Efficient relaxation scheme for the SIR and related compartmental models

In this paper, we introduce a novel numerical approach for approximating the SIR model in epidemiology. Our method enhances the existing linearization procedure by incorporating a suitable relaxation term to tackle the transcendental equation of nonlinear type. Developed within the continuous framework, our relaxation method is explicit and easy to implement, relying on a sequence of linear differential equations. This approach yields accurate approximations in both discrete and analytical forms. Through rigorous analysis, we prove that, with an appropriate choice of the relaxation parameter, our numerical scheme is non-negativity-preserving and globally strongly convergent towards the true solution. These theoretical findings have not received sufficient attention in various existing SIR solvers. We also extend the applicability of our relaxation method to handle some variations of the traditional SIR model. Finally, we present numerical examples using simulated data to demonstrate the effectiveness of our proposed method.Comment: 17 pages, 21 figures, 2 table

A hierarchical size-structured population model.

A model is considered for the dynamics of a size-structured population in which the birth, death and growth rates of an individual of size s are functions of the total population biomass of all individuals of size larger or smaller than s. The dynamics of the size distribution is governed by the McKendrick equations. An existence/uniqueness theorem for this equation is proved using an equivalent pair of partial and ordinary differential equations. The asymptotic dynamics of the density function is studied and some applications of the model to intraspecific predation and certain types of intraspecific competitions are given