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

Ricardo Almeida,Sania Qureshi

doi:10.3390/fractalfract3040053

Ricardo Almeida, Sania Qureshi

Open Access

https://doi.org/10.3390/fractalfract3040053

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Abstract

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 R 0 < 1 , 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.

Highlights

Diseases are very much around and the burden for some of the infectious diseases has always remained substantially high in human society
Classical epidemiological models when studied in the domain of fractional calculus, wherein infinite degrees of freedom are available for the order of differentiation, are proven to have better capability to capture the more accurate behavior for the transmission dynamics of the epidemic under consideration while yielding comparatively smaller amount of error associated with the nonlinear parameter estimation
A number of mathematical models related with epidemics of different kinds are being proposed to provide health organizations to decide effective strategies to control, eliminate and eradicate the infectious diseases

Summary

Introduction

Diseases are very much around and the burden for some of the infectious diseases has always remained substantially high in human society. Classical epidemiological models when studied in the domain of fractional calculus, wherein infinite degrees of freedom are available for the order of differentiation, are proven to have better capability to capture the more accurate behavior for the transmission dynamics of the epidemic under consideration while yielding comparatively smaller amount of error associated with the nonlinear parameter estimation (see, e.g., [4,5,6,7,8,9,10]) One of those concerns is the behavior of measles propagation among human society.

Analysis of the Model Consider the set

Numerical Simulations

Findings

Conclusions