Traveling waves in an SEIR epidemic model with the variable total population

In the present paper, we propose a simple diffusive SEIR epidemic model 
where the total population is variable. We first give the explicit formula of the 
basic reproduction number $\mathcal{R}_0$ for the model. And hence, we show that if 
$\mathcal{R}_0>1$, then there exists a constant $c^*>0$ such that for any $c>c^*$, the 
model admits a nontrivial traveling wave solution, and if $\mathcal{R}_0 0$ 
(or, $\mathcal{R}_0>1$ and $c\in(0,c^*)$), then the model has no nontrivial traveling 
wave solution. Consequently, we obtain the full information about the existence and 
non-existence of traveling wave solutions of the model by determined by the constants 
$\mathcal{R}_0$ and $c^*$. The proof of the main results is mainly based on Schauder 
fixed point theorem and Laplace transform.


Introduction. In 1927, Kermack and McKendrick
proposed the well-known deterministic susceptible-infected-removed (SIR) model describing the transmission of infectious diseases, and established a systematic "threshold theory" that determines whether a disease will become prevalent or not. These results gave information on prevalence of diseases, the transmission mechanisms and the effect of interventions [4,8,13]. Therefore, many researchers utilized ordinary differential equations to explore mechanisms and dynamical behaviors of communicable diseases qualitatively and quantitatively [4,8]. When individuals move randomly, the corresponding reaction-diffusion model takes into the following diffusive SIR model ∂ ∂t S(t, x) = d 1 ∆S(t, x) − βS(t, x)I(x, t), ∂ ∂t I(t, x) = d 2 ∆I(t, x) + βS(t, x)I(x, t) − γI(t, x), ∂ ∂t R(t, x) = d 3 ∆R(t, x) + γI(t, x), (1.1) where S(t, x), I(t, x) and R(t, x) denote respectively the number of the susceptible, infected and removed individuals at time t and space x ∈ R, ∆ = ∂ 2 ∂x 2 . The constant β is the transmission coefficient, γ is the recovery (or remove) rate, and d 1 , d 2 and d 3 are the diffusion rates.
Note that R does not appear in the first two equations of (1.1), it suffices to consider the two dimensional system for (S, I) as following  Hosono and Ilyas [9] showed that if S(0, x) = S 0 > 0 and βS 0 > γ, then for each c ≥ c * = 2 d 2 (βS 0 − γ) there exists a positive constant ε < S 0 such that model (1.2) has a traveling wave solution (S(x + ct), I(x + ct)) satisfying S(−∞) = S 0 , S(+∞) = ε, I(±∞) = 0. On the other hand, there is no traveling wave solution for (1.2) when βS 0 ≤ γ. There are substantial recent developments on the existence and non-existence of traveling wave solutions for Kermack-Mckendrick SIR model, see [1,2,7,12,17,19,20,24,25,27,28] and references therein. In particular, Wang and Wang [17] introduced the following general diffusive Kermack-McKendrick SIR model − γI(t, x) − δI(t, x), ∂ ∂t R(t, x) = d 3 ∆R(t, x) + γI(t, x), (1.3) in which N = S + I + R is the total population, d 1 , d 2 and d 3 are the diffusion rates of the susceptible (S), infective (I) and recovered (R) individuals, respectively. γ ≥ 0 is the recovery rate and δ ≥ 0 is the death (or quarantine) rate of infective individuals. Considered the mobility of individuals, they assumed that the total population N (t, x) is not fixed. Based on Schauder fixed point theorem and Laplace transform, Wang and Wang [17] established some similar results as correspond to Hosono and Ilyas' theorems for system (1.3). In a very recent work, Yang, Li and Wang [28] have extended the results in [17] to a nonlocal dispersal SIR model.
In this paper, we will incorporate exposed individuals into model (1.3) and consider the following SEIR epidemic model with spatial diffusion   (1.4) in which, the total varying individuals at time t and space x ∈ R, denoted by N (t, x), is sub-divided 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 (x, t). d i > 0 (i = 1, 2, 3, 4) are the diffusion rates, respectively. The constant β > 0 is the transmission coefficient, κ > 0 is the rate constant for exposed individuals becoming infectious, α > 0 is the rate constant that the infective individuals leave the infected class I. We assume that the fraction f of the αI members leaving the infective class recover and the remaining fraction (1 − f ) die of disease. In contrast, for the detailed epidemiological description of the corresponding ODE model, we refer to [4,8,13].
The purpose of the current paper is to study the existence and non-existence of traveling wave solutions of (1.4). To prove the existence theorem (Theorem 3.1), that is, there exists a constant c * > 0 such that (1.4) has a traveling wave solution if c > c * and β > α, we will employ Schauder fixed point theorem to the nonmonotone operator used in a suitable invariant convex set. In order to construct the appropriate invariant convex set, we should use the ideas of the iteration process [3,17,19,20] to construct the upper-lower solutions. One important feature of our method, which is different to [3,17,19,20], is that we need to construct the vectorvalue upper-lower solutions for (2.1) (see, Section 2.2) developed by [21] since system (2.1) consists of four equations. Moreover, we establish that (1.4) has no traveling wave for any c > 0 and β < α (Theorem 3.2). And also, we conclude the nonexistence of traveling wave solutions for (2.1) for c ∈ (0, c * ) and β > α (Theorem 3.3). Here the critical method is based on the two-side Laplace transform. As we know that the application of the Laplace transform requires the prior estimate of the exponential decay of the traveling wave solutions [5,17,19,20]. However, it seems that the analytical method in [5,17,19,20] cannot give the prior estimate due to the four dimensional system (2.1). Instead, we approve the Stable Manifold Theorem [14] to get the prior estimate. The approach in this paper provides a promising method to deal with high dimensional reaction-diffusion systems.
This paper is organized as follows. In Section 2, we establish some preliminary results. Sections 3 is devoted to the study of the existence and non-existence of traveling waves for system (1.4). Finally, we give a brief discussion in Section 4.

Preliminaries.
In this section, we should give some preliminary results such as the basic reproduction number and the eigenvalue problems for the wave profile equation (2.1), constructing a pair of upper-lower solutions for system (2.1) and verifying the conditions of the Schauder fixed point theorem.
2.1. The basic reproduction number and eigenvalue problem. First of all, we give the basic reproduction number R 0 for system (1.4). By similar arguments to those in [18,Theorem 2.3], we can show that the basic reproduction number R 0 equals the spectral radius of the following 2 × 2 matrix Hence, R 0 = β α . For the definition of the basic reproduction number R 0 for the reaction-diffusion models and its biological interpretation, we refer the readers to [18] for details.
Next we deal with the eigenvalue problems for the wave profile, which is obtained substituting S(t, x) = S(x + ct), E(t, x) = E(x + ct), I(t, x) = I(x + ct), and R(t, x) = R(x + ct) into (1.4). Here ( S, E, I, R) is called the wave profile, ξ := x + ct the wave coordinate and c the speed. For the sake of convenience, we still use S, E, I, R, t instead of S, E, I, R, ξ, and then get the following wave profile equation In the sequel, we always assume that the initial free equilibrium is (S 0 , 0, 0, 0) with S 0 > 0. We now consider the eigenvalue problem at (S 0 , 0, 0, 0). Linearizing of the second and third equations of (2.1) at (S 0 , 0, 0, 0) gives Plugging E(t) = v E e λt and I(t) = v I e λt into the above equations, we get the following eigenvalue problem Then the characteristic equation detA(λ) = 0 has at least one positive root.

Construction of the upper and lower solutions.
To establish existence of traveling wave solutions of (1.4), we will construct a convex invariant. For this, noting that system (2.1) consists of four equations, we use the iteration process [3,17,19,20] to construct a pair of vector-value upper-lower solutions for (2.1). The idea constructing a pair of vector-value upper-lower solutions is motivated by Weng and Zhao [21], see also Fang and Zhao [6], Wang [16] and Xu and Ai [26].
In the following, we always assume that R 0 > 1, i.e., β > α, and c > c * . Let λ 1 be the smallest eigenvalue defined as in Lemma 2.1(1) and (v E , v I ) 0 its associating eigenvector, which satisfies Also by Lemma 2.1(1), for a sufficient small ∈ (0, λ 2 − λ 1 ), we get Then we can choose a constant h(λ 1 + ) > 0 such that Now, for t ∈ R, we define eight continuous functions as follows and in which σ, , q 1 , q 2 are positive constants determined in the following lemmas.
Lemma 2.2. The following inequalities hold.
Proof. Note that the functions S(t) = S 0 , E(t) and R(t) are positive, and I(t) is nonnegative. Then (2.4) holds clearly. Next, we show the inequality (2.5) holds. Indeed, note that I(t) ≤ v I e λ1t for all t ∈ R, and E(t) = v E e λ1t when t < t 1 . Then, by (2.2), for t < t 1 , When t > t 1 , E(t) = α κ β α − 1 S 0 , and it follows from the facts S(t) = S 0 and Thus, the inequality (2.5) holds. Similar to the proof of (2.5), we can prove (2.6) holds. Finally, we show the inequality (2.7) holds. When t < t 2 , I(t) = v I e λ1t , one gets For any t ≥ t 2 , Hence, if we take holds. This completes the proof.
Then the following inequalities hold.
where t 3 := 1 σ ln(σS 0 ), and q 2 > v E is a sufficiently large constant such that Here the function (S(t), E(t), I(t), R(t)) is called a lower solution of (2.1).
2.3. The verification of Schauder fixed point theorem. In this subsection, we will use the upper and lower solutions (S(t), E(t), I(t), R(t)) and (S(t), E(t), I(t), R(t)) constructed in section 2.2 to verify that the conditions of Schauder fixed point theorem hold. Let α 1 > β, α 2 > κ, α 3 > α and α 4 > 0 be four constants, and let With aid of the upper and lower solutions, we define a convex set Γ as Since µ > λ 1 , it is easily see that Γ is uniformly bounded with respect to the norm | · | µ . Furthermore, we define an operator F : Γ → C(R, R 4 ). And for a given where and , , Proof. For any u = (S, E, I, R) ∈ Γ, it is obvious that we only need to show that the following inequalities hold for all t ∈ R. We now consider F 1 (u)(t). Note that H 1 (u)(t) ≤ α 1 S(t) ≤ α 1 S 0 for all t ∈ R. Thus By (2.8), for t = t 3 , Then, when t > t 3 , Similarly, when t < t 3 , by (2.8), we also show F 1 (u)(t) ≥ S(t) holds. Thus, we have shown that S(t) ≤ F 1 (u)(t) ≤ S 0 holds for all t ∈ R. The proofs of the following inequalities are similar to that of S(t) ≤ F 1 (u)(t) ≤ S 0 for all t ∈ R, and the proof is completed. Proof. We first show that F = (F 1 , F 2 , F 3 , F 4 ) : Γ → Γ is continuous with respect to the norm | · | µ . Indeed, for any u i = (S i , E i , I i , R i ) ∈ Γ, i = 1, 2, it is easy to see that there exists a constant L 1 > 0 such that Therefore, where The direct calculations show that that is, C(t) is uniformly bounded on R, which follows from (2.12) that the operator F 1 is continuous with respect to the norm | · | µ . Similarly, we also can show that operator F i : Γ → Γ, i = 2, 3, 4, is continuous with respect to the norm | · | µ . Consequently, F is a continuous operator on Γ with respect to the norm | · | µ . Next, we use the similar arguments as in [17] to prove the compactness of F , that is, we shall make use of Arzalà-Ascoli theorem and a standard diagonal process. Let I k = [−k, k] with k ∈ N be a compact interval on R and temporarily we regard Γ as bounded subset of C(I k , R 4 ) equipped with the norm | · | µ . Since F maps Γ into Γ, it is obvious that F is uniformly bounded on I k . In the following, we shall show that F is equi-continuous on I k . To this end, we first establish four inequalities for the derivative of F . In fact, note that u = (S, E, I, R) ∈ Γ, it is easy to see that there is a constant H 0 > 0 such that Consequently, for u = (S, E, I, R) ∈ Γ,

ZHITING XU
Similarly, we have, for all t ∈ R, Let {v n } be a sequence of Γ, which can be also viewed as a bounded subset of C(I k ). Since F is uniformly bounded and equi-continuous on I k , by the Arzelà-Ascoli theorem and the standard diagonal process, we can extract a subsequence {v n k } such that v n k := F vn k converges in C(I k ) for any k ∈ N. Let v := lim k→∞ v n k . It is readily seen that v ∈ C(R, R 4 ). Furthermore, since F (Γ) ⊂ Γ (by Lemma 2.4 and Γ is closed), it follows that v ∈ Γ. Note that µ > λ 1 , it follows from the definition of R(t) that e −µ|t| R(t) are uniformly bounded on R. Thus, Γ is uniformly bounded with respect to the norm | · | µ . Consequently, the norm |v n k − v| µ is uniformly bounded for all k ∈ N. For any ε > 0, we can find an integer N 0 > 0 independent of v n k such that |v n k (t) − v(t)|e −µ|t| < ε for any |t| > N 0 and k ∈ N. Since v n k converges to v on the compact interval [−N 0 , N 0 ] with respect to the maximum norm, there exists K ∈ N such that |v n k (t) − v(t)|e −µ|t| < ε holds for any |t| ≤ N 0 and k > K. The above two inequalities imply that v n k converges to v with respect to the norm | · | µ . This proves the compactness of the map F . Therefore, we complete the proof.
3. Existence and non-existence of traveling wave solutions.

3.1.
Existence of traveling wave solutions. In this subsection, we establish the existence of traveling waves for system (1.4). To begin with, we first give two propositions.
Proof. In view of Lemmas 2.4 and 2.5, it follows from Schauder foxed point theorem that there exists a pair of u = (S, E, I, R) ∈ Γ, which is a fixed point of the operator F . Consequently, the solution (S(x+ct), E(x+ct), I(x+ct), R(x+ct)) is a traveling wave solution of system (1.4), and for any t ∈ R, which follow Thus, recall that (3.1), by L'Höspital rule again, Similarly, we also show lim Proof. We first show the conclusion (1) holds. Indeed, in view of the facts lim If not, note that the fact 0 ≤ S(t) ≤ S 0 for all t ∈ R, by (3.3), we then conclude that there exists δ 0 > 0 such that S (t) > δ 0 for all large t > 0, which implies that lim t→+∞ S(t) = +∞, this is contradiction. Hence, the improper integral Integrating the last equality from t to +∞ yields which, together with the fact S(t) ≥ 0 and I(t) ≥ 0 are continuous and not identically zero in t ∈ R, implies S (t) < 0 for all t ∈ R. Thus, S(t) is monotonically decreasing in t ∈ R, and let S 0 := lim t→+∞ S(t), consequently, S 0 > S 0 ≥ 0. Note that E(t) satisfies the second equation of (2.1). Then here, Note that for (S, E, I, R) ∈ Γ, it is clear that Then Thus, E (t) is uniformly bounded, which, together with that E(t) ≥ 0 is integrable on R (by (3.6)), implies lim Combining (3.6) with (3.7), we derive that On the other hand, note that I(t) satisfies the third equation of (2.1). Then here, Then, by Fubini's theorem again and (3.9), As the same as the proof of the before, we also get I (t) is uniformly bounded, and then, together with (3.10), we get lim t→+∞ I(t) = 0. Integrating the third equation of (2.1) from −∞ to t follows Letting t → +∞ in the above, which, together with  hold. (2) S(t) and E(t) are monotonically decreasing in t ∈ R, and (3) R(t) is monotonically increasing in t ∈ R, and Proof. To prove Theorem 3.1, by Propositions 3.1 and 3.2, it suffices to show that holds. To this end, we first show 0 < S(t) < S 0 for all t ∈ R. In fact, in view of Lemmas 2.4 and 2.5, it follows from Schauder fixed point theorem that there exists a pair of u = (S, E, I, R) ∈ Γ, which is a fixed point of the operator F . As a result, the solution (S(x + ct), E(x + ct), I(x + ct), R(x + ct)) is a traveling wave solution of system (1.4), and 0 ≤ S(t) ≤ S 0 . We next show that the strict inequalities hold. Indeed, note that u = (S, E, I, R) ∈ Γ is a fixed point of the operator F , then S(t) = F 1 (S, E, I, R)(t). Consequently, Similarly, we can show that the inequalities S(t) < S 0 , E(t) > 0, I(t) > 0 and R(t) > 0 hold for all t ∈ R.
Next, we show E(t) < S 0 − S 0 for any t ∈ R. In fact, define which, by L'Höspital's rule, together with (3.12), implies that (3.14) By differentiating (3.13) once, Furthermore, by differentiating (3.15) once, and noting that E(t) satisfies the second equation of (2.1), one gets which follows that Consequently, G(t) is increasing in t ∈ R. Further, by (3.14), Similarly, we also get Hence, we complete the proof.
4. Discussion. In this paper, we present a diffusive SEIR epidemic model (1.4) with standard incidence rate where the total population is variable. Applying [18, Theorem 2.3], we give the explicit formula of the basic reproduction number R 0 for system (1.4). For the model under consideration, the traveling wave solutions describes the disease propagation into the susceptible individuals from an initial disease-free equilibrium to the final, also disease-free equilibrium. In this paper, we first use the iteration process to construct the vector-value upper-lower solutions for (2.1). Together with the Schauder fixed point theorem, we can establish existence of such a traveling wave solution. Second, we use the two-sides Laplace transform to establish the non-existence of such a traveling wave solution. These results could formulate the possible propagation models of the disease. Theorem 3.1 gives some asymptotic behaviors of the traveling wave solution (S(x+ct), E(x+ct), I(x+ct), R(x+ct)) of system (1.4) with speed c > c * . Note that, in the condition, S 0 > 0 is a constant representing the number of the susceptible individuals before being infected. Clearly, at any fixed x ∈ R, Theorem 3.1(1) and (2) describe that all the individuals were susceptible a long time ago (t → −∞) and all the susceptible individuals will be decreasing to S 0 after a long time (t → +∞). In particular, if S 0 = 0 (due to S 0 may be zero), then all the susceptible individuals will become the removed individuals also after a long time (t → +∞). Hence, the natural question arises. Can we know the value of S(+∞) = S 0 ? As pointed in [19], for general system such as (1.4) with nonzero diffusion terms, it seems impossible to obtain the value of S 0 , see [19] for a brief discussion on related works for this problem. We conjecture S 0 = 0, but unfortunately, we do not know to prove it. How to overcome technical problems that prevent a full analysis of relations of them should be a challenging work and leave it as our further project.
As a final remark, we would like to comment Theorems 3.1 and 3.3 by stressing that no existence or non-existence of wave has been derived for the wave speed c * (in the case R 0 > 1). Let us mention that a limiting argument [15] (looking at the convergence of a sequence of a traveling wave with speed {c n } such that lim n→∞ c n = c * ) is complicated because in this paper we can not show whether the waves are monotone. We expect that the constant c * is the minimum wave speed but this question remains an open problem.