In this paper, daily confirmed cases of COVID-19 in different countries are modelled using different mathematical regression models. The curve fitting is used as a prediction tool for modeling both past and upcoming coronavirus waves. According to virus spreading and average annual temperatures, countries under study are classified into three main categories. First category, the first wave of the coronavirus takes about two-year seasons (about 180 days) to complete a viral cycle. Second category, the first wave of the coronavirus takes about one-year season (about 90 days) to complete the first viral cycle with higher virus spreading rate. These countries take stopping periods with low virus spreading rate. Third category, countries that take the highest virus spreading rate and the viral cycle complete without stopping periods. Finally, predictions of different upcoming scenarios are made and compared with actual current smoothed daily confirmed cases in these countries.

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In this paper, we investigate the stochastic nature of the COVID-19 temporal dynamics by generating a fractional-order dynamic model and a fractional-order-stochastic model. Initially, we considered the first and second vaccination doses as multiple vaccinations were initiated worldwide. The concerned models are then tested for the Saudi Arabia second virus wave, which is assumed to start on 1st March 2021. Four daily vaccination scenarios for the first and second dose are assumed for 100 days from the wave beginning. One of these scenarios is based on function optimization using the invasive weed optimization algorithm (IWO). After that, we numerically solve the established models using the fractional Euler method and the Euler-Murayama method. Finally, the obtained virus dynamics using the assumed scenarios and the real one started by the government are compared. The optimized scenario using the IWO effectively minimizes the predicted cumulative wave infections with a 4.4 % lower number of used vaccination doses.

This work predicts the dynamics of the COVID-19 under widespread vaccination to anticipate the virus's current and future waves. We focused on establishing two population-based models for predictions: the fractional-order model and the fractional-order stochastic model. Based on dose efficacy, which is one of the main imposed assumptions in our study, some vaccinated people will probably be exposed to infection by the same viral wave. We validated the generated models by applying them to the current viral wave in Egypt. We assumed that the Egyptian current wave began on 10 th September 2021. Using current actual data and varying our models’ fractional orders, we generate different predicted wave scenarios. The numerical solution of our models is obtained using the fractional Euler method and the fractional Euler Maruyama method. At the end, we compared the current predicted wave under a high vaccination rate with the previous viral wave. Through this comparison, the vaccination control effect is quantified.

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