Lie-Hamilton systems and their role in the current Covid pandemic - - PowerPoint PPT Presentation

Lie-Hamilton systems and their role in the current Covid pandemic

Cristina Sard´

• n

UPM-ICMAT, Spain Online Friday Fish Seminar August 2020

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Contents

• Why are we interested in Lie systems?
• How are Lie systems related to the current pandemic?
• Brief recall of geometric fundamentals for Lie systems
• What is a Lie system? Their solutions as nonlinear superposition rules
• Which Lie systems are Hamiltonian?
• Remarkable geometric properties of Lie–Hamilton systems. Retrieving a

solution as a (nonlinear) superposition rule through the coalgebra method.

• Can we write down a Hamiltonian formulation for a pandemic model

using the Lie–Hamilton theory?

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Motivation

Properties and applications

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Why Lie systems?

... because Lie systems are first-order ODEs that admit general solutions in form of (generally nonlinear) superposition rules or functions Φ : Nm × N → N

• f the form x = Φ(x(1), . . . , x(m); k) allowing us to write the general solution as

x(t) = Φ(x(1)(t), . . . , x(m)(t); k), where x(1)(t), . . . , x(m)(t) is a generic family of particular solutions and k ∈ N. They enjoy a plethora of geometric properties:

• Finite dimensional Lie algebras
• Lie group actions
• The Poisson coalgebras
• They are compatible with multiple geometric structures (Dirac, Jacobi,

contact...)

• Superposition rules can be interpreted as zero-curvature connections

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Why Lie systems? II

Lie systems appear in the study of:

• Relevant physical models
• Mathematics
• Control theory
• Quantum Mechanics
• Biology and ecology. Predator-prey systems, viral dynamics...

Are there any epidemic models that are Lie systems? Lie systems and superposition rules have been extrapolated to higher-order systems of ODEs.

• Higher-order Riccati equation,
• The second- and third-order Kummer–Schwarz,
• Milne–Pinney and dissipative Milne–Pinney equations.

Lie systems have also been extended to the realm of PDEs, the so-called PDE-Lie systems.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie systems

Geometric Fundamentals

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Definitions

Lie algebra is a pair (V , [·, ·]), where V is a linear space equipped with Lie bracket [·, ·] : V × V → V . We define by Lie(B, V , [·, ·]), B ⊂ V , the smallest Lie subalgebra of (V , [·, ·]) containing B , namely the linear space generated by B and [B, B], [B, [B, B]], [B, [B, [B, B]]], [[B, B], [B, B]], . . . . A t-dependent vector field is the map X : R × N → TN such that the following diagram is commutative TN

π

• R × N

X

• π2

N

X(t, x) ∈ π−1(x) = TxN Thus, the maps Xt : x ∈ N → X(t, x) ∈ TN are {Xt}t∈R.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Definitions

We call integral curve of X, the integral curve of its suspension. For every γ : t ∈ R → (t, x(t)) ∈ R × N we have an associated system d(π2 ◦ γ) dt (t) = (X ◦ γ)(t) So, X determines a single first-order ODE. Conversely, given such system, there exists a unique X whose integral curves (t, x(t)) are its particular solutions. Minimal Lie algebra of a t-dependent vect. field X is the smallest real Lie algebra of vector fields V X containing {Xt}t∈R, namely, V X = Lie({Xt}t∈R, [·, ·]).

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie–Scheffers theorem

Theorem

(The Lie–Scheffers Theorem) A first-order system dx dt = F(t, x), x ∈ N, admits a superposition rule if and only if X can be written as Xt =

r

• α=1

bα(t)Xα for a certain family b1(t), . . . , br(t) of t-dependent functions and a family X1, . . . , Xr of vector fields on N spanning an r-dimensional real Lie algebra of vector fields V X. A Vessiot–Guldberg Lie algebra (VG henceforth).

Theorem

(The abbreviated Lie–Scheffers Theorem) A system X admits a superposition rule if and only if V X is finite-dimensional.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Examples of Lie systems

The Riccati equation on the real line dx dt = a0(t) + a1(t)x + a2(t)x2, where a0(t), a1(t), a2(t) are arbitrary t-dependent functions, admits the superposition rule Φ : (x(1), x(2), x(3); k) ∈ R3 × R → x ∈ R given by x(t) = x(1)(t)(x(3)(t) − x(2)(t)) + kx(2)(t)(x(1)(t) − x(3)(t)) (x(3)(t) − x(2)(t)) + k(x(1)(t) − x(3)(t)) . The first-order Riccati equation X = a0(t)X1 + a1(t)X2 + a2(t)X3, where X1 = ∂ ∂x , X2 = x ∂ ∂x , X3 = x2 ∂ ∂x span a VG isomorphic to sl(2, R). Then, Φ : (x(1), x(2), x(3); k) ∈ R3 × R such that it provides us with a solution x ∈ R

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie systems on Lie groups

Every Lie system X associated with a VG gives rise by integrating V X to a (generally local) Lie group action ϕ : G × N → N whose fundamental vector fields are the elements of V and such that TeG ≃ V with e being the neutral element of G. x(t) = ϕ(g1(t), x0), x0 ∈ Rn, with g1(t) being a particular solution of dg dt = −

r

• α=1

bα(t)X R

α (g),

where X R

1 , . . . , X R r

is a certain basis of right-invariant vector fields on G such that X R

α (e) = aα ∈ TeG, with α = 1, . . . , r, and each aα is the element of TeG

associated with the fundamental vector field Xα. Since X R

1 , . . . , X R r

span a finite-dimensional real Lie algebra, the Lie–Scheffers Theorem guarantees there exists a superposition rule and becomes a Lie system.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

A contemporary application

Lie systems in viral infection dynamics

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

A primitive viral infection

Finally, let us consider a simple viral infection model given by      dx dt = (α(t) − g(y))x, dy dt = β(t)xy − γ(t)y, where g(y) is a real positive function taking into account the power of the

• infection. Note that if a particular solution satisfies x(t0) = 0 or y(t0) = 0 for a

t0 ∈ R, then x(t) = 0 or y(t) = 0, respectively, for all t ∈ R. As these cases are trivial, we restrict ourselves to studying particular solutions within R2

x,y=0 = {(x, y) ∈ R2 | x = 0, y = 0}.

The simplest possibility consists in setting g(y) = δ, where δ is a constant. Then, (21) describes the integral curves of the t-dependent vector field Xt = (α(t) − δ)X1 + γ(t)X2 + β(t)X3, on R2

x,y=0, where the vector fields

X1 = x ∂ ∂x , X2 = −y ∂ ∂y , X3 = xy ∂ ∂y , close a finite-dimensional Lie algebra. So, X is a Lie system related to a Vessiot–Guldberg Lie algebra V ≃ R ⋉ R2 where X1 ≃ R and X2, X3 ≃ R2.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

The SIS model with fluctuations

This is the susceptible-infectious-susceptible (SIS) epidemic model. There are

• nly two states: infected or susceptible (no immunization).

The instantaneous density of infected individuals ρ(t) taking values in [0, 1] and the fluctuations have been neglected. The density of infected individuals decreases with rate γρ, where γ is the recovery rate, and the rate of growth of new infections is proportional to αρ(1 − ρ), where the intensity of contagion is given by the transmission rate α. dρ dt = αρ(1 − ρ) − γρ. (1) One can redefine the timescale as τ ≡ αt and ρ0 ≡ 1 − γ/α, so we can rewrite (1) as dρ dτ = ρ(ρ0 − ρ) (2) The equilibrium density is reached if ρ = 0 or ρ = ρ0. This model involves random mixing and large population assumptions. To add fluctuations, we need to introduce stochastic mechanics, since temporal fluctuations can drastically alter the prevalence of pathogens and spatial heterogeneity.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

“Stochastic” considerations

In Hamiltonian dynamics of the SIS epidemic model with stochastic fluctuations, (G.M. Nakamura, A.S. Martinez), the spreading of the disease is assumed as a Markov chain in discrete time δt in which at most one single recovery

• r transmission occurs in the duration of this infinitesimal interval.

dPµ dt = −

2N −1

• ν=0

HµνPν (3) The instantaneous probability of finding the system in the µ-configuration is expressed by Pµ(t) and the configuration label follows the rule µ = n0 · 20 + n1 · 21 + · · · + nN−1 · 2N−1, where nk = 1 if the k-th agent is infected and nk = 0 if it is not. The matrix elements Hµν express the transition rates from configuration ν to configuration µ. From these considerations, to evaluate the time evolution of relevant statistical moments of ρ(t), we take the average density of infected agents ρ(t) = 1 N

2N −1

• µ=0

N−1

• k=0

µ|nk |µPµ(t) (4) The first two equations for instantaneous mean dentisity of infected people ρ and variance σ2 = ρ2 − ρ2 are dρ dτ = ρ (ρ0 − ρ) − σ2(τ), dσ2 dτ = 2σ2 (ρ0 − ρ) − ∆3(τ) − 1 N ρ(1 − ρ) + γ Nα ρ (5) where ∆3(τ) = ρ3(τ) − ρ(τ)3.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Hamiltonization of SIS w/ fluctuations

We can ”Hamiltonize” the previous eq. when σ(τ) becomes irrelevant compared to ρ. We only require mean and variance, neglecting higher statistical moments and(N ≫ 1). d ln ρ dτ = ρ0 − ρ − σ2 ρ, 1 2 d ln σ2 dτ = ρ0 − 2ρ. (6) We can define dynamical variables that describe a Hamiltonian system when q = ρ and p = 1/σ while preserving the independent variable τ. So, dq dτ = qρ0 − q2 − 1 p2 , dp dτ = −pρ0 + 2pq. (7) System (8) is a Hamiltonian system coming from the Hamiltonian function H = qp (ρ0 − q) + 1

p .

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Solutions SISf

We employ the abbreviation SISf for system (8) to differentiate it from the classical SIS model in (2). The letter “f” accounts for “fluctuations”. dq dt = qρ0 − q2 − 1 p2 , dp dt = −pρ0 + 2pq. (8) We have computed the general solution to this system as q(t) = ρeρt(C1ρ2 − 4)eρt + 2C1C2ρ2 (C 2

1 ρ2 − 4)e2ρt + 4C2ρ2(C1eρt + C2),

(9) p(t) = C1 + C 2

1 ρ2 − 4

4ρ2 − C2 + C2e−ρt (10) We present three different choices of three particular solutions and their corresponding graphs according to the change of variables q =< ρ > and p = 1/σ.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Particular solution 1

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Particular solution 2

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Particular solution 3

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

SISf Lie system

The previous equations can be generalised to a model represented by a time-dependent vector field Xt = ρ0(t)X1 + X2, X1 = q ∂ ∂q − p ∂ ∂p , X2 =

• −q2 − 1

p2 ∂ ∂q + 2qp ∂ ∂p . The generalisation comes from the fact that ρ0(t) is no longer a constant, but it can evolve in time. A direct calculation shows that [X1, X2] = X2, which means that the Lie algebra spanned by X1, X2 is a finite dimensional Lie algebra. This means that the pandemic SIS model admits a solution in terms of a superposition rule. In the next slides we will derive its solution in terms of a superposition principle using the geometric properties of Lie systems. Let us start with it in the next section.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie systems

and superposition rules

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Derivation of superposition rules

Definition

Given a t-dependent vector field X on N, its diagonal prolongation X to N(m+1) is the unique t-dependent vector field on N(m+1) such that

• Given pr : (x(0), . . . , x(m)) ∈ N(m+1) → x(0) ∈ N, we have that pr∗

Xt = Xt ∀t ∈ R.

X is invariant under the permutations x(i) ↔ x(j), with i, j = 0, . . . , m. In coordinates, we have that Xj =

n

• i=1

X i(t, x) ∂ ∂xi ⇒ X =

m+1

• j=0

Xj =

m+1

• j=0

n

• i=1

X i(t, x) ∂ ∂xi .

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Derivation of superposition rules

• Take a basis X1, . . . , Xr of a Vessiot–Guldberg Lie algebra V associated

with the Lie system.

• Choose the minimum integer m so that the diagonal prolongations to Nm
• f X1, . . . , Xr are linearly independent at a generic point.
• Obtain n functionally independent first-integrals F1, . . . , Fn common to all

the diagonal prolongations, X1, . . . , Xr, to N(m+1), for instance, by the method of characteristics. We require such functions to hold that ∂(F1, . . . , Fn) ∂

• (x1)(0), . . . , (xn)(0)

= 0.

• Assume that these integrals take certain constant values, i.e., Fi = ki

with i = 1, . . . , n, and employ these equalities to express the variables (x1)(0), . . . , (xn)(0) in terms of the variables of the other copies of N within N(m+1) and the constants k1, . . . , kn. The obtained expressions constitute a superposition rule in terms of any generic family of m particular solutions and n constants.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Applications of superposition rules in:

SISf pandemic systems

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Superposition rule for SISf model

The model (8) can be generalized to a model represented by a time-dependent vector field Xt = ρ0(t)X1 + X2 (11) where the constitutive vector fields are computed to be X1 = q ∂ ∂q − p ∂ ∂p , X2 =

• −q2 − 1

p2 ∂ ∂q + 2qp ∂ ∂p . (12) Let us apply the steps introduced in the previous section one by one to arrive at the general solution. Step 1. For the vector fields in (12), a direct calculation shows that the Lie bracket [X1, X2] = X2 (13) is closed within the Lie algebra. This implies that the SISf model (8) is a Lie

• system. The Vessiot-Guldberg algebra spanned by X1, X2 is an imprimitive Lie

algebra of type I14

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Step 2. If we copy the configuration space twice, we will have four degrees of freedom (q1, p1, q2, p2) and we will archieve precisely two first-integrals in vinicity of the Fr¨

• benius theorem. A first-integral for Xt has to be a

first-integral for X1 and X2 simultaneously. We define the diagonal prolongation

• X1 of the vector field X1 and we look for a first integral F1 such that

X1[F1] vanishes identically. Notice that if F1 is a first-integral of X1 then it is of X2 due to the commutation relation.

• X1 = q1 ∂

∂q1 + q2 ∂ ∂q2 − p1 ∂ ∂p1 − p2 ∂ ∂p2 (14) through the following characteristic system dq1 q1 = dq2 q2 = dp1 −p1 = dp2 −p2 . (15) Fix the dependent variable q1 and obtain a new set of dependent variables, say (K1, K2, K3), which are computed to be K1 = q1 q2 , K2 = q1p1, K3 = q1p2. (16)

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Step 3. This induces the following basis in the tangent space ∂ ∂K1 = q2 ∂ ∂q1 − q2p1 q1 ∂ ∂p1 − q2p2 q1 ∂ ∂p1 , ∂ ∂K2 = 1 q1 ∂ ∂p1 , ∂ ∂K3 = 1 q1 ∂ ∂p2 . (17) provided that q1 is not zero. Introducing the coordinate changes exhibited in (16) into the diagonal projection X2 of the vector field X2, we arrive at the following expression

• X2 =
• 2K1 −
• 1 + 1

K 2

1

• ∂

∂K1 + 1 K 2

2

+ 1 K 2

3

• K 2

2 −

• 1 + 1

K 2

1

• K2
• ∂

∂K2 +

• 2K3

K2 −

• 1 + 1

K 2

1

• K3
• ∂

∂K3 . To integrate the system once more, we use the method of characteristics again and obtain d ln |K1| 1 −

1 K2

1

= d ln |K2|

1 K2 + K2 K3 − 1 − 1 K2

1

= d ln |K3|

2 K2 −

• 1 +

1 K2

1

. (18)

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Superposition principle SISf pandemic model

Exact solution. We obtain two first integrals by integrating in pairs (K1, K2) and (K1, K3), where we have fixed K1. After some cumbersome calculations we

• btain

K2 = K1

• 4k2

2K 2 1 + 4k1k2K1 + k2 1 − 4

• 2(K1 + 1)(K1 − 1)k2(2k2K1 + k1),

K3 = K1

• k2K 2

1 + k1K1 + k2

1 −4

4k2

• (K1 + 1)(K1 − 1)

. (19) By substituting back the coordinate transformation (16) into the solution (19) (please notice the difference between capitalized constants (K1, K2, K3) and lower case constants (k1, k2), we arrive at the following implicit equations q1 = − q2

• k1k2 ±
• 4k2

2p2 2q2 2 + k2 1k2p2q2 − 4k3 2p2q2 − 4k2p2q2 + 4k2 2

• 2k2(−p2q2 + k2)

p1 = 4q2

1k2 2 + 4q1q2k1k2 + q2 2k2 1 − 4q2 2

2k2(2q3

1k2 + q2 1k1q2 − 2q1k2q2 2 − k1q3 2).

(20) Let us notice that the equations (20) depend on a particular solution (q2, p2) and two constants of integration (k1, k2) which are related to initial conditions.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Let us show now the graphs and values of the initial conditions for which the solution reminds us of sigmoid behavior, which is the expected growth of ρ(t). As particular solution for (q2, p2), we have made use of particular solution 2 given in some previous slide through its corresponding values of q, p through the change of variables q =< ρ > and p = 1/σ. The error graph since it gives a constant zero graph because k1 → 0

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie–Hamilton systems

Superposition rules with the coalgebra method

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie–Hamilton systems

Many instances of relevant Lie systems possess VG of Hamiltonian vector fields with respect to a Poisson structure. Such Lie systems are hereafter called Lie–Hamilton systems. Every Lie–Hamilton system can be interpreted as a curve in finite-dimensional Lie algebra of functions (with respect to a certain Poisson structure). Additionally, Lie–Hamilton systems appear in the analysis of relevant physical and mathematical problems,

• second-order Kummer–Schwarz equations
• Riccati equations
• Some viral infections
• Lotka-Volterra, Predator-Prey models and many others...

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

A geometric reminder

A Poisson algebra is (A, ⋆, {·, ·}), A linear space, ⋆ : A × A → A, (A, ⋆) is an associative R-algebra and {·, ·} a Lie bracket on A, the so-called Poisson bracket with (A, {·, ·}) a Lie algebra and also the Leibnitz rule. {b ⋆ c, a} = b ⋆ {c, a} + {b, a} ⋆ c, ∀a, b, c ∈ A. Given a manifold N, the pair (N, {·, ·}) is a Poisson manifold such that (C ∞(N), ·, {·, ·}) is a Poisson algebra. As {·, f } with f ∈ C ∞(N) is a derivation on (C ∞(N), ·), there exists an unique vector field Xf on N the hamiltonian vector field assoc. with f such that Xf g = {g, f }, ∀g ∈ C ∞(N). The Jacobi identity for the Poisson structure entails X{f ,g} = −[Xf , Xg]. This is a Lie algebra antihomorphism between (C ∞(N), {·, ·}) and (Γ(TN), [·, ·]). As every Poisson structure is a derivation in each entry, it determines an unique Λ ∈ Γ(Λ2TN) {f , g} = Λ(df , dg) ∀f , g ∈ C ∞(N) We call Λ the Poisson bivector st. [·, ·]SN = 0 being the Schouten-Nijenhuis Lie bracket.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

An example of Lie–Hamilton system

Consider the system of differential equations dx dt = a0(t) + a1(t)x + a2(t)(x2 − y 2), dy dt = a1(t)y + a2(t)2xy, with a0(t), a1(t), a2(t) being arbitrary t-dependent real functions. By writing z = x + iy, we find that it is equivalent to dz dt = a0(t) + a1(t)z + a2(t)z2, z ∈ C, which is a particular type of complex Riccati equations. Let us show that it is a Lie system on R2

y=0. This is related to the t-dependent vector field

Xt = a0(t)X1 + a1(t)X2 + a2(t)X3, where X1 = ∂ ∂x , X2 = x ∂ ∂x + y ∂ ∂y , X3 = (x2 − y 2) ∂ ∂x + 2xy ∂ ∂y span a Vessiot–Guldberg real Lie algebra V ≃ sl(2) with commutation relations [X1, X2] = X1, [X1, X3] = 2X2, [X2, X3] = X3. Hence, {Xt}t∈R ⊂ V X ⊂ V is finite-dimensional and X is a Lie system.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

We search for a symplectic form, ω = f (x, y)dx ∧ dy, turning V = X1, X2, X3 into a Lie algebra of Hamiltonian vector fields with respect to it. We impose LXi ω = 0 (i = 1, 2, 3). In coordinates, these conditions read ∂f ∂x = 0, x ∂f ∂x + y ∂f ∂y + 2f = 0, (x2 − y 2) ∂f ∂x + 2xy ∂f ∂y + 4xf = 0. From the first equation, we obtain f = f (y). Using the second, f = y −2 is a particular solution of both equations. This leads to a closed and non-degenerate two-form on R2

y=0, namely

ω = dx ∧ dy y 2 . Using the relation ιXω = dh between a Hamiltonian vector field X and one of its corresponding Hamiltonian functions h, we observe that X1, X2 and X3 are Hamiltonian vector fields with Hamiltonian functions h1 = − 1 y , h2 = −x y , h3 = −x2 + y 2 y ,

• respectively. Obviously, the remaining vector fields of V become also
• Hamiltonian. Thus, planar Riccati equations admit a Vessiot–Guldberg Lie

algebra of Hamiltonian vector fields relative to ω = dx∧dy

y2

.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

The Poisson structure

Definition

A Lie–Hamiltonian structure is a triple (N, Λ, h), where (N, Λ) stands for a Poisson manifold and h represents a t-parametrized family of functions ht : N → R such that Lie({ht}t∈R, {·, ·}Λ) is a finite-dimensional real Lie algebra.

Definition

A t-dependent vector field X is said to admit a Lie–Hamiltonian structure (N, Λ, h) if Xt is the Hamiltonian vector field corresponding to ht for each t ∈ R. Lie({ht}t∈R, {·, ·}Λ) is called a Lie–Hamilton algebra for X.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Example of Lie–Hamiltonian

Reconsider the Planar Riccati equations, dx dt = a0(t) + a1(t)x + a2(t)(x2 − y 2), dy dt = a1(t)y + a2(t)2xy, whose vector fields are Hamiltonian w.r.t the symplectic form ω = dx∧dy

y2

and form a basis for a VG isomorphic to V ≃ sl(2, R). If {·, ·}ω : C ∞(R2

y=0) × C ∞(R2 y=0) → C ∞(R2 y=0) stands for the Poisson bracket

induced by ω, then {h1, h2}ω = −h1, {h1, h3}ω = −2h2, {h2, h3}ω = −h3. Hence, the planar Riccati equation X possesses a Lie–Hamiltonian structure of the form

• R2

y=0, ω, h = a0(t)h1 + a1(t)h2 + a2(t)h3

• .

We have that (HΛ, {·, ·}ω) ≡ (h1, h2, h3, {·, ·}ω) is a Lie–Hamilton algebra for X isomorphic to sl(2).

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Lie–Hamiltonian viral infection

Recall the simple viral infection model dx dt = (α(t) − g(y))x, (21) dy dt = β(t)xy − γ(t)y, (22) restricting ourselves to studying particular solutions within R2

x,y=0 = {(x, y) ∈ R2 | x = 0, y = 0} and g(y) = δ, and with δ a constant.

The vector fields X1 = x ∂ ∂x , X2 = −y ∂ ∂y , X3 = xy ∂ ∂y , are Hamiltonian with respect to ω = dx∧dy

xy

. Then, the vector fields X1, X2 and X3 have Hamiltonian functions h1 = ln y, h2 = ln x, h3 = −x, which along h0 = 1 define a Lie–Hamilton algebra (R ⋉ R2) with the associated t-dependent Hamiltonian ht = (α(t) − δ)h1 + γ(t)h2 + β(t)h3 giving rise to a Lie–Hamiltonian structure (R2

x,y=0, ω, h) for X.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

The Lie–Hamiltonian SISf model

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

The Lie–Hamiltonian SISf model

Consider now the canonical symplectic form ω = dq ∧ dp. It is easy to check that the vector fields X1 and X2 Xt = ρ0(t)X1 + X2, X1 = q ∂ ∂q − p ∂ ∂p , X2 =

• −q2 − 1

p2 ∂ ∂q + 2qp ∂ ∂p . are Hamiltonian with respect to the canonical symplectic form and Hamiltonian functions h1 = −qp, h2 = −q2p + 1 p , (23)

• respectively. It is easy to see that the Poisson bracket of these two functions

reads {h1, h2} = h2. It means that the Hamiltonian functions form a finite dimensional Lie algebra and it is isomorphic to the one defined by vector fields X1, X2. The time-dependent vector field of the Lie system Xt retrieves the Hamiltonian function as Xt = − Λ(dht). Therefore, the Hamiltonian of the system is h = ρ0(t)h1 + h2 = −q2p + 1 p − ρ0(t)qp (24) and it is exactly the Hamiltonian proposed in Hamiltonian dynamics of the SIS epidemic model with stochastic fluctuations, (G.M. Nakamura, A.S. Martinez),

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

A different derivation of superposition rules

Using the coalgebra method

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Geometry for the coalgebra method

Two examples of Poisson algebras are the symmetric Lie algebra Sg and the universal Lie algebra Ug. If we consider a finite-dim. real Lie algebra (g, [·, ·]), since g ≃ (g∗)∗, we can consider g as linear functions on g∗. We define Sg as the quotient of set of polynomials Tg on g by the ideal generated by u ⊗ w − w ⊗ u with u, w ∈ g, we refer to it as (Sg, ·, {·, ·}Sg). And Ug is obtained from the quotient Tg/R of the tensor algebra (Tg, ⊗) of g by the bilateral ideal R spanned by v ⊗ ω − ω ⊗ v − [v, ω] with v, ω ∈ g.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Geometry for the coalgebra method

If we have two Poisson algebras (A, ⋆A, {·, ·}A) and (B, ⋆B, {·, ·}B), the space A ⊗ B becomes a Poisson algebra (A ⊗ B, ⋆A⊗B, {·, ·}A⊗B) by defining (a ⊗ b) ⋆A⊗B (c ⊗ d) = (a ⋆A c) ⊗ (b ⋆B d) a, c ∈ A b, d ∈ B {a ⊗ b, c ⊗ d} = {a, c}A ⊗ b ⋆B d + a ⋆A c ⊗ {b, d}B We say that (A, ⋆A, {·, ·}A) is a Poisson coalgebra if the so-called coproduct ∆ : (A, ⋆A, {·, ·}A) → (A ⊗ A, ⋆A⊗A, {·, ·}A⊗A) is an coassociative Poisson algebra homomorphism, (∆ ◦ Id) ◦ ∆ = (Id ◦ ∆) ◦ ∆. Similarly, we can have Poisson structures on the m-th tensor product A(m) ≡ A ⊗ · · · ⊗ A. The m-coproduct map ∆m : A → A ⊗ · · · ⊗ A can be defined as ∆m(v) = v ⊗ Id ⊗ · · · ⊗ Id + · · · + Id ⊗ · · · ⊗ v (k) ⊗ · · · ⊗ Id + Id ⊗ · · · ⊗ v (25) which is the primitive coproduct.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

The coalgebra method for s.r.

Lemma

Given a Lie-Hamilton system X with Lie–Hamiltonian structure (N, Λ, h), the space (Sg, ·, {·, ·}g, ∆) with g ≃ (HΛ, {·, ·}Λ) is a Poisson coalgebra with a coproduct ∆ : Sg → Sg ⊗ Sg satisfying ∆(v) = v ⊗ Id + Id ⊗ v. If X is a Lie–Hamilton system with a Lie–Hamiltonian structure (N, Λ, h), then the diagonal prolongation ˜ X to each Nm+1 is also a Lie–Hamilton system endowed with a Lie–Hamilton structure (Nm+1, Λ(m+1), ˜ h) given by Λ(m+1)(x(0), . . . , x(m)) =

m

• a=0

Λ(x(a)) using the vector bundle isomorphism Λ2TNm+1 ≃ Λ2TN ⊕ · · · ⊕ Λ2TN and ˜ ht = D(m)(∆(m)(ht)) where D(m) the representation morphism.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Coalgebra method

• The map ∆(m) : Sg → S(m)

g

≡ Sg ⊗ · · · ⊗ Sg defined by recursion as ∆(m+1) = (Id ⊗ · · · ⊗ Id ⊗ ∆(2)) ◦ ∆(m), m > 2 is a P.A. morphism from (Sg, ·, {·, ·}Sg) to (S(m)

g

, ·, {·, ·}Sg(m) ) {P1⊗· · ·⊗Pm, Q1⊗· · ·⊗Qm}S(m)

g

=

m

• i=1

P1Q1⊗· · ·⊗{Pi, Qi}⊗· · ·⊗PmQm

• The Lie algebra morphism g ֒

→ C ∞(N) gives rise to a family of P.A. morphisms D(m) : Sg ⊗ · · · ⊗ Sg ֒ → C ∞(N) ⊗ · · · ⊗ C ∞(N) ⊂ C ∞(N × · · · × N) satisfying

• D(m)(v1 ⊗ · · · ⊗ vm)
• (x(1), . . . , x(m)) = [D(v1)](x(1)) . . . [D(vm)](x(m))

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Coalgebra method

Theorem

If X is a Lie–Hamilton system with a Lie–Hamiltonian structure (N, Λ, h) and C is a Casimir element of the Poisson algebra (Sg, ·, {·, ·}Sg) then,

• 1. The functions defined as

F (k) = D(k)(∆(k)(C)), k = 2, . . . , m are t-indp. constants of motion for diagonal prolon. ˜ X to Nm. Furthermore, m − 1 functionally functions in involution.

• 2. The functions given by

F (k)

ij

= Sij(F (k)), 1 ≤ i < j ≤ k, k = 2, . . . , m where Sij is the permutation of variables x(i) ↔ x(i), are t-indp. const. of motion for the diagonal prolong. ˜ X to Nm.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Example: coalgebra method for iso(2)

Over the plane, consider the following vector fields X1 = ∂ ∂x , X2 = ∂ ∂y , X3 = y ∂ ∂x − x ∂ ∂y (26) with commutation relations [X1, X2] = 0, [X1, X3] = −X2, [X2, X3] = X1. Step 1. With respect to the canonical symplectic structure ω = dx ∧ dy, this corresponds Lie algebra, denoted by iso(2), determined by a basis h1 = y, h2 = −x, h3 = 1 2(x2 + y 2), h0 = 1 (27) satisfying commutation relations {h1, h2}ω = h0, {h1, h3}ω = h2, {h2, h3}ω = −h1, {h0, ·}ω = 0, (28)

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Example: coalgebra method for iso(2)

The symmetric Poisson algebra S

• iso(2)
• has a non-trivial Casimir invariant

given by C = v3v0 − 1

2(v 2 1 + v 2 2 ).

Choosing the representation given in (27), we obtain a trivial constant of motion on (x, y) ≡ (x1, y1) F = D(C) = φ(v3)φ(v0) − 1

2

• φ2(v1) + φ2(v2)
• = h3(x1, y1)h0(x1, y1) − 1

2

• h2

1(x1, y1) + h2 2(x1, y1)

• = 1

2(x2 1 + y 2 1 ) × 1 − 1 2(y 2 1 + x2 1) = 0.

Introducing the coalgebra structure in S

• iso(2)
• through the coproduct, we
• btain nontrivial first integrals.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

F (2) = D(2)(∆(C)) = (h3(x1, y1) + h3(x2, y2)) (h0(x1, y1) + h0(x2, y2)) − 1

2

• ((h1(x1, y1) + h1(x2, y2))2 + (h2(x1, y1) + h2(x2, y2))2

= 1

2(x1 − x2)2 + 1 2(y1 − y2)2,

F (3) = D(3)(∆(C)) =

3

• i=1

h3(xi, yi)

3

• j=1

h0(xj, yj) − 1

2

• 3
• i=1

h1(xi, yi) 2 +

• 3
• i=1

h2(xi, yi) 2 = 1

2 3

• 1≤i<j
• (xi − xj)2 + (yi − yj)2

. (29) Furthermore, using the property of permutating subindices (??), we find more first integrals F (2)

12 = S12(F (2)) ≡ F (2),

F (2)

13 = S13(F (2)) = 1 2(x3 − x2)2 + 1 2(y3 − y2)2,

F (2)

23 = S23(F (2)) = 1 2(x1 − x3)2 + 1 2(y1 − y3)2

(30)

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Coalgebra superposition rule

for SISf Pandemic model

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Coalgebra method for SIS pandemic model

We need to find a Casimir function for the Poisson algebra I r=1

14A ≃ R ⋉ R, but

unfortunately, there exists no nontrivial Casimir. We can circumvent this problem by considering an inclusion of I r=1

14A as a Lie

subalgebra of a Lie algebra of another class admitting a Lie–Hamiltonian algebra with a nontrivial Casimir. For example, consider the I8 ≃ iso(1, 1) due to the simple form of its Casimir. If we obtain the superposition rule for I8, we simultaneously obtain the superposition for I r=1

14A as a byproduct.

The Lie–Hamilton algebra iso(1, 1) has the commutation relations {h1, h2} = h0, {h1, h3} = −h1, {h2, h3} = h2, {h0, ·} = 0, (31) with respect to ω = dx ∧ dy in the basis {h1 = y, h2 = −x, h3 = xy, h0 = 1}. Applying the coalgebra method to the Casimir associated to this Lie–hamilton algebra C = h1h2 + h3h0 and mapping the representation without coproduct, the first iteration is trivial, i.e., F = 0, but F (2) = (x1 − x2)(y1 − y2) = k1, F (2)

23 = (x1 − x3)(y1 − y3) = k2,

F (2)

13 = (x3 − x2)(y3 − y2) = k3.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

From them, we can choose two functionally independent constants of motion. In this case, F (2) = k1, F (2)

23 = k2 can be understood as the equations on R2.

The introduction of k3 again simplifies the final result which reads x1(x2, y2, x3, y3, k1, k2, k3) = 1 2(x2 + x3) + k2 − k1 ± B 2(y2 − y3) , y1(x2, y2, x3, y3, k1, k2, k3) = 1 2(y2 + y3) + k2 − k1 ∓ B 2(x2 − x3) , (32) where B =

• k2

1 + k2 2 + k2 3 − 2(k1k2 + k1k3 + k2k3).

In the case that matters to us, I r=1

14A , the third constant k3 is a function

k3 = k3(x2, y2, x3, y3) and B ≥ 0. We see there is a change of coordinates between this system and SIS, x = −qp, y = q − 1 qp2 (33)

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

So, the superposition principle for SISf would read

q = 1

2 q2+ 1 2 q3+(k2−k1±B) (2p2−2p3)

2

1 2 p2 + 1 2 p3 + (k2−k1∓B) (2q2−2q3)

• 1

2 q2+ 1 2 q3+(k2−k1±B) (2p2−2p3)

2 − 1 (34) p = − 1

2 q2+ 1 2 q3+(k2−k1±B) (2p2−2p3)

2 − 1

1 2 q2 + 1 2 q3 + (k2−k1±B) (2p2−2p3)

• 1

2 p2 + 1 2 p3 + (k2−k1∓B) (2q2−2q3)

• (35)

Here, (q2, p2) and (q3, p3) are two particular solutions and k1, k2, k3 are constants of integration.

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Here, (q2, p2) and (q3, p3) are two particular solutions and k1, k2, k3 are constants of integration. Now, we show the graphics for < ρ >= q(t) and σ2 = 1/p2 using the two particular solutions provided in the introduction. Notice that we have renamed c = (k2 − k1 ± B) and k = (k2 − k1 ∓ B).

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Conclusions about Lie systems and Covid

One may wonder how the current pandemic of COVID19 could be related to a SISf-pandemic model. Let us state clear some points.

• The SISf model is a very first approximation for a trivial infection process,

in which there is only two possible states: infected or susceptible. Hence, this model does not provide the possibility of adquiring immunity. It seems that COVID19 provides some 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., immune) is not a proper model for the current

• situation. One should have a model contemplating immune and

nonimmunized individuals.

• Unfortunately, we are still in search of a Lie system including potential

immunity and nonimmunity.

• There exists a stochastic theory of Lie systems. In the present work we

were lucky to find a theory with fluctuations that happened to match a stochastic expansion, but this is rather more of an exception than a rule. Maybe the stochastic Lie system approach could lead us to Hamiltonian Lie systems?

Motivation Geometric background Superposition rules Lie–Hamilton systems Superposition rules with the coalgebra method Lie

Thanks for the attention!

A Guide to Lie Systems with Compatible Geometric Structures.

• J. de Lucas, C. Sard´
• n. World Scientific (2020).

Part of the content free at: https://arxiv.org/abs/1508.00726 Preprint for SISf : arXiv:2008.02484