Dynamical behavior of a hepatitis B virus transmission model with vaccination

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.

Dynamic Vaccine Allocation for Control of Human-Transmissible Disease

During pandemics, such as COVID-19, supplies of vaccines can be insufficient for meeting all needs, particularly when vaccines first become available. Our study develops a dynamic methodology for vaccine allocation, segmented by region, age, and timeframe, using a time-sensitive, age-structured compartmental model. Based on the objective of minimizing a weighted sum of deaths and cases, we used the Sequential Least Squares Quadratic Programming method to search for a locally optimal COVID-19 vaccine allocation for the United States, for the period from 16 December 2020 to 30 June 2021, where regions corresponded to the 50 states in the United States (U.S.). We also compared our solution to actual allocations of vaccines. From our model, we estimate that approximately 1.8 million cases and 9 thousand deaths could have been averted in the U.S. with an improved allocation. When case reduction is prioritized over death reduction, we found that young people (17 and younger) should receive priority over old people due to their potential to expose others. However, if death reduction is prioritized over case reduction, we found that more vaccines should be allocated to older people, due to their propensity for severe disease. While we have applied our methodology to COVID-19, our approach generalizes to other human-transmissible diseases, with potential application to future epidemics.

A Mathematical Model to Study the Potential Hepatitis B Virus Infections and Effects of Vaccination Strategies in China

MOTIVATIONS

Hepatitis B is a potentially life-threatening infectious disease caused by the hepatitis B virus (HBV). Approximately 390,000 people in China die from HBV-related diseases each year. Around 86 million individuals suffer from infections of the hepatitis B virus, accounting for about 6% of the total population in the region. There are approximately 30 million chronic infections. From 2002 to 2007, China's government took part in "The Global Alliance for Vaccines and Immunization (GAVI)" initiative, which helped reduce cases of chronic HBV infections among children. However, incidences of hepatitis B remain persistently high in China. Accurately estimating the number of potential HBV infections is crucial for preventing and controlling the transmission of the hepatitis B virus. Up until now, there were no studies of potentially infectious hepatitis B virus infections.

METHODS

this study was based on data from the National Bureau of Statistics of China from 2003 to 2021; a dynamic model was built, which included a compartment for potentially infectious hepatitis B virus infections. The parameters in the model were fitted using a combination of nonlinear least-squares and genetic algorithm methods.

RESULTS

the calculated reproduction number for hepatitis B virus transmission within the population is Rc = 1.741. Considering the existing vaccine inefficiency rate of 0.1, the model estimates there are 449,535 (95%CI [415,651, 483,420]) potentially infectious hepatitis B virus infections, constituting 30.49% of total hepatitis B cases. Date fitting using MATLAB reveals that increasing the rate of hepatitis B vaccinations can effectively reduce the number of infections.

CONCLUSIONS

the results reveal that the number of potential infectious hepatitis B virus infections is so high that the number of hepatitis B patients persistently rises in China. To better control the transmission of the hepatitis B virus, an optional prevention and control strategy is needed to increase the vaccination of different age groups, and it is necessary to help the public correctly understand the transmission of hepatitis B and ensure adequate protection.

Mathematical model analysis of effective intervention strategies on transmission dynamics of hepatitis B virus

Hepatitis B is one of the world's most common and severe infectious diseases. Worldwide, over 350 million people are currently estimated to be persistent carriers of the hepatitis B virus (HBV), with the death of 1 million people from the chronic stage of HBV infection. In this work, developed a nonlinear mathematical model for the transmission dynamics of HBV. We constructed the mathematical model by considering vaccination, treatment, migration, and screening effects. We calculated both disease-free and endemic equilibrium points for our model. Using the next-generation matrix, an effective reproduction number for the model is calculated. We also proved the asymptotic stability of both local and global asymptotically stability of disease-free and endemic equilibrium points. By calculating the sensitivity indices, the most sensitive parameters that are most likely to affect the disease's endemicity are identified. From the findings of this work, we recommend vaccination of the entire population and screening all the exposed and migrants. Additionally, early treatment of both the exposed class after screening and the chronically infected class is vital to decreasing the transmission of HBV in the community.

Mathematical Model of Hepatitis B Disease with Optimal Control and Cost-Effectiveness Analysis

In this paper, a mathematical model of hepatitis B disease with a two-dose vaccine series has been formulated and analyzed. We demonstrated that the model’s disease-free equilibrium is globally asymptotically stable when the basic reproduction number $\left(R_0\right)$ is less than one, whereas the model’s endemic equilibrium is locally asymptotically stable when $R_0$ is greater than one. Sensitivity analysis is performed, and based on its results, the model is extended to an optimal control problem by incorporating two control interventions, namely, prevention and enhanced newborn vaccination. Finally, simulation analyses of the model are conducted to illustrate the theoretical findings and effectiveness of each strategy, which indicates that the use of prevention efforts is the most cost-saving strategy.

Modeling the Adaptive Immune Response in an HBV Infection Model with Virus to Cell Transmission in Both Liver with CTL Immune Response and the Extrahepatic Tissue

The objective of this paper is to investigate a mathematical model describing the infection of hepatitis B virus (HBV) in intrahepatic and extrahepatic tissues. Additionally, the model includes the effect of the cytotoxic T cell (CTL) immunity, which is described by a linear activation rate by infected cells. The positivity and boundedness of solutions for non-negative initial data are proven, which is consistent with the biological studies. The local stability of the equilibrium is established. In addition to this, the global stability of the disease-free equilibrium and the endemic equilibrium is fulfilled by using appropriate Lyapanov functions. Finally, numerical simulations are performed to support our theoretical findings. It has been revealed that the fractional-order derivatives have no influence on the stability but only on the speed of convergence toward the equilibria.

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.

Global stability analysis of hepatitis B virus dynamics

This paper considers the impact of an acute individual's spontaneous clearance, recovery of a chronic individual with full immunity, and risk factor reduction on a hepatitis B virus (HBV) model. The existence and the positivity solution of the model are established. The model threshold quantity is defined and sensitivity analysis is analyzed to demonstrate the effect of various parameters on the spread of the virus. The global stability analysis of the equilibrium is shown using Lyapunov and comparison theorem methods. Finally, computational simulation is presented to validate the analytical solution. The results show that treatment, spontaneous clearance and reduction of the risk factor are highly successful in transmitting and regulating HBV transmission. The effective measure of these parameters as substantiated by our simulations, providing an excellent control method of the transmissible infection of HBV.

The impact of testing and treatment on the dynamics of Hepatitis B virus

Despite the intervention of WHO on vaccination for reducing the spread of Hepatitis B Virus (HBV), there are records of the high prevalence of HBV in some regions. In this paper, a mathematical model was formulated to analyze the acquisition and transmission process of the virus with the view of identifying the possible way of reducing the menace and mitigating the risk of the virus. The models' positivity and boundedness were demonstrated using well-known theorems. Equating the differential equations to zero demonstrates the equilibria of the solutions i.e., the disease-free and endemic equilibrium. The next Generation Matrix method was used to compute the basic reproduction number for the models. Local and global stabilities of the models were shown via linearization and Lyapunov function methods respectively. The importance of testing and treatment on the dynamics of HBV were fully discussed in this paper. It was discovered that testing at the acute stage of the virus and chronic unaware state helps in better management of the virus.

Modelling homosexual and heterosexual transmissions of hepatitis B virus in China

Studies have shown that sexual transmission, both heterosexually and homosexually, is one of the main ways of HBV infection. Based on this fact, we propose a mathematical model to study the sexual transmission of HBV among adults by classifying adults into men and women and considering both same-sex and opposite-sex transmissions of HBV in adults. Firstly, we calculate the basic reproduction number R0 and the disease-free equilibrium point E0. Secondly, by analysing the sensitivity of R0 in terms of model parameters, we find that the infection rate among people who have same-sex partners, the frequency of homosexual contact and the immunity rate of adults play important roles in the transmission of HBV. Moreover, we use our model to fit the reported data in China and forecast the trend of hepatitis B. Our results demonstrate that popularizing the basic knowledge of HBV among residents, advocating healthy and reasonable sexual life style, reducing the number of adult carriers, and increasing the immunization rate of adults are effective measures to prevent and control hepatitis B.

Optimal control application to the epidemiology of HBV and HCV co-infection

It is very important to note that a mathematical model plays a key role in different infectious diseases. Here, we study the dynamical behaviors of both hepatitis B virus (HBV) and hepatitis C virus (HCV) with their co-infection. Actually, the purpose of this work is to show how the bi-therapy is effective and include an inhibitor for HCV infection with some treatments, which are frequently used against HBV. Local stability, global stability and its prevention from the community are studied. Mathematical models and optimality system of nonlinear DE are solved numerically by RK4. We use linearization, Lyapunov function and Pontryagin’s maximum principle for local stability, global stability and optimal control, respectively. Stability curves and basic reproductive number are plotted with and without control versus different values of parameters. This study shows that the infection will spread without control and can cover with treatment. The intensity of HBV/HCV co-infection is studied before and after optimal treatment. This represents a short drop after treatment. First, we formulate the model then find its equilibrium points for both. The models possess four distinct equilibria: HBV and HCV free, and endemic. For the proposed problem dynamics, we show the local as well as the global stability of the HBV and HCV. With the help of optimal control theory, we increase uninfected individuals and decrease the infected individuals. Three time-dependent variables are also used, namely, vaccination, treatment and isolation. Finally, optimal control is classified into optimality system, which we can solve with Runge–Kutta-order four method for different values of parameters. Finally, we will conclude the results for implementation to minimize the infected individuals.

Optimal Voluntary Vaccination of Adults and Adolescents Can Help Eradicate Hepatitis B in China

Hepatitis B (HBV) is one of the most common infectious diseases, with a worldwide annual incidence of over 250 million people. About one-third of the cases are in China. While China made significant efforts to implement a nationwide HBV vaccination program for newborns, a significant number of susceptible adults and teens remain. In this paper, we analyze a game-theoretical model of HBV dynamics that incorporates government-provided vaccination at birth coupled with voluntary vaccinations of susceptible adults and teens. We show that the optimal voluntary vaccination brings the disease incidence to very low levels. This result is robust and, in particular, due to a high HBV treatment cost, essentially independent from the vaccine cost.

Investment Case for a Comprehensive Package of Interventions Against Hepatitis B in China: Applied Modeling to Help National Strategy Planning

Background

In 2016, the first global viral hepatitis elimination targets were endorsed. An estimated one-third of the world’s population of individuals with chronic hepatitis B virus (HBV) infection live in China and liver cancer is the sixth leading cause of mortality, but coverage of first-line antiviral treatment was low. In 2015, China was one of the first countries to initiate a consultative process for a renewed approach to viral hepatitis. We present the investment case for the scale-up of a comprehensive package of HBV interventions.

Methods

A dynamic simulation model of HBV was developed and used to simulate the Chinese HBV epidemic. We evaluated the impact, costs, and return on investment of a comprehensive package of prevention and treatment interventions from a societal perspective, incorporating costs of management of end-stage liver disease and lost productivity costs.

Results

Despite the successes of historical vaccination scale-up since 1992, there will be a projected 60 million people still living with HBV in 2030 and 10 million HBV-related deaths, including 5.7 million HBV-related cancer deaths between 2015 and 2030. This could be reduced by 2.1 million by highly active case-finding and optimal antiviral treatment regimens. The package of interventions is likely to have a positive return on investment to society of US$1.57 per US dollar invested.

Conclusions

Increases in HBV-related deaths for the next few decades pose a major public health threat in China. Active case-finding and access to optimal antiviral treatment are required to mitigate this risk. This investment case approach provides a real-world example of how applied modeling can support national dialog and inform policy planning.

Analysis of fractional COVID-19 epidemic model under Caputo operator

The article deals with the analysis of the fractional COVID-19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease-free equilibrium using the method of Castillo-Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first-order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system.

An application of a novel geometric criterion to global-stability problems of a nonlinear SEIVS epidemic model

This work applies a novel geometric criterion for nonlinear autonomous differential equations developed by Lu and Lu (NARWA 36:20-43, 2017) to a nonlinear SEIVS epidemic model with temporary immunity and achieves its threshold dynamics. Specifically, global-stability problems for the SEIVS model of Cai and Li (AMM 33:2919-2926, 2009) are effectively solved. The corresponding optimal control system with vaccination, awareness campaigns and treatment is further established and four different control strategies are compared by numerical simulations to contain hepatitis B. It is concluded that joint implementation of these measures can minimize the numbers of exposed and infectious individuals in the shortest time, so it is the most efficient strategy to curb the hepatitis B epidemic. Moreover, vaccination for newborns plays the core role and maintains the high level of population immunity.

A numerical and analytical study of SE(Is)(Ih)AR epidemic fractional order COVID-19 model

This article describes the corona virus spread in a population under certain assumptions with the help of a fractional order mathematical model. The fractional order derivative is the well-known fractal fractional operator. We have given the existence results and numerical simulations with the help of the given data in the literature. Our results show similar behavior as the classical order ones. This characteristic shows the applicability and usefulness of the derivative and our numerical scheme.

Stability analysis and optimal control of covid-19 with convex incidence rate in Khyber Pakhtunkhawa (Pakistan)

The dynamic of covid-19 epidemic model with a convex incidence rate is studied in this article. First, we formulate the model without control and study all the basic properties and results including local and global stability. We show the global stability of disease free equilibrium using the method of Lyapunov function theory while for disease endemic, we use the method of geometrical approach. Furthermore, we develop a model with suitable optimal control strategies. Our aim is to minimize the infection in the host population. In order to do this, we use two control variables. Moreover, sensitivity analysis complemented by simulations are performed to determine how changes in parameters affect the dynamical behavior of the system. Taking into account the central manifold theory the bifurcation analysis is also incorporated. The numerical simulations are performed in order to show the feasibility of the control strategy and effectiveness of the theoretical results.

A stochastic SACR epidemic model for HBV transmission

In this article, a stochastic SACR hepatitis B epidemic model is taken to be under consideration. We develop a stochastic epidemic model by considering the effect of environmental fluctuation on the hepatitis B dynamics and distribute the transmission rate by white noise. Using the stochastic Lyapunov function theory, we have shown the existence and uniqueness of the global positive solution. The extinction and persistence for our proposed model are derived with sufficient conditions. The numerical simulations are carried out using first-order Itô-Taylor stochastic scheme in the last section for the verification of our theoretical results.

Fractal-Fractional Mathematical Model Addressing the Situation of Corona Virus in Pakistan

This work is the consideration of a fractal fractional mathematical model on the transmission and control of corona virus (COVID-19), in which the total population of an infected area is divided into susceptible, infected and recovered classes. We consider a fractal-fractional order SIR type model for investigation of Covid-19. To realize the transmission and control of corona virus in a much better way, first we study the stability of the corresponding deterministic model using next generation matrix along with basic reproduction number. After this, we study the qualitative analysis using "fixed point theory" approach. Next, we use fractional Adams-Bashforth approach for investigation of approximate solution to the considered model. At the end numerical simulation are been given by matlab to provide the validity of mathematical system having the arbitrary order and fractal dimension.

Progress towards elimination of mother-to-child transmission of hepatitis B virus infection in China: a modelling analysis

Objective

To determine the projected burden of hepatitis B virus (HBV) in China, the intervention strategies that can eliminate mother-to-child transmission (MTCT) by 2030 or earlier and the measurable parameters that can be used to monitor progress towards this target.

Methods

We developed a dynamic, sex- and age-stratified model of the HBV epidemic in China, calibrated using hepatitis B surface antigen (HBsAg) and e antigen (HBeAg) prevalence data from sequential national serosurveys (1979-2014) and the numbers of HBV-related cancer deaths (2012). We determined whether China can achieve elimination of MTCT of HBV by 2030 under current prevention interventions. We modelled various intervention scenarios to represent different coverage levels of birth-dose HBV vaccination, hepatitis B immunoglobulin to newborns of HBsAg-positive mothers and antiviral therapy (tenofovir) to HBeAg-positive pregnant women.

Findings

We project that, if current levels of prevention interventions are maintained, China will achieve the elimination target by 2029. By modelling various intervention scenarios, we found that this can be brought forward to 2025 by increasing coverage of birth-dose vaccination, or to 2024 by the administration of tenofovir to HBeAg-positive pregnant women. We found that achievement of the target by 2025 would be predicted by a measurement of less than 2% MTCT in 2020.

Conclusion

Our results highlight how high-quality national data can be combined with modelling in monitoring the elimination of MTCT of HBV. By demonstrating the impact of increased interventions on target achievement dates, we anticipate that other high-burden countries will be motivated to strengthen HBV prevention policies.

Existence, uniqueness, and stability of fractional hepatitis B epidemic model

This paper describes the existence and stability of the hepatitis B epidemic model with a fractional-order derivative in Atangana-Baleanu sense. Some new results are handled by using the Sumudu transform. The existence and uniqueness of the equilibrium solution are presented using the Banach fixed-point theorem. Moreover, sensitivity analysis complemented by simulations is performed to determine how changes in parameters affect the dynamical behavior of the system. The numerical simulations are carried out using a predictor-corrector scheme to demonstrate the obtained results.

Global dynamics of a model of hepatitis B virus infection in a sub-Saharan African rural area

We formulate and systematically study a deterministic compartmental model of Hepatitis B. This model has some important and novel features compared with the well-known basic model in the literature. Specifically, it takes into account the differential susceptibility that follows the vaccine formulation employing three-doses schedule. It points up the HbeAg status of carriers, their levels of viral replication, the fact that treatment being not curative is recommended only to a small proportion of chronic carriers, and finally the fact that only inactive carriers are able to recover from disease. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number [Formula: see text] and an endemic equilibrium when [Formula: see text]. By the use of Lyapunov functions, when it exists, we prove the global asymptotic stability of the endemic equilibrium under some conditions. Using data from Tokombere, a rural area in Cameroon, numerical simulations are performed. These numerical simulations first confirm analytical results, second they suggest that a policy based on treatment could not significantly impact the course of the infection. Third, they show as it is well known that vaccination is a very effective measure to control the infection. Furthermore, they show that neonatal vaccination influences more the course of infection than mass vaccination strategy. Nevertheless, they picture how much loss between consecutive doses of vaccine could be harmful. Finally, it is suggested that for a Sub-saharan African rural area, two-thirds of expected incidence of Hepatitis B virus infection and one third of expected prevalence of chronic carriers could be averted by 2030 if the birth dose vaccination becomes systematic and if mass vaccination rate increases to up 10%.

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.

Game-Theoretical Model of Retroactive Hepatitis B Vaccination in China

Hepatitis B (HepB) is one of the most common infectious diseases affecting over two billion people worldwide. About one third of all HepB cases are in China. In recent years, China made significant efforts to implement a nationwide HepB vaccination program and reduced the number of unvaccinated infants from 30 to 10%. However, many individuals still remain unprotected, particularly those born before 2003. Consequently, a catch-up retroactive vaccination is an important and potentially cost-effective way to reduce HepB prevalence. In this paper, we analyze a game theoretical model of HepB dynamics that incorporates government-provided vaccination at birth coupled with voluntary retroactive vaccinations. Given the uncertainty about the long-term efficacy of the HepB vaccinations, we study several scenarios. When the waning rate is relatively high, we show that this retroactive vaccination should be a necessary component of any HepB eradication effort. When the vaccine offers long-lasting protection, the voluntary retroactive vaccination brings the disease incidence to sufficiently low levels. Also, we find that the optimal vaccination rates are almost independent of the vaccination coverage at birth. Moreover, it is in an individual's self-interest to vaccinate (and potentially re-vaccinate) at a rate just slightly above the vaccine waning rate.

Ergodic Stationary Distribution of a Stochastic Hepatitis B Epidemic Model with Interval-Valued Parameters and Compensated Poisson Process

Hepatitis B epidemic was and is still a rich subject that sparks the interest of epidemiological researchers. The dynamics of this epidemic is often modeled by a system with constant parameters. In reality, the parameters associated with the Hepatitis B model are not certain, but the interval in which it belongs to can readily be determined. Our paper focuses on an imprecise Hepatitis B model perturbed by Lévy noise due to unexpected environmental disturbances. This model has a global positive solution. Under an appropriate assumption, we prove the existence of a unique ergodic stationary distribution by using the mutually exclusive possibilities lemma demonstrated by Stettner in 1986. Our main effort is to establish an almost perfect condition for the existence of the stationary distribution. Numerical simulations are introduced to illustrate the analytical results.

Stochastic mathematical model for the spread and control of Corona virus

This work is devoted to a stochastic model on the spread and control of corona virus (COVID-19), in which the total population of a corona infected area is divided into susceptible, infected, and recovered classes. In reality, the number of individuals who get disease, the number of deaths due to corona virus, and the number of recovered are stochastic, because nobody can tell the exact value of these numbers in the future. The models containing these terms must be stochastic. Such numbers are estimated and counted by a random process called a Poisson process (or birth process). We construct an SIR-type model in which the above numbers are stochastic and counted by a Poisson process. To understand the spread and control of corona virus in a better way, we first study the stability of the corresponding deterministic model, investigate the unique nonnegative strong solution and an inequality managing of which leads to control of the virus. After this, we pass to the stochastic model and show the existence of a unique strong solution. Next, we use the supermartingale approach to investigate a bound managing of which also leads to decrease of the number of infected individuals. Finally, we use the data of the COVOD-19 in USA to calculate the intensity of Poisson processes and verify our results.

A stochastic model for the transmission dynamics of hepatitis B virus

In this paper, we formulate a stochastic model for hepatitis B virus transmission with the effect of fluctuation environment. We divide the total population into four different compartments, namely, the susceptible individuals in which the disease transmission rate is distributed by white noise, the acutely infected individuals in which the same perturbation occur, the chronically infected individuals and the recovered individuals. We use the stochastic Lyapunov function theory to construct a suitable stochastic Lyapunov function for the existence of positive solution. We also then establish the sufficient conditions for the hepatitis B extinction and the hepatitis B persistence. At the end numerical simulation is carried out by using the stochastic Runge-Kutta method to support our analytical findings.

CONTROL STRATEGIES of HEPATITIS B WITH THREE CONTROL VARIABLES

In this paper, we present a compartmental mathematical model of hepatitis B virus with optimal control strategies. First, we formulate the model applying the optimal control techniques which use control variables in the form of isolation, educational campaign and vaccination. We derive the conditions under which it is optimal to eradicate the disease and examine the impact of possible vaccination treatment strategies on disease transmission. When such an elimination is impossible, we use the techniques of Pontryagin’s Maximum Principle to derive the necessary conditions for the optimal control problem. The numerical results show that some effective vaccination and control can reduce the disease spread in the community.

A Literature Review of Mathematical Models of Hepatitis B Virus Transmission Applied to Immunization Strategies From 1994 to 2015

A mathematical model of the transmission dynamics of infectious disease is an important theoretical epidemiology method, which has been used to simulate the prevalence of hepatitis B and evaluate different immunization strategies. However, differences lie in the mathematical processes of modeling HBV transmission in published studies, not only in the model structure, but also in the estimation of certain parameters. This review reveals that the dynamics model of HBV transmission only simulates the spread of HBV in the population from the macroscopic point of view and highlights several main shortcomings in the model structure and parameter estimation. First, age-dependence is the most important characteristic in the transmission of HBV, but an age-structure model and related age-dependent parameters were not adopted in some of the compartmental models describing HBV transmission. In addition, the numerical estimation of the force of HBV infection did not give sufficient weight to the age and time factors and is not suitable using the incidence data. Lastly, the current mathematical models did not well reflect the details of the factors of HBV transmission, such as migration from high or intermediate HBV endemic areas to low endemic areas and the kind of HBV genotype. All of these shortcomings may lead to unreliable results. When the mathematical model closely reflects the fact of hepatitis B spread, the results of the model fit will provide valuable information for controlling the transmission of hepatitis B.

The transmission dynamic and optimal control of acute and chronic hepatitis B

In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem and develop an optimal control strategy to control the spread of hepatitis B in a community. In order to do this, first we present the model formulation and find the basic reproduction number [Formula: see text]. We show that if [Formula: see text] then the disease-free equilibrium is both locally as well as globally asymptotically stable. Then, we prove that the model is locally and globally asymptotically stable, if [Formula: see text]. To control the spread of this infection, we develop a control strategy by applying three control variables such as isolation of infected and non-infected individuals, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. Finally, we present numerical simulation to illustrate the feasibility of the control strategy.

Numerical solution of diffusive HBV model in a fractional medium

Evolution systems containing fractional derivatives can result to suitable mathematical models for describing better and important physical phenomena. In this paper, we consider a multi-components nonlinear fractional-in-space reaction–diffusion equations consisting of an improved deterministic model which describe the spread of hepatitis B virus disease in areas of high endemic communities. The model is analyzed. We give some useful biological results to show that the disease-free equilibrium is both locally and globally asymptotically stable when the basic reproduction number is less than unity. Our findings of this paper strongly recommend a combination of effective treatment and vaccination as a good control measure, is important to record the success of HBV disease control through a careful choice of parameters. Some simulation results are presented to support the analytical findings.

Recognizing the impact of endemic hepatitis D virus on hepatitis B virus eradication

BACKGROUND

Hepatitis delta virus (HDV) in conjunction with hepatitis B virus (HBV) increases adult morbidity and mortality. A number of studies have performed cost-benefit analyses for HBV interventions, but they have ignored the impact of HDV on these outcomes.

METHODS

Using a mathematical model of HBV-HDV epidemiology, we compare health benefits and cost outcomes of four interventions: testing with HBV adult vaccination (diagnosis), diagnosis with antiviral treatment for HBV infections (mono-infections), diagnosis with antiviral treatment for HBV-HDV infections (dual-infections), and awareness programs. The relationship between optimal levels and outcomes of each of these interventions and HDV prevalence in HBV infected individuals ranging from 0 to 50% is determined.

RESULTS

Over a 50 year period under no intervention, HBV prevalence, per capita total cost and death toll increase by 2.25%, -$11 and 2.6-fold respectively in moderate HDV endemic regions compared to mono-infected regions; the corresponding values for high HDV endemic regions are 4.2%, -$21 and 3.9-fold. Optimal interventions can be strategized similarly in mono and dually endemic regions. Only implementation of all four interventions achieves a very low HBV prevalence of around 1.5% in a moderate HDV endemic region such as China, with 2.8 million fewer deaths compared to no intervention. Although the policy of implementation of all four interventions costs additional $382 billion compared to no intervention, it still remains cost-effective with an incremental cost-effectiveness ratio of $1400/QALY. Very high efficacy awareness programs achieve less prevalence with fewer deaths at a lower cost compared to treatment and/or vaccination programs.

CONCLUSION

HDV substantially affects the performance of any HBV-related intervention. Its exclusion results in over-estimation of the effectiveness of HBV interventions.

Comment on "Transmission Model of Hepatitis B Virus with Migration Effect"

We show the erroneous assumptions and reasoning by introducing the migration effect of individuals in the transmission model of Hepatitis B virus. First, some false results related to the eigenvalues and reproductive number in the recent literature in mathematical biology will be presented. Then, it will be proved that the product of the matrices in the next generation method to obtain the reproductive number R₀ is not correct and the local and global stability results based on the reproductive number R₀ are considered false.

Modeling and Analyzing the Transmission Dynamics of HBV Epidemic in Xinjiang, China

Hepatitis B is an infectious disease caused by the hepatitis B virus (HBV) which affects livers. In this paper, we formulate a hepatitis B model to study the transmission dynamics of hepatitis B in Xinjiang, China. The epidemic model involves an exponential birth rate and vertical transmission. For a better understanding of HBV transmission dynamics, we analyze the dynamic behavior of the model. The modified reproductive number σ is obtained. When σ < 1, the disease-free equilibrium is locally asymptotically stable, when σ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. In the simulation, parameters are chosen to fit public data in Xinjiang. The simulation indicates that the cumulated HBV infection number in Xinjiang will attain about 600,000 cases unless there are stronger or more effective control measures by the end of 2017. Sensitive analysis results show that enhancing the vaccination rate for newborns in Xinjiang is very effective to stop the transmission of HBV. Hence, we recommend that all infants in Xinjiang receive the hepatitis B vaccine as soon as possible after birth.

The independent impact of newborn hepatitis B vaccination on reducing HBV prevalence in China, 1992-2006: A mathematical model analysis

OBJECTIVE

To evaluate the independent impact of newborn hepatitis B vaccination on reducing HBV prevalence in China, from its introduction in 1992 to 2006.

METHODS

An age- and time-dependent discrete dynamic model was developed to simulate HBV transmission in China under the assumptions of no any change in interventions and only with newborn vaccination introduction, respectively. The initial conditions of the model were determined according to the national serosurvey in 1992. The simulated results were compared with the observed results of the national serosurvey in 2006, and the contribution rate of newborn vaccination on reducing HBV prevalence was calculated overall and by birth cohort.

RESULTS

The total HBV prevalence would remain stable through the 14-year period if no any change in interventions, but decrease year by year if only with newborn vaccination introduction. Newborn vaccination could account for more than 50% of the reduction of the total HBV prevalence, although the full 3-dose and timely birth dose vaccination coverage rates were low in the early years. The results by birth cohort showed that the higher the two coverage rates, the higher contribution rate on reducing HBV prevalence. For the 2005 birth cohort which had high levels in the two coverage rates, the contribution rate could reach more than 95%.

CONCLUSION

Newborn hepatitis B vaccination from 1992 to 2006 in China had played the most important role in reducing HBV prevalence. Newborn vaccination with high full 3-dose and timely birth dose coverage rates is the decisive factor in controlling hepatitis B in China.

A THEORETICAL ASSESSMENT OF THE EFFECTS OF DISTRIBUTED DELAY ON THE TRANSMISSION DYNAMICS OF HEPATITIS B

In this paper, we investigate the global dynamics of a system of delay differential equations which describes the interaction of hepatitis B virus (HBV) with both liver and blood cells. The model has two distributed time delays describing the time needed for infection of cell and virus replication. We also include the efficiency of drug therapy in inhibiting viral production and the efficiency of drug therapy in blocking new infection. We compute the basic reproduction number and find that increasing delays will decrease the value of the basic reproduction number. We study the sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Our analysis reveals that the model exhibits the phenomenon of backward bifurcation (where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium when the basic reproduction number is less than unity). Numerical simulations are presented to evaluate the impact of time-delays on the prevalence of the disease.

The impact of vaccination and antiviral therapy on hepatitis B and hepatitis D epidemiology

The major cause of liver cancer around the globe is hepatitis B virus (HBV), which also contributes to a large number of deaths due to liver failure alone. Hepatitis delta virus (HDV) is as potentially alarming as HBV since life threatening cases are 10 times more likely with HBV-HDV dual infection compared to HBV monoinfection. So far, there is no established effective treatment against HDV and the only preventive action suggested by the World Health Organization is to introduce HBV vaccination for children immediately after birth (newborns) and thus reduce the available pool for HDV infection. Here the main objective is to understand the complex dynamics of HBV-HDV infection in a human population that can inform public health policy makers on the level of different preventive measures required to eliminate HBV and HDV infections. Model simulations suggest that HBV vertical transmission and HBV vaccination rates for newborns are instrumental in determining HBV and HDV prevalence. A decrease in HBV prevalence is observed as vaccination coverage increases and it is possible to eradicate both HBV and HDV using high vaccination coverage of ≥80% in the long term. We further found that HDV presence results in lower HBV prevalence. An application of our model to China revealed that vaccinating every newborn in China will further prevent 1.69 million new infections by 2028 as compared to the current 90% vaccination coverage. Although, higher vaccination coverage of newborns should eliminate both HBV and HDV over a long time period, any short term strategy to eradicate HDV must include additional preventive measures such as HBV adult vaccination. Implementation of HBV adult vaccination programs at a rate of 10% per year over 15 years will further prevent 39 thousand new HDV infections in China by 2028 as compared to HBV vaccination programs solely for newborns.

Computational modeling of interventions and protective thresholds to prevent disease transmission in deploying populations

Military personnel are deployed abroad for missions ranging from humanitarian relief efforts to combat actions; delay or interruption in these activities due to disease transmission can cause operational disruptions, significant economic loss, and stressed or exceeded military medical resources. Deployed troops function in environments favorable to the rapid and efficient transmission of many viruses particularly when levels of protection are suboptimal. When immunity among deployed military populations is low, the risk of vaccine-preventable disease outbreaks increases, impacting troop readiness and achievement of mission objectives. However, targeted vaccination and the optimization of preexisting immunity among deployed populations can decrease the threat of outbreaks among deployed troops. Here we describe methods for the computational modeling of disease transmission to explore how preexisting immunity compares with vaccination at the time of deployment as a means of preventing outbreaks and protecting troops and mission objectives during extended military deployment actions. These methods are illustrated with five modeling case studies for separate diseases common in many parts of the world, to show different approaches required in varying epidemiological settings.

Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus

Hepatitis B virus (HBV) infection is a worldwide public health problem. In this paper, we study the dynamics of hepatitis B virus (HBV) infection which can be controlled by vaccination as well as treatment. Initially we consider constant controls for both vaccination and treatment. In the constant controls case, by determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady-state solutions of the model. Next, we take the controls as time and formulate the appropriate optimal control problem and obtain the optimal control strategy to minimize both the number of infectious humans and the associated costs. Finally at the end numerical simulation results show that optimal combination of vaccination and treatment is the most effective way to control hepatitis B virus infection.

An optimized Nash nonlinear grey Bernoulli model based on particle swarm optimization and its application in prediction for the incidence of Hepatitis B in Xinjiang, China

In this paper, by using a particle swarm optimization algorithm to solve the optimal parameter estimation problem, an improved Nash nonlinear grey Bernoulli model termed PSO-NNGBM(1,1) is proposed. To test the forecasting performance, the optimized model is applied for forecasting the incidence of hepatitis B in Xinjiang, China. Four models, traditional GM(1,1), grey Verhulst model (GVM), original nonlinear grey Bernoulli model (NGBM(1,1)) and Holt-Winters exponential smoothing method, are also established for comparison with the proposed model under the criteria of mean absolute percentage error and root mean square percent error. The prediction results show that the optimized NNGBM(1,1) model is more accurate and performs better than the traditional GM(1,1), GVM, NGBM(1,1) and Holt-Winters exponential smoothing method.

Transmission model of hepatitis B virus with the migration effect

Hepatitis B is a globally infectious disease. Mathematical modeling of HBV transmission is an interesting research area. In this paper, we present characteristics of HBV virus transmission in the form of a mathematical model. We analyzed the effect of immigrants in the model to study the effect of immigrants for the host population. We added the following flow parameters: "the transmission between migrated and exposed class" and "the transmission between migrated and acute class." With these new features, we obtained a compartment model of six differential equations. First, we find the basic threshold quantity Ro and then find the local asymptotic stability of disease-free equilibrium and endemic equilibrium. Furthermore, we find the global stability of the disease-free and endemic equilibria. Previous similar publications have not added the kind of information about the numerical results of the model. In our case, from numerical simulation, a detailed discussion of the parameters and their numerical results is presented. We claim that with these assumptions and by adding the migrated class, the model informs policy for governments, to be aware of the immigrants and subject them to tests about the disease status. Immigrants for short visits and students should be subjected to tests to reduce the number of immigrants with disease.

Modelling the epidemiology of hepatitis B in New Zealand

Hepatitis B is a vaccine preventable disease caused by the hepatitis B virus (HBV) that can induce potentially fatal liver damage. It has the second highest mortality rate of all vaccine preventable diseases in New Zealand. Vaccination against HBV was introduced in New Zealand in 1988, and the country is now categorised with overall low endemicity but with areas of both high and medium endemic levels. We present an SECIR compartmental mathematical model, with the population divided into age classes, for the transmission of HBV using local data on incidence of infection and vaccination coverage. We estimate the basic reproduction number, R(0), to be 1.53, and show that the vaccination campaign has substantially reduced this below one. However, a large number of carriers remain in the population acting as a source of infection.