Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations

The theory of Lie–Hamilton systems is used to construct generalized time-dependent SIS epidemic Hamiltonians with a variable infection rate from the ‘book’ Lie algebra. Although these are characterized by a set of non-autonomous nonlinear and coupled differential equations, their corresponding exact solution is explicitly found. Moreover, the quantum deformation of the book algebra is also considered, from which the corresponding deformed SIS Hamiltonians are obtained and interpreted as perturbations in terms of the quantum deformation parameter of previously known SIS systems. The exact solutions for these deformed systems are also obtained.


Introduction
In this contribution, we make use of the theory of Lie-Hamilton (LH) systems [1][2][3][4][5] in order to construct generalized time-dependent SIS epidemic models with a variable infection rate.We recall that LH systems form a particular class within Lie systems [5][6][7][8][9], due to the existence of Hamiltonian vector fields with respect to a Poisson structure.
In particular, we shall consider in the following the two-dimensional book Lie algebra b 2 = span{v A , v B } with Lie bracket [v A , v B ] = −v B , corresponding to the class I r=1 14A ≃ R ⋉ R in the classification of LH systems on R 2 [3,4].It is thus ensured that any Lie system with two vector fields satisfying the above Lie bracket belongs to this class in the classification and that the Lie system is locally diffeomorphic to b 2 .In this sense, we stress that the non-trivial issue is to obtain explicit local diffeomorphisms between LH systems belonging to the same class.It is worth mentioning that b 2 -LH systems have already been studied in various mathematical, physical and biological contexts such as • Generalised Buchdahl equations, i.e., second-order differential equations arising in the study of relativistic fluids [10][11][12], were shown in [3,4] to be b 2 -LH systems.These systems have also been studied from a Lagrangian approach in [13].
• Complex Bernoulli differential equations with t-dependent real coefficients [16]; these are particular cases of the non-autonomous complex Bernoulli differential equations with complex coefficients [17,18].In an LH framework, these have been analyzed in [4,19,20].
• Quite recently, b 2 -LH systems have also been shown to underlie the description of generalized stochastic SIS epidemic models with variable infection rates [21,22], which is the case that we consider in the following.
In the next section, we construct the LH systems coming from the book algebra in a 'canonical' form.Furthermore, we obtain their exact solution that depends on two arbitrary t-dependent coefficients.In section 3, these results are applied to t-dependent SIS models by deducing an appropriate explicit diffeomorphism (change of variables) with the canonical expressions.In particular, we review the main results presented in [22], thus generalizing the SIS Hamiltonians previously studied in [21] and [23].
In section 4 , we apply the formalism of Poisson-Hopf deformations of LH systems proposed in [19,24,25] to the book algebra.We consider the quantum deformation of b 2 and obtain the corresponding deformed LH systems, deducing their exact solution.In section 4.1, as a new result that complements those of [22], we obtain deformed SIS differential equations together with their general solution.

Lie-Hamilton systems from the book algebra
Let us consider the so-called two-dimensional book Lie algebra b We can interpret v A as a dilation generator, while v B corresponds to a translation one.As happens with any Lie algebra, b 2 can be endowed with a trivial Hopf algebra structure [26,27] For any algebra A, this map must be an algebra homomorphism and satisfy the coassociativity condition in A ⊗ A ⊗ A: such that the pair (A, ∆) defines a coalgebra.These properties are trivially fulfilled for any Lie algebra with non-deformed coproduct (1).
We introduce a symplectic representation D of b 2 in terms of canonical variables (x, y) and the standard symplectic form ω = dx ∧ dy defined by [19] h where the Hamiltonian functions fulfill the following Poisson bracket (with respect to ω): Using (3), we can easily deduce a differential representation of b 2 in terms of Cartesian coordinates (x, y) ∈ R 2 through the contraction or inner product operation, ι X i ω = dh i , that provides the following vector fields By construction, these verify the invariance condition for the Lie derivative: From the covariant or contravariant approach, we obtain either a t-dependent Hamiltonian or a t-dependent vector field depending on two real arbitrary parameters b A (t) and b B (t): that both lead to the same system of non-autonomous ODEs on R 2 : It is straightforward to verify that, due to the relation ( 5), the ODEs (7) possess the structure of a Lie system [5][6][7][8][9] with associated t-dependent vector field X t (6) and Vessiot-Guldberg Lie algebra isomorphic to b 2 .By the Lie-Scheffers Theorem, this guarantees that the system always admits a fundamental system of solutions, that is, a superposition rule.As (7) is separable in the coordinates (x, y), the latter property is not required, and the system can be solved explicitly by quadratures, with the exact solution given by where c 1 and c 2 are the two constants of integration, determined by the initial conditions, while a is a real number that ensures the existence of the integrals over the interval [a, t].Now, for any (local) diffeomorphism, i.e., a change of variables, such that a b 2 -LH system is expressed in variables different from (x, y), the general solution can be derived directly using formula (8).This strategy was used in [22] in order to obtain exact solutions for generalized stochastic SIS epidemic models, which, in the proper basis, correspond to a set of non-autonomous nonlinear and coupled differential equations.Additionally, the very same procedure has also been applied to complex Bernoulli differential equations with t-dependent real coefficients deducing their exact solution in [20].

Applications to generalized time-dependent SIS Hamiltonian models
In this section we briefly review the main results concerning the application of the 'abstract' b 2 -LH algebra described in section 2 to Hamiltonian systems that generalize SIS epidemic models with a time-dependent infection rate (see [22] for details).

A time-independent SIS Hamiltonian model
The starting point is the SIS model introduced by Nakamura and Martinez in [23].For more details on the use of differential equations to study the population dynamics of infectious diseases we refer to [28][29][30][31][32][33].In these models, all individuals are still susceptible to the infection after recovery, which means that they do not acquire immunity.Therefore, the model can be described only using two compartmental variables.The first one, 'I', corresponds to the number of infected individuals, meanwhile the second compartment 'S' describes the number of individuals susceptible to the infection at a given time.Under certain assumptions [22,23], the differential equation that determines the density ρ = ρ(t) of infected individuals can be written as where ρ 0 is a constant that corresponds to the equilibrium density.Then, following [23], we introduce fluctuations in the SIS model in such a manner that the spreading of the disease is interpreted as a Markov chain in discrete time, with at most one single recovery or transmission occurring in each infinitesimal interval.By introducing the mean density of infected individuals ρ and the variance σ 2 = ρ 2 − ρ 2 [34], assuming that σ becomes negligible when compared to ρ and ignoring higher statistical moments, for a sufficiently large number of individuals we are led to the system which corresponds to a stochastic expansion, according to [35].In this situation, we assume that the density ρ = ρ + η is properly described by the instantaneous average, as well as some noise function η.For consistency, it is further assumed that η = 0 and η 2 = σ 2 .The system (10) allows a Hamiltonian formulation, with the phase space variables given by the mean density of infected individuals ρ and the variance σ 2 .Defining as dynamical variables [23], the system (10) now adopts the form These are the canonical equations associated to the Hamiltonian given by Both the systems ( 12) and ( 10) can be explicitly integrated; the exact solution and a detailed analysis of all of the above results can be found in [23].

Time-dependent SIS Hamiltonian models
A generalization of the Hamiltonian (13) was proposed in [21], by considering a t-dependent infection rate through a smooth function ρ 0 (t).The latter amounts to introduce a t-dependent basic reproduction number R 0 (t), a fact that is actually observed in more accurate epidemic models [36][37][38].The remarkable point is that the resulting t-dependent Hamiltonian inherits the structure of an LH system [1][2][3][4][5].Furthermore, it is possible to extend such t-dependent SIS Hamiltonian by considering a second arbitrary t-dependent parameter b(t) in the form [22] leading to the following system of differential equations: The associated t-dependent vector field is given by fulfilling the commutation rule (5).We conclude that, for any choice of the parameters ρ 0 (t) and b(t), the differential equations (15) determine an LH system with associated b 2 -LH algebra, including the SIS epidemic models studied in [21] for b(t) ≡ 1, as well as the model (12) introduced in [23] for ρ 0 (t) ≡ ρ 0 and b(t) ≡ 1.
The classification of LH systems on the plane [3,4] implies that there must exist a canonical transformation between the variables (q, p) in ( 15) and (x, y) in (7).This reads as (see [22]) while keeping the canonical symplectic form ω = dx ∧ dy = dq ∧ dp.Observe that the change of variables considered in [21] is not canonical, so that the variables q and p used there do not correspond to the mean density of infected individuals ρ and the variance σ 2 (11).
Taking into account the results presented in section 2, we directly obtain the exact solution for the book SIS Hamiltonian model (15).Explicitly, if we start with the general solution for any book LH system (8), apply the transformation (17), and identify the 'abstract' t-dependent coefficients as b A (t) ≡ ρ 0 (t) and b B (t) ≡ b(t), we find that Hence, the exact solution of the t-dependent SIS system considered in [21] arises by simply setting b(t) ≡ 1 and keeping a variable ρ 0 (t) which, in fact, was not presented in that work.Moreover, if we fix b(t) ≡ 1, consider a constant ρ 0 (t) ≡ ρ 0 , and choose a = 0 in the integral Θ(t) (i.e.Θ(t) = ρ 0 t), we recover the solution of the initial t-independent SIS model ( 12), namely 4 Deformed Lie-Hamilton systems from the quantum book algebra In this section, we apply the formalism of Poisson-Hopf deformations of LH systems proposed in [19,24,25] to the book algebra.This will allow us to obtain all deformed LH systems coming from a quantum deformation of b 2 .We will later particularize these results to obtain novel deformed SIS differential equations, that by the preceding discussion admit an exact solution.
We consider the coboundary quantum deformation of the book Lie algebra b 2 coming from the classical r-matrix r = zv A ∧ v B , which is a solution of the classical Yang-Baxter equation, and where z is the quantum deformation parameter such that q = e z .For quantum deformations of the three-dimensional book algebra b 3 , we refer to [39].The quantum book algebra U z (b 2 ) ≡ b z,2 , is defined by the following deformed coproduct, fulfilling the coassociativity property (2), and compatible deformed commutation relation: A deformed symplectic representation D z of b z,2 (20), in terms of the canonical variables (x, y) of section 2 and keeping the canonical symplectic form ω, is given by with the following deformed Poisson bracket with respect to ω: The corresponding vector fields turn out to be spanning a smooth distribution in the sense of Stefan-Sussmann [40][41][42] Hence, we arrive at a deformed t-dependent Hamiltonian and vector field depending on two real arbitrary parameters b A (t) and b B (t) as follows (compare with (6)): Either one gives rise to the following system of non-autonomous nonlinear and coupled ODEs on generalizing (7).Therefore, this result can be seen as the general 'canonical' system of differential equations for the set of deformed LH systems based in the quantum book algebra b z,2 (20).
Observe that the expressions ( 21)-( 26) reduce to (3)-( 7) under the (classical) non-deformed limit z → 0. The presence of the quantum deformation parameter z can be regarded as the introduction of a perturbation in the initial LH system (7), in such a manner that a nonlinear interaction or coupling between the variables (x, y) in the deformed LH system (26) arises through the term e zx y.This fact can be clearly appreciated by taking a power series expansion in z of ( 26) and truncating at the first-order, namely which holds for a small value of z.In this approximation, we find that z introduces a quadratic term x 2 into the first equation, so becoming a standard Bernoulli equation, while it introduces a nonlinear interaction with the xy term in the second equation.The exact solution for (26) can be derived by solving the first differential equation, in which only x is involved, and then substituting this result into the second equation, yielding (see [20]) where, as in ( 8), c 1 and c 2 are the two constants of integration and a is a real number that ensures the existence of the integrals over the interval [a, t].We stress that this result encompasses the solution of the quantum deformation of all the specific b 2 -LH systems mentioned in the introduction, provided that a proper diffeomorphism/change of variables is known.

Applications to deformed time-dependent SIS Hamiltonian models
As a new result, not covered in [22], we apply the above quantum deformation to the t-dependent SIS Hamiltonian (14), obtaining its generalization in terms of the quantum deformation parameter z.We introduce the canonical change of variables (17), which preserves the interpretation (11), into the deformed Hamiltonian vector fields (21), finding that The deformed Hamiltonian (25) with b A (t) ≡ ρ 0 (t) and b B (t) ≡ b(t) leads to the system of differential equations given by As a byproduct, this system covers the quantum deformation of the SIS Hamiltonian studied in [21] for b(t) ≡ 1, together with the deformed counterpart of (12) given in [23] for ρ 0 (t) ≡ ρ 0 and b(t) ≡ 1.The associated deformed vector fields X z,i can be easily deduced from (30).
The exact solution of the system (30) is explicitly deduced from the change of variables (17) and the solution (28) for deformed book systems in the coordinates (x, y); the precise expressions are given by q(t) = e −Γ(t) c 2 + Despite the cumbersome expressions for the novel deformed SIS system (30) and its exact solution (31), one can always interpret z as a 'small' perturbation parameter, as in (27), which yields a first-order z-perturbation of the t-dependent SIS system (15) in the form dq dt ≃ ρ 0 (t)q − b(t) q 2 + 1 p 2 + zρ 0 (t)q 3 p, To end with, it is worth stressing that the same approach followed here for t-dependent SIS models with a b 2 -LH algebra symmetry, and its Poisson-Hopf deformation b z,2 , can be applied to other book LH systems such as generalized Buchdahl equations [10][11][12][13] and Lotka-Volterra systems with t-dependent coefficients [14,15], as well as certain higher dimensional algebras that contain b 2 , such as the oscillator algebra h 4 , for which exact solutions can also be derived, along the same lines.A detailed case by case analysis of exact solutions for these systems and their applications is currently in progress.