THE DYNAMIC OF AN ECO-EPIDEMIOLOGICAL MODEL INVOLVING FEAR AND HUNTING COOPERATION

: In the present paper, an eco-epidemiological model consisting of diseased prey consumed by a predator with fear cost, and hunting cooperation property is formulated and studied. It is assumed that the predator doesn’t distinguish between the healthy prey and sick prey and hence it consumed both. The solution’s properties such as existence, uniqueness, positivity, and bounded are discussed. The existence and stability conditions of all possible equilibrium points are studied. The persistence requirements of the proposed system are established. The bifurcation analysis near the non-hyperbolic equilibrium points is investigated. Numerically, some simulations are carried out to validate the main findings and obtain the critical values of the bifurcation parameters, if any. It is obtained that the existence of fear controls the disease outbreak and the system's persistence. While in the case of a rising hunting cooperation rate, the induced fear may control the outbreak of disease.


INTRODUCTION
Ecology research has long focused on the interaction between predators and their prey, which is a very important aspect of the field. A significant and well-known topic of study in population dynamics and applied mathematical modeling is the prey-predator relationship. Such relationships are one of many types of inter-species interactions that are important in determining how complex ecological systems in our diverse planet behave [1]. According to studies, it has an impact on the ecosystem as a whole and not just the species of predator and prey that interact [2].
However, some recent theoretical and experimental research has challenged conventional wisdom. Studies have shown how crucial indirect effects (panic or fear) are in influencing both the dynamics of prey-predator relationships and the ecosystem as a whole. Although Cannon first proposed the concept of fear in 1915 [3], it is still a relatively new concept in the world of mathematical modeling. Prey individuals have been seen to alter their typical foraging activity in the presence of predator species as a result of the psychological stress of being captured and murdered by predators. In some ways, this helps the prey species at that specific time by boosting their chances of surviving, but in the long run, it could result in a significant loss. In addition to affecting their foraging habits, this perceived predation risk lowers both their birth rate and the likelihood that their children will survive more than typical adults. Several recent field trials and theoretical analyses back up the aforementioned assertions. According to certain paradoxical findings from studies [4] and the references therein, the influence of indirect fear may occasionally outweigh the effect of direct predation. Since direct predation is relatively simple to detect in nature, it is typically believed in traditional prey-predator models that predators only have an impact on prey populations by direct killing. However, the presence of a predator may drastically alter prey physiology and behavior to the point where it may have a greater impact on the prey population than direct predation [5][6]. Numerous mathematical models examined how fear affected the relationship between prey and predator, see for example [7][8][9][10][11][12][13][14][15]. Recently, a tri-trophic food web with a fear reaction for the base prey and a Lotka-Volterra functional response for predation by both a specialist predator and a superpredator was recently developed and studied by Fakhry et al. 3 THE DYNAMIC OF AN ECO-EPIDEMIOLOGICAL MODEL [16]. They discovered a surprising result of the prey's fear of its expert predator, which is the potential extinction of the superpredator.
Even while epidemiology is a significant subject of research in and of itself, there has been a recent movement toward combining it with ecology to better understand how species interact in ecosystems under the influence of epidemiological causes. Because no species lives alone in nature but interacts with many other species directly or indirectly, studying the impact of disease in the context of interspecies interactions is more realistic than the one without it. As a result, it gave birth to a new branch of science called eco-epidemiology. This innovative approach is motivated by a curiosity to understand the impact of disease in prey-predator scenarios. The first to introduce eco-epidemiological modeling was Anderson and May [17]. In order to create a new essence of nature, scholars are becoming more and more interested in combining these two crucial fields of study. Eco-epidemiology is a new field of mathematical biology that addresses both ecological and epidemiological concerns. Eco-epidemiological systems, which are used to explain how illnesses interact with predators and prey in one population or both populations, must become crucial instruments in studying the transmission and management of infectious diseases. Therefore, several researchers examined ecological systems where the disease affects prey, predator, or both populations in eco-epidemiology systems [18][19][20][21][22]. On the other hand, others studies focused the eco-epidemiological systems in the existence of fear, see for example [23][24][25].
In the prey-predator concept, group hunting is also prevalent. Animals frequently engage in cooperative hunting, which helps predators survive by ensuring they have access to enough food [26]. The cooperative hunting strategy has been widely researched mathematically. Consequently, several researchers have recently included cooperative hunting strategies in their studies; see for example [25][26][27].
The analyses mentioned above inspired the development of a generic prey-predator model with fear cost, disease in the prey population, and hunting cooperation strategy. The prey population was split into two classes, susceptible prey, and diseased prey, with the former playing a substantial 4 NABAA HASSAIN FAKHRY, RAID KAMEL NAJI mathematical role. Predators are said to be unable to tell the difference between healthy and sick prey, therefore they both end up in their stomachs.

MODEL FORMULATION
In this section, an eco-epidemiological system incorporating a prey-predator with an infective disease in the prey population is proposed and studied. It is believed that there are two population classes that make up the entire population of prey: the susceptible prey class and the infected prey class, whose population densities are given by ( ) and ( ) , respectively. While ( ) represents the predator population density. Therefore, to formulate the described system mathematically the following hypotheses are adopted.
1. It is assumed that the disease is spread among the prey population exclusively, that only the susceptible prey may reproduce, while the sick prey competes for the resource only, and that the disease is not genetically inherited.
2. It is assumed that the predator consumes both populations of the prey according to the Lotka-Volterra functional response. However, the prey population grows logistically in the absence of the predator.
3. It is thought that predation anxiety changes the foraging behavior of the prey population, which in turn reduces the risk of disease transmission among prey. 4. As the predator has a hunting cooperation capability, it will profit and successfully acquire prey. As a result, the predator population's attack rate, say 1 > 0, can be increased by the cooperation term to become ( 1 + 2 ), where 2 ≥ 0 describes the predator cooperation in hunting [26].
Accordingly, the dynamic of the described eco-epidemiological system can be represented using the following set of nonlinear first-order differential equations.
The level of fear that reduces the growth of the prey The level of fear that reduces the disease transmission

> 0
The disease transmission rate The attack rate of the predator on the prey The predator cooperation in hunting The death rates of the infected prey populations The death rates of the predator populations 1 ∈ (0,1] The conversion efficiency from susceptible prey biomass to predator biomass 2 ∈ (0,1] The conversion efficiency from infected prey biomass to predator biomass To non-dimensionalize the system (1), the following transformation is used.

PROPERTIES OF THE SOLUTION
This section treats the properties of the solution of system (2), such as positivity and bounded as presented in the next theorems.
Theorem 1: All system (2)'s solutions with the initial conditions belong to . ℝ + 3 are positively invariant.
Proof. From the first equation of the system (2), it is obtained: Then integrating the above equation within the limit [0, t], gives that: Theorem 2: All system (2)'s solutions with the initial conditions belonging to ℝ + 3 are uniformly bounded Proof. From system (2), it is easy to verify that Then according to the lemma (2.2) (Chen, 2005), it is obtained that Hence for → ∞, it is obtained that 1 ( ) ≤ 1.
Then according to the lemma (2.1) [28], it is obtained that Therefore, for → ∞, it is obtained that: That completes the proof.

EXISTENCE OF EQUILIBRIUM POINTS AND STABILITY ANALYSIS
The examination of each potential equilibrium point's stability is determined in this section.
Obviously, this equation has a unique positive root provided that It may have two positive roots or zero positive roots provided that 8 NABAA HASSAIN FAKHRY, RAID KAMEL NAJI 4 < 1 7 > 1 6 }.
The positive equilibrium point (PEP) 5 While ̃3 represents a positive root of the following fifth-order polynomial equation.
Accordingly, due to the discarding rule of signs, equation (10) has at least one positive root provided that one set of the following sets of conditions occurs.
Therefore, the eigenvalues of ( 2 ) are given by Hence, 2 is locally asymptotically stable (LAS) provided that the following two conditions are met.
However, the point 2 is a saddle point when at least one of these inequalities given is reflected.
Finally, the AEP becomes a non-hyperbolic point if any one of these inequalities becomes equality.
, 0) the JM can be written as: .
Hence, the characteristic equation of ( 3 ) can be written as: Direct computation gives the following roots Hence, as the 31 and 32 have negative real parts, the point 3 is LAS provided that Otherwise, the PFEP will be saddle point if the condition (23) is reflected and becomes a nonhyperbolic point when the inequality of the condition (23) transfers to quality.
At 4 = (̂1, 0,̂3) the JM can be written as where 11 THE DYNAMIC OF AN ECO-EPIDEMIOLOGICAL MODEL Hence, the characteristic equation of ( 4 ) can be written as: Consequently, the eigenvalues of the ( 4 ) can be written as Direct computation shows that all the eigenvalues given by equation (26) However, violating any one of these two conditions makes the DFEP unstable.
According to the Routh-Hurwitz criterion [1] the characteristic equation (30) has three eigenvalues with negative real parts if the following conditions are satisfied 1 > 0; 3 > 0, and ∆= 1 2 − 3 > 0. Therefore, the following theorem for local stability of the PEP is follows.

Theorem 3:
The PEP of the system (2) is LAS if and only if the following set of conditions is met.
Proof. Direct with the application of the Routh-Hurwitz criterion.

PERSISTENCE
This section studies an eco-epidemiological model's persistence and extinction property involving fear and hunting cooperation. The objective is to investigate the influence of fear and hunting cooperation within a diseased prey-predator system, on the persistence and extinction of system species. In order to determine the conditions that ensure the continuity, the dynamics at the boundary levels of the system must be understood.
It is clear that system (2) has two subsystems; the first subsystem can be representing in case of the absence of predator, and the second subsystem can be representing in the absence of disease from the system. Therefore, these two subsystems can be written in the following forms 13 THE DYNAMIC OF AN ECO-EPIDEMIOLOGICAL MODEL respectively.
The second subsystem is The first subsystem (35)  (1,0), and 23 = (̂1,̂3), where ̂1 is given by equation (5) and ̂3 exists uniquely under the condition (7). Obviously, the equilibrium points of the above two subsystems coincide with the boundary equilibrium points of the system (2). Therefore, they have the same local stability conditions. Now, to investigate the possibilities of non-existence of periodic dynamics in the interior of positive quadrants corresponding to these two subsystems, Dulac-Bendixon criterion is applied.

Theorem 4:
There are no periodic dynamics fall entirely: 1. In the interior of positive quadrant of 1 2 −plane.
2. In the interior of positive quadrant of 1 3 − plane, provided that the following condition is met.  ). Then, it is obtained that

LOCAL BIFURCATION
The occurrence of local bifurcation is investigated in this section using the Sotomayor theorem [29]. Recall that a non-hyperbolic equilibrium point represents a necessary but not sufficient condition for a local bifurcation to occur. Therefore, in the following theorems, the bifurcation parameter is selected so that the equilibrium point becomes a non-hyperbolic point.
Therefore, PB takes place near PFEP, and the proof is complete.
Therefore, PB takes place near PFEP, and the proof is complete.
Theorem 9: Assume that condition (31) is met along with the following condition Then system (2) undergoes a saddle-node bifurcation (SNB) near PEP when the parameter 4 passes through the value 4 * , provided that the following conditions hold Also, by using equation (43) Thus, due to condition (50) the following is obtained.
It is observed that, for the dataset given by equation (51), system (2) starting from different initial values approach asymptotically to the PEP as shown in figure (1). In all the following figures, the star represents the attracting equilibrium point, and the magenta color is used for expressing the trajectory of the system (2). In contrast, the blue, green, and red colors are used to describe the trajectory of 1 , 2 , and 3 respectively in the time series.   According to figure (6), as 5 increases, the population density of the species 2 approaches zero. On the other hand, figure (7) shows the approaching of the population density of the species 3 to zero when the parameter 6 decreases, while the approaching of the population density of the species 2 approaching to zero as 6 increases.
It is observed that, the parameters 7 , and 1 have similar influence as that for 2 , and 5 respectively on the dynamics of the system (2). Finally, the influence of the parameter 2 can be detected fron the figure (8). Obviously from figure (8), as 2 decreases, the population density of the species 3 approaches zero.

DISCUSSION AND CONCLUSIONS
This paper suggested and researched the use of an eco-epidemiological system with a preypredator and an infectious disease in the prey population. Investigated were the effects of predation-related fear and the predator's hunting cooperation. All the properties of the solution of the system (2) were studied. The local stability of all the biologically feasible equilibrium points was investigated along with their existing requirements. The persistence conditions of the system were established. The local bifurcation near the non-hyperbolic points was studied. Finally, all the analytical findings were confirmed using the numerical simulation depending on the hypothetical dataset (51), and the obtained results are summarized as follows.
Increasing the fear rate that reduces the growth of the prey (or fear rate that reduces the disease transmission) above a vital value leads to extinction in the population of the predator (infected prey) and hence the system losses the persistence. Decreasing the infection rate (or predator death rate) below a specific value or increasing the infection rate (or predator death rate) above a vital