Perturbed Tumor Immunotherapy Domain of Attraction Estimation via the Arc-Length Function

Abstract

This paper aims at the estimation of the Domain of Attraction (DoA) of the free tumor equilibrium point of perturbed tumor immunotherapy model via the Arc-Length Function (ALF). The ALFs are categorized among the maximal Lyapunov functions which are able to provide a more accurate estimation of the DoA in comparison to their other counterparts such as Rational Lyapunov Functions (RLFs), Sum Of Square (SOS) polynomial Lyapunov functions, and Optimal Quadratic Lyapunov Functions (OQLFs). There is no analytical method to construct the ALFs, however, some numerical methods have been proposed in the literature. Based on the existing method, one can approximate the ALF with a certain degree of a polynomial function. That the system under study has a polynomial structure was the main basis of the previously proposed method to estimate the DoA via the ALFs. However, the intended model in this paper describing the tumor-immune system competition dynamics contains non-polynomial terms. To cope with the aforementioned problem, the Taylor expansion of the non-polynomial terms are considered and by solving an optimization problem, one can calculate the corresponding lower boundary of the level set with the approximated ALF as an estimation of the DoA. In order to represent the performance of the employed method, the obtained result is compared with the reported result in the literature.