ABSTRACT
In this manuscript we use triangular norms to model contact between susceptible and infected individuals in the susceptible-infected-recovered (SIR) epidemiological model. In the classical SIR model, the encounter between susceptible and infected individuals is traditionally modeled by the product of their densities ([Formula Omitted]). That is, the encounter is modeled by the product t-norm. We use the COVID-19 data and extended versions of the SIR model whose encounters are modeled by four triangular norms, namely, product, minimum, and Frank and Hamacher t-norms, to analyze the scenario in three countries: 1) Germany;2) Italy;3) Switzerland. We compare all versions of the SIR model based on these triangular norms, and we analyze their effectiveness in fitting data and determining important parameters for the pandemic, such as the basic and effective reproduction number. In addition, Frank and Hamacher triangular norms present an auxiliary parameter that can be interpreted as an indicator of control measure, which we show to be important in the current pandemic scenario.
ABSTRACT
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. 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 this compartmental model to study the dynamics of the spread of Covid-19 in some countries, one version with memory and one without memory.