Modeling and analysis of the fractional-order epidemic model to investigate mutual influence in HIV/HCV co-infection

BACKGROUND AND OBJECTIVE
Hepatitis virus infections are affecting millions of people worldwide, causing death, disability, and considerable expenditure. Chronic infection with hepatitis C virus (HCV) can cause severe public health problems because of their high prevalence and poor long-term clinical outcomes. Thus a fractional-order epidemic model of the hepatitis C virus involving partial immunity under the influence of memory effect to know the transmission patterns and prevalence of HCV infection is studied. Investigating the transmission dynamics of HCV makes the issue more interesting. The HCV epidemic model and worldwide dynamics are examined in this study. Calculate the basic reproduction number for the HCV model using the next-generation matrix technique. We determine the model's global dynamics using reproduction numbers, the Lyapunov functional approach, and the Routh-Hurwitz criterion. The model's reproduction number shows how the disease progresses.


METHODS
A fractional differential equation model of HCV infection has been created. Maximum relevant parameters, such as fractional power, HCV transmission rate, reproduction number, etc., influencing the dynamic process, have been incorporated. The model's numerical solutions are obtained using the fractional Adams method. Finally, numerical simulations support the theoretical conclusions, showing the great agreement between the two.


RESULTS
In the fractional-order HCV infection model, the memory effect, which is not seen in the classical model, was shown on graphs so that disease dynamics and vector compartments could be seen. We found that the fractional-order HCV infection model has more stages of freedom than regular derivatives. Fractional-order derivations, which may be the best and most reliable, explained bodily approaches better than classical order.


CONCLUSION
The current study modeled and analyzed a fractional-order HCV infection model. The current approach results in a much better understanding of HCV transmission in a population, which leads to important insights into its spread and control, such as better treatment dosage for different age groups, identifying the best control measure, improving health, prolonging life, reducing the risk of HCV transmission, and effectively increasing the quality of life of HCV patients. The creation of a fractional-order HCV infection model, which provides a better understanding of HCV transmission dynamics and leads to significant insights for better treatment dosages, identification of optimal control measures, and ultimately improvement of the quality of life for HCV patients, is the study's major outcome.

Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment

This research study proposes a novel hyperchaotic finance system with profit margin and then utilizes the Adomian Decomposition Method (ADM) to tackle the solution of the associated Caputo-derivative based fractional-order hyperchaotic finance system with profit margin. Numerical simulations and analyses, such as the evaluation of Lyapunov exponents (LE), the creation of bifurcation diagrams (BD), the complexity analysis (CA), and the 0-1 test, are used to get a full picture of the system. The results of our study demonstrate the occurrence of periodic, chaotic, and hyperchaotic dynamics in the system. Furthermore, the overall criteria for the existence and the uniqueness of the exact solutions for Caputo fractional-order models are presented. In addition, we present a control methodology for the fractional-order hyperchaotic financial system utilizing the Laplace transform and linear feedback control. Significantly, our research showcases a noteworthy correlation between the analytical results and numerical simulations, emphasizing the soundness and effectiveness of our methodology. 

Investigation of the hyperchaos and control in the fractional order financial system with profit margin

The objective of current investigation is to propose a solution to predict the interest rate, investment demand, and price index with optimal control in a fractional financial 4D chaotic model. He-Laplace method (HLM) is introduced with fractional derivative in Caputo sense to characterize the memory effect of the 4D chaotic model. For validation and comparison purposes, the given financial model is also solved through fractional residual power series algorithm. Analysis revealed that HLM provide improved results as compared to RPSA. Model is also analyzed graphically for interest rate, investment demand, price index and input control in fractional environment to understand the physical behavior of the model. The impact of variations in saving amount, cost per investment, and elasticity in demand are also presented through contours. It is reported that initially the interest rate, investment demand and price index are uniform, but later on drastic increase have been observed. Analysis also revealed that proposed methodology is stable and performed exceptionally well in chaotic scenarios, and hence can be extended to other complex models. 

New solutions of fractional 4D chaotic financial model with optimal control via He-Laplace algorithm

Abstract Background Cancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies. Methods In this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data. Results We propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results. Conclusion The findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems. 

A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

Physical phenomena and natural disasters, such as tsunamis and floods, are caused due to dispersive water waves and shallow waves caused by earthquakes. In order to analyze and minimize damaging effects of such situations, mathematical models are presented by different researchers. The Wu–Zhang (WZ) system is one such model that describes long dispersive waves. In this regard, the current study focuses on a non-linear (2 + 1)-dimensional time-fractional Wu–Zhang (WZ) system due to its importance in capturing long dispersive gravity water waves in the ocean. A Caputo fractional derivative in the WZ system is considered in this study. For solution purposes, modification of the homotopy perturbation method (HPM) along with the Laplace transform is used to provide improved results in terms of accuracy. For validity and convergence, obtained results are compared with the fractional differential transform method (FDTM), modified variational iteration method (mVIM), and modified Adomian decomposition method (mADM). Analysis of results indicates the effectiveness of the proposed methodology. Furthermore, the effect of fractional parameters on the given model is analyzed numerically and graphically at both integral and fractional orders. Moreover, Caputo, Caputo–Fabrizio, and Atangana–Baleanu approaches of fractional derivatives are applied and compared graphically in the current study. Analysis affirms that the proposed algorithm is a reliable tool and can be used in higher dimensional fractional systems in science and engineering.

/pdf/homotopy-perturbation-method-based-soliton-solutions-of-the-26krxv71.pdf

Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

Numerical analysis of Bazykin–Berezovskaya model

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Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

Numerical analysis of Bazykin–Berezovskaya model