Furaha Michael Chuma; Gasper Godson Mwanga

Stability Analysis of Equilibrium Points of Newcastle Disease Model of Village Chicken in the Presence of Wild Birds Reservoir

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Author(s)

Furaha Michael Chuma ^1,* Gasper Godson Mwanga ^1,2

1. The Nelson Mandela African Institution of Science and Technology, P.O Box 447, Arusha-Tanzania

2. University of Dar es salaam, P.O. Box 2329, Dar es salaam, Tanzania

* Corresponding author.

DOI: https://doi.org/10.5815/ijmsc.2019.02.01

Received: 10 Oct. 2018 / Revised: 28 Dec. 2018 / Accepted: 15 Feb. 2019 / Published: 8 Apr. 2019

Index Terms

Stability analysis, Equilibrium points, Newcastle disease, Village chicken

Abstract

Newcastle is a viral disease of chicken and other avian species. In this paper, the stability analysis of the disease free and endemic equilibrium points of the Newcastle disease model of the village chicken in the absence of any control are studied. The Hurwitz matrix criterion is applied to study the stability of the Newcastle disease free equilibrium point,Q⁰. The result shows that the disease free equilibrium point is locally asymptotically stable iff the principle leading minors of the Hurwitz Matrix, (for n∈ℝ⁺) are all positive. Using the Castillo Chavez Theorem we showed that, the disease free equilibrium point is globally asymptotically when R₀<1. Furthermore, using the logarithmic function and the LaSalle’s Theorem, the endemic equilibrium point is found globally asymptotically stable for R₀>1. Finally the numerical simulations confirm the existence and stability of the equilibrium points of the model. This reveals that, proper interventions are needed so as to decrease the frequently occurrence of the Newcastle disease in the village chicken population.

Cite This Paper

Furaha Michael Chuma, Gasper Godson Mwanga,"Stability Analysis of Equilibrium Points of Newcastle Disease Model of Village Chicken in the Presence of Wild Birds Reservoir", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.5, No.2, pp.1-18, 2019. DOI: 10.5815/ijmsc.2019.02.01

Reference

[1]Alexander D. J., Bell J.G., and Alders R.G., A Technology Review: Newcastle Disease, with Special Emphasis on its Effect on Village Chick-ens. Food & Agriculture Org., 2004, no. 161.

[2]Anggriani N., Supriatna A., and Soewono E., “The existence and stability analysis of the equilibria in dengue disease infection model,” in Journal of Physics: Conference Series, vol. 622, no. 1. IOP Publishing, 2015, p. 012039.

[3]Anishchenko V.S., Vadivasova T.E., and Strelkova G.I., “Stability of dynamical systems: Linear approach,” in Deterministic Nonlinear Systems. Springer, 2014, pp.23–35.

[4]Brin M. and Stuck G., Introduction to Dynamical Systems. Cambridge university press, 2002.

[5]Castillo-Chavez C., Feng Z., and Huang W., “On the computation of and its role on global stability in mathematical approaches for emerging and re-emerging infectious diseases,” Mathematical Approaches for Emerging and Reemerging Infectious Diseases: an introduction, vol. 1, p. 229, 2002.

[6]Chuma F., Mwanga G.G., and Kajunguri D., “Modeling the role of wild birds and environment in the dynamics of newcastle disease in village chicken,” Asian Journal of Mathematics and Application, vol. 2018, no. 446, p. 23, 2018.

[7]Daut E.F., Lahodny G. Jr, Peterson M.J., and Ivanek R., “Interacting effects of Newcastle disease transmission and illegal trade on a wild population of white-winged parakeets in Peru: A modeling approach,” PloS One, vol. 11, no. 1, 2016.

[8]Dortmans J.C., Koch G., Rottier P. J., and Peeters B.P., “Virulence of newcastle disease virus: what is known far?” Veterinary Research, vol. 42, no. 1, p. 1, 2011.

[9]Hugo A., Makinde O.D., Kumar S., and Chibwana F.F., “Optimal control and cost effectiveness analysis for newcastle disease eco-epidemiological model in Tanzania,” Journal of Biological Dynamics, vol. 11, no. 1, pp. 190–209, 2017.

[10]Hunter J.K., “Introduction to dynamical systems,” UCDavis Mathematics MAT A, vol. 207, p. 2011, 2011.

[11]Kahuru J., Luboobi L., and Nkansah-Gyekye Y., “Stability analysis of the dynamics of tungiasis transmission in endemic areas,” Asian Journal of Mathematics and Applications, vol. 2017, 2017.

[12]La Salle J.P., The Stability of Dynamical Systems. SIAM, 1976.

[13]Lawal J., Jajere S., Mustapha M., Bello A., Wakil Y., Geidam Y., Ibrahim U., and Gulani I., “Prevalence of Newcastle disease in Gombe, northeastern Nigeria: A ten-year retrospective study (2004-2013),” British Microbiology Research Journal, vol. 6, no. 6, p. 367, 2015.

[14]Lucchetti J., Roy M., and Martcheva M., “An avian influenza model and its fit to human avian influenza cases,” Advances in Disease Epidemiology, Nova Science Publishers, New York, pp.1–30, 2009.

[15]Mafuta P., Mushanyu J., S. Mushayabasa, and Bhunu C.P., “Trans-mission dynamics of trichomoniasis in bisexuals without the E,” World Journal of Modelling and Simulation, vol. 9, no. 4, pp. 302–320, 2013.

[16]Mwanga G. G., “Mathematical modeling and optimal control of malaria,” A PhD Thesis, Acta Lappeenranta University, 2014.

[17]Mpeshe S.C., Luboobi L.S., and Nkansah-Gyekye Y., “Stability analysis of the rift valley fever dynamical model,” Journal of Mathematical and Computational Science, vol. 4, no. 4, p. 740, 2014.

[18]Nyerere N., Luboobi L., and Nkansah-Gyekye Y., “Bifurcation and stability analysis of the dynamics of tuberculosis model incorporating, vaccination, screening and treatment,” Communications in Mathematical biology and Neuroscience, vol. 2014, pp. Article–ID, 2014.

[19]Olaniyi S., Lawal M.A., and Obabiyi O.S., “Stability and sensitivity analysis of a deterministic epidemiological model with pseudo-recovery,” IAENG International Journal of Applied Mathematics, vol. 46, no. 2, pp. 160–167, 2016.

[20]Perry B., Kalpravidh W., Coleman P., Horst H., McDermott J., Randolph T., and Gleeson L., “The economic impact of foot and mouth disease and its control in South- East Asia: a preliminary assessment with special reference to Thailand.” Technical scientific Review, offce of Epizootic diseases, vol. 18, no. 2, pp. 478–497, 1999.

[21]Selemani M.A., Luboobi L.S., and Nkansah Gyekye Y., “On stability of the in-human host and in mosquito dynamics of malaria parasite,” Asian Journal of Mathematics and Applications, vol. 2016, no. 353, p. 23, 2016.

[22]Sharma S and Samanta G., “Stability analysis and optimal control of an epidemic model with vaccination,” International Journal of Biomathematics, vol.8, no.03, p. 1550030, 2015.

[23]Sharif A., Ahmad T., Umer M., Rehman A., Hussain Z.., “Prevention and control of newcastle disease,” International Journal of Agriculture Innovations and Research, vol. 3, no. 2, pp. 454–460, 2014.

[24]Tumwiine J., Mugisha J., and Luboobi L., “A host-vector model for malaria with infective immigrants,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 139–149, 2010.

[25]Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer Science & Business Media, 2003, vol. 2.

International Journal of Mathematical Sciences and Computing (IJMSC)