Mathematical model and analysis of hepatitis B virus transmission dynamics

Hepatitis B is a liver infection induced by the hepatitis B virus (HBV). In this paper, the dynamics involved in the transmission of HBV is mathematically formulated with considerations of different populations of individuals. The role of HBV vaccination of new born babies and the treatment of infected individuals in controlling the transmission are factored into the model. The model in this study is based on the standard SEIR model.

Emerenini BO and Inyama SC.A vaccine against HB has been available since 1982, nevertheless there is still an increase in its transmission and spread.Key facts from 1 reveal that HBV can survive outside the human body for at least 7 days, and during this period HBV can still cause infection if it enters any unimmunized human body.Most HB carriers are asymptomatic during the acute infection phase, nonetheless some people experience acute illness that can last for several days with variations in the progression.

Reviewer Status
The use of mathematical models in scientific research has improved our understanding of contributing factors.Mathematical model of HBV has ranged from simple models 2,3 to more complex models involving the contributions of controls (e.g.vaccines) 4 , and analysis of the impact of immigrant 5 .
Motivated by other HB studies, we use an infectious disease model to understand the impact of HB vaccination and treatment on the dynamics of HBV transmission and prevalence using an SEIR format.

Mathematical model
A variety of mathematical models exist, such as the SIR, SIS, SIRS, and their variations; where S=Susceptible class, I=Infective class, and R=Recovered class.The model used in this study takes the form of SEIR based on ordinary differential equations, which shall be solved to obtain the disease-free equilibrium (DFE) state.

Governing equations
Table 1 lists the parameters used.Figure 1 shows a schematic presentation of the model.
We formulate the HB transmission model as follows: where M = immunized individuals, S = susceptible, L = latently infected/exposed, I = infectious individuals, and R = recovered.

Equilibrium solutions
Let E(M, S, L, I, R) be the equilibrium point of the system described by ( 1)-( 6), At the equilibrium state, we have In order to obtain the DFE state we solve equation (7)equation (15) simultaneously.

Stability analysis of the DFE state
To determine the stability of the DFE state E o , we examine the behavior of the model population near this equilibrium solution.Here, we determine condition(s) that must be met if the disease is to be totally eradicated.Recall that the system of equations in this model at equilibrium state is: cP We now linearize the system of equations to get the Jacobian matrix J. where ), the Jacobian Matrix becomes where The characteristic equation |J o − Iλ| = 0 is obtained from the Jacobian determinant with the Eigen values λ i (i = 1, 2, 3, 4, 5) From Equation (26), either From Equation (27), we deduce and For the DFE to be asymptotically stable, trace(A) < 0 and det A > 0.
And the trace of A is Obviously, trace(A) < 0 since all the parameters q, t, β, , β and η are positive.
For the determinant of A to be positive, we ) From equation (29)-equation (31), λ 1 , λ 2 , λ 3 of (25) all have negative real parts.We now establish the necessary and sufficient conditions for the remaining two Eigen values of (25) to have negative real part.
The inequality (32) gives the condition, necessary and sufficient for the DFE state of the model to be stable (asymptotically).This means that the product of total contraction and total breakdown of latent class given by must be less than the total removal rate from both latent and infectious classes given by (q + µ + β + λ)( + β + η).
Alternatively, the inequality (32) can also be expressed as The inequality (33) also gives the condition necessary and sufficient for the stability of DFE state, thus sum of the rate of recovery of latently infected people, the rate at which latently infected individuals progress to active infection and the rate of natural death of individuals (in the population, i.e. total removal rate from the latent class) must have a lower bound given by

Conclusion
Presented in this paper is a mathematical model of the role of vaccination and treatment on HB transmission dynamics.The proportion dynamics of the classes is described using five differential equations.We conclude that the trivial equilibrium The benefits of publishing with F1000Research: Your article is published within days, with no editorial bias You can publish traditional articles, null/negative results, case reports, data notes and more The peer review process is transparent and collaborative Your article is indexed in PubMed after passing peer review Dedicated customer support at every stage For pre-submission enquiries, contact research@f1000.com /doi.org/10.12688/f1000research.15557.1 17 Aug 2018, :1312 ( Latest published: 7 ) https://doi.org/10.12688/f1000research.15557.1 v1 Introduction Hepatitis B (HB) is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV), which is a DNA virus classified in the virus family of Hepadnaviridae.The World Health Organization (WHO) in 1 reported that more than 0.25 billion people are living with HBV infection, most of which resulted in several deaths.

Schematic presentation of the model.
Existence of a trivial equilibrium state (TES) Let E o (M o , S o , L o , I o , R o ) be TES of (1)-(6) of the model, ∃ no TES since the population cannot be extinct, so long as new babies are born into the population (i.e.cP ≠ 0 and (1 − c)P ≠ 0).That is, E o (M o , S o , L o , I o , R o ) ≠ (0, 0, 0, 0, 0) DFE state DFE state is the state of total eradication of disease.Let E o (M o , S o , L o , I o , R o ) be the DFE state.Suppose, both I and L must be zero.That is, for DFE state: state E o (M o , S o , L o , I o , R o ) is unstable; this is the state where there is no individual in the population.The DFE state, E o (M o , S o , L o , I o , R o ), was determined and its stability analysed using Routh-Hurwitz theorem.