Abstract
This manuscript investigates fractal-fractional order smoking models with relapse and harmonic mean type incidence rate under the Caputo derivative. We derive the existence and unique results about the solution for the considered model via fixed point theory. For the stability of the considered system, Ulam-Hyers (UH) approach is used. We compute the numerical solution by using fractional Adams-Bashforth method. For the simulation of the model, we consider different values of fractional order δ and fractal dimension θ by using some real values of the parameters. The proposed scheme is used to simulate the available data for some smoking community including potential, light, and quit smokers. Various graphical presentations are given to understand the dynamics of the model at various fractional orders.
Highlights
The first biological model that describes the dynamics of infectious disease was presented in 1927
Smoking is similar to infectious diseases by spreading its behavior in the population
Mahdy et al [20] found the approximate solution for a smoking model by utilizing the Sumudu transform with Caputo derivative
Summary
The first biological model that describes the dynamics of infectious disease was presented in 1927. Compared with integer-order model, fractional-order models have better fitting degree with different experimental results in signal processing, mathematical biology and engineering [16, 17,18,19] In this regard, Mahdy et al [20] found the approximate solution for a smoking model by utilizing the Sumudu transform with Caputo derivative. Motivated from the above work and from, we consider the model presented in [48] to fractal-fractional (FF) order in sense of Caputo operator which has various advantages. This model consists of four compartments, namely, people vulnerable to smoking PðtÞ, light smokers LðtÞ, smoker class SðtÞ, and quit smokers QðtÞ.
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