Multi-Hamiltonian structure of the epidemics model accounting for vaccinations and a suitable test for the accuracy of its numerical solvers

Abstract

We derive a generalized Hamiltonian formalism for a modified suscepti ble–infectious–recovered/removed (SIR) epidemic model taking into account the population V of vaccinated persons. The resulting SIRV model is shown to admit three possible functionally independent Hamiltonians and hence three associated Poisson structures. The reduced case of vanishing vaccinated sector shows a complete correspondence with the known Poisson structures of the SIR model. The SIRV model is shown to be expressible as an almost Nambu sys tem, except for a scale factor function breaking the divergenceless property. In the autonomous case with time-independent stationary ratios k and b, the SIRV model is shown to be a maximally super-integrable system. For this case we test the accuracy of numerical schemes that are suited to solve the stiff set of SIRV differential equations.