Some properties of the Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence

Abstract

Abstract Kermack-McKendrick epidemic model is considered as the basis from which many other compartmental models were developed. But the development of fractional calculus applied to mathematical epidemiology is still ongoing and relatively recent. We provide, in this article, some interesting and useful properties of the Kermack-McKendrick epidemic model with nonlinear incidence and fractional derivative order in the sense of Caputo. In the process, we used the generalized mean value theorem (Odibat and Shawagfeh in Appl. Math. Comput. 186:286-293, 2007) extended to fractional calculus to conclude some of the properties. A model of the Kermack-McKendrick with zero immunity is also investigated, where we study the existence of equilibrium points in terms of the nonlinear incidence function. We also establish the condition for the disease free equilibrium to be asymptotically stable and provide the expression of the basic reproduction number. Finally, numerical simulations of the monotonic behavior of the infection are provided for different values of the fractional derivative order α (

0 ≤ α < 1

). Comparing to the model with first-order derivative, there is a similar evolution for close values of α. The results obtained may help to analyze more complex fractional epidemic models.