ABSTRACT
In this paper, a fractional-order two-strain epidemic model with vaccination and general non-monotonic incidence rate is analyzed. The studied problem is formulated using susceptible, infectious and recovered compartmental model. A Caputo fractional operator is incorporated in each compartment to describe the memory effect related to an epidemic evolution. First, the global existence, positivity and boundedness of solutions of the proposed model are proved. The basic reproduction numbers associated with studied problem are calculated. Four steady states are given, namely the disease-free equilibrium, the strain 1 endemic equilibrium, the strain 2 endemic equilibrium, and the endemic equilibrium associated with both strains. By considering appropriate Lyapunov functions, the global stability of the equilibrium points is proven according to the model parameters. Our modeling approach using a generalized non-monotonic incidence functions encloses a variety of fractional-order epidemic models existing in the literature. Finally, the theoretical findings are illustrated using numerical simulations.
ABSTRACT
Mass-media coverage is one of the most widely used government strategies on influencing public opinion in times of crisis. Awareness campaigns are highly influential tools to expand healthy behavior practices among individuals during epidemics and pandemics. Mathematical modeling has become an important tool in analyzing the effects of media awareness on the spread of infectious diseases. In this paper, a fractional-order epidemic model incorporating media coverage is presented and analyzed. The problem is formulated using susceptible, infectious and recovered compartmental model. A long-term memory effect modeled by a Caputo fractional derivative is included in each compartment to describe the evolution related to the individuals' experiences. The well-posedness of the model is investigated in terms of global existence, positivity, and boundedness of solutions. Moreover, the disease-free equilibrium and the endemic equilibrium points are given alongside their local stabilities. By constructing suitable Lyapunov functions, the global stability of the disease-free and endemic equilibria is proven according to the basic reproduction number R 0 . Finally, numerical simulations are performed to support our analytical findings. It was found out that the long-term memory has no effect on the stability of the equilibrium points. However, for increased values of the fractional derivative order parameter, each solution reaches its equilibrium state more rapidly. Furthermore, it was observed that an increase of the media awareness parameter, decreases the magnitude of infected individuals, and consequently, the height of the epidemic peak.
