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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Zhenguo Bai

  2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Xiao-Qiang Zhao

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  1. Zhenguo Bai
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  2. Xiao-Qiang Zhao
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Correspondence to Zhenguo Bai.

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Zhenguo Bai’s research was funded by NSF of China (11971369, 11671315), NSF of Shaanxi Province (2019JM-241) and the Fundamental Research Funds for the Central Universities (JB190705); and Xiao-Qiang Zhao’s research was funded by NSERC of Canada.

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Bai, Z., Zhao, XQ. Basic reproduction ratios for periodic and time-delayed compartmental models with impulses. J. Math. Biol. 80, 1095–1117 (2020). https://doi.org/10.1007/s00285-019-01452-2

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  • DOI: https://doi.org/10.1007/s00285-019-01452-2

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