 Research
 Open access
 Published:
Classification and sensitivity analysis of the transmission dynamic of hepatitis B
Theoretical Biology and Medical Modelling volumeÂ 14, ArticleÂ number:Â 22 (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.
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) [4,5,6,7]. 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. [6,7,8].
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 individualâ€™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
With the different stages of hepatitis B, the total population is classified into seven different compartments with three infectious epidemiological classes, namely acute infected, chronically infected, and carrier individuals. First we develop the model, then we investigate the equilibria. For a biologically feasible region, we show the boundedness. Further, we find the basic reproduction number by using the nextgeneration matrix approach. In addition, we prove the local and global asymptotic stability of the proposed model. For the local stability, we use the method of linearization and Routhâ€“Herwitz criteria. The global asymptotic stability is retrieved by using the method proposed by CastilloChÃ¡vez et al. [15] and a geometrical approach. Finally, the numerical simulations are carried out by using a fourthorder Rungeâ€“Kutta method to show the feasibility of the obtained results. Moreover, a sensitivity analysis of some important parameters is also presented. Our work provides a coherent platform for studying the full dynamics of hepatitis B and an effective direction for theoretical work. In view of the characteristics of hepatitis B, we develop an epidemic model of hepatitis B by dividing the total population into seven epidemiological subclasses: susceptible S(t), latent L(t), acute infected A(t), chronic infected B(t), carrier C(t), recovered with permanent immunity R(t), and vaccinated V(t). We place the following assumptions on the model.

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.
Thus, the mathematical model can be presented by the following system of seven ordinary differential equations,
In the model (1), b represents the birth rate, Î¾ represents the proportion of births without successful vaccination, Î· represents the proportion of perinatally infected individuals, Ï† represents the rate of waning vaccineinduced immunity, Î² represents the transmission rate from susceptible to infected, Î³ and Î¶ represent the reduced transmission rate of chronic and carrier individuals infected with hepatitis B, respectively. The natural death rate is represented by Î¼ _{0}. We use v to denote the vaccination rate, Ïƒ represents the moving rate from latent class to acute class, Î³ _{1} represents the moving rate from acute to chronic and carrier, Ïˆ represents the recovery rate from acute class to recovered, Î³ _{2} represents the moving rate of chronic carrier to immune, Î³ _{3} represents the moving rate of carrier to immune, Î¼ _{1} and Î¼ _{2} represent the death rates occurring from hepatitis B, and p represents the average probability of an individualâ€™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
The diseasefree equilibrium point of the model (1) is denoted by E _{0} and defined as E _{0}â€‰=â€‰(S _{0},â€‰0,â€‰0,â€‰0,â€‰0,â€‰0,â€‰V _{0}), where
Similarly, the endemic equilibrium point is denoted by E _{1}â€‰=â€‰(S _{1},â€‰L _{1},â€‰A _{1},â€‰B _{1},â€‰C _{1},â€‰R _{1},â€‰V _{1}), where
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)â€‰=â€‰S(t)â€‰+â€‰L(t)â€‰+â€‰A(t)â€‰+â€‰B(t)â€‰+â€‰C(t)â€‰+â€‰R(t)â€‰+â€‰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 tâ€‰â†’â€‰âˆž, 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 Î¾â€‰>â€‰0 and tâ€‰â†’â€‰â€‰âˆžâ€‰, which completes the proof.
Basic reproduction number
The threshold quantity that determines whether an epidemic arises or the infection dies out is called the basic reproduction number of the disease, which is a key concept [11, 17]. It represents the expected average number of new infections produced directly and indirectly by a single infected individual, when introduced into a completely susceptible population. To find the basic reproduction number for the proposed model (1), we use the method of Driessche and Watmough [18]. Let Ï‡â€‰=â€‰(L(t),â€‰A(t),â€‰B(t),â€‰C(t))^{T}, so from the model (1), we have
In eq. (4), \( \overline{F} \) and \( \overline{V} \) are the matrices that contain the nonlinear and linear terms, respectively, such that
Now, we find the Jacobian matrix of \( \overline{F} \) and \( \overline{V} \) at the diseasefree equilibrium E _{0}, which becomes
where a _{11}â€‰=â€‰Ïƒâ€‰+â€‰Î¼ _{0}, a _{22}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î³ _{1}â€‰+â€‰Ïˆ, a _{32}â€‰=â€‰pÎ³ _{1}, a _{33}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î¼ _{1}â€‰+â€‰Î³ _{2}, a _{41}â€‰=â€‰(1â€‰âˆ’â€‰p)Î³ _{1}, and a _{44}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î¼ _{0}â€‰+â€‰Î³ _{3}â€‰âˆ’â€‰bÎ¾Î·. Thus, the basic reproduction number R _{0} is the spectral radius of the nextgeneration matrix \( \overline{K}={FV}^{1}, \) that is, \( {R}_0=\rho \left(\overline{K}\right)=\rho \left({FV}^{1}\right)=\max \left\{{\lambda}_1,\dots, {\lambda}_4\right\}, \) where Î» _{ i } for iâ€‰=â€‰1,â€‰2,â€‰3,â€‰4 are the eigenvalues of \( \overline{K} \). Hence, the basic reproduction number R _{0} for our proposed model (1) becomes
where
Local stability analysis
In this subsection, we discuss the local asymptotic satiability of the proposed model (1) at diseasefree equilibrium E _{0} and endemic equilibrium E _{1}. To show the local asymptotic stability, we reduce the proposed model, because R appears only in the sixth equation of the model. Thus, the reduced model is given by
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}â€‰>â€‰1, then the model (1) is locally asymptotically stable at the endemic equilibrium point E _{1} , and if R _{0}â€‰<â€‰1, then it is unstable.
Proof: The Jacobian matrix of model (6) at the endemic equilibrium point E _{1} is
where h _{11}â€‰=â€‰Î²S _{1}â€‰+â€‰Î³Î²B _{1}â€‰+â€‰Î¶Î²C _{1}â€‰âˆ’â€‰(Î¼ _{0}â€‰+â€‰v), h _{21}â€‰=â€‰Î²S _{1}â€‰+â€‰Î³Î²B _{1}â€‰+â€‰Î¶Î²C _{1}, h _{22}â€‰=â€‰Ïƒâ€‰+â€‰Î¼ _{0}, h _{33}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î³ _{1}â€‰+â€‰Ïˆ, h _{44}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î¼ _{1}â€‰+â€‰Î³ _{2}, h _{53}â€‰=â€‰(1â€‰âˆ’â€‰p)Î³ _{1}, h _{55}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î¼ _{2}â€‰+â€‰Î³ _{3}, h _{55}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î¼ _{2}â€‰+â€‰Î³ _{3} and h _{66}â€‰=â€‰Î¼ _{0}â€‰+â€‰Ï†. Using an elementary row operation to reduce the above matrix to echelon form, we obtain the following matrix
where K _{11}â€‰=â€‰â€‰âˆ’â€‰h _{11}, K _{22}â€‰=â€‰h _{11} h _{22}, K _{23}â€‰=â€‰Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{24}â€‰=â€‰Î³Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{25}â€‰=â€‰Î¶Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{33}â€‰=â€‰â€‰âˆ’â€‰h _{11} h _{22} h _{33}â€‰+â€‰ÏƒÎ²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{34}â€‰=â€‰ÏƒÎ³Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{35}â€‰=â€‰ÏƒÎ¶Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), \( {K}_{44}=\frac{\sigma \beta {S}_1{h}_{44}}{p{\gamma}_1}\left(\left({h}_{11}{h}_{21}\right)+{h}_{11}{h}_{22{h}_{33}}\right), \) K _{45}â€‰=â€‰Î¶Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{55}â€‰=â€‰K _{44}â€‰+â€‰ÏƒÎ¶Î²S _{1}(h _{11}â€‰âˆ’â€‰h _{21}), K _{66}â€‰=â€‰K _{44}â€‰âˆ’â€‰L, and L is defined as
Thus, the eigenvalues of the Jacobian matrix J _{1} are
All eigenvalues except Î» _{5} have negative real parts and Î» _{5} is negative, if Î³â€‰>â€‰Î¶. Hence, all eigenvalues of the Jacobian matrix J _{1} have negative real part, if Î³â€‰>â€‰Î¶. Therefore, for R _{0}â€‰>â€‰1, the model (1) is locally asymptotically stable at the endemic equilibrium point E _{1}, if Î³â€‰>â€‰Î¶.
Global stability analysis
In this section, the global asymptotic stability of the proposed model for both diseasefree as well as at endemic equilibrium is shown. The method of CastilloChÃ¡vez et al. [15] is used to prove the global asymptotic stability at diseasefree equilibrium. While to show that the model (1) is globally asymptotically stable at endemic equilibrium, the geometrical approach is implemented, which is a generalization of Lyapunov theory [19]. Here, we give a brief analysis of the CastilloChÃ¡vez et al. method and geometrical approach to prove the global stability of the model (1) at diseasefree equilibrium and endemic equilibrium. Thus, by using the method of CastilloChÃ¡vez et al. [15], to the following two subsystems given by
In the system (9), Ï‡ _{1} and Ï‡ _{2} represent the number of uninfected and infected (latent, acute infected, and chronic carrier) individuals, respectively, that is, Ï‡ _{1}â€‰=â€‰(S(t),â€‰V(t),â€‰R(t))â€‰âˆˆâ€‰R ^{3} and Ï‡ _{2}â€‰=â€‰(L(t),â€‰A(t),â€‰B(t),â€‰C(t))â€‰âˆˆâ€‰R ^{4}. The diseasefree equilibrium is denoted by E _{0} and defined as \( {E}_0=\left({\chi}_1^0,0\right). \) Thus, the existence of global stability at the diseasefree equilibrium point depends on the following two conditions.

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 (Ï‡ _{1},â€‰Ï‡ _{2})â€‰âˆˆâ€‰Î”..
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 Î” is the feasible region. Then the following statement holds.
Lemma 1 For R _{0}â€‰<â€‰1, the equilibrium point E _{0}â€‰=â€‰(Ï‡ ^{0},â€‰0) of the system (9) is said to be globally asymptotically stable, if the above conditions are satisfied.
Similarly, to prove the global stability of the model (1) at endemic equilibrium E _{1}, we use the geometrical approach [19]. According to this method, we investigate the sufficient condition through which the endemic equilibrium point is globally asymptotically stable. To do this, we consider a system of differential equations given by
where fâ€‰:â€‰Uâ€‰â†’â€‰R ^{n}, Uâ€‰âŠ‚â€‰R ^{n} is an open set simply connected, and fâ€‰âˆˆâ€‰C ^{1}(U). Let us assume that the solution to eq. (10) is f(x ^{âˆ—})â€‰=â€‰0 and for x(t,â€‰x _{0}), the following hypotheses hold.

There exists a compact absorbing set Kâ€‰âˆˆâ€‰U.

System (10) has a unique equilibrium.
The solution x ^{âˆ—} 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 ^{âˆ—}. For nâ€‰â‰¥â€‰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)â€‰<â€‰0 for nâ€‰=â€‰2 is robust under C ^{1} (see [19]). Further, a point x _{0}â€‰âˆˆâ€‰U is wandering for eq. (10), if there exists a neighborhood N of x _{0} and Ï„â€‰>â€‰0, such that Nâ€‰âˆ©â€‰x(t,â€‰N) is empty for all tâ€‰>â€‰Ï„. 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 ^{âˆ—} is globally asymptotically stable in U, provided that it is stable.
Now to prove the robustness required for Lemma 2, let us define a function, such that
Eq. (11) is a matrix valued function on U. Further, assume that P ^{âˆ’1} exists and is continuous for xâ€‰âˆˆâ€‰K. Now define a quantity, such that
where Bâ€‰=â€‰P _{ f } P ^{âˆ’1}â€‰+â€‰PJ ^{[2]} P ^{âˆ’1} and J ^{[2]} is the second additive compound matrix of the Jacobian matrix J, that is, J(x)â€‰=â€‰Uf(x). Let â„“(B) be the Lozinski measure of the matrix B with respect to the norm âˆ£â€‰âˆ£â€‰.â€‰âˆ£â€‰âˆ£ in R ^{n} (see [20]) defined by
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 ^{âˆ—} 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}â€‰<â€‰1, the proposed model (1) is globally asymptotically stable at diseasefree equilibrium E _{0} and unstable otherwise.
Proof: Let Ï‡ _{1}â€‰=â€‰(S(t),â€‰V(t)) and Ï‡ _{2}â€‰=â€‰(L(t),â€‰A(t),â€‰B(t),â€‰C(t)) represent the number of uninfected and infected individuals, respectively. Define E _{0}â€‰=â€‰(Ï‡ _{0},â€‰0), where
Now using the proposed model (1), we have
where w(t)â€‰=â€‰bÎ¾(1â€‰âˆ’â€‰Î·C(t))â€‰+â€‰Ï†V(t)â€‰âˆ’â€‰(Î²A(t)â€‰+â€‰Î³Î²B(t)â€‰+â€‰Î¶Î²B(t))S(t)â€‰âˆ’â€‰(Î¼ _{0}â€‰+â€‰v)S(t). Thus, for Sâ€‰=â€‰S _{0}, Vâ€‰=â€‰V _{0}, and G(Ï‡ _{1},â€‰0)â€‰=â€‰0, eq. (11) becomes
Thus, from eq. (16) as tâ€‰â†’â€‰âˆž, \( {\chi}_1\to {\chi}_1^0 \). Thus, \( {\chi}_1={\chi}_1^0 \) is globally asymptotically stable.
Now to prove the second condition, that is \( H\left({\chi}_1,{\chi}_2\right)=B{\chi}_1\overline{H}\left({\chi}_1,{\chi}_2\right) \), we have
where c _{11}â€‰=â€‰Î¼ _{0}â€‰+â€‰Ïƒ, c _{22}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î³ _{1}â€‰+â€‰Ïˆ, c _{32}â€‰=â€‰(1â€‰âˆ’â€‰p)Î³ _{1}, c _{33}â€‰=â€‰(Î¼ _{0}â€‰+â€‰Î¼ _{1}â€‰+â€‰Î³ _{2}), c _{44}â€‰=â€‰Î¼ _{0}â€‰+â€‰Î¼ _{2}â€‰+â€‰Î³ _{3}â€‰+â€‰bÎ·Î¾, and Ï–(t)â€‰=â€‰Î²S _{0} L(t)â€‰+â€‰Î³Î²S _{0} A(t)â€‰+â€‰Î¶Î²S _{0} C(t)â€‰âˆ’â€‰(Î²SL(t)â€‰+â€‰Î³Î²SA(t)â€‰+â€‰Î¶Î²SC(t)). Thus, matrix B and \( \overline{H}\left({\chi}_1,{\chi}_2\right) \) are given by
From the model (1), the total population is bounded by S _{0}, that is, S,â€‰L,â€‰A,â€‰B,â€‰Câ€‰â‰¤â€‰S _{0}, so Î²SLâ€‰â‰¤â€‰Î²S _{0} I, Î²SAâ€‰â‰¤â€‰Î²S _{0} A, Î²SBâ€‰â‰¤â€‰Î²S _{0} B, and Î²SCâ€‰â‰¤â€‰Î²S _{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}â€‰>â€‰1, the model (1) is globally asymptotically stable at endemic equilibrium E _{1} and unstable otherwise.
Proof: To prove the global asymptotic stability of the proposed model (1) at endemic equilibrium E _{1}, let us consider the subsystem of (1), such that
Obviously, the endemic equilibrium point E _{1} of the system (1) is locally asymptotically stable. Let J _{2} be the variational matrix of the system (19) given by
where
The second additive compound matrix of J _{2} is denoted by \( {J}_2^{\mid 2\mid }, \) which becomes
Now choose a function \( P\left(\chi \right)=P\left(S,L,A\right)=\mathit{\operatorname{diag}}\left\{\frac{S}{L},\frac{S}{L},\frac{S}{L}\right\}, \) which implies that \( {P}^{1}\left(\chi \right)=\mathit{\operatorname{diag}}\left\{\frac{L}{S},\frac{L}{S},\frac{L}{S}\right\}, \) Then, taking the time derivative, that is, P _{ f }(Ï‡), we obtain
Now \( {P}_f{P}^{1}=\mathit{\operatorname{diag}}\left\{\frac{\dot{S}}{S}\frac{\dot{L}}{L},\frac{\dot{S}}{S}\frac{\dot{L}}{L},\frac{\dot{S}}{S}\frac{\dot{L}}{L}\right\} \) and PJ ^{âˆ£2âˆ£} P ^{âˆ’1}â€‰=â€‰J ^{âˆ£2âˆ£}. Thus, we take Bâ€‰=â€‰P _{ f } P ^{âˆ’1}â€‰+â€‰PJ ^{âˆ£2âˆ£} P ^{âˆ’1}, which can be written as
where
with \( {x}_{11}=\frac{\dot{S}}{S}\frac{\dot{L}}{L}\beta A(t)\gamma \beta B(t)\zeta \beta C(t)2{\mu}_0v{\gamma}_1\psi, \) x _{21}â€‰=â€‰Î²A(t)â€‰+â€‰Î³Î²B(t)â€‰+â€‰Î¶Î²C(t), and \( {x}_{22}=\frac{\dot{S}}{S}\frac{\dot{L}}{L}2{\mu}_0\sigma {\gamma}_1\psi . \) Let (a _{1},â€‰a _{2},â€‰a _{3}) be a vector in R ^{3} and its norm âˆ£â€‰âˆ£â€‰.â€‰âˆ£â€‰âˆ£ defined by
Let â„“(B) be the Lozinski measure with respect to the above norm described by Martin [20], then we choose
where âˆ£â€‰âˆ£â€‰B _{12}â€‰âˆ£â€‰âˆ£ and âˆ£â€‰âˆ£â€‰B _{21}â€‰âˆ£â€‰âˆ£ are matrix norms, then
where \( \ell \left({B}_{11}\right)=\frac{\dot{S}}{S}\frac{\dot{L}}{L}\beta A(t)\gamma \beta B(t)\zeta BC(t)2{\mu}_0v\sigma \), âˆ£â€‰âˆ£â€‰B _{12}â€‰âˆ£â€‰â€‰âˆ£â€‰â€‰=â€‰Î²S(t), \( \ell \left({B}_{22}\right)=\max \left\{\frac{\dot{S}}{S}\frac{\dot{L}}{L}2{\mu}_0v{\gamma}_1\psi, \frac{\dot{S}}{S}\frac{\dot{L}}{L}2{\mu}_0\sigma {\gamma}_1\psi \right\} \) \( =\frac{\dot{S}}{S}\frac{\dot{L}}{L}2{\mu}_0{\gamma}_1\psi \min \left\{v,\sigma \right\} \) and âˆ£â€‰âˆ£â€‰B _{21}â€‰âˆ£â€‰â€‰âˆ£â€‰â€‰=â€‰maxâ€‰{Ïƒ,â€‰0}â€‰=â€‰Ïƒ. Therefore, g _{1} and g _{2} becomes
which implies that
Using eq. (27) in eq. (25), we obtain
Hence, \( \ell (B)\le \frac{\dot{S}}{S}2{\mu}_0. \) Now integrating the Lozinski measure â„“(B) with respect to t in the interval [0,â€‰t] and taking lim_{ tâ€‰â†’â€‰âˆž}, we obtain
From eq. (29), we have
Thus, the system containing the first three equations of the model (1) is globally asymptotically stable around its interior equilibrium (S _{1},â€‰L _{1},â€‰A _{1}). Now consider the subsystem of the model (1), such that
By taking the limit of the system (31), we obtain
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, tâ€‰â†’â€‰âˆž, B(t)â€‰â†’â€‰B _{1}, C(t)â€‰â†’â€‰C _{1}, R(t)â€‰â†’â€‰R _{1}, and V(t)â€‰â†’â€‰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 Rungeâ€“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: bâ€‰=â€‰0.0121, Î¾â€‰=â€‰0.8, Î·â€‰=â€‰0.11, Î²â€‰=â€‰0.012, Î³â€‰=â€‰0.46, Î¶â€‰=â€‰0.0123, Ïƒâ€‰=â€‰0.0012, Ï†â€‰=â€‰0.01, Ïˆâ€‰=â€‰0.012, vâ€‰=â€‰0.6, pâ€‰=â€‰0.6, Î³ _{1}â€‰=â€‰0.33, Î³ _{2}â€‰=â€‰0.009, Î³ _{3}â€‰=â€‰0.025, Î¼ _{0}â€‰=â€‰0.069, Î¼ _{1}â€‰=â€‰0.000532, and Î¼ _{2}â€‰=â€‰0.000532. Some of these parameters, the birth rate b, natural death rate Î¼ _{0}, and proportion of perinatally infected individuals Î·, are taken from [13, 21, 22] and the remaining parameters are assumed with biologically feasible values.
Fig. 1 represents the dynamical behavior of susceptible, recovered, vaccinated, latent, acute infected, chronically infected, and vaccinated individuals, respectively. Moreover, the time interval is taken 0â€“50, while the initial population size for the compartmental population susceptible, latent, acute infected, chronically infected, carriers, recovered, and vaccinated individuals are taken to be 100, 10, 70, 60, 50, 0, and 30, respectively. The simulation of our proposed model shows that the susceptible, acute infected, and chronically infected individuals decrease sharply, while the latent, carrier recovered, and vaccinated increase at the beginning and then decrease, as shown in Fig. 1.
Sensitivity analysis
In the study of biological dynamics, the transmission dynamics of infectious disease sensitivity analysis play an especially important role. Using sensitivity analysis, we can investigate the role of each parameter used in the model and can easily develop a strategy to control the spread of infection in the community. To do this, local sensitivity analysis of the proposed model (1) has been carried out by varying parameters such as the birth rate, birth rate without successful vaccination, proportion of perinatally infected individuals, interaction rate of infected and susceptible individuals, vaccination, and the average probability of those individuals who fail to recover in acute stage and develop the chronic stage. Thus, Figs 2â€“5 represents the sensitivity analysis of our proposed model (1) with respect to birth rate without successful vaccination, proportion of perinatally infected individuals, the interaction rate of susceptible and infected individuals, and vaccination.
Figure 2 shows that the birth rate without successful vaccination is directly proportional to carrier and inversely proportional to susceptible and vaccinated individuals, while having no impact on acute and chronically infected individuals, which shows that the inflow of newborns without successful vaccination will increase the risk of carrier individuals. Fig. 3 represents that the rate of perinatally infected individuals is directly proportional to carrier and inversely proportional to latent and vaccinated individuals, while it has no impact on susceptible, acute infected, or and chronically infected individuals. Similarly to the inflow of newborns without successful vaccination, perinatally infected individuals will also increase the risk of the carrier population. Fig. 4 shows that the transmission/contact rate is directly proportional to the number of infected individuals including the latent, acute infected, chronically infected, and carrier individuals, while inversely proportional susceptible, recovered, and vaccinated individuals, which shows that the increasing contact rate of infected and noninfected will increase the risk of the infected population. Fig. 5 shows that the vaccination rate is directly proportional to recovered and vaccinated individuals and inversely proportional to susceptible and latent individuals, which illustrates that increasing vaccination will decrease the risk of an infected population. Thus, from the above discussion it is clear that for the control of hepatitis B, we need to pay more attention to the above risk factors.
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 Routhâ€”Herwitz criteria were used, while global stability was retrieved by using the method of CastilloChÃ¡vez 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.
References
WHO. Hepatitis B. Fact Sheet NÂ°204. http://www.who.int/mediacentre/factsheets/fs204/en/. Accessed 30 Apr 2014.
Shepard CW, Finelli L, Bell BP. Hepatitis B virus infection: epidemiology and vaccination. Epidemiol Rev. 2006;28:112â€“25.
Mann J, Roberts M. Modelling the epidemiology of hepatitis B in New Zealand. J Theor Biol. 2011;269:266â€“72.
Lavanchy D. Hepatitis B virus epidemiology, disease burden, treatment, and current and emerging prevention and control measures. J Vir Hep. 2004;11:97â€“107.
Lok AS, Heathcote EJ, Hoofnagle JH. Management of hepatitis B, 2000 summary of a workshop. Gastroen. 2001;120:1828â€“53.
McMahon BJ. Epidemiology and natural history of hepatitis B. Semin Liver Dis. 2005;25:3â€“8.
Chang MH. Hepatitis virus infection. Semen Fetal Neonatal Med. 2007;12:160â€“7.
Thornley S, Bullen C, Roberts M. Hepatitis B in a high prevalence New Zealand population a mathematical model applied to infection control policy. J Theor Biol. 2008;254:599â€“603.
Zaman G, Kang YH, Jung IH. Stability and optimal vaccination of an SIR epidemic model. BioSys. 2008;93:240â€“9.
Zaman G, Kang YH, Jung IH. Optimal treatment of an SIR epidemic model with time delay. Biosys. 2009;98:43â€“50.
Anderson RM, May RM. Infectious disease of humans. Dynamics and control. Oxford UK: Oxford University Press; 1991.
Zhao SJ, Xu ZY, Lu Y. A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China. Intern J Epidemiol. 2000;29:744â€“52.
Zou L, Zhang W, Ruan S. Modeling the transmission dynamics and control of hepatitis B virus in China. J Theor Biol. 2010;262:330â€“8.
Khan T, Zaman G, Chohan M. The transmission dynamic and optimal control of acute and chronic hepatitis B. J Biol Dynamic. 2016;11:172â€“89.
CastilloChavez C, Yakubu AA. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction (IMA vol 125). Berlin: Springer; 2002. pp. 153â€“63.
Birkoff, Rota GC. Ordinary Differential Equation. Ginn Boston. 1982.
Driessche PVD, Watmough J. Reproduction number and subthreshold endemic equilibria for compartmental models of disease transmission. Math Biosci. 2002;180
Brauer F, van den Driessche P, Wu J, editors. Mathematical epidemiology. Berlin: SpringerVerlag; 2008.
Li MY, Muldowney JS. A geometric approach to global stability problems. SIAM J Math Anal Appl. 1996;27:1070â€“83.
Martin JR. Logarithmic norms and projections applied to linear differential system. J Math Anal Appl. 1974;45:4332â€“454.
Zou L, Ruan S, Zhang W. On the sexual transmission dynamic of hepatitis B virus in China. J Theor Biol. 2015;369:1â€“12.
Edmunds WJ, Medley GF, Nokes DJ. The transmission dynamics and control of hepatitis B virus in the Gambia. Stat Med. 1996;15:2215â€“33.
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. 20â€“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.
Author information
Authors and Affiliations
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.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
About this article
Cite this article
Khan, T., Jung, I.H., Khan, A. et al. Classification and sensitivity analysis of the transmission dynamic of hepatitis B. Theor Biol Med Model 14, 22 (2017). https://doi.org/10.1186/s1297601700683
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1297601700683