Infectious disease epidemics are challenging for medical and public health practitioners. They require prompt treatment, but it is challenging to recognize and define epidemics in real time. Knowing the prediction of an infectious disease epidemic can evaluate and prevent the disease's impact. Mathematical models of epidemics that work in real time are important tools for preventing disease, and data-driven deep learning enables practical algorithms for identifying parameters in mathematical models. In this paper, the SIR model was reduced to a logistic differential equation involving a constant parameter and a time-dependent function. The time-dependent function leads to constant, rational, and birational models. These models use several constant parameters from the available data to predict the time and number of people reported to be infected with the COVID-19 Omicron variant. Two out of these three models, rational and birational, provide accurate predictions for countries that practice strict mitigation measures, but fail to provide accurate predictions for countries that practice partial mitigation measures. Therefore, we introduce a time-series model based on neural networks to predict the time and number of people reported to be infected with the COVID-19 Omicron variant in a given country that practices both partial and strict mitigation measures. A logistics-informed neural network algorithm was also introduced. This algorithm takes as input the daily and cumulative number of people who are reported to be infected with the COVID-19 Omicron variant in the given country. The algorithm helps determine the analytical solution involving several constant parameters for each model from the available data. The accuracy of these models is demonstrated using error metrics on Omicron variant data for Portugal, Italy, and China. Our findings demonstrate that the constant model could not accurately predict the daily or cumulative infections of the COVID-19 Omicron variant in the observed country because of the long series of existing data of the epidemics. However, the rational and birational models accurately predicted cumulative infections in countries adopting strict mitigation measures, but they fell short in predicting the daily infections. Furthermore, both models performed poorly in countries with partial mitigation measures. Notably, the time-series model stood out for its versatility, effectively predicting both daily and cumulative infections in countries irrespective of the stringency of their mitigation measures.