Mathematical model of population evacuation taking into account time characteristics

Abstract

Modern urban areas are characterized by high population density, complex transportation networks, and limited road capacity, which complicate evacuation during emergencies and increase the risk of congestion, panic, and casualties. A pressing scientific and practical challenge is the development of formalized mathematical models to optimize evacuation under realistic urban constraints. This article proposes a dynamic mathematical model of population evacuation that explicitly accounts for temporal characteristics. The problem is formulated as a discrete-time transport optimization using linear programming. The model includes assembly and evacuation points, planning intervals, the number of evacuees, and the permissible capacity of evacuation points. The objective function minimizes total evacuation costs, reducing overall time or other generalized measures while satisfying all constraints. The novelty lies in integrating temporal dynamics with scenario-based modeling, with potential extensions to stochastic and intelligent decision-support systems. Numerical experiments show that the model reduces congestion peaks and improves evacuation efficiency by 20–30% compared to static models. The model can be used by civil defense agencies for developing evacuation plans, territorial safety passports, and emergency response scenarios.