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The monkeypox virus (MPXV) is what causes monkeypox (MPX) disease, which is comparable to both smallpox and cowpox. Using classical, fractional-order and complex order differential equations, we offer a deterministic mathematical model of the monkeypox virus in this study to research its possible breakouts in United States. The complex order derivative makes the fractional order derivative and the integer order derivative more common when the imaginary part of the complex order equals zero and when the real part in complex order derivatives is zero in this case, the new behaviour appears that doesn't appear in integer and fractional order derivatives. Eight nonlinear complex order differential equations make up this model consisting of humans and rodents population sizes. The population of humans N h is divided into five different classes. The population of rodent N r is divided into three classes, and the derivatives are described in the Atangana-Baleanu-Caputo sense and Mittag-Leffler kernels are employed. Numerical methods to simulate complex order systems such as The standard and nonstandard two-step Lagrange interpolation methods are utilised to fit the model. The basic reproduction number of the model is given. The stability of the suggested model's disease-free equilibrium point is shown. Finally, we study the duration and monkeypox outbreak's transmission pattern in the United States from June 13 to Sep 16, 2022, numerical simulations to illustrate our findings are presented. The results show that keeping diseased people apart from the general population decreases the spread of disease. [ABSTRACT FROM AUTHOR]

In this artical, we present a novel complex order nonlinear mathematical model of HIV/AIDS and Malaria co-infection with multi-delay. The proposed model is formulated as a system of twelve complex order differential equations. The complex order derivative is defined in the Atangana-Baleanu-Caputo sense. Two numerical methods are utilized for simulating the determined system; standard two-step Lagrange interpolation method and the non-standard two step Lagrange interpolation method. In order to establish numerical simulations, the theoretical results and comparative studies are given. [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]

In this article, optimal control for variable order fractional multi-delay mathematical model for the co-infection of HIV/AIDS and malaria is presented. This model consists of twelve differential equations, where the variable order derivative are in the sense of Caputo. Three control variables are presented in this model to minimize the number of the co-infected individuals showing no symptoms of AIDS, the infected individuals with malaria, and the individuals asymptomatically infected with HIV/AIDS. Necessary conditions for the control problem are derived. The Grünwald-Letnikov nonstandard finite difference scheme is constructed to simulating the proposed optimal control system. The stability of the proposed scheme is proved. In order to validate the theoretical results numerical simulations and comparative studies are given. [ABSTRACT FROM AUTHOR]

In this work, optimal control for fractional order model of malaria transmission dynamics with modified parameters is presented. The fractional derivative is defined in the Atangana-Beleanu sense. Two control variables are presented in this model to minimize the number of the population of low-risk infectious humans and high-risk infectious humans. Necessary conditions for the control problem are drived. Two types of nonstandard finite difference method for simulating the proposed optimal system with Mittag-Leffler kernels are presented. In order to validate the theoretical results numerical simulations and comparative studies are given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we apply the dynamical systems method (DSM) proposed by A. G. Ramm, and the variational regularization method (VRM), to obtain numerical solution to some singularly perturbed ill-posed problems contaminated by noise. The results obtained by these methods are compared to the exact solution for the model problems. It is found that the dynamical systems method is preferable because it is easier to apply, highly stable, robust, and it always converges to the solution even for large size models. [Copyright &y& Elsevier]

• Optimal control of the hybrid variable-order fractional model of Coronavirus is presented. • Existence, uniqueness, boundedness, positivity, local and global stability of the solutions of the proposed model problem are proved. • Two control variables are considered to reduce the transmission of infection into healthy people. • Necessary conditions for the control problem are given. • Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to study the optimality systems. • Stability analysis and convergence for the proposed method are proved. • Comparative studies between the obtained approximate solutions and the WHO real data for Egypt are given. • Numerical simulations are given. A novel coronavirus is a serious global issue and has a negative impact on the economy of Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 were foreigners. Recently, many mathematical models have been considered to better understand coronavirus infection. Most of these models are based on classical integer-order derivatives which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using new variable-order fractional derivatives. This paper presents an optimal control problem of the hybrid variable-order fractional model of Coronavirus. The variable-order fractional operator is modified by an auxiliary parameter in order to satisfy the dimensional matching between the both sides of the resultant variable-order fractional equations. Existence, uniqueness, boundedness, positivity, local and global stability of the solutions are proved. Two control variables are considered to reduce the transmission of infection into healthy people. To approximate the new hybrid variable-order operator, Grünwald-Letnikov approximation is used. Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to solve the optimality system. Numerical examples and comparative studies for testing the applicability of the utilized methods and to show the simplicity of these approximation approaches are presented. Moreover, by using the proposed methods we can concluded that, the model given in this paper describes well the confirmed real data given by WHO about Egypt. [ABSTRACT FROM AUTHOR]