Asymptotic Analysis of Poverty Dynamics via Feller Semigroups

Poverty is a multifaceted phenomenon impacting millions globally, defined by a deficiency in both material and immaterial resources, which consequently restricts access to satisfactory living conditions. Comprehensive poverty analysis can be accomplished through the application of mathematical and modeling techniques, which are useful in understanding and predicting poverty trends. These models, which often incorporate principles from economics, stochastic processes, and dynamic systems, enable the assessment of the factors influencing poverty and the effectiveness of public policies in alleviating it. This paper introduces a mathematical compartmental model to investigate poverty within a population ($\psi \left(t\right)$), considering the effects of immigration, crime, and incarceration. The model aims to elucidate the interconnections between these factors and their combined impact on poverty levels. We begin the study by ensuring the mathematical validity of the model by demonstrating the uniqueness of a positive solution. Next, it is shown that under specific conditions, the probability of poverty persistence approaches certainty. Conversely, conditions leading to an exponential reduction in poverty are identified. Additionally, the semigroup associated with our model is proven to possess the Feller property, and its distribution has a unique invariant measure. To confirm and validate these theoretical results, interesting numerical simulations are performed.