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Journal of Dynamics and Differential Equations

, Volume 31, Issue 2, pp 883–901

# Traveling Waves in Epidemic Models: Non-monotone Diffusive Systems with Non-monotone Incidence Rates

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## Abstract

We study the existence and nonexistence of traveling waves of diffusive epidemic models with general incidence rates. The model systems are non-monotone because of the intrinsic predator–prey interaction between the susceptible and infective compartments in epidemic systems. Moreover, the incidence rate may not be monotone in the infected population because social behaviors and collective activities may change in response to the prevalence of disease. To find positive traveling solutions of the non-monotone system with a non-monotone incidence function, we will construct a suitable convex set in a weighted function space, and then apply Schauder fixed point theorem. It turns out that the basic reproduction number of the corresponding ordinary differential equations plays an important role in the existence theory of traveling waves. Moreover, the critical wave speed can be explicitly obtained in terms of the  diffusion coefficient, recovery rate and death rate for the infected group, and partial derivative of incidence function at the disease-free equilibrium. Finally, we prove that the positive traveling wave solution does not exist if the basic reproduction number is no more than one, or the wave speed is less than the critical value.

## Keywords

Traveling waves Diffusive epidemic models Schauder fixed point theorem Non-monotonicity

## Mathematics Subject Classification

92D30 35K57 34B40

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## Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

## Authors and Affiliations

• Hongying Shu
• 1
• Xuejun Pan
• 1
• 2
• Xiang-Sheng Wang
• 3
• Jianhong Wu
• 4
1. 1.Department of MathematicsTongji UniversityShanghaiPeople’s Republic of China
2. 2.Tongji Zhejiang CollegeJiaxingPeople’s Republic of China
3. 3.Department of MathematicsUniversity of Louisiana at LafayetteLafayetteUSA
4. 4.Laboratory for Industrial and Applied MathematicsYork UniversityTorontoCanada