A viral propagation model with a nonlinear infection rate and free boundaries

Abstract

In this paper we put forward a viral propagation model with a nonlinear infection rate and free boundaries and investigate the dynamical properties. This model is composed of two ordinary differential equations and one partial differential equation, in which the spatial range of the first equation is the whole space ℝ, and the last two equations have free boundaries. As a new mathematical model, we prove the existence, uniqueness and uniform estimates of the global solution, and provide the criteria for spreading and vanishing, and the long time behavior of the solution components u, v and w. Comparing this model with the corresponding ordinary differential systems, the basic reproduction number \({{\cal R}_0}\) plays a different role. We find that when \({{\cal R}_0} \le 1\), the virus cannot spread successfully; when \({{\cal R}_0} > 1\), the successful spread of virus depends on the initial value and varying parameters.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11771110 and 11971128). The authors thank the reviewers for their helpful comments and suggestions that significantly improve the initial version of this paper.

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Correspondence to Mingxin Wang.

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Li, L., Liu, S. & Wang, M. A viral propagation model with a nonlinear infection rate and free boundaries. Sci. China Math. (2020). https://doi.org/10.1007/s11425-020-1680-0

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Keywords

  • viral propagation model
  • free boundaries
  • basic reproduction number
  • spreading-vanishing
  • long time behavior

MSC(2010)

  • 35K57
  • 35B40
  • 35R35
  • 92D30