Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is to attest, through a statistical approach, how the hysteresis phenomenon, which describes a type of memory effect present in biological systems, can be treated by fractional calculus. We also analyse the contribution of the historical values of a function in the evaluation of fractional operators according to their order. To illustrate the efficiency of this non-integer order calculus, we consider the SIR (susceptible-infected-recovered) compartmental model which is widely used in epidemiology. We employ this compartmental model to study the dynamics of the spread of COVID-19 in some countries, one version with memory and one without memory.

Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is, through a statistical approach, to attest how the hysteresis phenomenon, which describes the memory effect present in biological systems, can be treated by fractional calculus, and to analyze the contribution of the historical values of a function in the evaluation of fractional operators according their order. In order to illustrate the efficiency of this non-integer order calculus, we consider the SIR (Susceptible-Infected-Recovered) compartmental model which is widely used in epidemiology. We employ SIR models to model the dynamics, with and without memory, of the spread of Covid-19 in some countries.

Introduction: This work aims to develop a biomathematical transmission model of COVID-19, in the State of Sergipe, Brazil, to estimate the distribution of cases over time and project the impact on the spread of the epidemic outbreak due to interventions and control measures over the local population. Methods: This is an epidemiological mathematical modeling study conducted to analyze the dynamics of the accumulated cases of COVID-19, which used a logistic growth model that adds a term of withdrawal of individuals as a control measure. Three possible COVID-19 propagation scenarios were simulated based on three different rates of withdrawal of individuals. They were adjusted with real data of the infected and measures of control over the population. Results: The lockdown would be the best scenario, with a lower incidence of infected people, when compared to the other measures. The number of infected people would grow slowly over the months, and the number of symptomatic individuals in this scenario would be 40,265 cases. We noticed that the State of Sergipe is still in the initial stage of the disease in the scenarios. It was possible to observe that the peak of cases and the equilibrium, in the current situation of social isolation, will occur when reaching the new support capacity, at the end of August in approximately 1,171,353 infected individuals. Conclusions: We established that lockdown is the intervention with the highest ability to mitigate the spread of the virus among the population.

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