Analyzing the stochastic dynamics of COVID-19 waves in Turkey using real data and piecewise sinusoidal functions

In this study, the SIR compartmental model with vital dynamics and standard incidence is used to investigate COVID-19 transmission dynamics in Turkey. The transmission rate of the original model is replaced with a piecewise sinusoidal wave function to model the infection waves of COVID-19 experienced around the world. Multiplicative stochastic noise is added to the deterministic system to represent the uncertainty in the spread dynamics. The positivity and the boundedness of the deterministic system is given and official data from Turkish state institutions is used to compare the model estimations with the real daily case numbers. The basic reproduction number of the deterministic system is also examined with the new sinusoidal transmission coefficient. The system of nonlinear stochastic differential equations is analyzed with Euler–Maruyama and Milstein methods and simulations of the stochastic model are used to analyze the effects of the parameters on the results. It is shown that stochastic system successfully models the largest COVID-19 wave which happened in the beginning of 2022. Results show that at least 50% deviations can occur around the number of average daily cases. The approach of using sinusoidal contact rates and white noise is shown to provide an elementary and swift improvement for simple compartmental models in modeling transmission waves of infectious diseases. The new stochastic model is able to estimate the peak of daily infection percentage in Turkey (2020 population ~ 83.6 M) with around 0.01 % errors. The updated SIR model with sinusoidal infection rate is also applied to data for USA to present the ability of the new model to correspond to the major COVID19 waves in early 2022 for other countries as well. This method could provide a basis for modeling various other diseases, especially those showing seasonal patterns.