Mathematical Modeling and Numerical Analysis of Population Dynamics: Unraveling the Migration Patterns of Ukrainians during the 2022-2023 War

Abstract

Russia's invasion of Ukraine in February 2022 caused a major displacement crisis, with nearly one third of Ukrainians, mostly women and children, forced to flee their country. Despite leaving their homes, many express a strong attachment to their homeland and a desire to return once the war is over. Reasons for their willingness to return include reuniting with family and friends, the challenges of living abroad, and a desire to rebuild their lives in familiar surroundings. This project focuses on the development of a mathematical model that describes the social dynamics of Ukrainian sub-populations using a system of ordinary differential equations. We study different types of migration for two groups of sub-populations including women and children as one group and men as the other. The proposed set of models explore the social dynamics between Ukrainians women, children, and men who have been displaced due to war and aim to capture the complex interactions and relationships among these groups within the context of displacement. ​​We conduct numerical studies of the proposed models using both Euler's method and the fourth-order Runge-Kutta method to investigate the influence of parameters in the models. As the initial step, we performed a numerical study of the proposed model using the scarce data available to date. This investigation encompasses various scenarios that consider the impact of policies and regulations. Different variations of the models - those that neglect the influence of policies and those that incorporate them using constant or periodic terms - are being examined. The obtained results suggest that different initial distributions of inhabitants from both groups do not have an influence on their cyclic migration. Additionally, investigating various scenarios with synthetic data yielded the following outcomes: (i) When no regulation model is applied, the studied equations exhibit periodic solutions; (ii) Introducing constant regulation impacts the short-term state of the studied populations, but they eventually adjust to regular periodic behavior; and (iii) The solution of the model with periodic regulations depends on the regulation frequency (lower frequencies disrupt the periodicity of the solution, while higher frequency regulations restore more regular periodic behavior). To the best of our knowledge, this important humanitarian issue has not been rigorously investigated in a mathematical manner prior to this study.