In this paper a five-dimensional mathematical model with distributed time delays is built and analysed. The model describes the labour market focusing on the number of unemployed persons U(t), the number of immigrants M(t), the number of underemployed persons T(t), the number of regularly employed persons R(t) and the number of available vacancies V(t) at time t. In the modeling process, we consider a constant growth rate of unemployed persons which enter the labor market.

The non-dimensional model is deduced and the positivity and boundedness of solutions are proved. The existence of two equilibrium points depends on the basic reproduction number. The local and global asymptotic stability properties of the equilibrium points are studied, with respect to time delays. Moreover, the occurrence of Hopf bifurcation phenomena is investigated. Numerical simulations are carried out confirming the theoretical results.