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In this paper, we formulate a new model of a particular type of influenza virus called AH1N1/09 in the framework of the four classes consisting of susceptible, exposed, infectious and recovered people. For the first time, we here investigate this model with the help of the advanced operators entitled the fractal–fractional operators with two fractal and fractional orders via the power law type kernels. The existence of solution for the mentioned fractal–fractional model of AH1N1/09 is studied by some special mappings such as ϕ − ψ -contractions and ϕ -admissibles. The Leray–Schauder theorem is also applied for this aim. After investigating the stability criteria in four versions, to approximate the desired numerical solutions, we implement Adams–Bashforth (AB) scheme and simulate the graphs for different data on the fractal and fractional orders. Lastly, we convert our fractal–fractional AH1N1/09 model into a fractional model via the generalized Liouville–Caputo-type (GLC-type) operators and then, we simulate new graphs caused by the new numerical scheme called Kumar–Erturk method. [ABSTRACT FROM AUTHOR]

We employ Mittag-Leffler type kernels to solve a system of fractional differential equations using fractal–fractional (FF) operators with two fractal and fractional orders. Using the notion of FF-derivatives with nonsingular and nonlocal fading memory, a model of three polluted lakes with one source of pollution is investigated. The properties of a non-decreasing and compact mapping are used in order to prove the existence of a solution for the FF-model of polluted lake system. For this purpose, the Leray–Schauder theorem is used. After exploring stability requirements in four versions, the proposed model of polluted lakes system is then simulated using two new numerical techniques based on Adams–Bashforth and Newton polynomials methods. The effect of fractal–fractional differentiation is illustrated numerically. Moreover, the effect of the FF-derivatives is shown under three specific input models of the pollutant: linear, exponentially decaying, and periodic. • Mittag-Leffler fractal–fractional (FF) modeling of polluted lakes. • Well posedness of the polluted lake system with nonsingular and nonlocal fading memory. • Stability of the polluted lakes system. • New numerical techniques based on Adams–Bashforth and Newton polynomials methods. • The effect of linear, exponentially decaying, and periodic pollutants. [ABSTRACT FROM AUTHOR]

Mumps is the most common cause of acquired unilateral deafness in children, in which hearing loss occurs at all auditory frequencies. We use a box model to model hearing loss in children caused by the mumps virus, and since the fractional-order derivative retains the effect of system memory, we use the Caputo–Fabrizio fractional derivative in this modeling. In the beginning, we compute the basic reproduction number R 0 and equilibrium points of the system and investigate the stability of the system at the equilibrium point. By utilizing the Picard–Lindelof technique, we prove the existence an unique solution for given fractional CF -system of hearing loss model and investigate the stability of iterative method by fixed point theory. The optimal control of the system is determined by considering the treatment as a control strategy to reduce the number of infected people. Using the Euler method for the fractional-order Caputo–Fabrizio derivative, the approximate solution of the system is calculated. We present a numerical simulation for the transmission of disease with respect to the transmission rate and the basic reproduction number in two cases R 0 < 1 and R 0 > 1. To investigate the effect of fractional order derivative on the behavior and value of each of the variables in Model 2, we calculate the results for several fractional order derivatives and compare the results. Also, considering the importance of reproduction number in the continuation of disease transmission, we analyze the sensitivity of R 0 respect to each of the model parameters and determine the impact of each parameter. [ABSTRACT FROM AUTHOR]