GLOBAL STABILITY OF A FRACTIONAL-ORDER ECO-EPIDEMIOLOGICAL MODEL WITH INFECTED PREDATOR: THEORETICAL ANALYSIS

. A theoretical knowledge of the global stability of an eco-epidemiological model is not only important in itself but is also important in understanding the results of numerical simulations. In this paper the global stability of a fractional-order eco-epidemiological model with infected predator and harvesting is investigated using the Lyapunov function.


INTRODUCTION
Mathematical models of the relationship between predator and prey in the presence of infectious diseases, which play an important role in the dynamics, are called eco-epidemiological models. There have been various studies of such models with disease being present in the constituent populations. These studies include [1]- [9].
Harvesting can influence the dynamics of eco-epidemiological models. In recent years the demand for greater resources has resulted in over-exploitation. Therefore there is a need for a sustainable strategy to protect ecosystems [10].
Mathematical models incorporating fractional differential equations have attracted much attention in recent years. Such models are believed to be more suitable for models that depend on past history [11,12]. Further, such models are more realistic and less prone to errors [13].
Studies on fractional-order eco-epidemiological models include [14]- [21]. Ghosh et al. [1] studied a fractional-order eco-epidemiological model incorporating fear, treatment, and hunting cooperation effects to explore the memory effect in an ecological system through Caputo-type fractional-order derivative. In the work by Mukherjee [21], the author investigated a fractionalorder predator-prey system with fear effect. Moustafa et al. [14] described the dynamical behavior of fractional-order Rosenzweig-MacArthur model allowing for a prey refuge. The influence of an infectious disease on a prey-predator model equipped with a fractional-order derivative is studied in [3]. The effect of fractional-order derivative on a prey-predator model with infection and harvesting is discussed by Moustafa et al. [16]. However, these papers did not deal with fractional-order eco-epidemiological model with infected predator and harvesting as such.
This paper is a theoretical study of the global stability of a fractional-order ecoepidemiological model with infected predator and harvesting. There has, so far as we are aware, been no theoretical studies of such a model.

MODEL DESCRIPTION
This paper investigates the global dynamic properties of a generalisation of the integer-order eco-epidemiological model introduced in [22]. The Caputo fractional derivative of order q ( c D q ) is introduced and harvesting (H) is included. This generalised(fractional) model can be written as: where q ∈ (0, 1). The population is divided into: prey population density (x), susceptible predator population density (y) and infected predator population density (z).

EQUILIBRIUM POINTS AND GLOBAL STABILITY
So as to evaluate the equilibrium points of model (1), let c D q x(t) = 0, c D q y(t) = 0 and c D q z(t) = 0.
and that x 4 needs to be a positive root of the following cubic polynomial: In accordance with Theorem 3.4 in [23], the analytical conditions about the existence of the equilibrium point E 4 can be illustrated in Table 1, Table 2 and Table 3.

Conditions
Equilibria of model (1) For The following theorems investigate the global stability of the equilibrium points E 1 , E 2 , E 3 and E 4 .
Proof. The following positive definite Lyapunov function can be considered: Calculating the q-order derivative of V along the solution of model (1) and using Lemma 3.1 in [24], and ζ > min c 2 k a , m 2 k a+k + λ . By Lemma 4.6 in [25], it is proof that the equilibrium point E 1 is globally asymptotically stable.
Proof. The following positive definite Lyapunov function is considered.
By calculating the q-order derivative of V along the solution of model (1) and using Lemma 3.1 in [24], Suppose L = am 1 c 1 (a+x 2 ) . Thus, c D q ≤ 0 when y 2 < r(a+x 2 )a c 1 k and λ y 2 b + m 2 k a+k + Lc 2 x 2 a < ζ . Hence the theorem is proved. Proof. It can be used the following positive definite Lyapunov function.
Computing the time derivative of V along the solution of model (1) and utilizing Lemma 3.1 in [24], Hence the theorem is proved. Proof. The following positive definite Lyapunov function can be used.

CONCLUSION
In this paper, a fractional-order eco-epidemiological model with infected predator and harvesting has been formulated and analyzed. The equilibrium points were identified and their global properties were investigated. The existence of transcritical bifurcation was shown using Sotomayor's theorem. The threshold parameters (ℜ 01 and ℜ 02 ) were used to determine the existence conditions of the equilibrium points.