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).