Compartmental Model for Unemployment Dynamics in Ghana using Caputo Fractional Derivative

Abstract

The study develops a compartmental model for unemployment dynamics in Ghana. The compartmental model is first developed through the method of compartmental designs under certain assumptions such as, all entrants into the labor force qualify for any job, some of the employed become unemployed with time, some of the unemployed go through skill training, among others. The compartmental model is further represented in terms of integral form, and the value of the kernel is thereafter substituted as a power law correlation function to achieve a Caputo fractional derivative model. The Caputo model is analyzed to ascertain its invariant and boundedness, its fixed points and its stability. The unemployment basic reproduction number \(\Re_0\) which determines the threshold of recruitment is analyzed using the Next Generation Method Approach, and the global stability for the unemployment-free equilibrium is also analyzed through the approach of Lyapunov function. The results from the analysis show that, there is attraction of all the solution of the model in \(\mathbb{R}_+^4\) since the region is positively invariant. Again, the results indicates that, the model has two fixed point. Thus, unemployment-free equilibrium and risky-unemployment equilibrium. Notwithstanding, the analysis of both the local and global stability of the model show that, the unemployment free equilibrium is local asymptotically stable (LAS) for \(\Re_0\) <1, and global asymptotically stable (GAS). The results indicate that, the model can examine the unemployment dynamics decision making.