scholarly journals Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 641 ◽  
Author(s):  
Kuilin Wu ◽  
Kai Zhou
Keyword(s):  
Incidence Rate ◽  
Traveling Wave ◽  
Epidemic Model ◽  
Reaction System ◽  
Traveling Wave Solutions ◽  
Sir Epidemic Model ◽  
Nonlocal Dispersal ◽  
Sir Epidemic ◽  
Wave Solutions ◽  
Minimal Wave Speed

In this paper, we study the traveling wave solutions for a nonlocal dispersal SIR epidemic model with standard incidence rate and nonlocal delayed transmission. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number of the corresponding reaction system and the minimal wave speed. To prove these results, we apply the Schauder’s fixed point theorem and two-sided Laplace transform. The main difficulties are that the complexity of the incidence rate in the epidemic model and the lack of regularity for nonlocal dispersal operator.

Wave propagation of a discrete SIR epidemic model with a saturated incidence rate

2019 ◽  
Vol 12 (03) ◽  
pp. 1950029 ◽  
Author(s):  
Qiu Zhang ◽  
Shi-Liang Wu
Keyword(s):  
Incidence Rate ◽  
Traveling Wave ◽  
Epidemic Model ◽  
Traveling Wave Solutions ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Wave Solutions ◽  
Saturated Incidence Rate ◽  
Minimal Wave Speed ◽  
Saturated Incidence

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number [Formula: see text] of the corresponding ordinary differential system and the minimal wave speed [Formula: see text]. More specifically, we first prove the existence of the traveling wave solutions for [Formula: see text] and [Formula: see text] via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed [Formula: see text] is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] is proved.

Minimal wave speed of a diffusive SIR epidemic model with nonlocal delay

2019 ◽  
Vol 12 (07) ◽  
pp. 1950081
Author(s):  
Fuzhen Wu ◽  
Dongfeng Li
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Nonlocal Delay ◽  
Spatial Nonlocality ◽  
Wave Solutions ◽  
Minimal Wave Speed

This paper is concerned with the minimal wave speed in a diffusive epidemic model with nonlocal delays. We define a threshold. By presenting the existence and the nonexistence of traveling wave solutions for all positive wave speed, we confirm that the threshold is the minimal wave speed of traveling wave solutions, which models that the infective invades the habitat of the susceptible. For some cases, it is proven that spatial nonlocality may increase the propagation threshold while time delay decreases the threshold.

scholarly journals Existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed

2021 ◽  
Vol 18 (6) ◽  
pp. 9357-9380
Author(s):  
Shiqiang Feng ◽  
◽  
Dapeng Gao ◽  
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Auxiliary System ◽  
Sir Epidemic Model ◽  
Nonlocal Dispersal ◽  
Sir Epidemic ◽  
Wave Solutions

<abstract><p>This paper is about the existence of traveling wave solutions for a delayed nonlocal dispersal SIR epidemic model with the critical wave speed. Because of the introduction of nonlocal dispersal and the generality of incidence function, it is difficult to investigate the existence of critical traveling waves. To this end, we construct an auxiliary system and show the existence of traveling waves for the auxiliary system. Employing the results for the auxiliary system, we obtain the existence of traveling waves for the delayed nonlocal dispersal SIR epidemic model with the critical wave speed under mild conditions.</p></abstract>

scholarly journals Existence of traveling wave solutions with critical speed in a delayed diffusive epidemic model

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yueling Cheng ◽  
Dianchen Lu ◽  
Jiangbo Zhou ◽  
Jingdong Wei
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Sir Epidemic Model ◽  
Open Problems ◽  
Sir Epidemic ◽  
Nonlinear Anal ◽  
Wave Solutions ◽  
Saturated Incidence

AbstractIn this paper, we prove the existence of a critical traveling wave solution for a delayed diffusive SIR epidemic model with saturated incidence. Moreover, we establish the nonexistence of traveling wave solutions with nonpositive wave speed for this model. Our results solve some open problems left in the recent paper (Z. Xu in Nonlinear Anal. 111:66–81, 2014).

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