Abstract The purpose of this paper is to develop and analyze a mathematical model that represents the problem of unemployment in poor countries with limited availability of jobs. We are interested in studying the effect of training programs on the model’s dynamics. These training programs aim to improve the skills of unemployed people, contributing to a reduction of the unemployment rate. The model consists of a system of four ordinary differential equations; two equilibria of the model are found: one of which is the employment free equilibrium and the other corresponds to the presence of employment. The existence and stability analysis of both equilibria depend on the basic reproduction number which is found using the method of next generation matrix. However, this number depends on the model parameters which are usually unknown. Hence, we propose to estimate them using the modulating function-based method. Moreover, we study the existence of transcritical bifurcation with the rate of training as a bifurcation parameter. Finally, we support our qualitative results using numerical simulations.

中文翻译：

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失业的数学建模和参数估计与培训计划的影响

摘要 本文的目的是开发和分析一个数学模型，该模型代表了就业机会有限的贫穷国家的失业问题。我们有兴趣研究训练计划对模型动力学的影响。这些培训计划旨在提高失业人员的技能，从而有助于降低失业率。该模型由四个常微分方程组组成；发现模型的两个均衡：一个是无就业均衡，另一个对应于就业的存在。两个平衡点的存在性和稳定性分析都依赖于使用下一代矩阵的方法找到的基本再生数。然而，这个数字取决于通常未知的模型参数。因此，我们建议使用基于调制函数的方法来估计它们。此外，我们以训练速率作为分岔参数研究了跨临界分岔的存在。最后，我们使用数值模拟支持我们的定性结果。