The Dynamics of a Prey-Predator Model with Infectious Disease in Prey: Role of Media Coverage

In this paper, an eco-epidemiological model with media coverage effect is proposed and studied. A prey-predator model with modified Leslie-Gower and functional response is studied. An -type of disease in prey is considered. The existence, uniqueness and boundedness of the solution of the model are discussed. The local and global stability of this system are carried out. The conditions for the persistence of all species are established. The local bifurcation in the model is studied. Finally, numerical simulations are conducted to illustrate the analytical results.


Introduction
The utilization of mathematical models for studying and understanding the spread and controlling infectious diseases has become a highly important tool. The scientists extensively studied the dynamics of ecological models in the existence of infectious diseases and provided important insights into complex biological processes. The study of the spread of infectious diseases within populations of ecological systems is resulting in a branch called eco-epidemiology. This subject is rapidly growing as a branch of theoretical ecology ([1]- [3]), Later on, several researchers were proposed and studied eco-epidemiological models involving many biological factors, see ( [4][5][6][7][8][9][10][11][12][13]) The modified Leslie-Gower prey-predator model, which is proposed by Leslie and Gower [14] and modified by May [15], is considered by many scientists [16][17][18][19]). In the modified ISSN: 0067-2904

Hussein and Abdul Satar
Iraqi Journal of Science, 2021, Vol. 62, No. 12, pp: 4930-4952 4931 Leslie-Gower model, the predator acts as a generalist predator because it avoids extinction by using an alternative source of food. Although in case of a severe scarcity of prey the predator population growth may still be limited by the fact that their favorite food is not available in abundance, some predator species can switch to another available food in the environment. The impact of media coverage is one of the most key factors to establish the prevention and control measure that affect the spread of infectious disease. The role of media coverage of disease outbreaks is therefore crucial and should be given prominence in the study of disease dynamics [20]. Liu and Cui [21] studied a container model that characterized the spread and control of infectious disease under the influence of media coverage. Tchuenche and Bauch [22] proposed and studied a susceptible-infected-hospitalized-recovered model with vital dynamics, where media coverage of disease incidence and prevalence can influence people to reduce their contact rates. Li and Cui [23] introduced constant and pulse vaccines in media coverage for SIS disease models. In recent years, attempts have been made to develop mathematical models for the transmission dynamics of infectious diseases within the ecoepidemiological model. Alwan and Abdul Satar [24] proposed and studied a prey-predator model having a disease in predator species and involving media coverage. They used it for describing the predation process as a Lotka-Volterra type of functional response. In this paper, an SIS-type of disease in prey is considered, so that a modified Leslie-Gower prey-predator model is proposed and the effect of media coverage on the dynamics of a proposed eco-epidemiological model is studied. Moreover, Lotka-Volterra type of functional response is used to describe the predation process. The organization of this paper is given as follows. Section (2) deals with the model formulation. Section (3) determines the equilibrium points (EPs) and analyzes their local stability. The global stability for the EPs is studied with the help of the Lyapunov method (LM) in section (4). While, the bifurcation analysis of the system is investigated in section (5). Section (6) deals with the numerical simulation of the system. Finally, the discussion and conclusions are addressed in section (7).

The mathematical model
In this section, the effect of media coverage on a modified Leslie-Gower prey-predator model is formulated mathematically. An infectious disease of type in prey species is included in the model. It is assumed that the prey is consumed by the predator according to Lotka-Volterra type of functional responses. Now, in order to represent the dynamics of such a real-world system, the following hypotheses are adopted. Let the variables ( ) ( ) ( ) represent the densities at time for the susceptible prey, infected prey, and predator, respectively. It is assumed that ( ) grows logistically with as an intrinsic growth rate, while ( ) cannot reproduce due to the disease, instead of that, it competes with the susceptible prey for environment carrying capacity . However, the predator ( ) grows logistically with as a growth rate by sexual reproduction and carrying capacity, depending on the prey and given by , where represents a residual loss in predator population. The predator species ( ) consumes both the prey species ( ) ( ) using Lotka-Volterra type of functional responses with maximum attack rates of respectively. The term . / represents the infection rate due to the direct contact between ( ) ( ), where is the contact rate before media coverage alert, while represents the reduced value in the contact due to media coverage alert, so that is the maximum transmission rate under the media coverage and is the non-response rate of individual to the media coverage. Furthermore, since it is well known that the media coverage cannot prevent the spreading of the disease completely, then from now onward we take . Also, the infected individuals may recover with a rate of . The disease-caused death rate of infected individuals is given by , while the parameter is the maximum value in which per capita reduction rate of predator species can be attained due to intra-specific. Finally, the infected prey causes predator death due to disease when feeding on it with a probability ( ). According to the above hypotheses, the dynamics of the above-described system, that is consisting of a diseased prey-predator system incorporating the media coverage, can be represented by the following set of differential equations: .
with ( ) ( ) ( ) as an initial condition. Therefore, the system (1) has the domain *( ) +. Clearly, the system (1) contains a functions; therefore, these functions are Lipschitzain. Hence, the solution of the system (1) exists and is unique. Further, the uniformly bounded of the solutions of the system (1) is proved in the following theorem. Theorem (1): The system (1) has uniformly bounded solutions.

Proof. Define
, then can be written as where * +. Then, direct computation shows that for goes to , we have where .
Since the third equation of predator is a logistic growth equation, then it is easy to verify that ( ) . Therefore, all the variables are bounded.

Existence of EPs and Their Local Stability Analysis
The existence of EPs of the system (1) and their local stability analysis are discussed. The existence conditions for each of these EPs are established. The trivial EP, represented by ( ), always exists. The first axial EP, represented by ( ), always exists as the susceptible prey population grows to carrying capacity in the absence of predation.
The second axial EP, represented by . /, always exists as the predator population that growth logistically grows to carrying capacity supplied by the environment in the absence of preferred prey. The predator-free EP is denoted by where ̅ represents a positive root for the third order polynomial equation: (4) where the point ( ) represents the positive intersection point of the following two isoclines: Obviously, as then the isoclines become .
Now, to establish the local stability, the Jacobian matrix (JM) of system (1) about arbitrary where . / , It is clear that the system (1) has JM at trivial EP, ( ) specified by (7a) Therefore, the eigenvalues of ( ) are: ( ) .
(7b) Hence, the trivial EP is unstable (saddle point). The JM of the system (1) at the first axial EP, (8a) Therefore,, the eigenvalues of ( ) are given by ( ) .
The JM of the system (1) at the second axial EP, Clearly, the eigenvalues of ( ) are given by , Hence, all the eigenvalues are negative, and the second axial EP is LAS provided that . (9c) Now the JM of the system (1) at the predator-free EP, ( ̅ ̅ ), can be written as: , and ̅ . Clearly, one of the eigenvalues is ̅ and the other two eigenvalues are the roots of the equation: , (10b) where and ( ). Note that the direct computation gives that the roots (eigenvalues) of the equation (10b) can be written as √ √ (10c) Hence, all the eigenvalues of ( ) have negative real parts and hence is LAS if and only if Now the JM of the system (1) at the infected prey-free EP, ( ̃ ̃) , can be written as: Clearly, one of the eigenvalues is ̃ ( ) ̃ and the other two eigenvalues are given by: Finally the JM evaluated at the positive EP, , is given by: Then the characteristic equation of ( ) can be written as: Accordingly, the local stability of the positive EP can be given in the following theorem. Theorem (2). The positive EP of the system (1) is locally LAS provided that the following conditions hold: .
(15d) Proof. According to the Routh-Hurwitz criterion, all roots of the characteristic equation given by Eq. (14b) have negative real parts roots, if and only if , , and . Straightforward computation shows that the conditions (15a)-(15d) satisfy the Routh-Hurwitz criterion conditions, and hence all the eigenvalues of the Eq. (14b) have negative real parts. Then,, the positive EP is LAS.
The persistence of the system (1) is studied. It is well known that the biological system is persistent if and only if all its species are persistent all the time. Now, according to the system (1), if the predator individuals disappear, then Clearly, subsystem (16) is a 2D space that has a unique positive point given by ( ̅ ̅ ), which are given by Eq. (2a), and exists uniquely in the plane under the condition (2c). By using Poincare Bendixon theorem, the solution of system (16) approaches either to EP ( ̅ ̅ ) or else to the periodic dynamics. Now, by using the continuous function ( ) , we obtain Therefore, according to the Dulac criterion, there is no periodic dynamics in the interior of the positive quadrant of plane. Hence, using Poincare Bendixon theorem, the positive EP of the subsystem (16) is globally asymptotically stable (GAS) whenever it exists. Then the system (1) has no periodic dynamics in the boundary plane. The following theorem explains the conditions that guarantee the persistence of the system.
(17c) Proof: Suppose that is a point in the interior of and ( ) is the orbit through , and let ( ) is the omega limit set of ( ). Further, since ( ) is bounded, due to the boundedness of the system (1), then we first show that ( ) . Assume the contrary, since is a saddle point, then by Butler-McGhee lemma [25], there is at least one other point such that ( ) ( ), where ( ) is the stable manifold of . Now, since the stable manifold of is given by direction and the entire orbit through , say ( ), is contained in ( ), then we obtain a contradiction to the boundedness of ( ), due to the containment of an unbounded positive axis in it. This shows that ( ). Now, to proof that ( ), we assume the converse. Since is a saddle point, then by Butler-McGhee lemma, there is another point, say , so that ( ) ( ). Now, since the stable manifold of is given by direction and the entire orbit through , say ( ), is contained in ( ), hence we obtain a contradiction to the boundedness of ( ), due to the containment of an unbounded positive axis in it. This shows that ( ). Now, since the points are saddle points under the conditions (17a), (17b), and (17c), respectively. Then by using similar argument as that given in the first part of the proof, we obtain that ( ).

Global Stability Analysis
In this section, the global stability (GS) is studied for all LS Eps. Lyapunov method is used to investigate the GS or specify the basin of attraction of each EP. Theorem 4. Assume that the second axial EP, ( ̿ ), of the system (1) is LAS in , and: where is the upper bound of . Then, it is GAS in . Proof: Consider the following function: . ̿ ̿ /. Then, is a real valued function, which is a positive definite. Now, the derivative can be calculated as: Therefore, by using the above set of conditions, it is obtained that: Obviously, is a negative definite, and since is radially unbounded function, then is GAS. Theorem 5. Assume that the predator-free EP, , of the system (1) is LAS in , and: , where all the symbols are defined in the proof. Then it is GAS in . Proof: Consider the following function: .
Clearly, is function that is positive definite. Then we have , and is the upper bound of . Therefore, by using the condition (19a), it is obtained that Obviously, under the condition (19b), we have is negative definite, and since is radially unbounded function, then is GAS. Theorem 6. Assume that the infected prey-free EP, , of the system (1) is LAS in , and:

. (20c)
Then, it is GAS in . Proof: Consider the following function: .
Clearly, is that is a positive definite real valued function.
where is the upper bound for . Using conditions (20a)-(20b), its obtained that: Obviously, under the condition (20c), it is obtained that is negative definite. Further, since is radially unbounded function, then is GAS. Theorem 7. Assume that the positive EP, ( ), of the system (1) is LAS in , and: where all the symbols are defined in the proof. Then, it is GAS in . Proof: Consider the following function: . / . / . /.
Clearly, is a function that is a positive definite. Then, we have , and . So, after using the given conditions (21), it is obtained that: Obviously, we have is negative definite. Also, since is radially unbounded function, then is GAS.

Local bifurcation
The effect of varying the parameters values on the dynamics of the system (1) is studied in this section using the local bifurcation analysis with the help of the Sotomayor'stheorem. Now, for simplifying the notations, rewrite the system (1) in the vector form as follows ( ), with ( ) and ( ) So, according to the JM of the system (1) at the point ( ), it is easy to verify that for any vector ( ) , we have that where .
We also have The occurrence of LB around the EPs, , is investigated respectively. Theorem 8. Assume that the parameter satisfies that (24) Then, the system (1) near the second axial EP, , has a transcritical bifurcation (TB) but saddlenode bifurcation (SNB) and a pitchfork bifurcation (PB) cannot occur. Proof: Note that, when , then the JM of the system (1) at can be written as So, has the following eigenvalues: ( ) and .
Hence, the second axial EP is a non-hyperbolic point, and then the necessary but not sufficient condition for bifurcation is satisfied.
where represents the second derivative of w.r.t. that is given by equation (22). Accordingly, by Sotomayor theorem [26], the system (1) near the EP, , with possesses a TB but not PB. Theorem 9. Assume that the conditions (11b)-(11d) hold and the parameter satisfies ̅ .
(28b) Then, as the parameter passes through the value Clearly, the eigenvalues and have negative real parts due to conditions (15a) and (15b). Hence, the Jacobian matrix of the system (1)  We have that ( ) , hence we obtain that ( ) ( ) .
Therefore, we obtain that . Consequently, the first condition of SNB in view of Sotomayor theorem is satisfied. Now, since: , Clearly, , ( )( )under the condition (28b), and hence the system (1) undergoes SNB near the coexistence equilibrium.

Numerical Simulation
In this section, the global dynamics of the system (1) is further investigated. To specify the control set of parameters, the system is solved numerically using Runge-Kutta of ordered six, followed by forth steps Predictor-Corrector method. Then, all the obtained numerical results are drawn in the form of phase portraits and time series using Matlab version 6. Therefore, in order to run simulations, the following hypothetical set of biological data is used in this section: It is observed, for this set of data, that the system (1) approaches asymptotically to the unique coexistence EP, ( ), starting from five different initial values, as shown in Figures 1 and 2.  (2,1.5,1.8) started at (3,2.5,3.5) started at (4,1.9,3) started at (5,3.5,4.5) 0 500 1000 1500 0 According to these two figures, the system (1) persists at the coexistence point in . Now, in order to discuss the effect of varying the values of parameters on the dynamical behavior of the system, the system is solved numerically for the data given in Eq. (30), with varying a specific parameter each time and then the obtained solutions are drawn as shown below. It is observed that, for the values of parameter in the range with the other parameters as in Eq. (30), the system (1) approaches asymptotically to infected prey-free EP in the interior of of plane; otherwise, it has a GAS coexistence EP; see Figures 3a and 3b for typical values of . It is observed that varying the parameters and has a similar effect to that shown with varying . Now, for the parameter in the range , it is observed that the system (1) approaches asymptotically to predator-free EP in the interior of plane, as shown in the below typical figures given by Figure 4. However, it approaches to otherwise. The system approaches asymptotically to ( ) when . It is observed that varying the parameters has a similar effect to that shown with varying . Now, for the parameter in the range it is observed that the system (1) approaches asymptotically to the coexistence EP, as shown typically in Figure 5. However, it approaches to otherwise. It is observed that varying the parameters has similar effects as those shown with varying . On the other hand, varying the parameters of the infection rate of the system (1) is also studied. It is observed that, for the system (1) approaches asymptotically to the infected prey-free EP; otherwise, it has a GAS coexistence EP that has a GAS at . However, for (maximum transmission rate under the media coverage alert), with increasing the response of individuals to the media coverage alert or decreasing the parameter the system (1) approaches gradually to as shown in Figure 6 for the values respectively.  ) when . (c) The system approaches asymptotically to ( ) when . (d) The system approaches asymptotically to ( ) when . Now, for the the parameter in the range with the rest of parameters as in Eq. (30), the system (1) approaches asymptotically to . However, the system (1) approaches asymptotically to the coexistence equilibrium points and for the ranges , respectively, as shown typically in Figure 7. Finally, for the parameter in the range with the rest of parameters as in Eq.(30), the system (1) approaches asymptotically to as shown in Figure 8. Otherwise, the system (1) still approaches to in the interior of .

Discussion and Conclusion
In this paper, the effect of media coverage alert on the dynamical behavior of the diseased modified Leslie-Gower prey-predator model involving disease in the prey is considered. The system is studied theoretically as well as numerically. It is observed that the system has at most six non-negative equilibrium points. Since the solution of the system is proved to be uniformly bounded, it is observed that the solution approaches asymptotically to one of its equilibrium points depending on determined conditions. According to the numerical simulation, it is observed that media coverage works as a control parameter for the spread of disease.