A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence

Abstract

The main objective of this research is to investigate a new fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. The new model involves two distinct populations, diabetics and nondiabetics, while each of them consists of seven tuberculosis states: susceptible, fast and slow latent, actively tuberculosis infection, recovered, fast latent after reinfection, and drug-resistant. The fractional operator is also considered a recently introduced one with Mittag–Leffler nonsingular kernel. The basic properties of the new model including non-negative and bounded solution, invariant region, and equilibrium points are discussed thoroughly. To solve and simulate the proposed model, a new and efficient numerical method is established based on the product-integration rule. Numerical simulations are presented, and some discussions are given from the mathematical and biological viewpoints. Next, an optimal control problem is defined for the new model by introducing four control variables reducing the number of infected individuals. For the control problem, the necessary and sufficient conditions are derived and numerical simulations are given to verify the theoretical analysis.

On the existence of solutions for a pointwise defined multi-singular integro-differential equation with integral boundary condition

“…We know that many researchers are working on fractional differential equarions from different point of view (see, for example, ( [1][2][3][4][5][6][7][8][9][10][11][12] and [13]). In 2015, a new fractional derivative introduced entitled Caputo-Fabrizio and some researchers tried to obtain new techniques for studying of distinct integro-differential equations via the new derivation (see, for example, [14][15][16][17][18][19]) and new fractional models and optimal controls of different phenomena with the non-singular derivative operator (see, for example, [20][21][22][23][24] and [25]). Also, there has been published a lot of work about physical studies on fractional calculus and new aspects of fractional different models with Mittag-Leffler law (see, for example, [26][27][28][29] and [30]).…”

A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods

“…Such models have been extensively studied in the literature. On the other hand, it was shown that in many applications, the use of fractional derivatives provide more realistic models than those obtained via standard derivative (see, eg, previous studies and the references therein). Because of the above reason, the study of fractional models received a great attention form many reserchers.…”

Analysis and dynamical behavior of fractional‐order cancer model with vaccine strategy

“…We analyze the interactions between various tumor cell populations and immune system via a system of fractional differential equations (FDEs), 17 mathematical model for a dengue fever outbreak based on a system of fractional differential equations, 18 and fractional mathematical model involving a nonsingular derivative operator to discuss the clinical implications of diabetes and tuberculosis coexistence. 19 We consider a recently introduced fractional operator with Mittag-Leffler nonsingular kernel. We indicate that the complex behavior of many physical systems is realistically demonstrated via the fractional calculus modeling and also developed the fractional quantum mechanics and proved the hermiticity property of the Hamilton operator in fractional sense, 20,21 and some others techniques used in other works 22,23 are very helpful to demonstrate the results.…”