Epidemic processes with vaccination and immunity loss studied with the BLUES function method

TL;DR

This paper extends the BLUES function method to coupled nonlinear ODEs and applies it to an epidemiological SIRS model with vaccination, providing accurate analytic approximations that outperform other methods in convergence and long-term behavior.

Contribution

The paper introduces an extension of the BLUES method to coupled nonlinear differential equations and demonstrates its effectiveness on an epidemiological model with vaccination and immunity loss.

Findings

BLUES method converges rapidly and globally.

Accurate long-term asymptotic behavior captured.

Analytic expressions match numerical solutions early in iterations.

Abstract

The Beyond-Linear-Use-of-Equation-Superposition (BLUES) function method is extended to coupled nonlinear ordinary differential equations and applied to the epidemiological SIRS model with vaccination. Accurate analytic approximations are obtained for the time evolution of the susceptible and infected population fractions. The results are compared with those obtained with alternative methods, notably Adomian decomposition, variational iteration and homotopy perturbation. In contrast with these methods, the BLUES iteration converges rapidly, globally, and captures the exact asymptotic behavior for long times. The time of the infection peak is calculated using the BLUES approximants and the results are compared with numerical solutions, which indicate that the method is able to generate useful analytic expressions that coincide with the (numerically) exact ones already for a small number of iterations.