2021/2023 UNC Greensboro PDE Conference, Electronic Journal of Differential Equations, Conference 26 (2022/2025), pp. 219–242. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DOI: 10.58997/ejde.conf.26.z1 TRAVELING WAVE SOLUTIONS FOR AN EPIDEMIC MODEL ZHENBU ZHANG Abstract. In this article, we consider a one-dimensional reaction-diffusion epidemic model, which is neither cooperative nor competitive. We study the possible impact of the spatial movement by investigating the existence of trav- eling wave solutions. We construct a pair of upper-lower solutions and then use Shauder’s fixed point theorem to prove the existence of nonnegative non- trivial bounded semi-traveling wave solution. This is done by introducing a critical wave speed depending on the diffusion coefficients and other parame- ters in the model such that, for the wave speed that is greater than the critical wave speed, the model has such a solution. We also derive a condition under which the model has no nonnegative nontrivial bounded semi-traveling wave solution. 1. Introduction Mathematical models play an important role in understanding the transmission and spread and estimating the impact of control measures of infectious disease like influenza, rabies, rubeola, malaria, dengue disease, COVID-19, etc. It turns out that the transmission and spread of almost all infectious diseases heavily depend on climate, rainfall, environment(temperature and humidity), social interactions, migration of population, and the spatial and genetic heterogeneity of host and parasite. In order to investigate the possible impact of the spatial movement of population, following [21], we consider one-dimensional reaction-diffusion epidemic model described by ∂ui ∂t = diuixx + fi(u), 1 ≤ i ≤ n, x ∈ R, t > 0, (1.1) where ui is the density of a population in compartment i and di is the diffusion coefficient of population i, fi is the reaction term in compartment i under the influences of demographic process and epidemic interactions. u = (u1, u2, . . . , un) T with ui ≥ 0 represents the state of individuals in all compartments. Traveling waves are a common phenomenon in biology. The existence of traveling wave solutions for a reaction-diffusion system determines the long-term behavior of other solutions of the system and reveals a lot of information for the spread and transmission of the disease. In this paper, we introduce a critical wave speed c̃ such that (1.1) 2020 Mathematics Subject Classification. 35C07, 35K57, 35Q92. Key words and phrases. Reaction-diffusion; traveling waves; equilibrium; wave speed; basic reproduction number; upper-lower solutions. ©2025 This work is licensed under a CC BY 4.0 license. Published May 13, 2025. 1 2 Z. ZHANG EJDE-2022/25/CONF/26 has a nonnegative nontrivial semi-traveling wave solution u(t, x) = U(z), where z = x + ct, if c ≥ c̃. In epidemiology, the existence and nonexistence of nontrivial traveling waves indicate whether an infectious disease could persist as a wave front of infection that travels geographically across vast distance [4]. Because of the general form of fi(u) in the model, (1.1) can be applied to a large class of compartmental epidemic models. It has been investigated from various aspects. The corresponding multi-dimensional version of model (1.1) on a bounded region with smooth boundary coupled with homogeneous Neumann boundary con- dition has been investigated in [21]. The corresponding reaction model has been investigated in [20]. One of the most important concerns about infectious disease is its ability to spread into a population. The minimal wave speed cmin for a traveling wave is a key parameter to characterize the speed at which the disease spreads in a spatial domain [4, 12, 24, 26]. Biologically speaking, epidemics can spread for c ≥ cmin while they cannot spread for c < cmin [4]. Therefore, estimating the minimal wave speed cmin is a very significant work both theoretically and practically. It was first conjectured by Fisher in [8] that for model ut = duxx + ρu(1− u) the minimal wave speed cmin equals the spreading speed c∗ at which the region {x : u ∼ 1} takes over the set {x : u(x, 0) = 0} (see [17]). The conjecture was proved in [14]. Under certain assumptions, it has been proved (e.g. [2, 3]) that, for some more general models, the minimal wave speed cmin is equal to the spreading speed c∗. But usually, it is difficult to estimate c∗. When c∗ is equal to the spreading speed c̄ of the linearized system of (1.1) around the unique disease-free equilibrium, the spreading speed c∗ of (1.1) is said to be linearly determined (see [15, 23]). If this is the case, we can estimate c∗ as well as cmin by estimating c̄. Usually, to estimate c̄ is much easier than to estimate c∗ or cmin. For some models, c̄ can be estimated quantitatively, see [15]. It is shown in [17, 23] that if (1.1) is cooperative, that is, fiuj ≥ 0 for j ̸= i, then under certain assumptions, (1.1) is linearly determined. This method has been applied to estimate the minimal wave speed cmin for a west nile virus model in [16] and a malaria model in [27]. If (1.1) is a competitive model, we can use an appropriate change of variables as done in [15] to change it to a cooperative system and then use the method in [17] to estimate the minimal wave speed cmin. Unfortunately, some systems are neither cooperative nor competitive like the models investigated in [4, 9, 10, 24, 25, 26]. For these models, the method proposed in [17] cannot be used to estimate the minimal wave speed cmin. In this article, we adapt the approach used in [4] (same method is also used in [10, 18, 19, 22, 26]) to introduce a critical wave speed c̃ and to derive some sufficient conditions under which semi-traveling wave solutions exist. The critical wave speed c̃ introduced here will play an important role in estimating the minimal wave speed cmin. For some special cases, this critical wave speed coincides with the minimal wave speed of traveling wave solutions under appropriate assumptions. Because of the generality of our model, we have to overcome some technical difficulties by making appropriate changes. EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 3 To distinguish the disease-free states and infected states, following [20, 21], we let uI = (u1, u2, . . . , um)T , dI = diag(d1, d2, . . . , dm), uS = (um+1, um+2, . . . , un) T , dS = diag(dm+1, dm+2, . . . , dn), where ui, 1 ≤ i ≤ m, represent the infected compartments, and ui, m+ 1 ≤ i ≤ n represent the uninfected compartments. Then we write (1.1) as ∂uI ∂t = dIuIxx + fI(u) x ∈ R, t > 0, ∂uS ∂t = dSuSxx + fS(u), x ∈ R, t > 0, (1.2) where fI(u) = (f1(u), f2(u), . . . , fm(u))T , fS(u) = (fm+1(u), fm+2(u), . . . , fn(u)) T . We assume that fi(u) has the following form, for i = 1, 2, . . . ,m, fi(u) = fi(u1, u2, . . . , un) = ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) − qiui = AiBi − qiui, (1.3) and for i = m+ 1,m+ 2, . . . , n, fi(u) = fi(u1, u2, . . . , un) = Qi − ( m∑ j=1 aijuj )( n∑ k=m+1 bikuk ) − qiui = Qi −AiCi − qiui, (1.4) where Qi(i = m + 1,m + 2, . . . , n) and qi(i = 1, 2, . . . , n) are positive constants, aij , bj0, bik (i = 1, 2, . . . , n, j = 1, 2, . . .m, k = m+1,m+2, . . . , n) are nonnegative constants and Ai = Ai(uI) = Ai(u1, u2, . . . , um) = m∑ j=1 aijuj , i = 1, 2, . . . , n, Bi = Bi(uS) = Bi(um+1, um+2, . . . , un) = bi0 + n∑ k=m+1 bikuk, i = 1, 2, . . . ,m, Ci = Ci(uS) = Ci(um+1, um+2, . . . , un) = n∑ k=m+1 bikuk i = m+ 1,m+ 2, . . . , n. Such a model includes the diffusive influenza model with multiple strains ∂I1 ∂t = d1I1xx + (1− f)β1(I1 + δI2)S − k1I1, x ∈ R, t > 0, ∂I2 ∂t = d2I2xx + f(1− r)β1(I1 + δI2)S − k2I2, x ∈ R, t > 0, ∂I3 ∂t = d3I3xx + [frβ1(I1 + δI2) + β2I3]S − k3I3, x ∈ R, t > 0, ∂S ∂t = dSSxx + Λ− [β1(I1 + δI2) + β2I3]S − µS, x ∈ R, t > 0 (1.5) 4 Z. ZHANG EJDE-2022/25/CONF/26 which was investigated in [4]. The differential susceptibility epidemic model ∂I ∂t = dIIxx + ηβI l∑ j=1 αjSj − (µ+ γ)I x ∈ R, t > 0, ∂Si ∂t = diSixx + µpiS 0 − ηβαiISi − µSi, x ∈ R, t > 0, i = 1, 2, . . . , l, (1.6) whose corresponding reaction model was investigated in [13]. The within-host HIV model with cell-to-cell transmission, ∂I1 ∂t = d1I1xx + (β1I1 + β2I2)S − rI1, x ∈ R, t > 0, ∂I2 ∂t = d2I2xx + δI1 − µI2, x ∈ R, t > 0, ∂S ∂t = dSSxx + λ− (β1I1 + β2I2)S − ηS, x ∈ R, t > 0 (1.7) investigated in [19]. By direct computations, for i, j = 1, 2, . . . ,m, we have ∂fi ∂uj = aijBi, j ̸= i, ∂fi ∂ui = aiiBi − qi ; for i = 1, 2, . . . ,m, j = m+ 1,m+ 2, . . . , n, we have ∂fi ∂uj = bijAi ; for i = m+ 1,m+ 2, . . . , n, j = 1, 2, . . . ,m, we have ∂fi ∂uj = −aijCi ; and for i, j = m+ 1,m+ 2, . . . , n, we have ∂fi ∂uj = −bijAi, j ̸= i, ∂fi ∂ui = −biiAi − qi. From these expressions we can see that model (1.2) is neither cooperative nor competitive. This article is organized as follows. In next section we summarize some results for the corresponding reaction model that will be used in the following sections. In Section 3, we introduce a critical wave speed and establsih the existence of semi-traveling wave solutions. In Section 4, we prove a theorem of nonexistence of traveling wave solutions. Finally, in Section 5, we summarize what we did in this paper and suggest some possible directions of future work. 2. Basic reproduction number for the reaction model and some examples When the spatial diffusion of population is omitted, the diffusive model (1.1) is reduced to the reaction model dui dt = fi(u), i = 1, 2, . . . , n, (2.1) EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 5 where ui = ui(t) is independent of x. Since (1.1) is autonomous, the dynamics of (2.1) can give us some insight into the model (1.1). Therefore, we will cite some results about (2.1) from other references (see [20] and the references therein) and look at some specific examples. It is easily seen that (2.1) has a disease-free equilibrium E0 = (0, . . . , 0, u0 m+1, . . . , u 0 n) T with u0 i = Qi qi > 0, m+ 1 ≤ i ≤ n. The disease-free equilibrium is the population before the transmission of the disease. Write the linearized system of (2.1) around E0 as du dt = Df(E0)(u− E0), where Df(E0) is the derivative [ ∂fi∂uj ]n×n evaluated at E0. More specifically, Df(E0) = [ E −Q 0 −J −MS ] , where E =  a11B 0 1 a12B 0 1 · · · a1mB0 1 a21B 0 2 a22B 0 2 · · · a2mB0 2 · · · · · · · · · · · · am1B 0 m am2B 0 m · · · ammB0 m  m×m corresponds to the new infection, Q =  q1 0 · · · 0 0 q2 · · · 0 · · · · · · · · · . . . 0 0 . . . qm  m×m corresponds to the remaining transfer terms, J =  a(m+1)1C 0 m+1 a(m+1)2C 0 m+1 · · · a(m+1)mC0 m+1 a(m+2)1C 0 m+2 a(m+2)2C 0 m+2 · · · a(m+2)mC0 m+2 · · · · · · · · · · · · an1C 0 n an2C 0 n · · · anmC0 n  (n−m)×m , MS =  qm+1 0 · · · 0 0 qm+2 · · · 0 · · · · · · · · · · · · 0 0 · · · qn  (n−m)×(n−m) , where B0 i = Bi(u 0 S) = Bi(u 0 m+1, . . . , u 0 n), i = 1, 2, . . . ,m, C0 i = Ci(u 0 S) = Ci(u 0 m+1, . . . , u 0 n), i = m+ 1,m+ 2, . . . , n. Recall that the basic reproduction number R0 is the expected number of sec- ondary infections generated by a single infectious individual during the infection period in an entirely susceptible population (see [1, 5, 6, 11, 20, 21]). To intro- duce R0, following [20, 21] (also see the references therein), consider a “typical” infectious individual introduced into a completely susceptible population, to see how many new infections will be produced. Let ϕi(0) be the number of infected individuals initially in compartment i. Without reinfection, at any time t, let 6 Z. ZHANG EJDE-2022/25/CONF/26 ϕ(t) = (ϕ1(t), ϕ2(t), . . . , ϕm(t))T be the total infective member which were devel- oped from those initially infected individuals ϕi(0), then ϕ(t) satisfies ϕ′(t) = −Qϕ(t), ϕ(0) = (ϕ1(0), ϕ2(0), . . . , ϕm(0))T . Therefore, ϕ(t) = e−Qtϕ(0). Thus, the new infection at time t is Eϕ(t) = Ee−Qtϕ(0). Consequently, the expected number of new infections produced by the initially in- fected individuals is∫ ∞ 0 Eϕ(t)dt = ∫ ∞ 0 Ee−Qtϕ(0)dt = EQ−1ϕ(0). EQ−1 is called the next generation matrix for the model (see [6]). Following [6, 19, 20, 21], we introduce the basic reproduction number R0 = ρ(EQ−1), the spectral radius of EQ−1. R0 is the threshold value for the local stability of the disease-free equilibria of (2.1) as stated in the following theorem from [20] (see [5] for a similar result). Theorem 2.1. Let E0 be a disease-free equilibrium of model (2.1), then E0 is locally asymptotically stable if R0 < 1, but unstable if R0 > 1. For model (1.5), m = 3, n = 4. For notation convenience, denote ω = Λ µ , α = (1− f)β1, β = f(1− r)β1, γ = frβ1, then E0 = (0, 0, 0, ω), Df(E0) = [ E −Q 0 −J −MS ] , where E = ω α αδ 0 β βδ 0 γ γδ β2  , Q = k1 0 0 0 k2 0 0 0 k3  , J = [ β1ω β1δω β2ω ] , MS = µ. It is easily seen that EQ−1 = ω  α k1 αδ k2 0 β k1 βδ k2 0 γ k1 γδ k2 β2 k3  and its two nonzero eigenvalues are λ1 = ω ( α k1 + βδ k2 ) , which is denoted by RSC in [4], representing the number of secondary sensitive cases that one individual infected with the sensitive strain initiates in a completely susceptible population where antiviral treatment is implemented, and λ2 = β2ω k3 , which is denoted by RRC in [4], representing the number of secondary resistant cases that one individual infected with the resistant strain initiates in a completely susceptible population. ρ(EQ−1) = max{λ1, λ2} EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 7 is denoted by RC in [4] and is called the control reproduction number of the corre- sponding reaction model, which is used to determine whether the epidemic can be contained when certain control measures are taken (also see [1]). For model (1.6), m = 1, n = l + 1. Denote ω = ηβS0, q = µ + γ, then E0 = (0, p1S 0, p2S 0, . . . , plS 0), Df(E0) =  ω ∑l j=1 αjpj − q 0 0 0 · · · 0 −ωα1p1 −µ 0 0 · · · 0 −ωα2p2 0 −µ 0 · · · 0 · · · · · · · · · · · · · · · · · · −ωαlpl 0 0 0 · · · −µ  . ρ(EQ−1) = EQ−1 = ω ∑l j=1 αjpj q = ηβS0 ∑l j=1 αjpj µ+ γ , which is the reproductive number R0 introduced in [13]. For model (1.7), m = 2, n = 3, we denote ω = λ µ . Then E0 = (0, 0, ω) and Df(E0) = [ E −Q 0 −J −MS ] , where E = [ β1ω β2ω δ 0 ] , Q = [ r 0 0 µ ] , J = [β1ω β2ω], MS = η. It is easily seen that EQ−1 = [β1ω r β2ω µ δ r 0 ] and its two eigenvalues are λ = β1λ± √ β2 1λ 2 + 4β2δrλ 2rµ . Thus, ρ(EQ−1) = β1λ+ √ β2 1λ 2 + 4β2δrλ 2rµ . 3. Semi-traveling waves for the diffusive model The existence of traveling wave solutions for reaction-diffusion models is one of the most important topics for the past several decades. To investigate the existence of traveling wave solutions for (1.1), we start with the existence of so-called semi- traveling waves as defined in the following definition from [4, 12, 26]. Definition 3.1. A solution u(t, x) of (1.2) of the form u(t, x) = U(z), where z = x+ ct, is called a semi-traveling wave solution connected to the disease- free equilibrium E0 if it satisfies lim z→−∞ U(z) = E0. (3.1) 8 Z. ZHANG EJDE-2022/25/CONF/26 From the definition we can see that a semi-traveling wave solution is a traveling wave solution starting from the disease-free equilibrium. Such a solution is of biological significance since we can get a lot of information from it, such as whether epidemics will spread, asymptotic speed of propagation, and the final state of the wavefront, etc. To prove the existence of semi-traveling wave solution for (1.1), we adapt the ap- proach used in [4, 10, 18, 19, 22, 24, 25, 26]. That is, first we introduce an auxiliary system, then construct a pair of upper-lower solutions for the auxiliary system by linearizing the corresponding wave equation at the disease-free equilibrium. Then, use the Schauder’s fixed point theorem to prove the existence of semi-traveling wave solution for the auxiliary system. Finally, by taking the limit, we obtain the existence of semi-traveling wave solution for (1.1). We consider the following auxiliary system related to (1.2), ∂uI ∂t = dIuIxx,+fI(u)− δImu2 I x ∈ R, t > 0, ∂us ∂t = dSuSxx + fS(u), x ∈ R, t > 0, (3.2) where u2 I = (u2 1, u 2 2, . . . , u 2 m)T , δ > 0 is a small number, and Im is the identity matrix of size m. Let u(t, x) = U(z) be a semi-traveling wave solution of (3.2), then U satisfies cU ′ I = dIU ′′ I + fI(U)− δImU2 I , cU ′ S = dSU ′′ S + fS(U), (3.3) where UI = (U1, U2, . . . , Um)T and US = (Um+1, Um+2, . . . , Un) T . The limiting equations of (3.3) as δ → 0 are the wave equations corresponding to (1.2), especially, the equation involving UI only is the equation cU ′ I = dIU ′′ I + fI(U) (3.4) Linearizing (3.4) around the disease-free equilibrium E0 leads to the linear system cΦ′ I = dIΦ ′′ I + EΦI −QΦI , (3.5) where ΦI = (ϕ1, ϕ2, . . . , ϕm)T . Next we are going to introduce a critical wave speed c̃. To do this, we adopt the method used in [4]. Let (ϕ1(z), ϕ2(z), . . . , ϕm(z)) = eλz(v1, v2, . . . , vm), then system (3.5) is equivalent to A(v1, v2, . . . , vm)T = cλ(v1, v2, . . . , vm)T , (3.6) where A = λ2dI + E −Q. Denoting V = (v1, v2, . . . , vm)T , we can write (3.6) as (λ2dI − cλIm −Q)V = −EV or (−λ2Q−1dI + cλQ−1 + Im)V = Q−1EV. By direct computations, we have B(λ, c) = −λ2Q−1dI + cλQ−1 + Im = diag (−d1λ 2 + cλ+ q1 q1 , . . . , −dmλ2 + cλ+ qm qm ) . EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 9 Assume λ > 0 such that −diλ 2 + cλ+ qi > 0. Then B(λ, c) is invertible and the above equation can be written as B(λ, c)−1(Q−1E)V = V. Denoting H(λ, c) = B(λ, c)−1(Q−1E), we have H(λ, c)V = V. (3.7) Let d = max1≤i≤m di > 0. Observe that, for c > 0, the two roots of−diλ 2+cλ+qi = 0 are λ1 = c− √ c2 + 4diqi 2di < 0, and λ2 = c+ √ c2 + 4diqi 2di > c di ≥ c d > 0, we know that, for λ ∈ [0, c d ], B(λ, c) is invertible and B(λ, c)−1 = diag ( q1 −d1λ2 + cλ+ q1 , . . . , qm −dmλ2 + cλ+ qm ) , and H(λ, c) is a nonnegative matrix. Let pci(λ) = −diλ 2 + cλ+ qi, then γi(c) = pci( c 2d ) = (2d− di)c 2 4d2 + qi > 0 and which is strictly increasing for c ∈ [0,∞). Thus, H( c 2d , c) is strictly decreasing for c ∈ [0,∞) and H(0, 0) = Q−1E. Denote by ρ(H(λ, c)) the principal eigenvalue of the nonnegative matrix H(λ, c) for λ ∈ [0, c 2d ]. Since ρ(H(λ, c)) is continuous and monotonically increasing with respect to the nonnegative matrix H(λ, c), ρ(H( c 2d , c)) is strictly decreasing for c ∈ [0,∞) with ρ(H(0, 0)) = ρ(Q−1E) and lim c→∞ ρ ( H ( c 2d , c )) = 0. It can be shown that ρ(Q−1E) = ρ(EQ−1)(see [4]). Therefore, ifR0 = ρ(EQ−1) = ρ(Q−1E) = ρ(H(0, 0)) > 1, by the continuity and monotonicity of ρ(H( c 2d , c)) with respect to c, there exists a unique c̃ > 0 such that ρ(H( c̃ 2d , c̃)) = 1 and for c ∈ [0, c̃), ρ(H( c 2d , c)) > 1. For c ∈ (c̃,∞), ρ(H( c 2d , c)) < 1. For any c > c̃ fixed, since p′ci(λ) = c − 2diλ > 0 for λ ∈ [ 0, c 2d ] , ρ(H(λ, c)) is strictly decreasing and nonnegative for λ ∈ [ 0, c 2d ] . But, ρ(H(0, c)) = ρ(H(0, 0)) = R0 > 1 and ρ(H( c 2d , c)) < 1, by the continuity and monotonicity of ρ(H(λ, c)) with respect to λ, there exists a unique λc ∈ ( 0, c 2d ) such that ρ(H(λc, c)) = 1 and for λ ∈ [0, λc), ρ(H(λ, c)) > 1, and for λ ∈ ( λc, c 2d ] , ρ(H(λ, c)) < 1. Thus we have the following lemma (see [4, Lemmas A.1 and A.2]). Lemma 3.2. If R0 = ρ(EQ−1) > 1, then, there exists a unique c̃ > 0 such that for any c > c̃, there exists a unique λc ∈ ( 0, c 2d ) and Vc = (v1, v2, . . . , vm)T with vi > 0, 1 ≤ i ≤ m, such that H(λc, c)Vc = Vc. 10 Z. ZHANG EJDE-2022/25/CONF/26 The vector ΦI = (ϕ1(z), ϕ2(z), . . . , ϕm(z))T with ϕi(z) = vie λcz, i = 1, 2, . . . ,m, satisfies (3.5). Now we adapt the approach in [4] with necessary changes to construct a pair of upper-lower solutions of (3.3). That is, we define US(z) = (u0 m+1, . . . , u 0 n) T , US(z) = (um+1(z), . . . , un(z)) T , with ui(z) = max{u0 i (1− ηeαz), 0} for m+ 1 ≤ i ≤ n, and UI(z) = (u1(z), . . . , um(z))T , with ui(z) = min{vieλcz, viω} for 1 ≤ i ≤ m, UI(z) = (u1(z), . . . , um(z))T , with ui(z) = max{vieλcz(1 − τeµz), 0} for 1 ≤ i ≤ m, where constants η, α, ω, τ, µ are to be determined later. Lemma 3.3. For ω > 1 large enough, UI(z) satisfies cUI ′ ≥ dIUI ′′ + fI(U)− δImUI 2 , (3.8) where U = U(z) = (UI(z), US(z)) T . Proof. From the definition of ui(z) = min{vieλcz, viω} for 1 ≤ i ≤ m, if z such that eλcz ≤ ω, i.e. z ≤ lnω λc ≜ z1, then ui(z) = vie λcz and UI(z) = ΦI . Then, from the definition of E, Q, and U , we have fI(U) = (E − Q)ΦI . Therefore, from Lemma 3.2 and (3.5), we have dIUI ′′ − cUI ′ + fI(U) = dIΦ ′′ I − cΦ′ I + EΦI −QΦI = 0 ≤ δImUI 2 . That is, (3.8) holds. If z > z1, then ui(z) = viω and UI(z) = ωVc. Then dIUI ′′ − cUI ′ + fI(U)− δImUI 2 = fI(U)− δIm(v21 , . . . , v 2 m)Tω2. But, for i = 1, 2, . . . ,m, fi(U)− δv2i ω 2 = m∑ j=1 aijvjωB 0 i − qiviω − δv2i ω 2 = ( m∑ j=1 aijvjB 0 i − qivi − δv2i ω)ω. Therefore, if we take ω > 1 such that ω > max 1≤i≤m ∑m j=1 aijvjB 0 i − qivi δv2i , then dIUI ′′ − cUI ′ + fI(U)− δImUI 2 ≤ 0 and (3.8) holds. This completes the proof. □ EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 11 Lemma 3.4. For 0 < α < min{ c D , λc} and η > max { 1, max m+1≤i≤n ( ∑m j=1 aijvj)( ∑n k=m+1 biku 0 k) qiu0 i } , US(z) satisfies cUS ′ ≤ dSUS ′′ + fS(Ũ), (3.9) where D = max{dm+1, . . . , dn} and Ũ = Ũ(z) = (UI(z), US(z)) T . Proof. Let z2 = − ln η α < 0. If z satisfies z ≥ z2, then ui(z) = 0 for m+ 1 ≤ i ≤ n, US(z) = (0, . . . , 0)T and Ũ = (UI(z), 0, . . . , 0) T . Thus, fS(Ũ) > 0. Therefore, (3.9) holds. If z ≤ z2, ui(z) = u0 i (1 − ηeαz) for m + 1 ≤ i ≤ n. Observe that z2 < 0 < z1, ui(z) = vie λcz for 1 ≤ i ≤ m. Thus, Ũ = (v1e λcz, . . . , vmeλcz, u0 m+1(1− ηeαz), . . . , u0 n(1− ηeαz))T , and dSUS ′′ − cUS ′ + fS(Ũ) = (−dSα 2 + cα)ηeαzΘ+ fS(Ũ), where Θ = (u0 m+1, . . . , u 0 n) T . Since, for m+ 1 ≤ i ≤ n, fi(Ũ) = Qi − ( m∑ j=1 aijvje λcz )( n∑ k=m+1 biku 0 k(1− ηeαz) ) − qiu 0 i (1− ηeαz) = Qi − eλcz(1− ηeαz) ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) − qiu 0 i + qiηu 0 i e αz = ηeλczeαz ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) − eλcz ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + qiηu 0 i e αz ≥ −eλcz ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + qiηu 0 i e αz, we have diu ′′ i − cu′ i + fi(Ũ) ≥ (−diα 2ηu0 i + cαηu0 i )e αz − eλcz ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + qiηu 0 i e αz = [ − diα 2ηu0 i + cαηu0 i + qiηu 0 i − e(λc−α)z ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k )] eαz ≥ [ (−diα 2 + cα+ qi)ηu 0 i − ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k )] eαz ≥ 0 for 0 < α < min{ c D , λc} and η > ( ∑m j=1 aijvj)( ∑n k=m+1 biku 0 k) qiu0 i . Therefore, (3.9) holds. This completes the proof. □ 12 Z. ZHANG EJDE-2022/25/CONF/26 Lemma 3.5. Let 0 < µ < α < λc small enough such that λc + µ < c 2d . Then for τ > 1 large enough, UI(z) satisfies cUI ′ ≤ dIUI ′′ + fI(U)− δImUI 2, (3.10) where U = U(z) = (UI(z), US(z)) T . Proof. If 1 − τeµz ≤ 0, that is, z ≥ − ln τ µ ≜ z3, UI = (0, . . . , 0). Then, we have fI(U) = 0 and (3.10) holds. For z < z3, ui(z) = vie λcz(1−τeµz) for 1 ≤ i ≤ m. For τ > 1 large enough (depending on α and η), we have z3 < z2 < 0. Thus, for z < z3, ui(z) = u0 i (1− ηeαz) for m+ 1 ≤ i ≤ n and U = (v1e λcz(1− τeµz), . . . , vmeλcz(1− τeµz), u0 m+1(1− ηeαz), . . . , u0 n(1− ηeαz))T . Thus, for 1 ≤ i ≤ m, ui ′(z) = vi[λc(1− τeµz)− τµeµz]eλcz, and ui ′′(z) = vi[λ 2 c(1− τeµz)− (2λcτµ+ τµ2)eµz]eλcz. Therefore, for 1 ≤ i ≤ m, we have cu′ i − diu ′′ i − fi(U) + δu2 i = cvi[λc(1− τeµz)− τµeµz]eλcz − divi[λ 2 c(1− τeµz)− (2λcτµ+ τµ2)eµz]eλcz − [ m∑ j=1 aijvje λcz(1− τeµz) ( bi0 + n∑ k=m+1 biku 0 k(1− ηeαz) )] + qivie λcz(1− τeµz) + δv2i e 2λcz(1− τeµz)2 = eλcz(1− τeµz) [ − diviλ 2 c + cviλc − ( m∑ j=1 aijvj )( bi0 + n∑ k=m+1 biku 0 k ) + qivi ] + [divi(2λcτµ+ τµ2)− cviτµ]e (λc+µ)z + ηeλcz(1− τeµz)eαz ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + δv2i e 2λcz(1− τeµz)2. Since ΦI = (v1e λcz, . . . , vmeλcz)T satisfies (3.5), we have −diviλ 2 ce λcz + cviλce λcz − ( m∑ j=1 aijvje λcz )( bi0 + n∑ k=m+1 biku 0 k ) + qivie λcz = 0, or −diviλ 2 c + cviλc − ( m∑ j=1 aijvj )( bi0 + n∑ k=m+1 biku 0 k ) + qivi = 0. Therefore, cu′ i − diu ′′ i − fi(U) + δu2 i = [divi(2λcτµ+ τµ2)− cviτµ]e (λc+µ)z + ηeλcz(1− τeµz)eαz ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + δv2i e 2λcz(1− τeµz)2 = e(λc+µ)z { [di(2λc + µ)− c]viτµ EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 13 + η(1− τeµz)e(α−µ)z ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + δv2i e (λc−µ)z(1− τeµz)2 } . Observe that, for z < z3 = − ln τ µ , we have 0 < 1− τeµz < 1 and ez < e− ln τ µ . Thus, for 0 < µ < α < λc, e(α−µ)z < e−(α−µ) ln τ µ , and e(λc−µ)z < e−(λc−µ) ln τ µ . Hence, cu′ i − diu ′′ i − fi(U) + δu2 i ≤ e(λc+µ)z { [di(2λc + µ)− c]viτµ + ηe−(α−µ) ln τ µ ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + δv2i e −(λc−µ) ln τ µ } . Since λc < c 2d , we can choose µ > 0 small enough such that λc + µ < c 2d . Then di(2λc + µ) < d(2λc + µ) < 2d(λc + µ) < c. Therefore, di(2λc + µ)− c < 0, and we can choose τ > 1 large enough so that [di(2λc + µ)− c]viτµ + ηe−(α−µ) ln τ µ ( m∑ j=1 aijvj )( n∑ k=m+1 biku 0 k ) + δv2i e −(λc−µ) ln τ µ ≤ 0. Thus, cu′ i−diu ′′ i −fi(U)+δu2 i ≤ 0, and (3.10) holds. This completes the proof. □ Now we use Schauder’s fixed point theorem to prove the existence of semi- traveling wave solutions of (3.2). To do this, we introduce Banach space Cσ(R,Rn) such that for U = (u1(z), . . . , un(z)) T ∈ Cσ(R,Rn), its norm is defined by ∥U∥ ≜ |U(·)|σ = max 1≤i≤n sup z∈R |ui(z)|e−σ|z|, where σ > 0 will be determined later. We are going to look for semi-traveling wave solutions U(z) = (UI(z), US(z)) T ∈ Cσ(R,Rn) of (3.2) satisfying UI(z) ≤ UI(z) ≤ UI(z), US(z) ≤ US(z) ≤ US(z). Consider the closed and convex subset Σ of Cσ(R,Rn) given by Σ = {U = (UI , US) T ∈ Cσ(R,Rn), UI ≤ UI ≤ UI , US ≤ US ≤ US}, and define map Γ = (ΓI ,ΓS) : Σ → Cσ(R,Rn) as follows: For U(z) = (UI(z), US(z)) T ∈ Σ: Γ(U) = (ΓI(U),ΓS(U)) = (Γ1(U), . . . ,Γm(U),Γm+1(U), . . . ,Γn(U)), ΓI(U) = FIUI + fI(U)− δImU2 I , ΓS(U) = FSUS + fS(U), 14 Z. ZHANG EJDE-2022/25/CONF/26 where FI =  ξ1 0 0 . . . 0 0 ξ2 0 . . . 0 . . . . . . . . . . . . 0 0 0 . . . ξm  m×m , FS =  ξm+1 0 0 . . . 0 0 ξm+2 0 . . . 0 . . . . . . . . . . . . 0 0 0 . . . ξn  (n−m)×(n−m) , with ξi > qi large enough such that for U ∈ Σ, Γi(U) ≥ 0, which will be determined later. Then system (3.3) can be written as −diu ′′ i (z) + cu′ i(z) + ξiui(z) = Γi(U)(z), (3.11) for i = 1, 2, . . . n. Let xi1 < 0 < xi2 be the two distinct roots of dix 2 i − cxi − ξi = 0, then xi = xi2 − xi1 > 0. We define map Π = (Π1, . . . ,Πn) T : Σ → Cσ(R,Rn) with Πi : Σ → Cσ(R,R) given by Πi(U)(z) = 1 dixi [ ∫ z −∞ exi1(z−s)Γi(U)(s)ds+ ∫ ∞ z exi2(z−s)Γi(U)(s)ds ] , then it holds −diΠ ′′ i (z) + cΠ′ i(z) + ξiΠi(z) = Γi(U)(z). (3.12) It is easy to see that any fixed point of Π is a solution of (3.11), which is a semi- traveling wave solution of (3.2). On the other hand, a solution of (3.11) is a fixed point of operator Π . Lemma 3.6. The operator Π maps Σ into Σ and it is continuous and compact with respect to the norm | · |σ in Bσ(R,Rn), where Bσ(R,Rn) = {U(z) ∈ Cσ(R,Rn) : |U(·)|σ < ∞}. Proof. We first prove that Π maps Σ into Σ. Indeed, if U(z) = (UI(z), US(z)) T ∈ Σ, that is, UI(z) ≤ UI(z) ≤ UI(z), US(z) ≤ US(z) ≤ US(z), we need to prove that ui(z) ≤ Πi(z) ≤ ui(z), i = 1, 2, . . . , n. (3.13) Recall that, for i = 1, 2, . . . ,m, ui(z) = { vie λcz, z ≤ z1 = lnω λc , viω, z > z1, and ui(z) = { vie λcz(1− τeµz), z ≤ z3 = − ln τ µ , 0, z > z3, where ω > 1, τ > 1, and 0 < µ < min{α, λc}, λc + µ < c 2d . Now we prove that for i = 1, 2, . . . ,m, the left-hand side of (3.13) holds. Indeed, if z > z3, we have ui(z) = 0. Since Γi(z) ≥ 0 implies that Πi(z) ≥ 0. Therefore, we have ui(z) ≤ Πi(z). EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 15 For z ≤ z3, from (3.10) in Lemma 3.5, we have −diui ′′ + cui ′ + ξiui ≤ ξiui + fi(U)− δui 2 = ξiui + ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) − qiui − δui 2 = [(ξi − qi)ui − δui 2] + ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) . Since h(x) = (ξi − qi)x− δx2 is increasing on (0, ξi−qi 2δ ) and for δ > 0 small enough, for z ≤ z3, ui(z) ≤ vie λcz < ξi − qi 2δ . Thus, (ξi − qi)ui − δui 2 ≤ (ξi − qi)ui − δu2 i . Hence, −diui ′′ + cui ′ + ξiui ≤ [(ξi − qi)ui − δu2 i ] + ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) = Γi(U)(z). It follows that Πi(U)(z) = 1 dixi [ ∫ z −∞ exi1(z−s)Γi(U)(s)ds+ ∫ ∞ z exi2(z−s)Γi(U)(s)ds ] ≥ 1 dixi [ ∫ z −∞ exi1(z−s) + ∫ ∞ z exi2(z−s) ] [−diui ′′(s) + cui ′(s) + ξiui(s)]ds = 1 dixi ∫ z −∞ exi1(z−s)[−diui ′′(s) + cui ′(s) + ξiui(s)]ds + ∫ z3 z exi2(z−s)[−diui ′′(s) + cui ′(s) + ξiui(s)]ds + ∫ ∞ z3 exi2(z−s)[−diui ′′(s) + cui ′(s) + ξiui(s)]ds = ui(z) + 1 xi exi2(z−z3) [ ui ′(z3 + 0)− ui ′(z3 − 0) ] ≥ ui(z). Thus, we have proved that, for i = 1, 2, . . . ,m, the left-hand side of (3.13) holds. Next we prove that, for i = 1, 2, . . . ,m, the right-hand side of (3.13) holds. From Lemma 3.3, we have −diui ′′ + cui ′ + ξiui ≥ ξiui + fi(U)− δui 2 = ξiui + ( m∑ j=1 aijuj ) (bi0 + n∑ k=m+1 bikuk)− qiui − δui 2 = [(ξi − qi)ui − δui 2] + ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) 16 Z. ZHANG EJDE-2022/25/CONF/26 ≥ [(ξi − qi)ui − δu2 i ] + ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) = Γi(U)(z). It follows that Πi(U)(z) = 1 dixi [ ∫ z −∞ exi1(z−s)Γi(U)(s)ds+ ∫ ∞ z exi2(z−s)Γi(U)(s)ds ] ≤ 1 dixi [ ∫ z −∞ exi1(z−s) + ∫ ∞ z exi2(z−s) ] [−diui ′′(s) + cui ′(s) + ξiui(s)]ds = 1 dixi ∫ z −∞ exi1(z−s)[−diui ′′(s) + cui ′(s) + ξiui(s)]ds + ∫ ∞ z exi2(z−s)[−diui ′′(s) + cui ′(s) + ξiui(s)]ds = ui(z). Similarly, using the definitions of US(z) and US(z) and Lemma 3.4, we can prove that for i = m+ 1,m+ 2, . . . , n, (3.13) holds. Therefore, Π(Σ) ⊂ Σ. Next we prove that the operator Π is continuous with respect to the norm | · |σ in Bσ(R,Rn). To do this, first, we take σ > 0 such that 0 < σ < min{−xi1, xi2, i = 1, 2, . . . n}. For U = (u1(z), u2(z), . . . , un(z)) ∈ Σ and W = (w1(z), w2(z), . . . , wn(z)) ∈ Σ and i = 1, 2, . . . ,m, |Γi(U)(z)− Γi(W )(z)|e−σ|z| = |ξiui + fi(U)− δu2 i − ξiwi − fi(W ) + δw2 i |e−σ|z| ≤ [ξi|ui − wi|+ δ|ui − wi||ui + wi|+ |fi(U)− fi(W )|]e−σ|z| ≤ [ξi|ui − wi|+ δ|ui − wi||ui + wi|+ qi|ui − wi| + ∣∣∣( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) − ( m∑ j=1 aijwj )( bi0 + n∑ k=m+1 bikwk )∣∣∣]e−σ|z| = [(ξi + qi)|ui − wi|+ δ|ui − wi||ui + wi| + ∣∣∣( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) − ( m∑ j=1 aijwj )( bi0 + n∑ k=m+1 bikuk ) + ( m∑ j=1 aijwj )( bi0 + n∑ k=m+1 bikuk ) − ( m∑ j=1 aijwj )( bi0 + n∑ k=m+1 bikwk )∣∣∣]e−σ|z| ≤ [ (ξi + qi)|ui − wi|+ δ|ui − wi||ui + wi| + ∣∣∣( m∑ j=1 aij(uj − wj)|| ( bi0 + n∑ k=m+1 bikuk ))∣∣∣ + ∣∣∣ m∑ j=1 aijwj ∣∣∣ ∣∣∣ n∑ k=m+1 bik(uk − wk)| ] e−σ|z| ≤ Li|U(·)−W (·)|σ, EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 17 where Li is a positive constant depending on Qi, qi, aij , bik, vi, ξi, and the constants in ui and ui, (i, j = 1, 2, . . . ,m, k = m+ 1, . . . , n). Then |Πi(U)(z)−Πi(W )(z)|e−σ|z| ≤ e−σ|z| dixi [ ∫ z −∞ exi1(z−s) + ∫ ∞ z exi2(z−s) ] |Γi(U)(z)− Γi(W )(z)|ds ≤ Lie −σ|z| dixi [ ∫ z −∞ exi1(z−s)+σ|s| ds+ ∫ ∞ z exi2(z−s)+σ|s| ds ] |U(·)−W (·)|σ = Li dixi [ ∫ z −∞ exi1(z−s)−σ|z|+σ|s| ds+ ∫ ∞ z exi2(z−s)−σ|z|+σ|s| ds ] |U(·)−W (·)|σ ≤ Li dixi [ ∫ z −∞ exi1(z−s)+σ|z−s| ds+ ∫ ∞ z exi2(z−s)+σ|z−s| ds ] |U(·)−W (·)|σ ≤ Gi|U(·)−W (·)|σ, where Gi = Li(xi1 − xi2 + 2σ) dixi(xi1 + σ)(xi2 − σ) > 0. This implies |Πi(U)(·)−Πi(W )(·)|σ ≤ Gi|U(·)−W (·)|σ. Hence, Πi : Σ → C(R,R) is continuous with respect to the norm | · |σ in Bσ(R,Rn). Similarly, we can prove that, for i = m + 1,m + 2, . . . , n, Πi : Σ → C(R,R) is also continuous with respect to the norm | · |σ in Bσ(R,Rn). Therefore, Π: Σ → Σ is continuous with respect to the norm | · |σ in Bσ(R,Rn). Finally, we show that the operator Π is compact with respect to the norm | · |σ in Bσ(R,Rn). Observe that, for U(z) = (u1(z), u2(z), . . . , un(z)) ∈ Σ, we have 0 ≤ ui(z) ≤ Ni, where Ni depends on vi, λc, ω, τ , η, α, µ, qi, Qi. It follows that there is an Mi depending on these parameters as well as ξi such that 0 ≤ Γi(U) ≤ Mi. Thus,∣∣ d dz Πi(U)(z) ∣∣ = 1 dixi ∣∣∣[xi1 ∫ z −∞ exi1(z−s)ds+ xi2 ∫ ∞ z exi2(z−s)ds ] Γi(U)(s) ∣∣∣ ≤ Mi dixi [ |xi1| ∫ z −∞ exi1(z−s)ds+ xi2 ∫ ∞ z exi2(z−s)ds ] = 2Mi dixi , which implies ∣∣ d dz Πi(U)(z) ∣∣ σ ≤ 2Mi dixi . That is, | d dzΠi(U)(z)|σ is bounded. This is true for all i = 1, 2, . . . , n. This means that Π(Σ) is uniformly bounded and equicontinuous with respect to the norm | · |σ. 18 Z. ZHANG EJDE-2022/25/CONF/26 For fixed positive integer k, define an operator Πk = (Πk 1 ,Π k 2 , . . . ,Π k n) by Πk(U)(z) =  Π(U)(−k) z ∈ (−∞,−k], Π(U)(z) z ∈ [−k, k], Π(U)(k) z ∈ [k,∞). By Arzela-Ascoli theorem, Πk : Σ → Σ is compact with respect to the norm | · |σ in Bσ(R,Rn). For i = 1, 2, . . . , n, we have |Πi(U)(z)| = 1 dixi ∣∣∣[ ∫ z −∞ exi1(z−s)ds+ ∫ ∞ z exi2(z−s)ds ] Γi(U)(s) ∣∣∣ ≤ Mi dixi [ ∫ z −∞ exi1(z−s)ds+ ∫ ∞ z exi2(z−s)ds ] = Mi di|xi1xi2| . Therefore, |Πk i (U)(·)−Πi(U)(·) ∣∣ σ = sup z∈R ∣∣Πk i (U)(z)−Πi(U)(z) ∣∣ e−σ|z| = sup |z|≥k |Πk i (U)(z)−Πi(U)(z)|e−σ|z| ≤ 2Mi di|xi1xi2| e−σk → 0 as k → ∞. This means that |Πk(U)(·)−Π(U)(·)|σ → 0 as k → ∞. That is, Πk converges to Π in Σ with respect to the norm |·|σ. Therefore, Π : Σ → Σ is compact with respect to the norm | · |σ in Bσ(R,Rn). □ Now we are ready to prove the existence of semi-traveling wave solution for system (3.2) Proposition 3.7. If R0 > 1, there exists c̃ > 0 such that for any c > c̃, system (3.2) has a nonnegative bounded semi-traveling wave solution U(z) satisfying lim z→−∞ U(z) = E0. Proof. For R0 > 1, from Lemma 3.2, there exists c̃ > 0 such that for any c > c̃, there exists λc ∈ (0, c 2d ) with d = max1≤i≤m di and Vc = (v1, . . . , vm) with vi > 0 such that ΦI = (v1e λcz, . . . , vmeλcz) satisfies (3.5). Then we can define U(z) = (UI(z), US(z)), U(z) = (UI(z), US(z)) so that they satisfy Lemmas 3.3-3.5. Then we define operator Π(U)(z) = (Π1(U)(z), . . . ,Πn(U)(z)) : Σ → Σ such that (3.12) holds. From Lemma 3.6, we know that Π(Σ) ⊂ Σ and it is continuous and compact with respect to the norm | · |σ in Bσ(R,Rn). Therefore, EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 19 from Shauder’s fixed point theorem, Π has a fixed point U(z) ∈ Σ, which satisfies (3.3). It is easily seen that lim z→−∞ U(z) = lim z→−∞ U(z) = E0. It follows that limz→−∞ U(z) = E0. This completes the proof. □ Next we prove the existence of semi-traveling wave solutions of (1.2). Theorem 3.8. If R0 > 1, then there exists c̃ > 0 such that for any c > c̃, system (1.2) has a nonnegative bounded semi-traveling solution U(z) = (u1(z), . . . , un(z)) satisfying lim z→−∞ U(z) = E0. If bii > 0 for i = m+1, . . . , n, and there is at least one 1 ≤ j ≤ m such that aij > 0, then ui(z) < u0 i . For i = 1, . . . ,m, ui(z) > 0 and lim z→−∞ ui(z)e −λcz = vi, lim z→−∞ u′ i(z)e −λcz = λcvi. (3.14) Proof. For each positive integer k, in (3.3), set δ = δk = 1 k , we have δk ↓ 0 as k → +∞. From Proposition 3.7, for each δk, (3.2) has a nonnegative bounded semi-traveling wave solution Uk(z) = (u1k(z), u2k(z), . . . , unk(z)) ∈ Σ satisfying limz→−∞ Uk(z) = E0. From the proof of Lemma 3.6 and system (3.3), {Uk(z)}∞k=1, {U ′ k(z)}∞k=1, and {U ′′ k (z)}∞k=1 are equicontinuous and uniformly bounded in R. By Arzela-Ascoli theorem, there exists a subsequence {δkj} such that for some U(z) = (U1(z), U2(z), . . . , Un(z)) ∈ Σ, Ukj (z) → U(z), U ′ kj (z) → U ′(z), U ′′ kj (z) → U ′′(z) as j → ∞. Since Ukj (z) is a solution of (3.3), by taking kj → ∞ so that δkj → 0, we know that U(z) satisfies the wave equations corresponding (1.2), that is, cU ′ I = dIU ′′ I + fI(U), cU ′ S = dSU ′′ S + fS(U). (3.15) Thus, U(z) is a nonnegative semi-traveling solution of (1.2) satisfying lim z→−∞ U(z) = E0. Next, we show that ui(z) > 0 for i = 1, 2, . . .m. Since ui(z) = max{vieλcz(1 − τeµz), 0}, for z < z3 = − ln τ µ , ui(z) > 0, we have ui(z) ≥ ui(z) > 0. If there is a z0 such that ui(z0) = 0, then there are a and b such that a < z3 < b and z0 ∈ (a, b). Thus, ui(z) attains its nonpositive minimum over [a, b] at z0. Notice that on (a, b), ui(z) satisfies −diu ′′ i + ciu ′ i = fi(U(z)), or −diu ′′ i + ciu ′ i + qiui = ( m∑ j=1 aijui )( bi0 + n∑ k=m+1 uk ) ≥ 0, 20 Z. ZHANG EJDE-2022/25/CONF/26 By the strong maximum principle for second order elliptic equations (see [7]), u ≡ 0 on (a, b). This is a contradiction since ui(z) > 0 for z ∈ (a, z3). Therefore, we must have ui(z) > 0 for all z ∈ R. For i = m + 1,m + 2, . . . , n, we already know that 0 ≤ ui(z) ≤ u0 i . If, at some point z∗, ui(z ∗) = u0 i , then ui(z) attains its positive maximum at z∗. Therefore, u′ i(z ∗) = 0, u′′ i (z ∗) ≤ 0. But, from (3.15), we have diu ′′ i (z ∗) = ( m∑ j=1 aijuj(z ∗) )( n∑ k=m+1 bikuk(z ∗) ) ≥ ( m∑ j=1 aijuj(z ∗) ) (biiui(z ∗)) > 0. This is a contradiction. Therefore, we must have ui(z) < u0 i . Finally, we prove (3.14). We know that the semi-traveling wave solution U(z) satisfies U(z) ≤ U(z) ≤ U(z). For i = 1, 2, . . .m, ui(z) = { vie λcz, z ≤ z1 = lnω λc , viω, z > z1, and ui(z) = { vie λcz(1− τeµz), z ≤ z3 = − ln τ µ , 0, z > z3, where ω > 1, τ > 1, and 0 < µ < min{α, λc}, λc + µ < c 2d . Therefore, for z < 0 with |z| large enough, we have vie λcz(1− τeµz) ≤ ui(z) ≤ vie λcz. That is, vi(1− τeµz) ≤ ui(z)e −λcz ≤ vi. Thus, lim z→−∞ ui(z)e −λcz = vi. To prove that lim z→−∞ u′ i(z)e −λcz = λcvi, let us consider a nonnegative semi-traveling wave solution Uk(z) = (UIk(z), USk(z)) of the auxiliary system (3.2) with δ = δk = 1 k . Using that Uk(z) is a fixed point of operator Πk and the definition of Π, it is easy to prove that lim z→−∞ u′ i(z) = 0, i = 1, 2, . . . , n. For i = 1, 2, . . .m, integrating both sides of the i-th equation of (3.3) from −∞ to z and using the facts that ui(−∞) = u′ i(−∞) = 0 reveal that diu ′ i(z) = cui(z)− ∫ z −∞ fi(U)(s)ds+ δk ∫ z −∞ u2 i (s)ds. Therefore, lim z→−∞ u′ i(z)e −λcz = lim z→−∞ 1 di ( cui(z)e −λcz − e−λcz ∫ z −∞ fi(U)(s)ds+ δke −λcz ∫ z −∞ u2 i (s)ds ) = 1 di ( cvi − lim z→−∞ ∫ z −∞ fi(U)(s)ds eλcz + lim z→−∞ δk ∫ z −∞ u2 i (s)ds eλcz ) EJDE-2022/2025/CONF/26 TRAVELING WAVE SOLUTIONS 21 = 1 di ( cvi − lim z→−∞ fi(U)(z) λceλcz + lim z→−∞ δku 2 i (z)dz λceλcz ) = 1 di ( cvi − lim z→−∞ e−λczfi(U)(z) λc + lim z→−∞ δke −λczu2 i (z)dz λc ) = 1 di ( cvi − ( ∑m j=1 aijvj)(bi0 + ∑n k=m+1 biku 0 k)− qivi λc ) = λcdivi di = λcvi In the last equality we used (3.6). This completes the proof. □ 4. Nonexistence of traveling wave solutions The existence of semi-traveling wave solutions starting from the disease-free equi- librium E0 means that the disease will spread among the population. The sufficient conditions to have these semi-traveling wave solutions that we proved are R0 > 1 and c > c̃ with c̃ > 0 depending on parameters in the system, especially on the diffusion coefficients of those infected subclasses. This implies that the speed of population movement will affect the outbreak of the disease. The question is what condition will guarantee the nonexistence of semi- traveling wave solutions? For this we have the following result. Theorem 4.1. If R0 < 1, then for any c > 0, model (1.2) has no nonnegative nontrivial bounded semi-traveling wave solution satisfying (3.1). Proof. If R0 < 1, suppose that (1.2) has a nonnegative nontrivial bounded semi- traveling wave solution U(z) = (u1(z), u2(z), . . . , un(z)) satisfying (3.1). Without loss of generality, we can assume that 0 ≤ ui(z) ≤ u0 i for i = m + 1, . . . , n. Oth- erwise, we can rescale the solution so that this is true. Since R0 = ρ(EQ−1) = ρ(Q−1E) < 1, by Perron-Frobenius theorem, there exists aW = (w1, w2, . . . , wm) ∈ Rm with wi > 0 (i = 1, 2, . . .m) such that (Q−1E)W = R0W. (4.1) Observe that, for i = 1, 2, . . .m, ui(z) satisfies cu′ i = diu ′′ i + fi(U). It can be written as diu ′′ i + cu′ i + qiui = Ai(z)Bi(z) = ( m∑ j=1 aijuj )( bi0 + n∑ k=m+1 bikuk ) . (4.2) Let λi 1,2 be the roots of −diλ 2 + cλ+ qi = 0, that is, λi 1,2 = c± √ c2 + 4diqi 2di , then (4.2) can be converted to ui(z) = 1 Λi [ ∫ z −∞ eλ i 2(z−s)Hi(s) ds+ ∫ ∞ z eλ i 1(z−s)Hi(s) ds ] , (4.3) 22 Z. ZHANG EJDE-2022/25/CONF/26 where Λi = λi 1 − λi 2 > 0 and Hi(s) = Ai(s)Bi(s). From (4.3) and the assumption 0 ≤ ui(z) ≤ u0 i , i = m+ 1,m+ 2, . . . , n, we obtain 0 ≤ ui(z) = qi Λi [ ∫ z −∞ eλ i 2(z−s) 1 qi Hi(s) ds+ ∫ ∞ z eλ i 1(z−s) 1 qi Hi(s) ds ] ≤ (Q−1E)i qi Λi [ ∫ z −∞ eλ i 2(z−s)uI(s) ds+ ∫ ∞ z eλ i 1(z−s)uI(s) ds ] , (4.4) where (Q−1E)i (i = 1, 2, . . . ,m) denotes the i-th row of the matrix Q−1E and uI(s) = (u1(s), u2(s), . . . , um(s))T . Let u∗ i = supz∈R ui(z) for i = 1, 2, . . . ,m. Then u∗ = (u∗ 1, u ∗ 2, . . . , u ∗ m)T ≥ 0 and u∗ ̸= 0. By taking supremum on the both sides of (4.4), we obtain u∗ ≤ (Q−1E)u∗ (4.5) Since W is positive, there exists a positive constant ν > 0 such that u∗ ≤ νW (4.6) Using (4.5), (4.6), and (4.1) gives us u∗ ≤ (Q−1E)νW = ν(Q−1E)W = νR0W. (4.7) Using (4.5), (4.7), and (4.1) leads to u∗ ≤ νR2 0W. Continuing this process results in 0 ≤ u∗ ≤ νRm 0 W for all positive integer m. Taking limit m → ∞, since R0 < 1, leads to u∗ = 0. This is a contradiction. This completes the proof. □ 5. Discussion To investigate the influence of the mobility of population on the spread of disease, we propose and study a very general epidemic model which inculdes a large class of epidemic models that possess a disease-free equilibrium. Inspired by some specific models, we introduced basic reproduction number R0. It turns out that the basic reproduction number R0 is a threshold in the sense that the disease vanishes if R0 < 1 and spreads if R0 > 1. We introduced a critical wave speed c̃ and established the existence of semi- traveling wave solutions that start from the disease- free equilibrium for c > c̃ > 0. In order to investigate the final state of the disease, we need to establish the exis- tence of traveling wave solution. To do this, we need to impose more restrictions on the model to guarantee the existence of another nontrivial steady state equilibrium and the semi-traveling wave solutions whose existence we established also connect this steady state. As we mentioned before, this model is neither cooperative nor competitive. For these models, as the one investigated in [19], the spreading speed may be greater than the minimal wave speed. In a future work, we will investigate the relationship among the critical wave speed, the minimal wave speed, and the spreading speed. This is to prove that the critical wave speed introduced here is the same as the minimal wave speed. 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Zhenbu Zhang Department of Mathematics and Statistical Sciences, Jackson State University, Jack- son, MS 39217, USA Email address: zhenbu.zhang@jsums.edu 1. Introduction 2. Basic reproduction number for the reaction model and some examples 3. Semi-traveling waves for the diffusive model 4. Nonexistence of traveling wave solutions 5. Discussion References