The Artificial Hamiltonian, First Integrals, and Closed-Form Solutions of Dynamical Systems for Epidemics

Abstract

Abstract The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mover accent="true"> <m:mi>q</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>i</m:mi> </m:msup> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mi>p</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mfrac> <m:mn>,</m:mn> <m:mtext> </m:mtext> <m:msup> <m:mover accent="true"> <m:mi>p</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>i</m:mi> </m:msup> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mi>q</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:msup> <m:mo>Γ</m:mo> <m:mi>i</m:mi> </m:msup> <m:mn>(</m:mn> <m:mi>t</m:mi> <m:mn>,</m:mn> <m:mtext> </m:mtext> <m:msup> <m:mi>q</m:mi> <m:mi>i</m:mi> </m:msup> <m:mn>,</m:mn> <m:mtext> </m:mtext> <m:msub> <m:mi>p</m:mi> <m:mi>i</m:mi> </m:msub> <m:mn>)</m:mn> </m:mrow> </m:math> ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.