Numerical modeling of NPZ and SIR models with and without diffusion

Abstract

In this paper, the two biological models i.e. Nitrogen, Phytoplankton and Zooplankton (NPZ) and whooping cough SIR models (Charpentier et al., 2010) are being modified and solved numerically by finite difference and meshless methods. Diffusion process has been added to the existing models (Charpentier et al., 2010) so that a unidimensional movement of three species can be incorporated in the models. The effects of diffusion has been studied in both the models. An operator splitting method coupled with the meshless and finite difference procedures, is being considered for numerical solution of the two biological models with and without diffusion. A one step explicit meshless procedure is also applied for the numerical solution of the nonlinear models. The NPZ model contains the concentration of Nitrogen, Phytoplankton and Zooplankton and the whooping cough model contains susceptible, infected, and recovered classes of the population. Equilibrium points of both models have been investigated. Stability of equilibrium points regarding SIR model has been studied. The basic reproduction number of SIR model is also determined. Due to non-availability of the exact solution, the numerical results obtained are mutually compared and their correctness is being verified by the theoretical results as well.