The impact of a power law-induced memory effect on the SARS-CoV-2 transmission

It is well established that COVID-19 incidence data follows some power law growth pattern. Therefore, it is natural to believe that the COVID-19 transmission process follows some power law. However, we found no existing model on COVID-19 with a power law effect only in the disease transmission process. Inevitably, it is not clear how this power law effect in disease transmission can influence multiple COVID-19 waves in a location. In this context, we developed a completely new COVID-19 model where a force of infection function in disease transmission follows some power law. Furthermore, different realistic epidemiological scenarios like imperfect social distancing among home-quarantined individuals, disease awareness, vaccination, treatment, and possible reinfection of the recovered population are also considered in the model. Applying some recent techniques, we showed that the proposed system converted to a COVID-19 model with fractional order disease transmission, where order of the fractional derivative ( $$ ) in the force of infection function represents the memory effect in disease transmission. We studied some mathematical properties of this newly formulated model and determined the basic reproduction number ( $$ ). Furthermore, we estimated several epidemiological parameters of the newly developed fractional order model (including memory index $$ ) by fitting the model to the daily reported COVID-19 cases from Russia, South Africa, UK, and USA, respectively, for the time period March 01, 2020, till December 01, 2021. Variance-based Sobol's global sensitivity analysis technique is used to measure the effect of different important model parameters (including $$ ) on the number of COVID-19 waves in a location ( $$ ). Our findings suggest that $$ along with the average transmission rate of the undetected (symptomatic and asymptomatic) cases in the community ( $$ ) are mainly influencing multiple COVID-19 waves in those four locations. Numerically, we identified the regions in the parameter space of $$ and $$ for which multiple COVID-19 waves are occurring in those four locations. Furthermore, our findings suggested that increasing memory effect in disease transmission ( $$ $$ 0) may decrease the possibility of multiple COVID-19 waves and as well as reduce the severity of disease transmission in those four locations. Based on all the results, we try to identify a few non-pharmaceutical control strategies that may reduce the risk of further SARS-CoV-2 waves in Russia, South Africa, UK, and USA, respectively.