Mathematical Modeling of Epidemic Disease Spread Considering Population Density, Vaccination Rate, and Human Mobility Patterns

Abstract

This study presents a quantitative mathematical model to analyze the spread of epidemic diseases in Multan, Pakistan, by integrating population density, vaccination rate, and human mobility within an extended Susceptible-Exposed-Infected-Recovered (SEIR) framework. The model aims to simulate disease dynamics under varying epidemiological scenarios, assess the impact of population density and mobility on transmission, evaluate the effectiveness of vaccination coverage, and identify key influencing parameters through sensitivity analysis. Secondary data were obtained from the Pakistan Bureau of Statistics, World Health Organization (WHO), and demographic and transportation databases. The enhanced SEIR model incorporates a mobility-scaling parameter and a vaccination-adjusted force of infection, implemented using Python libraries such as SciPy and Matplotlib. Results indicate that population density significantly influences the basic reproduction number (R₀), increasing from 1.8 in low-density settings to 3.4 in observed urban conditions. High vaccination coverage (≥70%) effectively reduces epidemic peaks and accelerates herd immunity, while mobility restrictions (50% reduction) lower cumulative infections by 38% over 180 days. Sensitivity analysis identifies transmission rate (β) and vaccination rate (ν) as the most critical parameters. The findings highlight the importance of integrated strategies combining targeted vaccination, mobility control, and density-aware interventions to effectively manage epidemics in densely populated urban areas.