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# Ecoepidemiological Model and Analysis of Prey-Predator System

Journal of Applied Mathematics

Hindawi Limited

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### Abstract

In this paper, the prey-predator model of five compartments is constructed with treatment given to infected prey and infected predator. We took predation incidence rates as functional response type II, and disease transmission incidence rates follow simple kinetic mass action function. The positivity, boundedness, and existence of the solution of the model are established and checked. Equilibrium points of the models are identified, and local stability analyses of trivial equilibrium, axial equilibrium, and disease-free equilibrium points are performed with the method of variation matrix and the Routh-Hurwitz criterion. It is found that the trivial equilibrium point $E o$ is always unstable, and axial equilibrium point $E A$ is locally asymptotically stable if $β k − t 1 + d 2 < 0 , q p 1 k − d 3 s + k < 0$ and $q p 3 k − t 2 + d 4 s + k < 0$ conditions hold true. Global stability analysis of an endemic equilibrium point of the model has been proven by considering the appropriate Lyapunov function. The basic reproduction number of infected prey and infected predators are obtained as $R 01 = q p 1 − d 3 2 k β d 3 s 2 / q p 1 − d 3 q p 1 − d 3 2 k s t 1 + d 2 + r s q p 2 k q p 1 − k d 3 − d 3 s$ and $R 02 = q p 1 − d 3 q p 3 d 3 k + α r s q k q p 1 − k d 3 − d 3 s / q p 1 − d 3 2 t 2 + d 4 k$ , respectively. If the basic reproduction number is greater than one, then the disease will persist in the prey-predator system. If the basic reproduction number is one, then the disease is stable, and if the basic reproduction number is less than one, then the disease dies out from the prey-predator system. Finally, simulations are done with the help of DEDiscover software to clarify results.

### Most cited references9

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### Analysis of a predator–prey model with herd behavior and disease in prey incorporating prey refuge

(2019)
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### A prey–predator system with disease in prey and cooperative hunting strategy in predator

(2020)
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• Record: found

### A predator-prey model with strong Allee effect and disease in prey population

(2019)
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### Author and article information

###### Contributors
(View ORCID Profile)
###### Journal
Journal of Applied Mathematics
Journal of Applied Mathematics
Hindawi Limited
1687-0042
1110-757X
February 20 2021
February 20 2021
: 2021
: 1-17
###### Affiliations
[1 ]Department of Mathematics, Wollega University, Nekemte, Ethiopia
###### Article
10.1155/2021/6679686