scholarly journals Traveling Wave Solutions in a Reaction-Diffusion Epidemic Model

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Sheng Wang ◽  
Wenbin Liu ◽  
Zhengguang Guo ◽  
Weiming Wang
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Monotone Iteration ◽  
Wave Solutions ◽  
Minimal Wave Speed

We investigate the traveling wave solutions in a reaction-diffusion epidemic model. The existence of the wave solutions is derived through monotone iteration of a pair of classical upper and lower solutions. The traveling wave solutions are shown to be unique and strictly monotonic. Furthermore, we determine the critical minimal wave speed.

TRAVELING WAVES FOR A SIMPLE DIFFUSIVE EPIDEMIC MODEL

1995 ◽  
Vol 05 (07) ◽  
pp. 935-966 ◽  
Author(s):  
YUZO HOSONO ◽  
BILAL ILYAS
Keyword(s):  
Traveling Wave ◽  
Epidemic Model ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Shooting Argument ◽  
Wave Solutions ◽  
Minimal Wave Speed ◽  
Manifold Theory

We investigate the existence of traveling wave solutions for the infective-susceptible two-component epidemic model. The model system is described by reaction-diffusion equations with the nonlinear reaction term of the classical Kermack-McKendric type. The diffusion coefficients of infectives and susceptibles are assumed to be positive constants d1 and d2 respectively. By the shooting argument with the aid of the invariant manifold theory, we prove that there exists a positive constant c* such that the traveling wave solutions exist for any c≥c*. The minimal wave speed c* is shown to be independent of d2 and to have the same value as that for d2=0.

scholarly journals Minimal Wave Speed in a Predator-Prey System with Distributed Time Delay

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Fuzhen Wu ◽  
Dongfeng Li
Keyword(s):  
Time Delay ◽  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Wave Solutions ◽  
Prey System ◽  
Minimal Wave Speed ◽  
Distributed Time Delay

This paper is concerned with the minimal wave speed of traveling wave solutions in a predator-prey system with distributed time delay, which does not satisfy comparison principle due to delayed intraspecific terms. By constructing upper and lower solutions, we obtain the existence of traveling wave solutions when the wave speed is the minimal wave speed. Our results complete the known conclusions and show the precisely asymptotic behavior of traveling wave solutions.

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 Traveling Wave Solutions for Epidemic Cholera Model with Disease-Related Death

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Tianran Zhang ◽  
Qingming Gou
Keyword(s):  
Laplace Transform ◽  
Traveling Wave ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Wave Solutions ◽  
Diffusion Systems ◽  
Prior Estimate ◽  
Minimal Wave Speed

Based on Codeço’s cholera model (2001), an epidemic cholera model that incorporates the pathogen diffusion and disease-related death is proposed. The formula for minimal wave speedc∗is given. To prove the existence of traveling wave solutions, an invariant cone is constructed by upper and lower solutions and Schauder’s fixed point theorem is applied. The nonexistence of traveling wave solutions is proved by two-sided Laplace transform. However, to apply two-sided Laplace transform, the prior estimate of exponential decrease of traveling wave solutions is needed. For this aim, a new method is proposed, which can be applied to reaction-diffusion systems consisting of more than three equations.

Minimal wave speed of a competitive integrodifference system

2019 ◽  
Vol 12 (03) ◽  
pp. 1950031
Author(s):  
Fuguo Zhu ◽  
Shuxia Pan
Keyword(s):  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Competitive System ◽  
Volterra Type ◽  
Wave Solutions ◽  
Minimal Wave Speed

This paper is concerned with the minimal wave speed of traveling wave solutions of a discrete competitive system with Lotka–Volterra type nonlinearity. By constructing upper and lower solutions, we confirm the existence of traveling wave solutions if the wave speed is the minimal wave speed. Our results complete the earlier conclusions.

scholarly journals Minimal Wave Speed in a Delayed Lattice Dynamical System with Competitive Nonlinearity

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Fuzhen Wu
Keyword(s):  
Dynamical System ◽  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Partial Ordering ◽  
Wave Solutions ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems ◽  
Minimal Wave Speed

This paper deals with the minimal wave speed of delayed lattice dynamical systems without monotonicity in the sense of standard partial ordering in R2. By constructing upper and lower solutions appealing to the exponential ordering, we prove the existence of traveling wave solutions if the wave speed is not smaller than some threshold. The nonexistence of traveling wave solutions is obtained when the wave speed is smaller than the threshold. Therefore, we confirm the threshold is the minimal wave speed, which completes the known results.

scholarly journals Minimal Wave Speed in a Competitive Integrodifference System without Comparison Principle

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 571
Author(s):  
Luping Li ◽  
Shugui Kang ◽  
Lili Kong ◽  
Huiqin Chen
Keyword(s):  
Comparison Principle ◽  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Large Wave ◽  
Wave Speeds ◽  
Wave Solutions ◽  
Minimal Wave Speed ◽  
Decay Behavior

We investigate the traveling wave solutions of a competitive integrodifference system without comparison principle. In the earlier conclusions, a threshold of wave speed is defined while the existence or nonexistence of traveling wave solutions remains open when the wave speed is the threshold. By constructing generalized upper and lower solutions, we confirm the existence of traveling wave solutions when the wave speed is the threshold. Our conclusion completes the known results and shows the different decay behavior of traveling wave solutions compared with the case of large wave speeds.

scholarly journals Minimal Wave Speed in an Integrodifference System of Predator-Prey Type

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Baoju Sun ◽  
Fuzhen Wu
Keyword(s):  
Traveling Wave ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Traveling Wave Solutions ◽  
Prey Type ◽  
Predator Prey ◽  
Wave Speeds ◽  
Wave Solutions ◽  
Minimal Wave Speed ◽  
Asymptotic Spreading

This article studies the minimal wave speed of traveling wave solutions in an integrodifference predator-prey system that does not have the comparison principle. By constructing generalized upper and lower solutions and utilizing the theory of asymptotic spreading, we show the minimal wave speed of traveling wave solutions modeling the invasion process of two species by presenting the existence and nonexistence of nonconstant traveling wave solutions with any wave speeds.

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.

scholarly journals Existence of Traveling Wave Solutions for Cholera Model

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Tianran Zhang ◽  
Qingming Gou ◽  
Xiaoli Wang
Keyword(s):  
Traveling Wave ◽  
Wave Speed ◽  
Shooting Method ◽  
Reproduction Number ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Traveling Wave Solution ◽  
Reaction Diffusion Model ◽  
Minimal Wave Speed ◽  
The Basic Reproduction Number

To investigate the spreading speed of cholera, Codeço’s cholera model (2001) is developed by a reaction-diffusion model that incorporates both indirect environment-to-human and direct human-to-human transmissions and the pathogen diffusion. The two transmission incidences are supposed to be saturated with infective density and pathogen density. The basic reproduction numberR0is defined and the formula for minimal wave speedc*is given. It is proved by shooting method that there exists a traveling wave solution with speedcfor cholera model if and only ifc≥c*.

Sign in / Sign up

Export Citation Format

Share Document