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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Development of Optimal Strategy for Controlling Transmission Dynamics of HBV Epidemic Model

Journal of Applied Nonlinear Dynamics 11(2) (2022) 415--425 | DOI:10.5890/JAND.2022.06.011

Eihab B. M. Bashier$^{1, 2}$, Hasim A. Obaid$^2$

$^1$ Faculty of Education and Arts, Sohar University, P.O. Box 44, Postal Code: 311, Sohar, Oman

$^2$ Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Khartoum, P.O. Box

321, Postal Code: 11115, Khartoum, Sudan

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Abstract

In this paper, the problem of controlling the transmission dynamics of HBV epidemics is formulated as an optimal control problem governed by a system of nonlinear differential equations. To reduce the HBV infection, we formulate two controls representing the increase of effort to immunize the new born individuals and isolating the infection carriers. The first order necessary conditions for optimal control are derived. The numerical simulations considered many scenarios and the controls are shown to be effective in reducing the number of infectious individuals. They showed that reducing the numbers of infected carriers can be achieved by applying the maximum controls for long periods of times and the immunization of new born individuals is more effective than isolating the infected individuals.

References

  1. [1]  Liaw, Y. and Zoulim, F. (2016), Hepatitis B Virus in Human Diseases, Springer International Publishing, Switzerland.
  2. [2]  World Health Organization: https://www.who.int/news-room/fact-sheets/detail/hepatitis-b.
  3. [3]  Tripathi, A., Naresh, R., and Sharma, D. (2007), Modelling the effect of screening of unaware infectives on the spread of HIV infection, Applied Mathematics and Computation, 184, 1053-1068.
  4. [4]  Ida, A., Oharub, S., and Oharu, Y. (2007), A mathematical approach to HIV infection dynamics, Journal of Computational and Applied Mathematics, 204, 172-186.
  5. [5]  Waziri, A.S., Massawe, E.S., and Makinde, O.D. (2012), Mathematical modelling of HIV/AIDS dynamics with treatment and vertical transmission, Applied Mathematics, 2(3), 77-89.
  6. [6]  Lutera, J., Mbete, D., and Wangila, S. (2018), Co-infection Model of HIV/AIDS-Pneumonia on the Effect of Treatment at Initial and Final Stages, IOSR Journal of Mathematics, 14(5), 56-81.
  7. [7]  Ngwa, G.A. and Shu, W.S. (2000), A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32, 747-763.
  8. [8]  Pandey, R., Singh, R.N., and Pandey, P.N. (2013), Mathematical model for malaria transmission and chemical control, Journal of International Academy of Physical Sciences, 17(4), 377-394.
  9. [9]  Nah, K., Nakata, Y., and R\"{o}st, G. (2014), Malaria dynamics with long incubation period in hosts, Computers and Mathematics with Applications, 68, 915-930.
  10. [10]  Koutou, O., Traor{e}, B., and Sangar{e}, B. (2018), Mathematical model of malaria transmission dynamics with distributed delay and a wide, Cogent Mathematics, 5, 1564531.
  11. [11]  Irving, T.J., Blyuss, K.B., Colijn, C., and C.L. Trotter (2012), Modelling meningococcal meningitis in the African meningitis belt, Epidemiology and Infection, 140, 897-905.
  12. [12]  Martcheva, M. and Crispino-O'Connell, G. (2003), The transmission of meningococcal infection: a mathematical study, Journal of Mathematical Analysis and Applications, 283, 251-275.
  13. [13]  Mart{\i}nez, M.J.F., Merino, E.G., S{a}nchez, E.G., S{a}nchez, J.E.G., del Rey, A.M., and S{a}nchez, G.R. (2013), A mathematical model to study the meningococcal meningitis, Procedia Computer Science, 18, 2492-2495.
  14. [14]  Stollenwerk, N. and Jansen, V.A.A. (2003), Evolution towards criticality in an epidemiological model for meningococcal disease, Physics Letters A, 317, 87-96.
  15. [15]  Adewale, S.O., Podder, C.N., and Gumel, A.B. (2009), Mathematical analysis of a tuberculosis transmission model with DOTS, Canadian Applied Mathematics Quarterly, 17(1), 1-36.
  16. [16]  Bhunua, C.P., Garira, W., Mukandavire, Z., and Zimba, M. (2008), Tuberculosis Transmission Model with Chemoprophylaxis and Treatment, Bulletin of Mathematical Biology, 70, 1163-1191.
  17. [17]  Ogundile, O.P., Edeki, S.O., and Adewale, S.O. (2018), A Modelling Technique for Controlling the Spread of Tuberculosis, International Journal of Biology and Biomedical Engineering, 12, 131-136.
  18. [18]  Ramamurthy, T. and Bhattacharya, S.K. (2011), Epidemiological and Molecular Aspects on Cholera, Springer, New York.
  19. [19]  Lenhart, S. and Workman, J. (2007), Optimal Control Applied to Biological Models, Chapman and Hall/CRC, London.
  20. [20]  Zhang, T., Wang, K., and Zhang, X. (2015), Modelling and analyzing the transmission dynamics of HBV epidemic in Xinjiang, China, PLoS ONE, 10(9), doi:10.1371/journal.pone.0138765.
  21. [21]  Zhang, S., Guo, H., and Smith, R. (2018), Dynamical analysis for a hepatitis B transmission model with immigration and infection age, Mathematical Biosciences and Engineering, 15(6), 1291-1313.
  22. [22]  Chinebu, T.I., Ezennorom, E.O., and Mbah, G.C.E. (2018), A Mathematical Model of the Dynamics of Hepatitis B Virus (HBV) Infection with Controls, International Journal of Trend in Scientific Research and Development, 2(5), 1654-1677.
  23. [23]  Zou, L., Zhang, W., and Ruan, S. (2010), Modeling the transmission dynamics and control of hepatitis B virus in China, Journal of Theoretical Biology, 262, 330-338.
  24. [24]  Kamyad, A.V., Akbari, R., Heydari, A.A., and Heydari, A. (2014), Mathematical modeling of transmission dynamics and optimal Control of vaccination and treatment for hepatitis B Virus, Computational and Mathematical Methods in Medicine, 2014, Article ID 475451.
  25. [25]  Khan, T., Zaman, G., and Chohan, M.I. (2017), The transmission dynamic and optimal control of acute and chronic hepatitis B, Journal of Biological Dynamics, 11(1), 172-189.
  26. [26]  Zhang, J. and Zhang, S. (2018), Application and Optimal Control for an HBV Model with Vaccination and Treatment, Discrete Dynamics in Nature and Society, 2018, Article ID 2076983.
  27. [27]  Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. (1962), The Mathematical Theory of Optimal Processes, Wiley-Interscience, Los Angeles.
  28. [28]  Fleming, W.H. and Rishel, R.W. (1975), Deterministic and Stochastic Optimal Control, Springer Verlag, New York.
  29. [29]  Jung, E., Lenhart, S., and Feng, Z. (2002), Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems- Series B, 2(4), 473-482.

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