Some properties of the Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence
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RESEARCH
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Some properties of the Kermack-McKendrick epidemic model with fractional derivative and nonlinear incidence Emile Franc Doungmo Goufo* , Riëtte Maritz and Justin Munganga *
Correspondence: [email protected] Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa
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. Keywords: fractional epidemic model; Kermack-McKendrick model; nonlinear incidence; existence of solutions; wellposedness
1 Introduction and important facts Considered as one of the first compartmental models, Kermack-McKendrick epidemic model was developed in the late s with the pioneering work of Kermack and McKendrick [, ]. The model is described as the SIR model for the spread of disease, which consists of a system of three ordinary differential equations characterizing the changes in the number of susceptible (S), infected (I), and recovered (R) individuals in a given population. The model is a good one for many infectious diseases, despite its simplicity. Ever since, numerous and more complex compartmental mathematical models have been developed. For instance in biology, modeling is particularly useful in studying organs like the lungs, heart, intestinal edema and cancer, etc. Almost all these models take their source on Kermack-McKendrick’s model and serve to help gain insights into the transmission and control mechanisms of diseases like HIV, TB, malaria and their interactions with others. Then most of the works done on modeling the dynamics of epidemiological diseases © 2014 Doungmo Goufo et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, an
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