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 $\dot{q}=\frac{{\partial H}}{{\partial p}}$, $\dot{p}=-\frac{{\partial H}}{{\partial q}}+\Gamma(t,\;q,\;p)$ (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.

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This article presents the closed-form solutions of two-sector human capital-based Romer growth model. The partial Hamiltonian approach is effectively applied to some growth models in order to compute the closed-form solutions for economic variables involved in the model. Pontryagin's maximum principle provides the set of first-order system of ODEs, which are regarded as an essential criteria for optimality. The partial Hamiltonian approach is utilized to construct three first integrals of the system using the current value Hamiltonian. With the aid of these first integrals, we computed two distinct exact solutions of Romer model under certain parametric restrictions. The closed-form expressions for control, state, and costate variables are presented explicitly as a function of t. We have graphically illustrated the solution curves and observed the effect of human capital parameter on control and state variables. The growth rates of all economic variables are evaluated, and their long-run behavior is predicted. KEYWORDSfactor tax incidence, first integrals, gauge function, partial Hamiltonian, Pontryagin's principle MSC CLASSIFICATION34H05; 49K15

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