TRAVELING WAVES FOR A NONLOCAL DISPERSAL SIR MODEL EQUIPPED DELAY AND GENERALIZED INCIDENCE

. In this paper, the existence and non-existence of traveling wave solutions are established for a nonlocal dispersal SIR model equipped delay and generalized incidence. In addition, the existence and asymptotic behaviors of traveling waves under critical wave speed are also contained. Especially, the boundedness of traveling waves is obtained completely without imposing additional conditions on the nonlinear incidence.

As is well known, for a long range diffusion such as population ecology, neurology and epidemiology, the flow of individuals is not only limited to the same one point, but is affected by other points around it. Therefore, the nonlocal dispersal is more realistic than the local diffusion [4,10,20], which can be expressed by a convolution term L[u](x, t) = J * u(x, t)−u(x, t) = R J(x−η)(u(η, t)dη−u(x, t))dη, where u(x, t) denotes the density of individuals and J(x − η) is the probability distribution of individuals which jump from location η to location x. Then J * u(x, t) = R J(x − η)u(η, t)dη denotes the rate at which individuals are arriving at location x from all other locations, while the term −u(x, t) = − R J(x − η)u(x, t)dη is the rate at which they are leaving location x to travel to all other locations. Thus, the nonlocal dispersal L[u](x, t) can be biologically interpreted as the net increasing rate of u(x, t). In 2014, by the same method in [13], Li and Yang [11] inspected the nonlocal dispersal situation of (1.1): where R J(x − η)u(η, t)dη := J * u denotes the normal convolution. For other related works, one can refer to [17,18].
Furthermore, the state of time delay exists universally in the objective material world [12,16]. In addition, the general incidence is more extensive to illustrate the disease spread process than the special standard incidence. For the above reasons, Zhang et al. [19] considered the following SIR model They showed that there is a number c * > 0 such that traveling wave solutions (U (x + ct), V (x + ct)) of (1.3) conforming to U (±∞) = U ±∞ , V (±∞) = 0 and U (−∞) > U (+∞) exist when V is a bounded function and R 0 := f (U−∞)g (0) ν > 1 with c > c * , but for R 0 ∈ (0, 1) and R 0 ∈ (1, +∞) with c ∈ (0, c * ), there are no traveling waves.
Although there have been many excellent results as mentioned above, it is necessary to indicate the core problem that (i) the boundedness of traveling waves is not obtained easily by constructing bounded invariant cones due to the shortage of natural upper bound of nonlinear incidence g(u 2 ), which is different from the standard incidence case u2 u1+u2 < 1. On the other hand, (ii) it is extremely tough to investigate the existence and asymptotic behaviors with c = c * because of the absence of order-preserving quality of semi-flow of (1.3) and the inferior smoothness of solutions for the import of nonlocal dispersal.
In order to solve the first problem (i), Zhang et al. [19] obtained the boundedness and asymptotic behaviors of traveling waves when c > c * and R 0 > 1 by assuming that f (U −∞ )g(V 0 ) ≤ νV 0 holds for some V 0 ∈ R. Similarly, owing to the same difficulties, Zou and Wu [21] only obtained the boundedness and asymptotic behaviors under the large wave speed and a specific assumption.
However, the scope of incidence functions is not extensive since the strict condition in [19] and there is still not result of existence of traveling waves under critical wave speed. Fortunately, Yang and Li [18] recently considered a SIR model equipped bilinear function αu 1 u 2 and established the boundedness and asymptotic behaviors of traveling waves for c ≥ c * and R 0 > 1 by some limit discussions and a series of analyses without imposing additional conditions upon incidence function.
Based on the above fact and motivated by the idea in [17,18], in this paper, we illustrate the existence, boundedness and asymptotic behaviors of traveling waves of system (1.3) for non-critical and critical wave speed, respectively, which complete and improve the works in [19,21]. In this sense, the above two difficulties we mentioned in (i) and (ii) are solved. Moreover, we extend the delay-free case in [17,18] to the case with time delay and generalize the bilinear incidence to a more general case.
Below, the following assumptions are always valid for the whole paper: R J(x)dx = 1; Moreover, J is compactly supported. The remaining part of this paper is designed as follows. In section 2, we complete the existence results of traveling waves when R 0 > 1 with c > c * in [21] by some analytical techniques. In addition, the boundedness of traveling waves is also included. In section 3, the existence and asymptotic behaviors of traveling waves when R 0 > 1 and c = c * are established by a prior estimate and some technical analyses. In section 4, a new way is given to derive the non-existence of traveling waves for R 0 > 1 and c < c * .

Boundedness and existence of traveling waves with c > c *
In this section, the boundedness and existence of traveling wave solutions of (1 Noticing that the first two equation of (1.3) are independent of the function u 3 , we focus only on the solutions with the profile of (U (x+ct), V (x+ct)) = (U (ξ), V (ξ)) of the following system where ξ = x + ct. Next, the following two important conclusions in [21] are needed: Then some positive pair of (c * , λ * ) exists for the following equations According to Proposition 2.2, lim sup To perfect Proposition 2.2, we complete the case of 3 2 d 2 k 1 > c * and give out the proof of lim sup For the proof, we first establish the following lemmas and the boundedness of V (ξ).
Proof. Firstly, from Lemma 2.5 and Theorem 2.1 in [21], we obtain that Finally, we can prove similarly that V > 0 and U < U −∞ for ξ ∈ R. This proof is complete.
. Then, K and ω are both bounded for c * < c ≤ c 1 .
Proof. According to (2.1), we have and thus Therefore, H is non-decreasing and lim By an integral process for (2.7) from −∞ to ξ, it holds that By a similar integral process for (2.7) from ξ − R 1 to ξ, we find that By (2.6), (2.8) and (2.10), we have On the other hand, it is obvious that This proof is complete.
This completes the proof.
Up to now, by constructing the boundedness of V (ξ), we obtain a more general existence result and thus improve and complete the results in [19,21]. Next, we illustrate the existence under critical speed for further improvement.

Existence of traveling waves with c = c *
In this section, an approximating method is applied to establish the existence of solutions of (2.1) when R 0 > 1 with c = c * . For this proof, a prior estimate is needed in the followings.
Lemma 3.1. Assume {c k } ⊂ (c * , c * + 1) is a decreasing sequence with c k → c * as k → +∞ and let (c k , U k , V k ) be a solution of (2.1) for k ∈ N . Then, U k C 1,1 (R) and V k C 1,1 (R) are both uniformly bounded.
Proof. Firstly, we prove the uniform boundedness of {U k } and {V k }. It is obvious that {U k } is uniformly bounded due to (2.3). Suppose that there is a sequence {ξ k } satisfying lim k→+∞ V k (ξ k ) = +∞ for a contradiction.
Secondly, according to the above discussions and (2.1), it holds that {U k } and {V k } are both uniformly bounded. By a similar discussion to Lemma 2.6 in [19], it can be derived that U k , V k , U k (ξ) and V k (ξ) are all Lipschitz continuous. The proof is finished.
Next, we demonstrate the existence, boundedness, positivity and asymptotic behavior of traveling waves as followings.
The rest of proofs are divided into the following three steps.
The remaining proofs in this step are similar to (ii) of Theorem 2.1 in [21] and Step 2 of Theorem 2.6 in this paper, so we omit them here.
Step 2. The functions U * and V * are both positive on R.
Assume that U * (γ 0 ) = U −∞ for some γ 0 ∈ R. By the fact that U * (ξ) ≤ U −∞ and (2.1), it follows that which is impossible since the positivity of U * and V * . This finishes the proof.

Nonexistence of traveling waves
In this section, we prove the nonexistence of solution of (2.1) by a different approach which depends closely on the conclusions in Section 2.