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Applied Mathematical Modelling 33 (2009) 626–632 www.elsevier.com/locate/apm

Existence of travelling waves in a modiﬁed vector-disease model q Jianming Zhang Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, PR China Received 20 December 2006; received in revised form 22 November 2007; accepted 26 November 2007 Available online 4 December 2007

Abstract In this paper, we consider travelling wave solutions for a modiﬁed vector-disease model. Special attention is paid to the model in which a susceptible vector can receive the infection not only from the infectious host but also from the infectious vector. For the strong generic delay kernel, we show that travelling wave solutions exist using the geometric singular perturbation theory. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Vector-disease model; Travelling waves; Geometric singular perturbation theory

1. Introduction In recent years, population models in various ﬁelds of mathematical biology have been suggested and studied extensively. As an important research aspect, a large number of papers are devoted to the existence of travelling wave solutions. Following Busenberg and Cooke [1], Cooke [2], Marcati and Pozio [3] and Vole [4], Ruan and Xiao [5] made a vector-disease model, in which they considered a host in a bounded region X 2 RN ðN 6 3Þ, where a disease is carried by a vector. The host was divided into two classes, susceptible and infectious, whereas the vector population was divided into three classes, infectious, exposed and susceptible. For the transmission of the disease, they assumed that a susceptible host could receive the infection only by contacting infected vectors, a susceptible vector could receive the infection only from the infectious host, and a susceptible vector became exposed when it received the infection from an infected host. The exposed vector remains exposed for some time and then becomes infectious. In [5] it was assumed that the total vector population is constant and homogenous in X. All three vector classes diﬀuse inside X and cannot cross the boundary of X. Then the following equation was obtained: ou ðt; xÞ ¼ dMuðt; xÞ auðt; xÞ þ a½1 uðt; xÞvðt; xÞ; ot q

Supported by the Natural Science Foundation of Zhejiang province, PR China (Y604359). E-mail address: [email protected]

0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.11.024

ð1:1Þ

J. Zhang / Applied Mathematical Modelling 33 (2009) 626–632

627

where d is the diﬀusion constant, a is the cure rate of the infected host, a denotes the host-vector contact rate. uðt; xÞ denotes the normalized spatial density of infectious host at time t in x and vðt; xÞ denotes the normalized spatial density of susceptible host at time t in x. In [5], the stability of the steady states was studied using the contacting-convex sets technique; when the spatial variable was one dimension and the delay kernel assumed some special form, they established the existence of travelling wave solutions by using the linear chain trick and geometric singular perturbation method. In fact, in the process of the transmission of a disease, a susceptible host can also receive infection by contacting the infected host. If we take into account the case, we can obtain the following equations: ou ðt; xÞ ¼ Muðt; xÞ auðt; xÞ þ a½1 uðt; xÞvðt; xÞ þ d½1 uðt; xÞuðt; xÞ; ot

ð1:2Þ

where d denotes the susceptible–infected host contact rate. In this paper, we employ the geometric singular perturbation theory developed by Fenichel [6] to prove that in (1.2) travelling waves exist when vðt; xÞ takes a strong delay kernel, i.e., Z t Z 1 ðxyÞ2 t s ðtsÞ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e 4ðtsÞ 2 e s uðy; sÞ dy ds: ð1:3Þ vðx; tÞ ¼ ðf uÞðx; tÞ ¼ s 4pðt sÞ 1 1 The spatial spread of the newly introduced diseases is very important to human being. If we do not understand the spread feature of the epidemic disease, of course, we impossibly know how to control them. So the spatial spread of the newly introduced diseases is always an interesting subject to both theoreticians and empiricists. The existence of travelling waves shows that there is a moving zone of transition from the disease-free state to the infective state (see[5]). Therefore, the existence of travelling waves has been widely studied. A travelling wave solution of Eq. (1.2) exists if there exists a heteroclinic orbit connecting these two critical points (see[7–9]). So in the section that follows, we show that there is a heteroclinic orbit connecting two critical points of Eq. (1.2). 2. The existence of travelling waves In this section, we will seek the existence of the travelling waves for Eq. (1.2) with the kernel 1 x2 t t f ðx; tÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ e 4t 2 es : s 4pt

ð2:1Þ

For this kind of kernel, the travelling wave equations will be recast as a system of six order ODES by the linear chain technique, and then dynamical system theory, especially geometric singular perturbation theory, will be applied to this system for the small delay s: Assume b þ d a > 0, then Eq. (1.2) has two steady-state solu: Our aim is to establish the existence of travelling waves of (1.2) connecting the tions, u0 0 and u bþda bþd two equilibria u0 and u1 , for a suﬃciently small delay s. By direct computation, we obtain o2 v o3 v o4 1 o2 v ov 1 2 ðu vÞ; ¼ 2 þ ð2:2Þ ot2 otox2 ox4 s ox2 ot s thus we reformulate the integro-diﬀerential equation (1.2) as system ( ut ¼ uxx au þ bð1 uÞv þ dð1 uÞv; vtt ¼ 2vtxx vxxxx þ 2s ðvxx vt Þ þ s12 ðu vÞ:

ð2:3Þ

When s ! 0, then vðt; xÞ ! uðt; xÞ: Hence, system (1.2) reduces to the non-delay version of the model ut ¼ uxx au þ bð1 uÞu þ dð1 uÞu:

ð2:4Þ

Substituting uðt; xÞ ¼ /ðzÞ; z ¼ x þ ct into (2.4), we obtain /00 c/0 a/ þ bð1 /Þ/ þ dð1 /Þ/ ¼ 0: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Clearly, following Theorem 4:1 in [5], if c 6 2 b þ d a, then (2.4) has a travelling wave solution.

ð2:5Þ

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J. Zhang / Applied Mathematical Modelling 33 (2009) 626–632

For system (2.4), if we set uðt; xÞ ¼ /ðzÞ; vðt; xÞ ¼ wðzÞ; z ¼ x þ ct, then /; w satisfy the following travelling wave system: (

/00 c/0 a/ þ bð1 wÞw þ dð1 /Þ/ ¼ 0; wð4Þ 2cw000 þ 2c2 w00 2s ðw00 cw00 Þ s12 ð/ wÞ ¼ 0:

ð2:6Þ

A travelling wave of system (1.2) is a solution of system (2.6) satisfying /ð1Þ ¼ 0; /ð1Þ ¼ bþda ; wð1Þ ¼ 0; wð1Þ ¼ bþda : To seek these form solutions, we convert system (2.6) into a six-dimensional bþd bþd system. Letting /0 ¼ /1 ; w0 ¼ w1 ;

w01 ¼ w2 ;

w02 ¼ w3 ;

System (2.6) can be written as 8 0 / ¼ /1 ; > > > > > > > /01 ¼ c/1 þ a/ bð1 wÞw dð1 /Þw; > > > > > < w0 ¼ w1 ; > > w01 ¼ w2 ; > > > > 0 > > w2 ¼ w3 ; > > > > : 0 w3 ¼ 2cw3 2c2 w2 þ 2s ðw2 cw1 Þ þ s12 ð/ wÞ:

ð2:7Þ

System (2.7) has two equilibria, both of which are independent of s, namely, E0 ¼ ð0; 0; 0; 0; 0; 0; 0Þ

and

E1 ¼ ðu ; 0; u ; 0; 0; 0Þ:

Furthermore, we introduce the small parameter pﬃﬃﬃ e¼ s and deﬁne u1 ¼ /; u2 ¼ /1 ; v1 ¼ w; v2 ¼ ew1 ; v3 ¼ e2 w2 ; v4 ¼ e3 w3 : Then system (2.6) can be recast into standard form for a singular perturbation problem 8 0 u ¼ u2 ; > > > 1 > > > > u02 ¼ cu2 þ au1 bð1 u1 Þv1 dð1 u1 Þu1 ; > > > > > < ev0 ¼ v ; 1

2

> ev02 ¼ v3 ; > > > > > > ev0 ¼ v ; > > 4 > > 3 > : 0 ev4 ¼ 2cev4 2c2 e2 v3 þ 2ðv3 cev2 Þ þ u1 v1 :

ð2:8Þ

When e ¼ 0; system (2.8) reduces into the second-order ODE (2.5). We know that system (2.8) with e ¼ 0 has travelling waves. When e > 0 is suﬃciently small, system (2.8) is referred to as the slow system. Note that when e–0 it did not deﬁne a dynamic system in R6 : This problem can be overcome by the transformation z ¼ eg

J. Zhang / Applied Mathematical Modelling 33 (2009) 626–632

under which the system becomes 8 0 u1g ¼ eu2 ; > > > > > > u02g ¼ eðcu2 þ au1 bð1 u1 Þv1 dð1 u1 Þu1 Þ; > > > > > < v01g ¼ v2 ; > v02g ¼ v3 ; > > > > > > v03g ¼ v4 ; > > > > : v04g ¼ 2cev4 2c2 e2 v3 þ 2ðv3 cev2 Þ þ u1 v1 :

629

ð2:9Þ

This is called the fast system. The slow system and fast system are equivalent when e > 0: In the slow system (2.8), if e ¼ 0; then the ﬂow of that system is deﬁned to the set M 0 ¼ fðu1 ; u2 ; v1 ; v2 ; v3 ; v4 Þ 2 R6 : v2 ¼ 0; v3 ¼ 0; v4 ¼ 0; and u1 ¼ v1 g; which is a two-dimensional invariant manifold for system (2.8) with e ¼ 0: If this invariant manifold is normally hyperbolic, then we can obtain an invariant manifold M e of system (2.8) when s > 0, which is close to M 0 . The restriction of (2.8) to this invariant manifold M e yields a two-dimensional system. From Fenichel [6], we know that verifying normal hyperbolicity of M 0 involves checking that the linearization of the fast system (2.9), restricted to M 0 , has precisely dim M 0 eigenvalues on the imaginary axis, with the remainders of the spectrum being hyperbolic. The linearization of (2.9) restricted to M 0 is given by the matrix 0 1 0 0 0 0 0 0 B0 0 0 0 0 0C B C B C B0 0 0 1 0 0C B C B 0 0 0 0 1 0 C; B C B C @0 0 0 0 0 1A 1

0

1 0

2

0

which has six eigenvalues k1 ¼ k2 ¼ 0; k3 ¼ k4 ¼ 1; k5 ¼ k6 ¼ 1. Obviously, we have exactly dim M 0 eigenvalues on the imaginary axis and the other eigenvalues are hyperbolic. Thus the invariant manifold M 0 is normally hyperbolic in the sense of Fenichel [6]. By the geometric singular perturbation theory, we know that slow system (2.8) has an invariant manifold M e , which can be written as M e ¼ fðu1 ; u2 ; v1 ; v2 ; v3 ; v4 Þ 2 R6 : v1 ¼ gðu1 ; u2 ; eÞ þ u1 ; v2 ¼ hðu1 ; u2 ; eÞ; v3 ¼ kðu1 ; u2 ; eÞ; v4 ¼ rðu1 ; u2 ; eÞg; where the functions g; h; k and r satisfy gðu1 ; u2 ; 0Þ ¼ hðu1 ; u2 ; 0Þ ¼ kðu1 ; u2 ; 0Þ ¼ rðu1 ; u2 ; 0Þ ¼ 0:

ð2:10Þ

Thus functions g; h; k and r can be expanded into the form of Taylor series about e gðu1 ; u2 ; eÞ ¼ eg1 þ e2 g2 þ ; hðu1 ; u2 ; eÞ ¼ eh1 þ e2 h2 þ ; kðu1 ; u2 ; eÞ ¼ ek 1 þ e2 k 2 þ ;

ð2:11Þ

rðu1 ; u2 ; eÞ ¼ er1 þ e2 r2 þ : The restriction of (2.8) to the M e is a two-dimensional system. We ﬁrst determine the functions g; h; k and r. Note that M e is the invariant manifold for the ﬂow of system (2.8). Thus diﬀerentiating v1 ¼ gðu1 ; u2 ; eÞ þ u1 with respect to z, we obtain

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J. Zhang / Applied Mathematical Modelling 33 (2009) 626–632

og 0 og 0 u þ 1þ u ¼ v01 : ou1 1 ou2 2

From (2.12) and (2.8), we have og og u2 þ ðcu2 þ au1 bð1 u1 Þðg þ u1 Þ dð1 u1 Þu1 Þ þ u2 ¼ h: e ou1 ou2 Similarly, from v2 ¼ hðu1 ; u2 ; eÞ and (2.8), we get oh oh u2 þ ðcu2 þ au1 bð1 u1 Þðg þ u1 Þ dð1 u1 Þu1 Þ ¼ k: e ou1 ou2 From v3 ¼ kðu1 ; u2 ; eÞ and (2.8), we get ok ok u2 þ ðcu2 þ au1 bð1 u1 Þðg þ u1 Þ dð1 u1 Þu1 Þ ¼ r: e ou1 ou2 From v4 ¼ rðu1 ; u2 ; eÞ and (2.8), we get or or u2 þ ðcu2 þ au1 bð1 u1 Þðg þ u1 ÞÞ ¼ 2cer 2e2 k þ 2ðk cehÞ þ 2k g: e ou1 ou2

ð2:12Þ

ð2:13Þ

ð2:14Þ

ð2:15Þ

ð2:16Þ

Substituting (2.11) into (2.13)–(2.16) and comparing the coeﬃcients of e and e2 we obtain g1 ðu1 ; u2 Þ ¼ 0;

g2 ðu1 ; u2 Þ ¼ 2k 2 2cu2 ;

h1 ðu1 ; u2 Þ ¼ u2 ; h2 ðu1 ; u2 Þ ¼ 0; k 1 ðu1 ; u2 Þ ¼ 0; k 2 ðu1 ; u2 Þ ¼ cu2 þ au1 bð1 u1 Þ dð1 u1 Þu1 ; r1 ðu1 ; u2 Þ ¼ 0;

ð2:17Þ

r2 ðu1 ; u2 Þ ¼ 0:

Therefore, the slow system (2.8) restricted to M e is given by 0 u1 ¼ u 2 ; u02 ¼ cu2 þ au1 bð1 u1 Þðg þ u1 Þ dð1 u1 Þu1 ;

ð2:18Þ

where g is given by (2.11) and (2.17). It is easy to verify that when e ¼ 0, system (2.18) reduces to the corresponding ODE (2.5) for travelling waves of the non-delay problem and for any e > 0, system (2.18) has two ; 0Þ: We now wish to establish the existence of a heteroclinic conequilibrium points ðu1 ; u2 Þ ¼ ð0; 0Þ and ðbþda bþd nection between these two critical points. We know that such a connection exists when e ¼ 0. For the suﬃciently small e > 0, we shall seek such a solution of (2.18) that is a perturbation of travelling waves of (2.5). Now we use the implicit function theorem which has been used in [5] to prove such travelling waves exist. Theorem. For any s > 0 sufficiently small, there exists a speed c such that the integro-differential equation (1.2) has a travelling wave solution which satisfies /ð1Þ ¼ 0 and wðþ1Þ ¼ bþda bþd .

Proof. Denote Uðu1 ; u2 ; c; Þ ¼ cu2 þ au1 bð1 u1 Þðu1 þ gÞ dð1 u1 Þu1 : Then we write (2.18) as 0 u1 ¼ u 2 ; u02 ¼ Uðu1 ; u2 ; c; Þ:

ð2:19Þ

When ¼ 0, we know (1.2) has a travelling wave solution. Therefore there exists a function u2 ¼ f ðu1 ; c0 Þ which satisﬁes du2 Uðu1 ; u2 ; c0 ; 0Þ ¼ du1 u2

ð2:20Þ

J. Zhang / Applied Mathematical Modelling 33 (2009) 626–632

631

and f ð0; c0 Þ ¼ f ðbþda ; c0 Þ ¼ 0. By continuous dependence of the solutions on parameters, there exist u2 ¼ bþd f1 ðu1 ; c0 ; Þ and u1 ¼ f1 ðu1 ; c0 ; Þ, which satisfy bþd a f1 ð0; c0 ; Þ ¼ f2 ; c0 ; ¼ 0 bþd and the graph of f1 and f2 must cross the line u1 ¼ bþda somewhere if is suﬃciently small. Deﬁne bþd bþd a bþd a F ðc0 ; Þ ¼ f1 ; c 0 ; f2 ; c0 ; : 2ðb þ dÞ 2ðb þ dÞ j –0, by the implicit function Clearly, F ðc0 ; 0Þ ¼ 0, and both f1 and f2 satisfy Eq. (2.20). If we can prove oF oc c¼c0 theorem, for suﬃciently small > 0, the equation F ðc; Þ ¼ 0 can just determine a function c ¼ cðÞ near c0 . at the same point. Therefore the theorem This implies that the manifolds f1 and f2 cross the line u2 ¼ bþda 2ðbþdÞ is completed. In fact, we have d of1 o df1 o Uðu1 ; f1 ðu1 ; c; 0Þ; c; 0Þ ðu1 ; c0 ; 0Þ ¼ ðu1 ; c; 0Þ ¼ du1 oc oc du1 oc f1 ðu1 ; c; 0Þ c¼c0 c¼c0 o au1 cf1 ðu1 ; c; 0Þ ðb þ dÞð1 u1 Þu1 ¼ f1 ðu1 ; c; 0Þ oc c¼c0 o au1 ðb þ dÞð1 u1 Þu1 ¼ c þ f1 ðu1 ; c; 0Þ oc c¼c0 ¼ 1

ðb þ d aÞu1 ðb þ dÞu21 of1 ðu1 ; c0 ; 0Þ: 2 oc f ðu1 ; c0 Þ

Integrating from ðb þ d aÞ=ð2ðb þ dÞÞ to ðb þ d aÞ=ðb þ dÞ, we have ! Z ðbþdaÞ=ðbþdÞ Z s of1 b þ d a ðb þ d aÞy ðb þ dÞy 2 dy ds: exp ; c0 ; 0 ¼ oc 2ðb þ dÞ f ðy; c0 Þ ðbþdaÞ=2ðbþdÞ ðbþdaÞ=2ðbþdÞ

ð2:21Þ

Similarly, we have d of2 d of2 ðu1 ; c0 ; 0Þ ¼ ðu1 ; c0 ; 0Þ : du1 oc du1 oc Integrating from 0 to ðb þ d aÞ=2ðb þ dÞ yields ! Z ðbþdaÞ=2ðbþdÞ Z s of2 b þ d a ðb þ d aÞy ðb þ dÞy 2 dy ds: ; c0 ; 0 ¼ exp oc 2ðb þ dÞ f ðy; c0 Þ 0 ðbþdaÞ=2ðbþdÞ

ð2:22Þ

Following (2.21) and (2.22), we have oF of1 b þ d a of2 b þ d a ðc0 ; 0Þ ¼ ; c0 ; 0 ; c0 ; 0 oc 2ðb þ dÞ oc 2ðb þ dÞ oc ! Z s Z ðbþdaÞ=ðbþdÞ ðb þ d aÞy ðb þ dÞy 2 dy ds–0: exp ¼ f ðy; c0 Þ 0 ðbþdaÞ=2ðbþdÞ Hence the Proof is completed.

h

3. Conclusions Disease spread is an important research area in mathematical biology and disease control ﬁeld. In this paper, following [5], a more practical background modiﬁed vector disease model, in which a susceptible vector can receive the infection not only from the infectious host but also from the infectious vector, was discussed.

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J. Zhang / Applied Mathematical Modelling 33 (2009) 626–632

We ﬁrst transformed the travelling equations into a ﬁnite-dimensional system of ordinary diﬀerential equation by using the linear chain trick, then we applied the geometric singular perturbation method to prove the existence of a heteroclinic orbit when s > 0 is suﬃciently small and b þ d a > 0, which is the travelling wave solution for the original equation. The existence of the travelling wave solution means there is a moving zone of transition from the disease-free state to the infective state. In general, a nonlinear dynamics system is very complicated. This paper only discussed the existence of the travelling wave solution. Other dynamics such as the minimal wave speed, the uniqueness and stability of the travelling wave solution, in the model are still some open problems. Of course, these problems are interesting and challenging. Acknowledgements The author is grateful to the editor M. Cross and the anonymous referees for their helpful suggestions, which greatly helped in improving the quality of this manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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