This paper presents infectious disease in prey-predator system. In the present work, a three Compartment mathematical eco-epidemiology model consisting of susceptible prey- infected prey and predator are formulated and analyzed. The positivity, boundedness, and existence of the solution of the model are proved. Equilibrium points of the models are identified. Local stability analysis of Trivial, Axial, Predator-free, and Disease-free Equilibrium points are done with the concept of Jacobian matrix and Routh Hourwith Criterion. Global Stability analysis of endemic equilibrium point of the model has been proved by defining appropriate Liapunove function. The basic reproduction number in this eco-epidemiological model obtained to be Ro=[β (μ3)2] ⁄ [qp2 (qp1Λ - μ1μ3)]. If the basic reproduction number Ro > 1, then the disease is endemic and will persist in the prey species. If the basic reproduction number Ro=1, then the disease is stable, and if basic reproduction number Ro < 1, then the disease is dies out from the prey species. Lastly, Numerical simulations are presented with the help of DEDiscover software to clarify analytical results.
| Published in | Mathematical Modelling and Applications (Volume 5, Issue 3) |
| DOI | 10.11648/j.mma.20200503.17 |
| Page(s) | 183-190 |
| 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), 2020. Published by Science Publishing Group |
Mathematical Ecoepidemiology, Prey- Predator System, Stability Analysis, Reproduction Number, Simulation Study
| [1] | Alfred Hugo, Estomih S. Massawe, and Oluwole Daniel Makinde (2012). An Eco-Epidemiological Mathematical Model with Treatment and Disease Infection in both Prey and Predator Population. Journal of Ecology and natural environment Vol. 4 (10), pp. 266-273. |
| [2] | Shashi Kant, Vivek Kumar, (2017) Dynamics of A Prey-Predator System With Infection In Prey, Electronic Journal of Deferential Equations, Vol. 2017, No. 209, pp. 1-27. |
| [3] | Lihong Wang, Fanghong Zhang, & Cuncheng Jin, (2017) Analysis of an eco-epidemiological model with disease in the prey and predator International Journal of Mathematical Research Vol. 6, No. 1, pp 22-28. |
| [4] | T. K. Kar, Prasanta Kumar Mondal, (June 20120 A Mathematical Study on the Dynamics of an Eco-Epidemiological Model in the Presence of Delay, Applications and Applied Mathematics: An International Journal (AAM), Vol. 7, Issue 1, pp 300–333. |
| [5] | Abayneh Fentie Bezabih, Geremew Kenassa Edessa, Koya Purnachandra Rao (2020), Mathematical Eco-Epidemic Model on Prey-Predator System. IOSR Journal of Mathematics (IOSR-JM), 16 (1), pp. 22-34. |
| [6] | Rald Kamel Naji (2012) The dynamics of prey-predator model with disease in prey, research gate, Available online at http://scik.org. |
| [7] | Xin-You Meng, Ni-Ni Qin & Hai-Feng Huo (2018) Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species, Journal of Biological Dynamics, 12: 1, 342-374, DOI: 10.1080/17513758.2018.1454515. |
| [8] | Rald Kamel Naji, Rasha Ali Hamodi, (August 2016) The dynamics of an ecological model with infectious disease, Global Journal of Engineering Science and Researches, DOI: 10.5281/zenodo.61221. |
| [9] | Geremew Kenassa Edessa, Boka Kumsa, Purnachandra Rao Koya (2018). Dynamical behavior of Susceptible prey – Infected prey – Predator Populations. IOSR Journal of Mathematics (IOSR-JM) 14 (4) PP: 31-41. |
| [10] | S. P. Bera, A. Maiti, G. Samanta (2015). A Prey-predator Model with Infection in both prey and predator, Filomat 29 (8) pp, 1753-1767. |
| [11] | Paritosh Bhattacharya, Susmita Paul and K. S. Choudhury (2015). Mathematical Modeling of Ecological Networks, Structure and Interaction Of Prey and Predator, Palestine Journal of Mathematics Vol. 4 (2), pp 335–347. |
| [12] | Asrul Sani, Edi Cahyono, Mukhsar, Gusti Arviana Rahman (2014). Dynamics of Disease Spread in a Predator-Prey System, Indonesia, Advanced Studies in Biology, Vol. 6,, No. 4, pp 169–179. |
| [13] | C. M. Silva (2017). Existence of periodic solutions for periodic eco-epidemic models with disease in the prey, J. Math. Anal. Appl. 453 (1), pp 383–397. |
| [14] | M. Haque (2010). A predator–prey model with disease in the predator species only, Nonlinear Anal., Real World Appl., 11 (4), 2224–2236. |
| [15] | M. Liu, Z. Jin and M. Haque, An impulsive predator-prey model with communicable disease in the prey species only, Nonlinear Anal. Real World Appl. 10 (2009), no. 5, 3098–3111. |
| [16] | K. P. Hadeler and H. I. Freedman (1989). Predator-prey populations with parasitic infection, J. Math. Biol., 27, pp 609–631. |
| [17] | M. S. Rahman and S. Chakravarty (2013). A predator-prey model with disease in prey. Nonlinear Analysis: Modeling and Control, 18 (2), pp 191–209. |
| [18] | Sachin Kumar and Harsha Kharbanda (Sep 2017). Stability Analysis Of Prey-Predator Model With Infection, Migration and Vaccination In Prey, arXiv: 1709.10319vl [math. DS], 29. |
APA Style
Abayneh Fentie Bezabih, Geremew Kenassa Edessa, Purnachandra Rao Koya. (2020). Mathematical Eco-Epidemiological Model on Prey-Predator System. Mathematical Modelling and Applications, 5(3), 183-190. https://doi.org/10.11648/j.mma.20200503.17
ACS Style
Abayneh Fentie Bezabih; Geremew Kenassa Edessa; Purnachandra Rao Koya. Mathematical Eco-Epidemiological Model on Prey-Predator System. Math. Model. Appl. 2020, 5(3), 183-190. doi: 10.11648/j.mma.20200503.17
AMA Style
Abayneh Fentie Bezabih, Geremew Kenassa Edessa, Purnachandra Rao Koya. Mathematical Eco-Epidemiological Model on Prey-Predator System. Math Model Appl. 2020;5(3):183-190. doi: 10.11648/j.mma.20200503.17
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Download@article{10.11648/j.mma.20200503.17,
author = {Abayneh Fentie Bezabih and Geremew Kenassa Edessa and Purnachandra Rao Koya},
title = {Mathematical Eco-Epidemiological Model on Prey-Predator System},
journal = {Mathematical Modelling and Applications},
volume = {5},
number = {3},
pages = {183-190},
doi = {10.11648/j.mma.20200503.17},
url = {https://doi.org/10.11648/j.mma.20200503.17},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20200503.17},
abstract = {This paper presents infectious disease in prey-predator system. In the present work, a three Compartment mathematical eco-epidemiology model consisting of susceptible prey- infected prey and predator are formulated and analyzed. The positivity, boundedness, and existence of the solution of the model are proved. Equilibrium points of the models are identified. Local stability analysis of Trivial, Axial, Predator-free, and Disease-free Equilibrium points are done with the concept of Jacobian matrix and Routh Hourwith Criterion. Global Stability analysis of endemic equilibrium point of the model has been proved by defining appropriate Liapunove function. The basic reproduction number in this eco-epidemiological model obtained to be Ro=[β (μ3)2] ⁄ [qp2 (qp1Λ - μ1μ3)]. If the basic reproduction number Ro > 1, then the disease is endemic and will persist in the prey species. If the basic reproduction number Ro=1, then the disease is stable, and if basic reproduction number Ro < 1, then the disease is dies out from the prey species. Lastly, Numerical simulations are presented with the help of DEDiscover software to clarify analytical results.},
year = {2020}
}
TY - JOUR T1 - Mathematical Eco-Epidemiological Model on Prey-Predator System AU - Abayneh Fentie Bezabih AU - Geremew Kenassa Edessa AU - Purnachandra Rao Koya Y1 - 2020/08/20 PY - 2020 N1 - https://doi.org/10.11648/j.mma.20200503.17 DO - 10.11648/j.mma.20200503.17 T2 - Mathematical Modelling and Applications JF - Mathematical Modelling and Applications JO - Mathematical Modelling and Applications SP - 183 EP - 190 PB - Science Publishing Group SN - 2575-1794 UR - https://doi.org/10.11648/j.mma.20200503.17 AB - This paper presents infectious disease in prey-predator system. In the present work, a three Compartment mathematical eco-epidemiology model consisting of susceptible prey- infected prey and predator are formulated and analyzed. The positivity, boundedness, and existence of the solution of the model are proved. Equilibrium points of the models are identified. Local stability analysis of Trivial, Axial, Predator-free, and Disease-free Equilibrium points are done with the concept of Jacobian matrix and Routh Hourwith Criterion. Global Stability analysis of endemic equilibrium point of the model has been proved by defining appropriate Liapunove function. The basic reproduction number in this eco-epidemiological model obtained to be Ro=[β (μ3)2] ⁄ [qp2 (qp1Λ - μ1μ3)]. If the basic reproduction number Ro > 1, then the disease is endemic and will persist in the prey species. If the basic reproduction number Ro=1, then the disease is stable, and if basic reproduction number Ro < 1, then the disease is dies out from the prey species. Lastly, Numerical simulations are presented with the help of DEDiscover software to clarify analytical results. VL - 5 IS - 3 ER -
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