Transmission Model of Hepatitis B Virus with the Migration Effect

Abstract

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.

1. Introduction

According to World Health Organization, about 350 million people are infected with the hepatitis B virus (HBV)¹ and about 170 million people are chronically infected with the hepatitis C virus (HCV)². The majority of those infected live in developing countries with few incidences in Western countries. Migration is one of the defining issues of our time. For example, more than 5 million Canadians migrate out of the country each year and over 250,000 new immigrants arrive in Canada each year. Countries of the world are increasingly connected through travel and migration, and thus, migration has health implications in one location for both local and global migrations, since infectious diseases do not remain isolated geographically. The UK Hepatitis Foundation estimated in 2007 that the number of hepatitis B cases in the UK doubled in the previous 6 years chiefly due to immigration of infected people, many from the new member states of the European Union where the prevalence of viral hepatitis is higher. The United Arab Emirates mandated that hepatitis C testing is to be done at the time of residence visa renewals and added hepatitis C to the list of diseases such as HIV and hepatitis B as diseases warranting deportation [1]. In China, hepatitis B virus infection is a major public health problem. Hepatitis B is the first one among the diseases with legal management measures. In China, an estimated 93 million people have been infected with the hepatitis B virus [2]. The seroepidemiological survey on HBV infection conducted in 2006 showed that HBsAg carrier rate was 7.18 percent in the overall dynamics of HBV. In this paper, we consider a system of ordinary differential equations which describes the transmission of HBV transmission in China. Several mathematical models have been formulated on the HBV transmission in China. Medley and coauthors used a mathematical model and developed the strategies to eliminate the HBV in New Zealand in 2008 [3, 4]. Anderson and May [5] used a mathematical model which illustrated the effects of carriers on the transmission of HBV. An age structure model was proposed by Zhao et al. [6] to predict the dynamics of HBV transmission and evaluate the long-term effectiveness of the vaccination program in China. Wang et al. [7] proposed and analyzed the hepatitis B virus infection in a diffusion model confined to a finite domain. A hepatitis B virus (HBV) model with spatial diffusion and saturation response of the infection rate is investigated by Xu and Ma [8]. Also, Zou et al. [9] proposed a mathematical model to understand the transmission dynamics and prevalence of HBV in mainland China. A model to describe waning of immunity after sometime has been studied by a number of authors [10–16]. In the context of rapid global migration, there is a potential for any disease to be transferred faster than was previously possible. These implications concerning the movement of HBV and HCV merit far more attention by countries and the international community than they have given the problem to date. This is especially important given that the scope and speed of migration is expected to grow in coming years. In the context of rapid global migration, there is a potential for any disease to be transferred faster than was previously possible. These implications concerning the movement of HBV and HCV merit far more attention by countries and the international community than they have given the problem to date. This is especially important given that the scope and speed of migration is expected to grow in coming years.

In this paper, we construct the compartmental model of hepatitis B transmission. We have categorized the model into six compartments: Susceptible-𝑆(𝑡), Exposed-𝐸(𝑡), Acute-𝐴(𝑡), Carrier-𝐶(𝑡), Vaccinated-𝑉(𝑡), and Migrated-𝑀(𝑡) individuals. The migrated class of individuals comes from different parts of the world to the host country, and their interaction occurs in the form of sexual interactions, blood transportation, and transfusion. We modify the model from Pang et al. [10] by adding some new transmission dynamics and introduce the migrated class in the model. Furthermore, some authors [11, 13, 15, 17] show that acute hepatitis B could be found today in newborns of infected mothers. Pang et al. [10] added the vertical transmission term to exposed class from chronic carriers class on the basis of the characteristics of HBV transmission. In this paper, we improved the model of [10], with these new features, by adding the migrated class 𝑀(𝑡) and the following parameters: (i)𝜇₁: the transmission rate from migrated class to exposed class, (ii)𝜇₂: the transmission rate from migrated to acute class, (iii)𝛿: the death rate at the migrated class. The paper is organized as follows. Section 2 is devoted to the mathematical formation of the model. In Section 3 we find the Basic Reproduction Number, the disease-free equilibrium, and endemic equilibrium of the proposed model. Local asymptotic stability of the disease-free equilibrium and endemic equilibrium is discussed in Section 4. In Section 5, we study the global asymptotic stability of disease-free and endemic equilibria using the Lyapunov function. In Section 6, we study the numerical results of the proposed model and present the results in the form of plots for illustrations. The conclusion and references are presented in Section 7.

2. Model Formulation

In this section, we present the mathematical formulation of the compartmental model of hepatitis B, which consists of a system of differential equations. The model is based on the characteristics of HBV transmission. We divide the total population into six compartments, that is, Susceptible individuals 𝑆(𝑡), exposed 𝐸(𝑡), Acute 𝐴(𝑡), Carrier 𝐶(𝑡), Immunity class 𝑉(𝑡), and Migrated 𝑀(𝑡). The flow diagram (Figure 1) and the system is given in the following: 𝑑𝑆(𝑡)(𝑡)(𝑡)(𝐴(𝑡)(𝑡𝑆(𝑡)𝑑𝑡=𝛿𝜋1−𝜂𝐶−𝛿𝑆−𝛽+𝜅𝐶))+𝛿_𝑜𝑉(𝑡)(𝑡),(𝑡)−𝑝𝑆𝑑𝐸(𝐴(𝑡)(𝑡𝑆(𝑡)(𝑡)(𝑡)𝑑𝑡=𝛽+𝜅𝐶))−𝛿𝐸+𝛿𝜋𝜂𝐶−𝛾₁𝐸(𝑡)+𝜇₁𝑀(𝑡),𝑑𝐴(𝑡)𝑑𝑡=𝛾₁𝐸(𝑡)−𝛿𝐴(𝑡)−𝑞𝛾₂𝐴(𝑡)−1−𝑞𝛾₁𝐴(𝑡)+𝜇₂𝑀(𝑡),𝑑𝐶(𝑡)𝑑𝑡=𝑞𝛾₂𝐴(𝑡)−𝛿𝐶(𝑡)−𝛾₃𝐶(𝑡),𝑑𝑉(𝑡)𝑑𝑡=𝛾₃𝐶(𝑡)+1−𝑞𝛾₁𝐴(𝑡)−𝛿_𝑜𝑉(𝑡)(𝑡)()(𝑡),(𝑡)−𝛿𝑉+𝛿1−𝜋+𝑝𝑆𝑑𝑀𝜇𝑑𝑡=−₁+𝜇₂𝑀(𝑡)−𝛿𝑀(𝑡). (1) Subject to the initial conditions, 𝑆(0)(0)(0)𝐶(0)(0)(0)≥0,𝐸≥0,𝐴≥0,≥0,𝑉≥0,𝑀≥0. (2)

Figure 1 

The complete flow diagram of hepatitis B virus transmission model.

The proportion of failure immunization is shown by 𝜋. 𝛿 represent both the death and birth rate. At the 𝛾₁ rate the exposed individuals become infectious and move to the Acute class. 𝛾₂ is the rate at which the individuals move to the carrier class, 𝛾₃ is the flow of carrier to vaccinated class, 𝛽 shows the transmission coefficient, 𝜅 represents the carrier infectiousness to acute infection, 𝑞 is the proportion of acute individuals that become carrier, 𝛿_𝑜 represent the loss of immunity rate and the individual become the susceptible again, 𝑝 represents the vaccination of susceptible individuals, 𝜇₁ represents the rate of flow from migrated class to exposed class, and 𝛾₂ is the rate of transmission from migrated class to acute class. 𝜂 is the unimmunized children born to carrier mothers, 𝛿(1−𝜋) measures the successful immunization of newborn babies, and the term 𝛿𝜋(1−𝜂𝐶(𝑡)) shows that the newborns are unimmunized and become susceptible again.

We assume that the total population size is equal to 1, and just for simplifications, 𝑆(𝑡) is the susceptible, 𝐸(𝑡) the exposed, 𝐴(𝑡) the acute, 𝐶(𝑡) the carrier, 𝑉(𝑡) the immunity, and 𝑀(𝑡) is the migrated class representing the state variables in our proposed population model. The sum of the total population is 𝑆(𝑡)+𝐸(𝑡)+𝐴(𝑡)+𝐶(𝑡)+𝑉(𝑡)+𝑀(𝑡)=1, holds. We just add (1), and we can easily get. We ignore the fifth equation in system (1); so, the new models become 𝑑𝑆(𝑡)(𝑡)(𝑡)(𝐴(𝑡)(𝑡𝑆(𝑡)𝑑𝑡=𝛿𝜋1−𝜂𝐶−𝛿𝑆−𝛽+𝜅𝐶))+𝛿_𝑜((𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡(𝑡),(𝑡)1−+𝐸+𝐴+𝐶+𝑀)))−𝑝𝑆𝑑𝐸(𝐴(𝑡)(𝑡𝑆(𝑡)(𝑡)(𝑡)𝑑𝑡=𝛽+𝜅𝐶))−𝛿𝐸+𝛿𝜋𝜂𝐶−𝛾₁𝐸(𝑡)+𝜇₁𝑀(𝑡),𝑑𝐴(𝑡)𝑑𝑡=𝛾₁𝐸(𝑡)−𝛿𝐴(𝑡)−𝑞𝛾₂𝐴(𝑡)−1−𝑞𝛾₁𝐴(𝑡)+𝜇₂𝑀(𝑡),𝑑𝐶(𝑡)𝑑𝑡=𝑞𝛾₂𝐴(𝑡)−𝛿𝐶(𝑡)−𝛾₃𝐶(𝑡),𝑑𝑀(𝑡)𝜇𝑑𝑡=−₁+𝜇₂𝑀(𝑡)−𝛿𝑀(𝑡). (3) Let Γ=(𝑆,𝐸,𝐴,𝐶,𝑀)∈ℜ⁵₊,∣𝑆(𝑡)≤𝛿𝜋+𝛿_𝑜𝛿+𝛿_𝑜,+𝑝𝑆+𝐸+𝐴+𝐶+𝑀≤𝛿𝜋+𝛿_𝑜𝛿+𝛿_𝑜. (4) Here, Γ is a positively invariant set. All the solutions lie inside Γ which is our main focus.

3. Basic Reproduction Number/Threshold Quantity

The basic reproduction number or the threshold quantity ℜ₀ for the proposed model gives an average number of secondary infection when an infection is introduced in a purely susceptible population. We use the idea developed by [18], and also for detail see [19]. We haveℱ=0𝛽𝑆^𝑜𝛽𝜅𝑆^𝑜000000,𝑉=𝛿+𝛾₁0−𝛿𝜋𝜂−𝛾₁𝛿+𝑞𝛾₂+1−𝑞𝛾₁00−𝑞𝛾₂𝛿+𝛾₃,𝑉⁻¹=^{𝛿+𝑞𝛾2+𝛾1−𝑞1𝛿+𝛾3𝑞𝛾2𝛿𝜋𝜂𝛿𝜋𝜂𝛿+𝑞𝛾2+𝛾1−𝑞1𝛾1𝛿+𝛾3𝛿+𝛾1𝛿+𝛾3𝛿𝜋𝜂𝛾1𝑞𝛾1𝛾2𝑞𝛾2𝛿+𝛾1𝛿+𝛾1𝛿+𝑞𝛾2+𝛾1−𝑞1}𝛿+𝛾₁𝛿+𝑞𝛾₂+1−𝑞𝛾₁𝛿+𝛾₃−𝛿𝜋𝜂𝑞𝛾₁𝛾₂, (5)where 𝑄₁=(𝛿+𝑞𝛾₂+(1−𝑞)𝛾₁) and𝐹𝑉⁻¹=0𝛽𝑆^𝑜𝛿𝜋𝜂𝑞𝛾₂𝛿+𝛾₁𝛿+𝑞𝛾₂+1−𝑞𝛾₁𝛿+𝛾₃−𝛿𝜋𝜂𝑞𝑟₁𝑟₂𝜅𝛽𝑆^𝑜𝛿+𝛾₁1−𝑞+𝑞𝛾₂𝛿+𝛾₁𝛿+𝑞𝛾₂+1−𝑞𝛾₁𝛿+𝛾₃−𝛿𝜋𝜂𝑞𝑟₁𝑟₂000.000 (6)So, the reproduction number given by 𝜌(𝐹𝑉⁻¹) is ℜ_𝑜=𝜅𝛽𝑆^𝑜𝛿+𝛾₁1−𝑞+𝑞𝛾₂+𝛿𝜋𝜂𝑞𝑟₁𝑟₂𝛿+𝛾₁𝛿+𝑞𝛾₂+1−𝑞𝛾₁𝛿+𝛾₃. (7) Here, 𝑆^𝑜 shows the disease-free equilibrium (DFE), and 𝐷_𝑜=(𝑆^𝑜,0,0,0,0), giving 𝑆^𝑜=(𝛿𝜋+𝛿_𝑜)/(𝛿+𝛿_𝑜+𝑝). The endemic equilibrium point 𝑇^∗=(𝑆^∗,𝐸^∗,𝐴^∗,𝐶^∗,𝑀^∗) for system (3), whose endemic equilibrium is given in the following subsection.

Endemic Equilibria. To find the endemic equilibria of the system (3), by setting 𝑆=𝑆^∗, 𝐸=𝐸^∗, 𝐴=𝐴^∗, 𝐶=𝐶^∗, and 𝑀=𝑀^∗, equating left side of the system (3) equal to zero, we obtained 𝑆^∗=𝑄₁𝛿+𝛾₁𝛿+𝛾₃1−𝑅₀+𝜅𝛽𝑆^𝑜𝑄₁𝛾₁𝛽𝛿+𝛾₃+𝜅𝑞𝛾₂−𝛿+𝛾₃𝛿+𝛾₁+𝜇₁𝛾₁𝑀^∗𝛾₁𝛽𝛿+𝛾₃+𝜅𝑞𝛾₂𝐴^∗,𝐸^∗=𝑄₁𝐴^∗−𝜇₂𝑀^∗𝛾₁,𝐶^∗=𝑞𝛾₂𝛿+𝛾₃𝐴^∗. (8)

4. Local Stability Analysis

In this section, we find the local stability of disease-free and endemic equilibria. First, we show the local asymptotical stability of DFE equilibrium, and then we find the local asymptotical stability of endemic equilibrium. Now, we show the local stability of DFE about the point 𝐷_𝑜=(𝑆^𝑜,0,0,0,0) in the following theorem.

Theorem 1. For 𝑅₀≤1, the disease-free equilibrium of the system (3) about an equilibrium point 𝐷_𝑜=(𝑆^𝑜,0,0,0,0) is locally asymptotically stable if ((𝑄₁(𝛿+𝛿_𝑜+𝑝)(𝛿+𝛾₁)>𝛽𝛾₁(𝛿𝜋+𝛿_𝑜)); otherwise, the disease-free equilibrium of the system (3) is unstable for 𝑅₀>1.

Proof. To show the local stability of the system (3), about the point 𝐷_𝑜, we set the left-hand side of the system (3) equating to zero, and we obtain the following Jacobian matrix 𝐽_𝑜(𝜁):𝐽_𝑜(𝜁)=−𝛿−𝛿_𝑜−𝑝𝛿_𝑜−𝛽𝑆^𝑜−𝛿_𝑜−𝛿𝜋𝜂−𝛽𝜅𝑆^𝑜−𝛿_𝑜00−𝛿+𝛾₁𝛽𝑆^𝑜𝛽𝜅𝑆^𝑜+𝛿𝜋𝜂𝜇₁0𝛾₁−𝑄₁0𝜇₂00𝑞𝛾₂−𝛿+𝛾₃00000−𝜇₁+𝜇₂+𝛿. (9)By the elementary row operation, we get the following matrix:𝐽_𝑜(𝜁)=−𝛿+𝛿_𝑜+𝑝−𝛿_𝑜−𝛽𝑆^𝑜−𝛿_𝑜−𝛿𝜋𝜂−𝛽𝜅𝑆^𝑜00−𝛿+𝛾₁𝛽𝑆^𝑜𝛽𝜅𝑆^𝑜+𝛿𝜋𝜂𝜇₁00−𝑄₁𝛿+𝛾₁+𝛾₁𝛽𝑆^𝑜𝛾₁𝛽𝜅𝑆^𝑜+𝛿𝜋𝜂𝛿+𝛾₁𝜇₂+𝜇₁𝛾₁000𝑇₁00000−𝜇₁+𝜇₂,+𝛿 (10)where 𝑄₁=(𝛿+𝑞𝛾₂+(1−𝑞)𝛾₁) and 𝑇₁=−(−𝑄₁(𝛿+𝛾₁)+𝛾₁𝛽𝑆^𝑜)−𝑞𝛾₁𝛾₂(𝛽𝜅𝑆^𝑜+𝛿𝜋𝜂). The characteristic equation to the previous Jacobian matrix is given by 𝜆+𝛿+𝛿_𝑜+𝑝𝜆+𝜇₁+𝜇₂+𝛿𝜆³+𝑎₁𝜆²+𝑎₂𝜆+𝑎₃=0. (11) The first two eigenvalues −(𝛿+𝛿_𝑜+𝑝) and −(𝜇₁+𝜇₂+𝛿) have negative real parts. For the rest of the eigenvalues, we get 𝑎₁=𝛿+𝛿_𝑜+𝑝𝑄₁𝛿+𝛾₃+1+𝑄₁1−𝑅₀+𝛽𝛿𝜋+𝛿_𝑜𝛾₁𝛿+𝛾₃+𝑞𝛾₁𝛾₂−𝑄₁𝜅+𝛾₁×𝛿+𝛿_𝑜+𝑝⁻¹,𝑎₂=𝑄₁𝛿+𝛾₁𝛿+𝛿_𝑜+𝑝−𝛽𝛾₁𝛿𝜋+𝛿_𝑜×𝛿+𝛿_𝑜+𝑝𝛿+𝛾₁+𝛿+𝛾₁²+𝛿+𝛾₁𝑄₁𝛿+𝛾₁𝛿+𝛿_𝑜+𝑝−𝛽𝛾₁𝛿𝜋+𝛿_𝑜+𝑞𝛾₁𝛾₂𝛿+𝛾₁𝛽𝜅𝛿𝜋+𝛿_𝑜×+𝛿𝜋𝜂𝛿+𝛿+𝑝𝛿+𝛿_𝑜+𝑝⁻²+𝑞𝛾₁𝛾₂𝛿+𝛾₁𝛽𝜅𝛿𝜋+𝛿_𝑜+𝛿𝜋𝜂𝛿+𝛿_𝑜×+𝑝𝛿+𝛿_𝑜+𝑝⁻¹,𝑎₃=𝛿+𝛾₁𝛿+𝛿_𝑜𝑄+𝑝¹𝛿+𝛿_𝑜+𝑝𝛿+𝛾₁−𝛽𝛾₁𝛿𝜋+𝛿_𝑜×𝑞𝛾₁𝛾₂𝛽𝜅𝛿𝜋+𝛿_𝑜+𝑄+𝛿𝜋𝜂₁𝛿+𝛾₁𝛿+𝛿_𝑜+𝑝−𝛽𝛾₁𝛿𝜋+𝛿_𝑜. (12) By the Routh-Hurwitz criteria, 𝑎₁>0, 𝑎₃>0, and 𝑎₁𝑎₂>𝑎₃. Here, 𝑎₁>0 when 𝑅₀≤1 and ((𝑄₁(𝛿+𝛿_𝑜+𝑝)(𝛿+𝛾₁)>𝛽𝛾₁(𝛿𝜋+𝛿_𝑜)). Also, 𝑎₂>0 and 𝑎₃>0, and then 𝑎₁𝑎₂>𝑎₃. So, according to the Routh-Hurwitz criteria, the Jacobian matrix has negative real parts if and only if 𝑅₀≤1. Thus by Routh-Hurwitz criteria, the DFE of the system (3) is locally asymptotically stable about the point 𝐷_𝑜=(𝑆^𝑜,0,0,0,0). The proof is completed.

The stability of the disease-free equilibrium for 𝑅₀≤1 means that the disease dies out from the population. Next we show that the endemic equilibrium of the system (3) is locally asymptotically stable for 𝑅₀>1. When the disease-free equilibrium is locally asymptotically stable for 𝑅₀≤1, then the endemic equilibrium does not exist, but we are interested to know about the properties of an endemic equilibrium for 𝑅₀>1.

4.1. Stability of Endemic Equilibrium (EE)

In this subsection, we find the local asymptotic stability of EE about 𝐷^∗=(𝑆^∗,𝐸^∗,𝐴^∗,𝐶^∗,𝑀^∗), and we prove the local stability of endemic equilibrium in the following.

Theorem 2. For 𝑅₀>1, the endemic equilibrium 𝐷^∗ of system (3) is locally asymptotical stable, if the following conditions hold: 𝛾₁𝑍₂𝛿_𝑜𝑆^∗2>𝛾₁𝛽²𝐺𝜅,₁>𝐺₂; (13) otherwise, the system is unstable.

Proof. Here, we prove the that the system (3) about the equilibrium point 𝐷^∗ is locally stable, and for this, we obtain the Jacobian matrix 𝐽^∗(𝜁) of the system (3), in the following:𝐽^∗(𝜁)=−𝛿+𝛿_𝑜+𝑝+𝛽𝐴^∗+𝜅𝐶^∗𝛿_𝑜−𝛽𝑆^∗−𝛿_𝑜−𝛿𝜋𝜂−𝛽𝜅𝑆^∗−𝛿_𝑜−𝛿_𝑜𝛽𝐴^∗+𝜅𝐶^∗−𝛿+𝛾₁𝛽𝑆^∗𝛽𝜅𝑆^∗+𝛿𝜋𝜂𝜇₁0𝛾₁−𝑄₁0𝜇₂00𝑞𝛾₂−𝛿+𝛾₃00000−𝛿+𝜇₁+𝜇₂. (14)By elementary row operation and after simplification, we get the following Jacobian matrix:𝐽^∗(𝜁)=−𝑍₁−𝛿_𝑜−𝛽𝑆^∗−𝛿_𝑜𝑍₃−𝛿_𝑜0−𝑍₁𝛿+𝛾₁−𝑍₂𝛿_𝑜𝑍₁𝛽𝑆^∗−𝑍₂𝛽𝑆^∗+𝛿_𝑜𝑍₁𝑍₄+𝑍₂𝑍₃𝑍₁𝜇₁−𝑍₂𝛿_𝑜00𝑍₅𝛾₁𝑍₁𝑍₄+𝑍₂𝑍₃𝑍₆000𝑍₇00000−𝜇₁+𝜇₂,+𝛿 (15)where 𝑍₁=𝛿+𝛿_𝑜+𝑝+𝛽𝐴^∗+𝜅𝐶^∗,𝑍₂=𝛽𝐴^∗+𝜅𝐶^∗,𝑍₃=−𝛿𝜋𝜂+𝛽𝜅𝑆^∗+𝛿_𝑜,𝑍₄=𝛽𝜅𝑆^∗,𝑍+𝛿𝜋𝜂₅=−𝑄₁−𝑍₁𝛿+𝛾₁−𝑍₂𝛿_𝑜+𝛾₁𝑍₄𝛽𝑆^∗−𝑍₂𝛽𝑆^∗+𝛿_𝑜,𝑍₆=−𝜇₂−𝑍₁𝛿+𝛾₁−𝑍₂𝛿_𝑜+𝛾₁𝑍₁𝜇₁−𝑍₂𝛿_𝑜,𝑍₇=−𝑍₅𝛿+𝛾₃−𝑍₅𝛾₁𝑍₁𝑍₄+𝑍₂𝑍₃. (16) The eigenvalue 𝜆₁=−(𝜇₁+𝜇₂+𝛿)<0, 𝜆₂=−𝑍₁, and using the value of 𝑍₁ and further 𝐶^∗, we get 𝜆₂=−(𝛿+𝛿_𝑜+𝑝+𝛽((𝛿+𝛾₃)+𝜅𝑞𝛾₂)𝐴^∗)<0. 𝜆₃=−(𝑍₁(𝛿+𝛾₁)+𝑍₂𝛿_𝑜)<0, as 𝑍₁>0 and 𝑍₂>0. 𝜆₄=𝑍₅, 𝜆₄<0, if and only if 𝑍₅<0. After the simplifications, we get 𝛽^∗𝐴^∗𝛾₁𝑍₂𝛿_𝑜−𝛾₁𝛽²𝜅𝑆^∗2+𝛾₁𝛽𝛿𝜋𝜂𝛽^∗∗∗𝑀^∗−𝛾₁𝛽𝛿𝜋𝜂𝛽^∗∗+𝛽^∗𝑄₁𝑍₁𝛿+𝛾₁+𝑄₁𝑍₂𝛿_𝑜𝛽^∗>0, (17) where 𝛽^∗=𝛾₁𝛽((𝛿+𝛾₃)+𝜅𝑞𝛾₂), 𝛽^∗∗=𝑄₁(𝛿+𝛾₁)(𝛿+𝛾₃)(1−𝑅₀)+𝜅𝛽𝑆^𝑜𝑄₁, and {𝛾₁𝛽𝛿𝜋𝜂𝛽^∗∗∗𝑀^∗>[(𝛾₁𝛽𝛿𝜋𝜂𝛽^∗∗+𝛽^∗𝑄₁)𝑍₁(𝛿+𝛾₁)+𝑄₁𝑍₂𝛿_𝑜𝛽^∗]}, say 𝐺₁=𝛾₁𝛽𝛿𝜋𝜂𝛽^∗∗∗𝑀^∗ and 𝐺₂=[(𝛾₁𝛽𝛿𝜋𝜂𝛽^∗∗+𝛽^∗𝑄₁)𝑍₁(𝛿+𝛾₁)+𝑄₁𝑍₂𝛿_𝑜𝛽^∗]𝛽^∗∗∗=(𝛿+𝛾₃)[(𝛿+𝛾₁)+𝛾₁𝜇₁]. So, 𝜆₄ has negative real part if (𝛾₁𝑍₂𝛿_𝑜>𝛾₁𝛽²𝜅𝑆^∗2). For 𝜆₅=𝑍₇, we obtained negative real parts, and by using the 𝑍₅ which is positive under the conditions described in 𝜆₄, and 𝑍₁>0, 𝑍₂>0, we get the negative real parts. Thus, all the eigenvalues have negative real parts, so by the Routh-Hurwitz criteria the endemic equilibrium point 𝐷^∗ is locally asymptotically stable when 𝑅₀>1. The proof is completed.

5. Global Stability of DFE

In this section, we present the global stability of disease-free equilibrium DFE of the system (3). For different biological model, the Lyapunov function was used by [20, 21] for the global stability. For our model, we define and construct Lyapunov function in the following for the global stability of DFE. Further the global stability of endemic equilibrium we use the Lyapunov function and find its global asymptotical stability.

Theorem 3. For 𝑅₀≤1, the disease-free equilibrium of the system (3) is stable globally asymptotically, if 𝑆=𝑆^𝑜 and unstable for 𝑅₀>1.

Proof. Here, we define the Lyapunov function for the global stability of disease-free equilibrium, given by 𝑉(𝑡)=𝑑₁𝑆−𝑆^𝑜+𝑑₂𝐸+𝑑₃𝐴+𝑑₄𝐶+𝑑₅𝑀. (18) Differentiating the previous function with respect to 𝑡 and using the system (3), 𝑉^′(𝑡)=𝑑₁𝑆^′+𝑑₂𝐸^′+𝑑₃𝐴^′+𝑑₄𝐶^′+𝑑₅𝑀^′,𝑉^′(𝑡)=𝑑₁𝛿𝜋1−𝜂𝐶(𝑡)−𝛿𝑆(𝑡)−𝛽(𝐴(𝑡)+𝜅𝐶(𝑡))𝑆(𝑡)+𝛿_𝑜(1−(𝑆(𝑡)+𝐸(𝑡)+𝐴(𝑡)+𝐶(𝑡)+𝑀(𝑡)))−𝑝𝑆(𝑡)+𝑑₂𝛽(𝐴(𝑡)+𝜅𝐶(𝑡))𝑆(𝑡)−𝛿𝐸(𝑡)+𝛿𝜋𝜂𝐶(𝑡)−𝛾₁𝐸(𝑡)+𝜇₁𝑀(𝑡)+𝑑₃𝛾₁𝐸(𝑡)−𝛿𝐴(𝑡)−𝑞𝛾₂𝐴(𝑡)−1−𝑞𝛾₁𝐴(𝑡)+𝜇₂𝑀(𝑡)+𝑑₄𝑞𝛾₂𝐴(𝑡)−𝛿𝐶(𝑡)−𝛾₃𝐶(𝑡)+𝑑₅−𝜇₁+𝜇₂𝑀(𝑡)−𝛿𝑀(𝑡), (19) where 𝑑_𝑖, 𝑖=1,2,…5, are some positive constants to be chosen later.
After the arrangement, we obtain 𝑉^′(𝑡)=𝑑₂−𝑑₁𝛽(𝐴+𝜅𝐶)𝑆+𝑑₂−𝑑₁+𝑑𝛿𝜋𝜂𝐶₃𝛾₁−𝑑₂𝛿+𝛾₁𝐸+𝑑₂𝜇₁−𝑑₅𝛿𝑀+𝑑₄𝑞𝛾₂−𝑑₃𝑄₁𝐴+𝑑₁𝛿𝜋−𝑑₁𝛿𝑆+𝑑₁𝛿_𝑜−𝑑₁𝛿_𝑜(𝑆+𝐸+𝐴+𝐶+𝑀)−𝑑₁𝑝𝑆−𝑑₄𝛿+𝛾₃𝐶. (20) Choosing the constants, 𝑑₁=𝑑₂=𝛾₁,  𝑑₃=(𝛿+𝛾₁),  𝑑₄=𝑄₁(𝛿+𝛾₁)/𝑞𝛾₂, and 𝑑₅=𝛾₁𝜇₁/𝛿.
After the simplification, we get 𝑉^′(𝑡)=−𝑆−𝑆^𝑜−𝛾₁𝛿_𝑜𝐸−𝛾₁𝛿_𝑜𝐴−𝛾₁𝛿_𝑜+𝛿+𝛾₃𝛿+𝛾₁𝑄₁𝑞𝛾₂𝐶−𝛾₁𝛿_𝑜+𝛿_𝑜+𝛾₁𝜇₂𝑀, (21) where 𝑆^𝑜=(𝛿𝜋+𝛿_𝑜)/(𝛿+𝛿_𝑜+𝑝). 𝑉^′(𝑡)=0 if and only if 𝑆=𝑆^𝑜 and 𝐸=𝐴=𝐶=𝑀=0. Also, 𝑉^′(𝑡) is negative for (𝑆>𝑆^𝑜). So, by [22], the DFE is globally asymptotically stable in Γ. The proof is completed.

5.1. Global Stability of Endemic Equilibrium

In this subsection, we show the global asymptotical stability of the system (3). To do this, we state and prove the following theorem for the global stability of endemic equilibrium.

Theorem 4. For 𝑅₀>1, system (3) is globally asymptotically stable, if 𝑆=𝑆^∗ and 𝛿_𝑜>𝜇₁, and unstable for 𝑅₀≤1.

Proof. To prove that system (3) is globally asymptotically stable, we define the Lyapunov in the following: 𝐿(𝑡)=𝜇₁+𝜇₂+𝜇₃𝛾₁𝛾₁+𝛿𝑆−𝑆^∗+𝜇₁+𝜇₂+𝜇₃𝛾₁𝛾₁𝐸+𝜇+𝛿₁+𝜇₂+𝜇₃𝜇𝐴+¹+𝜇₂+𝜇₃𝑄₁𝑞𝛾₂𝐶+𝜇₂𝑀. (22) Taking the derivative with respect to time 𝑡, using the system (3), 𝐿^′(𝑡)=𝜇₁+𝜇₂+𝜇₃𝛾₁𝛾₁×(𝑡)(𝑡)(𝐴(𝑡)(𝑡𝑆(𝑡)+𝛿𝛿𝜋1−𝜂𝐶−𝛿𝑆−𝛽+𝜅𝐶))+𝛿_𝑜((𝑆(𝑡)(𝑡)(𝑡)(𝑡)(𝑡(𝑡+𝜇1−+𝐸+𝐴+𝐶+𝑀)))−𝑝𝑆)₁+𝜇₂+𝜇₃𝛾₁𝛾₁𝛽(𝐴(𝑡)(𝑡𝑆(𝑡)(𝑡)(𝑡)+𝛿+𝜅𝐶))−𝛿𝐸+𝛿𝜋𝜂𝐶−𝛾₁𝐸(𝑡)+𝜇₁𝑀(𝑡+𝜇₁)+𝜇₂+𝜇₃𝛾₁𝐸(𝑡)−𝛿𝐴(𝑡)−𝑞𝛾₂𝐴(𝑡)−1−𝑞𝛾₁𝐴(𝑡)+𝜇₂𝑀(𝑡+𝜇₁)+𝜇₂+𝜇₃𝑄₁𝑞𝛾₂𝑞𝛾₂𝐴(𝑡)−𝛿𝐶(𝑡)−𝛾₃𝐶(𝑡)+𝜇₂−𝜇₁+𝜇₂+𝛿𝑀. (23) Simplifying, we obtained 𝐿^′(𝑡)𝜇=−¹+𝜇₂+𝜇₃𝛾₁𝛾₁+𝛿𝛿+𝛿_𝑜+𝑝𝑆−𝑆^∗−𝜇₁+𝜇₂+𝜇₃𝛾₁𝛿_𝑜𝛾₁[]−𝜇+𝛿𝐸+𝐴+𝐶₁+𝜇₂+𝜇₃𝛾₁𝛾₁𝛿+𝛿^𝑜−𝜇₁𝑀. (24) The endemic equilibrium of the system (3) is globally asymptotically stable for 𝑅₀>1, if 𝑆=𝑆^∗ and 𝛿_𝑜>𝜇₁. So, the endemic equilibrium of the system (3) is globally asymptotically stable. The proof is completed.

6. Numerical Simulations

In this section, we present the numerical simulation of the proposed model (3), by using the Runge-Kutta order four scheme. For different values of the parameters, the numerical results are presented in Figures 2, 3, 4, 5, 6, 7, and 8. The variation of migration parameters 𝜇₁ and 𝜇₂, with different values, is presented. In the numerical solution of the model, the parameters and their values are presented in Table 1. In our simulation, the susceptible individuals are shown by dashed line, the exposed individuals by dotted dashed, the acute individuals by bold line, the carrier by dashed line and the migrated individuals by red bold line. Figures 2 to 8 represent the compartmental population of hepatitis B individuals with migration effect. The values presented in Table 1 are fixed except for 𝜇₁ and 𝜇₂. In Figure 2, by the values for 𝜇₁=0.9 and 𝜇₂=0.9, we see that the population of exposed and acute individuals is increasing. In Figure 3, we set 𝜇₁=0.8 and 𝜇₂=0.9, and the population of exposed and acute individuals decreased. Decreasing the values of 𝜇₁ and 𝜇₂, we obtain different results; see Figures 2 to 8. When we decrease the population of immigrants who have the HBV virus, we see the decrease in the population of exposed and acute individuals. The parameters presented in Table 1 were used by different authors, for example, the natural death rate equally birth rate by [4], the rate at which the latent individuals become infectious by [3], the rate at which the individuals move to the carrier class by [3], the rate at which the individuals move to the carrier class [3], the transmission coefficient 𝛽 by [24], 𝛿_𝑜 the loss of immunity rate by [13], the value of 𝜂 by [13], the value of 𝜅 by [10], and 𝜋 and 𝑞 from [23]. We assume the values for the parameters 𝛾₃, 𝑝, 𝜇₁, and 𝜇₂ in our simulation.

Table 1 

Parameter values used in numerical simulations.

Figure 2 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.90 and 𝜇2=0.90.

Figure 3 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.80 and 𝜇2=0.90.

Figure 4 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.70 and 𝜇2=0.80.

Figure 5 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.50 and 𝜇2=0.60.

Figure 6 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.20 and 𝜇2=0.30.

Figure 7 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.20 and 𝜇2=0.40.

Figure 8 

The plot shows the HBV transmission model of hepatitis B, with 𝜇1=0.10 and 𝜇2=0.20.

7. Conclusion

A compartmental model of HBV transmission virus has been presented. A mathematical model has been obtained by adding (1) the migrated class, (2) HBV transmission rate between migrated class and exposed class, (3) transmission rate between migrated class and acute class, and (4) death rate of individuals in the migrated class. By adding these new features, we have obtained a compartmental model of HBV with migration effect. First, we obtained the basic reproduction number for the proposed model. The disease-free equilibrium is locally as well as globally asymptotically stable for 𝑅₀≤1. We obtained the local and global asymptotical stability for the endemic equilibrium. For the reproduction number 𝑅₀>1, the endemic equilibrium is, locally as well as globally, asymptotically stable. Furthermore, we have solved the compartment model numerically, and the results are presented in Figures 2 to 8. By changing the values of 𝜇₁ and 𝜇₂, different results have been obtained. It is concluded that when the value of 𝜇₁ and 𝜇₂ decreases, the population of (exposed, acute, and carrier) individuals also decrease. The proportion of infected individuals decreases when the proportion of migrated individuals (who have the HBV virus) decreases. So, the number of infected individuals is directly proportional to the number of migrated individuals.

Last yet not the least, the authors of this work have agreed to devise in a course of time a more advanced model on restraining HBV transmission through migration.

Acknowledgment

The authors would like to thank the Kanury V. S. Rao for their careful reading of the original manuscripts and their many valuable comments and suggestions that greatly improve the presentation of this work.