Ecoepidemiological Model and Analysis of Prey-Predator System

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msub> <a:mrow> <a:mi>E</a:mi> </a:mrow> <a:mrow> <a:mi>o</a:mi> </a:mrow> </a:msub> </a:math> is always unstable, and axial equilibrium point <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:msub> <c:mrow> <c:mi>E</c:mi> </c:mrow> <c:mrow> <c:mi>A</c:mi> </c:mrow> </c:msub> </c:math> is locally asymptotically stable if <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mi>β</e:mi> <e:mi>k</e:mi> <e:mo>−</e:mo> <e:mfenced open="(" close=")"> <e:mrow> <e:msub> <e:mrow> <e:mi>t</e:mi> </e:mrow> <e:mrow> <e:mn>1</e:mn> </e:mrow> </e:msub> <e:mo>+</e:mo> <e:msub> <e:mrow> <e:mi>d</e:mi> </e:mrow> <e:mrow> <e:mn>2</e:mn> </e:mrow> </e:msub> </e:mrow> </e:mfenced> <e:mo>&lt;</e:mo> <e:mn>0</e:mn> <e:mo>,</e:mo> <e:mi> </e:mi> <e:mi>q</e:mi> <e:msub> <e:mrow> <e:mi>p</e:mi> </e:mrow> <e:mrow> <e:mn>1</e:mn> </e:mrow> </e:msub> <e:mi>k</e:mi> <e:mo>−</e:mo> <e:msub> <e:mrow> <e:mi>d</e:mi> </e:mrow> <e:mrow> <e:mn>3</e:mn> </e:mrow> </e:msub> <e:mfenced open="(" close=")"> <e:mrow> <e:mi>s</e:mi> <e:mo>+</e:mo> <e:mi>k</e:mi> </e:mrow> </e:mfenced> <e:mo>&lt;</e:mo> <e:mn>0</e:mn> </e:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M4"> <k:mi>q</k:mi> <k:msub> <k:mrow> <k:mi>p</k:mi> </k:mrow> <k:mrow> <k:mn>3</k:mn> </k:mrow> </k:msub> <k:mi>k</k:mi> <k:mo>−</k:mo> <k:mfenced open="(" close=")"> <k:mrow> <k:msub> <k:mrow> <k:mi>t</k:mi> </k:mrow> <k:mrow> <k:mn>2</k:mn> </k:mrow> </k:msub> <k:mo>+</k:mo> <k:msub> <k:mrow> <k:mi>d</k:mi> </k:mrow> <k:mrow> <k:mn>4</k:mn> </k:mrow> </k:msub> </k:mrow> </k:mfenced> <k:mfenced open="(" close=")"> <k:mrow> <k:mi>s</k:mi> <k:mo>+</k:mo> <k:mi>k</k:mi> </k:mrow> </k:mfenced> <k:mo>&lt;</k:mo> <k:mn>0</k:mn> </k:math> 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 <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M5"> <q:msub> <q:mrow> <q:mi>R</q:mi> </q:mrow> <q:mrow> <q:mn>01</q:mn> </q:mrow> </q:msub> <q:mo>=</q:mo> <q:msup> <q:mrow> <q:mfenced open="(" close=")"> <q:mrow> <q:mi>q</q:mi> <q:msub> <q:mrow> <q:mi>p</q:mi> </q:mrow> <q:mrow> <q:mn>1</q:mn> </q:mrow> </q:msub> <q:mo>−</q:mo> <q:msub> <q:mrow> <q:mi>d</q:mi> </q:mrow> <q:mrow> <q:mn>3</q:mn> </q:mrow> </q:msub> </q:mrow> </q:mfenced> </q:mrow> <q:mrow> <q:mn>2</q:mn> </q:mrow> </q:msup> <q:mi>k</q:mi> <q:mi>β</q:mi> <q:msub> <q:mrow> <q:mi>d</q:mi> </q:mrow> <q:mrow> <q:mn>3</q:mn> </q:mrow> </q:msub> <q:msup> <q:mrow> <q:mi>s</q:mi> </q:mrow> <q:mrow> <q:mn>2</q:mn> </q:mrow> </q:msup> <q:mo>/</q:mo> <q:mfenced open="(" close=")"> <q:mrow> <q:mi>q</q:mi> <q:msub> <q:mrow> <q:mi>p</q:mi> </q:mrow> <q:mrow> <q:mn>1</q:mn> </q:mrow> </q:msub> <q:mo>−</q:mo> <q:msub> <q:mrow> <q:mi>d</q:mi> </q:mrow> <q:mrow> <q:mn>3</q:mn> </q:mrow> </q:msub> </q:mrow> </q:mfenced> <q:mfenced open="{" close="}"> <q:mrow> <q:msup> <q:mrow>