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  3. Volume 11, Number 2, 2024
  4. Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration

Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration

MMC.
2024;
Volume 11, Number 2
: pp. 492–504
https://doi.org/10.23939/mmc2024.02.492
Received: April 27, 2023
Revised: November 03, 2023
Accepted: November 08, 2023

El Baz O., Khaloufi I., Kouidere A., Laarabi H., Rachik M.  Analysis and optimal control problem for a fractional mathematical model of tuberculosis with smoking consideration.  Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 492–504 (2024)

Authors:
  1. O. El Baz
  2. I. Khaloufi
  3. A. Kouidere
  4. H. Laarabi
  5. M. Rachik
1
Laboratory of Analysis Modeling and Simulation, Casablanca, Morocco
2
Laboratory of Analysis, Modeling and Simulation, Casablanca, Morocco
3
Laboratory of Analysis Modeling and Simulation, Casablanca, Morocco
4
Laboratory of Analysis, Modeling and Simulation, Casablanca, Morocco
5
Laboratory of Analysis, Modeling and Simulation, Casablanca, Morocco

This article studies a mathematical model of the fractional order of tuberculosis (TB).  It describes the dynamics of the spread of tuberculosis among smokers.  The purpose of this research is to protect vulnerable people against the virus.  According to the survey results, the required model has an equilibrium point: the disease-free equilibrium point $E_f$.  We also analyze the local stability of this equilibrium point of the model, using the basic reproduction number $\mathcal{R}_{0}$ calculated according to the new generation method.  In our model, we include three controls that represent: restricting individual contact, treatment, and sensitization.  This article aims at reducing the number of infected smokers and non-smokers using an optimal control strategy and a fractional derivation.  The maximum principle of Pontryagin is used to describe optimal controls with Caputo-derived fractional over time and the optimal system is resolved iteratively.  The numerical simulation is presented according to the method presented by Matlab.

Caputo fractional derivative
optimal control
tuberculosis
smoking
contagious virus
local stability
dynamic system
infectious diseases
stability
free equilibrium
Pontryagin maximum
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