Global dynamics of a virus model with invariant algebraic surfaces

In this paper by using the Poincaré compactification in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^3$$\end{document} we make a global analysis for the virus system x˙=λ-dx-βxz,y˙=-ay+βxzz˙=ky-μz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\dot{x}}=\lambda -dx-\beta xz, \quad {\dot{y}} = -ay+\beta xz \quad {\dot{z}} = ky-\mu z \end{aligned}$$\end{document}with (x,y,z)∈R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x, y, z) \in {{\mathbb {R}}}^3$$\end{document}, β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >0$$\end{document}, λ,a,d,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda , a, d, k$$\end{document} and μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} 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 R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^3$$\end{document} with the sphere S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^2$$\end{document} of the infinity).


Introduction and statement of the main results
In this work we do a global analysis of the virus systeṁ x = P(x, y, z) = λ − dx − βxz, y = Q(x, y, z) = −ay + βxz, where the state variables are (x, y, z) ∈ R 3 , β > 0, λ, a, d, k and μ are nonnegative parameters due to their biological meaning. As usual, the dots denote derivative with respect to the time t. Note that β cannot be zero otherwise the differential system becomes linear, and consequently its study is trivial. System (1) was proposed in the book [7]. For more studies related with virus models, including the importance of our model (1), we refer the readers to [4,8].
Let R[x, y, z] be the ring of real polynomials in the variables the variables x, y and z. We say that F = F(x, y, z) ∈ R[x, y, z] is a Darboux polynomial of system (1) if it satisfies (∇ F) · (P, Q, R) = K F, where K = K (x, y, z) is a real polynomial of degree at most 1, called the cofactor of F(x, y, z). If the cofactor is zero, then F(x, y, z) is a polynomial first integral of system (1). If F(x, y, z) is a Darboux polynomial, then the algebraic surface F(x, y, z) = 0 is called an invariant algebraic surface; i.e. if an orbit of system (1) has a point on this surface, then the whole orbit is contained in it. See an example of a polynomial differential system with many invariant planes in [6].
We say that a C 1 function I (x, y, z, t) is an invariant of the differential system (1) if d I /dt = 0 on the trajectories of the system. When an invariant function is independent of the time t, then it is a first integral. When a system has a Darboux polynomial F with a constant factor k = k 0 ∈ R, then the function I (x, y, z, t) = F(x, y, z)e −k 0 t is called a Darboux invariant of that system, see Chapter 8 of [3].
We say that the Darboux polynomials F 1 , . . . , F n generate all the Darboux polynomials of system (1) if any Darboux polynomial F of system (1) can be written as F = F α 1 1 · · · F α n n with α k ∈ N for k = 1, . . . , n.
The following theorem proved in [9] summarizes the results on the existence of invariant algebraic surfaces for system (1). Theorem 1 When βk(λ 2 +d 2 ) = 0, a set of generators for the set of all Darboux polynomials of system (1) are given in Table 1.
As any polynomial differential system the virus system (1) can be extended to an analytic system on a closed ball of radius one, whose interior is diffeomorphic to R 3 and its boundary, the 2-dimensional sphere S 2 , plays the role of the infinity because in R 3 one can go or come from infinity in as many directions as points have S 2 . This closed ball is denoted by B and called the Poincaré ball. The technique for doing such an extension is precisely the Poincaré compactification for a polynomial differential system in R 3 , which is described in details Table 1 Invariant algebraic surfaces of system (1) Darboux polynomial Cofactor Parameters x y z q q Fig. 1 Phase portrait of the virus system (1) at the infinity S 2 of the Poincaré ball. The circles of S 2 contained in x = 0 and z = 0 are filled of singular points. Additionally there are two diammetrally opposite singular points on z = 0 of nodal type in [2], see also [1]. By using this compactification technique the dynamics of system (1) at infinity can be determined.

Theorem 2
The phase portrait of the Poincaré compactification of system (1) at the infinity S 2 is topologically equivalent to the one described in Fig. 1.
We go further and describe the global dynamics of system (1) for fixed values of the parameters having the invariant planes F 1 = 0 or F 2 = 0 described in Table 1.
The singular points on F 1 (x, y, z) = 0 are described in Table 2. The phase portraits of system (1) restricted to the compactified invariant plane F 1 (x, y, z) = 0 are described in Fig. 7.
The singular points on F 2 (x, y, z) = 0 are described in Table 2. The phase portraits of system (1) restricted to the compactified invariant plane F 2 (x, y, z) = 0 are described in Fig. 7. Table 2 Finite singular points for the differential system (8) and for the parameters a 1 = βkλ/d 3  a 2 > 0, a 1 = a 2 P 0 = P 1 saddle-node Figure 7c a 2 > 0, a 1 < a 2 P 0 stable node, P 1 saddle Figure 7d a 2 > 0, a 1 > a 2 P 0 saddle, P 1 stable node Figure 7d We remark that system (1) restricted to the invariant plane F 0 = x = 0 with λ = 0 is linear.
We say that a set V ⊂ B is invariant by the flow of system (1) if for any p ∈ V the whole orbit passing through p is contained in V . The sphere of the infinity always is an invariant set. Let ϕ(t) = ϕ(t, p) be the solution of the compactified system (1) passing through the point p ∈ B when t = 0, defined on its maximal interval I p = R (because B is compact). Then the α-limit set of ϕ is the invariant set In a similar way the ω-limit set of ϕ is the invariant set The following theorem gives a complete description of the αand ω-limit sets in the Poincaré ball, including its boundary S 2 , for system (1) having the invariant algebraic surfaces of Table 1. (a) The α-limit set of γ is some of the four infinite singular points q, q , E 2 , E 2 contained in the boundary of the plane F i = 0 for i = 1, 2. (b) The ω-limit set of γ is contained in the closure of the invariant plane F i = 0 in the Poincaré ball. More precisely, Here P 0 , P 1 are singular points of system (1), contained in the plane F 1 = 0 (see Table 2), E 2 is the origin of the local chart U 2 , q = (−1, 0, 0) is a singular point in the local chart U 1 , and E 2 and q are the diametrally opposite points to E 2 and q in the Poincaré sphere.
The paper is organized as follows. In Sect. 2 we prove Theorem 2 by using the Poincaré compactification for a polynomial vector field in R 3 . In Sect. 3 we prove Theorem 3 by using the Poincaré compactification for a polynomial vector field in R 2 . For precise definitions of all these notions see Chapter 5 of [3]. Finally the proof of Theorem 4 is given in Sect. 4. In this section we analyze the flow of system (1) at infinity. Consider system (1) or equivalently its associated polynomial vector field X = (λ − dx − βxz, −ay + βxz, ky − μz).
In the next three subsections we will study the Poincaré compactification, p(X ), of system (1) in the local charts U i and V i , i = 1, 2, 3 in order to understand the global behavior of the solutions near infinity. See [1] for more details on these charts.

In the local chart U 3
Using the results of [1] we have that the expression of the Poincaré compactification Z 3 = p(X ) of system (1) in the local chart U 3 is given bẏ In the points of the sphere S 2 that correspond to the points at infinity we have z 3 = 0, and so system (2) becomesż After eliminating the common factor z 1 (by rescaling of the time) we do not obtain any singular point in the local chart U 3 which is distinguished. From this system we see that system (2) has at infinity a continuum of singular points given by (0, z 2 , 0). Moreover D Z 3 (0, z 2 , 0) has, for each z 2 , the eigenvalues −β and 0 with multiplicity 2. By the normal hyperbolicity theory (see [5] for details) to each point (0, z 2 , 0) it arrives two orbits, see Fig. 2. Note that the equilibria (0, z 2 , 0) with z 2 ∈ R represent half of a great circle through the north pole of S 2 . The orbits of the system in the local chart U 3 at infinity have the phase portrait given in Figs. 1 and 2.

In the local chart U 1
In the same way the Poincaré compactification Z 1 = p(X ) of system (1) in the local chart U 1 is Fig. 3 The phase portrait in the local chart U 1 at infinity System (3) restricted to infinity (that is with z 3 = 0 ) becomeṡ z 1 = βz 2 (z 1 + 1),ż 2 = βz 2 2 ,ż 3 = 0. So there is a line of singular points which is z 2 = 0. These singular points are such that D Z 1 (z 1 , 0, 0) has, for each z 1 the eigenvalue 0 with multiplicity 3. After eliminating the common factor z 2 (by a rescaling of the time) we get that the unique singular point is q = (−1, 0, 0) which is an unstable node. Note that the equilibria (z 1 , 0, 0) for z 1 ∈ R represent in the Poincaré sphere S 2 half of the equator. The orbits of the system in the local chart U 1 at infinity have the phase portrait given in Figs. 1 and 3.

In the local chart U 2
The expression of the Poincaré compactification Z 2 = p(X ) in the local chart U 2 iṡ Now the unique point of the local chart U 2 which is not covered by the local charts U 1 , V 1 , U 3 and V 3 is the origin of coordinates of U 2 . System (4) restricted at the infinity (that is with z 3 = 0 ) becomesż 1 = −βz 1 z 2 (z 1 + 1),ż 2 = −βz 1 z 2 2 ,ż 3 = 0. There are two lines of singular points z 1 = 0 and z 2 = 0. After removing these two lines of singular points, there is a singular point q = (−1, 0, 0) which is a stable node and its local phase portrait is the one given in Fig. 4.
We observe that the flows in the V i charts for i = 1, 2, 3 are the same as the ones in the respective U i charts for i = 1, 2, 3 but with the time reversed because the compactified vector field p(X ) in V i coincides with the vector field in U i multiplied by −1 for each i = 1, 2, 3. See [1].
From Sects. 2.1, 2.2 and 2.3 it follows the proof of Theorem 2.

Proof of Theorem 3
In Theorem 1 it was proved that the virus system (1) has the invariant algebraic surfaces where i = {1, 2}. Therefore I d is a Darboux invariant of system (1). This completes the proof of the first part of statements (a) and (b) of Theorem 3. System (1) has two isolated singular points in R 3 , namely Note that these two singular points are on the two invariant algebraic surfaces F 1 (x, y, z) = 0 when a = d = 0 and F 2 (x, y, z) = 0 when μ = d = 0. In what follows we shall analyze the flow of the system in these planes and relate their dynamics with the dynamics at infinity.

Phase portrait on {F 1 (x, y, z) = 0}
We can consider in this subsection system (1) with a = d = 0 and βkd(λ 2 +d 2 ) = 0. System (1) restricted to the invariant plane F 1 (x, y, z) = 0 is given by the differential systeṁ Doing the change of variables (with a rescaling of time) system (6) becomesẋ where a 1 = βkλ/d 3 , a 2 = μ/d ∈ [0, ∞), and we have renamed the new variables (X , Z ) by (x, z), and the dot denotes derivative with respect to the new time T . We now study the infinite singular points of system (8). We will use the Poincaré compactification for a polynomial vector field in R 2 which is described in Chapter 5 of [3].

Infinite singular points
System (8) in the local chart U 1 becomeṡ When z 2 = 0 the only infinite singular point is the origin. Since the origin is nilpotent, using Theorem 3.5 of [3] we obtain that the phase portrait of the origin consists of one hyperbolic, one elliptic and two parabolic sectors, see Fig. 5.
By a linear change of coordinates this system can be written aṡ Therefore (0, 0) is a semi-hyperbolic point. By Theorem 2.19 of [3] if a 2 = 0 we have that (0, 0) is a saddle-node point and going back to system (9) its local phase portrait is the one described in Fig. 6.
On the other hand, if a 2 = 0 but a 1 = 0, (0, 0) is a stable topological node. Finally, when a 2 = a 1 = 0, the system becomeṡ so there is a line of singular points which is z 2 = 0. After eliminating the common factor z 2 (by a rescaling of the time) we do not obtain any singular point in the local chart U 2 which is distinguished.

Finite singular points
In what follows we shall study the local stability at the singular points of system (8) for the parameters a 1 and a 2 . Case 1 a 2 = 0. In this case system (8) becomeṡ When a 1 = 0 the system has a continuum of singular points (0, z) on the straight line x = 0. When a 1 = 0, the unique singular point is (0, 0). The eigenvalues are As a 1 > 0, the origin is a saddle. Case 2 a 2 = 0. In this case system (8) has exactly the two finite singular points: The eigenvalues of the linear part at these singular points are −(a 2 + 1) ± (a 2 + 1) 2 + 4(a 1 − a 2 ) 2 for P 0 , and When a 2 < a 1 , P 0 is a saddle and P 1 is an attractor node. On the other hand, when a 1 < a 2 , P 0 is an attractor node and P 1 is a saddle. Finally, when a 1 = a 2 , we have that P 0 = P 1 . It is semi-hyperbolic and using Theorem 2.19 of [3] we conclude that it is a saddle-node. In short we have Table 2.
Taking into account the local information on the finite and infinite singular points of system (8) we get that the global phase portraits of systems (8) are topologically equivalent to those described in the Fig. 7.

Phase portrait on F 2 (x, y, z) = 0
In this subsection we consider system (1) with μ = d = 0 and βkd = 0. System (1) restricted to the invariant plane F 2 (x, y, z) = 0 is given by the differential systeṁ Doing the change of variables in (7) system (10) can be written aṡ where a 1 = βkλ/d 3 and a 2 = a/d and again we have renamed (X , Z ) as (x, z), and the dot means derivative with respect to the new time T . Systems (8) and (11) are the same and so their global phase portraits are exactly the same. In short, taking into account the results of this subsection we have proved Theorem 3. In this section we describe the αand the ω-limit sets of all orbits of system (1) contained in B and outside the invariant planes F 1 = 0 and F 2 = 0.
In Sect. 3 we proved that I d (x, y, z, t) = F i (x, y, z)e dt is an invariant of system (1) when be an orbit of the compactified system (1) outside the planes F i = 0 for i = 1, 2. Therefore Taking the limit in (12) when t → −∞, we obtain that lim t→∞ F i (x(t), y(t), z(t)) = ±∞.
Therefore the α-limit is contained in the boundary of the plane F i = 0 at infinity. Then from Fig. 1 these α-limit sets can only be one of the four singular points contained in the boundary of F i = 0 at infinity, i.e. q, q , E 2 and E 2 (see Fig. 7). So, statement (a) is proved. Now taking the limit in (12) when t → ∞, we obtain that lim t→∞ F i (x(t), y(t), z(t)) = 0.
Therefore the ω-limit is contained in the closed plane F i = 0 in the Poincaré ball. To complete the study of the ω-limit sets we first study the finite singular points in R 3 whenever either d = a, or d = μ. Since the study is the same we will focus in the case in which d = a. We consider different cases.
Case 1 λ = μ = 0. In this case the singular points are located at x = y = 0. Computing the eigenvalues of the Jacobian matrix at the points of this line, we get that they are 0, −a and −a − zβ. It follows from the normal hyperbolicity theory (see again [5]) that if −a − z 0 β < 0 then the singular point (0, 0, z 0 ) has a 2-dimensional stable manifold, whereas if −a − z 0 β > 0 then the singular point (0, 0, z 0 ) has a 1-dimensional stable manifold and 1-dimensional unstable invariant manifold.
Case 2 μ = 0 and λ = 0. In this case the unique singular point is (λ/a, 0, 0). Computing the eigenvalues of the Jacobian matrix at this point we get −a, −a 2 ± a 4 + 4akβλ 2a .
Case 3 μ = kβλ/a 2 . In this case the unique singular point is again (λ/a, 0, 0). Computing the eigenvalues of the Jacobian matrix at this point we obtain 0, −a, − a 3 + kβλ a 2 So by the normal hyperbolicity theory the singular point (λ/a, 0, 0) has either a 2-dimensional stable manifold.