Abstract
We study epidemic Susceptible–Infected–Susceptible (SIS) models in the fractional setting. The novelty is to consider models in which the susceptible and infected populations evolve according to different fractional orders. We study a model based on the Caputo derivative, for which we establish existence results of the solutions. Furthermore, we investigate a model based on the Caputo–Fabrizio operator, for which we provide existence of solutions and a study of the equilibria. Both models can be framed in the context of SIS models with time-varying total population, in which the competition between birth and death rates is macroscopically described by the fractional orders of the derivatives. Numerical simulations for both models and a direct numerical comparison are also provided.
Highlights
The interest of the scientific community in mathematical modeling for epidemiology has grown considerably in recent years
The use of mathematical models for epidemiology is useful to predict the behavior of an infection and make strategic decisions in emergency situations to limit the spread of the disease, which is microscopically modeled by the fractional order of the derivative
One model is based on the Caputo derivative, for which we establish the existence of results of the solutions and provide numerical simulations
Summary
The interest of the scientific community in mathematical modeling for epidemiology has grown considerably in recent years. One model is based on the Caputo derivative, for which we establish the existence of results of the solutions and provide numerical simulations. We refer to [19] for a SIR-type model using the Caputo–Fabrizio fractional operator with assumptions about births and deaths: the paper contains numerical simulations for a selected set of initial conditions. The peculiarity of this fractional operator is the presence of a non-singular kernel in contrast with the Caputo derivative, in which a singular kernel appears in the definition.
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