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

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.

On approximating a new generalization of traveling salesman problem.

The current best-known performance guarantees for the extensively studied Traveling Salesman Problem (TSP) of determinate approximation algorithms is 32, achieved by Christofides' algorithm 47 years ago. This paper investigates a new generalization problem of the TSP, termed the Minimum-Cost Bounded Degree Connected Subgraph (MBDCS) problem. In the MBDCS problem, the goal is to identify a minimum-cost connected subgraph containing n=|V| edges from an input graph G=(V,E) with degree upper bounds for particular vertices. We show that for certain special cases of MBDCS, the aim is equivalent to finding a minimum-cost Hamiltonian cycle for the input graph, same as the TSP. To appropriately solve MBDCS, we initially present an integer programming formulation for the problem. Subsequently, we propose an algorithm to approximate the optimal solution by applying the iterative rounding technique to solution of the integer programming relaxation. We demonstrate that the returned subgraph of our proposed algorithm is one of the best guarantees for the MBDCS problem in polynomial time, assuming P≠NP. This study views the optimization of TSP as finding a minimum-cost connected subgraph containing n edges with degree upper bounds for certain vertices, and it may provide new insights into optimizing the TSP in future research.

Physiological and chaos effect on dynamics of neurological disorder with memory effect of fractional operator: A mathematical study.

BACKGROUND AND OBJECTIVE: To study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases that are spread worldwide. The main objective of our work is to examine neurological disorders by early detection and treatment by taking asymptomatic. The central nervous system (CNS) is impacted by the prevalent neurological condition known as multiple sclerosis (MS), which can result in lesions that spread across time and place. It is widely acknowledged that multiple sclerosis (MS) is an unpredictable disease that can cause lifelong damage to the brain, spinal cord, and optic nerves. The use of integral operators and fractional order (FO) derivatives in mathematical models has become popular in the field of epidemiology. METHOD: The model consists of segments of healthy or barian brain cells, infected brain cells, and damaged brain cells as a result of immunological or viral effectors with novel fractal fractional operator in sight Mittag Leffler function. The stability analysis, positivity, boundedness, existence, and uniqueness are treated for a proposed model with novel fractional operators. RESULTS: Model is verified the local and global with the Lyapunov function. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium so that solutions are bounded in the feasible domain. To ensure the existence and uniqueness of the solutions to the suggested model, it makes use of Banach's fixed point and the Leray Schauder nonlinear alternative theorem. For numerical simulation and results the steps Lagrange interpolation method at different fractional order values and the outcomes are compared with those obtained using the well-known FFM method. CONCLUSION: Overall, by offering a mathematical model that can be used to replicate and examine the behavior of disease models, this research advances our understanding of the course and recurrence of disease. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies from our justified outcomes.

Stability and bifurcation analysis of a discrete predator-prey system of Ricker type with refuge effect.

The refuge effect is critical in ecosystems for stabilizing predator-prey interactions. The purpose of this research was to investigate the complexities of a discrete-time predator-prey system with a refuge effect. The analysis investigated the presence and stability of fixed points, as well as period-doubling and Neimark-Sacker (NS) bifurcations. The bifurcating and fluctuating behavior of the system was controlled via feedback and hybrid control methods. In addition, numerical simulations were performed as evidence to back up our theoretical findings. According to our findings, maintaining an optimal level of refuge availability was critical for predator and prey population cohabitation and stability.

Forced convection of non-darcy flow of ethylene glycol conveying copper(II) oxide and titanium dioxide nanoparticles subject to lorentz force on wedges: Non-newtonian casson model.

The topic of two-dimensional steady laminar MHD boundary layer flow across a wedge with non-Newtonian hybrid nanoliquid (CuO-TiO2/C2H6O2) with viscous dissipation and radiation is taken into consideration. The controlling partial differential equations have been converted to non-linear higher-order ordinary differential equations using the appropriate similarity transformations. It is demonstrated that a number of thermo-physical characteristics govern the transmuted model. The issue is then mathematically resolved. When the method's accuracy is compared to results that have already been published, an excellent agreement is found. While the thermal distribution increases with an increase in Eckert number, radiation and porosity parameters, the velocity distribution decreases as porosity increases.

Modeling the effects of the contaminated environments on COVID-19 transmission in India.

COVID-19 is an infectious disease caused by the SARS-CoV-2 virus that caused an outbreak of typical pneumonia first in Wuhan and then globally. Although researchers focus on the human-to-human transmission of this virus but not much research is done on the dynamics of the virus in the environment and the role humans play by releasing the virus into the environment. In this paper, a novel nonlinear mathematical model of the COVID-19 epidemic is proposed and analyzed under the effects of the environmental virus on the transmission patterns. The model consists of seven population compartments with the inclusion of contaminated environments means there is a chance to get infected by the virus in the environment. We also calculated the threshold quantity R 0 to know the disease status and provide conditions that guarantee the local and global asymptotic stability of the equilibria using Volterra-type Lyapunov functions, LaSalle's invariance principle, and the Routh-Hurwitz criterion. Furthermore, the sensitivity analysis is performed for the proposed model that determines the relative importance of the disease transmission parameters. Numerical experiments are performed to illustrate the effectiveness of the obtained theoretical results.

Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan.

Coronaviruses are a large family of viruses that cause different symptoms, from mild cold to severe respiratory distress, and they can be seen in different types of animals such as camels, cattle, cats and bats. Novel coronavirus called COVID-19 is a newly emerged virus that appeared in many countries of the world, but the actual source of the virus is not yet known. The outbreak has caused pandemic with 26,622,706 confirmed infections and 874,708 reported deaths worldwide till August 31, 2020, with 17,717,911 recovered cases. Currently, there exist no vaccines officially approved for the prevention or management of the disease, but alternative drugs meant for HIV, HBV, malaria and some other flus are used to treat this virus. In the present paper, a fractional-order epidemic model with two different operators called the classical Caputo operator and the Atangana-Baleanu-Caputo operator for the transmission of COVID-19 epidemic is proposed and analyzed. The reproduction number R 0 is obtained for the prediction and persistence of the disease. The dynamic behavior of the equilibria is studied by using fractional Routh-Hurwitz stability criterion and fractional La Salle invariant principle. Special attention is given to the global dynamics of the equilibria. Moreover, the fitting of parameters through least squares curve fitting technique is performed, and the average absolute relative error between COVID-19 actual cases and the model's solution for the infectious class is tried to be reduced and the best fitted values of the relevant parameters are achieved. The numerical solution of the proposed COVID-19 fractional-order model under the Caputo operator is obtained by using generalized Adams-Bashforth-Moulton method, whereas for the Atangana-Baleanu-Caputo operator, we have used a new numerical scheme. Also, the treatment compartment is included in the population which determines the impact of alternative drugs applied for treating the infected individuals. Furthermore, numerical simulations of the model and their graphical presentations are performed to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary-order derivative.

Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control.

In this paper, a nonlinear fractional order epidemic model for HIV transmission is proposed and analyzed by including extra compartment namely exposed class to the basic SIR epidemic model. Also, the infected class of female sex workers is divided into unaware infectives and the aware infectives. The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV transmission. The exposed class contains those susceptible males in the population who have sexual contact with the female sex workers and are exposed to the infection directly or indirectly. The Caputo type fractional derivative is involved and generalized Adams-Bashforth-Moulton method is employed to numerically solve the proposed model. Model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. Analysis of the model demonstrates that the population is free from the disease if R 0 < 1 and disease spreads in the population if R 0 > 1 . Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Furthermore, for the fractional optimal control problem associated with the control strategies such as condom use for exposed class, treatment for aware infectives, awareness about disease among unaware infectives and behavioral change for susceptibles, we formulated a fractional optimality condition for the proposed model. The existence of fractional optimal control is analyzed and the Euler-Lagrange necessary conditions for the optimality of fractional optimal control are obtained. The effectiveness of control strategies is shown through numerical simulations and it can be seen through simulation, that the control measures effectively increase the quality of life and age limit of the HIV patients. It significantly reduces the number of HIV/AIDS patients during the whole epidemic.

Modeling and simulation of spatial-temporal calcium distribution in T lymphocyte cell by using a reaction-diffusion equation.

T lymphocytes are white blood cells that play a central role in cell-mediated immunity. Ca2+ has its major signaling function when it is elevated in the cytosolic compartment. The free cytosolic Ca2+ dynamics plays a very important role in the activation, and fate decision process in the T lymphocytes. Here, we develop a quantitative spatio-temporal Ca2+ dynamic model which includes, the Ca2+ releasing channels ER leak and voltage-gated Ca2+ channel, buffering and re-uptaking mechanism in the T lymphocytes. In this model, the cell is represented as a circular-shaped geometrical domain. This representation introduces modeling flexibility needed for detailed representation of the properties of Ca2+ dynamics in the cell including important parameters. The proposed mathematical model is solved using a finite difference method and the finite element method. Appropriate initial and boundary conditions are incorporated in the model based on biophysical conditions of the problem. Computer simulations in MATLAB R2010a are employed to investigate mathematical models of reaction-diffusion equation. The estimation is based on reaction-diffusion equation associated with biophysical and biochemical reactions taking place in the cell. From our results, it is observed that, the coordinated combination of the incorporated parameters plays a significant role in Ca2+ regulation in T lymphocytes. ER leak and voltage-gated Ca2+ channel provides the necessary Ca2+ to the cell when required for its proper functioning, while on the other side buffers and Na+/Ca2+ exchanger makes balance in the Ca2+ concentration, so as to prevent the cell from death as higher concentration for longer time is harmful for the cell and can cause cell death. These results have been used to study the relationship of Ca2+ concentration with parameters like VGCC, Na+/Ca2+ exchanger, ER leak and buffers. The significance of the study reveals that there is a significant variation in Ca2+ profiles due to the effect of VGCC, Na+/Ca2+ exchanger, ER leak, and buffers. The results give us better insights of coordinated effect of VGCC, Na+/Ca2+ exchanger, ER leak, and buffers on Ca2+ distribution in T lymphocytes. T lymphocytes are the primary host cells to receive the viral infections which transmits the signal then to other cell types. The proper quantity of Ca2+ concentration makes T lymphocytes more active and healthier to fight the infection properly and can protect the immune system from various fatal viral infections. Thus, the application of the study lies in the field of immunology to protect a susceptible from various viral infectious diseases like HIV, HBV, HINI, etc. by strengthening the immune system. The outcomes of the study reveal that the applied finite element method is computationally very strong and effective to analyze differential equations that arise in Ca2+ dynamics.