Bulletin of Mathematical Biology

, Volume 74, Issue 10, pp 2403–2422

Sliding Mode Control of Outbreaks of Emerging Infectious Diseases

Original Article

DOI: 10.1007/s11538-012-9758-5

Cite this article as:
Xiao, Y., Xu, X. & Tang, S. Bull Math Biol (2012) 74: 2403. doi:10.1007/s11538-012-9758-5


This paper proposes and analyzes a mathematical model of an infectious disease system with a piecewise control function concerning threshold policy for disease management strategy. The proposed models extend the classic models by including a piecewise incidence rate to represent control or precautionary measures being triggered once the number of infected individuals exceeds a threshold level. The long-term behaviour of the proposed non-smooth system under this strategy consists of the so-called sliding motion—a very rapid switching between application and interruption of the control action. Model solutions ultimately approach either one of two endemic states for two structures or the sliding equilibrium on the switching surface, depending on the threshold level. Our findings suggest that proper combinations of threshold densities and control intensities based on threshold policy can either preclude outbreaks or lead the number of infecteds to a previously chosen level.


Threshold policy SIR epidemic model Outbreaks Sliding mode 

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsXi’an Jiaotong UniversityXi’anP.R. China
  2. 2.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anP.R. China

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