An Optimal Family of Block Techniques to Solve Models of Infectious Diseases: Fixed and Adaptive Stepsize Strategies

Abstract

The contemporary scientific community is very familiar with implicit block techniques for solving initial value problems in ordinary differential equations. This is due to the fact that these techniques are cost effective, consistent and stable, and they typically converge quickly when applied to solve particularly stiff models. These aspects of block techniques are the key motivations for the one-step optimized block technique with two off-grid points that was developed in the current research project. Based on collocation points, a family of block techniques can be devised, and it is shown that an optimal member of the family can be picked up from the leading term of the local truncation error. The theoretical analysis is taken into consideration, and some of the concepts that are looked at are the order of convergence, consistency, zero-stability, linear stability, order stars, and the local truncation error. Through the use of numerical simulations of models from epidemiology, it was demonstrated that the technique is superior to the numerous existing methodologies that share comparable characteristics. For numerical simulation, a number of models from different areas of medical science were taken into account. These include the SIR model from epidemiology, the ventricular arrhythmia model from the pharmacy, the biomass transfer model from plants, and a few more.