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

Abstract

This paper introduces a quantitative mathematical model to understand the spread of epidemic diseases in Multan, Pakistan, through the combination of the population density, the vaccination rate, and human mobility factors in a long Susceptible-Exposed-Infected-Recovered (SEIR) model. The main aims of the study consisted in the simulation of the dynamics of the disease propagation in different epidemiological conditions, in the determination of how population density and mobility impacts infection propagation, in the determination of the effect of vaccination coverage on the containment of the outbreak, and sensitivity analysis with a view to establishing the most important parameters according to which the disease spreads. Secondary data sources were retrieved using publicly available sources such as national health reports issued by the Pakistan Bureau of Statistics, the world health organization (WHO) and demographic and transportation data bases. The long SEIR model added a mobility-scaling parameter and a force-of-infection modifier of vaccination enabling the dynamic modeling of disease trajectories in a variety of scenarios. Calculations were carried out on Python with the help of the SciPy and Matplotlib packages. These main results show that the urban Multan population density has a significant effect on the basic reproduction number (R0) whereby, in the scenario of low-density, the number is about 1.8 whereas, in the case of observed densities, it is 3.4. A high-vaccination rate (70 percent and above) showed a strong ability to reduce the peaks of epidemics and hasten reaching herd immunity levels. Mobility restriction scenarios, simulated based on a 50 percent cut in the movements between districts, resulted in a 38 percent reduction in the cumulative infections in a 180-day simulation. The sensitivity analysis showed that the most significant parameters that influence the outcomes of the epidemic are the transmission rate (b) and the rate of vaccination (n). The research findings are that comprehensive and reciprocal approaches to the control of the epidemics through specific measures include targeted vaccination campaigns, mobility, and population density-sensitive response strategies can be used effectively to combat the epidemics in the densely populated cities like Multan.