Global dynamics of a virus model with invariant algebraic surfaces

Abstract

In this paper by using the Poincaré compactification in \({{\mathbb {R}}}^3\) we make a global analysis for the virus system

$$\begin{aligned} {\dot{x}}=\lambda -dx-\beta xz, \quad {\dot{y}} = -ay+\beta xz \quad {\dot{z}} = ky-\mu z \end{aligned}$$

with \((x, y, z) \in {{\mathbb {R}}}^3\), \(\beta >0\), \(\lambda , a, d, k\) and \(\mu \) are nonnegative parameters due to their biological meaning. We give the complete description of its dynamics on the sphere at infinity. For two sets of the parameter values the system has invariant algebraic surfaces. For these two sets we provide the global phase portraits of the virus system in the Poincaré ball (i.e. in the compactification of \({{\mathbb {R}}}^3\) with the sphere \({\mathbb {S}}^2\) of the infinity).

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Acknowledgements

The first author is partially supported by FAPEMIG Grants APQ-01086-15 and APQ-01158-17. The second author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.

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Correspondence to Fabio Scalco Dias.

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Dias, F.S., Llibre, J. & Valls, C. Global dynamics of a virus model with invariant algebraic surfaces. Rend. Circ. Mat. Palermo, II. Ser 69, 535–546 (2020). https://doi.org/10.1007/s12215-019-00417-0

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Keywords

  • Invariant algebraic surfaces
  • Poincaré compactification
  • Phase portrait
  • Dynamics at infinity
  • Virus model

Mathematics Subject Classification

  • 34A34
  • 34C05
  • 34C14