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In this work, we present a model that uses the fractional order Caputo derivative for the novel Coronavirus disease 2019 with different hospitalization strategies for severe and mild cases and incorporate an awareness program. We generalize the SEIR model of the spread of COVID-19 with a private focus on the transmissibility of people who are aware of the disease and follow preventative health measures and people who are ignorant of the disease and do not follow preventive health measures. Moreover, individuals with severe, mild symptoms and asymptomatically infected are also considered. The basic reproduction number (R 0 ) and local stability of the disease-free equilibrium (DFE) in terms of R 0 are investigated. Also, the uniqueness and existence of the solution are studied. Numerical simulations are performed by using some real values of parameters. Furthermore, the immunization of a sample of aware susceptible individuals in the proposed model to forecast the effect of the vaccination is also considered. Also, an investigation of the effect of public awareness on transmission dynamics is one of our aim in this work. Finally, a prediction about the evolution of COVID-19 in 1000 days is given. For the qualitative theory of the existence of a solution, we use some tools of nonlinear analysis, including Lipschitz criteria. Also, for the numerical interpretation, we use the Adams-Moulton-Bashforth procedure. All the numerical results are presented graphically.

In the present paper, we investigate the controllability, and observability of a non-homogeneous continuous-time fractal dynamical system (NC-TFDS). We show that the controllability is equivalent to a controllability matrix that has a full rank. We also show a relationship between controllability and fractal differential Lyapunov equation (F-DLE). We give some theorems for the observability of a NC-TFDS. Finally, we offer two examples are provided to illustrate the effectiveness of our results and compare it with the two papers Al-Zhour (2022) and ( Sadek et al., 2022 ).

Titanium dioxide is a compound of oxygen and titanium with the formula TiO2 present in nature and manufactured on an industrial scale. It is used in several fields and applications such as cosmetics, paint, food, photocatalyst, electrodes in lithium batteries, dye solar cells (DSSC), biosensors, etc., given its importance and its various fields of application, there are several methods of synthesis of TiO2 such as the sol–gel method widely used to obtain nanoparticles. In our study, on the one hand we synthesized titanium dioxide nanopowders crystallized in the anatase phase at a crystal size of 49.25 nm with success using titanium tetraisopropoxide (TTIP) as precursor by the sol–gel method. The powders obtained were analyzed by X-ray diffraction (XRD) with CuKα radiation (λ=0.15406 nm) and Fourier transform infrared spectroscopy (FTIR) in the wave number range 4000−400 cm−1, and on the other hand we present a mathematical model for the prediction of the TiO2 concentration as a function of time and the concentration of reactants by using the fractional order derivative more precise than the whole order derivative, we study the existence and the uniqueness of the solutions. In addition, we determine the points of equilibrium. Numerical simulations and their graphical representations are made to visualize the efficiency of this model.