First integrals and exact solutions of some compartmental disease models

The notions of artificial Hamiltonian (partial Hamiltonian) and partial Hamiltonian operators are used to derive the first integrals for the first order systems of ordinary differential equations (ODEs) in epidemiology, which need not be derived from standard Hamiltonian approaches. We show that every system of first order ODEs can be cast into artificial Hamiltonian system <(q)over dot> = partial derivative H/partial derivative p, <(q)over dot> = -partial derivative H/partial derivative q vertical bar Gamma(t, q, p) (see [1]). Moreover, the second order equations and the system of second order ODEs can be written in the form of artificial Hamiltonian system. Then, the partial Hamiltonian approach is employed to derive the first integrals for systems under consideration. These first integrals are then utilized to find the exact solutions of models from the epidemiology for a distinct class of population. For physical insights, the solution curves of the closed-form expressions obtained are interpreted in order for readers understand the disease dynamics in a much deeper way. The effects of various pertinent parameters on the prognosis of the disease are observed and discussed briefly.