## 1 Introduction  
Considering the literature referring to applications of the Monte Carlo method in the health sector, we found works involving the use of the method in relation to the COVID-19 pandemic, applied to the subject of the contagion. Thus, it has been used to analyze if the work activities are suspended when the contagion rate is of high risk. For this estimation, a descriptive model is analyzed to define the operation of the system designed in Excel, the infection rate or the basic number of reproduction within a certain range is considered as a random variable, and the dynamic transmission rate and the constant recovery rate are also considered. The results of the simulation indicate a significant difference between the historical data and the data obtained in the simulation. In the case of the infected variable, a positive slope is maintained, considering a minimum average infection rate. In order to be able to suspend the work, an average infection rate should be considered, considering that in all cases preventive measures should be considered [1]. When we analyze the transmission parameters of epidemic contagion, using deterministic models, we try to understand the epidemiological growth using multi-peak structures, using Monte Carlo model calculations to understand the spread of COVID-19, through contact interactions. The model considers the restrictions to be able to contain the contagion in the population with a stochastic interaction. The model considers populations susceptible to infection and recovery over time, with daily infection rate data. The resultsof the simulations consider four countries, India, Serbia, South Africa, and the USA. In all cases, the results reflect observed infection growth rates, the most important contribution being that the model predicts secondary and tertiary waves of infections, which can be used to plan mitigation strategies [2]. We found studies by sequential mathematical reduction of the three-parameter model (Susceptible-Infected-Recovered-Removed), in the Verhulst equation, together with the parameters presented in the epidemiological processes, by analyzing data from the European Centre for Disease Prevention and Control, applied to COVID-19. The model demonstrates a high degree of correlation, allowing for use in predictive estimates [3]. Continuing with the analysis of epidemiological models, Susceptible--Infected Unidentified--Confirmed (SUC) was used to estimate the infected population for COVID-19, specifically which people who are infected and not hospitalized, using time series of confirmed cases in China reported by the World Health Organization, using Monte Carlo simulation, with the need to prepare mitigation strategies, such as protection mechanisms and diagnostic kits, and to estimate hospital beds, medical personnel, etc. The proposed model can be used as a mathematical equation to estimate the infected population [4]. The studies dedicated to the prediction of the spread of COVID-19 are very helpful in epidemiological studies as well as in the mechanisms to mitigate the contagion, a time-dependent model is presented where a discretized method is considered: susceptible-exposed-infected-recovered-dead (SEIRD) through the time domain. The proposed model is applied to simulate and analyze the effects of COVID-19 in China and the USA based on the daily reported cases, where the prediction of the spread is demonstrated [5]. The most widely used COVID-19 mitigation mechanism is vaccines. A model of uncertainty quantification and sensitivity analysis is presented, using a mathematical model of transmission, by means of mass vaccination strategies, added to social distancing, through a spectral approach, through the expansion of cases in a polynomial type model based on statistical data, in order to have an output variable in a given time to perform sensitivity analysis. The results show that the spectral approach is a useful tool for analyzing uncertainty and analyzing the sensitivity of epidemiological models [6].