Traveling wave solutions in a two-group SIR epidemic model with constant recruitment

  • Lin Zhao
  • Zhi-Cheng Wang
  • Shigui Ruan


Host heterogeneity can be modeled by using multi-group structures in the population. In this paper we investigate the existence and nonexistence of traveling waves of a two-group SIR epidemic model with time delay and constant recruitment and show that the existence of traveling waves is determined by the basic reproduction number \(R_{0}.\) More specifically, we prove that (i) when the basic reproduction number \(R_{0}>1,\) there exists a minimal wave speed \(c^*>0,\) such that for each \(c \ge c^*\) the system admits a nontrivial traveling wave solution with wave speed c and for \(c<c^*\) there exists no nontrivial traveling wave satisfying the system; (ii) when \(R_{0} \le 1,\) the system admits no nontrivial traveling waves. Finally, we present some numerical simulations to show the existence of traveling waves of the system.


Two-group epidemic model Basic reproduction number Time delay Constant recruitment Traveling wave solutions 

Mathematics Subject Classification

35C07 35B40 35K57 92D30 



The authors are grateful to the two anonymous reviewers and the handling editor (Professor Klaus Dietz) for their helpful comments and suggestions which helped us in improving the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsLanzhou University of TechnologyLanzhouPeople’s Republic of China
  3. 3.Department of MathematicsUniversity of MiamiCoral GablesUSA

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