DYNAMICAL ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL EXPERIENCING THE CROWDING EFFECT OF INFECTED PREY

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INTRODUCTION
The dynamic relationship between predator and prey is considered one of the most fundamental relationships in ecology, as this relationship is one of the top-studied subjects by ecologists.The Lotka-Volterra system is the first fundamental prey-predator system; it goes back to the First World War [1].
The Lotka-Volterra simple mathematical model can be described as the following: The previous system (1) and its variants were extensively studied by many researchers aiming to understand the various ways in which predator-prey interactions take place [2]- [5].However, the representation in (1) does not explicitly incorporate the impact of transmittable illnesses on predator-prey interdependence.While valuable insights into interaction dynamics have been gleaned, the lack of consideration for infectious factors represents a limitation.Infectious diseases represent significant determinants shaping populace progression, as Research has shown that lowmagnitude perturbations originating from environmental sources can induce oscillations in population levels around an average threshold.In contrast, high-intensity disturbances emanating from the surrounding habitat may result in total elimination of the population [6].The work of [7] and [8] studied a predator-prey model where the prey population is subjected to disease, and the transmission rate of the infection follows the law of mass action.The study shows that the disease in the prey population can induce chaotic interactions.
To expand upon the classic Lotka-Volterra predator-prey model, various formulations of functional responses have been proposed which refine how consumption rates relate to prey abundance levels.One such example is the Holling type II functional response, which depicts a saturating applicable form as prey becomes more plentiful.Generalized variants of this specific DYNAMICAL ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL functional response have also been devised to capture additional nuances [9]- [11].Accounting for non-linearities in the predation relationship through customized functional responses allows for more sophisticated representations of harvesting effects than the simple linear assumption of the basic Lotka-Volterra equations.Continued exploration of diverse functional responses remains an intriguing avenue for developing an increasingly refined understanding of predator-prey population dynamics.
Other studies have also explored the dynamics of a predator-prey system incorporating a disease in either the prey or the predator [12]- [14].Nevertheless, these studies use the bilinear incidence rate to describe the transmission of the disease.Nonetheless, the bilinear incidence rate doesn't take into account the crowding effect of the infected individuals.Gumel and Moghadas [15] proposed the nonlinear incidence rate, which accounts for how the crowding effect, thereby more realistically portraying pathogen transmission dynamics as population densities fluctuate over space and time.The added nuance of factoring density effects represents an advantageous enhancement over the more straightforward law of mass action formulation for capturing important epidemiological subtleties.
Accordingly, we propose a predator-prey model that includes two key elements: 1) Holling type II functional response to describe the susceptible prey predation rate.
2) Nonlinear incidence rate for the infection transmission.
Accounting for such delays provides a more complete characterization of fundamental ecological interactions that inevitably involve consumption history effects.Incorporating time delays into our proposed predator-prey model with a nonlinear incidence rate could yield additional insights into how delays interact with and amplify the impact of other complexity features in the system.This represents a promising direction for future research to generate a highly sophisticated and representative theoretical framework, The nonlinear incidence rate was studied in [16], where the predator is considered to consume the infected prey only, and the predation followed Crowley-Martin-type functional response.
In this study, we formulate an eco-epidemiological model characterizing the interaction between a predator population and a prey population impacted by disease.The proposed model adopts a Holling type II functional response to represent the predation dynamics between predators and susceptible prey, aiming to realistically capture prey defense behaviors that can impede consumption rates.Meanwhile, a simple mass-action scheme governs the predator-infected prey interaction, as the infection is presumed to render exposed prey less able to escape or elude capture by predators.We assume disease transmission occurs via direct contact between susceptible and infected prey.Specifically, disease spread follows a nonlinear incidence rate first established by Gumel and Moghadas in [15] that accounts for inhibitory crowding effects among infected individuals.By integrating different functional response assumptions tied to disease state, along with density-dependent transmission dynamics, the model aims to offer a more nuanced depiction of predator-prey-pathogen interaction compared to mass-action formulations.
This paper offers a thorough analysis of the dynamics of the eco-epidemiological while still considering the crowding effect; such a model can be applied in a wide range of ecosystems where fatal diseases appear in the prey population.

THE MATHEMATICAL MODEL
We seek to develop an eco-epidemiological model mainly consisting of a prey and predator.The prey specie is divided into two sub-species 1) susceptible prey (S) and 2) infected prey (I).The predators are considered to be healthy at all times.
1.When there is no disease, the prey population follows the logistical growth function with a carrying capacity of  > 0 and a substantial growth rate constant of  > 0.
2. the prey population consists of two groups; susceptible prey () and infected prey ().The total number of preys at any given time , denoted as (), can be expressed as the sum of these subgroups: () = () + ().Additionally, it is posited that only susceptible prey retains the capacity for reproduction, allowing their numbers to rise up to environmental carrying capacity.
We also presume infected prey to be incapable of recovery, reproduction or resource competition during the afflicted stage.These assumptions aim to realistically capture the anticipated functional limitations on prey impacted by the disease factor 3. The disease transmission between susceptible and infected prey is modeled using a nonlinear incidence rate formula of the form  1+ . This formulation was originally proposed by Gumel and Moghadas in 2003 and has since been widely adopted.In this expression,  represents the infection force exerted by infected individuals, while the term accounts for inhibition caused by crowding of infected prey.This approach is deemed more realistic than solely relying on a basic bilinear rate βSI, as it considers how crowding of infected prey can hamper transmission in addition to preventing unlimited contact rates through judicious parameter selection fitting the modeled DYNAMICAL ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL system.By integrating density dependence effects, the incidence rate provides a reasonable depiction of disease spread dynamics versus simplistic mass action assumptions.
4. We assumed the predator consumes both the infected and susceptible prey as it is unable of identifying the infected prey from healthy.
5. We presumed that infected prey would be physically weakened by illness, making them easier targets.Accordingly, we considered that predators could immediately overtake infected prey, for simplicity, we treated the time spent by predators handling or subduing infected individuals as essentially negligible.This zero-handling time assumption resulted in the predation equation following a Volterra functional response dynamic, consistent with a Holling Type I functional response.
6.The model stipulates that the predators' consumption of the healthy(susceptible) prey is happening according to a Holling type II functional response.

POSITIVITY AND BOUNDEDNESS
In this section we aim to ensure that our model is well-posed by proving its positivity and boundness Existence and uniqueness of the solution Theorem 3.1.All solutions of system (6) corresponding to the initial conditions (0), (0), (0) ≥ 0 exist and are unique in the period [0, ] where 0 <  < ∞ and (), (), () ≥ 0. DYNAMICAL ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL ( 8) From (6) we write: (9) The phase space for the system ( 9) is are continuos functions and their partial derivatives exist and are also continuous on the phase space  3 ;therefor these functions are Lipschitzian on  + 3 and so, the system ( 9) with the nonnegative initial condition ( 8) has a unique solution on [0,  ], where 0 <  < ∞ [17].

Uniformly boundedness
The boundedness suggests that our eco-epidemiological model is biologically well-behaved.The boundedness of the system ( 6) is guaranteed by theorem 3.2.

Theorem 3.2.
All the solutions of the system (6) which starts in  + 3 are uniformly bounded.
Proof: for the proof of this theorem, we study two different cases for the initial value of
following the help of the theory of differential inequality [18], we obtain Where Γ(0) denotes the initial value of total population.as the time t grows to infinity.Thus, from (17) and (18) it can be concluded that Γ() is bounded as As in (19), we can confirm that 1+  is an upper bound of Γ().Therefore, the feasible solution for our system (6) stays in the positively invariant region Ω, where Ω={, , ) ∈  + 3 : Γ ≤ 1+  +  ∀ > 0. Thus, our system in ( 6) is biologically meaningful and mathematically well-posed in the domain Ω.Examining the population fluctuations within the designated theoretical parameters demonstrates the model's properties in that conceptual space proving the theorem.
This verification substantiates the theoretical proposition.In essence, one may consolidate the key consequence of this proposition as the model maintains finite values for all ( ≥ 0).

Feasibility
Obviously, the equilibria  0 ,  1 are feasible while  2 is not feasible so there will be no need to further analyze it,  3 is feasible under the conditions  3 (1 +  1 ) <  1  1 DYNAMICAL ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL To determine the feasibility of  4 we can say that the predator-free equilibrium points (p-free equilibrium points)  4 ( * ,  * , 0)exists if and only if there is a solution where  * ,  * > 0 and  * = 0 to the following algebraic nonlinear system: ( 20) Substituting  =  * ,  =  *   = 0 to the system (20) we get Therefor  4 is feasible under the condition For the interior equilibrium point  ̃(, ,  ̃) is feasible if and only if there is a solution where , ,  ̃> 0 to the following algebraic nonlinear system: From  3 we get From Therefor  ̃ is feasible under the either the conditions .

LOCAL STABILITY ANALYSIS
In this section we shall discuss the local stability of the equilibrium points we determined in the previous section.We will use the notation LAS instead of the term locally asymptotically stable.

BIFURCATION ANALYSIS
In this section we aim to explore some of the bifurcations that might occur in our system (6) using two theories 1) Sotomayor's theorem [19] and 2) the Hopf Bifurcation Theorem [1].To apply TALEB GABER, RATNA HERDIANA, WIDOWATI Sotomayor's theorem, we must first verify that one of the eigenvalues associated with the Jacobian matrix evaluated at a bifurcation equilibrium point is equal to zero.
For  1 (1,0,0) we have the following eigenvalues, are the eigenvalues can be written as, are − 2 ,  3 = 0 and the Jacobean matrix can be written as, From (33), ( 34) and ( 35) and according to Sotomayor's theory a transactional bifurcation occurs .

Hopf Bifurcation at 𝑬 𝟑
To study the Hopf bifurcation at  3 we will first assume that  is a bifurcation parameter for some system with the following characteristic equation corresponding to some equilibrium point say (, , ) of the system ( 6) is (36)  3 +  1 ()  2 +  2 () +  3 = 0 … We can now state the Hopf Bifurcation Theorem as it applies to our analysis: ( 1 *  1 + 3 ) on the condition that  1 is positive.

4)
Therefore, we can say that the Hopf bifurcation theorem conditions are all satisfied.Hence the Theorem 7.3 is proved.

GLOBAL STABILITY
Through this section we will discuss the global asymptotically stability (GAS) of the equilibrium points which we proved their locally asymptotically stability in section (5).Theorem 8.1.If  1 (1,0,0) is LAS then it is Globally asymptotically stable (GAS) in Π 1 : We took a Lyapunov function used by many other researchers such as [20], [21] to analyse the global stability of various forms predator-prey models, and modified it to fit our model as the following: Here  1 (, , ), is a positive definite function for all (, , ) other than (1,0,0) Computing the time derivative of L along the solutions of the system (6) will give us; And from the local stability conditions we have Also  1 (1,0,0) = 0.As this equilibrium point  1 = (1, 0, 0) is the lone solution to model (6) satisfying the condition  = 1, LaSalle's invariance principle [22] entails GAS.Theorem 8.2.If  3 (̅ , 0, ̅ ) exists and is LAS then it is Globally asymptotically stable (GAS) in Π 3 : Let us consider a suitable Lyapunov function And from the feasibility conditions we have  3 (1 +  1 ) <  1  1 We get Also  3 (̅ , 0, ̅ ) = 0.As this equilibrium point  3 (̅ , 0, ̅ ) is the lone solution to model (6) satisfying the condition  = ̅ and  = ̅ , LaSalle's invariance principle [22] entails GAS.
For the mentioned set of parameters, we notice that the trajectories start from (0.65,0.4,0.2) and converges to the equilibrium point in which only the susceptible prey survives while the infected prey and the predator wash out of the system, demonstrating a stable equilibrium.
We can notice that while the infected prey is still in the system the predator population increases but, as soon as the infected prey population is about to wash out of the system the predator As a result of the transcritical bifurcation that the system undergoes  1 ,  3 change their stability statues for  1 = 1.257142857142857where the system converges toward a stable state around  3 for  1 >  1  where both the predator and the susceptible prey survives and the infected prey is washed out of the system (see Fig. 3).When we raise the value of  1 for a certain value say  1 > 1.4 the system goes into unstable state and a limit cycle is born as we can see in fig. 4. values satisfy the stability conditions of  3 .In the absences of the infected prey, the system experiences a Hopf bifurcation as the parameter  1 crosses the critical value  1 =  1 = 0.101123595 (see Figure 6) and a limit cycle is born around  3 , (see Figure 7).existence equilibrium with the infected prey population number dropping down to a stable state of  ≈ 0.0345, it can be noticed that the system reaches equilibria very fast around ( ≈ 13) .

RESULTS
In this study, we analysed an eco-epidemiological model characterizing the interaction between a predator population and prey afflicted by disease.Rather than assume infection transmission followed a simple mass action formulation proportional to susceptible and infected prey populations, we adopted a nonlinear incidence rate of the form  1+ rooted in density dependence effects.We also accounted for infection potentially rendering prey more vulnerable by stipulating predators consumed infected prey according to mass action dynamics, while predation upon uninfected prey obeyed a Holling Type II functional response.Overall, this work aimed to generate a more nuanced understanding of predator-prey interactions complicated by disease spread according to the nonlinear incidence rate.
We proved that our model is ecologically well-posed as we showed the positiveness and boundness of the proposed model, determined the equilibrium points where we found: 1-one equilibrium at the origin  0 which is feasible but unstable 2-one axial equilibrium  1 which is feasible and globally asymptotically stable we also established the conditions for the trancritical bifurcation which occur at  1 3-three planar equilibria  2 ,  3 ,  4 where  2 is not feasible while  3 and  4 are feasible and GAS for some conditions which we established.We also established the conditions at which the  3 go through a Hopf bifurcation resulting in a limit cycle around  3 4-Finally, one interior equilibrium point at least which is feasible and stable for some parameters' values as it was shown through the numerical analysis Our findings align with the findings of [16], [23], as we have shown the impact of infection and predation rates on our model.We also conducted thoroughly analysis to our proposed model exploring interesting bifurcations occurring around  1 and  3 .

DISCUSSION
We found that the predation rate highly affects our proposed model as it could to a different type of bifurcations as we showed that for high enough predation rate a transcritical bifurcation occurs where  1 and  3 change stability and the solution converges into a state where both the susceptible prey and the predator survives instead of only the susceptible prey surviving the system.
While raising the predation rate for a value higher than a certain limit destabilize the system leading to the born of a limit cycle.We can also notice the direct relation between the Hopf bifurcation constant and the predation rate, showing the important role of the predation rate in creating a limit cycle and presenting a Hopf bifurcation to the proposed model highlighting the important role of the predation rate.
The infection rate on the other hand plays an important role in the dynamic of the system where for different values of the infection rate  and for some set of the parameters' values as we seen in the numeric simulations the system can converge into one of three states, i.e.If we decreased the value of the infection rate less than a certain value say  1 it can lead to the extinction of the infected prey and two stability cases one where only the susceptible prey survives the system and the second where both the susceptible prey and the predator survives; When the infection rate reaches the value  1 and in the period  1 <  <  2 the infected prey survives the extinction leading to state of co-existence stability between the three species; Finally, when the infection rate hits a certain value  2 and in period say  2 <  <  3 it leads to a high decrease in the susceptible prey numbers causing the predator to extinct and stabilizing the system.Furthermore, when the predation parameter goes higher than a  3 it destabilizes the system creating a limit cycle.
size of prey population P: The size of predator population T: Time α: Prey's growth rate m: The predation rate θ: Conversion factor which denotes the number of newly born predators for each killed prey d: Predator's natural death rate Which indicates that the total population size Γ() takes of the function  that starts with the initial value Γ(0) at the initial time  = 0 is bounded by the value 1+