BioMed Research International
Volume 2015 (2015), Article ID 492513, 4 pages
http://dx.doi.org/10.1155/2015/492513

Letter to the Editor

Comment on “Transmission Model of Hepatitis B Virus with Migration Effect”

Anwar Zeb¹ and Gul Zaman²

¹Department of Mathematics, COMSATS Institute of Information Technology Abbottabad, Khyber Pakhtunkhwa 22060, Pakistan
²Department of Mathematics, University of Malakand Chakdara, Dir(L), Khyber Pakhtunkhwa 18800, Pakistan

Received 5 November 2015; Accepted 17 November 2015

Academic Editor: Kanury V. S. Rao

Copyright © 2015 Anwar Zeb and Gul Zaman. 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.

Abstract.

We show the erroneous assumptions and reasoning by introducing the migration effect of individuals in the transmission model of Hepatitis B virus. First, some false results related to the eigenvalues and reproductive number in the recent literature in mathematical biology will be presented. Then, it will be proved that the product of the matrices in the next generation method to obtain the reproductive number is not correct and the local and global stability results based on the reproductive number are considered false.

1. Model Formulation

In this section, we consider the model developed by Khan et al. [1] taking into account migration class in the population by extending the work of Pang et al. [2]. In their work [1] they have categorized the model into six compartments: susceptible , exposed , acute , carrier , vaccinated , and migrated individuals. On the basis of their assumptions what are the parameters in the model that bring out the migrated class? They have considered the following assumptions:

(i)the transmission rate from migrated class to exposed class,(ii)the transmission rate from migrated to acute class,(iii)the death rate of the migrated class.

The model presented in [1] is given by

They have claimed that the migrated class of individuals come from different parts of the world to the host country, and their interaction occurs in the form of sexual interactions, blood transportation, and transfusion. What are the parameters in the model that bring out the migrated class? The authors have used in the new class (migrated) only three parameters: the transmission rate from migrated class to exposed class, the transmission rate from migrated to acute class, and the death rate of the migrated class. There is a serious flaw that how the migrated individuals move to different classes and there is no region and parameters defined in the model in which the population of migrated individuals have been generated. All the terms on the right hand side are negative in the migrated class of system (1); however, for large time it tends to zero. Since population is always positive (nonnegative), the said class (migrated) is not biologically feasible. Keeping these flaws in view the extended model of Khan et al. [1] is considered false and the goal is unlikely to be met if the conclusions of the model are based on false mathematical assumptions and reasoning.

2. Falseness of the Basic Reproduction Number in [1]

To find the basic reproductive number the authors have used the concept of next generation method introduced by Van Den Driessche and Watmough [3]. Several authors have used this method to find the basic reproductive number for different epidemic models; see, for example, [2, 4, 5].

For the basic reproductive number the authors in [1] considered that and are given by

The authors have used elementary matrix operation to find that is given by

The authors have defined but they did not use this value in the matrix.

The blunder of this paper is that the matrix multiplication is given by

The authors have obtained the above matrix by using matrix multiplication which is not correct. For simple matrix multiplication, see the following example.

Example 1. Let us consider the following two matrices:

Then simple matrix multiplication of matrices and becomes

Comparing the corresponding elements of and , we conclude that the elements of are not correct and the correct one will be of the following form:

where

Now if we follow matrix of the authors, then all the eigenvalues become zero because all the terms on main diagonal of are zeros. This means that there exists no such reproductive number which the authors have mentioned throughout the paper. However, if we follow , then the reproductive number will be

which is different from the reproductive number on page  4 in [1].

Also if we use the same method to find and , then the correct matrices become

Thus the reproductive number will be

If we follow this reproductive number, then the local stability will be completely determined by this reproductive number and no further condition is required.

Theorem 2. The disease-free equilibrium of the model (3) in [1] is locally asymptotically stable if and unstable if .

For the endemic equilibria of the proposed model the authors have obtained by considering that the right hand side of each equation of system in [1] is equal to zero. In order to follow this technique for endemic equilibria equation in system presented in [1]:

All parameters are nonnegative so their sum should always be positive. Thus, which is biologically not feasible and the objective of the paper [1] is based on baseless assumption. Therefore, all the results related to local and global stability presented in [1] are based on biologically not feasible and baseless assumption.

As every theorem about the local and global stability is based on and their is not correct, so this leads to incorrect proofs.

Beside this, the authors have used in [1] the Jacobian matrices for the local stability in ; the second element of first row is positive and last element of first row is zero while these should be in both.

Similarly in equation (10) in [1] last element of first row should be and last term of fourth row is not zero that should be

Also for the Jacobian matrix mentioned by equation in [1] the authors have represented that

while this should be

In the same way one can follow and in [1].

3. Conclusion

The above discussion and reasoning indicates that conclusions based on the model studied by Khan et al. [1] may not be valid and migration of individuals may be possible to the host area but they should be in different forms. It may be possible that the migrated individuals already have got this type of infection, which means coming to the infected class, or may be possible to come to susceptible class and they do not make any type of interaction occuring in the form of sexual interactions, blood transportation, and transfusion. On the basis of this reasoning, migrated class mentioned in [1] is totally wrong, if someone wants to introduce the migrated people in the host area that should be based on the abovementioned assumptions.

Furthermore, the authors have mentioned in [1] on page 2 thrice, “we introduced the migrated class" which shows that the English presentation also needs correction. On page 3 the authors have written, “ is the rate at which the individuals move to the carrier class,” while it should have written “ be the rate at which the acute individuals move to the carrier class.” On the same page the authors have written, “ is the rate of transmission from migrated class”; instead they should have written “ which is the rate of transmission from migrated class to acute class.”

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

M. A. Khan, S. Islam, M. Arif, and Z. Ul Haq, “Transmission model of hepatitis B virus with the migration effect,” BioMed Research International, vol. 2013, Article ID 150681, 10 pages, 2013.

J. Pang, J. A. Cui, and X. Zhou, “Dynamical behavior of a hepatitis B virus transmission model with vaccination,” Journal of Theoretical Biology, vol. 265, no. 4, pp. 572–578, 2010.

P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.

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A. Zeb, G. Zaman, O. J. Algahtani, and S. Momani, “A special case in giving up smoking model with relapse class,” Wulfenia Journal, vol. 22, no. 10, pp. 90–104, 2015.