Modeling of the propagation of infectious diseases: mathematics and population

Mathematical models and computer simulations provide an opportunity to study the behavior of a population by helping to understand the characteristics of the transmission of infectious diseases, determining the trends, and general predictions. This study was aimed to investigate the efforts made to epidemic modeling, especially compartmental models, and how to evaluate vaccination in models.

This review article was conducted using international databases such as PabMed, Google Scholar, and Science Direct, and with the keywords "epidemiology", "vaccination", "basic reproduction number", and "Covid-19".

A simple epidemic model without birth and death and also a model, including vital cases of natural death, disease-caused death, birth, and even immigration, are studied along with a vaccination program. Moreover, an advanced mathematical model for Covid-19 pandemic is reviewed based on the known aspects of the new coronavirus.

The basic reproduction number was introduced for these models, and it was observed that the behavior of the population is completely determined by this threshold: for those parameter values that this quantity is less than unity, the disease will be wiped out, while when it is greater than unity, the disease will persist in population. In addition, those parameters with most impact on spread of disease can be found by computational  simulations and sensitivity analysis. By managing these parameters, it is possible to significantly reduce the number of infected people, and therefore, control the disease.