Wave propagation in a diffusive SEIR epidemic model with nonlocal transmission and a general nonlinear incidence rate

We introduce a diffusive SEIR model with nonlocal delayed transmission between the infected subpopulation and the susceptible subpopulation with a general nonlinear incidence. We show that our results on existence and nonexistence of traveling wave solutions are determined by the basic reproduction number R0 = ∂IF(S0, 0)/γ of the corresponding ordinary differential equations and the minimal wave speed c∗. The main difficulties lie in the fact that the semiflow generated here does not admit the order-preserving property. In the present paper, we overcome these difficulties to obtain the threshold dynamics. In view of the numerical simulations, we also obtain that the minimal wave speed is explicitly determined by the time delay and nonlocality in disease transmission and by the spatial movement pattern of the exposed and infected individuals.


Introduction
In this paper, we consider the following diffusive SEIR model with nonlocal delayed transmission and a general nonlinear incidence rate: K(xy, ts)I(y, s) dy ds , (1.1) (i = 1, 2, 3, 4) are diffusion rates for the susceptible, exposed, infected, and removed individuals, respectively, α is the rate of the exposed turning infected, and γ stands for the recovery rate of the infective individuals. Obviously, 1/α is the average period for the exposed population to become infected. We use the assumption that the exposed individuals are of no infectiousness and their diffusive rate d 2 is not the same as d 3 . The reaction kernel K(xy, ts) ≥ 0 describes the interaction between the infective and susceptible individuals at spatial location x and present time t, which occurred at location y and at earlier instance s, and is assumed to satisfy the following hypotheses: (A1) K(x, t) is a nonnegative integrable function such that  [18]. If F(S, I) = βSI/(1 + ζ I q ) with ζ > 0 and q ≥ 1, then the incidence rate describes the saturated effects of the prevalence of infectious diseases; see [4,19,20]. For the case F(S, I) = βSI/(S + I), the incidence rate describes the outbreak disease model [26]. For more detail about the nonlinear incidence rates, we refer to [1,2,7,8,10,13,14,16,22,32].
In the study of population dynamics and the spread of infectious diseases, the reactiondiffusion equations with spatio-temporal delay are often used to describe biological and physical evolution processes. The spatial spread of infectious diseases is an important subject in mathematical epidemiology. Compartmental models describing the transmission of infectious diseases have been extensively studied in the literature. Usually, an infectious case is first found at one location, and then the disease spreads to other areas. Consequently, an important question for infectious diseases is what is the spreading speed?
Traveling wave solution is an important tool used in the study of the spreading speed of infectious diseases; see [3,11,15]. The existence and nonexistence of nontrivial traveling wave solutions indicate whether or not the disease can spread in the population and how fast a disease invades geographically. Also, theoretical results can help people make decisions on the disease control and prevention.
Some mathematical models may be described by (1.1)-(1.4) with appropriate choices of F and K . If we omit the exposed individuals E(x, t), then (1.1)-(1.4) reduce to (1.7) Taking F(S, I) = f (S)g(I) and K(x, t) = δ(x)δ(tτ ) with δ being the Dirac delta function, (1.5)-(1.7) become to the following model with delay: which was studied by Bai and Wu [2]. Choosing F(S, I) = βSI, (1.5)-(1.7) can be reduced to the following diffusive SIR model with spatio-temporal delay derived by Wang and Wu [27]: The constant β is the transmission rate between the infected and susceptibles. The convolution t -∞ +∞ -∞ K(xy, ts)I(y, s) dy ds shows the effects of spatial heterogeneity (geographical movement), nonlocal interaction, and time delay such as latent period on the transmission of diseases. They proved the existence and nonexistence of traveling wave solutions for system (1.11)-(1.12) by Schauder's fixed point theorem and Laplace transform.
Since equations (1.1)-(1.3) form a closed system, we omit equation (1.4) and study the following system only: Our purpose is to look for the nontrivial and nonnegative traveling wave solutions (S(x + ct), E(x + ct), I(x + ct)) of system (1.21)-(1.23) satisfying the following asymptotic boundary conditions at infinity: where S 0 is a positive constant representing the size of the susceptible individuals before being infected. Let ξ = x + ct ∈ R. Then the system describing traveling wave solutions is as follows:  4) introduces nonlinear incidence rate and nonlocal delayed transmission and is more complicated. Also, the loss of orderpreserving property for system (1.1)-(1.4) makes some classic methods fail to apply, for example, the shooting method [6], connection index theory [9], the general theory of traveling waves for monotone semiflows [17], the geometric singular perturbation method [21], monotone iteration combined with upper-lower solutions [29]. Fortunately, inspired by Zhao and Wang [33] that I can be presented by E and our system can be reduced to a twodimensional problem, we construct an auxiliary system with parameter κ, use a proper iteration scheme to construct a pair of upper and lower solutions, apply the Schauder fixed point theorem and obtain the existence of traveling wave solutions of the auxiliary system. By using a limit discussion we can see that the solution of the auxiliary system converges to the solution of the original system as κ → 0. Finally, we make use of the two-side Laplace transform in the proof of the nonexistence of traveling wave solutions.
We should point out that the exposed individuals have their own spacial diffusion rate, which plays an important part in the dynamics of transmission of disease. The exposed individuals play an important part in the transmission of diseases for their greater mobility compared to the infected. Therefore it is meaningful to study how the diffusion rate d 2 of the exposed influences the minimal wave speed c * , and in this work, we will show that the minimal traveling wave speed c * depends not only on d 2 , but also on d 3 . Moreover, c * is dependent on the pattern of nonlocal interaction between the infected and susceptible individuals and on the latent period of disease.
The remainder of this paper is organized as follows. In Sect. 2, we first prove some useful lemmas, which will be used in the proof of our main result. Later, we construct an invariant convex closed set, apply the Schauder fixed point theorem to prove the existence of traveling wave solutions for an auxiliary system, and then extend the result to the original system by a limiting argument. In Sect. 3, we prove the nonexistence theorem for two different cases: (i) ∂ I F(S 0 , 0)/γ > 1 and c < c * ; (ii) ∂ I F(S 0 , 0)/γ ≤ 1. In Sect. 4, we provided some examples to illustrate the main results. Finally, we carry out numerical simulations and give a brief discussion in Sect. 5.

Existence of traveling waves
To obtain the existence of traveling wave solutions of system (1.21)-(1.23) satisfying the asymptotic boundary conditions (1.24), we construct the auxiliary system where κ is a positive constant. Define h(x)e -μ + x < +∞ .
For i = 1, 2, we will give the definitions of a second-order linear differential operator D i and its inverse D -1 The roots of the equation We introduce the new symbol The differential operator D i , i = 1, 2, is defined by Furthermore, by a simple calculation we get Now we state some properties of the operators D i and D -1 i proved in [26].
for any M > 0, > 0, and λ such that - Given any A > 0, we have Remark 2.1 Although ϕ(ξ ) and ψ(ξ ) have certain points at which they are not differentiable, the integrals D -1 i (D i ϕ) and D -1 i (D i ψ) are well defined in the sense of distribution.
Define the convex closed set Proof For any given (S, E) ∈ , we first show that Moreover, Coupling the above two inequalities and using Lemma 2.3 yield In view of Lemma 2.3, we obtain from the above two inequalities that A combination of the above two inequalities and Lemma 2.3 yields This ends the proof.
Choose μ satisfying We set the functional space equipped with the norm Since μ > λ 1 > 0, it is easy to see that is uniformly bounded under the norm | · | μ in B μ (R, R 2 ). Before applying the Schauder fixed point theorem, we should verify that G is continuous and compact on .
Furthermore, from the above argument we obtain
Proof For any (S 1 , E 1 ) ∈ and (S 2 , E 2 ) ∈ , by the mean-value theorem we have whereĨ lies between [I(E 1 ) * K] and [I(E 2 ) * K],S lies between S 1 and S 2 , and Then where L κ = max{β 1 +m, m}. Thus we obtain that Since λ 1 < μ < min{--1 , --2 }, applying L'Hospital rule to the above formula gives Thus C(ξ ) is uniformly bounded on R. By Lemma 2.5 we conclude that G 1 is continuous with respect to the norm | · | μ . Similarly, we can show that G 2 is also continuous.
To prove the compactness of G, we will use the Ascoli-Arzelà theorem and the diagonal process. Denote I k := [-k, k], k ∈ N, and consider as a bounded subset of C(I k , R 2 ) with the maximum norm. It is easy to see that G( ) is uniformly bounded.
Next, we will show that G( ) is equicontinuous.
Let {u n } be a sequence of viewed as a bounded subset of C(I k ). Since {G(u n )} is uniformly bounded and equicontinuous on I k , by the Ascoli-Arzelà theorem and the diagonal process we can choose a subsequence {u n k } such that v n k := Gu n k converges in C(I k ) for all k ∈ N. Let v = lim k→+∞ v n k . Obviously, v ∈ C(R, R 2 ). Since G( ) ⊆ and is closed, we have v ∈ . It follows from μ > λ 1 > 0 that |E + (·)| μ is bounded and is uniformly bounded with respect to the norm | · | μ . Thus |v n k -v| μ is uniformly bounded for all k ∈ N. Given any > 0, we can choose a constant M > 0 independent of v n k such that e -μ|ξ | |v n k (ξ )v(ξ )| < for any |ξ | > M and k ∈ N. On the other hand, v n k converges to v on the compact interval [-M, M] with respect to the maximum norm, and thus there exists K ∈ N such that e -μ|ξ | |v n k (ξ )v(ξ )| < for all |ξ | ≤ M and k > K . Hence v n k converges to v with respect to the norm | · | μ . This proves the compactness of the map G. Due to S, E ∈ C -μ,μ (R) andi < -μ < μ < + i for i = 1, 2, we have Next, we will verify the asymptotic boundary conditions (1.24). Note that S -(ξ ) ≤ S(ξ ) ≤ S 0 and E -(ξ ) ≤ E(ξ ) ≤ E + (ξ ). It follows from the squeeze theorem that S(ξ ) → S 0 and E(ξ ) ∼ e λ 1 ξ as ξ → -∞. Recall the integral form of I(E): By the L'Hospital rule we have that Applying the L'Hospital rule to the maps G 1 and G 2 , it is easy to show that S (-∞) = 0 and E (-∞) = 0. From (2.21) we obtain Using the L'Hospital rule again, we get Integrating the above equality from ξ to +∞ gives Hence S is nonincreasing. Since (S, E) ∈ , for ξ < 0 with |ξ | sufficiently large, we have  Thus there exists ξ * < 0 such that S (ξ ) < 0 for all ξ < ξ * , which implies 0 ≤ S(+∞) = S ∞ < S 0 . Integrating (2.2) from -∞ to ξ yields By the boundedness of E and E on ξ ∈ R we obtain +∞ -∞ E(ξ ) dξ < +∞ and E(+∞) = 0, which implies I(+∞) = 0.
Similarly to the argument of (2.23), we have By applying the L'Hospital rule we obtain E (+∞) = 0, which yields I (+∞) = 0. Then from (2.2) we have E (+∞) = 0. Applying the L'Hospital rule to S again, we have that S (+∞) = 0. Therefore from (2.1) we obtain S (+∞) = 0. As a consequence, from ( Finally, we intend to prove the inequalities E(ξ ) ≤ S 0 -S ∞ and I(ξ ) < S 0 -S ∞ for all ξ ∈ R. Since E(ξ ) ∼ e λ 1 ξ as ξ → -∞ and E(ξ ) → 0 as ξ → +∞, we can define By the properties of E(ξ ) and the L'Hospital rule we obtain By differentiating (2.25) we get It is easy to see that An integration of the above equation from ξ to +∞ gives Here we have used the fact that H (+∞) = 0. Hence H(ξ ) is nondecreasing on R. Since H(+∞) = S 0 -S ∞ by the asymptotic formula obtained from equation (2.25), we obtain from the above equality that for ξ ∈ R. We can check that it satisfies the equation Obviously, Furthermore, we have that Hence we get that I(ξ ) ≤Ĥ(ξ ) < S 0 -S ∞ for all ξ ∈ R. Proof For c > c * , let {τ k } +∞ k=1 be a sequence such that 0 < τ k+1 < τ k < 1 and τ k → 0 as k → +∞. By Theorem 2.1, for any κ = τ k , there exists a solution (S k (ξ ), E k (ξ ), I k (ξ )) of (2.1)-(2.3) satisfying the results of Theorem 2.1. From (2.23) we have Similarly, we can show that By ( Thus (S k , E k , I k ),(S k , E k , I k ) and (S k , E k , I k ) are equicontinuous and uniformly bounded on R. By the Ascoli-Arzelà theorem it follows that there exists a subsequence of {τ k }, still denoted by {τ k }, such that, as k → +∞, uniformly on every bounded closed interval and pointwise on R. Noting that I k (ξ ) is bounded on R, we have By Lebesgue's dominated convergence theorem and the continuity of F we get that

Numerical simulations and discussion
In this paper, we consider a diffusive SEIR model with nonlocal delayed transmission and a general nonlinear incidence rate. By using the Schauder fixed point theorem and twoside Laplace transform we prove the existence and nonexistence of traveling wave solutions in terms of R 0 and c * . The results show that c * is the minimal wave speed, but it requires further research to show that c * is the asymptotic speed of propagation. This minimal wave speed c * is defined by Lemma 2.1, from which it is easy to see that c * depends on the diffusion rate d 2 of the exposed individuals, the diffusion rate d 3 of the infected individuals, the pattern of nonlocal interaction between the susceptible and infected individuals, and the latent period of disease. For c > 0 and λ ∈ [0, min{λ c , λ + }), c * is determined by the following equations: By calculation we get = -αλ 2 0 ∂ I F(S 0 , 0)e -λ 0 c * 1 τ λ 0 A 2 (λ 0 , c * 1 ) + αλ 2 0 τ ∂ I F(S 0 , 0)e -λ 0 c * 1 τ A(λ 0 , c * 1 ) + αλ 0 ∂ I F(S 0 , 0)R(λ 0 , c * 1 ) > 0, dc * 2 (d 2 , d 3 , ρ) dρ = αλ 0 c * 2 ∂ I F(S 0 , 0)e ρλ 2 0 A(λ 0 , c * 2 ) λ 0 A 2 (λ 0 , c * 2 ) + αλ 0 ∂ I F(S 0 , 0)R(λ 0 , c * 2 ) > 0, dc * 2 (d 2 , d 3 , ρ) dd 2 = -λ 2 0 A 2 (λ 0 , c * 2 ) λ 0 A 2 (λ 0 , c * 2 ) + αλ 0 ∂ I F(S 0 , 0)R(λ 0 , c * 2 ) > 0, dc * 2 (d 2 , d 3 , ρ) dd 3 = -αλ 2 0 ∂ I F(S 0 , 0)e ρλ 2 0 λ 0 A 2 (λ 0 , c * 2 ) + αλ 0 ∂ I F(S 0 , 0)R(λ 0 , c * 2 ) > 0, where A(λ, c) = -d 3 λ 2 + cλ + γ . By the above argument we can obtain that the latent period of disease can slow down the speed of the disease and the nonlocal interaction between the infective and susceptible individuals and that the diffusion rate d 2 of the exposed individuals and the diffusion rate d 3 of the infected individuals can increase the speed of the spread of the disease. To further illustrate our conclusions, we simulate how the latent period of disease, the nonlocal interaction between the infective and susceptible individuals and the spatial movement of expositive individuals and infective individuals affect the speed of the spread of the disease.
Third Model: K is Gaussian distribution N(0, ρ), that is,  We can see that in this case, the nonlocal reaction term degenerates to the local reaction.

Figure 4
The surface of c * (d 2 , d 3 ) for the second model. Let α = 1, β = 3, S 0 = 1, k = 0.2, and γ = 1.5 be fixed. Three surfaces from bottom to top are generated by taking τ = 4, 1, 0.2, respectively. We can see that the minimal wave speed c * increases with respect to d 2 and d 3 . Moreover, the surface of c * decreases as τ increases Figure 5 The surface of c * (d 2 , d 3 ) for the third model. Let α = 1, β = 3, S 0 = 1, γ = 1.5, and τ = 1 be fixed. Three surfaces from top to bottom are generated by taking ρ = 3, 2, 0.2, respectively. We can see that the surface of c * increases as ρ increases Remark 5.2 By numerical calculation we show how the nonlocal interaction between the infective and susceptible individuals affects the minimal wave speed c * ; see Fig. 1.
Remark 5. 3 We show how the latent period of disease affects the minimal wave speed c * in Fig. 2.
Furthermore, we are interested in the relation between the minimal wave speed c * and the diffusion rates of the exposed and the infected (see Fig. 3). For the second and third models, we have similar results given in Figs. 4 and 5.