þ x3 ðd þ cÞ ¼ 4x1 dx3 ðd þ cÞ < 0; hence gðxðtÞ; zðtÞÞ is negative. From (4.2) we obtain that V 0 ðut Þ is negative definite which implies the equilibrium (0,0,0) of system (3.6) is globally asymptotically stable by Lemma 2.1. Thus, the equilibrium E of system (1.2) is locally asymptotically stable. This completes the proof of Theorem 4.1. h Finally, we study the permanence of the disease of system (1.2), It is clear that the limit set of system (1.2) is on the plane S þ I þ R ¼ S0 . Thus, we focus on the reduced system
8 ds sÞ > < S0 ðtÞ ¼ b dSðtÞ ke SðtÞIðt þ cðS0 SðtÞ IðtÞÞ; 2 1þaI ðtsÞ
ds > sÞ : I0 ðtÞ ¼ ke SðtÞIðt ðd þ lÞIðtÞ: 2
ð4:3Þ
1þaI ðtsÞ
Consequently, the permanence of the disease of system (4.3) is equivalent to the permanence of the disease of system (1.2). we define
X1 ¼ fðS; IÞ : S P 0; I P 0; S þ I 6 S0 g: Clearly, X1 is positively invariant with respect to system (4.3). We have the following Theorem. Theorem 4.2. If R0 > 1 (i.e. 0 6 s < s ), then the disease of system (4.3) is permanent in X1 (i.e. the disease of system (1.2) is permanent in X). Proof. As the first step, we verify that the boundary planes of R2þ ¼ fðS; IÞ : S P 0; I P 0g repel the positive solution to sys2 2 tem (4.3) uniformly. Let C þ 1 ð½s; 0; Rþ Þ denote the space of continuous functions mapping ½s; 0 into Rþ . We choose
C 1 ¼ fðu1 ; u2 Þ 2 C þ1 ð½s; 0; R2þ Þ : u1 ðhÞ > 0; u2 ðhÞ 0; h 2 ½s; 0g; þ 2 0 2 0 Denote X ¼ C þ 1 ð½s; 0; Rþ Þ, and X ¼ IntC 1 ð½s; 0; Rþ Þ, then C 1 ¼ @X . It is easy to see that system (4.3) possesses constant 0 e solution in C 1 ¼ @X : E 2 C 1 with
e E ¼ fðu1 ; u2 Þ 2 C þ1 ð½s; 0; R2þ Þ : u1 ðhÞ S0 ;
u2 ðhÞ 0; h 2 ½s; 0g:
We verify below that the conditions of Lemma 2.3 are satisfied. By the definition of X 0 and @X 0 and system (4.3), it is easy to see that conditions (i) and (ii) of Lemma 2.3 are satisfied and that X 0 and @X 0 are invariant. Hence (H1) is also satisfied (See Fig. 1. Consider condition (iii) of Lemma 2.3. We have
S0 ðtÞjðu1 ;u2 Þ2C 1 0; thus SðtÞjðu1 ;u2 Þ2C 1 0 for all t P 0. Hence we have
I0 ðtÞjðu1 ;u2 Þ2C1 ¼ ðd þ lÞIðtÞ 6 0;
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tau=0.6
4
4
3.5
3.5
3
3
2.5
2
1.5
1.5
1
1
0.5
0.5
0
5
10
15
20
time t
25
30
35
5
0
40
3.5
3.5
3
3
solution
4
2.5
2
1.5
1.5
1
1
0.5
0.5
5
10
15
20
25
30
35
time t
10
15
20
time t
25
30
35
40
40
S(t) I(t) R(t)
tau=600
2.5
2
0
5
4.5
4
0
0
5
S(t) I(t) R(t)
tau=60
4.5
solution
2.5
2
0
S(t) I(t) R(t)
tau=6
4.5
solution
solution
4.5
5
S(t) I(t) R(t)
0
0
5
10
15
20
time t
25
30
35
40
Fig. 1. When b ¼ 4; d ¼ 1; k ¼ 0:8; a ¼ c ¼ l ¼ 1; s ¼ 0:47; s > s (i.e. R0 < 1), E0 is globally asymptotically stable, the disease dies out. We can see from figure that increase of time delay s ðs > s Þ does not affect global asymptotic stability of E0 .
e@ ¼ e from which follows that all points in C 1 approach e E, i.e. C 1 ¼ W s ð e EÞ. Hence A E and clearly it is isolated. Then the flow in e @ is acyclic, satisfying condition (iii) of Lemma 2.3. A EÞ \ X 0 ¼ /, assume the contrary, i.e. W s ð e EÞ \ X 0 –/. Then there exists a positive solution ðSðtÞ; IðtÞÞ Now we show that W s ð e in X1 to system (4.3) with limt!þ1 ðSðtÞ; IðtÞÞ ¼ ðS0 ; 0Þ, and for sufficiently small positive constant e with (See Fig. 2 ds
e<
ke
S0 ðd þ lÞ
ds
þ aðd þ lÞÞ
2ðke
;
there exists a positive constant T ¼ TðeÞ such that
pffiffiffi
pffiffiffi
e > 0; IðtÞ < e for all t P T:
SðtÞ > S0
By the second equation of (4.3) we have ds
I0 ðtÞ >
ke
ðS0 eÞIðt sÞ
1 þ aI2 ðt sÞ
ðd þ lÞIðtÞ; t P T þ s:
ð4:4Þ
Consider the equation
(
ds
u0 ðtÞ ¼ ke
ðS0 eÞuðtsÞ 1þau2 ðtsÞ
ðd þ lÞuðtÞ;
uðtÞ ¼ IðtÞ; t 2 ½T; T þ s:
ð4:5Þ
By (4.4) and the comparison theorem, we have IðtÞ P uðtÞ for all t P T. On the other hand, using Theorem 4.9.1 of [5], [ p. ffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds eÞðdþlÞ 159], we have limt!þ1 uðtÞ ¼ u for all solutions to system (4.5), where u ¼ ke ðSa0ðdþ > e is the unique positive equilÞ
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H.-F. Huo, Z.-P. Ma / Commun Nonlinear Sci Numer Simulat 15 (2010) 459–468 5
tau=0.47
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
5
10
15
20
25
30
35
0
40
S(t) I(t) R(t)
tau=0.47
4.5
solution
solution
4.5
5
S(t) I(t) R(t)
0
50
100
150
time t, t=0 to 40
200
250
300
350
400
time t, t=0 to 400
Fig. 2. When b ¼ 4; d ¼ 1; k ¼ 0:8; a ¼ c ¼ l ¼ 1; s ¼ s ¼ 0:47 (i.e. R0 ¼ 1), SðtÞ approaches to its steady state value while IðtÞ and RðtÞ approach zero as time goes to infinity, which indicate that E0 is globally attractive, the disease dies out.
5
tau=0.1
4
4
3.5
3.5
3 2.5
3 2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
10
20
30
40
50
60
70
80
90
0
100
0
10
20
30
40
5
tau=0.4
3.5
3
3
solution
solution
3.5
2.5
2
1.5
1.5
1
1
0.5
0.5
20
30
40
50
time t
80
90
100
60
70
80
90
100
S(t) I(t) R(t)
2.5
2
10
70
tau=0.45
4.5 4
0
60
5
S(t) I(t) R(t)
4
0
50
time t
time t
4.5
S(t) I(t) R(t)
tau=0.25
4.5
solution
solution
4.5
5
S(t) I(t) R(t)
0
0
10
20
30
40
50
time t
60
70
80
90
100
Fig. 3. When b ¼ 4; d ¼ 1; k ¼ 0:8; a ¼ c ¼ l ¼ 1; s ¼ 0:47; 0 6 s < s (i.e. R0 > 1), the disease is permanent. We can see from figure that SðtÞ approaches to S0 ¼ 4 while IðtÞ and RðtÞ approach zero as time delay s ð0 6 s < s Þ tends to the finite value s , which means that the disease will always exist until s P s (i.e. R0 6 1).
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H.-F. Huo, Z.-P. Ma / Commun Nonlinear Sci Numer Simulat 15 (2010) 459–468 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
alpha=0.5
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
5
10
15
20
25
30
35
40
0
alpha=10
0
5
10
time t
15
20
25
30
35
40
time t Fig. 4. The dependence of I on the parameter a.
pffiffiffi pffiffiffi librium of system (4.5). Hence we get limt!þ1 IðtÞ P u > e, contradicting IðtÞ < e as t P T. Thus we have W s ð e EÞ \ X 0 ¼ /. Now we get that system (4.3) satisfies all conditions of Lemma 2.3, thus the disease of system (4.3) is permanent in X1 . This completes the proof of Theorem 4.2. h Remark 4.2. Theorem 4.2 shows that time delay (1.2).
s ð0 6 s < s Þ does not affect the permanence of the disease of system
Remark 4.3. Computer observations (See Fig. 3 in Section 5) suggest that the endemic equilibrium is also globally stable. So, we give an interesting open problem: whether we can also obtain that the endemic equilibrium E is globally stable when R0 > 1. 5. Numerical simulation Throughout this paper, we considered a delayed epidemic model with non-monotonic incidence rate and obtained some results. From Theorems 3.1, 3.2 and 4.2 we have the following cases: 1. When R0 < 1 (i.e. s > s ), the disease-free equilibrium of system (1.2) is globally asymptotically stable, the disease will die out. 2. When R0 ¼ 1 (i.e. s ¼ s ), the disease-free equilibrium of system (1.2) is globally attractive, the disease will die out. 3. When R0 > 1 (i.e. 0 6 s < s ), the disease of system (1.2) is permanent, the disease will always exist. In the following, we present some numerical simulation of examples which validate these theoretical results obtained in this paper. 6. Discussions Recall that the parameter a describes the psychological effect of the general public toward the infectives. Though the basic reproduction number R0 does not depend on a explicitly, numerical simulations shown that when the disease is endemic, the steady value I of the infectives decreases as a increases (see Fig. 4). From the expression of I we can see that I approaches zero as a goes to infinity. In [1], Xiao and Ruan also mentioned the above properties of a. References [1] [2] [3] [4] [5] [6] [7]
Xiao D, Ruan S. Global analysis of an epidemic model with nonmonotone incidence rate. Mathematical Biosciences 2007;208:419–29. Capasso V. In: Mathematical Structure of Epidemic Systems. Lecture notes in biomathematics, vol. 97. Berlin: Springer; 1993. Hale JK. Theory of functional differential equations. New York: Springer; 1977. Hethcote HW, Tudor DW. Integral equation models for endemic infectious diseases. J Math Biol 1980;9:37–47. Kuang Y. Delay differential equations with applications in population dynamics. New York: Academic Press; 1993. Hale JK, Waltman P. Persistence in infinite-dimensional systems. SIAM J Math Anal 1989;20:388–95. Zhang T, Teng Z. Global behavior and permanence of SIRS epidemic model with time delay. Nonlin Anal Real World Appl 2007. doi:10.1016/ j.nonrwa.2007.03.010.
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[8] Faina B, Georgy K, Song B, Castillo-Chavez C. A simple epidemic model with surprising dynamics. Math Biosci Eng 2005;2:133–52. [9] Song B, Castillo-Chavez C, Aparicio JP. Tuberculosis models with fast and slow dynamics: the role of close and casual contacts. Math Biosci 2002;180:187–205. [10] Wang H, Li J, Kuang Y. Mathematical modeling and qualitative analysis of insulin therapies. Math Biosci 2007;210:17–33. [11] Zhen J, Ma Z, Han M. Global stability of an SIRS epidemic model with delays. Acta Mathematica Scientia 2006;26B(2):291–306. [12] Zhen J, Ma Z. The stability of an SIR epidemic model with time delays. Math Biosci 2006;3:101–9. [13] Capasso V. In: Mathematical structure of epidemic systems. Lecture notes in biomathematics, vol. 97. Berlin: Springer; 1993. [14] Levin SA, Hallam TG, Gross LJ. Applied mathematical ecology. New York: Springer; 1989. [15] Capasso V, Serio G. A generalization of the Kermack–McKendrick deterministic epidemic model. Math Biosci 1978;42:43. [16] Lin J, Andreasen V, Levin SA. Dynamics of influenza A drift: the linear three-strain model. Math Biosci 1999;162:33. [17] Liu WM, Hethcote HW, Levin SA. Dynamical behavior of epidemiological models with nonlinear incidence rates. J Math Biol 1987;25:359. [18] Yorke JA, London WP. Recurrent outbreaks of measles,chickenpox and mumps II. Am J Epidemiol 1973;98:469.