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Dynamics of Unemployment with Discouraged Workers: A Nonlinear Mathematical Model

Poushali Das, Shraddha Sachan, Mukul Chauhan, Narendra Kumar, Amit K. Verma

TL;DR

This paper develops a nonlinear five-variable dynamical system to model unemployment with a discouraged workforce, capturing interactions among unskilled unemployed, skilled unemployed, discouraged individuals, employed workers, and vacancies. It derives three equilibria, analyzes local stability via eigenvalues and the Routh–Hurwitz criterion, and validates the results through numerical simulations using RK4, including a detailed sensitivity analysis. The interior equilibrium is shown to be locally asymptotically stable under appropriate parameter regimes, and simulations illustrate how skill development and vacancy creation influence the distribution across groups. The findings emphasize policy relevance of skill-building programs and targeted vacancy strategies to mitigate discouragement and unemployment, supported by both analytical and numerical evidence.

Abstract

In this article, we formulate and analyze a new non-linear mathematical model to describe the dynamics of unemployment with a discouraged working population. We consider five dynamic variables, namely, unskilled unemployed individuals, skilled unemployed individuals, discouraged individuals, employed persons, and job vacancies. Furthermore, we determine the equilibrium points of the dynamic system and investigate their local stability. To demonstrate the results, we conduct numerical simulations by presenting solution trajectories and analyzing how variations in key parameters influence the states of the dynamical variables.

Dynamics of Unemployment with Discouraged Workers: A Nonlinear Mathematical Model

TL;DR

This paper develops a nonlinear five-variable dynamical system to model unemployment with a discouraged workforce, capturing interactions among unskilled unemployed, skilled unemployed, discouraged individuals, employed workers, and vacancies. It derives three equilibria, analyzes local stability via eigenvalues and the Routh–Hurwitz criterion, and validates the results through numerical simulations using RK4, including a detailed sensitivity analysis. The interior equilibrium is shown to be locally asymptotically stable under appropriate parameter regimes, and simulations illustrate how skill development and vacancy creation influence the distribution across groups. The findings emphasize policy relevance of skill-building programs and targeted vacancy strategies to mitigate discouragement and unemployment, supported by both analytical and numerical evidence.

Abstract

In this article, we formulate and analyze a new non-linear mathematical model to describe the dynamics of unemployment with a discouraged working population. We consider five dynamic variables, namely, unskilled unemployed individuals, skilled unemployed individuals, discouraged individuals, employed persons, and job vacancies. Furthermore, we determine the equilibrium points of the dynamic system and investigate their local stability. To demonstrate the results, we conduct numerical simulations by presenting solution trajectories and analyzing how variations in key parameters influence the states of the dynamical variables.
Paper Structure (8 sections, 1 theorem, 46 equations, 8 figures)

This paper contains 8 sections, 1 theorem, 46 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Proposed Mathematical Model
  3. Equilibrium Analysis
  4. Stability Analysis
  5. Numerical Simulation
  6. Solution Plots
  7. Sensitivity Analysis
  8. Conclusion

Key Result

Theorem 2.1

The set where $\gamma=\min\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\}$ is a region of attraction for the model in system Eq1 and attracts all solutions initiated in the interior of the positive octant.

Figures (8)

  • Figure 1: Schematic diagram of the model
  • Figure 2: Numerical simulation of the solution plots of $U_{u}(t), U_{s}(t), E(t), D(t),$ and $V(t)$ with respect to time (in days) with various initial conditions $(U_{u_{0}}, U_{s_{0}}, E_{0}, D_{0}, V_{0})$.
  • Figure 3: Temporal dynamics of the $U_{u}(t), U_{s}(t), E(t), D(t),$ and $V(t)$ with respect to time (in days) with various values of $\beta_{1}.$
  • Figure 4: Temporal dynamics of the $U_{u}(t), U_{s}(t), E(t), D(t),$ and $V(t)$ with respect to time (in days) with different values of $\beta_{2}.$
  • Figure 5: Temporal dynamics of the $U_{u}(t), U_{s}(t), E(t), D(t),$ and $V(t)$ with respect to time (in days) with different values of $\beta_{3}.$
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • proof
  • Remark 3.1