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

Dynamic modelling of hepatitis C virus transmission among people who inject drugs: a methodological review

Equipment sharing among people who inject drugs (PWID) is a key risk factor in infection by hepatitis C virus (HCV). Both the effectiveness and cost-effectiveness of interventions aimed at reducing HCV transmission in this population (such as opioid substitution therapy, needle exchange programs or improved treatment) are difficult to evaluate using field surveys. Ethical issues and complicated access to the PWID population make it difficult to gather epidemiological data. In this context, mathematical modelling of HCV transmission is a useful alternative for comparing the cost and effectiveness of various interventions. Several models have been developed in the past few years. They are often based on strong hypotheses concerning the population structure. This review presents compartmental and individual-based models in order to underline their strengths and limits in the context of HCV infection among PWID. The final section discusses the main results of the papers

Simulation of A Mathematical Model Of Hepatitis B Virus Transmission Dynamics In The Presence Of Vaccination And Treatment

In this paper, a mathematical model for the transmission dynamics of hepatitis B virus (HBV) infection incorporating vaccination and treatment as control parameters is presented. The basic reproduction number, , as a threshold parameter, was constructed, in terms of the given model parameters, by the next generation method.   was numerically assessed for its sensitivity to vaccination and treatment parameters. A unique disease-free equilibrium state was determined, indicating possibility of control of HBV disease. The model was solved numerically using Runge-Kutta method of order four to evaluate the effects of vaccination and treatment parameters on the prevalence of the disease. The numerical results of the sensitivity analysis show that increasing either vaccination or treatment rate has the potential of reducing  below unity. The results of the numerical simulations of the model show that effective vaccination, treatment or a combination of both of them as a control strategy can eradicate HBV disease, with the combination being far better than either of them. Finally, these findings strongly suggest that high coverage of vaccination and treatment are crucial to the success of HBV disease control

Modelling Hepatitis B Virus Transmission Dynamics In The Presence of Vaccination and Treatment

Hepatitis B is a global threat as over a billion people have been infected and about 300 million people die annually across the world. In this paper, a mathematical model for the transmission dynamics of hepatitis B virus infection considering vaccination and treatment as control parameters in the host population is presented. First, the disease-free equilibrium state of the model was determined. The next generation method was used to determine the basic reproduction number,  as a threshold parameter, in terms of the given model parameters.  was analytically evaluated for its sensitivity to vaccination and treatment parameters. It was proved that the disease-free equilibrium state is locally asymptotically stable if the  is below unity, otherwise, it is unstable. Local stability of the endemic equilibrium state was established using the centre manifold theory. The analytical results of the  show that increasing the proportion of people who receive vaccines, either at birth or later in life, reduces it below unity. Similarly, increasing the proportion of carriers who receive treatment achieves the same purpose. The result of the local stability analysis of the disease-free equilibrium state shows that the disease can be eliminated if  is below unity. The result of the centre manifold theory on the endemic equilibrium state shows that the disease can persist as the value of  increases above one. The findings of this study strongly suggest a combination of effective vaccination and treatment as a control strategy is crucial to the success of HBV disease control

Solving an optimal control problem of hepatitis B virus dynamics: Efficacy of fuzzy logic strategy

This work aims at using fuzzy logic strategy to solve a hepatitis B virus (HBV) optimal control problem. To test the efficacy of this numerical method, we compare numerical results with those obtained using direct method. We consider a patient under treatment during 12 months where the two drugs are taken as controls. The results are rather satisfactory. In particular, the reaction of HBV to drugs can be modeled and a feedback can be approximated by the solution of a linear quadratic problem. The drugs reduce the risk of HBV. Furthermore, results of both numerical methods are in good agreement with experimental data and this justifies the efficacy of fuzzy logic strategy in solving optimal control problems

Hepatitis B Virus Disease: A Mathematical Model for Vertical Transmission with Treatment Strategy

This paper presents a mathematical model that captures some essential information about the impacts of treatment on hepatitis B vertical transmission. The treatment induced reproduction number is compared with the basic reproduction number to assess the possible benefits to be obtained from this control measure. Numerical results and sensitivity analysis are carried out to support the analytical results and determine the parameters influencing the dynamics of the disease. It is indicated that in the presence of treatment, transmission of infection decreases, implying that the number of acute and chronic infected adult women decrease as well, resulting into fewer infected newborn babies.   Mathematics Subject Classification: 34D20 Key words: Hepatitis B virus, vertical transmission, treatment, reproduction numbe

An Age Structure Mathematical Model Analysis on the Dynamics of Chronic and Hyper Toxic Forms of Hepatitis B Virus by SVEA_1 A_2 CR Model with Vaccination Intervention Control Strategy in Ethiopia

In our work we considered nonlinear ordinary differential equations to study the dynamics of hepatitis B virus (HBV) epidemics in Ethiopia. We proved that the invariant and bounded ness of the solution of the dynamical system. We used a nonlinear stability analysis method for proving the local and global stability of the existing equilibrium points. We have got that the diseases free equilibrium point and endemic equilibrium point exist for some conditions. We proved that the disease free equilibrium point is locally asymptotically stable and also globally asymptotically stable. We found that the effective reproduction number for the system is   which depends on fifteen parameters. On the other hand, the basic reproduction number is   Using standard parameter estimation we found that the numerical value of the effective reproduction number is and   From this numerical value we conclude that the disease spreads in the community and vaccination intervention strategy reduces the spread. Out of these fifteen parameters we identified five parameters which contribute significant role in control of the disease; and these are the rate of moving from exposed to acutely infected class with age below or equal to 5 years , the rate of moving from exposed to acutely infected class with age above 5 years the proportion of leaving acutely infected class with age below or equal to 5 years and progressing to chronically infected class ,the proportions of vaccinated newborns  and the ratio of vaccinated newborn  to total population H; which influence the effective reproduction number. We also conduct numerical simulations which support the finding in the sensitivity analysis

Ebola Model and Optimal Control with Vaccination Constraints

The Ebola virus disease is a severe viral haemorrhagic fever syndrome caused by Ebola virus. This disease is transmitted by direct contact with the body fluids of an infected person and objects contaminated with virus or infected animals, with a death rate close to 90% in humans. Recently, some mathematical models have been presented to analyse the spread of the 2014 Ebola outbreak in West Africa. In this paper, we introduce vaccination of the susceptible population with the aim of controlling the spread of the disease and analyse two optimal control problems related with the transmission of Ebola disease with vaccination. Firstly, we consider the case where the total number of available vaccines in a fixed period of time is limited. Secondly, we analyse the situation where there is a limited supply of vaccines at each instant of time for a fixed interval of time. The optimal control problems have been solved analytically. Finally, we have performed a number of numerical simulations in order to compare the models with vaccination and the model without vaccination, which has recently been shown to fit the real data. Three vaccination scenarios have been considered for our numerical simulations, namely: unlimited supply of vaccines; limited total number of vaccines; and limited supply of vaccines at each instant of time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Industrial and Management Optimization' (JIMO), ISSN 1547-5816 (print), ISSN 1553-166X (online). Submitted February 2016; revised November 2016; accepted for publication March 201