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´ on 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 : N m × N → N of 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 X X ( t , x ) ∈ π − 1 ( x ) = T x N π π 2 � N R × N Thus, the maps X t : x ∈ N → X ( t , x ) ∈ TN are { X t } 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 ( π 2 ◦ γ ) ( t ) = ( X ◦ γ )( t ) dt 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 { X t } t ∈ R , namely, V X = Lie ( { X t } 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 � r X t = b α ( t ) X α α =1 for a certain family b 1 ( t ) , . . . , b r ( t ) of t-dependent functions and a family X 1 , . . . , X r 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 = a 0 ( t ) + a 1 ( t ) x + a 2 ( t ) x 2 , where a 0 ( t ) , a 1 ( t ) , a 2 ( t ) are arbitrary t -dependent functions, admits the superposition rule Φ : ( x (1) , x (2) , x (3) ; k ) ∈ R 3 × 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 = a 0 ( t ) X 1 + a 1 ( t ) X 2 + a 2 ( t ) X 3 , where X 1 = ∂ X 2 = x ∂ X 3 = x 2 ∂ ∂ x , ∂ x , ∂ x span a VG isomorphic to sl (2 , R ). Then, Φ : ( x (1) , x (2) , x (3) ; k ) ∈ R 3 × 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 T e G ≃ V with e being the neutral element of G . x 0 ∈ R n , x ( t ) = ϕ ( g 1 ( t ) , x 0 ) , with g 1 ( t ) being a particular solution of � r dg b α ( t ) X R dt = − α ( g ) , α =1 where X R 1 , . . . , X R is a certain basis of right-invariant vector fields on G such r that X R α ( e ) = a α ∈ T e G , with α = 1 , . . . , r , and each a α is the element of T e G associated with the fundamental vector field X α . Since X R 1 , . . . , X R span a finite-dimensional real Lie algebra, the Lie–Scheffers r 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  d x   d t = ( α ( t ) − g ( y )) x ,  d y  d t = 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 ( t 0 ) = 0 or y ( t 0 ) = 0 for a t 0 ∈ 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 x , y � =0 = { ( x , y ) ∈ R 2 | x � = 0 , y � = 0 } . R 2 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 X t = ( α ( t ) − δ ) X 1 + γ ( t ) X 2 + β ( t ) X 3 , on R 2 x , y � =0 , where the vector fields X 1 = x ∂ X 2 = − y ∂ X 3 = xy ∂ ∂ x , ∂ y , ∂ y , close a finite-dimensional Lie algebra. So, X is a Lie system related to a Vessiot–Guldberg Lie algebra V ≃ R ⋉ R 2 where � X 1 � ≃ R and � X 2 , X 3 � ≃ R 2 .

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 only 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 . d ρ dt = αρ (1 − ρ ) − γρ. (1) One can redefine the timescale as τ ≡ α t and ρ 0 ≡ 1 − γ/α , so we can rewrite (1) as d ρ d τ = ρ ( ρ 0 − ρ ) (2) The equilibrium density is reached if ρ = 0 or ρ = ρ 0 . 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.