Abstract

We study Lie-(Hamilton) systems on the plane, i.e. systems of first-order (nonlinear) ODEs describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar (Hamiltonian) vector fields with respect to a geometric structure. Lie-(Hamilton) systems enjoy a plethora of properties, e.g. they admit their general solution expressed as a (nonlinear) function through the so-called superposition rule of a finite set of particular solutions and some constants. Lie-Hamilton systems are important because of their appearance in the physics, mathematics and biology literature. For example, they can be used to study Milne–Pinney, second-order Kummer–Schwarz, complex Riccati and Buchdahl equations, which occur in cosmology, relativity and classical mechanics. They also appear in the investigation of Lotka–Volterra, predator-prey or growth of a viral infection models. We are particulary interested in the latter. In this talk, I present the geometrical properties of Lie-(Hamilton) systems and their application to SIS-pandemic models. We derive complete solutions for a SIS-pandemic model with fluctuations through the intrinsic properties of Lie systems and through the coalgebra method. We will present graphic representations of the solutions, and we will see how the number of infected individuals grows accordingly with a sigmoid-like function. This is precisely the expected behavior, and it is retrieved in two different geometric ways, from a symplectic point of view, and from a Poisson framework. We finish this talk discussing whether this model is applicable to the current Covid pandemic.