Modeling and Stability Analysis of the Unemployment Model with Caputo

Abstract

The problem of unemployment is acute, joblessness affects millions of people worldwide, which is why governments are constantly looking for efficient solutions to combat this social and economic ill. In response, we construct a model that encompasses the nonlinear nature of the unemployment rate. A fractional-order system of differential equations is employed in the structure of the considered model to produce a more accurate analysis of the unemployment dynamics of equilibrium positions and their stability or instability. In the model, there are three main dynamical parameters whose time evolution is described by fractional-order differential equations involving Caputo derivatives. The existence and uniqueness of the solutions are established by using the fixed point indices and the stability of the model is determined by applying the Hyers-Ulam stability test. A Newton polynomial approach is used for numerical simulation and investigates the results at fractional orders ω = 0.85 and ω = 1. The results presented in this paper suggest that there is one and only one stable positive equilibrium that is locally and globally asymptotically stable under some conditions. Computations demonstrate that for different values of ω, obtained with Caputo derivatives coinciding with the classical sense at ω = 1. In this study, by determining the employment rate and job creation rate that produce an unemployment rate of 7% the model complies with the governments policy objectives of low unemployment. These results provide important implications for the dynamic employment policies suggested and point to the use of fractional-order methodologies in other economic-social systems.