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Multiple Equilibria in a Non-smooth Epidemic Model with Medical-Resource Constraints

Bulletin of Mathematical Biology volume 81, pages 963–994 (2019)Cite this article

Abstract

The issue of medical-resource constraints has the potential to dramatically affect disease management, especially in developing countries. We analyze a non-smooth epidemic model with nonlinear incidence rate and resource constraints, which defines a vaccination program with vaccination rate proportional to the number of susceptible individuals when this number is below the threshold level and constant otherwise. To better understand the impact of this non-smooth vaccination policy, we provide a comprehensive qualitative analysis of global dynamics for the whole parameter space. As the threshold value varies, the target model admits multistability of three regular equilibria, bistability of two regular equilibria, that of one disease-free equilibrium and one generalized endemic equilibria, and that of one disease-free equilibrium and one crossing cycle. The steady-state regimes include healthy, low epidemic and high epidemic. This suggests the key role of the threshold value, as well as the initial infection condition in disease control. Our findings demonstrate that the case number can be contained at a satisfactorily controllable level or range if eradicating it proves to be impossible.

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Acknowledgements

The authors are grateful to an anonymous reviewer, whose comments greatly improved the manuscript. AW was supported by the National Natural Science Foundation of China (NSFC, 11801013), Scientific research plan projects of Shaanxi Education Department (16JK1047) and the funding from Baoji University of Arts and Sciences (ZK1048). YX was supported by the National Natural Science Foundation of China (NSFC, 11571273 and 11631012) and Fundamental Research Funds for the Central Universities (GK 08143042). RS? was supported by an NSERC Discovery Grant. For citation purposes, note that the question mark in “Smith?” is part of his name.

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Affiliations

  1. School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721013, People’s Republic of China

    Aili Wang

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yanni Xiao

  3. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Robert Smith

Authors
  1. Aili Wang
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  2. Yanni Xiao
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  3. Robert Smith
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Correspondence to Robert Smith.

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Wang, A., Xiao, Y. & Smith, R. Multiple Equilibria in a Non-smooth Epidemic Model with Medical-Resource Constraints. Bull Math Biol 81, 963–994 (2019). https://doi.org/10.1007/s11538-018-00544-2

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  • DOI: https://doi.org/10.1007/s11538-018-00544-2

Keywords

  • Non-smooth epidemic model
  • Resources constraints
  • Nonlinear incidence
  • Generalized equilibrium
  • Crossing cycle
  • Multistability