A multi-layer neural network approach for the stability analysis of the Hepatitis B model

In the present study, we explore the dynamics of Hepatitis B virus infection, a significant global health issue, through a newly developed dynamics system. This model is distinguished by its inclusion of asymptomatic carriers and the impact of vaccination and treatment strategies. Compared to Hepatitis A, Hepatitis B poses a more serious health risk, with some cases progressing from acute to chronic. To diagnose and predict disease recurrence, the basic reproduction number (R0) is calculated. We investigate the stability of the disease's dynamics under different conditions, using the Lyapunov function to confirm our model's global stability. Our findings highlight the relevance of vaccination and early treatment in reducing Hepatitis B virus spread, making them a useful tool for public health efforts aiming at eradicating Hepatitis B virus. In our research, we investigate the dynamics of a specific model that is characterized by a system of differential equations. This work uses deep neural networks (DNNs) technique to improve model accuracy, proving the use of DNNs in epidemiological modeling. Additionally, we want to find the curves that suit the target solutions with the minimum residual errors. The simulations we conducted demonstrate our methodology's capability to accurately predict the behavior of systems across various conditions. We rigorously test the solutions obtained via the DNNs by comparing them to benchmark solutions and undergoing stages of testing, validation, and training. To determine the accuracy and reliability of our approach, we perform a series of analyses, including convergence studies, error distribution evaluations, regression analyses, and detailed curve fitting for each equation.

Global dynamics and computational modeling for analyzing and controlling Hepatitis B: A novel epidemic approach

Hepatitis B virus (HBV) infection is a global public health issue. We offer a comprehensive analysis of the dynamics of HBV, which can be successfully controlled with vaccine and treatment. Hepatitis B virus (HBV) causes a significantly more severe and protracted disease compared to hepatitis A. While it initially presents as an acute disease, in approximately 5 to 10% of cases, it can develop into a chronic disease that causes permanent damage to the liver. The hepatitis B virus can remain active outside the body for at least seven days. If the virus penetrates an individual's body without immunization, it may still result in infection. Upon exposure to HBV, the symptoms often last for a duration ranging from 10 days to 6 months. In this study, we developed a new model for Hepatitis B Virus (HBV) that includes asymptomatic carriers, vaccination, and treatment classes to gain a comprehensive knowledge of HBV dynamics. The basic reproduction number [Formula: see text] is calculated to identify future recurrence. The local and global stabilities of the proposed model are evaluated for values of [Formula: see text] that are both below and above 1. The Lyapunov function is employed to ensure the global stability of the HBV model. Further, the existence and uniqueness of the proposed model are demonstrated. To look at the solution of the proposed model graphically, we used a useful numerical strategy, such as the non-standard finite difference method, to obtain more thorough numerical findings for the parameters that have a significant impact on disease elimination. In addition, the study of treatment class in the population, we may assess the effectiveness of alternative medicines to treat infected populations can be determined. Numerical simulations and graphical representations are employed to illustrate the implications of our theoretical conclusions.

A fractional order model for the co-interaction of COVID-19 and Hepatitis B virus

Fractional differential equations are beginning to gain widespread usage in modeling physical and biological processes. It is worth mentioning that the standard mathematical models of integer-order derivatives, including nonlinear models, do not constitute suitable framework in many cases. In this work, a mathematical model for COVID-19 and Hepatitis B Virus (HBV) co-interaction is developed and studied using the Atangana-Baleanu fractional derivative. The necessary conditions of the existence and uniqueness of the solution of the proposed model are studied. The local stability analysis is carried out when the reproduction number is less than one. Using well constructed Lyapunov functions, the disease free and endemic equilibria are proven to be globally asymptotically stable under certain conditions. Employing fixed point theory, the stability of the iterative scheme to approximate the solution of the model is discussed. The model is fitted to real data from the city of Wuhan, China, and important parameters relating to each disease and their co-infection, are estimated from the fitting. The approximate solutions of the model are compared using the integer and fractional order derivatives. The impact of the fractional derivative on the proposed model is also highlighted. The results proven in this paper illustrate that HBV and COVID-19 transmission rates can greatly impact the dynamics of the co-infection of both diseases. It is concluded that to control the co-circulation of both diseases in a population, efforts must be geared towards preventing incident infection with either or both diseases.

A study on fractional HBV model through singular and non-singular derivatives

The current study's aim is to evaluate the dynamics of a Hepatitis B virus (HBV) model with the class of asymptomatic carriers using two different numerical algorithms and various values of the fractional-order parameter. We considered the model with two different fractional-order derivatives, namely the Caputo derivative and Atangana-Baleanu derivative in the Caputo sense (ABC). The considered derivatives are the most widely used fractional operators in modeling. We present some mathematical analysis of the fractional ABC model. The fixed-point theory is used to determine the existence and uniqueness of the solutions to the considered fractional model. For numerical results, we show a generalized Adams-Bashforth-Moulton (ABM) method for Caputo derivative and an Adams type predictor-corrector (PC) algorithm for Atangana-Baleanu derivatives. Finally, the models are numerically solved using computational techniques and obtained results graphically illustrated with a wide range of fractional-order values. We compare the numerical results for Caputo and ABC derivatives graphically. In addition, a new variable-order fractional network of the HBV model is proposed. Considering the fact that most communities interact with each other, and the rate of disease spread is affected by this factor, the proposed network can provide more accurate insight for the modeling of the disease.

Optimal control strategies for dengue fever spread in Johor, Malaysia

Background Dengue is a vector-borne viral disease endemic in Malaysia. The disease is presently a public health issue in the country. Hence, the use of mathematical model to gain insights into the transmission dynamics and derive the optimal control strategies for minimizing the spread of the disease is of great importance. Methods A model involving eight mutually exclusive compartments with the introduction of personal protection, larvicide and adulticide control strategies describing dengue fever transmission dynamics is presented. The control-induced basic reproduction number (R˜₀) related to the model is computed using the next generation matrix method. Comparison theorem is used to analyse the global dynamics of the model. The model is fitted to the data related to the 2012 dengue outbreak in Johor, Malaysia, using the least-squares method. In a bid to optimally curtail dengue fever propagation, we apply optimal control theory to investigate the effect of several control strategies of combination of optimal personal protection, larvicide and adulticide controls on dengue fever dynamics. The resulting optimality system is simulated in MATLAB using fourth order Runge-Kutta scheme based on the forward-backward sweep method. In addition, cost-effectiveness analysis is performed to determine the most cost-effective strategy among the various control strategies analysed. Results Analysis of the model with control parameters shows that the model has two disease-free equilibria, namely, trivial equilibrium and biologically realistic disease-free equilibrium, and one endemic equilibrium point. It also reveals that the biologically realistic disease-free equilibrium is both locally and globally asymptotically stable whenever the inequality R˜₀<1holds. In the case of model with time-dependent control functions, the optimality levels of the three control functions required to optimally control dengue disease transmission are derived. Conclusion We conclude that dengue fever transmission can be curtailed by adopting any of the several control strategies analysed in this study. Furthermore, a strategy which combines personal protection and adulticide controls is found to be the most cost-effective control strategy.

OPTIMAL CONTROL ANALYSIS OF THE EFFECT OF TREATMENT, ISOLATION AND VACCINATION ON HEPATITIS B VIRUS

Hepatitis B infection is a serious health issue and a major cause of deaths worldwide. This infection can be overcome by adopting proper treatment and control strategies. In this paper, we develop and use a mathematical model to explore the effect of treatment on the dynamics of hepatitis B infection. First, we formulate and use a model without control variables to calculate the basic reproduction number and to investigate basic properties of the model such as the existence and stability of equilibria. In the absence of control measures, we prove that the disease free equilibrium is locally asymptotically stable when the basic reproduction number is less than unity. Also, using persistent theorem, it is shown that the infection is uniformly persistent, whenever the basic reproduction number is greater than unity. Using optimal control theory, we incorporate into the model three time-dependent control variables and investigate the conditions required to curtail the spread of the disease. Finally, to illustrate the effectiveness of each of the control strategies on disease control and eradication, we perform numerical simulations. Based on the numerical results, we found that the first two strategies (treatment and isolation strategy) and (vaccination and isolation strategy) are not very effective as a long term control or eradication strategy for HBV. Hence, we recommend that in order to effectively control the disease, all the control measures (isolation, vaccination and treatment) must be implemented at the same time.

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results.