Basic reproduction ratios for periodic and time-delayed compartmental models with impulses

  • Zhenguo BaiEmail author
  • Xiao-Qiang Zhao


Much work has focused on the basic reproduction ratio \({\mathcal {R}}_0\) for a variety of compartmental population models, but the theory of \({\mathcal {R}}_0\) remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of \({\mathcal {R}}_0\) for a class of such impulsive models. We first introduce \({\mathcal {R}}_0\) and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of \({\mathcal {R}}_0\). Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.


Impulsive models Time delay Basic reproduction ratio Computer virus Threshold dynamics 

Mathematics Subject Classification

34A37 92D30 37N25 



Zhenguo Bai would like to thank the China Scholarship Council (201606965021) for financial support during the period of his overseas study and to express his gratitude to the Department of Mathematics and Statistics, Memorial University of Newfoundland, for its kind hospitality. We are also grateful to two anonymous referees for careful reading and valuable comments which led to improvements of our original manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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