A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease

• Main Authors: Ricardo Almeida, Sania Qureshi
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Fractal and Fractional
• Subjects: non-integer calculus; epidemiological models; numerical simulations
• Online Access: https://www.mdpi.com/2504-3110/3/4/53

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model.