Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

Abstract

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

1. Introduction

Recently, great attention has been paid to the study of the traveling wave solutions in reaction-diffusion models [1–17]. In the sense of epidemiology, the traveling wave solutions describe the transition from a disease-free equilibrium to an endemic equilibrium; the existence and nonexistence of nontrivial traveling wave solutions indicate whether or not the disease can spread [11]. The results contribute to predicting the developing tendency of infectious diseases, to determining the key factors of the spread of infectious disease, and to seeking the optimum strategies of preventing and controlling the spread of the infectious diseases [18–21].

Some methods have been used to derive the existence of traveling wave solutions in reaction-diffusion models, and the monotone iteration method has been proved to be an effective one. Such a method reduces the existence of traveling wave solutions to that of an ordered pair of upper-lower solutions [6, 7, 9, 10, 14, 15].

In [22], Berezovsky and coworkers introduced a simple epidemic model through the incorporation of variable population, disease-induced mortality, and emigration into the classical model of Kermack and McKendrick [23]. The total population (𝑁) is divided into two groups of susceptible (𝑆) and infectious (𝐼); that is to say, 𝑁=𝑆+𝐼. The model describing the relations between the state variables is 𝑑𝑆𝑁𝑑𝑡=𝑟𝑁1−𝐾−𝛽𝑆𝐼𝑁−𝜇+𝑚𝑆,𝑑𝐼𝑑𝑡=𝛽𝑆𝐼𝑁−𝜇+𝑑𝐼, (1) where the reproduction of susceptible follows a logistic equation with the intrinsic growth rate 𝑟 and the carrying capacity 𝐾, 𝛽 denotes the contact transmission rate (the infection rate constant), 𝜇 is the natural mortality; 𝑑 denotes the disease-induced mortality, and 𝑚 is the per-capita emigration rate of uninfected.

For model (1), the epidemic threshold, the so-called basic reproduction number 𝑅₀, is then computed as 𝑅₀=𝛽/(𝜇+𝑑). The disease will successfully invade when 𝑅₀>1 but will die out if 𝑅₀<1. 𝑅₀=1 is usually a threshold whether the disease goes to extinction or goes to an endemic. Large values of 𝑅₀ may indicate the possibility of a major epidemic [19]. In addition, the basic demographic reproductive number 𝑅_𝑑 is given by 𝑅_𝑑=𝑟/(𝜇+𝑚). It can be shown that if 𝑅_𝑑>1 the population grows, while 𝑅_𝑑≤1 implies that the population does not survive [22].

For simplicity, rescaling model (1) by letting 𝑆→𝑆/𝐾, 𝐼→𝐼/𝐾, and 𝑡→𝑡/(𝜇+𝑑) leads to the following model: 𝑑𝑆𝑑𝑡=𝜈𝑅_𝑑(𝑆+𝐼)[1−(𝑆+𝐼)]−𝑅₀𝑆𝐼𝑆+𝐼−𝜈𝑆,𝑑𝐼𝑑𝑡=𝑅₀𝑆𝐼𝑆+𝐼−𝐼, (2) where 𝜈=(𝜇+𝑚)/(𝜇+𝑑) is defined by the ratio of the average life span of susceptibles to that of infectious.

For details, we refer the reader to [20, 22].

In this paper, we are interested in the existence of traveling wave solutions in the following reaction-diffusion epidemic model [20]: 𝜕𝑆𝜕𝑡=𝜈𝑅_𝑑(𝑆+𝐼)[1−(𝑆+𝐼)]−𝑅₀𝑆𝐼𝜕𝑆+𝐼−𝜈𝑆+𝑑²𝑆𝜕𝑥²,𝜕𝐼𝜕𝑡=𝑅₀𝑆𝐼𝜕𝑆+𝐼−𝐼+𝑑²𝐼𝜕𝑥²,𝑆(𝑥,0)=𝑆₀(𝑥),𝐼(𝑥,0)=𝐼₀(𝑥), (3) where 𝜈,𝑅₀,𝑅_𝑑 are all positive constants, 𝑑 is the diffusion coefficient, and (𝑥,𝑡)∈𝑅×𝑅⁺.

We are looking for the traveling wave solutions of model (3) with the following form: 𝑆(𝑥,𝑡)=𝑆(𝜉),𝐼(𝑥,𝑡)=𝐼(𝜉),𝜉=𝑥+𝑐𝑡, (4) satisfying the following boundary value conditions: (𝑆(−∞),𝐼(−∞))^𝑇=𝐸₁,(𝑆(+∞),𝐼(+∞))^𝑇=𝐸₂, (5) where 𝐸₁,𝐸₂ are the equilibrium points of model (3).

This paper is arranged as follows. In Section 2, we construct a pair of ordered upper-lower solutions of model (3) and establish the uniqueness and strict monotonicity of the traveling wave solutions.

2. Existence and Asymptotic Decay Rates

In this section, we will establish the existence of traveling wave solutions of model (3) by constructing a pair of ordered upper-lower solutions. The definition of the upper solution and the lower solution is standard. We assume that the inequality between two vectors throughout this paper is componentwise.

Setting 𝑅𝑆=^𝑑−1𝑅_𝑑−𝑆,𝐼̂=𝐼, (6) then model (3) can be written as 𝜕𝑆𝜕𝑡=−𝜈𝑅_𝑑𝑅_𝑑−1𝑅_𝑑−𝐼̂𝑅𝑆+1−^𝑑−1𝑅_𝑑−𝑆+𝐼̂+𝑅₀𝑅_𝑑−1/𝑅_𝑑−𝑆𝐼̂𝑅_𝑑−1/𝑅_𝑑−𝐼̂𝑅𝑆++𝜈^𝑑−1𝑅_𝑑−𝑆𝜕+𝑑²𝑆𝜕𝑥²,𝜕𝐼̂𝜕𝑡=𝑅₀𝑅_𝑑−1/𝑅_𝑑−𝑆𝐼̂𝑅_𝑑−1/𝑅_𝑑−𝐼̂−̂𝜕𝑆+𝐼+𝑑²𝐼̂𝜕𝑥²,𝑆,𝐼̂^𝑇(−∞)=(0,0)^𝑇,𝑆,𝐼̂^𝑇(+∞)=𝑆^∗,𝐼^∗𝑇. (7)

For model (3), the equilibria are 𝐸₁=((𝑅_𝑑−1)/𝑅_𝑑,0) and 𝐸₂=(𝑆^∗,𝐼^∗), where 𝑆^∗=𝜈𝑅₀𝑅_𝑑−𝑅₀−𝜈+1𝜈𝑅²₀𝑅_𝑑,𝐼^∗=𝑅₀−1𝑆^∗, (8) and for model (7), the equilibria are 𝐸₁=(0,0) and 𝐸₂=(𝑆^∗,𝐼^∗), where 𝑆^∗=𝑅₀−1𝜈𝑅₀𝑅_𝑑−𝜈𝑅₀−𝜈+1𝜈𝑅²₀𝑅_𝑑,𝐼^∗=𝑅₀−1𝜈𝑅₀𝑅_𝑑−𝑅₀−𝜈+1𝜈𝑅²₀𝑅_𝑑. (9) Obviously, 𝐼^∗−𝑆^∗=(𝜈−1)𝑅₀−1𝜈𝑅₀𝑅_𝑑,𝑆^∗=𝑅_𝑑−1𝑅_𝑑−𝑆^∗,𝐼^∗=𝐼^∗. (10) For simplicity, we define the following functions and constants: 𝛼₀=𝑅_𝑑−1𝑅_𝑑,𝜙(𝐼)=𝛼₀𝐼^∗+𝐼^∗−𝑆^∗𝛽𝐼;₀=𝛼₀𝐼^∗2𝑅₀−1𝑅₀𝑅₀;𝛾−𝜈+1₀=2𝜈𝑅_𝑑−1𝐼^∗−𝑆^∗−𝜈𝐼^∗+𝑅₀−1𝑆^∗;𝜓(𝐼)=𝜈𝑅_𝑑𝐼^∗−𝑆^∗2𝐼²+𝛾₀𝐼^∗𝐼+𝛽₀;𝜂₀𝛾=−⁰𝐼^∗2𝜈𝑅_𝑑𝐼^∗−𝑆^∗2;𝜑(𝐼)(𝐼)𝐼=1,𝐼>0;𝜑=−1,𝐼≤0;𝐵=^∗2𝐼1+𝜑^∗2−𝜂₀. (11) And we will always assume the following hypotheses throughout the rest of this paper:  [H1] 𝑅₀>1,1<𝑅_𝑑<2𝑅²₀+2𝑅₀−23𝑅²₀−2𝑅₀,max27𝑅₀𝑅_𝑑−1²𝑅³_𝑑,𝑅₀−1𝑅₀𝑅_𝑑−1<𝜈<−1𝑅₀𝑅_𝑑−𝑅₀.−1 (12) [H2] 𝑅𝜈≥max⁰2−𝑅_𝑑,𝑅³₀−2𝑅²₀+4𝑅₀−22𝑅²₀+2𝑅₀−2−3𝑅²₀−2𝑅₀𝑅_𝑑,𝜓(𝐵)≤0. (13)

Then we can obtain the following.

Lemma 1. If [H1] holds, then 𝐸₂ and 𝐸₂ are endemic points of model (3) and model (7), respectively.

Lemma 2. For model (7), if [H1] holds, then 𝐸₁ is unstable, and 𝐸₂ is stable.

For the sake of convenience, let 𝑥=√𝑑𝑥. For simplicity, we still use the variables 𝑆, 𝐼, and 𝑥 instead of 𝑆, 𝐼̂, and 𝑥, respectively, then model (7) could be rewritten as 𝜕𝑆𝜕𝑡=−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀𝛼−𝑆+𝐼+𝜈₀+𝜕−𝑆²𝑆𝜕𝑥²,𝜕𝐼𝜕𝑡=𝑅₀𝛼₀−𝑆𝐼𝛼₀𝜕−𝑆+𝐼−𝐼+²𝐼𝜕𝑥²,(𝑆,𝐼)^𝑇(−∞)=(0,0)^𝑇,(𝑆,𝐼)^𝑇(+∞)=𝑆^∗,𝐼^∗𝑇. (14)

Following the definition of quasi-monotonicity [17], we can obtain the following results.

Lemma 3. Model (14) is a quasi-monotone decreasing system in (𝑆,𝐼)∈[𝐸₁,𝐸₂].

Proof. Let 𝐹₁(𝑆,𝐼)=−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀𝛼−𝑆+𝐼+𝜈₀,𝐹−𝑆₂(𝑆,𝐼)=𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼. (15)
From [17], we can know that the functions 𝐹₁(𝑆,𝐼) and 𝐹₂(𝑆,𝐼) are said to possess a quasi-monotone nonincreasing system, if the sign of 𝜕𝐹₁(𝑆,𝐼)/𝜕𝐼 and 𝜕𝐹₂(𝑆,𝐼)/𝜕𝑆 are both nonpositive.
Since 𝜕𝐹₂(𝑆,𝐼)𝜕𝑆=−𝑅₀𝐼𝛼₀−𝑆+𝐼²≤0,𝜕𝐹₁(𝑆,𝐼)𝜕𝐼=𝑅₀𝛼₀−𝑆𝛼₀−𝑆+𝐼²+2𝜈𝑅_𝑑𝛼₀−𝑆+𝐼−𝜈𝑅_𝑑,𝜕𝜕𝑆𝜕𝐹₁(𝑆,𝐼)𝜕𝐼=−2𝜈𝑅_𝑑−2𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼³≤−2𝜈𝑅_𝑑<0. (16) Then, 𝜕𝐹₁(𝑆,𝐼)𝜕𝐼≤𝑅₀𝛼₀𝛼₀+𝐼²+2𝜈𝑅_𝑑𝛼₀+𝐼−𝜈𝑅_𝑑. (17) Let 𝐺(𝑧)=𝛼²₀𝑅₀𝑧²+2𝜈𝑅_𝑑𝑧−𝜈𝑅_𝑑,𝑧∈𝛼₀,𝛼₀+𝐼^∗, (18) then 𝐺^′(𝑧)=2𝜈𝑅_𝑑−2𝛼²₀𝑅₀𝑧³=0, (19) obviously, 𝑧^∗=³𝛼²₀𝑅₀/𝜈𝑅_𝑑 is the unique real root of 𝐺^′(𝑧).
Since 𝜈>27𝑅₀(𝑅_𝑑−1)²/𝑅³_𝑑, consider 𝛼₀=(𝑅_𝑑−1)/𝑅_𝑑, then we can get 𝐺𝑧^∗=𝛼²₀𝑅₀^2/327𝛼²₀𝑅₀^1/3−𝜈𝑅_𝑑^1/3(𝑧^∗)²<0. (20) And lim_𝑧→0+𝐺(𝑧)=lim_𝑧→+∞𝐺(𝑧)=+∞; (21) hence, 𝐺(𝑧) has two positive roots.
Since 𝜈≥𝑅₀/(2−𝑅_𝑑), thus 𝐺(𝛼₀)=𝑅₀+𝜈𝑅_𝑑−2𝜈≤0.
According to conditions [𝐻1] and [𝐻2], we can get 𝐺𝛼₀+𝐼^∗=𝜈𝑅_𝑑−2𝜈+2𝜈𝑅_𝑑𝐼^∗+𝑅₀𝛼₀𝛼₀+𝐼^∗2<𝜈𝑅_𝑑−2𝜈+2𝜈𝑅_𝑑𝐼^∗+𝑅₀=3𝑅²₀𝑅_𝑑−2𝑅²₀−2𝑅₀𝑅_𝑑−2𝑅₀+2𝜈𝑅²₀+𝑅³₀−2𝑅²₀+4𝑅₀−2𝑅²₀≤0. (22) Then, 𝐺([𝛼₀,𝛼₀+𝐼^∗])≤0. Hence, 𝜕𝐹₁(𝑆,𝐼)/𝜕𝐼≤0.
That is to say, model (14) is a quasi-monotone system in (𝑆,𝐼)∈[𝐸₁,𝐸₂].

Since the traveling wave solution of model (14) has the following form 𝑆(𝜉)=𝑆(𝑥+𝑐𝑡),𝐼(𝜉)=𝐼(𝑥+𝑐𝑡),𝜉=𝑥+𝑐𝑡,𝑐>0; (23) substituting (23) into model (14), we can get the following model: 𝑆^′′−𝑐𝑆^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀𝛼−𝑆+𝐼+𝜈₀𝐼−𝑆=0,_′′−𝑐𝐼^′+𝑅₀𝛼₀−𝑆𝐼𝛼₀()−𝑆+𝐼−𝐼=0,𝑆,𝐼^𝑇(−∞)=(0,0)^𝑇,(𝑆,𝐼)^𝑇(+∞)=𝑆^∗,𝐼^∗𝑇. (24)

Obviously, we can know the following.

Remark 4. Model (24) is also a quasi-monotone system in (𝑆,𝐼)∈[𝐸₁,𝐸₂].

Now we establish the existence of traveling wave solutions of model (24) through monotone iteration of a pair of smooth upper and lower solutions. Following [17], we give the definitions of the upper and lower solutions of model (24) as follows, respectively.

Definition 5. A smooth function (𝑆(𝜉),𝐼(𝜉))^𝑇  (𝜉∈ℝ) is an upper solution of model (24) if its derivatives (𝑆^′,𝐼^′)^𝑇 and (𝑆^′′,𝐼^′′) are continuous on ℝ, and (𝑆,𝐼)^𝑇 satisfies 𝑆^′′−𝑐𝑆^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀𝛼−𝑆+𝐼+𝜈₀𝐼−𝑆≤0,_′′−𝑐𝐼^′+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼≤0, (25) with the following boundary value conditions 𝑆𝐼()=00,𝑆𝐼()≥𝑆−∞+∞^∗𝐼^∗. (26)

Definition 6. A smooth function (𝑆(𝜉),𝐼(𝜉))^𝑇  (𝜉∈ℝ) is a lower solution of model (24) if its derivatives (𝑆^′,𝐼^′)^𝑇 and (𝑆^′′,𝐼^′′) are continuous on ℝ, and (𝑆,𝐼)^𝑇 satisfies 𝑆^′′−𝑐𝑆^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀𝛼−𝑆+𝐼+𝜈₀𝐼−𝑆≥0,_′′−𝑐𝐼^′+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼≥0, (27) with the following boundary value conditions 𝑆𝐼()=00,𝑆𝐼()≤𝑆−∞+∞^∗𝐼^∗. (28)

The construction of the smooth upper-lower solution pair is based on the solution of the following KPP equation: 𝑤^′′−𝑐𝑤^′(𝑤)𝑤()()+𝑓=0,−∞=0,𝑤+∞=𝑏, (29) where 𝑓∈𝐶²([0,𝑏]) and 𝑓>0 in the open interval (0,𝑏) with 𝑓(0)=𝑓(𝑏)=0, 𝑓^′(0)=𝑎₁>0, and 𝑓^′(𝑏)=−𝑏₁<0 [15]. First, let us recall the following result.

Lemma 7 (see [1, 15]). Corresponding to every 𝑐≥2√𝑎₁, model (29) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution 𝑤(𝜉) for 𝜉∈𝑅. The traveling wave solution 𝑤 has the following asymptotic behaviors.(i) For the wave solution with noncritical speed 𝑐>2√𝑎₁, one has 𝑤(𝜉)=𝑎_𝜔𝑒^{((𝑐−𝑐2−4𝑎1)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4𝑎1)/2)𝜉}𝑤(𝜉)𝑎𝑠𝜉⟶−∞,=𝑏−𝑏_𝜔𝑒^{((𝑐−𝑐2+4𝑏1)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝑏1)/2)𝜉}𝑎𝑠𝜉⟶+∞, (30)where 𝑎_𝜔 and 𝑏_𝜔 are positive constants.(ii) For the wave with critical speed 𝑐=2√𝑎₁, one has 𝑤(𝜉)=𝑎_𝑐+𝑑_𝑐𝜉𝑒^√𝑎1𝜉+𝑜𝜉𝑒^√𝑎1𝜉𝑤(𝜉)𝑎𝑠𝜉⟶−∞,=𝑏−𝑏_𝑐𝑒^{(√𝑎1−𝑎1+𝑏1)𝜉}+𝑜𝑒^{(√𝑎1−𝑎1+𝑏1)𝜉}𝑎𝑠𝜉⟶+∞, (31) where the constant 𝑑_𝑐 is negative, 𝑏_𝑐 is positive, and 𝑎_𝑐∈𝑅.

For constructing the upper solution of the model (24), we start with the following model: 𝐼^′′−𝑐𝐼^′+𝐼𝐼^∗𝛼−𝐼⁰𝑅₀−1𝛼₀𝐼^∗+𝐼^∗−𝑆^∗𝐼𝐼()()=𝐼=0,−∞=0,𝐼+∞^∗. (32)

Define 𝑓(𝐼)=𝐼(𝐼^∗−𝐼)(𝛼₀(𝑅₀−1)/𝜙(𝐼)), 𝐼∈[0,𝐼^∗], one can verify that all of the following conditions are satisfied:(i)𝑓(𝐼)=𝐼(𝐼^∗−𝐼)(𝛼₀(𝑅₀−1)/𝜙(𝐼))∈𝐶²([0,𝐼^∗]);(ii)𝑓(𝐼)>0,for all 𝐼∈(0,𝐼^∗) and 𝑓(0)=𝑓(𝐼^∗)=0;(iii)𝑓^′(0)=𝑅₀−1>0, 𝑓^′(𝐼^∗)=−𝛼₀(𝑅₀−1)²/𝑅₀𝐼^∗<0.

From Lemma 7, we know that, for each 𝑐≥2𝑅₀−1, equation (32) has a unique traveling wave solution 𝐼(𝜉) (up to a translation of the origin), satisfying the given boundary value conditions (26).

Define 𝑆(𝜉)𝐼(𝜉)=𝑆^∗𝐼^∗𝐼(𝜉)𝐼(𝜉),𝜉∈𝑅, (33) then we can get the following result.

Lemma 8. For each 𝑐≥2𝑅₀−1, (33) is a smooth upper solution of model (24).

Proof. On the boundary, 𝑆𝐼()=00−∞,𝑆𝐼()≥𝑆+∞^∗𝐼^∗. (34) As for the 𝐼 component, we have 𝐼^′′−𝑐𝐼^′+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼=−𝐼𝐼^∗−𝐼𝛼₀𝑅₀−1𝜙𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼=−𝐼𝐼^∗−𝐼𝛼₀𝑅₀−1𝜙𝐼+𝑅₀𝛼₀𝐼^∗−𝑆^∗𝐼𝜙𝐼𝐼−𝐼=−𝐼𝐼^∗−𝐼𝛼₀𝑅₀−1𝜙𝐼+𝐼𝛼₀𝑅₀−1𝐼^∗−𝑅₀𝑆^∗−𝑆^∗+𝐼^∗𝐼𝜙𝐼=−𝐼𝐼^∗−𝐼𝛼₀𝑅₀−1𝜙𝐼+𝐼𝛼₀𝑅₀−1𝐼^∗−𝛼₀𝑅₀−1𝐼𝜙𝐼=0. (35) As for the 𝑆 component, since 𝜈>1, then 𝐼^∗−𝑆^∗=(𝜈−1)(𝑅₀−1)/𝜈𝑅₀𝑅_𝑑>0. And(i)if 𝜂₀<𝐼^∗/2, then max_{𝐼𝜉∈[0,}^∗_]𝜓(𝐼)=𝜓(𝐼^∗)=𝜓(𝐵);(ii)if 𝜂₀≥𝐼^∗/2, then max_{𝐼𝜉∈[0,}^∗_]𝜓(𝐼)=𝜓(0)=𝜓(𝐵).
Thus we can get: 𝑆^′′−𝑐𝑆^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆=𝑆^∗𝐼^∗𝐼^′′−𝑐𝐼^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆=𝑆^∗𝐼^∗−𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝐼−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼×1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆=𝑆1−^∗𝐼^∗𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝑆^∗𝐼^∗𝐼−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼×1−𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆=𝐼^∗−𝑆^∗𝐼^∗𝑅₀𝛼₀𝐼^∗−𝑆^∗𝐼𝜙𝐼𝑆𝐼+^∗𝐼^∗𝐼−𝜈𝑅_𝑑𝜙𝐼𝐼^∗𝜙1−𝐼𝐼^∗𝛼+𝜈⁰𝐼^∗−𝑆^∗𝐼𝐼^∗=𝐼^∗−𝑆^∗𝐼𝜓𝐼𝐼^∗2𝜙𝐼≤0. (36) Hence, (𝑆,𝐼) forms a smooth upper solution for model (24).

For constructing the lower solution of the model (24), we start with the following model: 𝐼^′′−𝑐𝐼^′+𝐼𝐼^∗−()𝐼𝛼1+𝜀⁰𝑅₀−1𝛼₀𝐼^∗+𝐼^∗−𝑆^∗𝐼𝐼()()=𝐼=0,−∞=0,𝐼+∞^∗.1+𝜀 (37)

Define 𝑔(𝐼)=𝐼[𝐼^∗−(1+𝜀)𝐼](𝛼₀(𝑅₀−1)/(𝛼₀𝐼^∗+(𝐼^∗−𝑆^∗)𝐼)), 𝐼∈[0,𝐼^∗/(1+𝜀)]. One can easily verify that all of the following conditions hold:(i)𝑔(𝐼)=𝐼[𝐼^∗−(1+𝜀)𝐼](𝛼₀(𝑅₀−1)/(𝛼₀𝐼^∗+(𝐼^∗−𝑆^∗)𝐼))∈𝐶²([0,𝐼^∗/(1+𝜀)]);(ii)𝑔(𝐼)>0, for all 𝐼∈(0,𝐼^∗/(1+𝜀)) and 𝑔(0)=𝑔(𝐼^∗/(1+𝜀))=0;(iii)𝑔^′(0)=𝑅₀−1>0, 𝑔^′(𝐼^∗/(1+𝜀))=−(1+𝜀)𝛼₀(𝑅₀−1)/(𝜀𝛼₀+(𝑅₀/(𝑅₀−1))𝐼^∗)<0.

From Lemma 7, we know that, for each fixed 𝑐≥2𝑅₀−1, model (37) has a unique traveling wave solution 𝐼(𝜉) (up to a translation of the origin), satisfying the given boundary value conditions (28).

Define 𝑆(𝜉)𝐼(𝜉)=𝑆^∗𝐼^∗𝐼(𝜉)𝐼(𝜉),𝜉∈𝑅, (38) then we have the following result:

Lemma 9. For each fixed 𝑐≥2𝑅₀−1, (38) is a lower solution of model (24).

Proof. On the boundary, 𝑆𝐼()=00,𝑆−∞𝐼()=𝑆+∞^∗𝐼1+𝜀_∗≤𝑆1+𝜀^∗𝐼^∗. (39) As for the 𝐼 component, we have 𝐼^′′−𝑐𝐼^′+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼=−𝐼𝐼_∗−(1+𝜀)𝐼𝛼₀𝑅₀−1𝜙𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼=−𝐼𝐼_∗−(1+𝜀)𝐼𝛼₀𝑅₀−1𝜙𝐼+𝐼𝐼^∗−𝐼𝛼₀𝑅₀−1𝜙𝐼=𝜀𝐼²𝛼₀𝑅₀−1𝜙𝐼≥0. (40) As for the 𝑆 component, we have 𝑆^′′−𝑐𝑆^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆=𝑆^∗𝐼^∗𝐼^′′−𝑐𝐼^′−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆=𝑆^∗𝐼^∗𝜀𝐼²𝛼₀𝑅₀−1𝜙𝐼−𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼+𝜈𝛼₀−𝑆𝑆=𝜀^∗𝐼^∗𝐼²𝛼₀𝑅₀−1𝜙𝐼+𝐼^∗−𝑆^∗𝐼𝜓𝐼𝐼^∗2𝜙𝐼≥0. (41) Thus (𝑆,𝐼) forms a smooth lower solution for model (24).

Next, we show that, by shifting the upper solution far enough to the left, then the upper-lower solution in Lemmas 8 and 9 are ordered.

Lemma 10. Let 𝑐≥2𝑅₀−1, (𝑆,𝐼)^𝑇 and (𝑆,𝐼)^𝑇 be the upper solution and the lower solution defined in (33) and (38), then there exists a positive number 𝑟, such that (𝑆,𝐼)^𝑇(𝜉+𝑟)≥(𝑆,𝐼)^𝑇(𝜉) for all 𝜉∈𝑅.

Proof. Our proof is only for 𝑐>2𝑅₀−1, and the proof for the case of 𝑐=2𝑅₀−1 is similar to it.
First, we derive the asymptotic behaviors of the upper solution and the lower solution at infinities.
According to Lemma 7, when 𝜉→−∞, we can obtain: 𝑆𝐼(𝜉)=𝑆^∗𝐼^∗𝐴₁𝐴₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉},𝑆𝐼(𝜉)=𝑆^∗𝐼^∗𝐵₁𝐵₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}. (42)
And let 𝜎₀=(1/2)(𝑐−𝑐²+4(𝛼₀(𝑅₀−1)²/𝑅₀𝐼^∗)<0, 𝛿₀=(1/2)(𝑐−𝑐²+4((1+𝜀)𝛼₀(𝑅₀−1)/(𝜀𝛼₀+(𝑅₀/(𝑅₀−1))𝐼^∗))<0, when 𝜉→+∞, we can get 𝑆𝐼(𝜉)=𝑆^∗𝐼^∗−𝑆^∗𝐼^∗𝐴₂𝐴₂𝑒^𝜎0𝜉+𝑜𝑒^𝜎0𝜉,𝑆𝐼(𝜉)=1𝑆1+𝜀^∗𝐼^∗−𝑆^∗𝐼^∗𝐵₂𝐵₂𝑒^𝛿0𝜉+𝑜𝑒^𝛿0𝜉, (43) where, 𝐴₁, 𝐴₂, 𝐵₁, 𝐵₂ are all positive constants.
Since for any 𝑟̃>0, 𝐼𝑟̃(𝜉)≡𝐼(𝜉+𝑟)̃ is also a solution of model (32). Thus, (𝑆𝑟̃,𝐼𝑟̃)^𝑇(𝜉) is an upper solution of model (24). So, according to Lemma 7, when 𝜉→−∞, we can get: 𝑆𝑟̃𝐼𝑟̃(𝜉)=𝑆^∗𝐼^∗𝐴₁𝐴₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)}𝑟𝑒̃_{((𝑐−𝑐}²_{−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}. (44)
Since (𝑐−𝑐²−4(𝑅₀−1))/2>0, we can choose a large enough number 𝑟̃≫0, such that 𝑆^∗𝐼^∗𝐴₁𝐴₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)}𝑟̃>𝑆^∗𝐼^∗𝐵₁𝐵₁, (45) hence, there exists a large number 𝑁₁≫1, such that 𝑆𝑟̃(𝜉)𝐼𝑟̃(𝜉)>𝑆(𝜉)𝐼(𝜉),𝜉∈−∞,−𝑁₁. (46)
By using a similar argument as above, there exists a large enough number 𝑁₂≫1, such that 𝑆𝑟̃(𝜉)𝐼𝑟̃(𝜉)>𝑆(𝜉)𝐼(𝜉),𝜉∈𝑁₂,+∞. (47)
Second, we show that 𝑆𝑟̃(𝜉)𝐼𝑟̃(𝜉)>𝑆(𝜉)𝐼(𝜉),𝜉∈−𝑁₁,𝑁₂. (48)
We deal with such two possible cases:
Case  1. If 𝑆𝑟̃(𝜉)𝐼𝑟̃(𝜉)>𝑆(𝜉)𝐼(𝜉),𝜉∈−𝑁₁,𝑁₂, (49) then, the proof is completed.
Case  2. If there exists a point 𝜉₀∈(−𝑁₁,𝑁₂), such that 𝑆𝑟̃𝜉₀𝐼𝑟̃𝜉₀≤𝑆𝜉₀𝐼𝜉₀ (50) satisfying  𝑆𝑟̃(𝜉₀)<𝑆(𝜉₀) or 𝐼𝑟̃(𝜉₀)<𝐼(𝜉₀).
In this case, we use the Sliding Domain method [15].
Step  1. we shift (𝑆𝑟̃,𝐼𝑟̃)^𝑇 to the left by increasing the number 𝑟̃ until finding a new number 𝑟₁>𝑟̃ such that (𝑆^𝑟1,𝐼^𝑟1)^𝑇>(𝑆,𝐼)^𝑇 on the smaller interval [−𝑁₁,𝑁₂−(𝑟₁−𝑟)̃].
Step  2. we shift (𝑆^𝑟1,𝐼^𝑟1)^𝑇 back to the right by decreasing 𝑟₁ to a smaller number 𝑟̃<𝑟₂<𝑟₁ such that one of the branches of the upper solution touches its counterpart of the lower solution at some point 𝜉₁ in the interval (−𝑁₁+(𝑟₁−𝑟₂),𝑁₂−(𝑟₁−𝑟)̃). On the endpoints of the interval (−𝑁₁+(𝑟₁−𝑟₂),𝑁₂−(𝑟₁−𝑟)̃), we still have (𝑆^𝑟2,𝐼^𝑟2)^𝑇>(𝑆,𝐼)^𝑇.
Let 𝑊^→(𝜉)=(𝑆^𝑟2,𝐼^𝑟2)^𝑇−(𝑆,𝐼)^𝑇 and ^→𝐹=(𝐹₁,𝐹₂)^𝑇, where 𝐹₁=−𝜈𝑅_𝑑𝛼₀−𝑆+𝐼1−𝛼₀−𝑆+𝐼+𝑅₀𝛼₀−𝑆𝐼𝛼₀𝛼−𝑆+𝐼+𝜈₀,𝐹−𝑆₂=𝑅₀𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼−𝐼. (51)
For 𝜉∈(−𝑁₁+(𝑟₁−𝑟₂),𝑁₂−(𝑟₁−𝑟)̃), we get that 𝑊^→′′−𝑐𝑊^→′+𝜕𝐹₁𝑆𝜕𝑆+𝜁₁𝜔₁,𝐼^𝑟2𝜕𝐹₁𝜕𝐼𝑆^𝑟2,𝐼+𝜁₂𝜔₂𝜕𝐹₂𝑆𝜕𝑆+𝜁₃𝜔₁,𝐼^𝑟2𝜕𝐹₂𝜕𝐼𝑆^𝑟2,𝐼+𝜁₄𝜔₂^→𝑊=0, (52) where 𝜁_𝑖∈[0,1], 𝑖=1,2,3,4. Since the above model is monotone and the cube [(0,0),(𝑆^∗,𝐼^∗)] is convex, thus we can deduce by Maximum Principle that 𝑊^→>0 for 𝜉∈[−𝑁₁+(𝑟₁−𝑟₂),𝑁₂−(𝑟₁−𝑟)̃]. So 𝜉₁ does not exist and we can decrease 𝑟₂ further to 𝑟̃. It is calculated that the point 𝜉₀ does not exist either. The proof of this lemma is completed.

To ease the burden of notations, we still use (𝑆,𝐼)^𝑇 to denote the shifted upper solution as given in Lemma 8. Let 𝐷₁₁𝑅=−²⁰+𝜈𝑅₀𝑅_𝑑+𝜈𝑅₀−4𝑅₀−2𝜈+3𝑅₀,𝐷₁₂=𝜈𝑅₀𝑅_𝑑−2𝑅₀−2𝜈+3𝑅₀,𝐷₂₁𝑅=−⁰−1²𝑅₀,𝐷₂₂𝑅=−⁰−1𝑅₀,𝜇₁=−𝐷₁₁+𝐷₂₂+𝐷₁₁−𝐷₂₂²+4𝐷₁₂𝐷₂₁2,𝜇₂=−𝐷₁₁+𝐷₂₂−𝐷₁₁−𝐷₂₂²+4𝐷₁₂𝐷₂₁2. (53)

With such constructed ordered upper-lower solution pair, we can get the following.

Theorem 11. For 𝑐≥2𝑅₀−1, model (24) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:(i) if 𝑐>2𝑅₀−1, when 𝜉→−∞, 𝑆𝐼(𝜉)=𝐴₁𝐴₂𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}. (54)when 𝜉→+∞, and if 𝜇₁≠𝜇₂, then 𝑆𝐼(𝜉)=𝑆^∗𝐼^∗−𝐴₁𝐴₂𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}, (55) while 𝜇₁=𝜇₂: 𝑆𝐼(𝜉)=𝑆^∗𝐼^∗−𝐴₁₁+𝐴₁₂𝜉𝐴₂₁+𝐴₂₂𝜉𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}, (56) where, 𝜇=min{𝜇₁,𝜇₂}>0, 𝐴₁₁,𝐴₂₁∈ℝ, 𝐴₁, 𝐴₂, 𝐴₁, 𝐴₂, 𝐴₁₂ and 𝐴₂₂ are all positive constants.(ii) if 𝑐=2𝑅₀−1, when 𝜉→−∞, 𝑆𝐼(𝜉)=𝐵₁₁+𝐵₁₂𝜉𝐵₂₁+𝐵₂₂𝜉𝑒^{𝑅0−1𝜉}+𝑜𝜉𝑒^{𝑅0−1𝜉}, (57)when 𝜉→+∞, and if 𝜇₁≠𝜇₂, then 𝑆𝐼(𝜉)=𝑆^∗𝐼^∗−𝐵₁₁𝐵₂₂𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}+𝑜𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}, (58) while 𝜇₁=𝜇₂, 𝑆𝐼(𝜉)=𝑆^∗𝐼^∗−𝐵₁₁+𝐵₁₂𝜉𝐵₂₁+𝐵₂₂𝜉𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}+𝑜𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}, (59) where 𝜇=min{𝜇₁,𝜇₂}>0, 𝐵₁₂,𝐵₂₂<0, 𝐵₁₁,𝐵₂₁,𝐵₁₁,𝐵₂₁∈ℝ, and 𝐵₁₁, 𝐵₂₂, 𝐵₁₂, 𝐵₂₂ are all positive constants.

Proof. From Lemma 3 and Remark 4, we know that model (24) is a quasi-monotone nonincreasing system in (𝑆,𝐼)∈[𝐸₁,𝐸₂], and by using the monotone iteration scheme given in [3, 13], we can obtain the existence of the solution (𝑆,𝐼)^𝑇 to the first two equations in model (24) for every 𝑐≥2𝑅₀−1, which satisfies 𝑆(𝜉)𝐼(𝜉)≤𝑆(𝜉)𝐼(𝜉)≤𝑆(𝜉)𝐼(𝜉). (60)
According to the above inequality, we can get that, on the boundary, the solution tends to (0,0)^𝑇 as 𝜉→−∞ and (𝑆^∗,𝐼^∗)^𝑇 as 𝜉→+∞.
To derive the asymptotic decay rate of the traveling wave solutions as 𝜉→±∞, we just let 𝑐>2𝑅₀−1 and 𝑈(𝜉)=(𝑆(𝜉),𝐼(𝜉))^𝑇,−∞<𝜉<+∞ (61) be the traveling wave solution of model (24) generated form the monotone iteration, since the case of (ii) 𝑐=2𝑅₀−1 is similar to it.
We differentiate model (24) with respect to 𝜉, and note that 𝑈^′(𝜉)=(𝜒₁,𝜒₂)^𝑇(𝜉) satisfies 𝜒₁^′′−𝑐𝜒^′₁+𝐶₁₁(𝑆,𝐼)𝜒₁+𝐶₁₂(𝑆,𝐼)𝜒₂𝜒=0,_2′′−𝑐𝜒^′₂+𝐶₂₁(𝑆,𝐼)𝜒₁+𝐶₂₂(𝑆,𝐼)𝜒₂=0, (62) where 𝐶₁₁(𝑆,𝐼)=𝜈𝑅_𝑑1−𝛼₀−𝑆+𝐼−𝜈𝑅_𝑑𝛼₀−𝑅−𝑆+𝐼₀𝐼𝛼₀+𝑅−𝑆+𝐼⁰𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼²𝐶−𝜈,₁₂(𝑆,𝐼)=−𝜈𝑅_𝑑1−𝛼₀−𝑆+𝐼+𝜈𝑅_𝑑𝛼₀+𝑅−𝑆+𝐼₀𝛼₀−𝑆𝛼₀−𝑅−𝑆+𝐼⁰𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼²,𝐶₂₁()𝑅𝑆,𝐼=−⁰𝐼𝛼₀+𝑅−𝑆+𝐼⁰𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼²,𝐶₂₂()=𝑅𝑆,𝐼⁰𝛼₀−𝑆𝛼₀−𝑅−𝑆+𝐼⁰𝛼₀−𝑆𝐼𝛼₀−𝑆+𝐼²−1. (63)
Now, we study the exponential decay rate of the traveling wave solution as 𝜉→−∞. The asymptotic model of model (62) as 𝜉→−∞ is 𝜆^′′−𝑐𝜆^′+𝐸₁₁𝜆+𝐸₁₂𝜇𝜇=0,_′′−𝑐𝜇^′+𝐸₂₁𝜆+𝐸₂₂𝜇=0, (64) where 𝐸₁₁=−𝜈𝑅_𝑑−1,𝐸₁₂=𝜈𝑅_𝑑+𝑅₀𝐸−2𝜈,₂₁=0,𝐸₂₂=𝑅₀−1. (65)
The second equation of model (64) has two independent solutions with the following form: 𝜇⁽¹⁾(𝜉)=𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉},𝜇⁽²⁾(𝜉)=𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜉}. (66)
Relating the second equation of model (62) with the second equation of model (64), we can deduce that 𝜒₂(𝜉) has the following property as 𝜉→−∞: 𝜒₂(𝜉)=𝛼[1+𝑜(1)]𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝛽[1+𝑜(1)]𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜉} (67) for some constants 𝛼 and 𝛽. Thus, we can obtain that 𝜒₂(𝜉)=𝛼𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝛽𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜉}+Υ₁(𝜉)+Υ₂(𝜉), (68) where lim_{𝜉→−∞}Υ₁(𝜉)𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}=0,lim_{𝜉→−∞}Υ₂(𝜉)𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜉}=0. (69) So we obtain that lim_{𝜉→−∞}𝜒₂(𝜉)−𝛼𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}=lim_{𝜉→−∞}Υ₁(𝜉)+Υ₂(𝜉)+𝛽𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜉}𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}=lim_{𝜉→−∞}Υ₁(𝜉)𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝛽lim_{𝜉→−∞}𝑒^{𝑐2−4(𝑅0−1)𝜉}+lim_{𝜉→−∞}Υ₂(𝜉)𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜉}lim_{𝜉→−∞}𝑒^{𝑐2−4(𝑅0−1)𝜉}=0. (70) Thus, 𝜒₂(𝜉)=𝛼[1+𝑜(1)]𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}.
Now, we consider the first equation of model (64). We rewrite it as 𝜆^′′−𝑐𝜆^′−𝜈𝑅_𝑑−1𝜆=−𝜈𝑅_𝑑+𝑅₀−2𝜈𝜇. (71)
One can verify that (𝑐−𝑐²−4(𝑅₀−1))/2 is not a characteristic of 𝜆^′′−𝑐𝜆^′−𝜈𝑅_𝑑−1𝜆=0. (72)
The above equation has two independent solutions of the following form: 𝜆⁽¹⁾(𝜉)=𝑒^{((𝑐−𝑐2+4𝜈(𝑅𝑑−1))/2)𝜉},𝜆⁽²⁾(𝜉)=𝑒^{((𝑐+𝑐2+4𝜈(𝑅𝑑−1))/2)𝜉}. (73) Thus, when 𝜉→−∞, 𝜒₁(𝜉) has the following property: 𝜒₁(𝜉)=𝛼[1+𝑜(1)]𝑒^{((𝑐+𝑐2+4𝜈(𝑅𝑑−1))/2)𝜉}+𝛽[1+𝑜(1)]𝑒^{((𝑐−𝑐2+4𝜈(𝑅𝑑−1))/2)𝜉}+𝛾[1+𝑜(1)]𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉} (74) for some constants 𝛼,𝛽; 𝛾≠0. Since 𝜒₁(−∞)=0, thus 𝛽=0. So, when 𝜉→−∞, we have the following formula: 𝜒₁(𝜉)𝜒₂(𝜉)=𝛾[1+𝑜(1)]𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}𝛼[1+𝑜(1)]𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}. (75)
Then, we study the exponential decay rate of the traveling wave solution as 𝜉→+∞. The asymptotic model of model (62) as 𝜉→+∞ is 𝜓₁^′′−𝑐𝜓^′₁+𝐷₁₁𝜓₁+𝐷₁₂𝜓₂𝜓=0,_2′′−𝑐𝜓^′₂+𝐷₂₁𝜓₁+𝐷₂₂𝜓₂=0. (76)
By setting (𝜓_𝑖)^′(𝜉)=𝜓_𝑖, 𝑖=1,2, we rewrite model (76) as a first order model of ordinary differential equation in the four components (𝜓₁,𝜓₁,𝜓₂,𝜓₂)^𝑇: 𝜓^′₁=𝜓₁,𝜓^′₁=𝑐𝜓₁−𝐷₁₁𝜓₁−𝐷₁₂𝜓₂,𝜓^′₂=𝜓₂,𝜓^′₂=𝑐𝜓₂−𝐷₂₁𝜓₁−𝐷₂₂𝜓₂. (77)
In the case of (i) 𝜇₁≠𝜇₂, we can obtain that the solution of model (77) has the following form: 𝜓₁,𝜓₁,𝜓₂,𝜓₂^𝑇=⁴_𝑖=1𝑐_𝑖ℎ_𝑖𝑒^𝜆𝑖𝜉, (78) where 𝜆₁=𝑐+𝑐²+4𝜇₁2,𝜆₂=𝑐−𝑐²+4𝜇₁2,𝜆₃=𝑐+𝑐²+4𝜇₂2,𝜆₄=𝑐−𝑐²+4𝜇₂2, (79) and ℎ_𝑖(𝑖=1,2,3,4) are the eigenvectors of the constant matrix with 𝜆_𝑖(𝑖=1,2,3,4) as the corresponding eigenvalues, 𝑐_𝑖(𝑖=1,2,3,4) are arbitrary constants. Since lim_𝜉→+∞𝜓₁,𝜓₁,𝜓₂,𝜓₂^𝑇=0, (80) thus (𝜓₁,𝜓₁,𝜓₂,𝜓₂)^𝑇=𝑐₂ℎ₂𝑒^𝜆2𝜉+𝑐₄ℎ₄𝑒^𝜆4𝜉, so when 𝜉→+∞, we can get that 𝜒₁(𝜉)𝜒₂(𝜉)=𝜅₁Λ₁+𝑜(1)𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}𝜅₁Γ₁+𝑜(1)𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}+𝜅₂Λ₂+𝑜(1)𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}𝜅₂Γ₂+𝑜(1)𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}. (81)
Furthermore, we can obtain that 𝜒₁(𝜉)=𝜅₁Λ₁𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}+𝜅₂Λ₂𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}+Ω₁₁(𝜉)+Ω₁₂(𝜉),𝜒₂(𝜉)=𝜅₁Γ₁𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}+𝜅₂Γ₂𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}+Ω₂₁(𝜉)+Ω₂₂(𝜉), (82) where lim_𝜉→+∞Ω₁₁(𝜉)𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}=0,lim_𝜉→+∞Ω₁₂(𝜉)𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}=0,lim_𝜉→+∞Ω₂₁(𝜉)𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}=0,lim_𝜉→+∞Ω₂₂(𝜉)𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}=0. (83)𝜅₁, 𝜅₂, Λ₁, Λ₂, Γ₁, and Γ₂ are all constants.
Let 𝜇=min{𝜇₁,𝜇₂}, then lim_𝜉→+∞𝜒₁(𝜉)𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}=𝜅₁Λ₁lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇1)/2)𝜉}+𝜅₂Λ₂lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇2)/2)𝜉}+lim_𝜉→+∞Ω₁₁(𝜉)𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇1)/2)𝜉}+lim_𝜉→+∞Ω₁₂(𝜉)𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇2)/2)𝜉}=Δ₁𝜅₁,𝜅₂,Λ₁,Λ₂,lim_𝜉→+∞𝜒₂(𝜉)𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}=𝜅₁Γ₁lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇1)/2)𝜉}+𝜅₂Γ₂lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇2)/2)𝜉}+lim_𝜉→+∞Ω₂₁(𝜉)𝑒^{((𝑐−𝑐2+4𝜇1)/2)𝜉}lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇1)/2)𝜉}+lim_𝜉→+∞Ω₂₂(𝜉)𝑒^{((𝑐−𝑐2+4𝜇2)/2)𝜉}lim_𝜉→+∞𝑒^{((𝑐2+4𝜇−𝑐2+4𝜇2)/2)𝜉}=Δ₂𝜅₁,𝜅₂,Γ₁,Γ₂, (84) where Δ₁𝜅₁,𝜅₂,Λ₁,Λ₂=𝜅₁Λ₁,𝜇₁<𝜇₂,𝜅₂Λ₂,𝜇₁>𝜇₂,Δ₂𝜅₁,𝜅₂,Γ₁,Γ₂=𝜅₁Γ₁,𝜇₁<𝜇₂,𝜅₂Γ₂,𝜇₁>𝜇₂; (85) thus, when 𝜉→+∞, we can get that 𝜒₁(𝜉)𝜒₂(𝜉)=Δ₁𝜅₁,𝜅₂,Λ₁,Λ₂(1+𝑜(1))𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}Δ₂𝜅₁,𝜅₂,Γ₁,Γ₂(1+𝑜(1))𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}. (86)
In the case of (ii) 𝜇₁=𝜇₂, we can obtain that the solution of model (77) has the following form: 𝜓₁,𝜓₁,𝜓₂,𝜓₂^𝑇=𝐺₁+𝐺₂𝜉𝐻_1,2𝑒^𝜆1𝜉+𝐺₃+𝐺₄𝜉𝐻_3,4𝑒^𝜆3𝜉, (87) where 𝐻_1,2 is the eigenvector of the constant matrix with 𝜆₁=𝜆₂ as the corresponding eigenvalues, 𝐻_3,4 is the eigenvector of the constant matrix with 𝜆₃=𝜆₄ as the corresponding eigenvalues, 𝐺_𝑖(𝑖=1,2,3,4) are arbitrary constants.
Since lim_𝜉→+∞(𝜓₁,𝜓₁,𝜓₂,𝜓₂)^𝑇=0, thus 𝜓₁,𝜓₁,𝜓₂,𝜓₂^𝑇=𝐺₃+𝐺₄𝜉𝐻_3,4𝑒^𝜆3𝜉. (88) So, when 𝜉→+∞, we can get that 𝜒₁(𝜉)𝜒₂(𝜉)=𝐺_1,3+𝐺_1,4𝜉𝑒^𝜆3𝜉+𝑜𝑒^𝜆3𝜉𝐺_2,3+𝐺_2,4𝜉𝑒^𝜆4𝜉+𝑜𝑒^𝜆4𝜉. (89)
By comparing the upper solution and roughness of the exponential dichotomy [24], we obtain the asymptotic decay rate of the traveling wave solutions at +∞ given in Theorem 11.
According to the monotone iteration process [3], the traveling wave solution 𝑈(𝜉) is increasing; thus 𝑈^′(𝜉)≥0 and 𝑈^′(𝜉)=(𝜒₁,𝜒₂)^𝑇(𝜉) hold 𝜒₁^′′−𝑐𝜒^′₁+𝜕𝐹₁𝜒𝜕𝑆¹+𝜕𝐹₁𝜒𝜕𝐼²𝜒=0,_2′′−𝑐𝜒^′₂+𝜕𝐹₂𝜒𝜕𝑆¹+𝜕𝐹₂𝜒𝜕𝐼²=0, (90) satisfying 𝜒₁,𝜒₂^𝑇(𝜉)≥0,𝜒₁,𝜒₂^𝑇(±∞)=0. (91)
The strong Maximum Principle implies that (𝜒₁,𝜒₂)^𝑇(𝜉)>0. So the strict monotonicity of the traveling wave solutions is concluded.
Now, we use the Sliding domain method to prove the uniqueness of the traveling wave solution. Let 𝑈₁(𝜉)=(𝑆₁,𝐼₁)^𝑇(𝜉) and 𝑈₂(𝜉)=(𝑆₂,𝐼₂)^𝑇(𝜉) be the traveling wave solution of model (24), with 𝑐>2𝑅₀−1. Thus, there are some positive numbers 𝐴_𝑖,𝐵_𝑖(𝑖=1,2,3,4), such that for a big enough number 𝑁≫1, when 𝜉<−𝑁, we have 𝑆₁(𝜉)𝐼₁(𝜉)=𝐴₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}𝐴₂𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉},𝑆₂(𝜉)𝐼₂(𝜉)=𝐴₃𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}𝐴₄𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}, (92) when 𝜉>𝑁, 𝑆₁(𝜉)𝐼₁(𝜉)=𝑆^∗−𝐵₁𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}𝐼^∗−𝐵₂𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉},𝑆₂(𝜉)𝐼₂(𝜉)=𝑆^∗−𝐵₃𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}𝐼^∗−𝐵₄𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}. (93) Since the traveling wave solutions of model (24) are translation-invariant, then for any 𝜃∈𝑅, 𝑈^𝜃₁(𝜉)≡𝑈₁(𝜉+𝜃)≡(𝑆₁(𝜉+𝜃),𝐼₁(𝜉+𝜃))^𝑇 is also a traveling wave solution of model (24). Thus, by using the same method as above, when 𝜉<−𝑁, we can get 𝑆₁()𝐼𝜉+𝜃₁()=𝐴𝜉+𝜃₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜃}𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}𝐴₂𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜃}𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}, (94) when 𝜉>𝑁, 𝑆₁()𝐼𝜉+𝜃₁()=𝑆𝜉+𝜃_∗−𝐵₁𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜃}𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}𝐼^∗−𝐵₂𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜃}𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}. (95) If 𝜃 is large enough, then we can obtain the following inequalities: 𝐴₁𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜃}>𝐴₃,𝐴₂𝑒^{((𝑐+𝑐2−4(𝑅0−1))/2)𝜃}>𝐴₄,𝐵₁𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜃}<𝐵₃,𝐵₂𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜃}<𝐵₄. (96) Thus, if 𝜃 is large enough, then 𝑈^𝜃₁(𝜉)>𝑈₂(𝜉), for all 𝜉∈𝑅⧵[−𝑁,𝑁].
Now, we consider model (24) on the interval [−𝑁,𝑁].
First, suppose that 𝑊(𝜉)≡𝑈^𝜃₁(𝜉)−𝑈₂(𝜉)≥0,𝜉∈[−𝑁,𝑁], (97) then 𝑊^′′−𝑐𝑊^′+𝜕𝐹₁𝑆𝜕𝑆²+𝜁₁𝜔₁,𝐼₁𝜕𝐹₁𝑆𝜕𝐼¹,𝐼₂+𝜁₂𝜔₂𝜕𝐹₂𝑆𝜕𝑆²+𝜁₃𝜔₁,𝐼₁𝜕𝐹₂𝑆𝜕𝐼¹,𝐼₂+𝜁₄𝜔₂𝑊()(𝑁)𝑊=0,−𝑁>0,𝑊>0, (98) where, 𝜁_𝑖∈(0,1)(𝑖=1,2,3,4), 𝜉∈(−𝑁,𝑁). Since the above model is monotone, by the Maximum Principle, we can deduce that 𝑊(𝜉)>0,𝜉∈[−𝑁,𝑁]. Consequently, we get that 𝑈^𝜃₁(𝜉)>𝑈₂(𝜉),𝜉∈𝑅.
Second, we suppose that there exists a point 𝜉_∗∈(−𝑁,𝑁) such that 𝑆^𝜃₁𝜉_∗<𝑆₂𝜉_∗ (99) or 𝐼^𝜃₁𝜉_∗<𝐼₂𝜉_∗. (100)
In this case, we increase 𝜃, that is shifting 𝑈^𝜃₁ to the left, so that 𝑈^𝜃₁(−𝑁)>𝑈₂(−𝑁) and 𝑈^𝜃₁(𝑁)>𝑈₂(𝑁). According to the monotonicity of 𝑈^𝜃₁ and 𝑈₂, we can find a number 𝜃∈(0,2𝑁) such that 𝑈^𝜃₁(𝜉+𝜃)>𝑈₂(𝜉), 𝜉∈(−𝑁,𝑁). Shifting 𝑈^𝜃₁(𝜉+𝜃) back until one component of 𝑈^𝜃₁(𝜉+𝜃) touches its counterpart of 𝑈₂(𝜉) at some point 𝜉∈(−𝑁,𝑁). Since 𝑈^𝜃₁(𝜉+𝜃) and 𝑈₂(𝜉) are strictly increasing, 𝜉∈(−𝑁,𝑁), thus, we get that 𝑈^𝜃₁(𝜉+𝜃)>𝑈₂(𝜉), 𝜉=±𝑁. However, by the Maximum Principle for that component again, we find that components of 𝑈^𝜃₁ and 𝑈₂ are identically equal for all 𝜉∈[−𝑁,𝑁] for a larger number 𝜃. This is a contradiction, thus 𝑈^𝜃₁(𝜉)>𝑈₂(𝜉), 𝜉∈𝑅. Here, 𝜃 is a new number which is chosen by the above mean.
Now, decrease the 𝜃 until one of the following happens.
Case (a). There is a 𝜃≥0, such that 𝑈^𝜃₁=𝑈₂(𝜉), 𝜉∈𝑅. In this case, we have finished the proof.
Case (b). There are a 𝜃 and a point 𝜉₁∈𝑅, such that one of the components of 𝑈^𝜃₁ and 𝑈₂ are equal. And 𝑈^𝜃₁≥𝑈₂, 𝜉∈𝑅. On 𝑅 for that component, according to the Maximum Principle, we find that 𝑈^𝜃₁ and 𝑈₂ must be identical on that component. We can return to Case (a).
Consequently, in either situation, their exists a number 𝜃≥0 such that 𝑈^𝜃₁(𝜉)=𝑈₂(𝜉),𝜉∈(−∞,+∞). (101)
This ends of the proof.

By Theorem 11, we can get the following theorem:

Theorem 12. For each 𝑐≥2𝑅₀−1, model (3) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing and has the following asymptotic properties:(i)𝑐>2𝑅₀−1: when 𝜉→−∞, 𝑆(𝜉)=𝑅_𝑑−1𝑅_𝑑−𝐴₁𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉},𝐼(𝜉)=𝐴₂𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2−4(𝑅0−1))/2)𝜉}. (102)when 𝜉→+∞, and if 𝜇₁≠𝜇₂, then 𝑆(𝜉)=𝑆^∗+𝐴₁𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉},𝐼(𝜉)=𝐼^∗−𝐴₂𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}, (103) if 𝜇₁=𝜇₂, 𝑆(𝜉)=𝑆^∗+𝐴₁₁+𝐴₁₂𝜉𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉},𝐼(𝜉)=𝐼^∗−𝐴₂₁+𝐴₂₂𝜉𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}+𝑜𝑒^{((𝑐−𝑐2+4𝜇)/2)𝜉}, (104) where 𝜇=min{𝜇₁,𝜇₂}>0, 𝐴₁₁,𝐴₂₁∈ℝ  𝐴₁, 𝐴₂, 𝐴₁, 𝐴₂, 𝐴₁₂ and 𝐴₂₂ are all positive constants.(ii)𝑐=2𝑅₀−1: when 𝜉→−∞, 𝑆(𝜉)=𝑅_𝑑−1𝑅_𝑑−𝐴₁₁+𝐴₁₂𝜉𝑒^{𝑅0−1𝜉}+𝑜𝜉𝑒^{𝑅0−1𝜉},𝐼(𝜉)=𝐴₁₁+𝐴₁₂𝜉𝑒^{𝑅0−1𝜉}+𝑜𝜉𝑒^{𝑅0−1𝜉}. (105) when 𝜉→+∞, and if 𝜇₁≠𝜇₂, then 𝑆(𝜉)=𝑆^∗+𝐵₁₁𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}+𝑜𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉},𝐼(𝜉)=𝐼^∗−𝐵₂₂𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}+𝑜𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}, (106) if 𝜇₁=𝜇₂, then 𝑆(𝜉)=𝑆^∗+𝐵₁₁+𝐵₁₂𝜉𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}+𝑜𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉},𝐼(𝜉)=𝐼^∗−𝐵₂₁+𝐵₂₂𝜉𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}+𝑜𝑒^{(𝑅0−1−𝑅0−1+𝜇)𝜉}, (107) where 𝜇=min{𝜇₁,𝜇₂}>0, 𝐵₁₂,𝐵₂₂<0, 𝐵₁₁,𝐵₂₁,𝐵₁₁,𝐵₂₁∈ℝ, 𝐵₁₁, 𝐵₂₂, 𝐵₁₂, 𝐵₂₂ are all positive constants.

Theorem 13. There is no monotone traveling wave solution of model (24) for any 0<𝑐<2𝑅₀−1. In other words, there is no monotone traveling wave solution of model (3) for any 0<𝑐<2𝑅₀−1.

Proof. Suppose there is a monotone traveling wave solution 𝕃(𝜉)=(𝑙₁(𝜉),𝑙₂(𝜉))^𝑇 of model (24) with the wave speed 𝑐₀, where 𝑐₀∈(0,2𝑅₀−1).
The asymptotic model of 𝕃(𝜉)=(𝑙₁(𝜉),𝑙₂(𝜉))^𝑇 as 𝜉→−∞ is 𝜆^′′−𝑐₀𝜆^′−𝜈𝑅_𝑑−1𝜆+𝜈𝑅_𝑑+𝑅₀−2𝜈𝜇=0,𝜇^′′−𝑐₀𝜇^′+𝑅₀−1𝜇=0. (108) The second function of (108) has two characteristics as the following ones: (𝑐₀+4(𝑅₀−1)−𝑐²₀𝑖)/2, (𝑐₀−4(𝑅₀−1)−𝑐²₀𝑖)/2. Thus it has two independent solutions of the following form: 𝜇₁₁=𝑒^{(𝑐0/2)𝜉}cos4𝑅₀−1−𝑐²₀2𝜉,𝜇₂₂=𝑒^{(𝑐0/2)𝜉}sin4𝑅₀−1−𝑐²₀2𝜉. (109)
Similar to the proof of Theorem 11, we can get that, when 𝜉→−∞, 𝜒₂(𝜉) can be described as the following equation: 𝜒₂(𝜉)=𝐾₁𝑒^{(𝑐0/2)𝜉}cos4𝑅₀−1−𝑐²₀2𝜉+𝐾₂𝑒^{(𝑐0/2)𝜉}sin4𝑅₀−1−𝑐²₀2𝜉=+h.o.t,𝐾²₁+𝐾²₂𝑒^{(𝑐0/2)𝜉}sin4𝑅₀−1−𝑐²₀2𝜉+𝜏(𝜉)+h.o.t, (110) where tan(𝜏(𝜉))=𝐾₁/𝐾₂, and h.o.t is the short notation for the higher order terms.
That is to say, 𝑙₂(𝜉) is oscillating. Thus, any solution of model (24) with 0<𝑐<2𝑅₀−1 is not strictly monotone.

Theorems 12 and 13 indicate that 𝑐=2𝑅₀−1 is the critical minimal wave speed.

Acknowledgments

The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to improvements of our original paper. This research was supported by the Natural Science Foundation of Zhejiang Province (LY12A01014, R1110261, and LQ12A01009), the National Science Foundation of China (61272018), and the National Basic Research Program of China (2012CB426510).