Archive
Special Issues

Volume 9, Issue 5, October 2020, Page: 165-174
Age-Infection Model and Control of Marek Disease
Uwakwe Joy Ijeoma, Department of Mathematics, Alvan Ikoku Federal College of Education, Owerri, Nigeria
Inyama Simeon Chioma, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Omame Andrew, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Received: Jun. 22, 2020;       Accepted: Jul. 7, 2020;       Published: Oct. 12, 2020
Abstract
We formulated three compartmental model of Marek Disease model. 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 Marek disease and we drew five graphs to demonstrate this. The first figure shows the effect of both vaccination (v) and biosecurity measures (u) on the latently infected birds. The population of infected birds increases speedily and then remains stable without the application of any control measure, with the controls, the population increases to about 145 and then begins to reduce from day 8 till it drops to 50 on day 20 and then remains stable. With this strategy, only bird vaccination (v) is applied to control the system while the other control is set to zero. In the second figure, the effect of bird vaccination and its’ positive impact is revealed, though there is an increase to about 160 before a decrease occurs. From the third figure, as the control (u) ranges from 0.2 to 0.9, we see that the bird population still has a high level of latently infected birds. This result from figure shows that the bird population is not free from the disease, hence, the biosecurity control strategy is not effective without vaccination of susceptible birds and hence it is not preferable as the only control measure for marek disease. The numerical result in the fourth figure shows that as the latently infected bird population increases without control, with vaccination it decreases as more susceptible birds are vaccinated. From the fifth figure we observe, that as the control parameter increases, the total deaths by infection reduces, also as the age of the infection increases to the maximum age of infection which is 6 months (thatis, T=24 weeks), the number of deaths increases to 30 in a day. Hence, control measures should be applied at the early ages of infection in order to avoid high mortality rate during the outbreak of the disease.
Keywords
Age-Infection Model, Marek Disease, Biosecurity Control Strategy, Vaccination, Compartmental Model of Marek Disease
Uwakwe Joy Ijeoma, Inyama Simeon Chioma, Omame Andrew, Age-Infection Model and Control of Marek Disease, Applied and Computational Mathematics. Vol. 9, No. 5, 2020, pp. 165-174. doi: 10.11648/j.acm.20200905.13
Reference
[1]
Andrew Omame, Reuben Andrew Umana, Nneka Onyinyechi Iheonu and Simeon Chioma Inyama (2015) On the Existence of a Stochastic Model of Typhoid Fever; Journal of Mathematical Theory and Modelling (www.iiste.orgISSN2224-5804 (Paper), ISSN2225-0522 (Online)) Vol. 5, No. 8, Pp 104-113.
[2]
Andrew, R. F., Atkins, K. E., Savill, N. J., Walken-Brown, S. W. & Woolhouse, M. E. (2013). The effectiveness of mass vaccination on Marek disease virus (MDV) out breaks and detection with enabroiler barn; A modelling study Epidemics, Dec, 5 (4): 208-217. Doi: 10.1016/j.epidem.2013.10.001.
[3]
A. Omame, R. A. Umana, D. Okuonghae and S. C. Inyama (2018) Mathematical analysis of a two-sex Human Papilloma virus (HPV) model, International Journal of Biomathematics Vol. 11, No. 7, 1850092 (43) Pages (c) World Scientific Publishing Company, DOI: 10.1142/S1793524518500924.
[4]
Atkins, K. E., Read, A. F., Savill, N. J., Renz, K. G., Islam, A. F., Walkden-Brown S. W. & Woolhouse, M. E. J. (2013). Vaccination and reduced cohortduration candriv evirulence evolution: Mareks disease virus and industrialized agriculture. Evolution, 67: 851–860.
[5]
Carly, R. (2016) Animplusive differential equation model for Marek disease. A thesis submitted to Department of Mathematics and Statistics; Queen’s University Kingston, Ontario Canada.
[6]
Dalloul, R. A. and Lillehoj, H. S. (2006). Poultry Coccidiosis: Recent advancements in control measures and vaccine development. Expertre view vaccines, Vol. 5, pp 143-163; PMID: 16451116.
[7]
Fenner, F. J., Gibbs, E., Paul, J., Murphy, F. A., Rott, R., Studdert, M. J. & White, D. O. (1993). Veterinary Virology (2nd ed.). Academic Press, Inc.
[8]
Hirai, K. (2001). Current Topics in Microbiology and Immunology: Marek's Disease (Current Topicsin Microbiology and Immunology). Springer: Berlin. ISBN3-540-67798-4.
[9]
Inyama, S. C. (2008) Mathematical Model of Fish Harvesting in a Common Access Fishery. Journal of Mathematical Sciences, Kolkata, India, Vol. 19, No. 1, 43-47.
[10]
Inyama, Simeon Chioma (2009) A Mathematical Model for Lassa Fever with Reserved Population; Advances in Mathematics: Proceedings of a Memorial Conference in Honour of Prof. C. O. A. Sowunmi, Department of Mathematics, University of Ibadan, Ibadan, Nigeria, Vol. 1, 67-81.
[11]
Inyama, S. C. (2013) Mathematical Model of the Transmission Dynamics of the East African Sleeping Sickness (Try panesomiasis) International Journal of Mathematical Sciences and Engineering Applications (IJMSEA) Vol. 7, No. 4, September, 377-384.
[12]
Inyama Simeon Chioma; Ekeamadi Godsgift Ugonna; Uwagboe Osazee Michael; Omame Andrew; Mbachu Hope Ifeyinwa; Uwakwe Joy Ijeoma (2019) Application of Homotopy Analysis Method for Solving an SEIRS Epidemic Model; Journal of Mathematical Modelling and Applications; Vol. 4, No. (3): 36-48; http://www.sciencepublishinggroup.com/j/mmadoi:10.11648/j.mma.20190403.11, ISSN:2575-1786 (Print); ISSN:2575-1794 (Online).
[13]
Joy I. Uwakwe; Blessing O. Emerenini and Simeon C. Inyama (2020) Mathematical Model, Optimal Control and Transmission Dynamics of Avian Spirochaetosis; Journal of Applied Mathematics and Physics, 8, 270-293 https://www.scirp.org/journal/jampISSNOnline:2327-4379 ISSNPrint:2327-4352.
[14]
Jwander, L. D., Abdu P. A., Ekong P. S., Ibrahim, N. D. G., Nok, A. J. (2012). Marek disease knowledge, attitudes and practices among veterinarians in Nigeria. Journal of veterinary Science, Vol. 9, Pp. 71-76.
[15]
Katherine, E. A., Andrew, F. R., Nicholas, J. S., Katrin, G., Stephen, W. B., Mark, E. J. W. (2011). Modelling Marek disease virus infecion, parameter estimates for mortality rate and Infectiousness; Veterinary Research, 2:52–55.
[16]
Mohamed, E. (2006). Analysis of an SIRS age-structured epidemic model with vaccination and vertical transmission of disease. Application and Applied Mathematics (AAM), Vol. 1, No. 1, Pp. 36-61. Available on line at http://pamu.edu/pages/398/asp.
[17]
Mpeshe, S. C., Luboobi, L. S., Nkansah-Gyekyr, Y. (2014). Optimal control strategies for the dynamics of rift valley fever. Communications in optimization theory, Vol. 5, ISSN 2051-2953.
[18]
Musa, I. W., SaiduL. andAbalaka, E. S. (2012). Economic impact of recurrent outbreaks gumboro disease in a commercial poultry farm in Kano, Nigeria. Asian Journal of poultry Science 6, 152-159.
[19]
Nath, M. (2015). A frame work for integrating molecular information in as to chastic genetic epidemiological model for Marek’s disease resistance in poultry. Central Avian Research Institute, Izatnagar 243122 India.
[20]
OIE (2008). Newcastle Disease. Office International desEpizootics (OIE) Manual of standard for Diagnostic Tests and Vaccines, Paris, France, Pp. 221-232.
[21]
Ojo, S. O. (2003). Productivity and technical efficiency of poultry egg production in Nigeria. International Journal of Poultry Science, Vol. 2, Pp. 459-464.
[22]
Okosun, K. O., Mukamari, M. and Makinde, D. O. (2016). Global stability analysis and control of leptospirosis, Open Mathematics, Vol. 14, Issue1, Pp. 567-585, ISSN online 2391-5455. Doi: https://doi.org/10.1515/math-2016-0053.
[23]
Okwor, E. C. and Eze, D. C. (2011). Outbreak and persistence of Marek disease in batches of bird reared in a poultry farm located at Nsukka, South-East Nigeria. International Journal of poultry sciences10 (8), 617-620, ISSN1682-8356.
[24]
Omame, A.; Umana, R. A. and Inyama, S. C. (2020), Analysis of a co-infection model for HPV-TB; Applied Mathematical Modelling, Vol. 77|, Pp. 881-901.
[25]
Onuoha, Joy Ijeoma and Inyama, Simeon Chioma (2014) Mathematical Model of thet ransmission dynamics of Swine Flu; International Journal of Engineering Research and Industrial Application (IJERIA), ISSN 0974-1518, Vol. 7, No. II, Pp. 85-97.
[26]
Onuoha, Joy Ijeoma, Inyama, Simeon Chioma and Udofia Sunday Ekere (2014) Mathematical Model of the Transmission Dynamics of Swine Flu with the Vaccination of Newborn; International Journal of Mathematical Science & Engineering Applications, (IJMSEA), Vol. 8, No. V (September), 217-229.
[27]
Onuoha, Joy Ijeoma; Inyama, Simeon Chioma; Udofia Sunday Ekere and Omame, Andrew (2015) Mathematical Model of the Transmission Dynamics of Swine Flu with the Vaccination of Non Newborns, International Journal of Mathematical Science & Engineering Applications (IJMSEA), (ISSN0973-9424), Vol. 9, NoI (March),259-275.
[28]
Uwakwe, J. I., Inyama, S. C., Nse, C. A. and Emerenini, B. O. (2018) Analysis And Control of Coccidiosis Disease in Poultry IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume14, Issue6Ver. I (Nov-Dec 2018), Pp. 31-40 www.iosrjournals.org, DOI: 10.9790/5728-1406013140
[29]
Uwakwe, Joy I; Inyama, Simeon C.; Emerenini, Blessing O.; and Nse, Celestine A. (2019) Mathematical and Control Model of Bursal Disease (Ibd), IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, P-ISSN: 2319-765X. Volume15, Issue 4, Ser. III, (Jul–Aug 2019), PP15-29.
[30]
Uwakwe Joy Ijeoma, Inyama, Simeon Chioma, Omame, Andrew (2020) Mathematical Model and Optimal Control of New-Castle Disease (ND); Applied and Computational Mathematics; Vol. 9, No. 3, 2020, pp. 70-84. Doi: 10.11648/j.acm.20200903.14 (June 4th).
[31]
Witter, R. L., Gimeno, I. M., Pandiri, A. R., Fadly, A. M. (2010). Tumor diagnosis manual: the differential diagnosis of lymphoid and myeloid tumors in the chicken. American Association of avian pathologists, Vol. 1, Pp. 7-84.