On Differential Equation Models with Memory Effects in Epidemiology

The paper has two main objectives. Firstly, it aims to identify appropriate fractional derivatives for epidemiological models and to investigate their advantages over ordinary derivatives, particularly in terms of the accuracy of epidemic simulations. The choice of a fractional derivative is motivated by its ability to represent the memory effects inherent in epidemiological processes. Different fractional derivatives may induce substantially different memory mechanisms; therefore, the relation between the choice of the derivative and the resulting memory effect is discussed in detail. The results are complemented by stability studies. The obtained uniqueness results provide a theoretical basis for the application of numerical methods, while numerical simulations demonstrate the impact of the fractional-order parameters. Secondly, the paper presents an SIQR epidemiological model incorporating quarantine, fractional derivatives, and multiple delays. The simultaneous use of fractional derivatives and delays makes it possible to account for both distributed memory effects and specific temporal lags within a unified model framework.