Journal of Elliptic and Parabolic Equations

, Volume 5, Issue 2, pp 359–381 | Cite as

Analysis of a vector-borne disease model with impulsive perturbation and reinfection

  • Suxia Zhang
  • Hongsen Dong
  • Xiaxia Xu
  • Xiaoqin ShenEmail author


We formulate a mathematical model for vector-borne disease with impulsive perturbation based on indoor residual spraying. The dynamical properties are studied theoretically and numerically. It is shown that with consideration of impulsive spraying at fixed times, there exists a disease-free periodic solution that is locally stable when the threshold \({\mathcal {R}}_0\) is less than unity, otherwise the disease is uniformly persistent if \({\mathcal {R}}_0>1\). Furthermore, the bifurcation analysis is performed, revealing the possible existence of nontrivial periodic solution bifurcated from the disease-free periodic solution at \({\mathcal {R}}_0=1\). In addition to simulations of parameter sensitivity, when implementing impulsive spraying once the number of infected humans exceeds a threshold level, the effectiveness of such state-dependent control is also conducted numerically.


Vector-borne disease Indoor residual spraying Impulsive perturbation Backward bifurcation State-dependent control 

Mathematics Subject Classification

92D30 35B20 34K45 



This research was funded by National Natural Science Foundation of China (#11501443, #11801439).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Orthogonal Publisher and Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Suxia Zhang
    • 1
  • Hongsen Dong
    • 1
  • Xiaxia Xu
    • 1
  • Xiaoqin Shen
    • 1
    Email author
  1. 1.School of ScienceXi’an University of TechnologyXi’anPeople’s Republic of China

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