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[Display omitted] • A novel variable-order nonlinear model of dengue virus is analyzed as an optimal control problem. • The bang-bang control is suggested to minimize the dose and duration of the intervention for model. • Necessary conditions for the control problem are derived. • Two numerical methods are constructed to solve the proposed optimal control problem. • Comparative studies and numerical simulations are implemented. Dengue and Malaria are the most important mosquito-borne viral diseases affecting humans. Fever is transmitted between human hosts by infected female aedes mosquitoes. The modeling study of viral infections is very useful to show how the virus replicates in an infected individual and how the human antibody response acts to control that replication, which antibody playing a key role in controlling infection. Optimal control of a novel variable-order nonlinear model of dengue virus is studied in the present work. Bang-bang control is suggested to minimize the viral infection as well as quick clearance of the virus from the host. Necessary conditions for the control problem are given. The variable-order derivatives are given in the sense of Caputo. Moreover, the parameters of the proposed model are dependent on the same variable-order fractional power. Two numerical schemes are constructed for solving the optimality systems. Comparative studies and numerical simulations are implemented. The variable-order fractional derivative can be describe the effects of long variable memory of time dependent systems than the integer order and fractional order derivatives. Both the nonstandard generalized fourth order Runge-Kutta and the nonstandard generalized Euler methods are presented. We have successfully applied a kind of Pontryagin's maximum principle with bang-bang control and were able to reduce the viraemia level by adding the dose of DI particles. The nonstandard generalized fourth order Runge-Kutta method has the best results than nonstandard generalized Euler method. The combination of the variable-order fractional derivative and bang-bang control in the Dengue mathematical model improves the dynamics of the model. The nonstandard generalized Euler method and the nonstandard generalized fourth order Runge-Kutta method can be used to study the variable order fractional optimal control problem simply. [ABSTRACT FROM AUTHOR]

[Display omitted] • A novel mathematical model of Corona virus with new hybrid fractional operator derivative are presented. • Three control variables are presented to minimize the number of infected population. • Necessary control conditions are derived. • Two numerical methods are constructed to study the behavior of the obtained fractional optimality system. • The stability of the proposed methods are proved. • Numerical simulations and comparative studies are given. Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann–Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method. The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists. [ABSTRACT FROM AUTHOR]

The aim of this work is to present numerical treatments to a complex order fractional nonlinear one-dimensional problem of Burgers' equations. A new parameter σ t is presented in order to be consistent with the physical model problem. This parameter characterizes the existence of fractional structures in the equations. A relation between the parameter σ t and the time derivative complex order is derived. An unconditionally stable numerical scheme using a kind of weighted average nonstandard finite-difference discretization is presented. Stability analysis of this method is studied. Numerical simulations are given to confirm the reliability of the proposed method. [ABSTRACT FROM AUTHOR]

• Optimal control problem for the fractional TB infection model is presented. • The nonstandard two-step Lagrange interpolation method is presented for numerically solving the optimality system. • Necessary and sufficient conditions that guarantee the existence and the uniqueness of the solution of the control problem are given. • Four controls variables are proposed to minimize the cost of interventions. • New numerical schemes for simulating fractional order optimality system with Mittag-Leffler kernel are given. The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]