Traveling waves for a diffusive SEIR epidemic model

In this paper, we propose a diffusive SEIR epidemic 
model with saturating incidence rate. We first study the well 
posedness of the model, and give the explicit formula of the basic 
reproduction number $\mathcal{R}_0$. And hence, we show that if 
$\mathcal{R}_0>1$, then there exists a positive constant $c^*>0$ such that for each 
$c>c^*$, the model admits a nontrivial traveling wave solution, and 
if $\mathcal{R}_0\leq1$ and $c\geq 0$ (or, $\mathcal{R}_0>1$ and 
$c\in[0,c^*)$), then the model has no nontrivial traveling wave 
solutions. Consequently, we confirm that the constant $c^*$ is indeed the 
minimal wave speed. The proof of the main results is mainly based on 
Schauder fixed theorem and Laplace transform.

1. Introduction. In order to describe the transmission of communicable diseases, Kermack and McKendrick [10] proposed the classic deterministic Susceptible-Infected -Removed (SIR) model where S(t), I(t) and R(t) denote the number of the susceptible, infected and removed individuals, respectively. The constant β is the transmission coefficient, and γ is the recovery rate. Given S(0) = S 0 > 0, I(0) > 0 and R(0) = 0. It is well known that, by the so-called basic reproduction number R 0 := βS 0 γ , Kermack and McKendrick [10] obtained full information about the transmission dynamics and epidemic potential: if R 0 > 1, then I(t) first increases to its maximum and then decreases to zero and hence an epidemic occurs; if R 0 < 1, then I(t) decreases to zero and epidemic does not happen.
In recent years, many researchers have attention to study the existence and nonexistence of traveling wave solutions for some diffusive epidemic models (see,e.g. [1, 12,20,22,23,24,25,30,33]). In particular, Wang and Wu [23] investigated the existence and non-existence of traveling wave solutions for a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission. In [22], Wang, Wang and Wu studied traveling waves of reaction-diffusion equations for a diffusive SIR model. In a very recent paper, for a three-dimensional reaction-diffusion systems, Zhang and Wang [33] established the existence of traveling wave solutions for influenza model with treatment.
In present paper, we will formulate a simple diffusive epidemic model incorporating exposed individuals into model (1.1). In the model considered, the total individuals at time t ≥ 0 and position x ∈ R, denoted by a constant N , is subdivided into four mutually exclusive compartments of susceptible S(x, t), exposed E(x, t), infectious I(x, t) and recovered R(x, t) individuals, so that S(x, t)+E(x, t)+ I(x, t) + R(x, t) = N . Here the model is called an SEIR model [2,17] where individuals are susceptible, then exposed (i.e., in the latent period), then infectious, then recovered from infectious individuals. In addition, we also assume that both the exposed and infected individuals are infected, they are distinguished by the absence/presence of their ability to infect other population [6,7]. On the other hand, it is believed that, in the modeling of infectious diseases, the incidence function plays a very important role, it can determine the rise and fall of epidemics [3,6,7]. Here we will use the saturating incidence rate denoted as βSI 1+aI (a > 0) [3] other than the bilinear incidence rate βSI. As reported in the literature, the saturating incidence rate is more realistic than the bilinear rate and can cause some interesting dynamic behaviors of infectious diseases, such as limit cycle, heteroclinic orbit, saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation, see, [7,14,18,28,29,31], for example.
In view of the above discussion, the proposed model is given by the following diffusive Susceptible-Exposed-Infected-Removed (SEIR) model with a saturating incidence rate ∂ ∂t S(t, x) = d 1 ∆S(t, x) − βS(t, x)g(I(t, x)), 2.1. The basic reproduction ratio and the well-posedness. Because R(t, x) does not appear in the first three equations of (1.3), and for the simplicity of notation, let (S, E, I) = (u 1 , u 2 , u 3 ), we consider the following new system where g(u) = u 1+au . Accompanied with (2.1), we consider the initial value conditions u i (0, x) = ϕ i (x) ≥ 0, x ∈ R, i = 1, 2, 3, but not identically zero. (2.2) In this subsection, we focus on the existence, uniqueness, invariance of solutions for the Cauchy problem to system (2.1) in R (see, Theorem 2.1). Firstly, we define the basic reproduction ratio R 0 for system (2.1). By similar arguments to those in [21,Theorem 2.3], we can show that the basic reproduction ratio R 0 equals the spectral radius of the following 2 × 2 matrix Hence, R 0 = βS 0 γ . For the definition of the basic reproduction ratio R 0 for the reaction-diffusion models and its biological interpretation, we refer the readers to [21] for details. In the following, we always assume that R 0 > 1.
In view of Theorem 2.1, the dynamics of system (2.1) can be analyzed in the bounded feasible region X M . Furthermore, the region X M is positively invariant with respect to model (2.1) and the model is well posedness.

2.2.
Eigenvalue problem. This subsection deals with the eigenvalue problems for the wave profile, which is obtained by substituting u i (t, x) = U i (x + ct) (i = 1, 2, 3) in (2.1). Here (U 1 , U 2 , U 3 ) is called the wave profile, ξ := x + ct the wave coordinate and c the speed. For the sake of convenience, we still use u i , t instead of U i , ξ. And for technical reasons, we let d 2 = d 3 =: d, and then consider the following wave profile equation We now consider the eigenvalue problem at (S 0 , 0, 0). Linearizing of the last two equations of (2.3) at (S 0 , 0, 0) implies that Plugging u i (t) = v i e λt , i = 2, 3, into the above equations, we get the following eigenvalue problem Clearly, the fact R 0 > 1 implies detA(0) = αγ(1 − R 2 0 ) < 0. Then the equation detA(λ) = 0 has at least one positive root.
When R 0 > 1, we set

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About the distribution of eigenvalue and their eigenvectors, we have the following results.
Lemma 2.1. Assume that R 0 > 1. Then we have the following statements.
The proofs of the conclusions (2) and (3) are easy and omitted the details.

Construction of the upper and lower solutions.
To establish existence of traveling wave solutions of (2.1), we will construct a convex invariant set. For this, we will use the iteration process to construct a pair of vector-value upper-lower solutions for (2.3). Namely, we first choose the upper solution u 1 (t) = S 0 for the u 1component (Lemma 2.2), which is then used to build the upper solutions u 2 (t) and u 3 (t) for the u 2 and u 3 -components (Lemma 2.3). The upper solutions u 3 (t) is in turn employed to produce the lower solutions u 1 (t) and u 3 (t) for the u 1 -component (Lemma 2.4). Finally, the lower solutions u 1 (t) and u 3 (t) are further builded to construct the lower solution u 2 (t) for the u 3 -components (Lemma 2.5). Here we also would like to stress that, different from that in [23], the pair of upper-lower solutions constructed in this paper are vector-value type and bounded on R.
In the following, we always assume that R 0 > 1, i.e., βS 0 > γ, and c > c * . Let λ 1 be the smallest eigenvalue defined as in Lemma 2.1(1) and (v 3 ) 0 its associating eigenvector. For practical reasons, it will be convenient to choose the vector (v for the calculations in the proofs of Lemmas 2.3 and 2.5.
For t ∈ R, we define six continuous functions as follows in which σ, M, are positive constants determined in the following lemmas.
It is easy to see the following Lemma 2.2 holds.
Lemma 2.2. The function u 1 (t) satisfies the inequality for all t ∈ R.
Lemma 2.3. The functions u 2 (t) and u 3 (t) satisfy the inequalities Proof. We only show the inequality (2.4) holds, since the proof of (2.5) is similar to that of (2.4). Indeed, when When t > t 1 , u 2 (t) = γ aα βS 0 γ − 1 . It follows from the fact that u 3 (t) ≤ 1 a βS 0 γ − 1 for all t ∈ R, and the function g(x) is increasing in x > 0. Then Thus, the inequality (2.4) holds. This completes the proof. (2.6) Then the functions u 1 (t) and u 3 (t) satisfy the inequality Proof. If t > t 3 , then the inequality (2.7) holds immediately since u which follows from the fact Noting that (2.6), we see (σS 0 ) which completes the proof.
2.4. The verification of Schauder fixed point theorem. In this subsection, we will use the upper and lower solutions (u 1 (t), u 2 (t), u 3 (t)) and (u 1 (t), u 2 (t), u 3 (t)) to verify that the conditions of Schauder fixed point theorem hold. Here, we use the usual Banach space B = C(R, R 3 ) of bounded continuous functions endowed with the maximum norm, see, [32], For c > c * , let where each α i is a large positive number such that α 1 > β a , α 2 > α and α 3 > γ. Then system (2.3) can be rewritten as Define the operator F : Γ → B by where u = (u 1 , u 2 , u 3 ), and in which It is easily verified that the operator F is well defined for u = (u 1 , u 2 , u 3 ) ∈ Γ, and satisfies Thus the fixed point of F is the solution of (2.12), which is a traveling wave solution of (2.1). Hence the existence of solution of (2.12) is reduced to verify that the operator F satisfies the conditions of Schauder fixed point theorem. Here we divide the proof into the following two lemmas.
Proof. Given u = (u 1 , u 2 , u 3 ) ∈ Γ. It is obvious that we only need to show that Based on the choice of the constants α 1 , α 2 and α 3 , it suffices to prove that, for any t ∈ R, Firstly, we prove (2.13) holds. In fact, for t > t 3 , then, by (2.7), Similarly, for t ≤ t 3 , we also get Therefore, (2.13) holds. The proofs of (2.14) and (2.15) are similar to that of (2.13) and are omitted. Consequently, we complete the proof. Proof. We first show that F = (F 1 , F 2 , F 3 ) : Γ → Γ is continuous with respect to the norm · . Indeed, for any u = (u 1 , u 2 , u 3 ), u = ( u 1 , u 2 , u 3 ) ∈ Γ, it is easy to see that there exists L > 0 such that Therefore,

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which indicates that the operator F 1 is continuous with respect to the norm · . Similarly, we also can show that operator F i , : Γ → Γ, i = 2, 3, is continuous with respect to the norm · .
3. Existence and non-existence of traveling wave solutions. In this section, we shall state precisely and prove the main results of this paper.

Existence of traveling wave solutions.
Here we establish the existence of traveling waves for system (2.1) with d 2 = d 3 =: d. To begin with, we first give some propositions.
Proof. In view of Lemmas 2.6 and 2.7, it follows from Schauder fixed point theorem that there exists a pair of u = (u 1 , u 2 , u 3 ) ∈ Γ, which is a fixed point of the operator F . Consequently, the solution (u 1 (x + ct), u 2 (x + ct), u 3 (x + ct)) is a traveling wave solution of system (2.1), and We claim that the strict inequalities hold. Indeed, note that u = (u 1 , u 2 , u 3 ) ∈ Γ is a fixed point of the operator F , then u 1 (t) = F 1 (u 1 , u 2 , u 3 )(t). Consequently, since u 1 (t) is continuous and is not identically zero, and Similarly, we can obtain that other inequalities are also strict ones. Thus, (3.1) holds.
On the other hand, in view of the following inequalities and e λ1t 1 − M e t ≤ u 3 (t) ≤ e λ1t , ∀ t ∈ R, which follows that Note that u = (u 1 , u 2 , u 3 ) ∈ Γ is a fixed point of the operator of F . Applying L'Höspital rule to the maps F 1 , F 2 and F 3 , it is easy to see that lim t→−∞ (u 1 (t), u 2 (t), u 3 (t)) = 0.
Hence, we have shown that (3.2) holds. Now, integrating both sides of the second equation of (2.3) from −∞ to t gives which implies, by the first one of (3.3), lim u3(t) = λ 1 . Hence, we complete the proof.
Proof. We first show the conclusion (1) holds. Indeed, in view of the fact (3.2), integrating the two sides of the first equation of (2.3) from −∞ to t follows We now claim that the integral Indeed, if not, note that the fact 0 < u 1 (t) < S 0 for all t ∈ R, by (3.5), we then conclude that there exists δ 0 > 0 such that u 1 (t) > δ 0 for all large t > 0, which implies that u 1 (t) → +∞ as t → +∞, this is contradiction. Hence, the improper integral +∞ −∞ u 1 (s)g(u 3 (s))ds converges, i.e., (3.6) holds. As a result that it follows u 1 (t) is uniformly bounded for all t ∈ R. Here it is clear from the first equation of (2.3) that Integrating the last equality from t to +∞ yields which, together with the fact u 1 (t) > 0 and u 3 (t) > 0 are continuous in t ∈ R, implies u 1 (t) < 0 for all t ∈ R. Thus, u 1 (t) is monotonically decreasing in t ∈ R, that is, the conclusion (1) holds. And let S 0 := lim t→+∞ u 1 (t), consequently, S 0 > S 0 ≥ 0.

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On the other hand, note that u 3 (t) satisfies the third equation of (2.3). Then here, Then, by Fubini's theorem again, by (3.11), As the same as the proof of the before, we also get u 3 (t) is uniformly bounded, together with (3.12), we get lim t→+∞ u 3 (t) = 0. Integrating the last equality of (2.3) from −∞ to t gives Letting t → +∞ in the above, which, together with (3.10) and (3.12), implies lim t→+∞ u 3 (t) = 0. Consequently, we have shown the conclusion (2) holds. Obviously, it follows from (3.9), (3.10) and (3.12) that the conclusion (3) holds. The proof is completed.
In view of Propositions 3.1 and 3.2, we are in a position to state and prove the existence of traveling wave solutions of system (2.1) with d 2 = d 3 =: d.

3.2.
Non-existence of traveling wave solutions. In this subsection, we will establish the nonexistence of traveling waves for system (2.1) either R 0 < 1 and c ≥ 0, or R 0 > 1 and c ∈ (0, c * ).
Theorems 3.1 and 3.3 combined establish a threshold condition for the existence and non-existence of traveling wave solutions in terms of the basic reproduction number R 0 of system (2.1). Namely, R 0 > 1 implies c * > 0. It follows from Theorems 3.1 that the propagation of the pathogen as a wave with a fixed shape and a fixed speed for any c > c * from the initial disease-free equilibrium (S 0 , 0, 0) to another disease-free equilibrium (S 0 , 0, 0). For every c < c * , Theorem 3.3 implies that system (2.1) has no nontrivial and nonnegative traveling wave connecting the two equilibria (S 0 , 0, 0) and (S 0 , 0, 0). The study of traveling wave solutions provides important insight into the spatial patterns of invading diseases.
As observed, the proofs of Theorem 3.1(1) and Theorem 3.2 depend heavily on the choice of the same diffusion rate d for exposed individuals and infected ones, i.e., the last two equations of system (2.1) have the same diffusion rate d. Hence, some results obtained here can not be extrapolated for a more general class of population models. However, in realistic communities, the infectious disease transmits in the spatial movement of the region by the different diffusion rates for exposed and infected individuals. From the view of mathematical biology, it is important problem to deal with this case and we will consider the topic in our further study.