 Research
 Open Access
Classification and sensitivity analysis of the transmission dynamic of hepatitis B
 Tahir Khan^{1},
 Il Hyo Jung^{2}Email author,
 Amir Khan^{1, 3} and
 Gul Zaman^{1}
https://doi.org/10.1186/s1297601700683
© The Author(s). 2017
 Received: 31 May 2016
 Accepted: 10 October 2017
 Published: 5 December 2017
Abstract
Background
Hepatitis B infection caused by the hepatitis B virus is one of the most serious viral infections and a global health problem. In the transmission of hepatitis B infection, three different phases, i.e. acute infected, chronically infected, and carrier individuals, play important roles. Carrier individuals are especially significant, because they do not exhibit any symptoms and are able to transmit the infection. Here we assessed the transmissibility associated with different infection stages of hepatitis B and generated an epidemic model.
Methods
To demonstrate the transmission dynamic of hepatitis B, we investigate an epidemic model by dividing the infectious class into three subclasses, namely acute infected, chronically infected, and carrier individuals with both horizontal and vertical transmission.
Results
Numerical results and sensitivity analysis of some important parameters are presented to show that the proportion of births without successful vaccination, perinatally infected individuals, and direct contact rate are highest risk factors for the spread of hepatitis B in the community.
Conclusion
Our work provides a coherent platform for studying the full dynamics of hepatitis B and an effective direction for theoretical work.
Keywords
 Hepatitis B epidemic model
 Basic reproduction number
 Stability analysis
 Lyapunov function theory
 Geometrical approach
 Numerical simulation
Background
Hepatitis implies the inflammation of liver. Hepatitis B infection caused by the hepatitis B virus is among the most serious viral infections. It is a global health problem and one of the leading causes of death around the world. Worldwide, 2 billion people are infected with hepatitis B virus and about 360 million individuals live with chronic hepatitis B infection [1, 2]. In addition, hepatitis B virus infection is responsible for about 80% of primary liver cancers [3]. Therefore, every year approximately 780,000 individuals die from chronic or acute hepatitis B virus infection [1]. Hepatitis B virus can be transmitted from one individual to another in different ways, such as transmission through blood (sharing of razors, blades, or toothbrushes), semen, and vaginal secretions (unprotected sexual contact) [4–7]. The other major transmission route is from an infected mother to her child during childbirth, which is called vertical transmission. However, hepatitis B virus cannot be transmitted through water, food, hugging, kissing, or causal contact such as in the work place, school, etc. [6–8].
Hepatitis B infection has multiple phases: acute, chronic, and carrier. Acute hepatitis B is a shortterm infection within the first 6 months after someone is infected with the virus. In this stage, the immune system is usually able to clear the virus from the body, and recover within a few months. Chronic hepatitis B refers to the illness that occurs when the virus remains in the individual’s body and, over time, the infection develops into a serious health problem. Individuals with chronic hepatitis often have no history of acute illness; however, it can cause liver scarring, which becomes the cause of liver failure and may also develop into liver cancer [3]. The phase at which the individuals do not exhibit any symptoms, but transmit the disease to others is known as the carrier phase, which plays an important role in the transmission of hepatitis B infection. This is the most dangerous and serious phase of hepatitis B, because it is difficult to control the hepatitis B virus infection when a large group of carriers exist, as they will be responsible for transmitting the disease to new individuals.
Mathematical modeling is a powerful tool to describe the dynamical behavior of different diseases in the real world [9, 10]. Several mathematicians and biologists have developed different epidemic models to understand and control the spread of transmissible diseases in the population. In the last two decades, the field of mathematical modeling has been used frequently for the study of transmission of different types of infectious diseases. Mann and Roberts [3] and Thornley et al. [8] used a mathematical model for eliminating hepatitis B virus in New Zealand. In 1991, Anderson and May [11] described the effect of carriers on the transmission of hepatitis B virus by using a simple deterministic model. Zhao et al. [12] presented an age structured model for the prediction of the dynamics of hepatitis B virus transmission and evaluated the longterm effectiveness of the vaccination program in China. In 2010, Zou et al. [13] presented a model for the transmission dynamics and control of the hepatitis B virus in China. Recently, a mathematical model for the transmission dynamics and optimal control of hepatitis B has been presented by Khan et al. [14].
The different phases of hepatitis B play a very important role in the transmission of hepatitis B infection, and have not yet been investigated collectively for their potential role in generating a hepatitis B epidemic model. We consider a hepatitis B epidemic model by identifying the different phases, acute, chronic, and carrier, of hepatitis B infection.
Methods

A _{ 1 } The initial populations S(0), L(0), A(0), B(0),C(0), V(0), and R(0) are all known and nonnegative.

A _{ 2 } Recovered individuals have permanent immunity.

A _{ 3 } The inflow of newborns with successful vaccination go into the vaccinated subclass.

A _{ 4 } The inflow of newborns with perinatal infection go into the carrier subclass.

A _{ 5 } The inflow of newborns without perinatal infection go into the susceptible subclass.

A _{ 6 } The population with successful vaccination go into the vaccinated subclass.
In the model (1), b represents the birth rate, ξ represents the proportion of births without successful vaccination, η represents the proportion of perinatally infected individuals, φ represents the rate of waning vaccineinduced immunity, β represents the transmission rate from susceptible to infected, γ and ζ represent the reduced transmission rate of chronic and carrier individuals infected with hepatitis B, respectively. The natural death rate is represented by μ _{0}. We use v to denote the vaccination rate, σ represents the moving rate from latent class to acute class, γ _{1} represents the moving rate from acute to chronic and carrier, ψ represents the recovery rate from acute class to recovered, γ _{2} represents the moving rate of chronic carrier to immune, γ _{3} represents the moving rate of carrier to immune, μ _{1} and μ _{2} represent the death rates occurring from hepatitis B, and p represents the average probability of an individual’s failure to clear an acute infection and going to the carrier state.
To represent the dynamics of our proposed model (1), we need to find the equilibria of the proposed model (1), which are diseasefree and endemic equilibria.
Equilibrium analysis
Boundedness
For the biologically feasible region, we prove the boundedness of the proposed model.
Theorem 1 The solution of the model (1) is bounded.
Proof: Let N(t) denote the total population, then N(t) = S(t) + L(t) + A(t) + B(t) + C(t) + R(t) + V(t). Differentiation of N(t) with respect to time and the use of model (1) yields \( \frac{dN(t)}{dt}= b\xi {\mu}_0N(t){\mu}_1B(t){\mu}_2C(t) \). Therefore, we can write \( \frac{dN(t)}{dt}+{\mu}_0N(t)\le b\xi . \) Integrating both sides and then using the theory of differential inequality [16], we obtain \( 0<N\left(S,L,A,B,C,R,V\right)\le \frac{b\xi}{\mu_0}\left(1{e}^{{\mu}_0t}\right)+{N}_0{e}^{{\mu}_0t}. \) Now let t → ∞, it becomes \( 0<N\left(S,L,A,B,C,R,V\right)\le \frac{b\xi}{\mu_0}. \) Hence, the solution of the model (1) initiating in \( {R}_{+}^7 \) is limited in the set \( \varDelta =\left\{\left(S,L,A,B,C,R,V\right)\in {R}_{+}^7:N=\frac{b\xi}{\mu_0}+\xi \right\} \) for any ξ > 0 and t → ∞ , which completes the proof.
Basic reproduction number
Local stability analysis
Regarding the local asymptotic stability of the proposed model at diseasefree and endemic equilibrium points, we have the following results.
Theorem 2 If R _{0} > 1, then the model (1) is locally asymptotically stable at the endemic equilibrium point E _{1} , and if R _{0} < 1, then it is unstable.
All eigenvalues except λ _{5} have negative real parts and λ _{5} is negative, if γ > ζ. Hence, all eigenvalues of the Jacobian matrix J _{1} have negative real part, if γ > ζ. Therefore, for R _{0} > 1, the model (1) is locally asymptotically stable at the endemic equilibrium point E _{1}, if γ > ζ.
Global stability analysis

If \( \frac{{d\chi}_1}{d t}=G\left({\chi}_1,0\right), \) \( {\chi}_1^0 \) is globally asymptotically stable.

We have \( H\left({\chi}_1,{\chi}_2\right)=B{\chi}_1\overline{H}\left({\chi}_1,{\chi}_2\right), \) where \( \overline{H}\left({\chi}_1,{\chi}_2\right)\ge 0 \) for (χ _{1}, χ _{2}) ∈ Δ..
In the second condition \( B={D}_{\chi_2}H\left({\chi}_1^0,0\right) \) is an Mmatrix, that is, the offdiagonal entries are positive and Δ is the feasible region. Then the following statement holds.
Lemma 1 For R _{0} < 1, the equilibrium point E _{0} = (χ ^{0}, 0) of the system (9) is said to be globally asymptotically stable, if the above conditions are satisfied.

There exists a compact absorbing set K ∈ U.

System (10) has a unique equilibrium.
The solution x ^{∗} is said to be globally asymptotically stable in U, if it is locally asymptotically stable and all trajectories in U converge to the equilibrium x ^{∗}. For n ≥ 2, a condition is satisfied for f, which precludes the existence of a nonconstant periodic solution of eq. (10) known as the Bendixson criterion. The classical Bendixson criterion divf(x) < 0 for n = 2 is robust under C ^{1} (see [19]). Further, a point x _{0} ∈ U is wandering for eq. (10), if there exists a neighborhood N of x _{0} and τ > 0, such that N ∩ x(t, N) is empty for all t > τ. Thus, the following global stability principle is established for an autonomous system in any finite dimension.
Lemma 2 If conditions 3 and 4 and the Bendixson criterion are satisfied for eq. (10), then it is robust under C ^{1} local perturbation of f at all nonequilibrium, nonwandering points for eq. (10). Then, x ^{∗} is globally asymptotically stable in U, provided that it is stable.
Hence, if \( \overline{q}<0 \), this shows that the presence of any orbit gives rise to a simple closed rectifiable curve, such as periodic orbits and heterocyclic cycles.
Lemma 3 Let U be simply connected, and conditions 3 and 4 be satisfied, then the unique equilibrium x ^{∗} of eq. (10) is globally asymptotically stable in U, if \( \overline{q}<0 \) .
Now we apply the above techniques to prove the global stability of model (1) at diseasefree equilibrium and endemic equilibrium, respectively. Thus, we have the following stability results.
Theorem 3 If R _{0} < 1, the proposed model (1) is globally asymptotically stable at diseasefree equilibrium E _{0} and unstable otherwise.
Thus, from eq. (16) as t → ∞, \( {\chi}_1\to {\chi}_1^0 \). Thus, \( {\chi}_1={\chi}_1^0 \) is globally asymptotically stable.
From the model (1), the total population is bounded by S _{0}, that is, S, L, A, B, C ≤ S _{0}, so βSL ≤ βS _{0} I, βSA ≤ βS _{0} A, βSB ≤ βS _{0} B, and βSC ≤ βS _{0} C, which implies that \( \overline{H}\left({\chi}_1,{\chi}_2\right) \) is positive definite. In addition, from eq. (18), it is clear that matrix B is an Mmatrix; that is, the offdiagonal elements are nonnegative. Thus, conditions 1 and 2 are satisfied, so by Lemma 1, the diseasefree equilibrium point E _{0} is globally asymptotically stable.
Theorem 4 If R _{0} > 1, the model (1) is globally asymptotically stable at endemic equilibrium E _{1} and unstable otherwise.
This solves the system (32) using the initial conditions B(0), C(0), R(0), and V(0). Thus, for large time t, that is, t → ∞, B(t) → B _{1}, C(t) → C _{1}, R(t) → R _{1}, and V(t) → V _{1}, which is sufficient to prove that the endemic equilibrium point E _{1} is globally asymptotically stable.
Results and discussions
Numerical results and discussion
In this section, the numerical simulations of the proposed model (1) are presented. The numerical results are obtained by using the fourthorder Runge–Kutta scheme [9, 10]. The simulation of our paper should be considered from a qualitative point of view, but not from the quantitative point of view. Therefore, for this purpose, some of the parameters are taken from published articles and some are assumed with feasible values. For our simulation, we consider the parameter values as follows: b = 0.0121, ξ = 0.8, η = 0.11, β = 0.012, γ = 0.46, ζ = 0.0123, σ = 0.0012, φ = 0.01, ψ = 0.012, v = 0.6, p = 0.6, γ _{1} = 0.33, γ _{2} = 0.009, γ _{3} = 0.025, μ _{0} = 0.069, μ _{1} = 0.000532, and μ _{2} = 0.000532. Some of these parameters, the birth rate b, natural death rate μ _{0}, and proportion of perinatally infected individuals η, are taken from [13, 21, 22] and the remaining parameters are assumed with biologically feasible values.
Sensitivity analysis
Conclusion
In this article, we established a model for the transmission dynamics of hepatitis B by taking into account the classification of different phases of individuals infected with hepatitis B. We studied different mathematical analyses, including equilibrium analysis and boundedness, and obtained the basic reproduction number by using the nextgeneration matrix. Moreover, we discussed the stability analysis and showed that the established model is both locally as well as globally asymptotically stable for the possible equilibria. To discuss the local stability, linearization and Routh—Herwitz criteria were used, while global stability was retrieved by using the method of CastilloChávez et al. and a geometrical approach. Finally, the numerical simulation and sensitivity analysis were presented to show the feasibility of the proposed work. Our work provides a coherent platform for studying the full dynamics of hepatitis B and an effective direction for theoretical work. The techniques used in this article are also applicable to other epidemic models.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments.
Funding
This work has been partially supported by the Higher Education Commission (HEC) of Pakistan under project No. 20–1983/R, D/HEC/11 and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2015R1D1A1A02062131).
Availability of data and materials
Not applicable.
Authors’ contributions
TK and GZ developed the model and showed the local as well as the global stability of the proposed model, while AK and IH Jung derived the numerical simulation of the proposed model. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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