Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission

Abstract

We obtain full information about the existence and non-existence of travelling wave solutions for a general class of diffusive Kermack-McKendrick SIR models with nonlocal and delayed disease transmission. We show that this information is determined by the basic reproduction number of the corresponding ordinary differential model, and the minimal wave speed is explicitly determined by the delay (such as the latent period) and non-locality in disease transmission, and the spatial movement pattern of the infected individuals. The difficulty is the lack of order-preserving property of the general system, and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying Schauder's fixed point theorem.

Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

“…We refer to [9,10,11,26,28,39] and references therein for more results on the development of the study of epidemic problems.…”

Traveling waves in a diffusive influenza epidemic model with vaccination

“…As we known that the application of the Laplace transform requires the prior estimate of the exponential decay of the traveling wave solutions [3,10,18,19]. However, it seems that the analytical method in [3,10,18,19] cannot give the prior estimate for the model because of the four dimensional system (3.1) again. Instead, we approve the approach recently introduced by [25] to get the prior estimate.…”

“…To prove the existence theorem (Theorem 4.1), we first introduce an auxiliary system (3.1) and employ Schauder fixed point theorem [8,9,12] to prove the existence of traveling wave solutions for the auxiliary system, and then by the limit arguments and Arzelà-Asocli's theorem, we extend the results for (3.1) to (1.2). Here, in order to apply Schauder fixed point theorem, we also use the idea of the iteration process [1,10,18,19] to construct the upper-lower solutions. One important feature of our method, which is different from the ones [1,10,18,19], is that we need to construct the vector-value upper-lower solutions for (3.1) (Section 3.1) since system (3.1) consists of four equations.…”

“…Here, in order to apply Schauder fixed point theorem, we also use the idea of the iteration process [1,10,18,19] to construct the upper-lower solutions. One important feature of our method, which is different from the ones [1,10,18,19], is that we need to construct the vector-value upper-lower solutions for (3.1) (Section 3.1) since system (3.1) consists of four equations. The idea of such a construction is motivated by Weng and Zhao [21], see also [17] for the related works.…”

Traveling waves for a nonlocal dispersal SIR model with delay and external supplies

“…As we know, since Bartlett [1] in 1956 predicted a wave of infection moving out from the initial source of infection with a constant speed, epidemic waves have been extensively studied, see for example [10,29,22,45,15,38,40,42,20,12,19,11,23,46]. See also the monograph [30], the survey papers [32,33] and the references therein for more details.…”