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Psoriasis is an immune-mediated genetic disease, characterized by its manifestation on the skin, the joints, or both. In this paper, our primary aim is to increase awareness about the intricate nature of this multifaceted condition, highlight the potential of therapeutic approaches, and examine the factors affecting the future course of psoriasis. This paper introduces a mathematical model for psoriasis, formulated by fractional order differential equations (FODE) with Caputo sense. The model also includes the drug effect of immune-boosting drugs on psoriasis. It has been shown that the solution of the proposed system exists and is unique, and its positivity and boundedness have been proven. Additionally, the local stability and global stability of the co-existing equilibrium point have been investigated. Numerical solutions have been conducted using the Adams Bashforth PECE method to analyze the influence of fractional order derivatives (FODs) and distinct parameters on population dynamics. The graphics have been acquired using the L1 scheme, incorporating a memory trace (MT) mechanism capable of comprehensively capturing and amalgamating historical system dynamics to visualize the memory trace in detail. One of the results deduced from this paper is that the MT disappears when α equals 1. Upon decreasing the fractional order α from 1, the MT experiences a non-linear augmentation starting from zero. This observed MT emphasizes the distinction between derivatives of fractional and integer orders. Within the proposed model, our findings suggest that introducing the immune booster drug efficiently controls psoriasis. Furthermore, when appropriate clinical patient data are available, the proposed results can be used for a specific psoriasis patient. Our study suggests that treatment with the drug may be a new insight into psoriasis treatment and may be proposed as a treatment policy for future clinical trials.

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This study presents a novel approach to investigating COVID-19 and Cholera disease. In this situation, a fractional-order model is created to investigate the COVID-19 and Cholera outbreaks in Congo. The existence, uniqueness, positivity, and boundedness of the solution are studied. The equilibrium points and their stability conditions are achieved. Subsequently, the basic reproduction number (the virus transmission coefficient) is calculated that simply refers to the number of people, to whom an infected person can make infected, as R 0 = 6 . 7442389 e - 10 by using the next generation matrix method. Next, the sensitivity analysis of the parameters is performed according to R 0 . To determine the values of the parameters in the model, the least squares curve fitting method is utilized. A total of 22 parameter values in the model are estimated by using real Cholera data from Congo. Finally, to find out the dynamic behavior of the system, numerical simulations are presented. The outcome of the study indicates that the severity of the Cholera epidemic cases will decrease with the decrease in cases of COVID-19, through the implementation and follow-up of safety measures that have been taken to reduce COVID-19 cases.

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In this study, a new approach to COVID-19 pandemic is presented. In this context, a fractional order pandemic model is developed to examine the spread of COVID-19 with and without Omicron variant and its relationship with heart attack using real data from the United Kingdom. In the model, heart attack is adopted by considering its relationship with the quarantine strategy. Then, the existence, uniqueness, positivity and boundedness of the solution are studied. The equilibrium points and their stability conditions are achieved. Subsequently, we calculate the basic reproduction number (the virus transmission coefficient) that simply refers to the number of people, to whom an infected person can make infected, as R 0 = 3.6456 by using the next generation matrix method. Next, we consider the sensitivity analysis of the parameters according to R 0 . In order to determine the values of the parameters in the model, the least squares curve fitting method, which is one of the leading methods in parameter estimation, is benefited. A total of 21 parameter values in the model are estimated by using real Omicron data from the United Kingdom. Moreover, in order to highlight the advantages of using fractional differential equations, applications related to memory trace and hereditary properties are given. Finally, the numerical simulations are presented to examine the dynamic behavior of the system. As a result of numerical simulations, an increase in the number of people who have heart attacks is observed when Omicron cases were first seen. In the future, it is estimated that the risk of heart attack will decrease as the cases of Omicron decrease.

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In the present paper, interactions between COVID-19 and diabetes are investigated using real data from Turkey. Firstly, a fractional order pandemic model is developed both to examine the spread of COVID-19 and its relationship with diabetes. In the model, diabetes with and without complications are adopted by considering their relationship with the quarantine strategy. Then, the existence and uniqueness of solution are examined by using the fixed point theory. The dynamic behaviors of the equilibria and their stability analysis are studied. What is more, with the help of least-squares curve fitting technique (LSCFT), the fitting of the parameters is implemented to predict the direction of COVID-19 by using more accurately generated parameters. By trying to minimize the mean absolute relative error between the plotted curve for the infected class solution and the actual data of COVID-19, the optimal values of the parameters used in numerical simulations are acquired successfully. In addition, the numerical solution of the mentioned model is achieved through the Adams-Bashforth-Moulton predictor-corrector method. Meanwhile, the sensitivity analysis of the parameters according to the reproduction number is given. Moreover, numerical simulations of the model are obtained and the biological interpretations explaining the effects of model parameters are performed. Finally, in order to point out the advantages of the fractional order modeling, the memory trace and hereditary traits are taken into consideration. By doing so, the effect of the different fractional order derivatives on the COVID-19 pandemic and diabetes are investigated.

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