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In this research, a new SICA model is proposed in a fractional framework for the HIV/AIDS transmission dynamics. The new model involves a Caputo‐type fractional derivative with a Mittag‐Leffler function as its nonsingular kernel. In addition, the resultant fractional equations avoid dimensional mismatching by using an auxiliary parameter. Furthermore, the nonnegativity of the solution, the equilibrium points, and their stability are studied. Additionally, we implement the model by a powerful approximation scheme based on the product‐integration rule. Comparative results with the real experimental observations, happened in Cape Verde Islands from 1987 to 2014, verify that the new fractional model is more efficient than the pre‐existent classical model with ordinary time derivatives. More to the point, the fractional order itself provides a degree of flexibility affecting the performance of the model, which is helpful to exhibit the hidden features of the disease transmission in an accurate, appropriate manner. [ABSTRACT FROM AUTHOR]

Zika virus infection is a vastly transmitted disease among humans. It was carried worldwide by international travelers. In 2016, Zika virus infection was present in more than 20 countries and territories in America. Thousands of cases were diagnosed in Cabo Verde, western Africa. Fifty-seven regions suffered from Zika virus in 2020, and the World Health Organization reported more than one hundred thousand cases worldwide. In this work, the modeling and transmission dynamics of Zika virus are studied dynamically and numerically. Positivity, boundedness, reproduction number, equilibria, and local stability are part of the numerical analysis. New nonstandard numerical techniques are examined for the said model. The primary purpose is to maintain the continuous model's behavior and dynamical properties. The proposed nonstandard finite approximation is studied according to the consistency and local stability of the solutions. Some numerical examples clearly show the improvement of the new schemes compared to other well-known methods. [ABSTRACT FROM AUTHOR]

This paper deals with a new mathematical model for a dengue fever outbreak based on a system of fractional differential equations. The equilibrium points and stability of the new system are studied. To simulate this model, a new and efficient numerical method is provided and its stability and convergence are proved. According to a real outbreak on the Cape Verde Islands occurred in year 2009, the new model is examined for a period of three months by using singular or nonsingular kernels in the definition of derivative operator. Simulation results show that the proposed formalism with exponential kernel agrees well with the real data in the early stage of the epidemic while the Mittag-Leffler kernel fits the reality for the later part of the time interval. Hence, the new framework in a hybrid manner can properly simulate the dynamics of the disease in the whole of the time interval. In order to stabilize the disease-free equilibrium point of the system under investigation, two control strategies are suggested. Numerical simulations verify that the proposed stabilizing controllers are efficient and provide significantly remarkable results. • A novel fractional model of the dengue fever outbreak is proposed. • A new numerical method is provided and its stability and convergence are proved. • Two stabilizing controllers are suggested for the disease-free equilibrium point. • The new framework can simulate the dynamics of the disease properly. • The stabilizing controllers are efficient and provide satisfactory results. [ABSTRACT FROM AUTHOR]