AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 369072 10.1155/2014/369072 369072 Research Article Existence of Traveling Waves for a Delayed SIRS Epidemic Diffusion Model with Saturation Incidence Rate Zhou Kai 1 Wang Qi-Ru 2 Wang Youyu 1 Department of Mathematics Chizhou University Chizhou, Anhui 247000 China czu.edu.cn 2 School of Mathematics and Computational Science Sun Yat-sen University Guangzhou 510275 China sysu.edu.cn 2014 3042014 2014 22 01 2014 11 04 2014 30 4 2014 2014 Copyright © 2014 Kai Zhou and Qi-Ru Wang. 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.

This paper is concerned with the existence of traveling waves for a delayed SIRS epidemic diffusion model with saturation incidence rate. By using the cross-iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By careful analyzsis, we derive the existence of traveling waves connecting the disease-free steady state and the endemic steady state through the establishment of the suitable upper-lower solutions.

1. Introduction

Since Kermack and Mckendrick  proposed an ordinary differential system to study epidemiology in 1927, various models have been used to describe various kinds of epidemics, and the dynamics of these systems have been investigated. Let S ( t ) represent the number of individuals who are susceptible to the disease, let I ( t ) represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and let R ( t ) represent the number of individuals who have been infected and then removed from the possibility of being infected again. Mena-Lorca and Hethcote  considered the following SIRS epidemic model: (1) S ˙ ( t ) = A - d S ( t ) - β S ( t ) I ( t ) + δ R ( t ) , I ˙ ( t ) = β S ( t ) I ( t ) - ( γ + μ + d ) I ( t ) , R ˙ ( t ) = γ I ( t ) - ( δ + d ) R ( t ) , where the parameters A , d , β , δ , γ , μ are positive constants and A is the recruitment rate of the population, d is the natural death rate of the population, β is the transmission rate, δ is the rate at which recovered individuals lose immunity and return to the susceptible class, γ is the recovery rate of the infective individuals, and μ is the death rate of the infective individuals due to disease. The SIRS model assumes that the recovered individuals have only temporary immunity, which is reasonable in the study of some communicable diseases.

However, due to the diseases latency or immunity, the presence of time delays in such models makes them more realistic. On the other hand, the environment in which an individual lives is actually heterogeneous and the mobility of people within a country or even worldwide is large; introducing the spatial diffusion in these epidemic models is unavoidable. In recent years, the dynamics of the delayed epidemic diffusion model have been widely studied by many researchers (see, e.g., ), and these studies are mainly focused on the global attractivity, basic reproductive number, and especially the epidemic waves. For example, Gan et al.  considered the following delayed SIRS epidemic model with spatial diffusion: (2) S t = D S 2 S x 2 + A - d S ( x , t ) mmm - β S ( x , t ) I ( x , t - τ ) + δ R ( x , t ) , I t = D I 2 I x 2 + β S ( x , t ) I ( x , t - τ ) mmm - ( γ + α + d ) I ( x , t ) , R t = D R 2 R x 2 + γ I ( x , t ) - ( δ + d ) R ( x , t ) , and obtained the existence of traveling wave solutions.

In systems (1) and (2), the terms β S ( t ) I ( t ) and β S ( x , t ) I ( x , t - τ ) are called incidence rate and both of them are bilinear. However, as the number of susceptible individuals is large, it is reasonable to consider the saturation incidence rate (see ) instead of the bilinear incidence rate. Motivated by the works mentioned above, we will consider the following delayed SIRS epidemic diffusion model with nonlinear saturation rate (3) S t = D S 2 S x 2 + A - d S ( x , t ) mmm - β S ( x , t ) I ( x , t - τ ) 1 + α I ( x , t - τ ) + δ R ( x , t ) , I t = D I 2 I x 2 + β S ( x , t ) I ( x , t - τ ) 1 + α I ( x , t - τ ) mmm - ( γ + μ + d ) I ( x , t ) , R t = D R 2 R x 2 + γ I ( x , t ) - ( δ + d ) R ( x , t ) and study its traveling wave solutions. The main tool is the upper-lower solutions coupled with cross-iteration method established by Ma . We point out that the nonlinear terms in (3) do not satisfy the common various (exponential) monotonicity conditions such as in ; thus the main difficulty is the construction and verification of the upper-lower solutions.

2. Preliminaries and Lemmas

Throughout this paper, we employ the usual notations for the standard ordering in 3 . That is, for u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ) , we denote u v if u i v i , i = 1,2 , 3 ; u < v if u v but u v ; and u v if u v but u i v i , i = 1,2 , 3 . Let · denote the Euclidean norm in 3 .

First, we assume that D S = D I = D R = D for (3). Denoting N = S + I + R , then (3) reduces to the following system: (4) N t = D 2 N x 2 + A - d N ( x , t ) - μ I ( x , t ) , I t = D 2 I x 2 + β ( N - I - R ) I ( x , t - τ ) 1 + α I ( x , t - τ ) mmm - ( γ + μ + d ) I ( x , t ) , R t = D 2 R x 2 + γ I ( x , t ) - ( δ + d ) R ( x , t ) .

By making changes of variables N ~ = A / d - N , I ~ = I , R ~ = R and dropping the tildes, (4) is converted to the following system: (5) N t = D 2 N x 2 - d N ( x , t ) + μ I ( x , t ) , I t = D 2 I x 2 + β ( A d - N - I - R ) I ( x , t - τ ) 1 + α I ( x , t - τ ) mmm - ( γ + μ + d ) I ( x , t ) , R t = D 2 R x 2 + γ I ( x , t ) - ( δ + d ) R ( x , t ) .

Consider the equilibrium equation of system (5): (6) μ I - d N = 0 , β ( A d - N - I - R ) I 1 + α I - ( γ + μ + d ) I = 0 , γ I - ( δ + d ) R = 0 . Obviously, system (5) often has a trivial equilibrium E 0 ( 0,0 , 0 ) . From the first and the third equation of (6), we know that N = ( μ / d ) I , R = ( γ / ( δ + d ) ) I . Substituting the expressions into the second equation of (6), if 0 : = A β / d ( γ + μ + d ) > 1 , we get a positive equilibrium E * ( k 1 , k 2 , k 3 ) of system (5), where (7) k 1 = μ ( δ + d ) [ A β - d ( γ + μ + d ) ] m m m × ( + d 2 α ( γ + μ + d ) ( δ + d ) d β [ δ ( μ + d ) + d ( γ + μ + d ) ] m m m m m + d 2 α ( γ + μ + d ) ( δ + d ) ) - 1 , k 2 = ( δ + d ) [ A β - d ( γ + μ + d ) ] β [ δ ( μ + d ) + d ( γ + μ + d ) ] + d α ( γ + μ + d ) ( δ + d ) , k 3 = γ [ A β - d ( γ + μ + d ) ] β [ δ ( μ + d ) + d ( γ + μ + d ) ] + d α ( γ + μ + d ) ( δ + d ) .

By calculating, we can obtain that k 1 + k 2 + k 3 < A / d , which is important in the following text. In fact, (8) k 1 + k 2 + k 3 = [ A β - d ( γ + μ + d ) ] β [ δ ( μ + d ) + d ( γ + μ + d ) ] + d α ( γ + μ + d ) ( δ + d ) mm × [ μ d ( δ + d ) + ( δ + d ) + γ ] < [ A β - d ( γ + μ + d ) ] β ( μ δ + μ d + δ d + γ d + d 2 ) mm × 1 d ( μ δ + μ d + δ d + γ d + d 2 ) = [ A β - d ( γ + μ + d ) ] β d < A d .

Now, we study the existence of traveling wave solutions for system (5) connecting E 0 and E * .

Substituting N ( x , t ) = ϕ ( x + c t ) , I ( x , t ) = φ ( x + c t ) , R ( x , t ) = ψ ( x + c t ) into (5), and denoting x + c t still by t , we derive the following wave profile system from (5): (9) D ϕ ( t ) - c ϕ ( t ) - d ϕ ( t ) + μ φ ( t ) = 0 , D φ ( t ) - c φ ( t ) + β ( A d - ϕ ( t ) - φ ( t ) - ψ ( t ) ) φ ( t - c τ ) 1 + α φ ( t - c τ ) - ( γ + μ + d ) φ ( t ) = 0 , D ψ ( t ) - c ψ ( t ) + γ φ ( t ) - ( δ + d ) ψ ( t ) = 0 .

Note that 0 > 1 imply A β / d - ( γ + μ ) > d . Moreover, we have (10) μ k 2 k 1 = d , γ k 2 k 3 - δ = d . We can select suitable M 1 , M 2 , M 3 such that M i > k i , i = 1,2 , 3 , which satisfy (11) A β d - ( γ + μ ) > μ M 2 M 1 > d , A β d - ( γ + μ ) > γ M 2 M 3 - δ > d , A d > M 1 + M 2 + M 3 .

Denote C [ 0 , M ] ( , 3 ) = { ( ϕ , φ , ψ ) C ( , 3 ) : 0 ( ϕ ( s ) , φ ( s ) , ψ ( s ) ) M } , where M = ( M 1 , M 2 , M 3 ) .

Denote f = ( f 1 , f 2 , f 3 ) : C [ 0 , M ] ( , 3 ) C ( , 3 ) : (12) f 1 ( ϕ , φ , ψ ) ( t ) = - d ϕ ( t ) + μ φ ( t ) , f 2 ( ϕ , φ , ψ ) ( t ) = β ( A d - ϕ ( t ) - φ ( t ) - ψ ( t ) ) m m m m m m m m n × φ ( t - c τ ) 1 + α φ ( t - c τ ) - ( γ + μ + d ) φ ( t ) , f 3 ( ϕ , φ , ψ ) ( t ) = γ φ ( t ) - ( δ + d ) ψ ( t ) .

For ( ϕ , φ , ψ ) C [ 0 , M ] ( , 3 ) , by a careful calculation, we have (13) | f 1 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) - f 1 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) | ( d + μ ) | Φ ( t ) - Ψ ( t ) | 3 , | f 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) - f 2 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) | L | Φ ( t ) - Ψ ( t ) | 3 , | f 3 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) - f 3 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) | ( γ + δ + d ) | Φ ( t ) - Ψ ( t ) | 3 , where L : = A β / d + β ( 1 + α M 2 2 + M 1 + M 3 + 2 M 2 + 2 M 2 / ( 1 + α M 2 ) ) + ( γ + μ + d ) , Φ = ( ϕ 1 , φ 1 , ψ 1 ) , Ψ = ( ϕ 2 , φ 2 , ψ 2 ) .

For the positive constants ρ 1 , ρ 2 , ρ 3 , we define H : C [ 0 , M ] ( , 3 ) C ( , 3 ) by (14) H 1 ( ϕ , φ , ψ ) ( t ) = f 1 ( ϕ , φ , ψ ) ( t ) + ρ 1 ϕ ( t ) , H 2 ( ϕ , φ , ψ ) ( t ) = f 2 ( ϕ , φ , ψ ) ( t ) + ρ 2 φ ( t ) , H 3 ( ϕ , φ , ψ ) ( t ) = f 3 ( ϕ , φ , ψ ) ( t ) + ρ 3 ψ ( t ) .

Then operators H 1 , H 2 , H 3 have the following properties.

Lemma 1.

For 0 ϕ 2 ( t ) ϕ 1 ( t ) M 1 , 0 φ 2 ( t ) φ 1 ( t ) M 2 , 0 ψ 2 ( t ) ψ 1 ( t ) M 3 , one has

(15) H 1 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) H 1 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) , H 3 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) H 3 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) ;

(16) H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) H 2 ( ϕ 1 , φ 2 , ψ 1 ) ( t ) , H 2 ( ϕ 2 , φ 1 , ψ 1 ) ( t ) H 1 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) , H 2 ( ϕ 1 , φ 1 , ψ 2 ) ( t ) H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) .

Proof.

According to the definitions of f and H , we have (17) H 1 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) - H 1 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) = ( ρ 1 - d ) ( ϕ 1 ( t ) - ϕ 2 ( t ) ) + μ ( φ 1 ( t ) - φ 2 ( t ) ) , H 3 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) - H 3 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) = ( ρ 3 - d - δ ) ( ψ 1 ( t ) - ψ 2 ( t ) ) + γ ( ϕ 1 ( t ) - ϕ 2 ( t ) ) . Let ρ 1 = d , ρ 3 = d + δ ; we obtain the properties for H 1 and H 3 .

For (ii), we have (18) H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) - H 2 ( ϕ 1 , φ 2 , ψ 1 ) ( t ) = β ( A d - ϕ 1 ( t ) - ψ 1 ( t ) ) × ( φ 1 ( t - c τ ) 1 + α φ 1 ( t - c τ ) - φ 1 ( t - c τ ) 1 + α φ 1 ( t - c τ ) ) - β ( φ 1 ( t ) φ 1 ( t - c τ ) 1 + α φ 1 ( t - c τ ) - φ 2 ( t ) φ 2 ( t - c τ ) 1 + α φ 2 ( t - c τ ) ) + [ ρ 2 - ( γ + μ + d ) ] ( φ 1 ( t ) - φ 2 ( t ) ) . Note that M 1 + M 3 < A / d , and x / ( 1 + α x ) is nondecreasing; we have that the first term of the last formula is nonnegative, and the second term is bigger than - ( β M 2 / ( 1 + α M 2 ) ) ( φ 1 ( t ) - φ 2 ( t ) ) . Let ρ 2 = β M 2 / ( 1 + α M 2 ) + γ + μ + d ; we have H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) H 2 ( ϕ 1 , φ 2 , ψ 1 ) ( t ) . Since (19) H 2 ( ϕ 2 , φ 1 , ψ 1 ) ( t ) - H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) = [ β φ 1 ( t - c τ ) 1 + α φ 1 ( t - c τ ) + ρ 2 ] ( φ 1 ( t ) - φ 2 ( t ) ) , H 2 ( ϕ 1 , φ 1 , ψ 2 ) ( t ) - H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) = [ β φ 1 ( t - c τ ) 1 + α φ 1 ( t - c τ ) + ρ 2 ] ( ψ 1 ( t ) - ψ 2 ( t ) ) , then, for any positive constant ρ 2 , we have H 2 ( ϕ 2 , φ 1 , ψ 1 ) ( t ) H 1 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) , H 2 ( ϕ 1 , φ 1 , ψ 2 ) ( t ) H 2 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) .

Remark 2.

For H 2 , we can further conclude that H 2 ( ϕ 2 , φ 1 , ψ 2 ) ( t ) H 2 ( ϕ 1 , φ 2 , ψ 1 ) ( t ) from Lemma 1(ii).

According to the definition of H , system (9) can be written as (20) D ϕ ( t ) - c ϕ ( t ) - ρ 1 ϕ ( t ) + H 1 ( ϕ , φ , ψ ) ( t ) = 0 , D φ ( t ) - c φ ( t ) - ρ 2 φ ( t ) + H 2 ( ϕ , φ , ψ ) ( t ) = 0 , D ψ ( t ) - c ψ ( t ) - ρ 3 ψ ( t ) + H 3 ( ϕ , φ , ψ ) ( t ) = 0 .

Define (21) λ 1 = c - c 2 + 4 ρ 1 D 2 D , λ 2 = c + c 2 + 4 ρ 1 D 2 D , λ 3 = c - c 2 + 4 ρ 2 D 2 D , λ 4 = c + c 2 + 4 ρ 2 D 2 D , λ 5 = c - c 2 + 4 ρ 3 D 2 D , λ 6 = c + c 2 + 4 ρ 3 D 2 D , and operator F = ( F 1 , F 2 , F 3 ) : C [ 0 , M ] ( , 3 ) C ( , 3 ) by (22) F 1 ( ϕ , φ , ψ ) ( t ) = 1 D ( λ 2 - λ 1 ) [ - t e λ 1 ( t - s ) H 1 ( ϕ , φ , ψ ) ( s ) d s mmmmmmmmm + t + e λ 2 ( t - s ) H 1 ( ϕ , φ , ψ ) ( s ) d s ] , F 2 ( ϕ , φ , ψ ) ( t ) = 1 D ( λ 4 - λ 3 ) [ - t e λ 3 ( t - s ) H 2 ( ϕ , φ , ψ ) ( s ) d s mmmmmmmmm + t + e λ 4 ( t - s ) H 2 ( ϕ , φ , ψ ) ( s ) d s ] , F 3 ( ϕ , φ , ψ ) ( t ) = 1 D ( λ 6 - λ 5 ) [ - t e λ 5 ( t - s ) H 3 ( ϕ , φ , ψ ) ( s ) d s mmmmmmmmm + t + e λ 6 ( t - s ) H 3 ( ϕ , φ , ψ ) ( s ) d s ] . It is easy to see that F i ( ϕ , φ , ψ ) ( t ) ( i = 1,2 , 3 ) satisfy system (20); thus the fixed point of operator F satisfies (9), which is a traveling wave solution of system (5). Therefore, we will use Schauder's fixed point theorem to find the fixed point of F , where the continuity of F is required. For this purpose, let ν > 0 ; we define a norm for Φ ( t ) = ( ϕ , φ , ψ ) ( t ) C ( , 3 ) by (23) | Φ | ν = sup t e - ν | t | Φ ( t ) .

Define (24) B ν ( , 3 ) = { Φ C ( , 3 ) : | Φ | ν < } . Then it is obvious that ( B ν ( , 3 ) , | · | ν ) is a Banach space. We also need the following definition of upper and lower solutions for system (9).

Definition 3.

A pair of continuous functions Φ ¯ = ( ϕ ¯ , φ ¯ , ψ ¯ ) and Φ _ = ( ϕ _ , φ _ , ψ _ ) are called an upper solution and a lower solution of (9), respectively, if there exist finite points T 1 , T 2 , , T m such that Φ ¯ , Φ _ are twice differentiable and bounded on { T i } , i = 1,2 , , m , and satisfy (25) D ϕ ¯ ( t ) - c ϕ ¯ ( t ) + f 1 ( ϕ ¯ , φ ¯ , ψ _ ) ( t ) 0 , D φ ¯ ( t ) - c φ ¯ ( t ) + f 2 ( ϕ _ , φ ¯ , ψ _ ) ( t ) 0 , D ψ - ( t ) - c ψ ¯ ( t ) + f 3 ( ϕ ¯ , φ ¯ , ψ ¯ ) ( t ) 0 , D ϕ _ ( t ) - c ϕ _ ( t ) + f 1 ( ϕ _ , φ _ , ψ _ ) ( t ) 0 , D φ _ ( t ) - c φ _ ( t ) + f 2 ( ϕ ¯ , φ _ , ψ ¯ ) ( t ) 0 , D ψ _ ( t ) - c ψ _ ( t ) + f 3 ( ϕ _ , φ _ , ψ _ ) ( t ) 0 , for t { T i } , respectively.

We assume that a pair of upper-lower solutions Φ ¯ = ( ϕ ¯ , φ ¯ , ψ ¯ ) and Φ _ = ( ϕ _ , φ _ , ψ _ ) are given such that

(26) ( 0,0 , 0 ) ( ϕ _ ( t ) , φ _ ( t ) , ψ _ ( t ) ) ( ϕ ¯ ( t ) , φ ¯ ( t ) , ψ ¯ ( t ) ) ( M 1 , M 2 , M 3 ) ;

(27) lim t - ( ϕ _ ( t ) , φ _ ( t ) , ψ _ ( t ) ) = lim t - ( ϕ ¯ ( t ) , φ ¯ ( t ) , ψ ¯ ( t ) ) = ( 0,0 , 0 ) , lim t + ( ϕ _ ( t ) , φ _ ( t ) , ψ _ ( t ) ) = lim t + ( ϕ ¯ ( t ) , φ ¯ ( t ) , ψ ¯ ( t ) ) = ( k 1 , k 2 , k 3 ) ;

(28) Φ ¯ ( t + ) Φ ¯ ( t - ) , Φ _ ( t + ) Φ _ ( t - ) for    t .

Define the set (29) Γ = { ( ϕ , φ , ψ ) ( t ) C [ 0 , M ] ( , 3 ) : ϕ _ ( t ) ϕ ( t ) ϕ ¯ ( t ) , m φ _ ( t ) φ ( t ) φ ¯ ( t ) , ψ _ ( t ) ψ ( t ) ψ ¯ ( t ) } . Then by the property of H , we have the property of F .

Lemma 4.

For 0 ϕ 2 ( t ) ϕ 1 ( t ) M 1 , 0 φ 2 ( t ) φ 1 ( t ) M 2 , 0 ψ 2 ( t ) ψ 1 ( t ) M 3 , one has (30) F 1 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) F 1 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) , F 2 ( ϕ 2 , φ 1 , ψ 2 ) ( t ) F 2 ( ϕ 1 , φ 2 , ψ 1 ) ( t ) , F 3 ( ϕ 1 , φ 1 , ψ 1 ) ( t ) F 3 ( ϕ 2 , φ 2 , ψ 2 ) ( t ) .

Similar to Lemmas 4.4–4.6 in , we have the following lemmas and omit their proofs.

Lemma 5.

F = ( F 1 , F 2 , F 3 ) is continuous with respect to the norm | · | ν in ( B ν ( , 3 ) ) .

Lemma 6.

Consider F : Γ Γ .

Lemma 7.

F : Γ Γ is compact with respect to the norm | · | ν .

Theorem 8.

Assume that A β / d ( γ + μ + d ) > 1 . If (9) has a pair of upper-lower solutions Φ ¯ = ( ϕ ¯ , φ ¯ , ψ ¯ ) and Φ _ = ( ϕ _ , φ _ , ψ _ ) and satisfies (P1), (P2), and (P3), then system (5) has a traveling wave solution.

Proof.

According to Lemmas 57 and applying Schauder’s fixed point theorem, operator F has a fixed point Φ * = ( ϕ * , φ * , ψ * ) Γ , which is traveling wave for system (5). Furthermore, by (P2) we have (31) lim t - ( ϕ * , φ * , ψ * ) = ( 0,0 , 0 ) , lim t + ( ϕ * , φ * , ψ * ) = ( k 1 , k 2 , k 3 ) . Therefore, the fixed point is a traveling wave solution for (5) connecting E 0 and E * . The proof is complete.

3. Existence of Traveling Waves

To prove the existence of traveling wave solutions for (5), we only need to construct a pair of upper-lower solutions.

Consider the following functions: (32) Δ 1 ( η , c ) = D η 2 - c η - d + μ M 2 M 1 , Δ 2 ( η , c ) = D η 2 - c η + A β d - ( γ + μ + d ) , Δ 3 ( η , c ) = D η 2 - c η - ( δ + d ) + γ M 2 M 3 . Note that (11) and 0 > 1 ; we know that there exist positive numbers c 1 * , c 2 * , c 3 * such that (33) Δ 1 ( η , c ) = 0    has    two    zeros    0 < η 1 < η 2 , for    c > c 1 * , Δ 2 ( η , c ) = 0    has    two    zeros    0 < η 3 < η 4 , for    c > c 2 * , Δ 3 ( η , c ) = 0    has    two    zeros    0 < η 5 < η 6 , for    c > c 3 * .

Denote c * = max { c 1 * , c 2 * , c 3 * } . According to [5, Lemma 3.8], we have η 1 < η 3 and η 5 < η 3 .

Assume that μ / d + γ / ( δ + d ) < 1 ; we can select ε i > 0 ( i = 1,2 , , 6 ) , ε 1 , ε 2 ( 0 , k 1 ) , ε 3 , ε 4 ( 0 , k 2 ) , ε 5 , ε 6 ( 0 , k 3 ) satisfying the following inequalities: (34) μ ( k 2 + ε 3 ) - d ( k 1 + ε 1 ) < 0 , β ( A d - k 1 + ε 2 - k 2 - ε 3 - k 3 + ε 6 ) - ( γ + μ + d ) < 0 , γ ( k 2 + ε 3 ) - ( δ + d ) ( k 3 + ε 5 ) < 0 , d ( k 1 - ε 2 ) - μ ( k 2 - ε 4 ) < 0 , ( γ + μ + d ) - β 1 + α ( k 2 - ε 4 ) m m m h m × ( A d - k 1 - ε 1 - k 2 + ε 4 - k 3 - ε 5 ) < 0 , ( δ + d ) ( k 3 - ε 6 ) - γ ( k 2 - ε 4 ) < 0 .

In fact, we first choose ε 3 , ε 4 ( 0 , k 2 ) such that (35) ( μ d + γ δ + d ) ε 3 < ε 4 , ( μ d + γ δ + d ) ε 4 < ε 3 .

For ε 3 ( 0 , k 2 ) , noting that k 1 = ( μ / d ) k 2 and k 3 = ( γ / ( δ + d ) ) k 2 , we can find ε 1 ( 0 , k 1 ) , ε 5 ( 0 , k 3 ) such that (36) k 1 > ε 1 > μ d ε 3 = μ d ( k 2 + ε 3 ) - k 1 mmmmmmm μ ( k 2 + ε 3 ) - d ( k 1 + ε 1 ) < 0 , k 3 > ε 5 > γ δ + d ε 3 = γ δ + d ( k 2 + ε 3 ) - k 3 mmmmmmmnm γ ( k 2 + ε 3 ) - ( δ + d ) ( k 3 + ε 5 ) < 0 .

For ε 4 ( 0 , k 2 ) , we can find ε 2 ( 0 , k 1 ) , ε 6 ( 0 , k 3 ) such that (37) k 1 > ε 2 > μ d ε 4 = k 1 - μ d ( k 2 - ε 4 ) mmmmmmm d ( k 1 - ε 2 ) - μ ( k 2 - ε 4 ) < 0 , k 3 > ε 6 > γ δ + d ε 4 = k 3 - γ δ + d ( k 2 - ε 4 ) mmmmmmmmn ( δ + d ) ( k 3 - ε 6 ) - γ ( k 2 - ε 4 ) < 0 .

Furthermore, for ε 1 > ( μ / d ) ε 3 , ε 5 > ( γ / ( δ + d ) ) ε 3 and (35), we can find suitable ε 1 , ε 5 satisfying ε 1 + ε 5 < ε 4 . Similarly, we can find suitable ε 2 , ε 6 satisfying ε 2 + ε 6 < ε 3 . Thus we have (38) β ( A d - k 1 + ε 2 - k 2 - ε 3 - k 3 + ε 6 ) - ( γ + μ + d ) < 0 , ( γ + μ + d ) - β 1 + α ( k 2 - ε 4 ) × ( A d - k 1 - ε 1 - k 2 + ε 4 - k 3 - ε 5 ) < 0 .

We define continuous functions Φ ( t ) = ( ϕ 1 ( t ) , φ 1 ( t ) , ψ 1 ( t ) ) and Ψ ( t ) = ( ϕ 2 ( t ) , φ 2 ( t ) , ψ 2 ( t ) ) as follows: (39) ϕ 1 ( t ) = { k 1 e η 1 t , t t 1 , k 1 + ε 1 e - η t , t > t 1 , ϕ 2 ( t ) = { 0 , t t 2 , k 1 - ε 2 e - η t , t > t 2 , φ 1 ( t ) = { k 2 e η 3 t , t t 3 , k 2 + ε 3 e - η t , t > t 3 , φ 2 ( t ) = { 0 , t t 4 , k 2 - ε 4 e - η t , t > t 4 , ψ 1 ( t ) = { k 3 e η 5 t , t t 5 , k 3 + ε 5 e - η t , t > t 5 , ψ 2 ( t ) = { 0 , t t 6 , k 3 - ε 6 e - η t , t > t 6 , where t 1 , t 3 , t 5 > 0 , t 2 , t 4 , t 6 < 0 , and η > 0 is a proper constant to be chosen later.

Furthermore, we can conclude that t 3 max { t 1 , t 5 } and t 4 min { t 2 , t 6 } , which can help us verify the upper-lower solution for system (9). We point out that Φ ( t ) and Ψ ( t ) satisfy (P1), (P2), and (P3) for proper parameters.

Lemma 9.

Suppose μ / d + γ / ( δ + d ) < 1 . Then the functions Φ ( t ) and Ψ ( t ) defined above are upper and lower solutions of (9), respectively.

Proof.

If t t 1 , ϕ 1 ( t ) = k 1 e η 1 t , φ 1 ( t ) = k 2 e η 3 t , we have (40) p 1 ( t ) : = D ϕ 1 ( t ) - c ϕ 1 ( t ) - d ϕ 1 ( t ) + μ φ 1 ( t ) ( D η 1 2 - c η 1 - d + μ M 2 M 1 ) k 1 e η 1 t = 0 . If t > t 1 , ϕ 1 ( t ) = k 1 + ε 1 e - η t , φ 1 ( t ) k 2 + ε 3 e - η t , we know (41) p 1 ( t ) I 1 ( η ) , where I 1 ( η ) = ( D ε 1 η 2 + c ε 1 η ) e - η t - d ( k 1 + ε 1 e - η t ) + μ ( k 2 + ε 3 e - η t ) . Then I 1 ( 0 ) = μ ( k 2 + ε 3 ) - d ( k 1 + ε 1 ) . It follows from (34) that I 1 ( 0 ) < 0 and there exists η 1 * > 0 such that p 1 ( t ) < 0 for all η ( 0 , η 1 * ) .

If t t 3 , φ 1 ( t ) = k 2 e η 3 t , we obtain that (42) p 2 ( t ) : = D φ 1 ( t ) - c φ 1 ( t ) mmmm + β ( A d - ϕ 2 ( t ) - φ 1 ( t ) - ψ 2 ( t ) ) φ 1 ( t - c τ ) 1 + α φ 1 ( t - c τ ) mmmm - ( γ + μ + d ) φ 1 ( t ) mmm ( D η 2 3 - c η 3 - γ - μ - d ) k 2 e η 3 t + A β d k 2 e η 3 ( t - c τ ) mmm [ D η 2 3 - c η 3 + A β d - γ - μ - d ] k 2 e η 3 t = 0 . If t > t 3 , φ 1 ( t ) = k 2 + ε 3 e - η t , ϕ 2 ( t ) = k 1 - ε 2 e - η t , ψ 2 ( t ) = k 3 - ε 6 e - η t , we have (43) p 2 ( t ) I 2 ( η ) , where (44) I 2 ( η ) = ( D ε 3 η 2 + c ε 3 η ) e - η t mmmm - ( γ + μ + d ) ( k 2 + ε 3 e - η t ) mmmm + β [ A d - k 1 + ε 2 e - η t - k 2 - ε 3 e - η t - k 3 + ε 6 e - η t ] mmmm × ( k 2 + ε 3 e - η ( t - c τ ) ) . It follows from (34) that I 2 ( 0 ) < 0 and there exists η 2 * > 0 such that p 2 ( t ) < 0 for all η ( 0 , η 2 * ) .

If t t 5 , ψ 1 ( t ) = k 3 e η 5 t , φ 1 ( t ) = k 2 e η 3 t , we have (45) p 3 ( t ) : = D ψ 1 ( t ) - c ψ 1 ( t ) + γ φ 1 ( t ) - ( δ + d ) ψ 1 ( t ) [ D η 5 2 - c η 5 - ( δ + d ) + γ M 2 M 3 ] k 3 e η 5 t = 0 . If t > t 5 , ψ 1 ( t ) = k 3 + ε 5 e - η t , φ 1 ( t ) k 2 + ε 3 e - η t , (46) p 3 ( t ) I 3 ( η ) , where I 3 ( η ) = ( D ε 5 η 2 + c ε 5 η ) e - η t + γ ( k 2 + ε 3 e - η t ) - ( δ + d ) ( k 3 + ε 5 e - η t ) . We can derive from (34) that there exists η 3 * > 0 such that p 3 ( t ) < 0 for all η ( 0 , η 3 * ) .

If t t 2 , ϕ 2 ( t ) = 0 , we have (47) q 1 ( t ) : = D ϕ 2 ( t ) - c ϕ 2 ( t ) - d ϕ 2 ( t ) + μ φ 2 ( t ) = μ φ 2 ( t ) 0 . If t > t 2 , ϕ 2 ( t ) = k 1 - ε 2 e - η t , φ 2 ( t ) = k 2 - ε 4 e - η t , we have (48) q 1 ( t ) I 4 ( η ) , where I 4 ( η ) = - ( D ε 2 η 2 + c ε 2 η ) e - η t - d ( k 1 - ε 2 e - η t ) + μ ( k 2 - ε 4 e - η t ) . It follows from (34) that I 4 ( 0 ) > 0 and there exists η 4 * > 0 such that q 1 ( t ) > 0 for all η ( 0 , η 4 * ) .

If t t 4 , φ 2 ( t ) = 0 , we obtain that (49) q 2 ( t ) : = D φ 2 ( t ) - c φ 2 ( t ) mmmm + β ( A d - ϕ 1 ( t ) - φ 2 ( t ) - ψ 1 ( t ) ) φ 2 ( t - c τ ) 1 + α φ 2 ( t - c τ ) mmmm - ( γ + μ + d ) φ 2 ( t ) = 0 . If t > t 4 , φ 2 ( t ) = k 2 - ε 4 e - η t , φ 2 ( t - c τ ) k 2 - ε 4 e - η ( t - c τ ) , ϕ 1 ( t ) k 1 + ε 1 e - η t , ψ 1 ( t ) k 3 + ε 5 e - η t , we have (50) q 2 ( t ) I 5 ( η ) , where (51) I 5 ( η ) = - ( D ε 4 η 2 + c ε 4 η ) e - η t - ( γ + μ + d ) ( k 2 - ε 4 e - η t ) + β ( A d - k 1 - ε 1 e - η t - k 2 + ε 4 e - η t - k 3 m m m - ε 5 e - η t A d ) k 2 - ε 4 e - η ( t - c τ ) 1 + α k 2 - α ε 4 e - η ( t - c τ ) . It follows from (34) that I 5 ( 0 ) > 0 and there exists η 5 * > 0 such that q 2 ( t ) > 0 for all η ( 0 , η 5 * ) .

If t t 6 , ψ 2 ( t ) = 0 , we have (52) q 3 ( t ) : = D ψ 2 ( t ) - c ψ 2 ( t ) + γ φ 2 ( t ) - ( δ + d ) ψ 2 ( t ) 0 . If t > t 6 , ψ 2 ( t ) = k 3 - ε 6 e - η t , φ 2 ( t ) = k 2 - ε 4 e - η t , we have (53) q 3 ( t ) I 6 ( η ) , where I 6 ( η ) = - ( D ε 6 η 2 + c ε 6 η ) e - η t + γ ( k 2 - ε 4 e - η t ) - ( δ + d ) ( k 3 - ε 6 e - η t ) . It follows from (34) that I 6 ( 0 ) > 0 and there exists η 6 * > 0 such that q 3 ( t ) > 0 for all η ( 0 , η 6 * ) .

Thus, taking η ( 0 , min 1 i 6 { η i * } ) , we prove that Φ ( t ) and Ψ ( t ) are upper and lower solutions of (9).

Now we obtain and state the main result in this paper.

Theorem 10.

Assume that A β / d ( γ + μ + d ) > 1 and μ / d + γ / ( δ + d ) < 1 ; then, for any c > c * , (5) has a traveling wave solution connecting two equilibria E 0 and E * . Furthermore, system (4) has a traveling wave solution with speed c , which connects two states ( A / d , 0,0 ) and ( A / d - k 1 , k 2 , k 3 ) .

Conflict of Interests

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

Acknowledgments

Kai Zhou is supported by Scientific Research Program of Anhui Provincial Education Department (no. KJ2013B173). Qi-Ru Wang is supported by the NNSF of China (no. 11271379).

Kermack M. Mckendrick A. Contributions to the mathematical theory of epidemics Proceedings of the Royal Society of London A 1927 115 4 700 721 10.1098/rspa.1927.0118 Mena-Lorca J. Hethcote H. W. Dynamic models of infectious diseases as regulators of population sizes Journal of Mathematical Biology 1992 30 7 693 716 10.1007/BF00173264 MR1175099 ZBL0748.92012 Wang X.-S. Wang H. Wu J. Traveling waves of diffusive predator-prey systems: disease outbreak propagation Discrete and Continuous Dynamical Systems A 2012 32 9 3303 3324 10.3934/dcds.2012.32.3303 MR2912077 ZBL1241.92069 Weng P. Zhao X.-Q. Spreading speed and traveling waves for a multi-type SIS epidemic model Journal of Differential Equations 2006 229 1 270 296 10.1016/j.jde.2006.01.020 MR2265628 ZBL1126.35080 Yu X. Wu C. Weng P. X. Traveling waves for a SIRS model with nonlocal diffusion International Journal of Biomathematics 2012 5 5 1250036 10.1142/S1793524511001787 MR2944181 Zhang S. Xu R. Travelling waves and global attractivity of an SIRS disease model with spatial diffusion and temporary immunity Applied Mathematics and Computation 2013 224 1 635 651 10.1016/j.amc.2013.09.007 MR3127651 Gan Q. T. Xu R. Yang P. H. Travelling waves of a delayed SIRS epidemic model with spatial diffusion Nonlinear Analysis: Real World Applications 2011 12 1 52 68 10.1016/j.nonrwa.2010.05.035 MR2728663 ZBL1202.35046 Yang J. Liang S. Zhang Y. Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion PLoS ONE 2011 6 6 2-s2.0-79959591138 10.1371/journal.pone.0021128 e21128 Ma S. Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem Journal of Differential Equations 2001 171 2 294 314 10.1006/jdeq.2000.3846 MR1818651 ZBL0988.34053 Huang J.-H. Zou X.-F. Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity Acta Mathematicae Applicatae Sinica (English Series) 2006 22 2 243 256 10.1007/s10255-006-0300-0 MR2215516 ZBL1106.34037 Li W.-T. Lin G. Ruan S. Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems Nonlinearity 2006 19 6 1253 1273 10.1088/0951-7715/19/6/003 MR2229998 ZBL1103.35049 Wang Q.-R. Zhou K. Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity Journal of Computational and Applied Mathematics 2010 233 10 2549 2562 10.1016/j.cam.2009.11.002 MR2577842 ZBL1184.35102