Applied and Computational Mathematics

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Mathematical Model and Optimal Control of New-Castle Disease (ND)

Received: 09 December 2019    Accepted: 10 January 2020    Published: 04 June 2020
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Abstract

We formulated a five compartmental model of ND for both the ordinary and control models. We first determined the basic Reproduction number and the existence of Steady (Equilibrium) states (disease-free and endemic). Conditions for the local stability of the disease-free and endemic steady states were determined. Further, the Global stability of the disease-free equilibrium (DFE) and endemic equilibrium were proved using Lyponav method. We went further to carry out the sensitivity analysis or parametric dependence on R0 and later formulated the optimal control problem. We finally looked at numerical Results on poultry productivity in the presence of Infectious Newcastle Disease (ND) and we drew six graphs to demonstrate this. We observe that in absence of any control measure, the number of latently infected birds will increase rapidly from the initial population size of 80 to 160 birds within 1-3 days, whereas in the presence of control measures the population size will reduces to about 30 birds and goes to a stable state. This shows that the control measures are effective. The effect of the three control measures on the infectious classes can be seen. The number of non-productive infectious birds reduces to zero with control whereas the number of infectious productive reduces to about 8 birds and goes to its stable state when control is applied. This shows that the application of all three control measures tends to be more effective in the non- productive infectious bird population. It was also establish that the combination of efficient vaccination therapy and optimal efficacy of the vaccines are significantly more effective in the infectious productive birds’ population, since the combination reduces the population size of the birds to zero with 9–10 days. From the simulation also we see that optimal efficacy of the vaccine and effort to increase the number of recovered birds increases the number of latently infected birds population to about 129 at the early days of the infection whereas from another graph, the infectious productive birds reduces to 15 while the non -productive birds reduces to zero. The results from the simulation also show clearly, the effect of vaccination therapy on the latently infected birds. We observe that this programme will reduce the number of latently infected birds even if it not done more often. From the simulation, we further observe that this programme has effect on the infectious classes especially the non-productive infectious bird population, which reduces to zero after about 4 days.

DOI 10.11648/j.acm.20200903.14
Published in Applied and Computational Mathematics (Volume 9, Issue 3, June 2020)
Page(s) 70-84
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Newcastle Disease (ND), Disease-free Equilibrium (DFE), Global Stability, Efficient Vaccination, Lyponav Method

References
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[2] Alabi, R. A., Tariuma, I. O., Onemolense, P. E. A., Mafimisebi, A., Isah, T. A., Esobhawa, A. O., Oviasogie, D. I. (2000). Risk management in poultry enterprises in Edo state insurance scheme proceedings of the 5th Annual conference of Animal Science Association of Nigeria, Sept. 19-22, Portharcourt. pp 182-184.
[3] Alders, R. G., Bagnol, B., Costa, R. & Young, M. P. (2012). Sustainable control of Newcastle Disease in village poultry. INFPD International Network for Family Poultry Development (FAO) GPFRP Note No. 05.
[4] Blake, D. P., Tomley F. M., (2014) Securing Poultry Production from the ever–present Eimeria Challenge; Trends in Parasitology, 30 (1): 12–19.
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[6] Buonomo, B., Lotignola D., Vargas De-Leon, C. (2014) Qualitative analysis and optimal control of an epidemic model with vaccination and treatment; Mathematics and Computers in Simulation, 100 pp 88–102.
[7] Cassidy, L. R., Calistus, N. N. & Mathew, H. B. (2015). Modelling the burden of poultry disease on the rural poor in Madagascar. Elsevier One Health, 1 (6065).
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[16] Ibu, O. J., Adulugba A., Adeleke M. A., Tijjani A. Y. (2000). Activity of Newcastle disease and Infectious bursal disease viruses in ducks and guinea fowls in Jos area, Nigeria. Journal of Veterinary Sciences, vol. 2, pp 45-46.
[17] Jake, M. F., Jessia B. L., Vincent L. C., Andres J. G., Elizabeth A. H., Maia M., Craig, W. O. (2014). Optimal sampling strategies for detecting Zoonotic disease epidemics; (Article on maternal Health task force year 5); Available online at http://dx.doi.org/10.1371/journal.pibi.1003668.
[18] Jing, L., Xu D., Zang J., Xiao J., Wang H. (2010). The comparison of ARMA exponential Smoothing and seasonal Index model for predicting incidence of Newcastle Disease; World Automation Congress (WAC).
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[20] Kapczynski, D. R., Afonso, C. L., Miller, P. J. (2013). Immune response to poultry to Newcastle disease virus. Dev. Comp. Immunology 41 (3): 447-533. doi: 10.1016/j.dci.2013.04.012.
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[22] Linda, P. H. (2009). Epidemiology and characterization of Newcastle disease in small holder poultry in Mozambique; Available online at http:/epsilon.slu.se, ISSN 1652-8697.
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[25] Omame, A., Okuonghae, D., Umana, R. A., Inyama, S. C., (2020). Analysis of a co-infection model for HPV-TB, Applied Mathematical Modelling, 77: 881-901
[26] Senne, D. A., King, D. J., Kapczynski D. R. (2004). Control of Newcastle disease by vaccination. Dev. Biology (Basel), 119: 165-170.
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Author Information
  • Department of Mathematics, Alvan Ikoku Federal College of Education, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

  • Department of Mathematics, Federal University of Technology, Owerri, Nigeria

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    Uwakwe Joy Ijeoma, Inyama Simeon Chioma, Omame Andrew. (2020). Mathematical Model and Optimal Control of New-Castle Disease (ND). Applied and Computational Mathematics, 9(3), 70-84. https://doi.org/10.11648/j.acm.20200903.14

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    Uwakwe Joy Ijeoma; Inyama Simeon Chioma; Omame Andrew. Mathematical Model and Optimal Control of New-Castle Disease (ND). Appl. Comput. Math. 2020, 9(3), 70-84. doi: 10.11648/j.acm.20200903.14

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    Uwakwe Joy Ijeoma, Inyama Simeon Chioma, Omame Andrew. Mathematical Model and Optimal Control of New-Castle Disease (ND). Appl Comput Math. 2020;9(3):70-84. doi: 10.11648/j.acm.20200903.14

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  • @article{10.11648/j.acm.20200903.14,
      author = {Uwakwe Joy Ijeoma and Inyama Simeon Chioma and Omame Andrew},
      title = {Mathematical Model and Optimal Control of New-Castle Disease (ND)},
      journal = {Applied and Computational Mathematics},
      volume = {9},
      number = {3},
      pages = {70-84},
      doi = {10.11648/j.acm.20200903.14},
      url = {https://doi.org/10.11648/j.acm.20200903.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20200903.14},
      abstract = {We formulated a five compartmental model of ND for both the ordinary and control models. We first determined the basic Reproduction number and the existence of Steady (Equilibrium) states (disease-free and endemic). Conditions for the local stability of the disease-free and endemic steady states were determined. Further, the Global stability of the disease-free equilibrium (DFE) and endemic equilibrium were proved using Lyponav method. We went further to carry out the sensitivity analysis or parametric dependence on R0 and later formulated the optimal control problem. We finally looked at numerical Results on poultry productivity in the presence of Infectious Newcastle Disease (ND) and we drew six graphs to demonstrate this. We observe that in absence of any control measure, the number of latently infected birds will increase rapidly from the initial population size of 80 to 160 birds within 1-3 days, whereas in the presence of control measures the population size will reduces to about 30 birds and goes to a stable state. This shows that the control measures are effective. The effect of the three control measures on the infectious classes can be seen. The number of non-productive infectious birds reduces to zero with control whereas the number of infectious productive reduces to about 8 birds and goes to its stable state when control is applied. This shows that the application of all three control measures tends to be more effective in the non- productive infectious bird population. It was also establish that the combination of efficient vaccination therapy and optimal efficacy of the vaccines are significantly more effective in the infectious productive birds’ population, since the combination reduces the population size of the birds to zero with 9–10 days. From the simulation also we see that optimal efficacy of the vaccine and effort to increase the number of recovered birds increases the number of latently infected birds population to about 129 at the early days of the infection whereas from another graph, the infectious productive birds reduces to 15 while the non -productive birds reduces to zero. The results from the simulation also show clearly, the effect of vaccination therapy on the latently infected birds. We observe that this programme will reduce the number of latently infected birds even if it not done more often. From the simulation, we further observe that this programme has effect on the infectious classes especially the non-productive infectious bird population, which reduces to zero after about 4 days.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Mathematical Model and Optimal Control of New-Castle Disease (ND)
    AU  - Uwakwe Joy Ijeoma
    AU  - Inyama Simeon Chioma
    AU  - Omame Andrew
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    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20200903.14
    AB  - We formulated a five compartmental model of ND for both the ordinary and control models. We first determined the basic Reproduction number and the existence of Steady (Equilibrium) states (disease-free and endemic). Conditions for the local stability of the disease-free and endemic steady states were determined. Further, the Global stability of the disease-free equilibrium (DFE) and endemic equilibrium were proved using Lyponav method. We went further to carry out the sensitivity analysis or parametric dependence on R0 and later formulated the optimal control problem. We finally looked at numerical Results on poultry productivity in the presence of Infectious Newcastle Disease (ND) and we drew six graphs to demonstrate this. We observe that in absence of any control measure, the number of latently infected birds will increase rapidly from the initial population size of 80 to 160 birds within 1-3 days, whereas in the presence of control measures the population size will reduces to about 30 birds and goes to a stable state. This shows that the control measures are effective. The effect of the three control measures on the infectious classes can be seen. The number of non-productive infectious birds reduces to zero with control whereas the number of infectious productive reduces to about 8 birds and goes to its stable state when control is applied. This shows that the application of all three control measures tends to be more effective in the non- productive infectious bird population. It was also establish that the combination of efficient vaccination therapy and optimal efficacy of the vaccines are significantly more effective in the infectious productive birds’ population, since the combination reduces the population size of the birds to zero with 9–10 days. From the simulation also we see that optimal efficacy of the vaccine and effort to increase the number of recovered birds increases the number of latently infected birds population to about 129 at the early days of the infection whereas from another graph, the infectious productive birds reduces to 15 while the non -productive birds reduces to zero. The results from the simulation also show clearly, the effect of vaccination therapy on the latently infected birds. We observe that this programme will reduce the number of latently infected birds even if it not done more often. From the simulation, we further observe that this programme has effect on the infectious classes especially the non-productive infectious bird population, which reduces to zero after about 4 days.
    VL  - 9
    IS  - 3
    ER  - 

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