Existence of Traveling Wave Solutions for Cholera Model

Abstract

To investigate the spreading speed of cholera, Codeço's cholera model (2001) is developed by a reaction-diffusion model that incorporates both indirect environment-to-human and direct human-to-human transmissions and the pathogen diffusion. The two transmission incidences are supposed to be saturated with infective density and pathogen density. The basic reproduction number [subscript]R0[/subscript] is defined and the formula for minimal wave speed [superscript]c*[/superscript] is given. It is proved by shooting method that there exists a traveling wave solution with speed c for cholera model if and only if c...5;[superscript]c*[/superscript] .

(ProQuest: ... denotes non-US-ASCII text omitted.)

Tianran Zhang 1 and Qingming Gou 2 and Xiaoli Wang 1

Academic Editor:Kaifa Wang

1, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2, College of Mathematics & Computer Science, Yangtze Normal University, Chongqing 408100, China

Received 28 December 2013; Accepted 25 January 2014; 26 March 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Cholera has been a serious threat to human health in the past and at present, which is an acute, diarrheal illness caused by infection of the intestine with the bacterium Vibrio cholera . An estimated 3-5 million cases and over 100,000 deaths occur each year around the world [1]. The cholera bacterium is usually found in water or food sources that have been contaminated by feces from a person infected with cholera. Cholera is most likely to be found and to spread in places with inadequate water treatment, poor sanitation, and inadequate hygiene. Therefore, cholera outbreaks have occurred in developing countries, for example, Iraq (2007-2008), Guinea Bissau (2008), Zimbabwe (2008-2009), Haiti (2010), Democratic Republic of Congo (2011-2012), and Sierra Leone (2012) [2].

To understand the propagation mechanism of cholera, many mathematical models were proposed, whose earlier one was established by Capasso and Paveri-Fontana [3] to study the 1973 cholera epidemic in the Mediterranean region as follows: [figure omitted; refer to PDF] where B(t) and I(t) denote the concentrations of the pathogen and the infective populations, respectively. In addition, Codeço [4] investigated the role of the aquatic pathogen in dynamics of cholera through the following susceptible-infective-pathogen model: [figure omitted; refer to PDF] where S(t) is the susceptible individuals. In this model, human is divided into two groups: the susceptible group and the infective group. As pointed out in [4-8], bacterium Vibrio cholera can spread by direct human-to-human and indirect environment-to-human modes. To understand the complex dynamics of cholera, model (2) is extended by [8-15] and so forth.

In all previous models the influences of space distribution of human on the transmission of cholera are omitted. Cholera usually spreads in spatial wave [16]. Cholera bacteria live in rivers and interact with the plankton on the surface of the water [17]. When individuals drink contaminated water and are infected, they will release cholera bacteria through excretion [18]. Capasso et al. [19-23] developed model (1) by incorporating the bacterium diffusion in a bounded area and studied the existence and stability of solutions. To deeply investigate the interaction of transmission modes and bacterium diffusion, Bertuzzo et al. [24, 25] incorporated patchy structure into model (2) and supposed that pathogen in water could diffuse among these patches. Furthermore, Mari et al. [26] studied the influence of diffusion of both human and pathogen on cholera dynamics through a patchy model.

Infectious case is usually found firstly at some location and then spreads to other areas. Consequently, the most important question for cholera is what the spreading speed of cholera is. However, the above spatial models mainly focus on the stability of solutions not the spreading speed. Traveling wave solution is an important tool used to study the spreading speed of infectious diseases [27-29]. Based on Capasso's model (1), Zhao and Wang [30], Xu and Zhao [31], Jin and Zhao [32], and Hsu and Yang [33] studied the influences of pathogen diffusion on the spread speed of cholera.

The studies of traveling wave solutions of Capasso's model (1) incorporating pathogen diffusion provide insight into the spreading speed of cholera. However, some pieces of information are omitted, such as the interaction of direct human-to-human and indirect environment-to-human transmissions. In this paper, a reaction-diffusion model with pathogen diffusion and both transmission paths is proposed by developing Codeço's model (2). Based on model (2) and ignoring the disease-related death, a general diffusive cholera model can be formulated as the following reaction-diffusion system: [figure omitted; refer to PDF] where S=S(x,t) and I=I(x,t) denote the concentrations of susceptible and infected individuals, respectively, and B=B(x,t) is the concentration of the infectious agents. N is the total human population, b stands for the natural birth and death rate, e denotes the contribution of each infected person to the concentration of cholera, and m is the net death rate of vibrio cholera . f(I) and g(B) are the human-to-human and environment-to-human transmission incidences, respectively. Similar to [10], we assume that f(I) and g(B) satisfy

(A1) f(0)=0 , ^{f[variant prime]} (I)...5;0 , ^{f[variant prime][variant prime]} (I)...4;0 ;

(A2) g(0)=0 , ^{g[variant prime]} (0)>0 , ^{g[variant prime]} (B)...5;0 , ^{g[variant prime][variant prime]} (B)...4;0 , and g(B) is strictly monotonously increasing in [0,+∞) .

It is easy to conclude that f(I)...4;^{f[variant prime]} (0)I , g(B)...4;^{g[variant prime]} (0)B , and f(I)/I and g(B)/B are nonincreasing. Obviously, hypotheses (A1) and (A2) imply that the two transmission paths are saturated. In Tian and Wang [10], f(I) and g(B) have the following expressions: [figure omitted; refer to PDF] Obviously, as a special case, such selections satisfy (A1) and (A2).

Shooting method is very important in proving the existence of traveling wave solutions, which was proposed by Dunbar [34, 35] and was applied to many models (e.g., [36-40]). In this paper, the existence of traveling wave solutions of system (3) will be proved by shooting method and the formula for minimal wave speed will be given.

This paper is organized as follows. In next section, the main theorem and the formula for minimal wave speed will be given. In Section 3, the nonexistence of the traveling wave solutions for c<^c* is proved by geometric method. Section 4 is devoted to shooting arguments and the construction of Wazewski set. In Section 5, we prove the existence of traveling wave solutions for c>^c* and then give the existence of traveling wave solution for c=^c* by limit arguments. The final section is devoted to the simulations.

2. Main Results

For convenience, we introduce dimensionless variables and parameters. By setting [figure omitted; refer to PDF] model (3) has the form [figure omitted; refer to PDF] where _f1 (_u2 )=f(bN_u2 ) and _g1 (_u3 )=g(ebN_u3 /m) .

Denote _R0 =[^{f1[variant prime]} (0)+^{g1[variant prime]} (0)]/^b2 , which is the basic reproduction number of (6). Then hypotheses (A1) and (A2) imply that system (6) has two nonnegative constant solutions _P1 (1/b,0,0) and _P2 (1/b-^u* ,^u* ,^u* ) if and only if _R0 >1 , where ^u* is the only one positive root of equation [figure omitted; refer to PDF] and 0<^u* <1/b . Biologically, _P1 corresponds to disease-free equilibrium and _P2 corresponds to endemic equilibrium. To study the spreading wave of cholera, it is assumed that _R0 >1 holds in this paper; that is [figure omitted; refer to PDF]

A traveling wave solution of system (6) with speed c is a nonnegative solution of the form [figure omitted; refer to PDF]

Substituting traveling profile (_u1 (s),_u2 (s),_u3 (s)) into system (6) yields the following equations: [figure omitted; refer to PDF] where [variant prime] denotes d/ds . To investigate invasion question by cholera, we will study the positive solutions of (10) such that [figure omitted; refer to PDF]

Before giving the main theorem, we introduce the equation for minimal wave speed [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Theorem 1.

There exists a constant ^c* >0 which is the greatest positive root of (12). When c...5;^c* , system (6) has a traveling wave solution satisfying boundary condition (11). When 0<c<^c* , system (6) has no traveling wave solutions satisfying boundary condition (11).

3. Nonexistence of Traveling Wave Solutions for c<^c*

From (10), we have [figure omitted; refer to PDF]

Consequently, if _u1 (0)+_u2 (0)...0;1/b , then [figure omitted; refer to PDF]

Hence, the traveling profile (_u1 (s),_u2 (s),_u3 (s)) with boundary condition (11) must satisfy [figure omitted; refer to PDF]

Therefore, to study traveling wave solutions we assume (16) satisfies. Setting ^{u3[variant prime]} =z in system (10) and noticing (16), it follows [figure omitted; refer to PDF]

If _u1 (s)=0 , then ^{u1[variant prime]} (s)=1/c>0 by system (10). Therefore, we suppose _u1 (s)=1/b-_u2 (s)>0 for any s ; that is, _u2 (s)<1/b .

Obviously, system (17) has two equilibria _E1 (0,0,0) and _E2 (^u* ,^u* ,0) . A profile solution of (10) which satisfies boundary condition (11) corresponds to the positive solution (_u2 (s),_u3 (s),z(s)) of system (17) which satisfies [figure omitted; refer to PDF] where u(s)=(_u2 (s),_u3 (s),z(s)) . Therefore, to study the solutions of (10), it is sufficient to study those of system (17) satisfying boundary condition (18).

Firstly, we investigate the dynamics near _E1 . Simple calculations show that the characteristic equation of the linearization of system (17) at _E1 is [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Because _a0 >0 (19) has a negative real root, which is denoted by _λ3 . Let _λ1 and _λ2 be the other two eigenvalues of (19) and suppose that Re_λ1 ...5;Re_λ2 . To investigate the distribution of roots of (19), denote [figure omitted; refer to PDF] and introduce the following lemma [41].

Lemma 2.

( a ) If _Δ0 >0 , (19) has one real root and two nonreal complex conjugate roots.

( b ) If _Δ0 =0 , (19) has a multiple root and all its roots are real.

( c ) If _Δ0 <0 , (19) has three distinct real roots.

Direct calculations show that _Δ0 =-Δ/(108^b4c4 ) , where Δ is defined by (12).

Lemma 3.

( a ) The real parts of _λ1 and _λ2 are positive.

( b ) Assume ^{f1[variant prime]} (0)...4;^b2 . Then, there exists ^c* >0 which is the only positive root of Δ(c)=0 . When c...5;^c* , _λ1 , and _λ2 are real. When 0<c<^c* , _λ1 , and _λ2 are complex and nonreal.

( c ) Assume that ^{f1[variant prime]} (0)>^b2 . Then, there exist two positive constants ^c1* <^c* which are all positive roots of Δ(c)=0 . _λ1 and _λ2 are complex and nonreal if and only if ^c1* <c<^c* . If c>^c* , then ^λ* <_λ2 <_λ1 ; if 0<c...4;^c1* , then _λ2 ...4;_λ1 <^λ* , where ^λ* =(^{f1[variant prime]} (0)-^b2 )/(bc) .

( d ) _λ1 =_λ2 if and only if c=^c* or ^c1* .

Proof.

Suppose λ=βi...0;0 is the root of (19). Substituting λ=βi into (19) and comparing real and imaginary parts show that _a1 =^β2 >0 and _a0 =_a1a2 . Since _a0 >0 , then _a2 >0 . However, it is impossible that _a1 >0 and _a2 >0 by the expressions of _a1 and _a2 . Therefore, the real parts of _λ1 and _λ2 are not zero. Furthermore, since it is impossible that _a1 >0 and _a2 >0 , Routh-Hurwitz theorem implies that it is impossible that the real parts of both _λ1 and _λ2 are negative. Consequently, there are two cases: (i) _λ1 and _λ2 are complex conjugate roots with positive real parts; (ii) _λ1 and _λ2 are real and at least one is positive. However, Descartes' rule of signs shows that the number of positive roots of (19) is zero or two. Thus, if case (ii) is true, both of _λ1 and _λ2 are real and positive. Therefore, (a) is proved.

In this paragraph, we consider the case ^{f1[variant prime]} (0)...4;^b2 . Firstly, suppose that ^{f1[variant prime]} (0)<^b2 . Obviously, _b0 <0 and _b3 >0 . By the expression of _b2 , we have [figure omitted; refer to PDF] since ...=^{f1[variant prime]} (0)-^b2 <0 . Now, assume ^{f1[variant prime]} (0)=^b2 ; that is, ...=0 . Then. _b3 >0 , _b2 >0 , _b1 <0 , and _b0 =0 . Then, if ^{f1[variant prime]} (0)...4;^b2 , Descartes' rule of signs shows that there exists ^c* >0 which is the only positive root of Δ(c)=0 , where Δ(c)<0 for 0<c<^c* and Δ(c)>0 for c>^c* . Using Lemma 2 completes the proof of (b).

Suppose that ^{f1[variant prime]} (0)>^b2 in this paragraph and, thus, ...>0 . Calculations show that [figure omitted; refer to PDF] and that ^{H[variant prime]} (λ)=0 has two roots ^λ1* and ^λ2* , where [figure omitted; refer to PDF] and ^λ1* >^λ2* . By letting _c0 [triangle, =].../b...+m^b2 and using trivial calculations, we get (see Figure 1) [figure omitted; refer to PDF]

Therefore, if c=_c0 , then H(^λ1* )=H(^λ* )>0 . Since ^λ1* is the only minimum-value point of H(λ) , and then H(λ)>0 for any λ>0 and both of _λ1 and _λ2 are not real. Lemma 2 shows that Δ(_c0 )<0 . Thus, since _b0 >0 and _b3 >0 , there exist two positive roots ^c1* <^c* for equation Δ(c)=0 such that ^c1* <_c0 <^c* . Then, using (25) and Lemma 2 completes the proof of (c) and (d).

Distribution of eigenvalues of (19) when ^{f1[variant prime]} (0)>^b2 , (a) for c>^c* and (b) for c<^c1* .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Direct calculations show that corresponding eigenvectors of eigenvalue _λi are [figure omitted; refer to PDF] where i=1,2,3 . Since [figure omitted; refer to PDF] and thus [figure omitted; refer to PDF]

Then, we have the following lemma.

Lemma 4.

If 0<c<^c* , there exist no traveling wave solutions which satisfy boundary condition (11).

Proof.

Assume that ^{f1[variant prime]} (0)...4;^b2 and 0<c<^c* . Then, (b) of Lemma 3 implies that _λ1 and _λ2 are complex conjugate eigenvalues and there exits locally unstable manifold ^...B2;u and locally stable manifold ^...B2;s . If a solution of (17) tends to _E1 when s[arrow right]-∞ , then it will be spiral on ^...B2;u . By the structures of _e1 and _e2 , _u2 (s)<0 at some time s<0 , which shows that there exist no traveling wave solutions departing from _E1 .

Suppose that ^{f1[variant prime]} (0)>^b2 . If ^c1* <c<^c* , (c) of Lemma 3 shows that _λ1 and _λ2 are complex conjugate eigenvalues and similar arguments to that of previous paragraph finish the proof. If 0<c...4;^c1* , (c) of Lemma 3 shows that _λ1 and _λ2 are real; however, _λ2 ...4;_λ1 <^λ* . If a solution of (17) tends to _E1 when s[arrow right]-∞ , structures of _e1 and _e2 indicate that there is an s<0 such that _u2 (s)<0 . The proof is completed.

From Section 4 to Section 5.2, we suppose that c>^c* , which implies ^λ* <_λ2 <_λ1 .

4. Shooting Method and Wazewski Set

To prove the existence of traveling wave, shooting method developed by Dunbar [34] is used. Firstly, we give the shooting arguments.

Consider the differential equation [figure omitted; refer to PDF] where f(y) from ^Rn to ^Rn satisfies Lipschitz condition about y . Let y(s;_y0 ) denote the unique solution of (29) with initial value y(0)=_y0 . It is convenient to give the notations _y0 ·s[triangle, =]y(s;_y0 ) and _y0 ·S[triangle, =]{_y0 ·s|"s∈S⊂R} . To describe the shooting method (or Wazewski theorem), some definitions are necessary.

Definition 5.

(a) For W⊆^Rn , define immediate exit set ^W- of W as [figure omitted; refer to PDF]

(b) For Σ⊆W , let ^Σ0 [triangle, =]{_y0 ∈Σ|"∃_s0 >0 such that _y0 ·_s0 ∉W} .

(c) Given _y0 ∈^Σ0 , define exit time T(_y0 ) of _y0 by [figure omitted; refer to PDF]

Then, Wazewski theorem is formulated as follows.

Lemma 6 (see [34]).

Suppose that

(1) if _y0 ∈Σ and _y0 ·[0,s]⊆cl...(W) , then _y0 ·[0,s]⊆W .

(2) If _y0 ∈Σ , _y0 ·s∈W and _y0 ·s∉^W- , then there exists an open set _Vs about _y0 ·s disjoint from ^W- .

(3) If Σ=^Σ0 , Σ is compact and Σ intersects a trajectory of (29) only once.

Then, the mapping H(_y0 )=_y0 ·T(_y0 ) is a homeomorphism from Σ to its image on ^W- .

A set W⊆^Rn satisfying conditions (1) and (2) of Lemma 6 is called a Wazewski set. In the following, we first construct the Wazewski set W . Fundamental idea to construct a Wazewski set is that the characteristic vectors corresponding eigenvalues with positive real parts should be removed from W and that those characteristic vectors corresponding eigenvalues with negative real parts should be included. Therefore, we set [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

It is obvious that ∂W=∂P∪∂Q . Firstly, we give the construction of ^W- , which is described in Figure 2.

Figure 2: The construction of W and ^W- .

[figure omitted; refer to PDF]

Lemma 7.

The construction of ^W- is as follows: [figure omitted; refer to PDF] where J={(_u2 ,_u3 ,z):0...4;_u2 ...4;^u* ,_u3 =0,z...4;0} .

Proof.

It is enough to analyze the behavior of solution on ∂P∪∂Q . We only study ∂Q and omit the proof of ∂P since the analysis of ∂P is similar to that of ∂Q and is simpler. In the process of this proof, we use some notations to simplify the proof. Set [figure omitted; refer to PDF]

From hypotheses (A1) and (A2), we find that _f1 (_u2 )/_u2 and _g1 (_u2 )/_u2 are monotonously decreasing, h(_u2 ) is strictly monotonously decreasing for _u2 ∈(0,1/b) , and ^u* is the only positive root of h(_u2 )=0 . The set ∂Q is classified into two cases according to variable z .

: (a) Case z<0 . This case is classified as follows.

: (1) Case 0=_u3 <_u2 <^u* . Then ^{u3[variant prime]} =z<0 and the solution of (17) will enter int...(W) .

: (2) Case 0<_u3 =_u2 <^u* . Then [figure omitted; refer to PDF]

: The solution of (17) will enter Q .

: (3) Case 0<_u3 <_u2 =^u* . Then [figure omitted; refer to PDF]

: The solution of (17) will enter Q .

: (4) Case 0=_u3 =_u2 <^u* . Then ^{u3[variant prime]} =z<0 and the solution of (17) will enter int...(W) .

: (5) Case 0=_u3 and _u2 =^u* . The solution of (17) will enter int...(W) .

: (6) Case _u3 =_u2 =^u* . Then ^{u2[variant prime]} =0 , [figure omitted; refer to PDF]

: and ^{(_u3 -_u2 )[variant prime]} =z<0 . Therefore, the solution of (17) will enter Q .

: (b) Case z=0 . This case is classified as follows.

: (1) Case 0<_u3 <_u2 <^u* . Then ^{z[variant prime]} =m(_u3 -_u2 )<0 and the solution of (17) will enter Q .

: (2) Case 0=_u3 <_u2 <^u* . Then ^{u3[variant prime]} =z=0 , ^{u3[variant prime][variant prime]} =^{z[variant prime]} =-m_u2 <0 . The solution of (17) will enter int...(W) .

: (3) Case 0<_u3 =_u2 <^u* . Then ^{(_u3 -_u2 )[variant prime]} =-h(_u2 )_u2 /c<0 , ^{z[variant prime]} =0 , and ^{z[variant prime][variant prime]} =c^{z[variant prime]} +m^{(_u3 -_u2 )[variant prime]} <0 . The solution of (17) will enter Q .

: (4) Case 0<_u3 <_u2 =^u* . Then ^{u2[variant prime]} <0 and ^{z[variant prime]} =m(_u3 -_u2 )<0 . The solution of (17) will enter Q .

: (5) Case 0=_u3 =_u2 <^u* . In this case, (0,0,0) is equilibrium and is constant.

: (6) Case 0=_u3 and _u2 =^u* . Then ^{u3[variant prime]} =z=0 and ^{u3[variant prime][variant prime]} =^{z[variant prime]} =-m_u2 <0 . The solution of (17) will enter int...(W) .

: (7) Case _u3 =_u2 =^u* . Then (^u* ,^u* ,0) is equilibrium and is constant.

The proof is completed.

5. Existence of Traveling Wave Solution for c...5;^c*

In this section, we prove the existence of traveling wave solution for c...5;^c* . Firstly, we study the behaviors of solutions near _E1 .

5.1. Behaviors of Solutions Near _E1

Lemma 8.

Suppose (_u2 (s),_u3 (s),z(s)) is a solution of (17) satisfying initial conditions [figure omitted; refer to PDF] where k=(_λ1 +_λ2 )/2 . Then, for every s>0 , we have [figure omitted; refer to PDF]

Proof.

From Lemma 3, we have (bck+^b2 -^{f1[variant prime]} (0))/^{g1[variant prime]} (0)>0 . To finish the proof, it is sufficient to prove that the set [figure omitted; refer to PDF] is positively invariant. It is obvious that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Suppose that (_u2 (_s0 ),_u3 (_s0 ),z(_s0 ))∈∂_Ψ1 . Then, z(_s0 )=k_u3 (_s0 ) and [figure omitted; refer to PDF]

The last inequality is given since _λ2 <k<_λ1 . Suppose that (_u2 (_s0 ),_u3 (_s0 ),z(_s0 ))∈∂_Ψ2 . If _u2 (_s0 )>0 , then [figure omitted; refer to PDF]

If _u2 (_s0 )=0 , we have [figure omitted; refer to PDF] Consequently, the solution of system (17) departing from Ψ cannot intersect ∂_Ψ1 ∪∂_Ψ2 . If (_u2 (_s0 ),_u3 (_s0 ),z(_s0 ))∈∂_Ψ3 , then ^{u2[variant prime]} (_s0 )=_g1 (_u3 (_s0 ))/(bc)>0 . Since _E1 is equilibrium, in summary, Ψ is positive invariant.

Since _λ1 >_λ2 >0 , stable manifold theorem implies that there exists a one-dimensional strong unstable manifold _...B2;1 tangent to _e1 at _E1 such that the point on _...B2;1 near _E1 can be expressed by [figure omitted; refer to PDF] Furthermore, there is a two-dimensional unstable manifold _...B2;2 tangent to span {_e1 ,_e2 } at _E1 such that _...B2;2 near _E1 can be expressed by [figure omitted; refer to PDF]

Lemma 9.

Suppose that u(s)[triangle, =](_u2 (s),_u3 (s),z(s)) is a solution of (17) such that u(0)∈_...B2;1 for small [straight epsilon]>0 . Then, u(s) will leave W and enter P .

Proof.

Obviously, u(s) satisfies initial condition (39) by the structure of _e1 , and Lemma 8 implies u(s)>0 (u(s)>0 means that _ui (s)>0 and z(s)>0 , i=2,3 ) for every s>0 .

Furthermore, Lemma 8 shows that ^{u3[variant prime]} (s)=z(s)>k_u3 (s) , implying _{lim...s[arrow right]+∞u3 (s)} =+∞ . Since _u2 (s)<1/b , it follows _{lim...s[arrow right]+∞} z(s)=+∞ . Suppose that _u2 (s)<^u* for every s>0 . Then [figure omitted; refer to PDF] for large s since _u3 (s) and _g1 (_u3 ) are strictly monotonous increasing with respect to s and _u3 , respectively. Thus, we have that _{lim...s[arrow right]+∞u2 (s)} =+∞ , contradicting _u2 (s)<1/b for any s∈R . Therefore, there exists _s1 >0 such that _u2 (_s1 )=^u* . Without losing generality, let _s1 =inf...{s>0:_u2 (s)=^u* } . Obviously, we have ^{u2[variant prime]} (_s1 )...5;0 . If _u3 (_s1 )<^u* , then [figure omitted; refer to PDF] which is a contradiction. Therefore, _u3 (_s1 )...5;^u* and u(_s1 )∈∂P . Then, the construction of ^W- shows that u(s) will leave W and enter P .

Let C be a small circle on _...B2;2 centered at _E1 . Then, points on C can be expressed in terms of local coordinate by [figure omitted; refer to PDF] where θ∈[_θ1 ,2π+_θ1 ) , [straight epsilon]>0 , and _θ1 is chosen such that F(_θ1 ) lies on _...B2;1 with z>0 . Then, stable manifold theorem shows that _θ1 [arrow right]0 when [straight epsilon][arrow right]0 . Denote F(θ)[triangle, =](_u-2 (θ),_u-3 (θ),z-(θ)) .

Lemma 10.

There exists a _θ2 ∈(π/2,3π/4) such that [figure omitted; refer to PDF] and that [figure omitted; refer to PDF] for θ∈[_θ1 ,_θ2 ) .

Proof.

From (51), we have [figure omitted; refer to PDF] where sin(_{[straight phi]0} )=_λ1 /^λ12 +^λ22 , cos...(_{[straight phi]0} )=_λ2 /^λ12 +^λ22 , and _{[straight phi]0} ∈(π/4,π/2) since _λ1 >_λ2 . Therefore, z-(_θ2 )=0 and _θ2 ∈[0,π] imply that _θ2 =π-_{[straight phi]0} +O([straight epsilon])∈(π/2,3π/4) . Obviously, z-(θ)>0 for any θ∈[_θ1 ,_θ2 ) . However, [figure omitted; refer to PDF]

Then, equality z-(_θ2 )=[straight epsilon][_λ1 cos..._θ2 +_λ2 sin_θ2 +O([straight epsilon])]=0 , together with the last of (55), reveals _u-2 (_θ2 )-_u-3 (_θ2 )=[straight epsilon][(_λ1 -_λ2 )_λ2 sin_θ2 +O([straight epsilon])]/m>0 ; that is, _u-2 (_θ2 )>_u-3 (_θ2 ) . For θ∈[_θ1 ,_θ2 ] , the first and second equalities of (55) imply that 0<_u-i (θ)<^u* where i=2,3 since _λ1 >_λ2 and 0<[straight epsilon]...a;1 .

Let [figure omitted; refer to PDF]

By Lemma 10, Σ is an arc of circle, Σ⊆W , and the solution of (17) with initial value being the endpoint F(_θ2 ) will enter Q since F(_θ2 )∈^W- ∩∂Q . From Lemma 9, the solution of (17) with initial value being the endpoint F(_θ1 ) will enter P .

5.2. Traveling Wave Solution for c > ^c*

Lemma 11.

Let u(s)=(_u2 (s),_u3 (s),z(s)) be a solution of (17) such that u(0)∈Λ . If u(s)∈W for any s...5;0 , then u(s)∈Λ for any s>0 , where [figure omitted; refer to PDF] and k=c+^c2 +4m .

Proof.

Set _s0 =inf...{s:u(s)∉Λ,s...5;0} . Suppose the conclusion is false; that is, _s0 <+∞ . Obviously, _s0 >0 and u(_s0 )∈∂Λ where [figure omitted; refer to PDF]

In Figure 3, we find ∂_Λ1 ={unbounded area B_E2 CD} , ∂_Λ2 ={triangle OA_E2 O} , ∂_Λ3 ={unbounded cone... _u3 OE} , ∂_Λ4 ={triangle A_E2 CA} , ∂_Λ5 ={unbounded area DCAOE} , ∂_Λ6 ={unbounded area B_E2 O_u3 } , and ∂_Λ7 ={segment OA} .

Since ∂_Λ1 ∪∂_Λ2 ⊂^W- , thus u(_s0 )∉∂_Λ1 ∪∂_Λ2 . If u(_s0 )∈∂_Λ3 , we have ^{u2[variant prime]} (_s0 )...4;0 because _u2 (s)>0 for 0<s<_s0 and _u2 (_s0 )=0 . However, ^{u2[variant prime]} (_s0 )=_g1 (_u3 (_s0 ))/(bc)>0 which is a contradiction. Therefore, u(_s0 )∉∂_Λ3 . If u(_s0 )∈∂_Λ4 , then [figure omitted; refer to PDF] contradicting ^{u2[variant prime]} (_s0 )...5;0 . If u(_s0 )∈∂_Λ5 , then [figure omitted; refer to PDF] since (c-k)k+m<0 , contradicting ^{[z(s)-k_u3 (s)]s=_s0 [variant prime]} ...5;0 . If u(_s0 )∈∂_Λ6 , then ^{z[variant prime]} (_s0 )=m[_u3 (_s0 )-_u2 (_s0 )]>0 which is a contradiction. In conclusion, u(_s0 )∉∂_Λ4 ∪∂_Λ5 ∪∂_Λ6 . If u(_s0 )∈∂_Λ7 , then _u3 (s)>0 and z(s)>0 for any 0<s<_s0 . Hence, ^{u3[variant prime]} (s)=z(s)>0 for any 0<s<_s0 , which implies that _u3 (_s0 )>_u3 (0)>0 . From this contradiction we find u(_s0 )∉∂_Λ7 . Because _E2 is a constant solution, we get u(_s0 )...0;_E2 . In summary, u(_s0 )∉∂Λ and _s0 =+∞ . The proof is completed.

Figure 3: The construction of ∂Λ .

[figure omitted; refer to PDF]

Lemma 12.

There exists a point _u0 =(_u20 ,_u30 ,_z0 )∈Σ such that the solution u(s;_u0 )=(_u2 (s),_u3 (s),z(s)) of (17) with initial value being _u0 will stay in W for any s>0 .

Proof.

It is sufficient to prove Σ...0;_Σ0 . Suppose that Σ=_Σ0 . Firstly, we verify Conditions (1) and (2) of Lemma 6. Condition (1) of Lemma 6 is valid since W is closed.

Suppose _u0 =(_u20 ,_u30 ,_z0 )∈Σ , s<T(_u0 ) and u(s;_u0 )∈W\^W- . Then, u(s;_u0 )∈int...W∪J and _u0 ...0;F(_θ2 ) since F(_θ2 )∈^W- . The structure of Σ implies that _u20 >0 , _u30 >0 , and _z0 >0 . By the proof of Lemma 11, we have that u(s;_u0 )>0 for s<T(_u0 ) . Therefore, u(s;_u0 )∉J and u(s;_u0 )∈int...W . Condition (2) of Lemma 6 holds.

Lemma 6 shows that Σ is homeomorphic to H(Σ) . Since [figure omitted; refer to PDF] and ^W- is disconnected, we have that H(Σ) is disconnected, contradicting the connection of Σ . Thus, Σ...0;_Σ0 and the proof is completed.

Lemma 13.

Let c>^c* . Then, there exists a positive solution u(s)=(_u2 (s),_u3 (s),z(s)) of (17) such that [figure omitted; refer to PDF]

Proof.

By Lemma 12 there exists a point _u0 =(_u20 ,_u30 ,_z0 )∈Σ such that the solution u(s;_u0 )=(_u2 (s),_u3 (s),z(s)) of (17) with initial value being _u0 will stay in W for any s>0 . Furthermore, Lemma 11 shows u(s;_u0 )>0 for any s...5;0 . Stable manifold theorem implies that u(s;_u0 )>0 for any s...4;0 and _{lim...s[arrow right]-∞} u(s;_u0 )=_E1 . Therefore, u(s;_u0 ) is a positive solution.

To complete the proof, it is sufficient to show that _{lim...s[arrow right]+∞} u(s;_u0 )=_E2 . By Lemma 11, we know that _u2 (s)<^u* for any s>0 since u(s;_u0 ) remains in W for all s . Because ^{u3[variant prime]} (s)=z(s)>0 , then the limit of _u3 (s) exists; that is, _{lim...s[arrow right]+∞u3} (s)=^u3* and 0<^u3* ...4;+∞ . Suppose that ^u* <^u3* ...4;+∞ . The first equation of (17) shows that [figure omitted; refer to PDF] for large s , which implies that there is an ^s* >0 such that _u2 (^s* )>^u* . This is a contradiction, and thus 0<^u3* ...4;^u* . From the first equation of (17), we have _{lim...s[arrow right]+∞u2} (s)=^u2* where ^u2* is the only positive root of algebra equation [figure omitted; refer to PDF] At the same time, the third equation of (17) implies _{lim...s[arrow right]+∞} z(s)=^z* and ^z* =m(^u2* -^u3* )/c or ±∞ . It is impossible that ^z* =±∞ due to the boundedness of _u3 (s) . In conclusion, the limit _{lim...s[arrow right]+∞} u(s)=(^u2* ,^u3* ,^z* ) exists and is finite. By [42], (^u2* ,^u3* ,^z* ) must be equilibrium. Since ^u3* >0 , then (^u2* ,^u3* ,^z* )=_E2 .

Noticing the relation of systems (17) and (10) completes the proof of Theorem 1 for case c>^c* .

5.3. Traveling Wave Solution for c = ^c*

Firstly, suppose c>^c* and let u(s;_u0 )=(_u2 (s),_u3 (s),z(s)) be the traveling wave solution of (17). Then, Lemma 11 implies that u(s;_u0 )∈Λ for all s . From the proof of Lemma 13, we find _u3 (s)...4;^u* . Therefore, for all s , we have u(s;_u0 )∈Π where [figure omitted; refer to PDF]

Let {_cn } be a sequence such that ^c* <_cn <_cn+1 for any n and _{lim...n[arrow right]∞cn} =^c* . Set _kn =_cn +^cn2 +4m and [figure omitted; refer to PDF] Then, _Πn ⊆_Π1 for any n .

Lemma 13 shows that there is a positive solution _wn (s)=(_u2,n (s),_u3,n (s),_zn (s)) for system [figure omitted; refer to PDF] satisfying boundary condition (62) such that _wn (s)∈_Πn ⊆_Π1 for any s .

Lemma 14.

Let c=^c* . Then, there exists a traveling wave solution for system (6) satisfying boundary condition (11).

Proof.

It is sufficient to prove that there exists a positive solution u(s)=(_u2 (s),_u3 (s),z(s)) of (17) satisfying boundary condition (62).

Firstly, we show that sequences {_u2,n } , {_u3,n } , {_zn } , {^{u2,n[variant prime]} } , {^{u3,n[variant prime]} } , and {^{zn[variant prime]} } are uniformly bounded and equicontinuous. The idea of Lemma 11 in [34] is used. Obviously, {_u2,n } , {_u3,n } , and {_zn } are uniformly bounded since _wn (s)⊆_Π1 for any s . Because _wn (s)=(_u2,n (s),_u3,n (s),_zn (s)) is the solution of (67), {^{u2,n[variant prime]} } , {^{u3,n[variant prime]} } , and {^{zn[variant prime]} } are also uniformly bounded. Since |_zn (_s1 )-_zn (_s2 )|=^{zn[variant prime]} (_s3 )|_s1 -_s2 | where _s1 <_s3 <_s2 , then {_zn } is equicontinuous. Similarly, {_u2,n } and {_u3,n } are also equicontinuous. By differentiating the equations of (67) and using the previous bounds, we can get that {^{u2,n[variant prime][variant prime]} } , {^{u3,n[variant prime][variant prime]} } , and {^{zn[variant prime][variant prime]} } are uniformly bounded, and hence {^{u2,n[variant prime]} } , {^{u3,n[variant prime]} } , and {^{zn[variant prime]} } are equicontinuous.

The previous paragraph and Arzelà-Ascoli theorem imply that there exist subsequences, again denoted by {_u2,n } , {_u3,n } , and {_zn } and functions _u2 , _u3 , and z such that [figure omitted; refer to PDF] uniformly on compact subsets of ... , thus pointwise on ... . Same arguments imply that {^{u2,n[variant prime]} } , {^{u3,n[variant prime]} } , and {^{zn[variant prime]} } are also uniformly convergent on compact subsets of ... and pointwise convergent on ... . Consequently, we get [figure omitted; refer to PDF] Since (_u2,n ,_u3,n ,_zn ) is the solution of (67), then u(s)=(_u2 (s),_u3 (s),z(s)) is the solution of (17) for c=^c* and u(s)∈cl...(_Π1 ) , where cl...(_Π1 ) is the closer of _Π1 . Because system (67) is autonomous and (_u2,n ,_u3,n ,_zn ) satisfies boundary condition (62), we can assume that _u3,n (0)=^u* /2 for all n ; thus, _u3 (0)>0 . Then, similar to the proof of Lemma 13, we have that the solution u(s) satisfies boundary condition (62).

6. Simulations

In this section, we present some simulations to confirm the theoretical results. Set [figure omitted; refer to PDF] and assign numerical values to parameters as follows: [figure omitted; refer to PDF]

Obviously, such selection for f(I) and g(B) satisfies (A1) and (A2). Then, the traveling wave solution is described in Figure 4.

Figure 4: The wave profiles for S and I and their movements.

[figure omitted; refer to PDF]

Acknowledgments

The authors are supported by the Fundamental Research Funds for the Central Universities (Grants nos. XDJK2012C042 and SWU113048) and NSFC (Grant no. 11201380).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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