Stability and Energy-Casimir Mapping for Integrable Deformations of the Kermack-McKendrick System

Integrable deformations of a Hamilton-Poisson system can be obtained altering its constants of motion. These deformations are integrable systems that can have various dynamical properties. In this paper, we give integrable deformations of the KermackMcKendrick model for epidemics, and we analyze a particular integrable deformation. More precisely, we point out two Poisson structures that lead to infinitely many Hamilton-Poisson realizations of the considered system. Furthermore, we study the stability of the equilibrium points, we give the image of the energy-Casimir mapping, and we point out some of its properties.


Introduction
Evolution equations represent models for describing phenomena that appear in physics, biology, chemistry, economy, and engineering.In many situations these evolution equations can be analyzed in the frame of the Lagrangian mechanics or the Hamiltonian mechanics.Furthermore, there are phenomena that are modeled by three-dimensional systems of differential equations, particularly Hamilton-Poisson systems.Such systems can be perturbed in order to obtain a desired behavior.A way to perturb a threedimensional Hamilton-Poisson systems consists in alteration of its constants of motion.This method leads to integrable deformations of the initial system.
In recent papers, integrable deformations of some particular Hamilton-Poisson systems were analyzed.In [1], observing that the constants of motion of the Euler top determine its equations, integrable deformations of the Euler top were given.In [2], integrable deformations of the three-dimensional real valued Maxwell-Bloch equations were obtained by altering the constants of motion of the considered system.In the same manner, in [3], integrable deformations of the Rikitake system were constructed.These integrable deformations can be viewed as controlled systems and, in consequence, a study of modifications in their dynamics can be performed.Moreover, the integrable deformations of the above systems are also Hamilton-Poisson systems.Consequently, they can be analyzed from some standard and nonstandard Poisson geometry points of view [4].
The study of a three-dimensional Hamilton-Poisson system from some standard and nonstandard Poisson geometry points of view tries to answer the following open problem formulated by Tudoran et al. [4]: "Is there any connection between the dynamical properties of a given dynamical system and the geometry of the image of the energy-Casimir mapping, and if yes, how can one detect as many as possible dynamical elements (e.g., equilibria, periodic orbits, homoclinic and heteroclinic connections) and dynamical behavior (e.g., stability, bifurcation phenomena for equilibria, periodic orbits, homoclinic and heteroclinic connections) by just looking at the image of this mapping?"Affirmative answers were given for some particular systems [5][6][7][8][9].In these cases the image of the energy-Casimir mapping EC = (, ), where  is the Hamiltonian and  is a Casimir function, is a closed subset of R 2 , namely, the convex hull of the images of the stable equilibrium points through EC.Furthermore, the images of the equilibrium points through the energy-Casimir mapping give an algebraic partition of the set Im(EC), and the orbits of these systems are bounded.On the other hand, the image of the energy-Casimir mapping can be R 2 ( [10,11]), and other connections were observed (for example, there are unbounded orbits).In [12], taking into account these facts, some questions regarding the connections between the dynamics of a Hamilton-Poisson systems and the associated energy-Casimir mapping were asked.We recall some of them: "are the observed properties in every case when Im(EC) ⊊ R 2 true?" or "can Im(EC) be a nonconvex set?If yes, do the observed properties remain true?"One of the goals of this paper is to give some answers to these questions.
The finding of some counterexamples assumes the study of many Hamilton-Poisson systems, and how we have already seen such systems can be obtained using integrable deformations of known integrable systems.Moreover, because the constants of motion  and  of the above-mentioned systems are polynomials, it is a good idea to analyze systems that have nonpolynomials constants of motion.Such a system is the well-known system introduced in 1927 by Kermack and McKendrick [13] and brought back in attention by Anderson and May, in 1979 [14].The Kermack-McKendrick system and its generalizations were widely investigated.We mention a very short list of works [15][16][17].We also notice the applications of such type of systems in health, networks, informatics, economics, and finance (see, for example, [18] and references therein).
The paper is organized as follows.In Section 2, we recall the Kermack-McKendrick model and we give integrable deformations of this system.In Section 3, we analyze a particular integrable deformation of the Kermack-McKendrick system.More precisely, we point out two Poisson structures and, in consequence, we obtain two Hamilton-Poisson realizations of the considered system.In addition, using these structures, we construct infinitely many Hamilton-Poisson realizations of our system.Furthermore, we study the stability of the equilibrium points, we give the image of the energy-Casimir mapping, and we point out some of its properties.The conclusions are presented in the last section.

Integrable Deformations of the Kermack-McKendrick System
Following [1], in this section, we give integrable deformations of the Kermack-McKendrick system.First, we recall the epidemic model introduced by Kermack and McKendrick [13] (for details, see also [19]).
In the mathematical theory of epidemics, a basic model is given by the Kermack-McKendrick system.This model intends to describe the spread of the infection within the population as a function of time.It is considered that the total population is constant, and it is divided into three distinct groups.First group is formed by individuals who can catch the disease, named the susceptibles.At a moment  their number is ().The second group, the infected population, consists in individuals who have the disease and can transmit it.Their number is ().Finally, the group of the removed subjects, in number of (), formed by those who had the disease, cannot be reinfected and cannot infect other individuals.In order to obtain the evolution equations, some assumptions were made.Firstly, the gain in the infective group is at a rate proportional to the number of infected subjects and susceptibles, that is, ()(), where  > 0 is the infection rate.The susceptibles are lost at the same rate.Furthermore, the rate of removal of the infected subjects to the removed group is proportional to the number of the infected subjects, that is, (), where  > 0 is the removal rate of the infected subjects.In addition, the incubation period is negligible, and every pair of individuals has equal probability of coming into contact with one another.Therefore, the following equations were deduced: We denote  fl ,  fl ,  fl .Then the Kermack-McKendrick system is written as follows: where ,  are positive constants.
It is obvious that a constant of motion is given by the total number of individuals; namely,  1 (, , ) =  +  + . ( We recall the second constant of motion Differentiating the above constants of motion we obtain and considering ż = , we get system (2).Therefore the constants of motion (3), (4) generate system (2).This property allows us to obtain integrable deformations of system (2) by alteration of its constants of motion [1].Consider as constants of motion the functions Ĩ1 (, , ) =  +  +  +  (, , ) , where ,  are arbitrary differentiable functions.As above, we obtain that these functions generate the following system: where    fl /.If  and  are constant functions, then (7) reduces to (2).Therefore, for any differential functions  and , system ( 7) is an integrable deformation of the Kermack-McKendrick system.
Remark 1.In order to maintain constant the total population, the function  vanishes.In this case system (7) becomes

Dynamical Properties of an Integrable Deformation of the Kermack-McKendrick System
In this section, we consider some particular deformation functions, and we give some dynamical properties of the corresponding integrable deformation of the Kermack-McKendrick system.First, we give Hamilton-Poisson realizations of this system that provides the geometric framework of our study.Furthermore, we study the stability of the equilibrium points.We also give some properties of the energy-Casimir mapping associated with the considered system.We consider the following deformation functions: where  ∈ R is the deformation parameter.Then system ( 7) becomes and its constants of motion are given by ( 6); namely, In what follows we need constants of motion defined on R 3 .We immediately get that the function  2 , is a constant of motion of system (10).

Hamilton-Poisson
Realizations.We recall that the dynamical system generated by the  1 vector field has the Hamilton-Poisson realization (, {⋅, ⋅}, ), if it can be put in the form where for every ,  ∈  ∞ (R 3 , R).In addition, see, for example, [20], using any smooth function ] on , a new Poisson bracket is given by In both cases, the function  is a Casimir of the Poisson structure; that is, {, } = 0 for every .In coordinates, the Poisson structure is given in matrix notation Moreover, if there is a smooth function  such that system (13) takes the form ẋ  = Π ]  ⋅ ∇, then ( 13) is a Hamilton-Poisson system.
In our case, let  ̸ = 0 and let We obtain the rescaling function ](, , ) =  ⋅  −(/) and the Poisson structure generated by  1 : Due to its linearity, the above Poisson structure is a Lie-Poisson structure on the dual vector space of a Lie algebra, namely, se(2) * .Indeed, consider the special Euclidean Lie group SE(2) of all orientation-preserving isometries (see, for example, [21]), given by The corresponding Lie algebra of SE( 2) is with the commutator bracket [, ] =  − .
Using the same notations as in Proposition 2, we similarly obtain the following results.
then system (10) has the Hamilton-Poisson realization where  1 is the Hamiltonian function.
and hence ( 10) is a bi-Hamiltonian system.Moreover, this pair of Hamilton-Poisson realizations gives rise to infinitely many Hamilton-Poisson realizations of system (10) (see Proposition 5).

Stability of the Equilibrium Points.
The equilibrium points of system (10) are given by the following families: We remark that the second family represents the set of all equilibrium points of Kermack-McKendrick system (2).Therefore, the existence of another family of equilibrium points produces changes in the dynamics of initial system (2).We are concerned with the study of these new equilibrium points.
Proof.(i) Let  be the matrix of the linear part of our system; that is, The characteristic roots of (  1 ) are given by Considering  such that ( 2 + 2 ) < 0, we conclude that the equilibrium point   1 is unstable.(ii) Now, let  be such that ( 2 + 2 ) > 0. We use the energy-Casimir method [23].Let   be the energy-Casimir function: where  : R → R is a smooth real valued function defined on R.
The first variation of   is given by where φ () fl /.We have which vanishes if and only if The second variation of   is given by Taking into account relation (42), we obtain If  > 0, then we choose a function  such that relation (42) holds and For example, let We get that  2   (  1 ) is positive definite.Therefore the equilibrium point   1 is nonlinearly stable.If  < 0, then the same function  has the property We obtain that  2   (  1 ) is negative definite, and, in consequence, the equilibrium point   1 is nonlinearly stable.
The eigenvalues of the characteristic polynomial associated with the linearization of system (10) at  , 2 are given by  1,2 = 0,  3 =  −  − .Therefore the equilibrium point  , 2 is unstable in the case  −  −  > 0.
We also consider the subsets For  > 0, we deduce that Γ +  is the graph of a function  = (ℎ), ℎ > −1/.Also, for  < 0, we have that Γ −  is the graph of a function  = (ℎ), ℎ < −1/.We define the sets The set Im(EC) is described in the next result.
Remark 9.The image of the energy-Casimir mapping is a nonconvex subset of R 2 .Moreover, it is not a closed set, and, clearly, it is not the convex hull of the set of the images of the stable equilibrium points of the system through the map EC.
Taking into account the results that have been reported in the papers [5][6][7][8][9], we notice that our example shows there is no a general result regarding the properties of the image of the energy-Casimir mapping.Furthermore, the answer to the question "can Im(EC) be a nonconvex set?" is affirmative.
Because one of the constants of motion is not a polynomial function, it remains an open problem to establish that the results observed in the above-mentioned papers are true in the cases when the constants of motion are polynomials.
Remark 10.Another property that has been reported is the following.As a closed set, the set Im(EC) has the boundary given by images of some stable equilibrium points of the system through the energy-Casimir mapping.In our case, this property is partially true, in the sense that only a part of the boundary of Im(EC), namely, the set Γ +  (51), is formed by the images of stable equilibrium points through EC ( > 0, see Figure 1).If  < 0, a similar result is obtained for the set Γ −  (52) (see Figure 2).
Remark 11.It is easy to see that the image through the energy-Casimir mapping of a family of equilibrium points that has the form E() = ((), (), ()),  ∈ R, is a curve included in Im(EC).In our case, for  > 0, we have EC(E 1 ) = Γ +  ∪ Γ −,  ∪ Γ −,  (Figure 1), where the superscripts  and  mean stable and unstable, respectively.On the other hand, the second family of equilibrium points depends of two parameters.It is natural to ask about the image of this family through the energy-Casimir mapping.In the next result we give the set of all images of the equilibrium points that belong to E 2 through the energy-Casimir mapping.Proposition 12. (i) Let  > 0.

Conclusions
In [4], Tudoran et al. have considered the energy-Casimir mapping associated with a Hamilton-Poisson system and have proposed an open problem regarding the connections between dynamical properties of a Hamilton-Poisson system and the corresponding energy-Casimir mapping.The observed properties remain true for some particular systems [5][6][7][8][9].It was natural to ask if there are other cases [12].In our paper, we have considered such a case, obtained by using integrable deformations of the Kermack-McKendrick model.We have given Hamilton-Poisson realizations of the considered system.We have also studied the stability of the new family of equilibrium points that has developed in the considered dynamics.Furthermore, we have pointed out some properties of the energy-Casimir mapping associated with the considered system.
In our case, the image of the energy-Casimir mapping has other properties than those reported for other systems, which leaves room for further studies such as the existence of the periodic orbits of the considered system around some nonlinearly stable equilibrium points that belong to the first family, as well as homoclinic and heteroclinic orbits.

Proposition 3 .
Let  be the rescaling function given by (, , ) = −](, , ) and let  2 be a Casimir function.If Π   2 is the Poisson structure generated by  2 and , given in matrix notation by

Remark 4 .
The above Poisson structures are compatible and