International Journal of Mathematical Research

January 2017, Volume 6, 1, pp 22-28

# Analysis of an Eco-Epidemiological Model with Disease in the Prey and Predator Lihong Wang, Fanghong Zhang , Cuncheng Jin

Lihong Wang 1

Fanghong Zhang 1 Cuncheng Jin 1
1. Longqiao College of Lanzhou University of Finance and Economics, Lanzhou, China 1

on PubMed

Article History:

Received: 23 December, 2016
Revised: 20 January, 2017
Accepted: 08 March, 2017
Published: 20 April, 2017

# Index

## Abstract

We analyze and formulate an Eco-Epidemiological model with disease in the prey and predator, study the existence of the non-negative equilibria, obtain the sufficient conditions of locally asymptotical stability of the equilibria, then analyze the global stability of the positive equilibria.

Keywords: Eco-epidemiological model, Asymptotically stable, Equilibria, Liapunov function, Prey, Predator .

Received: 23 December 2016 / Revised: 20 January 2017 / Accepted: 8 March 2017 / Published: 20 April 2017

#### Contribution/ Originality

: This study contributes in the existing literature of Eco-Epidemiological model. We get the conditions of local asymptotic and the existence of the boundary balance, and we proved the positive balance point is global asymptotical stability by constructing Liapunov function.

## 1. INTRODUCTION

Mathematical ecology and mathematical epidemiology are major ﬁelds of study. Since transmissible disease in ecological situation can’t be ignored, it is very important from both the ecological and the mathematical points of view to study ecological systems subject to epidemiological factors. A number of studies have been performed in this field; However, all these papers available only discussed the disease spread  in a species ,seeing [1-3 ] deal with the disease is  spread among the  predator population only ,but in literatures [4-7 ] the disease is spread among the preys population considered. In our common life ,the disease may spread among the prey and the predator. On the basic of this ,this paper deals with the prey-predator model with diseases in the prey and predator ,  and we suppose the predator with disease dose not  capture on the preys, the susceptible predator capture both on the susceptible an on the infected prey, but the  capture rate  is different , which much closer to the actual  situation . This paper consider the model as follows:  All the parameters are assumed to be positive.

## 2. EQUILBRIA ANALYSIS

Let      ## 3. STABLE ANALYSIS

The Jacobi matrix of the system is   The corresponding characteristic polynomial is    All roots have negative real parts, by Hurwitz criterion.

Hence, we have the following main theorems: ## 4. GLOBAL STABILITY    Let This paper mainly discusses the prey-predator model with disease in the preys and predators ,we get the conditions of local asymptotic and the existence of the boundary balance .We prove the positive balance  point Funding: This study received no specific financial support. Competing Interests: The authors declare that they have no competing interests. Contributors/Acknowledgement: All authors contributed equally to the conception and design of the study.

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