BioMed Research International
Volume 2016 (2016), Article ID 9306329, 9 pages
http://dx.doi.org/10.1155/2016/9306329

Corrigendum

Corrigendum to “Transmission Model of Hepatitis B Virus with the Migration Effect”

Muhammad Altaf Khan,¹ Saeed Islam,¹ Muhammad Arif,¹ and Zahoor ul Haq²

¹Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan
²Department of Management Sciences, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan

Received 13 March 2016; Accepted 3 October 2016

Copyright © 2016 Muhammad Altaf Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Hepatitis B is one of the major public health problems in the world. It is an infection that causes the liver disease. The aim of this note is to provide corrections to the basic reproduction number in [1] as described in the letter to the editor [2, 3]. In view of [2, 3], the model is reconstructed and all their mathematical results are computed.

2. Model Correction

In view of [2, 3], we decided that the product of and , that is, , is incorrect. If the product is incorrect, then the obtained basic reproduction number in [1] is incorrect. The incorrectness of the basic reproduction number affects the stability (local and global), which is to be addressed again. Further, in light of [2], the model is valid biologically. By observing all these assumptions in [2, 3], we decide to reformulate the model by making the following changes to the model revised; we add the parameter , a rate from migrated class to susceptible, and is the rate of flow from exposed to migrated classes to the model published in [1]. The complete flow diagram of our new improved model can be seen in Figure 1. For complete details of each class with their parameters description, we refer the reader to [1], except and . Adding the above parameters we obtain the following improved model:

being subject to the initial conditions

The model published in [1] should be replaced by (1) as well as the results. Assume ; we obtain the following reduced model for (1):

Let

Here is a positively invariant set. All the solutions inside are our main focus.

Figure 1: The complete flow diagram of hepatitis B virus transmission model.

3. Equilibria and Basic Reproduction Number

System (3) has the disease-free equilibrium at ; that is, . At , the endemic equilibria of system (3) are given by

The basic reproduction number for system (3) can be obtained by using the method in [4]. We obtain the following matrices:

The basic reproduction number for (3) is given by

Define

where and .

4. Stability of Disease-Free Equilibrium (DFE)

In this section, we discuss the local and global stability of system (3) at disease-free equilibrium. We have the following results.

Theorem 1. The disease-free equilibrium of system (3) is locally asymptotically stable if and the conditions and are satisfied; otherwise, they become unstable.

Proof. At the disease-free equilibrium the corresponding Jacobian matrix of system (3) is computed as follows:

where . The root of is clearly negative; the other roots can be obtained from the following equation:

where

The eigenvalues of the characteristics equation have negative real part if the Routh-Hurwitz condition is satisfied; that is, , with . Therefore, if and and , we get for . Thus, at the disease-free equilibrium of system (3) is locally asymptotically stable if the above conditions are satisfied.

Next, we show the global stability of DFE by using the method in [5].

Theorem 2. The disease-free equilibrium of system (3) is globally asymptotically stable if .

Proof. Consider , , and . The invariant domain is clearly a compact positive set. We can present the subsystem as

System (3) represents a linear system which is globally asymptotically stable at . The hypotheses in [5] are satisfied. The matrix is given by

Following the hypothesis in [5], for any the above matrix is Metzler and irreducible. Further, we check the fourth condition in [5]. There is a maximum in if . This corresponds to disease-free equilibrium and the maximum is given by

The last hypothesis requires that We write in block form as

where

Clearly is stable Metzler matrix, so we can write Let .

The characteristics equation of are given by

where

The Routh-Hurwitz criteria ensure that the above characteristics equation has three negative eigenvalues if and the condition . All the conditions in [5] are satisfied. Thus, we conclude that system (3) at disease-free equilibrium is globally asymptotically stable.

5. Stability EE

In this section, we determine the local and global stability of (3) at endemic equilibrium.

Theorem 3. If , then model (3) at endemic equilibrium is locally asymptotically stable.

Proof. The Jacobian matrix computed at is given by

The characteristics equation at is

where

where , , , , , , , and . The characteristics equation will give negative roots if the following conditions are satisfied:

(i),(ii),(iii),(iv)

Conditions (i) and (ii) are easy to satisfy. If conditions (iii) and (iv) are satisfied; then the characteristics equation given above will give negative eigenvalues. Thus, it follows from Routh-Hurwitz criteria that system (3) is locally asymptotically stable at .

Next result shows the global stability of the endemic equilibrium of system (3) for special case when and . We have the following result.

Theorem 4. The endemic equilibrium of system (3) is globally asymptotically stable if condition (30) holds.

Proof. We define the Lyapunov function in the following form:

The time durative of is

The coefficients are positive and will be determined later. At endemic steady state the first equation of (3) is given by

Using (25) and the equation (3) we obtain

Similarly,

Using the values of (26) and (27) and the last three equations of system (3) and substituting in (24) we obtain

After implication we obtain

where the constants are chosen as , , , , and , .
Equation (29) if the following inequalities are satisfied:

The endemic equilibrium of system (3) is said to be globally asymptotically stable if condition (30) holds.

6. Numerical Simulation

This section deals with the numerical solution of model (3). The numerical results for model (3) are presented in Figures 2–8. In this paper, the value for and is taken between 0 and 1. The rest of the parameters values are taken from [1], except those mentioned in the figures. Figure 1 shows the population behavior of incidentals when and . Figure 3 represents the population of incisiveness when and . In Figures 2 and 3, the individuals of carriers decrease sharply. For and , we present Figure 4. Figure 5 is population behavior of individuals when and . Figures 6 and 7, respectively, represent the population of carriers individuals for different values of parameters. The population behavior of susceptible, exposed, acute, and carriers individuals for different parameters is presented in Figure 8. The numerical results from Figures 2 to 8 show when there is a decrease in the value of suggested parameters, and the population of individuals in the host decreases sharply.

Figure 2: Population behavior of individuals when , .

Figure 3: Population behavior of individuals when , .

Figure 4: Population behavior of individuals when ,  .

Figure 5: Population behavior of individuals when , .

Figure 6: Population behavior of carriers individuals: solid line, , ; dashed , .

Figure 7: Population behavior of carriers individuals: solid line , , and ; dashed , , and .

(a)

(b)

(c)

(d)

Figure 8: Population behavior of susceptible, exposed, acute, and carriers individuals: solid line , ; dashed , .

7. Conclusion

In this corrigendum, we make all the necessary changes to the published paper [1], which are highlighted in the comment papers [2, 3]. We added the parameter , a rate from migrated class to susceptible, and is the rate of flow from exposed to migrated classes. Further, the basic reproduction number has been investigated. The mathematical results for the revised model are presented successfully.

References

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