International Journal of Discrete Mathematics

Submit a Manuscript

Publishing with us to make your research visible to the widest possible audience.

Propose a Special Issue

Building a community of authors and readers to discuss the latest research and develop new ideas.

Mathematical Analysis of Varicella Zoster Virus Model

Chicken Pox (also called Varicella) is a disease caused by a virus known as Varicella Zoster Virus (VZV) also known as human herpes virus 3 (HHV -3). Varicella Zoster Virus (VZV) is a DNA virus of the Herpes group, transmitted by direct contact with infective individuals. In this work, a deterministic mathematical model for transmission dynamics of Varicella Zoster Virus (VZV) with vaccination strategy was solved, using Adomian Decomposition Method (ADM) and Fourth-Fifth Rungekutta Felhberg Method and Approximate solutions were realized. ADM, yields analytical solution in terms of rapidly convergent infinite power series with easily computed terms. This solution was realized by applying Adomian polynomials to the nonlinear terms in the system. Similarly, fourth-fifth-order Runge-Kutta Felberg method with degree four interpolant (RK45F) was used to compute a numerical solution that was used as a reference solution to compare with the semi-analytical approximations. The main advantage of the ADM is that it yields an approximate series solution in close form with accelerated convergence. The effect of Varicella was considered in five compartments: The Susceptible, the Vaccinated, the Exposed, the Infective and the Recovered class. The Varicella Zoster virus model which is a nonlinear system can only be solved conveniently using powerful semi-analytic tool such as the ADM. Numerical simulations of the model show that, the combination of vaccination and treatment is the most effective way to combat the epidemiology of VZV in the community.

Varicella, Zoster, Adomian Decomposition, Modeling, Sensitivity, Vaccination, Epidemiology

APA Style

Anebi Elisha, Terhemen Aboiyar, Anande Richard Kimbir. (2021). Mathematical Analysis of Varicella Zoster Virus Model. International Journal of Discrete Mathematics, 6(2), 23-37. https://doi.org/10.11648/j.dmath.20210602.11

ACS Style

Anebi Elisha; Terhemen Aboiyar; Anande Richard Kimbir. Mathematical Analysis of Varicella Zoster Virus Model. Int. J. Discrete Math. 2021, 6(2), 23-37. doi: 10.11648/j.dmath.20210602.11

AMA Style

Anebi Elisha, Terhemen Aboiyar, Anande Richard Kimbir. Mathematical Analysis of Varicella Zoster Virus Model. Int J Discrete Math. 2021;6(2):23-37. doi: 10.11648/j.dmath.20210602.11

Copyright © 2021 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Babolian, E., Biazar J. and Vahidi, A. R. (2004). The Decomposition Method Applied to Systems of Fredholm Integral Equations of the Second Kind. Applied Mathematics and Computation, 148 (2): 443–452.
2. Dehghan M. and Tatari M., (2006). “The Use of Adomian Decomposition Method for Solving Problems in Calculus of Variations,” Mathematical Problems in Engineering, volume 2006, Article ID 65379, 12 pages.
3. Adomian, G., 1986. Nonlinear Stochastic Operator Equations. Academic Press, New York, ISBN: 9780120443758, pp: 287.
4. Mustafa, I. (2004). On Numerical Solutions of Partial Differencial Equations by the Decomposition Method. Kragujevac Journal of Mathematics, 26: 153-164.
5. Adjedj, B. (1999). Application of the Decomposition Method to the Understanding of HIV Immune Dynamics. Kybernetes, 28 (3): 271-283.
6. Sanchez, F. Abbaoui, K. and Cherruault, Y. (2000). Beyond the Thin-sheet Approximation: Adomian's Decomposition. Optics Communication, 173: 397-401.
7. Cherruault, Y. and Adomian, G. (1993). Decomposition Methods: A New Proof of Convergence. Mathematical and Computer Modelling, 18: 103-106.
8. Lesnic, D. (2002). Convergence of Adomian's Method: Periodic Temperatures. Applied and Computational Mathematics, 44: 13-24.
9. John, T. and Edward, J. (1997). A comparison of Adomian Decomposition Methods for Approximate Solution of Some Predator Prey Model Equations Numerical Analysis Report no, 309 a report association with university college chester.
10. Biaza J. and Ebrahimi, H. (2005). An Approximation to the Solution of Hyperbolic Equations by Adomian Decomposition Method and Comparison with Characteristic Method. Applied mathematics and Computation, 163: 633-638.
11. Abbaoui, K. and Cherruault, Y. (1995). New ideas for proving convergence of decomposition methods, Applied Mathematics and Computational 29 (7): 103-108.
12. Duan, J.-S., Rach, R., Baleanu, D., Wazwaz, A.-M.: A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc. 3 (2), 73–99 (2012)
13. Hamoud, A. A., Ghadle, K., Atshan, S.: The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method. Khayyam J. Math. 5 (1), 21–39 (2019).
14. Odibat, Z.: An optimized decomposition method for nonlinear ordinary and partial differential equations. Physica A 541, Article ID 123323 (2019).
15. Turkyilmazoglu, M.: Accelerating the convergence of Adomian decomposition method (ADM). J. Comput. Sci. 31, 54–59 (2019). https://doi.org/10.1016/j.jocs.2018.12.014.
16. Li, W., Pang, Y. Application of Adomian decomposition method to nonlinear systems. Adv Differ Equ 2020, 67 (2020). https://doi.org/10.1186/s13662-020-2529-y.
17. Gershon, M., and Gershon, A. (2018). Varicella-zoster virus and the enteric nervous system. J. Infect. Dis. 218 (Suppl._2), S113–S119. doi: 10.1093/infdis/ jiy407.
18. Edward, Stephen., Dmitry, K. and Silas, M. (2014). Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Applied and Computational Mathematics, 3: 150-162.
19. Gilden, D., Nagel, M., Cohrs, R., Mahalingam, R., and Baird, N. (2015). Varicella zoster virus in the nervous system. F1000Res 4: F1000FacultyRev–1356. doi: 10.12688/f1000research.7153.1.
20. Nagel, M. A., Jones, D., and Wyborny, A. (2017). Varicella zoster virus vasculopathy: the expanding clinical spectrum and pathogenesis. J. Neuroimmunol. 308, 112–117. doi: 10.1016/j.jneuroim.2017. 03.014.
21. Sorel, O., and Messaoudi, I. (2018). Varicella virus-host interactions during latency and reactivation: lessons from simian varicella virus. Front. Microbiol. 9: 3170. doi: 10.3389/fmicb.2018.03170.
22. Cohrs, R. J., and Gilden, D. H. (2003). Varicella zoster virus transcription in latently-infected human ganglia. Anticancer Res. 23, 2063–2069.
23. Ouwendijk, W. J., Choe, A., Nagel, M. A., Gilden, D., Osterhaus, A. D., Cohrs, R. J., et al. (2012). Restricted varicella-zoster virus transcription in human trigeminal ganglia obtained soon after death. J. Virol. 86, 10203–10206. doi: 10.1128/JVI.01331-12.
24. Zerboni, L., Sobel, R. A., Lai, M., Triglia, R., Steain, M., Abendroth, A., et al. (2012). Apparent expression of varicella-zoster virus proteins in latency resulting from reactivity of murine and rabbit antibodies with human blood group a determinants in sensory neurons. J. Virol. 86, 578–583. doi: 10.1128/JVI. 05950-11.
25. Depledge, D. P., Ouwendijk, W. J. D., Sadaoka, T., Braspenning, S. E., Mori, Y., Cohrs, R. J., et al. (2018a). A spliced latency-associated VZV transcript maps antisense to the viral transactivator gene 61. Nat. Commun. 9: 1167. doi: 10.1038/ s41467-018-03569-2.