الفهرس 
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Abstract
In this thesis which consists of three chapters, A novel hybrid variable-order fractional multi-vaccination model for COVID-19 is presented in chapter 2, in order to further explore the spread of Covid-19. The main advantage of the hybrid variable-order fractional operator is, it can be defined as a linear combination of the variable-order integral of Riemann-Liouville and the variable-order Caputo derivative, it is one of the most effective and reliable operators and it is more general than Caputo fractional operator. The proposed model’s dynamics are improved, and its complexity is increased by employing variable order fractional derivatives. Furthermore, the variable order fractional Caputo operator can be derived as a special case from the CPC operator. For the compatible with the physical model, a new parameter σ is added. The proposed model is numerically studied using CPC-θFDM and GRK4M. CPC-θFDM depends on the values of the factor θ. It can be explicit θ=1 or fully implicit θ=0 with a large stability region. We compared our results with the real data from the state of Texas in the United States. Moreover, the results obtained from the CPC-θFDM are more stable than the results obtained from the proposed method in. As a result, some graphs are provided for various linear and non-linear variable order derivatives. In Chapter 3, special attention is given to studying the hybrid variable-order FracInt mathematical model of vaccination Covid-19 and their optimal control. The optimality conditions are numerically derived. CPC-GLFDM is developed to study the optimality system. Generally, the numerical results show that the proposed FracInt system is effective in controlling the disease. Generally, the numerical findings demonstrate that the FOC system performs better only during a portion of the time period. In order to combat infection, we, therefore, suggest a system in which the derivative order changes over the course of the time interval, becoming fractional or integer as appropriate. This FracInt-named variable-order fractional model appears to be the most successful at controlling the illness. In the future, the presented studies can be extended to FracInt delay optimal control and examine the impact of multiple vaccination strategies on the dynamics of COVID-19 in a population. Also, we can study mathematical models of Cancer and their optimal control.