Geometry and solutions of an epidemic SIS model permitting fluctuations and quantization
Oğul Esen, Eduardo Fernández-Saiz, Cristina Sardón, Marcin Zając
GGEOMETRY AND SOLUTIONS OF AN EPIDEMIC SIS MODELPERMITTING FLUCTUATIONS AND QUANTIZATION
O˘gul Esen † , Eduardo Fern´andez-Saiz ‡ , Cristina Sardón ∗ , Marcin Zając ∗∗ Department of Mathematics † ,Gebze Technical University, 41400 Gebze, Kocaeli, [email protected] de ´Algebra, Geometr´ıa y Topolog´ıa ‡ Universidad Complutense de Madrid, Pza. Ciencias 3, E-28040 Madrid, [email protected] of Applied Mathematics ∗ Universidad Polit´ecnica de MadridC/ Jos´e Guti´errez Abascal, 2, 28006, Madrid. [email protected] of Mathematical Methods in Physics ∗∗ University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, [email protected]
Abstract.
Some recent works reveal that there are models of differential equations for themean and variance of infected individuals that reproduce the SIS epidemic model at somepoint. This stochastic SIS epidemic model can be interpreted as a Hamiltonian system, there-fore we wondered if it could be geometrically handled through the theory of Lie–Hamiltonsystems, and this happened to be the case. The primordial result is that we are able to ob-tain a general solution for the stochastic/ SIS-epidemic model (with fluctuations) in form ofa nonlinear superposition rule that includes particular stochastic solutions and certain con-stants to be related to initial conditions of the contagion process. The choice of these initialconditions will be crucial to display the expected behavior of the curve of infections duringthe epidemic. We shall limit these constants to nonsingular regimes and display graphics ofthe behavior of the solutions. As one could expect, the increase of infected individuals followsa sigmoid-like curve.Lie–Hamiltonian systems admit a quantum deformation, so does the stochastic SIS-epidemicmodel. We present this generalization as well. If one wants to study the evolution of an SISepidemic under the influence of a constant heat source (like centrally heated buildings), onecan make use of quantum stochastic differential equations coming from the so-called quantumdeformation.
Contents
1. Introduction 22. Lie Analysis of the SISf model 82.1. Time-dependent vector fields 82.2. Lie Systems 92.3. The SISf is a Lie system 113. Lie–Hamilton analysis of the SISf model 153.1. Lie–Hamilton systems 15 a r X i v : . [ q - b i o . P E ] A ug .2. The SISf model is Lie-Hamiltonian 174. A (quantum) deformation of the SISf model 204.1. The Poisson—Hopf algebra deformations of Lie–Hamilton systems 204.2. The deformed SISf Model 215. Conclusions 246. Acknowledgment 26References 26Appendix A. 29A.1. Lie Algebras 29A.2. Tangent Bundle 29A.3. Cotangent Bundle - Canonical Symplectic Space 31A.4. Poisson Algebras 32A.5. Lie coalgebras 33A.6. Poisson–Hopf algebras 341. Introduction
Epidemic models try to predict the spread of an infectious disease afflicting a specific popu-lation, see for example [14, 39, 42]. These models are rooted in the works of Bernoulli in the18th century, when he proposed a mathematical model to defend the practice of inoculatingagainst smallpox [27]. This was the start of germ theory.At the beginning of the 20th century, the emergence of compartmental models was starting todevelop. Compartmental models are deterministic models in which the population is dividedinto compartments, each representing a specific stage of the epidemic. For example, S repre-sents the susceptible individuals to the disease, I designates the infected individuals, whilst R stands for the recovered ones. The evolution of these variables in time is represented bya system of ordinary differential equations whose independent variable, the time, is denotedby t . Some of these first models are the Kermack–McKendrick [28] and the Reed–Frost [1]epidemic models, both describing the dynamics of healthy and infected individuals amongother possibilities. There are several types of compartmental models [25, 26, 40], as it canbe the SIS model, in which after the infection the individuals do not adquire immunity, theSIR model, in which after the infection the individuals adquire immunity, the SIRS model,for which immunity only lasts for a short period of time, the MSIR model, in which infantsare born with immunity, etc. In this present work our focus is on the SIS model. The SIS model.
The susceptible-infectious-susceptible (SIS) epidemic model assumes a pop-ulation of size N and one single disease disseminating. The infectious period extends through-out the whole course of the disease until the recovery of the patient with two possible states,either infected or susceptible. This implies that there is no immunization in this model. In his approach the only relevant variable is the instantaneous density of infected individuals ρ = ρ ( τ ) depending on the time parameter τ , and taking values in [0 , γρ , where γ is the recovery rate, and the rate of growthof new infections is proportional to αρ (1 − ρ ), where the intensity of contagion is given by thetransmission rate α . These two processes are modelled through the compartmental equation(1.1) dρdτ = αρ (1 − ρ ) − γρ. One can redefine the timescale as t := ατ and introduce the constant ρ := 1 − γ/α , so wecan rewrite (1.1) as(1.2) dρdt = ρ ( ρ − ρ ) . Clearly, the equilibrium density is reached if ρ = 0 or ρ = ρ . Although compartmentalequations have proven their efficiency for centuries, they are still based on strong hypotheses.For example, the SIS model works more efficiently under the random mixing and large pop-ulation assumptions. The first assumption is asking homogeneous mixing of the population,that is, individuals contact with each other randomly and do not gather in smaller groups,as abstaining themselves from certain communities. This assumption is nevertheless rarelyjustified. The second assumption is the rectangular and stationary age distribution, whichmeans that everyone in the population lives to an age L , and for each age up to L , which isthe oldest age, there is the same number of people in each subpopulation. This assumptionseems feasible in developed contries where there exists very low infant mortality, for example,and a long live expectancy. Nonetheless, it looks reasonable to implement probability at somepoint to permit random variation in one or more inputs over time. Some recent experimentsprovide evidence that temporal fluctuations can drastically alter the prevalence of pathogensand spatial heterogeneity also introduces an extra layer of complexity as it can delay thepathogen transmission [21, 49]. The SIS model with fluctuations.
It is needless to point out that fluctuations shouldbe considered in order to capture the spread of infectious diseases more closely. Nonetheless,the introduction of these fluctuations is not trivial. One way to account for fluctuations isto consider stochastic variables. On the other hand, it seems that in the case of SIS modelsthere exist improved differential equations for the mean and variance of infected individuals.Recently, in [43], the model assumes the spreading of the disease as a Markov chain in discretetime in which at most one single recovery or transmission occurs in the duration of thisinfinitesimal interval. As a result [29], the first two equations for instantaneous mean densityof infected people h ρ i and the variance σ = h ρ i − h ρ i are d h ρ i dt = h ρ i ( ρ − h ρ i ) − σ ,dσ dt = 2 σ ( ρ − h ρ i ) − ∆ − N h ρ (1 − ρ ) i + γN α h ρ i (1.3) here ∆ = h ρ i − h ρ i . This system finds excellent agreement with empirical data [43]. Equa-tions (1.2) and (1.3) are equivalent when σ becomes irrelevant compared to h ρ i . Therefore, ageneralization of compartmental equations only requires mean and variance, neglecting higherstatistical moments. The skewness coefficient vanishes as a direct consequence of this assump-tion, so that ∆ := 3 σ h ρ i . For a big number of individuals ( N (cid:29) d ln h ρ i dt = ρ − h ρ i − σ h ρ i , d ln σ dt = ρ − h ρ i . (1.4)The system right above can be obtained from a stochastic expansion as it is given in [54], aswell. Hamiltonian character of the model (1.4) . The investigation of the geometric and/or thealgebraic foundations of a system permits to employ several powerful techniques of geometryand algebra while performing the qualitative analysis of the system [3, 5, 33]. This even resultsin an analytical/general solution of the system in our case. For example, we cite [32] and [44]for Lie symmetry approach to solve the classical SIS model. Particularly, the Hamiltoniananalysis of a system plays an important role in the geometrical analysis of a given system.For instance, in compartmental theories we can cite an early work [45] that classifies theSIR model as bihamiltonian. We also refer to [23] for conformal Hamiltonian analysis of theKermack-McKendrick Model. In a very recent study [7], it is shown that all classical epidemiccompartmental models admit Hamiltonian realizations.In [43], the SIS system (1.4) involving fluctuations has been recasted in Hamiltonian form inthe following way: the dependent variables are the mean h ρ i and the variance σ , and theyboth depend on time. Then, we define the dynamical variables q = h ρ i and p = 1 /σ , so thesystem (1.4) turns out to be dqdt = qρ − q − p ,dpdt = − pρ + 2 pq. (1.5)We employ the abbreviation SISf for system (1.5) to differentiate it from the classical SISmodel in (1.2). The letter “f” accounts for “fluctuations”.We have computed the general solution to this system, finding a more general solution thanthe one provided by Nakamura and Mart´inez in [43]. Indeed, we have found obstructions intheir model solution. We shall comment this in the last section gathering all our new results. ur general solution for this system reads: q ( t ) = ρe ρt ( C ρ − e ρt + 2 C C ρ ( C ρ − e ρt + 4 C ρ ( C e ρt + C ) ,p ( t ) = C + C ρ − ρ − C + C e − ρt . (1.6)In order to develop a geometric theory for this system of differential equations, we need tochoose certain particular solutions that we shall make use of. Here we present three differentchoices and their corresponding graphs according to the change of variables q = h ρ i and p = 1 /σ . Figure 1.
The first particular solution igure 2. The second particular solution
Figure 3.
The third particular solutionLet us turn now to interpret these equations geometrically on a symplectic manifold. Thesymplectic two-form ω = dq ∧ dq is a canonical skew-symmetric tensorial object in two-dimensions. For a chosen (real-valued) Hamiltonian function h = h ( q, p ), the dynamics isgoverned by a Hamiltonian vector field X h defined through the Hamilton equation(1.7) ι X h ω = dh, where ι X h is the contraction operator in the tensor algebra, and dH is the (exterior) derivativeof h . In terms of the coordinates ( q, p ), the Hamilton equations (1.7) become(1.8) dqdt = ∂h∂p , dpdt = − ∂h∂q . t is possible to realize that the SISf system (1.5) is a Hamiltonian system since it fulfills theHamilton equations (1.7). To see this, consider the Hamiltonian function(1.9) h = qp ( ρ − q ) + 1 p . and substitute it into (1.8). A direct calculation will lead us to (1.5). The skew-symmetryof the symplectic two-form implies that the Hamiltonian function is constant all along themotion. In classical mechanics, where the Hamiltonian is taken to be the total energy, thiscorresponds to the conservation of energy. The goal of the present work.
Our focus is the SISf model permitting fluctuations given in(1.5). We shall perform both the Lie analysis and the Lie-Hamilton analysis of this system inorder to explore the geometric/algebraic foundations of the model as well as to arrive at theanalytical general solution. Further, we shall introduce a deformation (the so-called quantumdeformation) of the model (1.5) to make it applicable to more complicated issues.To get involved in the mathematical theories needs some special emphasis or/and some ex-pertise. For that, we shall try to present both the theoretical and the practical results in themost accessible forms for a general audience as far as we can. Our plan to achieve this is toexhibit itinerary maps, graphs, and a comprehensive appendix.The paper is organized into three main sections, and an appendix. Sections 2 and 3 concernthe Lie and the Lie-Hamilton analysis of the SISf model (1.5), respectively. Section 4 proposesthe so-called quantum deformation of the model and Appendix A is a brief review of somealgebraic constructions, such as Lie algebras, Poisson algebras, Lie coalgebras and PoissonHopf algebras, that we refer to in the main body of the paper. . Lie Analysis of the SISf model
A (nonautonomous) system of ordinary differential equations (ODEs) can be written as a time-dependent vector field and vice versa. Such system is called Lie system if the associated time-dependent vector field takes values in a finite-dimensional Lie algebra of vector fields [15, 16].Equivalently, one can also define a Lie system as a system of ODEs admitting a(nonlinear)superposition principle, that is, a map allowing us to express the general solution of thesystem of ODEs in terms of a family of particular solutions and a set of constants related toinitial conditions. Accordingly, in this section, we shall show that the SISf model (1.5) is aLie system. For the completeness of this work, we start with a brief summary of some basicnotions concerning the theory of Lie systems. We refer to Appendix A.1 for the definitions ofLie (sub)algebras, and to Appendix A.2 for the definitions of (co)tangent bundle and vectorfields. For some further reading on Lie systems especially about its relevant role in Physics,Mathematics, Biology, Economics, and other fields of research, see [18, 35, 51] and extensivereference lists there in.2.1.
Time-dependent vector fields.
Let N be a manifold. A time-dependent vector field on N is a differentiable mapping(2.1) X : R × N −→ T N, τ N ◦ X = pr where pr is the projection to the second factor in R × N . In this regard, we can considera time-dependent vector as a set (family) of standard vector fields { X t } t ∈ R , depending ona single parameter. The converse of this assertion is also true, that is a set of vector fieldsdepending smoothly on a single parameter can be written as a time-dependent vector field[18]. We plot the following diagram to summarize the discussions.(2.2) T N τ N (cid:15) (cid:15) R × N X (cid:58) (cid:58) pr (cid:47) (cid:47) N A time dependent vector field determines a non-autonomous ODE system(2.3) dxdt = X ( t, x ) , x ∈ N. The suspension of a time-dependent vector field X on N is the vector field on R × N definedto be X + ∂/∂t , where t is the global coordinate for R . An integral curve of a time-dependentvector field X is an integral curve of the suspension [3]. Conversely, every system of first-orderdifferential equations in normal form describes the integral curves of a unique time-dependentvector field. onsider the case that N is a two dimensional manifold with local coordinates ( q, p ), andthe extended space R × N with induced coordinates ( t, q, p ). A time dependent vector field iswritten as(2.4) X = k ( t, q, p ) ∂∂q + g ( t, q, p ) ∂∂q where k and g are real valued functions on R × N . In this coordinate system, the dynamicalequations governed by X determine a system of nonautonomous ordinary differential equations(2.5) ˙ q = k ( t, q, p ) , ˙ p = g ( t, q, p ) . Notice that, the SISf model (1.5) fits this picture with even the generalization ρ = ρ ( t ). Weshall turn to this discussion in the following subsections. Vessiot-Guldberg algebra.
Start with a time-dependent vector field X , and determine theset { X t } t ∈ R of vector fields. We denote the smallest Lie subalgebra containing the set { X t } t ∈ R by V X . The associated distribution D X for a time-dependent vector field X is the generalizeddistribution spanned by the vector fields in V X that is(2.6) D Xx = { Y x | Y ∈ V X } ⊂ T x N. The associated codistribution ( D X ) ◦ for a time-dependent vector field X is then defined tobe the annihilator of D X that is(2.7) ( D Xx ) ◦ = { ϑ ∈ T ∗ x N | ϑ ( Z x ) = 0 , ∀ Z x ∈ D Xx } . is a subbundle of the cotangent bundle T ∗ N . Our interest relies in a distribution D X deter-mined by a finite-dimensional V X and hence D X becomes integrable on the whole N . In thiscase, V X is called Vessiot-Guldberg Lie algebra. It is worth noting that even in this case,( D X ) ◦ does not need to be a differentiable codistribution.A function f on an open neighbourhood U is a local t -independent constant of motion for asystem X if and only if df ( x ) in ( D Xx ) ◦ | U for all x in N . This statement reads that (locallydefined) time-independent constants of motion of time-dependent vector fields are determinedby (locally defined) exact one-forms taking values in the associated codistribution. Therefore,( D X ) ◦ is a crucial object in the calculation of such constants of motion for a system X .2.2. Lie Systems.
Let X be a time-dependent vector field defined on a manifold N . A superposition rule de-pending on m particular solutions of X is a function(2.8) φ : N m × N → N, x = φ ( x (1) , . . . , x ( m ) , λ ) uch that the general solution x ( t ) of X can be written as x ( t ) = φ ( x (1) ( t ) , . . . , x ( m ) ( t ) , λ ) forany generic family x (1) ( t ) , . . . , x ( m ) ( t ) of particular solutions. Here, λ is a point of N relatedwith the initial conditions. A system of equations admitting superposition rule is called Liesystem. Lie–Scheffers Theorem.
A modern statement of this result is described in [16]. A first-ordersystem (2.3) admits a superposition rule if and only if the associated time-dependent vectorfield X can be written as(2.9) X t = r X α =1 b α ( t ) X α for a certain family b ( t ) , . . . , b r ( t ) of time-dependent functions and a family X , . . . , X r ofvector fields on N spanning an r -dimensional real Lie subalgebra of vector fields. We referto [35] for details, see [58, 50] for some further discussions and the first examples. The Lie–Scheffers Theorem yields that a system X admits a superposition rule if and only if V X isfinite-dimensional, [18]. An itinerary map to derive superposition rules.
General solutions of Lie systems canalso be investigated through superposition rules. There exist various procedures to derivethem [4, 57], but we hereafter use the method devised in [16], which is based on the notion ofdiagonal prolongation [18]. Let us denote the m + 1-times Cartesian product of N by itself as N ( m +1) . We denote by(2.10) pr : N ( m +1) −→ N, ( x (0) , x (1) , . . . , x ( m ) ) x (0) the projection onto the first factor. Given a time-dependent vector field X on N , the diagonalprolongation e X of X to the product space N ( m +1) is a unique time-dependent vector field on N ( m +1) projecting to X by the projection pr that is pr ∗ e X t = X t for all t . We also require e X to be invariant under the permutations x ( i ) ↔ x ( j ) with i, j = 0 , . . . , m . The procedure todetermine superposition rules described in [16] goes as follows: Step 1.
Take a basis X , . . . , X r of a Vessiot–Guldberg Lie algebra associated with the Liesystem X . Step 2.
Choose the minimum integer m so that the diagonal prolongations e X , . . . , e X r , of X , . . . , X r to N m are linearly independent at a generic point. Step 3.
Obtain, for instance, by the method of characteristics, n functionally independentfirst-integrals F , . . . , F n common to all the diagonal prolongations, e X , . . . , e X r , to N ( m +1) . e require such functions to satisfy ∂ ( F , . . . , F n ) ∂ (cid:16) ( x ) (0) , . . . , ( x n ) (0) (cid:17) = 0 . Assume that these integrals take certain constant values, i.e., F i = k i where i = 1 , . . . , n , andemploy these equalities to express the variables ( x ) (0) , . . . , ( x n ) (0) in terms of the variablesof the other copies of N within N ( m +1) and the constants k , . . . , k n .The obtained expressions constitute a superposition rule in terms of any generic family of m particular solutions and n constants. Let us apply these steps in the realm of the SISf model.2.3. The SISf is a Lie system.
The model (1.5) can be generalized to a model represented by a time-dependent vector field(2.11) X t = ρ ( t ) X + X where the constitutive vector fields are computed to be(2.12) X = q ∂∂q − p ∂∂p , X = (cid:18) − q − p (cid:19) ∂∂q + 2 qp ∂∂p . The generalization comes from the fact that ρ ( t ) is no longer a constant, but it can evolvein time. Let us apply the steps introduced in the previous section one by one to arrive at thegeneral solution. Step 1.
For the vector fields in (2.12), a direct calculation shows that the Lie bracket(2.13) [ X , X ] = X is closed within the Lie algebra. This implies that the SISf model (1.5) is a Lie system.The Vessiot-Guldberg algebra spanned by X , X is an imprimitive Lie algebra of type I according to the classification presented in [8]. Step 2.
If we copy the configuration space twice, we will have four degrees of freedom( q , p , q , p ) and we will archieve precisely two first-integrals in vinicity of the Fr¨obeniustheorem. A first-integral for X t has to be a first-integral for X and X simultaneously. Wedefine the diagonal prolongation e X of the vector field X in the decomposition (2.13). Thenwe look for a first integral F such that e X [ F ] vanishes identically. Notice that if F is afirst-integral of the vector field e X then it is a first integral of e X due to the commutationrelation. For this reason, we start by integrating the prolonged vector field(2.14) e X = q ∂∂q + q ∂∂q − p ∂∂p − p ∂∂p hrough the following characteristic system(2.15) dq q = dq q = dp − p = dp − p . Fix the dependent variable q and obtain a new set of dependent variables, say ( K , K , K ),which are computed to be(2.16) K = q q , K = q p , K = q p . Step 3.
This induces the following basis in the tangent space ∂∂K = q ∂∂q − q p q ∂∂p − q p q ∂∂p , ∂∂K = 1 q ∂∂p , ∂∂K = 1 q ∂∂p . (2.17)provided that q is not zero. Introducing the coordinate changes exhibited in (2.16) into thediagonal projection e X of the vector field X , we arrive at the following expression e X = (cid:18) K − (cid:18) K (cid:19)(cid:19) ∂∂K + (cid:18)(cid:18) K + 1 K (cid:19) K − (cid:18) K (cid:19) K (cid:19) ∂∂K + (cid:18) K K − (cid:18) K (cid:19) K (cid:19) ∂∂K . To integrate the system once more, we use the method of characteristics again and obtain(2.18) d ln | K | − K = d ln | K | K + K K − − K = d ln | K | K − (cid:16) K (cid:17) . Exact solution.
We obtain two first integrals by integrating in pairs ( K , K ) and ( K , K ),where we have fixed K . After some cumbersome calculations we obtain(2.19) K = K (cid:0) k K + 4 k k K + k − (cid:1) K + 1)( K − k (2 k K + k ) , K = K (cid:16) k K + k K + k − k (cid:17) ( K + 1)( K − . By substituting back the coordinate transformation (2.16) into the solution (2.19) (pleasenotice the difference between capitalized constants ( K , K , K ) and lower case constants( k , k ), we arrive at the following implicit equations q = − q (cid:18) k k ± q k p q + k k p q − k p q − k p q + 4 k (cid:19) k ( − p q + k ) p = 4 q k + 4 q q k k + q k − q k (2 q k + q k q − q k q − k q ) . (2.20)Let us notice that the equations (2.20) depend on a particular solution ( q , p ) and twoconstants of integration ( k , k ) which are related to initial conditions.Let us show now the graphs and values of the initial conditions for which the solution remindsus of sigmoid behavior, which is the expected growth of ρ ( t ). As particular solution for ( q , p ), e have made use of particular solution 2 given in Figure 2 through its corresponding valuesof q, p through the change of variables q = h ρ i and p = 1 /σ . Figure 4.
Superposition rule for exact solutionNotice that we have not included the error graph since it gives a constant zero graph because k → Step 3 revisited - linear approximation.
Since the solution (2.20) is quite complicated,one may look for a solution of a linearized model. We first employ the following change ofcoordinates(2.21) { u = ln | K | , v = ln | K | , w = ln | K |} . In terms of these new variables, the system (2.18) reads(2.22) du − e − u = dve − v + e v − w − − e − u = dw e − v − (1 + e − u ) . One can solve the system above by introducing a linear approximation1 − e − u ’ u,e − v + e v − w − − e − u ’ u − w, e − v − (1 + e − u ) ’ u − v, (2.23)after which (2.22) reads(2.24) du u = dv u − w = dw u − v . e can solve now v and w in terms of u and obtain(2.25) v ( u ) = k u −√ / + k u √ / + u, w ( u ) = √ (cid:16) k u −√ / − k u √ / (cid:17) . According to the algorithm for deriving superposition rule, we need to isolate the constantsof integration k and k . Hence, the two first integrals read now(2.26) k = u √ (cid:0) √ v − √ u + w (cid:1) / √ , k = u − √ (cid:0) √ v − √ u − w (cid:1) / √ . Now, if we substitute the coordinate changes in (2.21) and in (2.16), we arrive at the followinggeneral solution(2.27) q = q exp (cid:16) − ln ( q p )1 + k + k (cid:17) , p = 1 q exp (cid:16) √
22 ( k − k ) ln ( q p )1 + k + k (cid:17) . which can be written as(2.28) q = q (cid:16) q p (cid:17) − k k , p = 1 q (cid:16) q p (cid:17) √ k − k k k . Notice that the solution depends on a particular solution ( q , p ) and two constants of inte-gration ( k , k ), as in (2.20).Let us show now the graphs and values of the initial conditions for which the solution remindsus of sigmoid behavior, which is the expected growth of ρ ( t ). As particular solution for ( q , p ),we have made use of particular solution 2 given in Figure 2 through its corresponding valuesof q, p through the change of variables q = < ρ > and p = 1 /σ . Figure 5.
Superposition rule for linear approximation . Lie–Hamilton analysis of the SISf model
In this section, we shall show that the SISf model (1.5) is a Lie-Hamilton system [10, 17, 35].Among the developed methods for Lie–Hamilton systems, we consider a very important re-cent method for the obtainance of solutions as superposition principles through the Poissoncoalgebra method [8, 9]. The traditional method for the computation of superposition prin-ciples for Lie systems relies in the integration of systems of ordinary or partial differentialequations [4, 57], but in the case of Lie–Hamilton systems, the nonlinear superposition rulecan be obtained straightforwardly through a Casimir of the Vessiot–Guldberg Lie algebra. Werefer to Appendices A.3 and A.4 for informal summaries of the symplectic and the Poissongeometries, and to Appendix A.5 for Lie coalgebras. We refer Appendix A.6 for the definitionof the symmetric algebra.3.1.
Lie–Hamilton systems.
A Lie-Hamiltonian structure is a triple ( N, Λ , h ), where ( N, Λ) stands for a Poisson manifoldand h represents a t -parametrized family of functions h t : N → R such that the Lie algebra H := Lie( { h t } t ∈ R ) generated by this family is finite-dimensional Poisson algebra. A time-dependent vector field X is said to possess a Lie-Hamiltonian structure ( N, Λ , h ) if X t is theHamiltonian vector field corresponding to h t , for each t ∈ R ,(3.1) X t = − ˆΛ ◦ d ( h t ) . In this case, X is called a Lie-Hamilton system, and H is called a Lie–Hamilton algebra for X . Now we examine superposition rules for the case of Lie–Hamilton systems. Itinerary map for superposition rules of the Lie–Hamilton Systems.
Let X be a Lie–Hamilton system with a Poisson algebra H spanned by the linearly independent Hamiltonianfunctions { h , . . . , h r } . Step 1.
Consider the injection g , → H of a Lie algebra g into H that turns the basis v i of g into the Hamiltonian functions φ ( v i ) := h i for i = 1 , . . . , r . Referring to this inclusion, we candefine Poisson algebra homomorphisms(3.2) D : S ( g ) → C ∞ ( N ) , D ( m ) : S ( m ) ( g ) → C ∞ ( N ) ( m ) ⊂ C ∞ ( N m ) , where S ( g ) is the symmetric algebra. Step 2. If C is a polynomial Casimir of the Poisson algebra S ( g ), say C = C ( v , . . . , v r ), then D ( C ) is a constant of motion for X , and so are the functions defined by(3.3) F ( k ) ( h , . . . , h r ) = D ( k ) h ∆ ( k ) ( C ( v , . . . , v r )) i , k = 2 , . . . , m. et us notice, that each F ( k ) can naturally be considered as a function of C ∞ ( M m ) for every m › k . Step 3.
From the functions F ( k ) , we can obtain other constants of motion in the form(3.4) F ( k ) ij = S ij ( F ( k ) ) , ‹ i < j ‹ k, k = 2 , . . . , m, where S ij is the permutation of variables x ( i ) ↔ x ( j ) on M m .Indeed, since the prolongation e X is invariant under the permutations x ( i ) ↔ x ( j ) , then the F ( k ) ij are also t -independent constants of motion for the diagonal prolongations e X to M m . Letus now illustrate this procedure in the following example. Example 1.
Over the plane, consider the following vector fields (3.5) X = ∂∂x , X = ∂∂y , X = y ∂∂x − x ∂∂y with commutation relations [ X , X ] = 0 , [ X , X ] = − X , [ X , X ] = X . Step 1.
With respect to the canonical symplectic structure ω = d x ∧ d y , this corresponds Liealgebra, denoted by iso (2) , determined by a basis (3.6) h = y, h = − x, h = 12 ( x + y ) , h = 1 satisfying commutation relations (3.7) { h , h } ω = h , { h , h } ω = h , { h , h } ω = − h , { h , ·} ω = 0 , Step 2.
The symmetric Poisson algebra S (cid:16) iso (2) (cid:17) has a non-trivial Casimir invariant givenby C = v v − ( v + v ) . Choosing the representation given in (3.6) , we obtain a trivial constant of motion on ( x, y ) ≡ ( x , y ) F = D ( C ) = φ ( v ) φ ( v ) − (cid:16) φ ( v ) + φ ( v ) (cid:17) = h ( x , y ) h ( x , y ) − (cid:0) h ( x , y ) + h ( x , y ) (cid:1) = ( x + y ) × − ( y + x ) = 0 . ntroducing the coalgebra structure in S (cid:16) iso (2) (cid:17) through the coproduct (A.34), we obtainnontrivial first integrals. F (2) = D (2) (∆( C ))= ( h ( x , y ) + h ( x , y )) ( h ( x , y ) + h ( x , y )) − (cid:16) (( h ( x , y ) + h ( x , y )) + ( h ( x , y ) + h ( x , y )) (cid:17) = ( x − x ) + ( y − y ) ,F (3) = D (3) (∆( C ))= X i =1 h ( x i , y i ) X j =1 h ( x j , y j ) − (cid:16)(cid:0) X i =1 h ( x i , y i ) (cid:1) + (cid:0) X i =1 h ( x i , y i ) (cid:1) (cid:17) =
12 3 X ‹ i Furthermore, using the property of permutating subindices (3.4), we find more firstintegrals F (2)12 = S ( F (2) ) ≡ F (2) , F (2)13 = S ( F (2) ) = ( x − x ) + ( y − y ) ,F (2)23 = S ( F (2) ) = ( x − x ) + ( y − y ) (3.9) Observe that F (3) = F (2)12 + F (2)13 + F (2)23 . We may consider as many first integrals as the numberof degrees of freedom of the system in order to integrate it. The combination of these functionsleads us to a superposition rule. The SISf model is Lie-Hamiltonian. The SISf epidemic system in (1.5) admits a Hamiltonian formulation as in (1.9), where thecorresponding Hamilton equations are (1.5). In this section we will retrieve the Hamiltonian(1.9) using the theory of Lie systems, in particular, the theory of Lie–Hamilton systems. Toretrieve the Hamiltonian, first we need to prove that equations in (1.5) form a Lie–Hamiltonsystem.We have already proven in (2.13) that (1.5) defines a Lie system. In order to see if it is a Lie–Hamilton system, we first need to check whether the vector fields in (2.13) are Hamiltonianvector fields. Consider now the canonical symplectic form ω = dq ∧ dp . It is easy to checkthat the vector fields X and X in (2.13) are Hamiltonian with respect to the Hamiltonianfunctions(3.10) h = − qp, h = − q p + 1 p , espectively. It is easy to see that the Poisson bracket of these two functions reads { h , h } = h . It means that the Hamiltonian functions form a finite dimensional Lie algebra, denoted inthe literature as I r =114 A ’ R (cid:110) R , and it is isomorphic to the one defined by vector fields X , X .The Hamiltonian function for the total system is(3.11) h = ρ ( t ) h + h = − q p + 1 p − ρ ( t ) qp and it is exactly the Hamiltonian function (1.9) proposed in [43].Lie-Hamilton systems can also be integrated in terms of a superposition rule, as it was ex-plained in the preliminary section. In order to do that, we need to find a Casimir function forthe Poisson algebra, but unfortunately, there exists no nontrivial Casimir in this particularcase. It is interesting to see how a symmetry of the Lie algebra { X , X } commutes with theLie bracket, i.e. the vector field(3.12) Z = − p ( C p q + 4 C pq + C )( pq − pq + 1) ∂∂p + C p q + C p q − C pq − C p ( pq − ( pq + 1) ∂∂q fulfills [ X , Z ] = 0 , [ X , Z ] = 0 . Notice too that Z is a conformal vector field, that is,(3.13) L Z ω = − ( C / ω. Since it is a Hamiltonian system, one would expect that a first integral for Z , let us say f ,would Poisson commute with the Poisson algebra { h , h } , since Z = − ˆΛ( df ). Nonetheless,this is not the case unless f = constant. This implies that the Casimir is a constant, hencetrivial and the coalgebra method can not be directly applied. However, there is a way inwhich we can circumvent this problem by considering an inclusion of the algebra I r =114 A as aLie subalgebra of a Lie algebra to another class admitting a Lie–Hamiltonian algebra with anon-trivial Casimir. In this case, we will consider the algebra, denoted by I ’ iso (1 , I , we simultaneouslyobtain the superposition for I r =114 A as a byproduct.The Lie–Hamilton algebra iso (1 , 1) has the commutation relations(3.14) { h , h } = h , { h , h } = − h , { h , h } = h , { h , ·} = 0 , with respect to ω = dx ∧ dy in the basis { h = y, h = − x, h = xy, h = 1 } . The Casimirassociated to this Lie–Hamilton algebra is C = h h + h h . Let us apply the coalgebra methodto this case. Mapping the representation without coproduct, the first iteration is trivial, i.e., F = 0. We could usethe second order coproduct and third order coproduct ∆ (2) and ∆ (3) , orthe second order coproduct ∆ (2) together with the permuting subindices property. We needthree constants of motion, this would be equivalent to integrating the diagonal prolongation X on ( R ) . Using the coalgebra method and subindex permutation, one obtains F (2) = ( x − x )( y − y ) = k ,F (2)23 = ( x − x )( y − y ) = k ,F (2)13 = ( x − x )( y − y ) = k . (3.15)From them, we can choose two functionally independent constants of motion. Our choice is F (2) = k , F (2)23 = k . The introduction of k simplifies the final result, with expression x ( x , y , x , y , k , k , k ) = 12 ( x + x ) + k − k ± B y − y ) ,y ( x , y , x , y , k , k , k ) = 12 ( y + y ) + k − k ∓ B x − x ) , (3.16)where(3.17) B = q k + k + k − k k + k k + k k ) . In the case that matters to us, I r =114 A , the third constant k is a function k = k ( x , y , x , y )and B › 0. Notice though that this superposition rule is expressed in the basis (3.14), there-fore, we need the change of coordinates between the present iso (1 , 1) and our problem (3.10).See that the commutation relation { h , h } = − h in (3.14) coincides with the commutationrelation { h , h } = h of our pandemic system (3.10). So, by comparison, we see there is achange of coordinates(3.18) x = − qp, y = q − qp . This way, introducing this change (3.18) in (3.16), the superposition principle for our Hamil-tonian pandemic system reads q = (cid:18) q + q +( k − k ± B )(2 p − p ) (cid:19) (cid:16) p + p + ( k − k ∓ B )(2 q − q ) (cid:17)(cid:18) q + q +( k − k ± B )(2 p − p ) (cid:19) − p = − (cid:18) q + q +( k − k ± B )(2 p − p ) (cid:19) − q + q + ( k − k ± B )(2 p − p ) (cid:16) p + p + ( k − k ∓ B )(2 q − q ) (cid:17) (3.19)Here, ( q , p ) and ( q , p ) are two particular solutions and k , k , k are constants of integra-tion.Now, we show the graphics for < ρ > = q ( t ) and σ = 1 /p using the two particular solutionsin Figure 2 and Figure 3 provided in the introduction. Notice that we have renamed c =( k − k ± B ) and k = ( k − k ∓ B ). igure 6. Superposition rule with two particular solutions4. A (quantum) deformation of the SISf model A Lie–Hamilton system admits a Poisson–Hopf deformation, see [11]. The interest of Poisson–Hopf deformations of these models resides in the fact that many of the outcome deformedsystems happen to be other systems that are already identified in the physics and mathematicsliterature. This is a highlight, since this basically implies that two different systems scattered inthe mathematical physics literature can be merely related by a Poisson–Hopf transformation.This could be a start for the classification of the fuzzy number of integrable systems.We have interpreted that the SISf model (1.5) is a Lie-Hamilton system. So, in this section wepropose a (quantum) deformation of the SISf model [8, 12, 13, 19, 35] which does not rely onthe complicated functional analysis considerations for unbounded quantum operators [30]. Theintroduction of the quantum approach will account for the existence of a possible interactionof the SISf model with a heat bath, which is effectively an infinite pool of thermal energy at agiven constant temperature. This could correspond with centrally heated buildings. We referAppendix A.6 for some details on Poisson-Hopf algebras.4.1. The Poisson—Hopf algebra deformations of Lie–Hamilton systems. To obtain a deformation of the Lie–Hamilton realization of the SISf model we make use ofdeformed Poisson–Hopf algebras. Following [11], we summarize the procedure for the planarsystems as follows: Itinerary map to arrive at a deformation of a Lie-Hamilton system. Consider aLie–Hamilton system given by a time-dependent vector field X t ∈ R describing a Lie system. here is a corresponding Lie algebra of Hamiltonian functions Lie( { h t } t ∈ R ) satisfying (3.1)that defines the Lie system as a Lie–Hamilton system. Step 1. Deform the space C ∞ ( { h t z } ∗ t ∈ R ) with a deformation parameter z ∈ R and accordinglythe Poisson bracket(4.1) { h z,i , h z,j } ω = F z,ij ( h z, , . . . , h z,l ) . Here, F z,ij are certain smooth functions also depending smoothly on the deformation param-eter z and such that(4.2) lim z → h z,i = h i , lim z → ∇ h z,i = ∇ h i , lim z → F z,ij ( h z, , . . . , h z,l ) = l X k =1 c kij h k , where ∇ is the gradient. Step 2. Deform the vector fields X z,i as(4.3) ι X z,i ω := d h z,i . Step 3. The deformed total dynamics is encoded in(4.4) X z := l X i =1 b i ( t ) X z,i . Notice that when the parameter goes to the zero, we have(4.5) lim z → { h z,i , h z,j } ω = { h i , h j } ω , lim z → X z,i = X i , and the deformed dynamics reduces to the initial one.4.2. The deformed SISf Model. For the SISf model (1.5), we start with the Vessiot–Guldberg algebra (2.13) labelled as I r =114 A .To obtain a deformation of a Lie algebra I r =114 A , we need to rely on a bigger Lie algebra, in thiscase, we make use of sl (2). To this end, consider the vector fields X and X in (2.12), andlet X be a vector field given by(4.6) X := p q ( − p q + c + 6) + c p q − ∂∂q − p q ( c + 2)( p q − ∂∂p , where c ∈ R . Then, { X , X , X } span a Vessiot–Lie algebra V isomorphic to sl (2) thatsatisfies the following commutation relations(4.7) [ X , X ] = X , [ X , X ] = − X , [ X , X ] = 2 X . his vector field X admits a Hamiltonian function, say h , with respect to the canonicalsymplectic form on R , so that we have the family(4.8) h = − qp, h = 1 p − q p, h = 2 p q + c − p q . Hence, { h , h , h } span a Lie–Hamilton algebra H ω ; isomorphic to sl (2) where the commu-tation relations with respect to the Poisson bracket induced by the canonical symplectic form ω on R are given by(4.9) { h , h } ω = h , { h , h } ω = − h , { h , h } ω = 2 h . Here is the deformation of the model by steps. Step 1. Applying the non-standard deformation of sl (2) in [11] we arrive at the Hamiltonianfunctions(4.10) h z ;1 = − shc (2 zh z ;2 ) qp, h z ;2 = 1 p − q p, h z ;3 = − p (cid:0) shc (2 zh z ;2 )2 q p + c (cid:1) shc (2 zh z ;2 )( q p − , Here, shc ( x ) is the cardinal hyperbolic sinus function. Accordingly, the Poisson brackets arecomputed to be(4.11) { h z ;1 , h z ;2 } ω = shc (2 zh z ;2 ) h z ;2 , { h z ;2 , h z ;3 } ω = 2 h z ;1 , { h z ;1 , h z ;3 } ω = − cosh (2 zh z ;2 ) h z ;3 , Step 2. The vector fields X z ;1 and X z ;2 associated to the Hamiltonian functions h z ;1 and h z ;2 exhibited in (4.10) are X z, = cosh (cid:16) z ( p − q p ) (cid:17) ( p q − (cid:20) (1 − p q ) ∂∂q + (2 p q − p q ) ∂∂p (cid:21) + shc (cid:16) z ( p − q p ) (cid:17) ( p q − (cid:20) q ∂∂q − p ( p q + 1) ∂∂p (cid:21) ,X z, = (cid:18) − q − p (cid:19) ∂∂q + 2 qp ∂∂p . (4.12)We do not write explicitly the expression of the vector field X z ;3 because it does not play arelevant role in our system. The deformed vector fields keep the commutation relations(4.13) [ X z ;1 , X z ;2 ] = cosh (cid:18) z (cid:18) p − q p (cid:19)(cid:19) X z ;2 . Step 3. The total Hamiltonian function for the deformed model is(4.14) h z = ρ ( t ) h z ;1 + h z ;2 = − ρ ( t ) shc (2 zh z ;2 ) qp + 1 p − q p. o that the deformed dynamics is computed to be dqdt = cosh (cid:16) z ( p − q p ) (cid:17) ( p q − (1 − p q ) + shc (cid:16) z ( p − q p ) (cid:17) ( p q − q ρ ( t ) − q − p ,dpdt = cosh (cid:16) z ( p − q p ) (cid:17) ( p q − (2 p q − p q ) − p shc (cid:16) z ( p − q p ) (cid:17) ( p q − 1) ( p q + 1) ρ ( t ) − qp. (4.15)This system describes a family of z-parametric differential equations that generalizes theSISf model (1.5), where the demographic interaction and both rates allow a more realisticrepresentation of the epidemic evolution. According to the kind of deformation, this may becalled a quantum family SISf model. Note that the SISf model can be recovered in the limitwhen z tends to zero. Constants of motion. For the present case, the constants of motion defined in (3.3) arecomputed to be(4.16) F (1) = c , F (2) = (cid:16) h (1)2 + h (2)2 (cid:17) (cid:16) h (1)3 + h (2)3 (cid:17) − (cid:16) h (1)1 + h (2)1 (cid:17) , after the quantization, the latter one becomes(4.17) F (2) z = shc (cid:16) zh (2) z ;2 (cid:17) h (2) z ;2 h (2) z ;3 − (cid:16) h (2) z ;1 (cid:17) , where h (2) z ; j := D (2) z (∆ z ( v j )). This coproduct ∆ z can be described as a follows∆ z ( v ) = v ⊗ ⊗ v , ∆ z ( v j ) = v j ⊗ e zv + e − zv ⊗ v j , j = 1 , . More explictly, using the expressions given in (4.10), we have h (2) z ; j = h z ; j ( q , p ) e zh z ;2 ( q ,p ) + h z ; j ( q , p ) e − zh z ;2 ( q ,p ) , j = 1 , h (2) z ;2 = h z ;2 ( q , p ) + h z ;2 ( q , p ) . (4.18)So, to retrieve another constant of motion we can apply the trick of permuting indices. Then,here we have a second constant of motion, writing it implicitly,(4.19) F (2) z, (23) = shc (cid:16) zh (2) z ;2(23) (cid:17) h (2) z ;2(23) h (2) z ;3(23) − (cid:16) h (2) z ;1(23) (cid:17) , where the subindex (23) means that the variables ( q , p ) are interchanged with ( q , p ) whenthey appear in the deformed Hamiltonian functions h z ; j and h (2) z ; j (23) = h z ; j ( q , p ) e zh z ;2 ( q ,p ) + h z ; j ( q , p ) e − zh z ;2 ( q ,p ) , j = 1 , h (2) z ;2(23) = h z ;2 ( q , p ) + h z ;2 ( q , p ) . n (4.17), we have h z ;2 ( q , p ) = − shc (2 zh z ;2 ) q p , h z ;2 = 1 p − q p , h z ;3 = − p (cid:0) shc (2 zh z ;2 )2 q p + c (cid:1) shc (2 zh z ;2 )( q p − h z ;2 ( q , p ) = − shc (2 zh z ;2 ) q p , h z ;2 = 1 p − q p , h z ;3 = − p (cid:0) shc (2 zh z ;2 )2 q p + c (cid:1) shc (2 zh z ;2 )( q p − h z ;2 ( q , p ) = − shc (2 zh z ;2 ) q p , h z ;2 = 1 p − q p , h z ;3 = − p (cid:0) shc (2 zh z ;2 )2 q p + c (cid:1) shc (2 zh z ;2 )( q p − h z ;2 ( q , p ) = − shc (2 zh z ;2 ) q p , h z ;2 = 1 p − q p , h z ;3 = − p (cid:0) shc (2 zh z ;2 )2 q p + c (cid:1) shc (2 zh z ;2 )( q p − F (2) z, (23) = k and F (2) z = k , with k , k ∈ R , one is able to retrieve a superposition principle for q = q ( q , q , p , p , k , k )and p = p ( q , q , p , p , k , k ). Notice that here ( q , p ) and ( q , p ) are two pairs of partic-ular solutions and k , k are two constants over the plane to be related to initial conditions.5. Conclusions Here we present a summary of our new results, we conclude how they agree with the experi-mental data and how they clash with other models and results by other authors. Let us listour major results. Our results. • We have achieved a more general solution (1.6) for the SISf system, a SIS modelpermitting fluctuations, presented in (1.5) than the one provided in [43]. • We have limited the range of C and C in the solution (1.6) for system (1.5), so thatwe obtain sigmoid-type or hyperbolic-type like behavior, which is the one that we couldexpect for the growth of infected individuals ρ ( t ). This range is 3 C ‹ C ‹ C .For simplification, we choose C = 1. If one exceeds these limits, it is easy to spotsingularities and unexpected behavior of the function. This is precisely what we haverealized as we plotted the solutions given in [43]. We are still unsure whether theirsolution is an actual solution or if their choice of initial conditions are not the idealto overcome singularities and find the real regime. • The error σ decays with time, as the density of infected individuals stabilizes asconstant. This is what one expects intuitively. We have been able to retrieve general solutions for equations (1.5) making use ofgeometric methods. The theory of Lie systems has permitted us to find a generalsolution in terms of a particular solution and a challenging choice of the constantsof integration ( k , k ), in both cases, the linear approximation and the unlinearizedversion. • We have been able to retrieve a general solution for (1.5) making use of the theory ofLie–Hamilton systems and it is expressed in terms of two particular solutions and avery challenging choice of two constants ( c, k ). One can check the choices of constantsfor every case in the legends of the graphics. • It is interesting to notice that using the geometric methods of the Lie–Hamilton theory,one retrieves the introduced hamiltonian (1.9) by [43] as performed in (3.11). • Furthermore, we have generalized this hamiltonian (1.9) by introducing a deforma-tion parameter, know as the quantum deformation parameter using the Poisson–Hopfalgebra method. The deformed hamiltonian (4.14) gives rise to a new integrable SISmodel in which one includes a quantum deformation parameter (4.15). Relation with Covid19 pandemic. One may wonder how the current pandemic of COVID19could be related to a SISf-pandemic model. The SISf model is a very first approximation fora trivial infection process, in which there is only two possible states of the individuals in thepopulation: they are either infected or susceptible to the infection. Hence, this model does notprovide the possibility of adquiring immunity at any point. It seems that COVID19 providessome certain type of immunity, but only to a thirty percent of the infected individuals, hence,a SIR model that considers “R” for recuperated individuals (not susceptible anymore, i.e., im-mune) is not a proper model for the current situation. One should have a model contemplatingimmune and nonimmunized individuals. Unfortunately, we are still in search of a stochasticHamiltonian model including potential immunity and nonimmunity. Future work. As future work, we would like to extend our study to more complicatedcomparmental models, although at a first glance we have not been able to identify more Liesystems, at least in their current PDE form. We suspect that the Hamiltonian description ofthese compartmental models could nonetheless behave as a Lie system, as it has happened inour presented case. This shall be part of our future endeavors. Moreover, one could inspect inmore meticulous detail how the solutions of the quantumly-deformed system (4.15) recover thenondeformed solutions when the introduced parameter tends to zero. We need to further studyhow this precisely models a heat bath and if this new integrable system could correspond toother models apart from infectious models. We would like to figure out whether it is possibleto modelize subatomic dynamics with the resulting deformed hamiltonian (4.14).Apart from that, there exists a stochastic theory of Lie systems developed in [31] that couldbe another starting point to deal with compartmental systems. In the present work we were ucky to find a theory with fluctuations that happened to match a stochastic expansion, butthis is rather more of an exception than a rule. 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Consider the multiplication map m anda unit element map ι , given by(A.1) m : A ⊗ A −→ A, ι : R −→ A. We say that the triple ( A, m, ι ) is an algebra. This algebra is said to be associative if themultiplication is associative, i.e.,(A.2) m ( x, m ( x , x )) = m ( m ( x, x ) , x )for all x , y , and z in A . Otherwise it is referred to as a nonassociative algebra. A mappingbetween two algebras is called homomorphism if it respects the multiplicatios and unit fromone algebra to the other.An algebra is called a Lie algebra, denoted by g , if the multiplication is skew-symmetric andsatisfies the Jacobi identity [6, 24]. In this case, the multiplication operation is a Lie bracket[ • , • ]. The Jacobi identity is(A.3) [[ x, x ] , x ] + [[ x , x ] , x ] + [[ x , x ] , x ] = for all x , x and x in g . If this identity is fulfilled, then the nonassociativity of the multipli-cation is followed. A mapping between two algebras is called a Lie algebra homomorphism ifit preserves the Lie algebra structure.Let B be an arbitrary subspace of a Lie algebra g . We define a new set [ B, B ] by collectingall possible pairings of the elements in B . If the set [ B, B ] precisely equals B , that is, if B isclosed under the Lie bracket, we have a Lie subalgebra. In general, we compute the (possibleinfinite) hierarchy [ B, B ] , [ B, [ B, B ]] , [[ B, B ] , [ B, B ]] . . . consisting of all possible pairings. Bycontinuing in this way, we arrive at a collection that turns out to be a Lie subalgebra of g .Evidently, it is the smallest Lie subalgebra containing B which we denote Lie( B ).A.2. Tangent Bundle. Let N be an n -dimensional manifold (a locally Euclidean space). A differentiable curve on N through the point x is a function γ : R → Q with, say, γ (0) = x . Let us define now anequivalence relation in the set of differentiable curves passing through x . Two curves γ and ˜ γ are equivalent if they take the same value at x and if the directional derivative of functions long them at x are the same, namely,(A.4) γ (0) = ˜ γ (0) , ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f ◦ γ ) ( t ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f ◦ ˜ γ ) ( t )for all functions f : N → R . A tangent vector v ( x ) at x is an equivalence class of curves at x . The set of all equivalence classes of vectors, that is, the set of all tangent vectors at x istangent space T x N at x in N . It is possible to show that T x N admits an n -dimensional vectorspace structure.The union of all tangent spaces T x N in x ∈ N is(A.5) T N = G x ∈ N T x N and it is a 2 n -dimensional manifold that is called tangent bundle of N . From the point of viewof applications, if a manifold is the configuration space of a physical system, then, the tangentbundle is the velocity phase space of the system. That is, it consists of all possible positionsand all possible velocities. There is a projection, called tangent fibration , τ N mapping atangent vector to its base point(A.6) τ N : T N −→ N, v ( x ) x. Vector fields. A section of the tangent fibration is a vector field X on N mapping a vectorto each point in N , that is a map Y : N → T N such that τ N ◦ Y : id N where id N is theidentity map on N . The set of vector fields X ( N ) on a manifold Q admits a module structureover the ring of functions.(A.7) X ( N ) = { Y : N T N | Y ◦ τ N = id N } . An integral curve of a vector field X with initial condition x , is a curve γ passing through γ (0) = x so that(A.8) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 γ ( t ) = Y ( x ) . From the theorem of existence and uniqueness of solutions for ODEs, we know that for every x there exists a unique integral curve γ satisfying (A.8). The flow of Y is a smooth oneparameter group of diffeomorphisms defined by means of integral curves as follows(A.9) γ t : N −→ N, x γ ( t ) , where γ (0) = x . This realization of vector fields enables us to define the directional derivativeof a function f along a vector field Y as(A.10) Y : F ( N ) −→ F ( N ) , f Y ( f ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ( f ◦ γ )( t )where γ is an integral curve of Y . Notice that, if f is a constant of motion, then, Y ( f ) = 0. ie Algebras. The space of all vector fields, denoted by X ( N ), is a Lie algebra if it is endowedwith a Lie bracket. For any two vector fields Y and Y on the base manifold N , the Lie bracketreads(A.11) [ Y , Y ]( f ) = Y ( Y ( f )) − Y ( Y ( f )) , for all smooth functions f on N . Here, the notation Y ( f ) is simply the directional derivativeof the function f in the direction of Y . A distribution generated by a finite set of vector fields,say Y , ... Y r , is the collection of subspaces D x , generated by the vectors Y ( x ), ... Y r ( x ), inall points. It is called an involutive distribution if the Lie algebra is closed for the set Y , ... Y r . Local realization in D . Assume now that N is a two dimensional manifold with localcoordinates ( q, p ), then the tangent bundle T N admits the induced coordinates ( q, p ; ˙ q, ˙ p ). Inthis case, a vector field is written as(A.12) X = k ( q, p ) ∂∂q + g ( q, p ) ∂∂q where k and g are real valued functions on N . In this coordinate system, the dynamicalequations governed by X determine a system of nonautonomous ordinary differential equations(A.13) ˙ q = k ( q, p ) , ˙ p = g ( q, p ) . An integral curve to the vector field X in (A.12) is a solution of the system (A.13) given by q = q ( t ) and p = p ( t ).A.3. Cotangent Bundle - Canonical Symplectic Space. A tangent space T q Q at point q admits a (finite dimensional) vector space structure. So, thereexists the linear algebraic dual of T q Q , which we denote by T ∗ q Q . We call T ∗ x N a cotangentspace. By collecting all of them, we define the cotangent bundle(A.14) T ∗ Q = G q ∈ Q T ∗ q Q. There is a projection π Q from T ∗ Q to Q , mapping a covector α ( q ) to its base point q . Sections θ of the cotangent bundle are one-forms on Q satisfying π Q ◦ θ = id Q .2 D Symplectic Manifold. Assume now that N is a two dimensional planar manifold withcoordinates ( q, p ). For this case, we can understand N = T ∗ Q for some one-dimensional Q .There is a canonical symplectic two-form ω on N . A symplectic two-form is a closed, skew-symmetric and nondegenerate bilinear mapping taking two vector fields to the space of scalars,i.e., ω takes two vector fields Y and Y on N to a unique function. In this case, the symplectic wo-form is computed to be(A.15) ω = dq ∧ dp. It is the nondegeneracy of the symplectic two-form what makes the Hamilton equations (1.7)be realized in their local form (1.8). It assigns uniquely a vector field X h to a chosen (Hamil-tonian) function h . From the point of view of applications in classical mechanics, if the Hamil-tonian function h is the total energy, then X h describes the motion.A.4. Poisson Algebras. An associative algebra ( A, m, ι ) is called Poisson if it admits a (Poisson) bracket {• , •} satisfy-ing the Jacobi identity, so that the triple ( A, {• , •} , ι ) becomes a Lie algebra, and the Leibnizidentity is satisfied [56].(A.16) { m ( x, y ) , z } = m ( x, { y, z } ) + m ( { x, y } , z ) , We denote a Poisson algebra by a quadruple ( A, m, ι, {• , •} ). An element of the algebra iscalled Casimir, and denoted by C , if it commutes with all other elements under the Poissonbracket, that is if(A.17) { C, x } = 0 , ∀ x ∈ A. Poisson manifolds. A manifold, say P , is called Poisson if it is equipped with a skew-symmetric (Poisson) bracket {• , •} on the space of smooth functions on P satisfying both theJacobi identity (A.3) and the Leibniz identity (A.16), see [55]. A Hamiltonian vector field X h on N associated with a smooth (Hamiltonian) function h , is a unique vector field such thatthe following identity holds(A.18) X h ( g ) := { g, h } . The Jacobi identity for the Poisson bracket therefore entails that h X h is a Lie algebraanti-homomorphism between a Poisson algebra C ∞ ( N ) endowed with a Poisson bracket andspace of vector fields X ( N ) endowed with a Lie bracket. Notice that every symplectic manifoldis a Poisson manifold and in this case the Hamilton equation (1.7) takes the form of (A.18). Lie-Poisson bracket. Let g be a Lie algebra, and g ∗ be its linear algebraic dual space. g ∗ isa Poisson manifold, known as a Lie-Poisson manifold with the Lie-Poisson bracket(A.19) { h, g } ( z ) = − (cid:10) z, (cid:2) δhδz , δgδz (cid:3)(cid:11) , for any z in g ∗ , and any two function(als) h and g on g ∗ , [37]. Here, the bracket on the righthand side is the Lie algebra bracket on g . Let us note that δh/δz and δg/δz denote the partial for infinite dimensional cases Fr´echet) derivatives of the function(al)s. Under the assumptionof the reflexivity, they are elements of g . Poisson bivectors. Being a derivation for each factor, a Poisson structure determines aunique bivector field Λ ∈ Γ( ∧ T N ), that we call a Poisson bivector, such that(A.20) Λ( dh, dg ) := { h, g } . As a manifestation of the Jacobi identity, the Schouten–Nijenhuis bracket of the bivector[Λ , Λ] S equals zero. Conversely, every bivector field Λ on N satisfying the null condition givesrise to a Poisson structure. A bivector induces a unique bundle morphism from the space ofone-forms Ω ( N ) to the space of vector fields X ( N ) on N , that is,(A.21) ˆΛ : Ω ( N ) → X ( N ) , h β, ˆΛ( α ) i := Λ( α, β ) . Referring to this mapping, we define a Hamiltonian vector field (A.18), as follows(A.22) X h = − ˆΛ ◦ dh, where dh is the (exterior) derivative of h . The space of Hamiltonian vector fields induces anintegrable generalized distribution F Λ on N associated to Λ. Pointwisely, each fiber F Λ x ofthis distribution is defined to be the image space of the mapping ˆΛ, that is { X x | X ∈ Im ˆΛ } .Here, the leaves are symplectic manifolds with respect to the restrictions of Λ [55].A.5. Lie coalgebras. In order to arrive the formal definition of a coalgebra, one simply reverse the directions of the(multiplication and unit) arrows in (A.1). Accordingly, a vector space A is called a coalgebraif it admits a comultiplication and a counit given by∆ : A −→ A ⊗ A, ∆ z = z [1] ⊗ z [2] ,(cid:15) : A −→ R , (cid:15) ( z ) = 1(A.23)respectively. We ask that these operations must satisfy the relation(A.24) (Id ⊗ (cid:15) )∆( z ) = ( (cid:15) ⊗ Id)∆( z ) = z, ∀ z ∈ A, where Id is being the identity mapping. We denote a coalgebra by a triple ( A, ∆ , (cid:15) ). A mappingis called a coalgebra homomorphism if it respects the coalgebra structures. Lie coalgebras. A coalgebra A is called a Lie coalgebra [36, 38] if the comultiplication, thistime called Lie cobracket, satisfies the following two conditions z [1] ⊗ z [2] = − z [2] ⊗ z [1] ,z [1] ⊗ z [2][1] ⊗ z [2][2] + z [2][1] ⊗ z [2][2] ⊗ z [1] + z [2][2] ⊗ z [1] ⊗ z [2][1] = 0 , (A.25) here we code the elements in A from left to the right with numbers, for example,(A.26) z [1] ⊗ z [2][1] ⊗ z [2][2] := z [1] ⊗ ∆ z [2] . Notice that the first condition in (A.25) is dual of the skew symmetry whereas the secondcondition in (A.25) is dual of the Jacobi identity. Dual of a Lie (co)algebra. The dual of a Lie coalgebra admits a Lie algebra structure. TheLie algebra bracket on the dual space is defined by means of the following equality(A.27) [ • , • ] A ∗ : A ∗ ⊗ A ∗ −→ A ∗ , h [ x, x ] G ∗ , z i := ∆ z ( x, x ) , where the pairing on the left hand side is the one between A ∗ and A whereas the pairing onthe right hand side is between the tensor products A ∗ ⊗ A ∗ and A ⊗ A .Inversely, the dual g ∗ of a Lie algebra g is an immediate example of Lie coalgebras. In thiscase the cobracket is defined to be(A.28) ∆ : g ∗ −→ g ∗ ⊗ g ∗ , ∆ z ( x, x ) := h z, [ x, x ] i , where the bracket [ • , • ] on the right hand side is the Lie algebra bracket on g .A.6. Poisson–Hopf algebras. Consider a vector space admitting both an algebra structure ( A, m, ι ) and a coalgebra struc-ture ( A, ∆ , (cid:15) ). If these two structures are compatible, that is, either ∆ and (cid:15) are algebrahomomorphisms or m and ι are coalgebra homomorphisms, then A is a called a bialgebra[2, 20, 36]. We denote a bialgebra by a quintuple ( A, m, ι, ∆ , (cid:15) ). Poisson bialgebra. Consider a bialgebra ( A, m, ι, ∆ , (cid:15) ) where there is an associated Poissonbracket {• , •} with A . If ∆ : A A ⊗ A preserves the Poisson structure as well, then, thetuple ( A, m, ι, {• , •} , ∆ , (cid:15) ) is a Poisson bialgebra. In this picture the Poisson bracket on thetensor space A ⊗ A is defined to be(A.29) { ( x ⊗ x ) , ( x ⊗ x † ) } = { x, x } ⊗ m ( x , x † ) + m ( x, x ) ⊗ { x , x † } . Poisson–Hopf algebra. A bialgebra ( A, m, ι, ∆ , (cid:15) ) is called a Hopf algebra if there exist anantihomomorphism, known as the antipode γ : A −→ A , such that for every a ∈ A one gets:(A.30) m ((Id ⊗ γ )∆( a )) = m (( γ ⊗ Id)∆( a )) = (cid:15) ( a ) ι, see [22, 41, 53]. We also cite [47, 48, 52] for some recent topological and cohomological dis-cussions. We denote a Hopf algebra by the tuple ( A, m, ι, ∆ , (cid:15), γ ). If a bialgebra ( A, m, ι, ∆ , (cid:15) ) s both a Poisson and a Hopf algebra, then it is a Poisson–Hopf algebra and it is denoted bythe tuple ( A, m, ι, {• , •} , ∆ , (cid:15), γ ).The space of smooth functions C ∞ ( g ∗ ) on the dual of a Lie algebra g is a Hopf algebra relativeto its natural associative algebra with unit provided that m ( h ⊗ g )( z ) : = h ( z ) g ( z ) , ι (1)( z ) := 1 , ∆( f )( z, z ) : = f ( z + z ) (cid:15) ( f ) := f (0) , γ ( f )( z ) := f ( − z ) , (A.31)for every z, z in g ∗ , and f, g, h in C ∞ ( g ∗ ). Further, considering the Lie-Poisson bracket (A.19), C ∞ ( g ∗ ) turns out to be a Poisson–Hopf algebra. Symmetric (co)algebra. The symmetric algebra S ( g ) of a (finite dimensional) Lie algebra g is the smallest commutative algebra containing g . To reach such algebra, we do the following.The second tensor power g ⊗ g of the Lie algebra is the space of real valued bilinear maps onthe dual space. Iteratively, the kth tensor power g ⊗ k is the space of real valued k-linear maps.Taking the direct sum of the tensor powers of all orders, we arrive at the tensor algebra Tg of g . Here, the multiplication is(A.32) Tg × Tg −→ Tg , ( v, u ) v ⊗ u. We consider a basis { x , . . . , x r } of the Lie algebra g . The space generated by the elements(A.33) x i ⊗ x j − x j ⊗ x i is an ideal, denoted by R , of the tensor algebra Tg . The quotient space Tg / R is called asymmetric algebra and denoted by S ( g ). The elements of S ( g ) can be regarded as polynomialfunctions on g ∗ , so, we can endow it with the Lie-Poisson bracket (A.19) that makes S ( g ) aPoisson algebra. One can show that S ( g ) can always be endowed with a coalgebra structureby introducing the comultiplication(A.34) ∆ : S ( g ) → S ( g ) ⊗ S ( g ) , ∆( x ) = x ⊗ ⊗ x, ∀ x ∈ g ⊂ S ( g ) , which is a Poisson algebra homomorphism. This makes S ( g ) a Poisson-Hopf algebra. Further-more, in the light of the coassociatity condition(A.35) ∆ (3) := (∆ ⊗ Id) ◦ ∆ = (Id ⊗ ∆) ◦ ∆ , we can define the third-order coproduct(A.36) ∆ (3) : S ( g ) → S ( g ) ⊗ S ( g ) ⊗ S ( g ) , ∆ (3) ( x ) = x ⊗ ⊗ ⊗ x ⊗ ⊗ ⊗ x for all x ∈ g , where g is understood as a subset of S ( g ). The m th-order coproduct map canbe defined, recursively, as(A.37) ∆ ( m ) : S ( g ) → S ( m ) ( g ) , ∆ ( m ) := ( ( m − − times z }| { Id ⊗ . . . ⊗ Id ⊗ ∆ (2) ) ◦ ∆ ( m − , m › , hich, clearly, is also a Poisson algebra homomorphism. Universal enveloping algebras. We consider once more the tensor algebra Tg of a Liealgebra g with a basis { x , . . . , x r } . We now define the space generated by the elements(A.38) x i ⊗ x j − x j ⊗ x i − [ x i , x j ] . This space is an ideal, denoted by L , of the tensor algebra. The quotient space Tg / L is calleduniversal enveloping algebra of g , and it is denoted by U ( g ). U ( g ) is the biggest associativealgebra containing all possible representations of g . See that, if the Lie algebra bracket on g is trivial, then, the universal enveloping algebra U ( g ) is equal to the symmetric algebra S ( g ).).