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An SEI–SI avian–human influenza model with diffusion and nonlocal delay
Applied Mathematics and Computation 247 (2014) 753–761
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An SEI–SI avian–human influenza model with diffusion and nonlocal delay q Qiulin Tang a,b, Jing Ge b, Zhigui Lin b,⇑ a b
School of Science, Nantong University, Nantong 226007, China School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
a r t i c l e
i n f o
Keywords: Reaction–diffusion systems SEI–SI model Avian influenza Nonlocal delay Stability
a b s t r a c t An avian–human influenza epidemic model with diffusion and nonlocal delay is investigated. This model describes the transmission of avian influenza among birds and human; especially the asymptomatic individuals in the latent period have infectious force. By analyzing the corresponding characteristic equation, the local stability of uniform steady state of the bird system is discussed. Sufficient conditions are given for the global asymptotical stability of the disease-free equilibrium of the bird system by using the method of upper–lower solutions and its associated monotone iteration scheme. For the full system, the global stability of disease-free equilibrium is studied by the comparison arguments. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Avian influenza (AI) or ‘‘bird flu’’ is a highly infectious disease of birds. AI viruses are negative single-stranded enveloped RNA viruses that belong to the influenza A genus of the Orthomyxoviridae family [9]. The most virulent form of AI is known as the highly pathogenic avian influenza (HPAI). This is a highly contagious disease of domestic fowl that was first identified as a serious disease of poultry by an Italian scientist, Edoardo Perroncito, in 1878 [2]. Usually, influenza viruses occur among birds. Wild birds worldwide carry the viruses in their intestines, but usually do not get sick from them. However, avian influenza is very contagious among birds and can make some domesticated birds, including chickens, ducks, and turkeys, very sick and kill them [5]. Infected birds shed influenza viruses from their saliva, nasal secretions, etc. When susceptible birds come in contact with the contaminated surfaces, they become infected. Domesticated birds may become infected with avian influenza viruses through direct contact with infected waterfowl or other infected poultry or through contact with surfaces (such as dirt or cages) or materials (such as water or food) that have been contaminated with the virus [18]. The highly pathogenic avian influenza H5N1 virus was first isolated in China in 1996, and the first human case was found in China in 1997. After that, infection to human of avian influenza occurred successively. It is known that already 133 humans have been infected in Asia since late 2003 and 68 killed [5]. Although the avian influenza virus can infect many animal species, it is of major concern for the poultry industry, as the virus can spread rapidly in and between flocks, causing high mortality and severe economic losses [16]. So it is important for us to take effective measure to prevent the pandemic and limit the damage by the pandemic at the minimum. q This work is supported by PRC Grant NSFC (11371311), the Ph.D. Programs Foundation of Ministry of Education of China (No. 20113250110005) and the NSF of Nantong University (10Z009). ⇑ Corresponding author. E-mail address: [email protected] (Z. Lin).
http://dx.doi.org/10.1016/j.amc.2014.09.042 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761
As we know, mathematical models are useful tools for building theories, assessing conjectures, answering questions, and determining sensitivities to parameter values. We can use mathematical models to compare, plan, implement, and evaluate various detection, prevention, and control programs [3]. In recent years, many epidemic models have been proposed in the study of transmission and control of avian or avian–human influenza (see, for example, [1,4–6,8,14,15,20,21]). In view of the fact that the spatial diffusion and environmental heterogeneity are important factors in modeling the spread of many diseases, in this paper, we propose an avian–human epidemic model which was described by the following reaction–diffusion equations:
8 @S1 > D1 DS1 ¼ A1 d1 S1 b1 S1 E1 b2 S1 I1 ; > @t > > > @E1 > > D2 DE1 ¼ b1 S1 E1 þ b2 S1 I1 ðd1 þ eÞE1 ; > < @t @I1 D3 DI1 ¼ eE1 ða1 þ d1 ÞI1 ; @t > R Rt > > @S2 > D4 DS2 ¼ A2 d2 S2 þ cI2 bS2 X 1 Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy; > @t > > R Rt > @I2 : D5 DI2 ¼ bS2 X 1 Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy ða2 þ d2 þ cÞI2 ; @t
ð1:1Þ
for x 2 X; t > 0 with homogeneous Neumann boundary conditions
@S1 @E1 @I1 @S2 @I2 ¼ ¼ ¼ ¼ ¼ 0; @g @g @g @g @g
x 2 @ X;
t>0
ð1:2Þ
and initial conditions
S1 ðx; 0Þ ¼ /1 ðxÞ P 0; S2 ðx; 0Þ ¼ /4 ðxÞ P 0;
E1 ðx; tÞ ¼ /2 ðx; tÞ P 0; I1 ðx; tÞ ¼ /3 ðx; tÞ P 0; I2 ðx; 0Þ ¼ /4 ðxÞ P 0; x 2 X;
ð1:3Þ
P @2 n where D ¼ ni¼1 @x 2 , is the Laplace operator, X is a bounded smooth domain in R ; @ X and X are the boundary and the closure i of X, respectively, g is the outward unit normal vector on @ X. The positive constants Di ði ¼ 1; . . . ; 5Þ are the coefficients of i € lder continuous and satisfy @/ diffusion. The initial functions /i are nonnegative, Ho ¼ 0 on the boundary. @g The first three equations of (1.1) form an SEI model describing the interactions among the birds. S1 ðx; tÞ; E1 ðx; tÞ; I1 ðx; tÞ represent respectively the number of susceptible, exposed and infected birds with avian influenza at position x at time t. These three equations are named as the bird system. It is assumed that the individuals in the latent period have infectious force. In fact, it is observed that poultry without any symptom can excrete much high pathogenic virus, which makes it more difficult to inhibit the H5N1 type virus from spreading. The parameter A1 is the recruitment rate of susceptible population; The natural death rate is assumed to be constant uniformly for each class (i.e. d1 ); the transmission coefficients of latent and infective are b1 and b2 , respectively; e is the rate constant at which the exposed individuals become infective; a1 is extra disease-related death rate constants of the infective. The last two equations of (1.1) form an SI model for humans, which consist of two kinds of individuals: susceptible ðS2 Þ humans and infected ðI2 Þ humans with avian influenza. The parameter A2 is the rate at which new humans are born. d2 is the natural death rate for susceptible and infected human. c is the recovery rate, and a2 is the disease-related death rate. b is the rate at which avian influenza is contracted from an average human individual infective. Thus bS2 E1 and bS2 I1 are the force of infection by exposed and infected birds with avian influenza, respectively. The term
Z Z X
t
Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy;
1
accounts for the infection of individuals to their present position at time t caused by the exposed and infected individuals from all possible positions at all previous times [17,19]. The kernel Kðx; y; tÞ depends on both the spatial and the temporal variables. Here we further assumed that
Kðx; y; tÞ ¼ Gðx; y; tÞkðtÞ P 0;
x; y 2 X;
t > 0;
where the function kðtÞ is called the delay kernel and used to weight the distributed time-delay, and satisfies
kðtÞ P 0;
8t P 0; tkðtÞ 2 L1 ðð0; þ1Þ; RÞ;
Z
1
kðtÞdt ¼ 1;
0
G is the weighting function describing the distribution at past times of the exposed individual E1 and the infected individual I1 at position x and time t, and satisfies
Z X
Gðx; y; tÞdx ¼
Z
Gðx; y; tÞdy ¼ 1:
X
For example, Gðx; y; tÞ is Green’s function of the operator condition, and kðtÞ ¼ 1s et=s with a constant s.
@ @t
D2 D or
@ @t
D3 D subject to homogeneous Neumann boundary
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The remaining of this paper is organized as follows. In the next section, uniform bounds for the solutions of (1.1)–(1.3) are given. In Section 3, local and global stabilities of the equilibria for the bird system are discussed. Section 4 deals with global asymptotical behaviors of solutions to the full system (1.1)–(1.3). A brief conclusion is made in Section 5. 2. Uniform bound Noting that the initial values are nonnegative and the growth functions in the right hand of (1.1) are assumed to be sufficiently smooth in R5þ , the solution of (1.1) is unique and continuous for all positive time in X by the standard PDE theory. Next we show that the solution of (1.1) is positive and uniformly bounded. We first give the following lemma which is so-called the positivity lemma [11]. T
C 2;1 ðX ð0; TÞÞ and satisfies 8 5 X > > @u > bij uj ðx; tÞ; x 2 X; 0 < t 6 T; > @ti Di Dui P > < j¼1
Lemma 2.1. Let ui 2 CðX ½0; TÞ
@ui > P 0; > > @g > > : ui ðx; 0Þ P 0;
x 2 @ X;
0 < t 6 T;
ð2:1Þ
x 2 X;
where T > 0; bij ðx; tÞ 2 CðX ½0; TÞ. If bij P 0 for j – i, then ui ðx; tÞ P 0 on X ½0; T. Moreover, ui ðx; tÞ > 0 or ui 0 in X ð0; T. As the consequence of Lemma 2.1, we have the following positivity result. Lemma 2.2. Any solution of problem (1.1)–(1.3) with nonnegative nontrivial initial value is positive. By means of Gagliardo–Nirenberg inequality and Moser iteration technique, we can prove the following lemma. Lemma 2.3. Let uðx; tÞ satisfies
8 @u > < @t DDu ¼ gðu; x; tÞ; x 2 X; 0 < t < 1; u @@ug 6 0; x 2 @ X; 0 < t < 1; > : x 2 X; uðx; 0Þ ¼ u0 ðxÞ;
ð2:2Þ
where D > 0; jgðu; x; tÞj 6 Ljuj. If there exists a p with 1 6 p < 1 such that kuðx; tÞkLp ðXÞ is uniformly bounded for t P 0, then kuðx; tÞkLq ðXÞ is uniformly bounded for t P 0, where q ¼ p 2N ; N ¼ 1; 2; . . .. In particular, kuðx; tÞkL1 ðXÞ is uniformly bounded for t P 0. The following theorem shows that the solution of (1.1)–(1.3) is uniformly bounded. T 5 Theorem 2.4. Let ðS1 ; E1 ; I1 ; S2 ; I2 Þ 2 ½CðX ½0; TÞÞ C 2;1 ðX ð0; TÞÞ be the solution of problem (1.1)–(1.3) with nonnegative nontrivial initial value. Then T ¼ 1 and there exist M 1 and M 2 such that
0 6 S1 ;
E1 ;
I1 6 M 1 ;
0 6 S2 ;
I2 6 M 2 ;
ðx; tÞ 2 X ð0; 1Þ;
ðx; tÞ 2 X ð0; 1Þ:
Proof. Clearly, we have
@ðS1 þ E1 þ I1 Þ DðD1 S1 þ D2 E1 þ D3 I1 Þ ¼ A1 d1 ðS1 þ E1 þ I1 Þ a1 I 6 A1 d1 ðS1 þ E1 þ I1 Þ: @t
ð2:3Þ
Integrating (2.3) over X yields
d dt Let M ¼
Z
ðS1 þ E1 þ I1 Þdx 6 A1 jXj d1
Z
X
ðS1 þ E1 þ I1 Þdx:
X
R P3 i¼1 /i ðxÞÞdx. Then Xð
A1 jXj kS1 þ E1 þ I1 kL1 6 max M; d1
ð2:4Þ
by comparison principle. From Lemma 2.3, we obtain the uniform bounds of S1 ; E1 and I1 . The uniform bounds of S2 and I2 can be proved in a similar way. h
3. The bird system Since the bird system is independent of the human system, we first study the bird system:
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8 @S1 > > < @t D1 DS1 ¼ A1 d1 S1 b1 S1 E1 b2 S1 I1 ; @E1 D2 DE1 ¼ b1 S1 E1 þ b2 S1 I1 ðd1 þ eÞE1 ; @t > > : @I1 D DI ¼ eE ða þ d ÞI : 3
@t
1
1
1
1
ð3:1Þ
1
It is clear that the system (3.1) always has the disease-free equilibrium P0 ¼ ðA1 =d1 ; 0; 0Þ which represents that no infected bird with avian influenza exists. Further, if the following conditions are satisfied
R0 ,
A1 b1 A1 eb2 > 1; þ d1 ðd1 þ eÞ d1 ðd1 þ eÞða1 þ d1 Þ
then the system also has a unique endemic equilibrium P ¼ ðS1 ; E1 ; I1 Þ, where
S1 ¼
ðd1 þ eÞða1 þ d1 Þ ; ða1 þ d1 Þb1 þ eb2
E1 ¼
A1 1 ð1 Þ; R0 d1 þ e
I1 ¼
e E : a1 þ d1 1
R0 is called the basic reproduction number of (3.1). In the below, we analyze the local stability at P 0 and P . We first set up the following notations, similarly as in [10,13]. Notation 3.1 (i) 0 ¼ l1 < l2 < l3 < . . . ! 1 are the eigenvalues of D on X under homogeneous Neumann boundary condition; (ii) Eðli Þ is the space of eigenfunction corresponding to li ; (iii) Xij ¼ fc /ij jc 2 R3 }, where f/ij g is an orthonormal basis of Eðli Þ for j ¼ 1; 2; . . . ; dim Eðli Þ; 3
Þ j (iv) X :¼ fu ¼ ðS1 ; E1 ; I1 Þ 2 ½C 1 ðX
@S1 @g
dim Eðli Þ
1 ¼ @E ¼ @I@g1 ¼ 0on @ Xg, and so X ¼ 1 i¼1 X i , where X i ¼ j¼1 @g
X ij .
^ Let D ¼ diagðD1 ; D2 ; D3 Þ; Z ¼ ðS1 ; E1 ; I1 Þ; LZ ¼ DDZ þ GðPÞZ, where
0 ^ ¼B GðPÞ @
d1 b1 E01 b2 I01
b1 S01
b2 S01
b1 E01 þ b2 I01
b1 S01 ðd1 þ eÞ
b2 S01
0
e
ðd1 þ a1 Þ
1 C A
^ ¼ ðS0 ; E0 ; I0 Þ represents any feasible uniform steady state of system (3.1). The linearization of system (3.1) at P ^ is of the and P 1 1 1 form Z t ¼ LZ. For each i P 1; X i is invariant under the operator L, and k is an eigenvalue of L if and only if it is an eigenvalue ^ of the matrix li D þ GðPÞ. The characteristic equation of li D þ GðP0 Þ takes the form
A1 þ d1 þ e k d1 A1 A1 ¼ 0: þ D2 li b1 þ d1 þ e D3 li þ a1 þ d1 eb2 d1 d1
ui ðkÞ , ðk þ li D1 þ d1 Þ k2 þ ðD2 þ D3 Þli þ ða1 þ d1 Þ b1
ð3:2Þ
Clearly, for any i P 1, Eq. (3.2) always have a negative real root: ðli D1 þ d1 Þ. Its other roots are determined by the following equation
A1 A1 A1 ¼ 0: k2 þ ðD2 þ D3 Þli þ ða1 þ d1 Þ b1 þ d1 þ e k þ D2 li b1 þ d1 þ e ðD3 li þ a1 þ d1 Þ eb2 d1 d1 d1
ð3:3Þ
If R0 < 1, then
ðD2 þ D3 Þli þ ða1 þ d1 Þ b1
A1 þ d1 þ e > 0; d1
A1 A1 D2 li b1 þ d1 þ e ðD3 li þ a1 þ d1 Þ eb2 > 0: d1 d1 So the two roots of Eq. (3.3) have negative real parts. Moreover, following the standard technique as in [10], we can obtain that the spectrum of L lies in fRek < dg for some positive d. Therefore P0 is locally asymptotically stable if R0 < 1. When R0 > 1, we take i ¼ 1, then l1 ¼ 0 and
A1 A1 A1 A1 D2 li b1 þ d1 þ e ðD3 li þ a1 þ d1 Þ eb2 ¼ b1 þ d1 þ e ða1 þ d1 Þ eb2 d1 d1 d1 d1
¼ ðd1 þ eÞða1 þ d1 Þ ½b1 ða1 þ d1 Þ þ eb2
A1 < 0: d1
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Hence, (3.3) has a positive root, that is to say, there is a characteristic root k with a positive real part in the spectrum of L. It implies that the disease-free equilibrium P 0 is unstable if R0 > 1. As for the equilibrium P , the coefficients Ai ; Bi and C i of the characteristic equation ui ðkÞ , jkI þ li GðP Þj ¼ k3 þ Ai k2 þ Bi k þ C i ¼ 0 are defined as follows:
Ai ¼ ðd1 þ D1 li Þ þ ðd1 þ e b1 S1 þ D2 li Þ þ ða1 þ d1 þ D3 li Þ þ b1 E1 þ b2 I1 ;
Bi ¼ ða1 þ d1 þ D3 li Þ 2d1 þ e b1 S1 þ b1 E1 þ b2 I1 þ ðD1 þ D2 Þli
þ ðd1 þ D1 li Þðd1 þ e b1 S1 þ D2 li Þ þ ðd1 þ e þ D2 li Þðb1 E1 þ b2 I1 Þ eb2 S1 ;
C i ¼ ða1 þ d1 þ D3 li Þ ðd1 þ D1 li Þðb1 S1 þ d1 þ e þ D2 li Þ þ ðd1 þ e þ D2 li Þðb1 E1 þ b2 I1 Þ ðd1 þ D1 li Þeb2 S1 : e E , we obtain Noting that ðb1 E1 þ b2 I1 ÞS1 ðd1 þ eÞE1 ¼ 0 and I1 ¼ a1 þd 1 1
b1 S1 ¼ ðd1 þ eÞ b2
I1 e S ¼ ðd1 þ eÞ b2 S : E1 1 a1 þ d1 1
Thus
d1 þ e b1 S1 ¼ b2
e S > 0; a1 þ d1 1
which in turn leads to Ai > 0; Bi > 0 and C i > 0 for i P 1. On the other hand, it is easy to see that
Hi , Ai Bi C i > 0: Therefore, according to the Routh–Hurwitz criterion, the three roots ki;1 ; ki;2 ; ki;3 of ui ðkÞ ¼ 0 all have negative real parts. Furthermore, the similar analysis as in [10] yields that there exists a positive constant e such that
Refki;1 g;
Refki;2 g;
Refki;3 g 6 e;
i P 1:
In summary, we have proved the following theorem. Theorem 3.1. For the bird system ð3:1Þ, if R0 < 1, then the disease-free equilibrium P 0 ðAd11 ; 0; 0Þ is locally asymptotically stable while no endemic equilibrium exists. if R0 > 1, then the endemic equilibrium P ðS1 ; E1 ; I1 Þ exists and is locally asymptotically stable while the disease-free equilibrium P0 is unstable. Next we consider the global stability of the disease-free equilibrium P0 ¼ ðAd11 ; 0; 0Þ, when R0 < 1. We use the method of upper and lower solutions, similar to the argument in [7]. We first investigate the asymptotic behavior of the solution of (3.1) in relation to its corresponding constant steady-state solution ðc1 ; c2 ; c3 Þ of (3.1) given by
8 A1 d1 c1 b1 c1 c2 b2 c1 c3 ¼ 0; > > < b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ¼ 0; > > : ec2 ða1 þ d1 Þc3 ¼ 0:
ð3:4Þ
Now, we deal with the algebraic equations in (3.4) as the following elliptic system:
8 > < Dc1 ¼ A1 d1 c1 b1 c1 c2 b2 c1 c3 ; Dc2 ¼ b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ; > : Dc3 ¼ ec2 ða1 þ d1 Þc3 :
ð3:5Þ
c ¼ ð~c1 ; ~c2 ; ~c3 Þ and ^ c ¼ ð^c1 ; ^c2 ; ^c3 Þ a pair of upper and lower solutions of (3.5) if ~ cP^ c P ð0; 0; 0Þ and We denote by ~
8 A1 d1 ~c1 b1 ~c1 ^c2 b2 ~c1 ^c3 6 0 6 A1 d1 ^c1 b1 ^c1 ~c2 b2 ^c1 ~c3 ; > > < b1 ~c1 ~c2 þ b2 ~c1 ~c3 ðd1 þ eÞ~c2 6 0 6 b1 ^c1 ^c2 þ b2 ^c1 ^c3 ðd1 þ eÞ^c2 ; > > : e~c2 ða1 þ d1 Þ~c3 6 0 6 e^c2 ða1 þ d1 Þ^c3 :
ð3:6Þ
Every pair of upper and lower solutions of (3.5) is also a pair of upper and lower solutions of (3.1) whenever ^ci 6 /i 6 ~ci in X for i ¼ 1; 2; 3. Obviously, we have the Lipschitz condition
j½A1 d1 S1 b1 S1 E1 b2 S1 I1 ½A1 d1 S01 b1 S01 E01 b2 S01 I01 j 6 K 0 ðjS1 S01 j þ jE1 E01 j þ jI1 I01 jÞ; j½b1 S1 E1 þ b2 S1 I1 ðd1 þ eÞE1 ½b1 S01 E01 þ b2 S01 I01 ðd1 þ eÞE01 j 6 K 0 ðjS1 S01 j þ jE1 E01 j þ jI1 I01 jÞ; j½eE1 ða1 þ d1 ÞI1 ½eE01 ða1 þ d1 ÞI01 j 6 K 0 ðjS1 S01 j þ jE1 E01 j þ jI1 I01 jÞ ðmÞ ðmÞ for ^ci 6 S1 ; E1 ; I1 ; S01 ; E01 ; I01 6 ~ci , where K 0 ¼ K 0 ðA1 ; d1 ; b1 ; b2 ; a1 ; e; ~ci ; ^ci Þ. Now we construct two sequences fcðmÞ g fc1 ; c2 ; ðmÞ
ðmÞ
ðmÞ
ðmÞ
c3 g; fcðmÞ g fc1 ; c2 ; c3 g from the iteration process
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8 ðmÞ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ c1 ¼ c1 þ K10 ½A1 d1 c1 b1 c1 c2 b2 c1 c3 ; > > > > > ðmÞ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ 1 > > c2 ¼ c2 þ K 0 ½b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ; > > > > > < cðmÞ ¼ cðm1Þ þ 1 ½ecðm1Þ ða1 þ d1 Þcðm1Þ ; 3 3 2 3 K0 ðmÞ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ > > c1 ¼ c1 þ K10 ½A1 d1 c1 b1 c1 c2 b2 c1 c3 ; > > > > > cðmÞ ¼ cðm1Þ þ 1 ½b cðm1Þ cðm1Þ þ b cðm1Þ cðm1Þ ðd þ eÞcðm1Þ ; > 1 > 1 1 2 1 2 2 2 3 2 K0 > > > : ðmÞ ðm1Þ ðm1Þ ðm1Þ 1 c3 ¼ c3 þ K 0 ½ec2 ða1 þ d1 Þc3 ;
ð3:7Þ
^, respectively, where K 0 is the above Lipschitz constant. It is easy to see that the c and cð0Þ ¼ c with initial iteration cð0Þ ¼ ~ ðmÞ ðmÞ sequences fc g; fc g posses the monotone property
^c 6 cðmÞ 6 cðmþ1Þ 6 cðmþ1Þ 6 cðmÞ 6 ~c;
m ¼ 1; 2; . . .
and the limits
c ¼ lim cðmÞ ; m!1
c ¼ lim cðmÞ
ð3:8Þ
m!1
exist and satisfy the equations
8 > < A1 d1 c1 b1 c1 c2 b2 c1 c3 ¼ 0 ¼ A1 d1 c1 b1 c1 c2 b2 c1 c3 ; b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ¼ 0 ¼ b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ; > : ec2 ða1 þ d1 Þc3 ¼ 0 ¼ ec2 ða1 þ d1 Þc3 :
ð3:9Þ
c; ~ c >. In general, the constant vectors c and c are The constant vectors c and c are called quasisolutions of problem (3.4) in < ^ c; ~ c >. Now we present the folnot true solutions of (3.4). However, if c ¼ c, then c (or c) is the unique solution of (3.4) in < ^ lowing relation between the solution of (3.1) and that of (3.4). c and ^ c be a pair of upper and lower solutions of (3.4) or (3.1). Then the Theorem 3.2 ([12], Theorems 2.1 and 2.2). Let ~ ðmÞ ðmÞ sequences fc g; fc g given by (3.7) converge monotonically to their respective limits c and c, which are the quasisolutions of (3.1) and satisfy (3.9). For any initial function satisfying ^ci 6 /i ðxÞ 6 ~ci in X for i ¼ 1; 2; 3, the corresponding solution u ¼ ðS1 ; E1 ; I1 Þ of (3.1) possesses the property
c 6 lim inf uðx; tÞ 6 lim supuðx; tÞ 6 c in X: t!1
t!1
Moreover, if c ¼ c, then c (or c) is the unique solution of (3.4) in < ^ c; ~ c > and the solution u ¼ ðS1 ; E1 ; I1 Þ of problem (3.1) converges to c as t ! 1 uniformly on X. Next we show that the time-dependent solution of problem (3.1) converges to ðA1 =d1 ; 0; 0Þ as t ! 1 if R0 < 1. Let ðS1 ðx; tÞ; E1 ðx; tÞ; I1 ðx; tÞÞ be a solution of (3.1). In order to apply Theorem 3.2, we first construct the upper and lower solutions ~ c. We define c and ^
~c1 ¼
A1 þ e1 ; d1
~c2 ¼ e2 ;
~c3 ¼ e3 ;
^c1 ¼ d1 ;
^c2 ¼ 0;
^c3 ¼ 0;
where
0 < e1
h i 8 9 <ða1 þ d1 Þðe þ d1 Þ eb2 Ad1 þ ða1 þ d1 Þb1 Ad1 d ðd þ eÞ A b = 1 1 1 1 1 1 < min ; ; : ; d1 b1 ða1 þ d1 Þb1 þ eb2
e a1 þ d1
e2 < e3 <
0 < d1 <
ðd1 þ eÞ b1 ðAd11 þ e1 Þ b2 ðAd11 þ e1 Þ
e2 ;
A1 : d1 þ b1 e2 þ b2 e3
Since S1 ðx; tÞ 6 S1 ðtÞ, where S1 ðtÞ is the solution of problem
dS1 ¼ A1 d1 S1 ; dt
S1 ð0Þ ¼ jj/ ðxÞjj ; 1 1
and limt!1 S1 ðtÞ ¼ Ad11 , we deduce that there exists a T 0 > 0 such that S1 6 Ad11 þ e1 for t P T 0 . Thus ~ c and ^ c are coupled upper and lower solutions of (3.5) in ½T 0 ; þ1Þ. Noticing that c1 P c1 P ^c1 ¼ d1 > 0; c2 ¼ 0 and c3 ¼ 0, (3.9) is reduced to
Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761
8 A1 d1 c1 ¼ 0; > > > < A d c b c c b c c ¼ 0; 1 1 1 1 1 2 2 1 3 > b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ¼ 0; > > : ec2 ða1 þ d1 Þc3 ¼ 0:
759
ð3:10Þ
It follows from (3.10) that
A1 A1 e c2 b1 þ b2 ðd1 þ eÞ ¼ 0: d1 d1 a1 þ d1 If we assume c2 – 0, then b1 Ad11 þ b2 Ad11
e a1 þd1
ðd1 þ eÞ ¼ 0. This leads to a contradiction to R0 < 1 and thus c2 ¼ 0. We then
have c3 ¼ 0 from the fourth equation of (3.10) and further, c1 ¼ c1 ¼ A1 =d1 from the first two equations of (3.10). Therefore ci ¼ ci for i ¼ 1; 2; 3. Applying Theorem 3.2 we have limt!1 ðS1 ðx; tÞ; E1 ðx; tÞ; I1 ðx; tÞÞ ¼ ðAd11 ; 0; 0Þ. The disease-free equilibrium is globally asymptotically stable if R0 < 1. Theorem 3.3. Assume that R0 < 1 holds. Then the disease-free equilibrium P 0 of problem ð3:1Þ is globally asymptotically stable. 4. The full system ~ 0 ðA1 ; 0; 0; A2 ; 0Þ which represents that no infected bird with The full system (1.1) always has the disease-free equilibrium P d1 d2 avian influenza exists and no infected human exists. If R0 > 1, then the full system (1.1) has the full-endemic equilibrium ~ ¼ ðS ; E ; I ; S ; I Þ, where S ; E ; I are defined above and P 1 2 2 1 1 1 1 1
A2 ða2 þ d2 ÞI2 ; d2
S2 ¼
I2 ¼
bða2 þ
bA2 ðI1 þ E1 Þ ; þ E1 Þ þ d2 ða2 þ d2 þ cÞ
d2 ÞðI1
which implies that birds and humans are infected with avian influenza. ~ 0 and P ~ which are As above, using linearization method and spectral analysis, we can obtain the local stability of P described as the following theorem. ~ 0 of the full system ð1:1Þ is locally asymptotically stable. If R0 > 1, the Theorem 4.1. If R0 < 1, the disease-free equilibrium P ~ is locally asymptotically stable. endemic equilibrium P ~ 0 ðA1 ; 0; 0; A2 ; 0Þ. By applying the comparison principle for the parabolic equation, Next we investigate the global stability of P d1 d2 we can easily prove the following lemma. ½0; 1ÞÞ T C 2;1 ðX ð0; 1ÞÞ be a nonnegative nontrivial solution of the scalar problem Lemma 4.2. Let w 2 CðX
8 @w > < @t DDw ¼ f ðx; tÞ Bwðx; tÞ; x 2 X; 0 < t < 1; @w ¼ 0; x 2 @ X; 0 < t < 1; @g > : wðx; 0Þ P 0; x2X
ð4:1Þ
if f ðx; tÞ ! A as t ! 1 uniformly where B > 0 and f ðx; tÞ is nonnegative continuous function. Then w ! AB as t ! 1 uniformly on X . on X The following lemma implies that the nonlocal integral term does not affect the long time behavior of solution. Lemma 4.3. If wðx; tÞ is a bounded function and limt!1 jjwðx; tÞ Ajj1 ¼ 0, then
Z Z X
t
Kðx; y; t sÞwðs; yÞdsdy ! A as t ! þ1
1
. uniformly on X ~ 0 of the full system ð1:1Þ is globally asymptotically stable. Theorem 4.4. If R0 < 1, then the disease-free equilibrium P Proof. When R0 < 1, it follows from Theorem 3.3 that the disease-free equilibrium of bird system P0 is globally stable. Thus we have limt!1 jjS1 ðx; tÞ A1 =d1 jj1 ¼ 0; limt!1 jjE1 ðx; tÞ 0jj1 ¼ 0 and limt!1 jjI1 ðx; tÞ 0jj1 ¼ 0. Using Lemma 4.3 yields
Z Z X
t
Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy ! 0 as t ! þ1
1
. By Lemma 4.1, we obtain uniformly on X
limjjI2 ðx; tÞ 0jj1 ¼ 0:
t!1
760
Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761
For the fourth equation of system (1.1), using again the Lemma 4.1 gives
limjjS2 ðx; tÞ A2 =d2 jj1 ¼ 0:
t!1
~ 0 is globally asymptotically stable. Therefore, P
h
Remark 4.1. For model ð1:1Þ, the basic reproduction number R0 ¼ 1 is the threshold, which determines when an infection can invade and persist in a new host population. Remark 4.2. Theorem 4.4 shows that the disease-free equilibrium of ð1:1Þ is globally asymptotically stable if the contact rate is small. Remark 4.3. The local stabilities of the full system in Theorem 4.1 can be obtained by using linearization method and spectral analysis. Here we show that Theorem 4.1 is also a consequence of Theorem 3.1, Lemmas 4.1 and 4.3. In fact, if R0 > 1 and the initial value near the full-endemic equilibrium ðS1 ; E1 ; I1 ; S2 ; I2 Þ, the local stability in Theorem 3.1 shows that limt!1 jjS1 ðx; tÞ S1 jj1 ¼ 0, limt!1 jjE1 ðx; tÞ E1 jj1 ¼ 0 and limt!1 jjI1 ðx; tÞ I1 jj1 ¼ 0 since the bird system is independent of the human system. Similarly as in the proof of Theorem 4.4, we have limt!1 jjS2 ðx; tÞ S2 jj1 ¼ 0 and limt!1 jjI2 ðx; tÞ I2 jj1 ¼ 0 by using Lemmas 4.1 and 4.3. 5. Conclusion The World Health Organization (WHO) has warned of a risk of pandemic of avian influenza and pointed out that outbreaks of avian influenza in poultry may raise global public health concerns in the future. Recently mathematical models have been studied to describe the transmission of avian influenza among birds and humans. For example, a SI-SIR model was proposed in [5], that is, two equations with susceptible (X) and infected (Y) birds are SI model describing the interactions among the birds, and four equations are SIR model for humans, which consist of four kinds of individuals: susceptible (S) humans, infected (B) humans with avian influenza, infected (H) humans with mutant avian influenza and recovered (R) ones from mutant avian influenza. The governing system is
8 _ > > > XðtÞ ¼ c bX xXY; > > _ > > > YðtÞ ¼ xXY ðb þ mÞY; > > < SðtÞ _ ¼ k lS ðb1 Y þ b2 HÞS; _ > BðtÞ ¼ b1 SY ðl þ d þ eÞB; > > > > > _ > HðtÞ ¼ b2 SH þ eB ðl þ a þ cÞH; > > > :_ RðtÞ ¼ cH lR:
ð5:1Þ
It was later extended to a diffusive epidemic model in [8]. In this paper, a similar reaction–diffusion model (1.1) is introduced to interpret the spreading of wild avian influenza from the bird world to the human world. A striking difference between (5.1) and (1.1) is that in (1.1), the asymptomatic bird individuals in the latent period have infectious force and the bird system is describing by SEI model. We first discuss the positivity and the uniform bounds of solutions to system (1.1), which are crucial for the investigation of stability of system (1.1). We then introduce a threshold value R0 and obtain that the avian influenza will be permanent when R0 > 1 and the avian influenza will be extinct when R0 < 1. It shows that eliminate and isolate the infected birds with wild avian influenza ~ 0 ðA1 ; 0; 0; A2 ; 0Þ of is useful to epidemic control. By comparison arguments, we also obtain that the disease-free equilibrium P d1
d2
the full system (1.1) is locally asymptotically stable when R0 < 1. It indicates that the most effective way to control disease spreads among human is to avoid contacting with the infected birds and any surfaces that appear to be contaminated with feces from poultry or other animals. References [1] L. Bourouiba, A. Teslya, J. Wu, Highly pathogenic avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling, J. Theor. Biol. 271 (2011) 181–201. [2] I. Capua, F. Mutinelli, A Colour Atlas and Text on Avian Influenza, Papi Editore, Bologna, Italy, 2001. [3] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000) 599–653. [4] S. Iwamia, Y. Takeuchia, A. Korobeinikovb, X. Liu, Prevention of avian influenza epidemic: what policy should we choose?, J Theor. Biol. 252 (2008) 732–741. [5] S. Iwami, Y. Takeuchi, X. Liu, Avian–human influenza epidemic model, Math. Biosci. 207 (2007) 1–25. [6] A. Kaddar, A. Abta, H.T. Alaoui, A comparison of delayed SIR and SEIR epidemic models, Nonlinear Anal. Model. Control 16 (2011) 181–190. [7] K.I. Kim, Z.G. Lin, Asymptotic behavior of an SEI epidemic model with diffusion, Math. Comput. Model. 47 (2008) 1314–1322. [8] K.I. Kim, Z.G. Lin, L. Zhang, Avian–human influenza epidemic model with diffusion, Nonlinear Anal. Real Word Appl. 11 (2010) 313–322. [9] H.K. Leong, C.S. Goh, S.T. Chew, et al, Prevention and control of avian influenza in Singapore, Ann. Acad. Med. Singap. 37 (2008) 504–509.
Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761 [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
761
Z.G. Lin, M. Pedersen, Stability in a diffusive food-chain model with Michaelis–Menten functional response, Nonlinear Anal. 57 (2004) 421–433. C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. C.V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. 48 (2002) 349–362. P.Y.H. Pang, M.X. Wang, Strategy and stationary pattern in a three-species predator–prey model, J. Differ. Equ. 200 (2004) 245–273. X. Qiang, Z. Kou, Prediction of interspecies transmission for avian influenza A virus based on a back-propagation neural network, Math. Comput. Model. 52 (2010) 2060–2065. G.P. Samanta, Permanence and extinction for a nonautonomous avian–human influenza epidemic model with distributed time delay, Math. Comput. Model. 52 (2010) 1794–1811. D. Spekreijse, A. Bouma, J.A. Stegeman, et al, The effect of inoculation dose of a highly pathogenic avian influenza virus strain H5N1 on the infectiousness of chickens, Vet. Microbiol. 147 (2011) 59–66. H.R. Thieme, X. Zhao, A non-local delayed and diffusive predator–prey model, Nonlinear Anal. Real Word Appl. 2 (2001) 145–160. R.K. Upadhyay, N. Kumaria, V.S.H. Rao, Modeling the spread of bird flu and predicting outbreak diversity, Nonlinear Anal. Real Word Appl. 9 (2008) 1638–1648. Z. Wang, W. Li, S. Ruan, Travelling wave fronts of reaction–diffusion systems with spatio–temporal delays, J. Differ. Equ. 222 (2006) 185–232. R. Xu, Global dynamics of an SEIS epidemic model with saturation incidence and latent period, Appl. Math. Comput. 218 (2012) 7927–7938. X. Ye, X. Li, Simple models for avian influenza, Rocky Mountain J. Math. 38 (2008) 1813–1828.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An SEI–SI avian–human influenza model with diffusion and nonlocal delay q Qiulin Tang a,b, Jing Ge b, Zhigui Lin b,⇑ a b
School of Science, Nantong University, Nantong 226007, China School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
a r t i c l e
i n f o
Keywords: Reaction–diffusion systems SEI–SI model Avian influenza Nonlocal delay Stability
a b s t r a c t An avian–human influenza epidemic model with diffusion and nonlocal delay is investigated. This model describes the transmission of avian influenza among birds and human; especially the asymptomatic individuals in the latent period have infectious force. By analyzing the corresponding characteristic equation, the local stability of uniform steady state of the bird system is discussed. Sufficient conditions are given for the global asymptotical stability of the disease-free equilibrium of the bird system by using the method of upper–lower solutions and its associated monotone iteration scheme. For the full system, the global stability of disease-free equilibrium is studied by the comparison arguments. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Avian influenza (AI) or ‘‘bird flu’’ is a highly infectious disease of birds. AI viruses are negative single-stranded enveloped RNA viruses that belong to the influenza A genus of the Orthomyxoviridae family [9]. The most virulent form of AI is known as the highly pathogenic avian influenza (HPAI). This is a highly contagious disease of domestic fowl that was first identified as a serious disease of poultry by an Italian scientist, Edoardo Perroncito, in 1878 [2]. Usually, influenza viruses occur among birds. Wild birds worldwide carry the viruses in their intestines, but usually do not get sick from them. However, avian influenza is very contagious among birds and can make some domesticated birds, including chickens, ducks, and turkeys, very sick and kill them [5]. Infected birds shed influenza viruses from their saliva, nasal secretions, etc. When susceptible birds come in contact with the contaminated surfaces, they become infected. Domesticated birds may become infected with avian influenza viruses through direct contact with infected waterfowl or other infected poultry or through contact with surfaces (such as dirt or cages) or materials (such as water or food) that have been contaminated with the virus [18]. The highly pathogenic avian influenza H5N1 virus was first isolated in China in 1996, and the first human case was found in China in 1997. After that, infection to human of avian influenza occurred successively. It is known that already 133 humans have been infected in Asia since late 2003 and 68 killed [5]. Although the avian influenza virus can infect many animal species, it is of major concern for the poultry industry, as the virus can spread rapidly in and between flocks, causing high mortality and severe economic losses [16]. So it is important for us to take effective measure to prevent the pandemic and limit the damage by the pandemic at the minimum. q This work is supported by PRC Grant NSFC (11371311), the Ph.D. Programs Foundation of Ministry of Education of China (No. 20113250110005) and the NSF of Nantong University (10Z009). ⇑ Corresponding author. E-mail address: [email protected] (Z. Lin).
http://dx.doi.org/10.1016/j.amc.2014.09.042 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
754
Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761
As we know, mathematical models are useful tools for building theories, assessing conjectures, answering questions, and determining sensitivities to parameter values. We can use mathematical models to compare, plan, implement, and evaluate various detection, prevention, and control programs [3]. In recent years, many epidemic models have been proposed in the study of transmission and control of avian or avian–human influenza (see, for example, [1,4–6,8,14,15,20,21]). In view of the fact that the spatial diffusion and environmental heterogeneity are important factors in modeling the spread of many diseases, in this paper, we propose an avian–human epidemic model which was described by the following reaction–diffusion equations:
8 @S1 > D1 DS1 ¼ A1 d1 S1 b1 S1 E1 b2 S1 I1 ; > @t > > > @E1 > > D2 DE1 ¼ b1 S1 E1 þ b2 S1 I1 ðd1 þ eÞE1 ; > < @t @I1 D3 DI1 ¼ eE1 ða1 þ d1 ÞI1 ; @t > R Rt > > @S2 > D4 DS2 ¼ A2 d2 S2 þ cI2 bS2 X 1 Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy; > @t > > R Rt > @I2 : D5 DI2 ¼ bS2 X 1 Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy ða2 þ d2 þ cÞI2 ; @t
ð1:1Þ
for x 2 X; t > 0 with homogeneous Neumann boundary conditions
@S1 @E1 @I1 @S2 @I2 ¼ ¼ ¼ ¼ ¼ 0; @g @g @g @g @g
x 2 @ X;
t>0
ð1:2Þ
and initial conditions
S1 ðx; 0Þ ¼ /1 ðxÞ P 0; S2 ðx; 0Þ ¼ /4 ðxÞ P 0;
E1 ðx; tÞ ¼ /2 ðx; tÞ P 0; I1 ðx; tÞ ¼ /3 ðx; tÞ P 0; I2 ðx; 0Þ ¼ /4 ðxÞ P 0; x 2 X;
ð1:3Þ
P @2 n where D ¼ ni¼1 @x 2 , is the Laplace operator, X is a bounded smooth domain in R ; @ X and X are the boundary and the closure i of X, respectively, g is the outward unit normal vector on @ X. The positive constants Di ði ¼ 1; . . . ; 5Þ are the coefficients of i € lder continuous and satisfy @/ diffusion. The initial functions /i are nonnegative, Ho ¼ 0 on the boundary. @g The first three equations of (1.1) form an SEI model describing the interactions among the birds. S1 ðx; tÞ; E1 ðx; tÞ; I1 ðx; tÞ represent respectively the number of susceptible, exposed and infected birds with avian influenza at position x at time t. These three equations are named as the bird system. It is assumed that the individuals in the latent period have infectious force. In fact, it is observed that poultry without any symptom can excrete much high pathogenic virus, which makes it more difficult to inhibit the H5N1 type virus from spreading. The parameter A1 is the recruitment rate of susceptible population; The natural death rate is assumed to be constant uniformly for each class (i.e. d1 ); the transmission coefficients of latent and infective are b1 and b2 , respectively; e is the rate constant at which the exposed individuals become infective; a1 is extra disease-related death rate constants of the infective. The last two equations of (1.1) form an SI model for humans, which consist of two kinds of individuals: susceptible ðS2 Þ humans and infected ðI2 Þ humans with avian influenza. The parameter A2 is the rate at which new humans are born. d2 is the natural death rate for susceptible and infected human. c is the recovery rate, and a2 is the disease-related death rate. b is the rate at which avian influenza is contracted from an average human individual infective. Thus bS2 E1 and bS2 I1 are the force of infection by exposed and infected birds with avian influenza, respectively. The term
Z Z X
t
Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy;
1
accounts for the infection of individuals to their present position at time t caused by the exposed and infected individuals from all possible positions at all previous times [17,19]. The kernel Kðx; y; tÞ depends on both the spatial and the temporal variables. Here we further assumed that
Kðx; y; tÞ ¼ Gðx; y; tÞkðtÞ P 0;
x; y 2 X;
t > 0;
where the function kðtÞ is called the delay kernel and used to weight the distributed time-delay, and satisfies
kðtÞ P 0;
8t P 0; tkðtÞ 2 L1 ðð0; þ1Þ; RÞ;
Z
1
kðtÞdt ¼ 1;
0
G is the weighting function describing the distribution at past times of the exposed individual E1 and the infected individual I1 at position x and time t, and satisfies
Z X
Gðx; y; tÞdx ¼
Z
Gðx; y; tÞdy ¼ 1:
X
For example, Gðx; y; tÞ is Green’s function of the operator condition, and kðtÞ ¼ 1s et=s with a constant s.
@ @t
D2 D or
@ @t
D3 D subject to homogeneous Neumann boundary
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755
The remaining of this paper is organized as follows. In the next section, uniform bounds for the solutions of (1.1)–(1.3) are given. In Section 3, local and global stabilities of the equilibria for the bird system are discussed. Section 4 deals with global asymptotical behaviors of solutions to the full system (1.1)–(1.3). A brief conclusion is made in Section 5. 2. Uniform bound Noting that the initial values are nonnegative and the growth functions in the right hand of (1.1) are assumed to be sufficiently smooth in R5þ , the solution of (1.1) is unique and continuous for all positive time in X by the standard PDE theory. Next we show that the solution of (1.1) is positive and uniformly bounded. We first give the following lemma which is so-called the positivity lemma [11]. T
C 2;1 ðX ð0; TÞÞ and satisfies 8 5 X > > @u > bij uj ðx; tÞ; x 2 X; 0 < t 6 T; > @ti Di Dui P > < j¼1
Lemma 2.1. Let ui 2 CðX ½0; TÞ
@ui > P 0; > > @g > > : ui ðx; 0Þ P 0;
x 2 @ X;
0 < t 6 T;
ð2:1Þ
x 2 X;
where T > 0; bij ðx; tÞ 2 CðX ½0; TÞ. If bij P 0 for j – i, then ui ðx; tÞ P 0 on X ½0; T. Moreover, ui ðx; tÞ > 0 or ui 0 in X ð0; T. As the consequence of Lemma 2.1, we have the following positivity result. Lemma 2.2. Any solution of problem (1.1)–(1.3) with nonnegative nontrivial initial value is positive. By means of Gagliardo–Nirenberg inequality and Moser iteration technique, we can prove the following lemma. Lemma 2.3. Let uðx; tÞ satisfies
8 @u > < @t DDu ¼ gðu; x; tÞ; x 2 X; 0 < t < 1; u @@ug 6 0; x 2 @ X; 0 < t < 1; > : x 2 X; uðx; 0Þ ¼ u0 ðxÞ;
ð2:2Þ
where D > 0; jgðu; x; tÞj 6 Ljuj. If there exists a p with 1 6 p < 1 such that kuðx; tÞkLp ðXÞ is uniformly bounded for t P 0, then kuðx; tÞkLq ðXÞ is uniformly bounded for t P 0, where q ¼ p 2N ; N ¼ 1; 2; . . .. In particular, kuðx; tÞkL1 ðXÞ is uniformly bounded for t P 0. The following theorem shows that the solution of (1.1)–(1.3) is uniformly bounded. T 5 Theorem 2.4. Let ðS1 ; E1 ; I1 ; S2 ; I2 Þ 2 ½CðX ½0; TÞÞ C 2;1 ðX ð0; TÞÞ be the solution of problem (1.1)–(1.3) with nonnegative nontrivial initial value. Then T ¼ 1 and there exist M 1 and M 2 such that
0 6 S1 ;
E1 ;
I1 6 M 1 ;
0 6 S2 ;
I2 6 M 2 ;
ðx; tÞ 2 X ð0; 1Þ;
ðx; tÞ 2 X ð0; 1Þ:
Proof. Clearly, we have
@ðS1 þ E1 þ I1 Þ DðD1 S1 þ D2 E1 þ D3 I1 Þ ¼ A1 d1 ðS1 þ E1 þ I1 Þ a1 I 6 A1 d1 ðS1 þ E1 þ I1 Þ: @t
ð2:3Þ
Integrating (2.3) over X yields
d dt Let M ¼
Z
ðS1 þ E1 þ I1 Þdx 6 A1 jXj d1
Z
X
ðS1 þ E1 þ I1 Þdx:
X
R P3 i¼1 /i ðxÞÞdx. Then Xð
A1 jXj kS1 þ E1 þ I1 kL1 6 max M; d1
ð2:4Þ
by comparison principle. From Lemma 2.3, we obtain the uniform bounds of S1 ; E1 and I1 . The uniform bounds of S2 and I2 can be proved in a similar way. h
3. The bird system Since the bird system is independent of the human system, we first study the bird system:
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Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761
8 @S1 > > < @t D1 DS1 ¼ A1 d1 S1 b1 S1 E1 b2 S1 I1 ; @E1 D2 DE1 ¼ b1 S1 E1 þ b2 S1 I1 ðd1 þ eÞE1 ; @t > > : @I1 D DI ¼ eE ða þ d ÞI : 3
@t
1
1
1
1
ð3:1Þ
1
It is clear that the system (3.1) always has the disease-free equilibrium P0 ¼ ðA1 =d1 ; 0; 0Þ which represents that no infected bird with avian influenza exists. Further, if the following conditions are satisfied
R0 ,
A1 b1 A1 eb2 > 1; þ d1 ðd1 þ eÞ d1 ðd1 þ eÞða1 þ d1 Þ
then the system also has a unique endemic equilibrium P ¼ ðS1 ; E1 ; I1 Þ, where
S1 ¼
ðd1 þ eÞða1 þ d1 Þ ; ða1 þ d1 Þb1 þ eb2
E1 ¼
A1 1 ð1 Þ; R0 d1 þ e
I1 ¼
e E : a1 þ d1 1
R0 is called the basic reproduction number of (3.1). In the below, we analyze the local stability at P 0 and P . We first set up the following notations, similarly as in [10,13]. Notation 3.1 (i) 0 ¼ l1 < l2 < l3 < . . . ! 1 are the eigenvalues of D on X under homogeneous Neumann boundary condition; (ii) Eðli Þ is the space of eigenfunction corresponding to li ; (iii) Xij ¼ fc /ij jc 2 R3 }, where f/ij g is an orthonormal basis of Eðli Þ for j ¼ 1; 2; . . . ; dim Eðli Þ; 3
Þ j (iv) X :¼ fu ¼ ðS1 ; E1 ; I1 Þ 2 ½C 1 ðX
@S1 @g
dim Eðli Þ
1 ¼ @E ¼ @I@g1 ¼ 0on @ Xg, and so X ¼ 1 i¼1 X i , where X i ¼ j¼1 @g
X ij .
^ Let D ¼ diagðD1 ; D2 ; D3 Þ; Z ¼ ðS1 ; E1 ; I1 Þ; LZ ¼ DDZ þ GðPÞZ, where
0 ^ ¼B GðPÞ @
d1 b1 E01 b2 I01
b1 S01
b2 S01
b1 E01 þ b2 I01
b1 S01 ðd1 þ eÞ
b2 S01
0
e
ðd1 þ a1 Þ
1 C A
^ ¼ ðS0 ; E0 ; I0 Þ represents any feasible uniform steady state of system (3.1). The linearization of system (3.1) at P ^ is of the and P 1 1 1 form Z t ¼ LZ. For each i P 1; X i is invariant under the operator L, and k is an eigenvalue of L if and only if it is an eigenvalue ^ of the matrix li D þ GðPÞ. The characteristic equation of li D þ GðP0 Þ takes the form
A1 þ d1 þ e k d1 A1 A1 ¼ 0: þ D2 li b1 þ d1 þ e D3 li þ a1 þ d1 eb2 d1 d1
ui ðkÞ , ðk þ li D1 þ d1 Þ k2 þ ðD2 þ D3 Þli þ ða1 þ d1 Þ b1
ð3:2Þ
Clearly, for any i P 1, Eq. (3.2) always have a negative real root: ðli D1 þ d1 Þ. Its other roots are determined by the following equation
A1 A1 A1 ¼ 0: k2 þ ðD2 þ D3 Þli þ ða1 þ d1 Þ b1 þ d1 þ e k þ D2 li b1 þ d1 þ e ðD3 li þ a1 þ d1 Þ eb2 d1 d1 d1
ð3:3Þ
If R0 < 1, then
ðD2 þ D3 Þli þ ða1 þ d1 Þ b1
A1 þ d1 þ e > 0; d1
A1 A1 D2 li b1 þ d1 þ e ðD3 li þ a1 þ d1 Þ eb2 > 0: d1 d1 So the two roots of Eq. (3.3) have negative real parts. Moreover, following the standard technique as in [10], we can obtain that the spectrum of L lies in fRek < dg for some positive d. Therefore P0 is locally asymptotically stable if R0 < 1. When R0 > 1, we take i ¼ 1, then l1 ¼ 0 and
A1 A1 A1 A1 D2 li b1 þ d1 þ e ðD3 li þ a1 þ d1 Þ eb2 ¼ b1 þ d1 þ e ða1 þ d1 Þ eb2 d1 d1 d1 d1
¼ ðd1 þ eÞða1 þ d1 Þ ½b1 ða1 þ d1 Þ þ eb2
A1 < 0: d1
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757
Hence, (3.3) has a positive root, that is to say, there is a characteristic root k with a positive real part in the spectrum of L. It implies that the disease-free equilibrium P 0 is unstable if R0 > 1. As for the equilibrium P , the coefficients Ai ; Bi and C i of the characteristic equation ui ðkÞ , jkI þ li GðP Þj ¼ k3 þ Ai k2 þ Bi k þ C i ¼ 0 are defined as follows:
Ai ¼ ðd1 þ D1 li Þ þ ðd1 þ e b1 S1 þ D2 li Þ þ ða1 þ d1 þ D3 li Þ þ b1 E1 þ b2 I1 ;
Bi ¼ ða1 þ d1 þ D3 li Þ 2d1 þ e b1 S1 þ b1 E1 þ b2 I1 þ ðD1 þ D2 Þli
þ ðd1 þ D1 li Þðd1 þ e b1 S1 þ D2 li Þ þ ðd1 þ e þ D2 li Þðb1 E1 þ b2 I1 Þ eb2 S1 ;
C i ¼ ða1 þ d1 þ D3 li Þ ðd1 þ D1 li Þðb1 S1 þ d1 þ e þ D2 li Þ þ ðd1 þ e þ D2 li Þðb1 E1 þ b2 I1 Þ ðd1 þ D1 li Þeb2 S1 : e E , we obtain Noting that ðb1 E1 þ b2 I1 ÞS1 ðd1 þ eÞE1 ¼ 0 and I1 ¼ a1 þd 1 1
b1 S1 ¼ ðd1 þ eÞ b2
I1 e S ¼ ðd1 þ eÞ b2 S : E1 1 a1 þ d1 1
Thus
d1 þ e b1 S1 ¼ b2
e S > 0; a1 þ d1 1
which in turn leads to Ai > 0; Bi > 0 and C i > 0 for i P 1. On the other hand, it is easy to see that
Hi , Ai Bi C i > 0: Therefore, according to the Routh–Hurwitz criterion, the three roots ki;1 ; ki;2 ; ki;3 of ui ðkÞ ¼ 0 all have negative real parts. Furthermore, the similar analysis as in [10] yields that there exists a positive constant e such that
Refki;1 g;
Refki;2 g;
Refki;3 g 6 e;
i P 1:
In summary, we have proved the following theorem. Theorem 3.1. For the bird system ð3:1Þ, if R0 < 1, then the disease-free equilibrium P 0 ðAd11 ; 0; 0Þ is locally asymptotically stable while no endemic equilibrium exists. if R0 > 1, then the endemic equilibrium P ðS1 ; E1 ; I1 Þ exists and is locally asymptotically stable while the disease-free equilibrium P0 is unstable. Next we consider the global stability of the disease-free equilibrium P0 ¼ ðAd11 ; 0; 0Þ, when R0 < 1. We use the method of upper and lower solutions, similar to the argument in [7]. We first investigate the asymptotic behavior of the solution of (3.1) in relation to its corresponding constant steady-state solution ðc1 ; c2 ; c3 Þ of (3.1) given by
8 A1 d1 c1 b1 c1 c2 b2 c1 c3 ¼ 0; > > < b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ¼ 0; > > : ec2 ða1 þ d1 Þc3 ¼ 0:
ð3:4Þ
Now, we deal with the algebraic equations in (3.4) as the following elliptic system:
8 > < Dc1 ¼ A1 d1 c1 b1 c1 c2 b2 c1 c3 ; Dc2 ¼ b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ; > : Dc3 ¼ ec2 ða1 þ d1 Þc3 :
ð3:5Þ
c ¼ ð~c1 ; ~c2 ; ~c3 Þ and ^ c ¼ ð^c1 ; ^c2 ; ^c3 Þ a pair of upper and lower solutions of (3.5) if ~ cP^ c P ð0; 0; 0Þ and We denote by ~
8 A1 d1 ~c1 b1 ~c1 ^c2 b2 ~c1 ^c3 6 0 6 A1 d1 ^c1 b1 ^c1 ~c2 b2 ^c1 ~c3 ; > > < b1 ~c1 ~c2 þ b2 ~c1 ~c3 ðd1 þ eÞ~c2 6 0 6 b1 ^c1 ^c2 þ b2 ^c1 ^c3 ðd1 þ eÞ^c2 ; > > : e~c2 ða1 þ d1 Þ~c3 6 0 6 e^c2 ða1 þ d1 Þ^c3 :
ð3:6Þ
Every pair of upper and lower solutions of (3.5) is also a pair of upper and lower solutions of (3.1) whenever ^ci 6 /i 6 ~ci in X for i ¼ 1; 2; 3. Obviously, we have the Lipschitz condition
j½A1 d1 S1 b1 S1 E1 b2 S1 I1 ½A1 d1 S01 b1 S01 E01 b2 S01 I01 j 6 K 0 ðjS1 S01 j þ jE1 E01 j þ jI1 I01 jÞ; j½b1 S1 E1 þ b2 S1 I1 ðd1 þ eÞE1 ½b1 S01 E01 þ b2 S01 I01 ðd1 þ eÞE01 j 6 K 0 ðjS1 S01 j þ jE1 E01 j þ jI1 I01 jÞ; j½eE1 ða1 þ d1 ÞI1 ½eE01 ða1 þ d1 ÞI01 j 6 K 0 ðjS1 S01 j þ jE1 E01 j þ jI1 I01 jÞ ðmÞ ðmÞ for ^ci 6 S1 ; E1 ; I1 ; S01 ; E01 ; I01 6 ~ci , where K 0 ¼ K 0 ðA1 ; d1 ; b1 ; b2 ; a1 ; e; ~ci ; ^ci Þ. Now we construct two sequences fcðmÞ g fc1 ; c2 ; ðmÞ
ðmÞ
ðmÞ
ðmÞ
c3 g; fcðmÞ g fc1 ; c2 ; c3 g from the iteration process
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8 ðmÞ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ c1 ¼ c1 þ K10 ½A1 d1 c1 b1 c1 c2 b2 c1 c3 ; > > > > > ðmÞ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ 1 > > c2 ¼ c2 þ K 0 ½b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ; > > > > > < cðmÞ ¼ cðm1Þ þ 1 ½ecðm1Þ ða1 þ d1 Þcðm1Þ ; 3 3 2 3 K0 ðmÞ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ ðm1Þ > > c1 ¼ c1 þ K10 ½A1 d1 c1 b1 c1 c2 b2 c1 c3 ; > > > > > cðmÞ ¼ cðm1Þ þ 1 ½b cðm1Þ cðm1Þ þ b cðm1Þ cðm1Þ ðd þ eÞcðm1Þ ; > 1 > 1 1 2 1 2 2 2 3 2 K0 > > > : ðmÞ ðm1Þ ðm1Þ ðm1Þ 1 c3 ¼ c3 þ K 0 ½ec2 ða1 þ d1 Þc3 ;
ð3:7Þ
^, respectively, where K 0 is the above Lipschitz constant. It is easy to see that the c and cð0Þ ¼ c with initial iteration cð0Þ ¼ ~ ðmÞ ðmÞ sequences fc g; fc g posses the monotone property
^c 6 cðmÞ 6 cðmþ1Þ 6 cðmþ1Þ 6 cðmÞ 6 ~c;
m ¼ 1; 2; . . .
and the limits
c ¼ lim cðmÞ ; m!1
c ¼ lim cðmÞ
ð3:8Þ
m!1
exist and satisfy the equations
8 > < A1 d1 c1 b1 c1 c2 b2 c1 c3 ¼ 0 ¼ A1 d1 c1 b1 c1 c2 b2 c1 c3 ; b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ¼ 0 ¼ b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ; > : ec2 ða1 þ d1 Þc3 ¼ 0 ¼ ec2 ða1 þ d1 Þc3 :
ð3:9Þ
c; ~ c >. In general, the constant vectors c and c are The constant vectors c and c are called quasisolutions of problem (3.4) in < ^ c; ~ c >. Now we present the folnot true solutions of (3.4). However, if c ¼ c, then c (or c) is the unique solution of (3.4) in < ^ lowing relation between the solution of (3.1) and that of (3.4). c and ^ c be a pair of upper and lower solutions of (3.4) or (3.1). Then the Theorem 3.2 ([12], Theorems 2.1 and 2.2). Let ~ ðmÞ ðmÞ sequences fc g; fc g given by (3.7) converge monotonically to their respective limits c and c, which are the quasisolutions of (3.1) and satisfy (3.9). For any initial function satisfying ^ci 6 /i ðxÞ 6 ~ci in X for i ¼ 1; 2; 3, the corresponding solution u ¼ ðS1 ; E1 ; I1 Þ of (3.1) possesses the property
c 6 lim inf uðx; tÞ 6 lim supuðx; tÞ 6 c in X: t!1
t!1
Moreover, if c ¼ c, then c (or c) is the unique solution of (3.4) in < ^ c; ~ c > and the solution u ¼ ðS1 ; E1 ; I1 Þ of problem (3.1) converges to c as t ! 1 uniformly on X. Next we show that the time-dependent solution of problem (3.1) converges to ðA1 =d1 ; 0; 0Þ as t ! 1 if R0 < 1. Let ðS1 ðx; tÞ; E1 ðx; tÞ; I1 ðx; tÞÞ be a solution of (3.1). In order to apply Theorem 3.2, we first construct the upper and lower solutions ~ c. We define c and ^
~c1 ¼
A1 þ e1 ; d1
~c2 ¼ e2 ;
~c3 ¼ e3 ;
^c1 ¼ d1 ;
^c2 ¼ 0;
^c3 ¼ 0;
where
0 < e1
h i 8 9 <ða1 þ d1 Þðe þ d1 Þ eb2 Ad1 þ ða1 þ d1 Þb1 Ad1 d ðd þ eÞ A b = 1 1 1 1 1 1 < min ; ; : ; d1 b1 ða1 þ d1 Þb1 þ eb2
e a1 þ d1
e2 < e3 <
0 < d1 <
ðd1 þ eÞ b1 ðAd11 þ e1 Þ b2 ðAd11 þ e1 Þ
e2 ;
A1 : d1 þ b1 e2 þ b2 e3
Since S1 ðx; tÞ 6 S1 ðtÞ, where S1 ðtÞ is the solution of problem
dS1 ¼ A1 d1 S1 ; dt
S1 ð0Þ ¼ jj/ ðxÞjj ; 1 1
and limt!1 S1 ðtÞ ¼ Ad11 , we deduce that there exists a T 0 > 0 such that S1 6 Ad11 þ e1 for t P T 0 . Thus ~ c and ^ c are coupled upper and lower solutions of (3.5) in ½T 0 ; þ1Þ. Noticing that c1 P c1 P ^c1 ¼ d1 > 0; c2 ¼ 0 and c3 ¼ 0, (3.9) is reduced to
Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761
8 A1 d1 c1 ¼ 0; > > > < A d c b c c b c c ¼ 0; 1 1 1 1 1 2 2 1 3 > b1 c1 c2 þ b2 c1 c3 ðd1 þ eÞc2 ¼ 0; > > : ec2 ða1 þ d1 Þc3 ¼ 0:
759
ð3:10Þ
It follows from (3.10) that
A1 A1 e c2 b1 þ b2 ðd1 þ eÞ ¼ 0: d1 d1 a1 þ d1 If we assume c2 – 0, then b1 Ad11 þ b2 Ad11
e a1 þd1
ðd1 þ eÞ ¼ 0. This leads to a contradiction to R0 < 1 and thus c2 ¼ 0. We then
have c3 ¼ 0 from the fourth equation of (3.10) and further, c1 ¼ c1 ¼ A1 =d1 from the first two equations of (3.10). Therefore ci ¼ ci for i ¼ 1; 2; 3. Applying Theorem 3.2 we have limt!1 ðS1 ðx; tÞ; E1 ðx; tÞ; I1 ðx; tÞÞ ¼ ðAd11 ; 0; 0Þ. The disease-free equilibrium is globally asymptotically stable if R0 < 1. Theorem 3.3. Assume that R0 < 1 holds. Then the disease-free equilibrium P 0 of problem ð3:1Þ is globally asymptotically stable. 4. The full system ~ 0 ðA1 ; 0; 0; A2 ; 0Þ which represents that no infected bird with The full system (1.1) always has the disease-free equilibrium P d1 d2 avian influenza exists and no infected human exists. If R0 > 1, then the full system (1.1) has the full-endemic equilibrium ~ ¼ ðS ; E ; I ; S ; I Þ, where S ; E ; I are defined above and P 1 2 2 1 1 1 1 1
A2 ða2 þ d2 ÞI2 ; d2
S2 ¼
I2 ¼
bða2 þ
bA2 ðI1 þ E1 Þ ; þ E1 Þ þ d2 ða2 þ d2 þ cÞ
d2 ÞðI1
which implies that birds and humans are infected with avian influenza. ~ 0 and P ~ which are As above, using linearization method and spectral analysis, we can obtain the local stability of P described as the following theorem. ~ 0 of the full system ð1:1Þ is locally asymptotically stable. If R0 > 1, the Theorem 4.1. If R0 < 1, the disease-free equilibrium P ~ is locally asymptotically stable. endemic equilibrium P ~ 0 ðA1 ; 0; 0; A2 ; 0Þ. By applying the comparison principle for the parabolic equation, Next we investigate the global stability of P d1 d2 we can easily prove the following lemma. ½0; 1ÞÞ T C 2;1 ðX ð0; 1ÞÞ be a nonnegative nontrivial solution of the scalar problem Lemma 4.2. Let w 2 CðX
8 @w > < @t DDw ¼ f ðx; tÞ Bwðx; tÞ; x 2 X; 0 < t < 1; @w ¼ 0; x 2 @ X; 0 < t < 1; @g > : wðx; 0Þ P 0; x2X
ð4:1Þ
if f ðx; tÞ ! A as t ! 1 uniformly where B > 0 and f ðx; tÞ is nonnegative continuous function. Then w ! AB as t ! 1 uniformly on X . on X The following lemma implies that the nonlocal integral term does not affect the long time behavior of solution. Lemma 4.3. If wðx; tÞ is a bounded function and limt!1 jjwðx; tÞ Ajj1 ¼ 0, then
Z Z X
t
Kðx; y; t sÞwðs; yÞdsdy ! A as t ! þ1
1
. uniformly on X ~ 0 of the full system ð1:1Þ is globally asymptotically stable. Theorem 4.4. If R0 < 1, then the disease-free equilibrium P Proof. When R0 < 1, it follows from Theorem 3.3 that the disease-free equilibrium of bird system P0 is globally stable. Thus we have limt!1 jjS1 ðx; tÞ A1 =d1 jj1 ¼ 0; limt!1 jjE1 ðx; tÞ 0jj1 ¼ 0 and limt!1 jjI1 ðx; tÞ 0jj1 ¼ 0. Using Lemma 4.3 yields
Z Z X
t
Kðx; y; t sÞðE1 þ I1 Þðs; yÞdsdy ! 0 as t ! þ1
1
. By Lemma 4.1, we obtain uniformly on X
limjjI2 ðx; tÞ 0jj1 ¼ 0:
t!1
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For the fourth equation of system (1.1), using again the Lemma 4.1 gives
limjjS2 ðx; tÞ A2 =d2 jj1 ¼ 0:
t!1
~ 0 is globally asymptotically stable. Therefore, P
h
Remark 4.1. For model ð1:1Þ, the basic reproduction number R0 ¼ 1 is the threshold, which determines when an infection can invade and persist in a new host population. Remark 4.2. Theorem 4.4 shows that the disease-free equilibrium of ð1:1Þ is globally asymptotically stable if the contact rate is small. Remark 4.3. The local stabilities of the full system in Theorem 4.1 can be obtained by using linearization method and spectral analysis. Here we show that Theorem 4.1 is also a consequence of Theorem 3.1, Lemmas 4.1 and 4.3. In fact, if R0 > 1 and the initial value near the full-endemic equilibrium ðS1 ; E1 ; I1 ; S2 ; I2 Þ, the local stability in Theorem 3.1 shows that limt!1 jjS1 ðx; tÞ S1 jj1 ¼ 0, limt!1 jjE1 ðx; tÞ E1 jj1 ¼ 0 and limt!1 jjI1 ðx; tÞ I1 jj1 ¼ 0 since the bird system is independent of the human system. Similarly as in the proof of Theorem 4.4, we have limt!1 jjS2 ðx; tÞ S2 jj1 ¼ 0 and limt!1 jjI2 ðx; tÞ I2 jj1 ¼ 0 by using Lemmas 4.1 and 4.3. 5. Conclusion The World Health Organization (WHO) has warned of a risk of pandemic of avian influenza and pointed out that outbreaks of avian influenza in poultry may raise global public health concerns in the future. Recently mathematical models have been studied to describe the transmission of avian influenza among birds and humans. For example, a SI-SIR model was proposed in [5], that is, two equations with susceptible (X) and infected (Y) birds are SI model describing the interactions among the birds, and four equations are SIR model for humans, which consist of four kinds of individuals: susceptible (S) humans, infected (B) humans with avian influenza, infected (H) humans with mutant avian influenza and recovered (R) ones from mutant avian influenza. The governing system is
8 _ > > > XðtÞ ¼ c bX xXY; > > _ > > > YðtÞ ¼ xXY ðb þ mÞY; > > < SðtÞ _ ¼ k lS ðb1 Y þ b2 HÞS; _ > BðtÞ ¼ b1 SY ðl þ d þ eÞB; > > > > > _ > HðtÞ ¼ b2 SH þ eB ðl þ a þ cÞH; > > > :_ RðtÞ ¼ cH lR:
ð5:1Þ
It was later extended to a diffusive epidemic model in [8]. In this paper, a similar reaction–diffusion model (1.1) is introduced to interpret the spreading of wild avian influenza from the bird world to the human world. A striking difference between (5.1) and (1.1) is that in (1.1), the asymptomatic bird individuals in the latent period have infectious force and the bird system is describing by SEI model. We first discuss the positivity and the uniform bounds of solutions to system (1.1), which are crucial for the investigation of stability of system (1.1). We then introduce a threshold value R0 and obtain that the avian influenza will be permanent when R0 > 1 and the avian influenza will be extinct when R0 < 1. It shows that eliminate and isolate the infected birds with wild avian influenza ~ 0 ðA1 ; 0; 0; A2 ; 0Þ of is useful to epidemic control. By comparison arguments, we also obtain that the disease-free equilibrium P d1
d2
the full system (1.1) is locally asymptotically stable when R0 < 1. It indicates that the most effective way to control disease spreads among human is to avoid contacting with the infected birds and any surfaces that appear to be contaminated with feces from poultry or other animals. References [1] L. Bourouiba, A. Teslya, J. Wu, Highly pathogenic avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling, J. Theor. Biol. 271 (2011) 181–201. [2] I. Capua, F. Mutinelli, A Colour Atlas and Text on Avian Influenza, Papi Editore, Bologna, Italy, 2001. [3] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000) 599–653. [4] S. Iwamia, Y. Takeuchia, A. Korobeinikovb, X. Liu, Prevention of avian influenza epidemic: what policy should we choose?, J Theor. Biol. 252 (2008) 732–741. [5] S. Iwami, Y. Takeuchi, X. Liu, Avian–human influenza epidemic model, Math. Biosci. 207 (2007) 1–25. [6] A. Kaddar, A. Abta, H.T. Alaoui, A comparison of delayed SIR and SEIR epidemic models, Nonlinear Anal. Model. Control 16 (2011) 181–190. [7] K.I. Kim, Z.G. Lin, Asymptotic behavior of an SEI epidemic model with diffusion, Math. Comput. Model. 47 (2008) 1314–1322. [8] K.I. Kim, Z.G. Lin, L. Zhang, Avian–human influenza epidemic model with diffusion, Nonlinear Anal. Real Word Appl. 11 (2010) 313–322. [9] H.K. Leong, C.S. Goh, S.T. Chew, et al, Prevention and control of avian influenza in Singapore, Ann. Acad. Med. Singap. 37 (2008) 504–509.
Q. Tang et al. / Applied Mathematics and Computation 247 (2014) 753–761 [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
761
Z.G. Lin, M. Pedersen, Stability in a diffusive food-chain model with Michaelis–Menten functional response, Nonlinear Anal. 57 (2004) 421–433. C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. C.V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. 48 (2002) 349–362. P.Y.H. Pang, M.X. Wang, Strategy and stationary pattern in a three-species predator–prey model, J. Differ. Equ. 200 (2004) 245–273. X. Qiang, Z. Kou, Prediction of interspecies transmission for avian influenza A virus based on a back-propagation neural network, Math. Comput. Model. 52 (2010) 2060–2065. G.P. Samanta, Permanence and extinction for a nonautonomous avian–human influenza epidemic model with distributed time delay, Math. Comput. Model. 52 (2010) 1794–1811. D. Spekreijse, A. Bouma, J.A. Stegeman, et al, The effect of inoculation dose of a highly pathogenic avian influenza virus strain H5N1 on the infectiousness of chickens, Vet. Microbiol. 147 (2011) 59–66. H.R. Thieme, X. Zhao, A non-local delayed and diffusive predator–prey model, Nonlinear Anal. Real Word Appl. 2 (2001) 145–160. R.K. Upadhyay, N. Kumaria, V.S.H. Rao, Modeling the spread of bird flu and predicting outbreak diversity, Nonlinear Anal. Real Word Appl. 9 (2008) 1638–1648. Z. Wang, W. Li, S. Ruan, Travelling wave fronts of reaction–diffusion systems with spatio–temporal delays, J. Differ. Equ. 222 (2006) 185–232. R. Xu, Global dynamics of an SEIS epidemic model with saturation incidence and latent period, Appl. Math. Comput. 218 (2012) 7927–7938. X. Ye, X. Li, Simple models for avian influenza, Rocky Mountain J. Math. 38 (2008) 1813–1828.