The Effects of Media Coverage on the Dynamics of Disease in Prey- Predator Model

In this paper, an eco-epidemiological model with media coverage effects is established and studied. An -type of disease in predator is considered. All the properties of the solution of the proposed model are discussed. An application to the stability theory was carried out to investigate the local as well as global stability of the system. The persistence conditions of the model are determined. The occurrence of local bifurcation in the model is studied. Further investigation of the global dynamics of the model is achieved through using a numerical simulation.


Introduction
The term eco-epidemiological models is used to describe the models that incorporate disease in ecological communities [1]. The first eco-epidemiological model including infectious diseases in the prey was introduced by Anderson and May [2]. Later on, a number of researchers proposed and studied eco-epidemiological models involving many biological factors [3][4][5][6][7][8][9][10]. It is observed that the spread of disease among the population is a main reason for the species extinction. Although many studies showed interest in eco-epidemiology. The impact of the media coverage, which has an important role in the outbreak of the infectious diseases, have been mostly neglected in the previous research papers.
The spreading and controlling of a susceptible-infected-recovered-susceptible (SIRS) disease with the media coverage was investigated by Liu and Cui [11]. Cui et al. [12] constructed a mathematical model that incorporates media coverage to understand its effects on the spread of infectious diseases in a given population. They concluded that more extensive media coverage in a given population leads to reduce the number of infected individuals. It is observed that the use of media coverage alert causes a reduction in the spread of AIDS, due to reducing the contact between human beings [13]. A similar

ISSN: 0067-2904
Alwan and Satar Iraqi Journal of Science, 2021, Vol. 62, No. 3, pp: 981-996 982 observation was obtained regarding the spread of the severe acute respiratory syndrome (SARS) during 2002 and 2004 [14,15]. Later on, some authors studied the effects of media coverage on the spread of infectious diseases [16]. Recently, Li et al. [17] proposed an epidemic model with media to describe the spread of infectious diseases in a given region. They found that media coverage plays an important role in controlling the spread of the disease. Al Basir [18] formulated and analyzed an epidemic model on the prevalence of infectious diseases using awareness campaign driven by media, with the aim of investigating the effects of awareness and delay on disease outbreak. These studies observed that effective media coverage can postpone the arrival of the infections peak and that a fewer number of individuals become infected in the course of transmission. Since the real-world system contains many species that interact with each other in different ways , we intended in this paper to study the influence of media coverage on the dynamics of infectious diseases in prey-predator model. Consequently, a prey-predator model having disease in predator species and involving media coverage is proposed. Lotka-Volterra functional response is used for describing the predation process. Moreover, this paper is organized as follows. Section 2 deals with the model formulation. Section 3 determines the equilibrium points (EPs) and their local stability analysis. The global stability for the EPs is studied with the help of Lyapunov method (LM) in section 4. 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 effects of media coverage on an eco-epidemiological model dynamics are studied. The model consists of an infectious disease of type in predator species that feeds on a prey. It is assumed that the prey is consumed by the predator according to Lotka-Voltera types of functional responses. Thus, in order to represent the dynamics of such a real-life system, the following hypotheses are adopted. Let the variables ( ) ( ) ( ) represent the densities at time for the prey, susceptible predator, and infected predator, respectively. It is assumed that ( ) grows logistically with as the intrinsic growth rate and as the carrying capacity. The species ( ) is consumed by the species ( ) ( ) using Lotka-Volterra functional responses, with maximum attack rates of and conversion rates of ( ) and ( ) respectively. The term .
/ represents the infection rate [11] 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's to the media coverage. Furthermore, since it is well known that the media coverage cannot prevent the spreading disease completely, then from now onward we take . Also, the infected individuals may recover with a rate of . Finally, the natural death rate of predator individuals is given by while the disease death rate is represented by . According to the above hypotheses, the dynamics of the above-described system, that consists of a prey-predator system, which incorporates the media coverage, can be described in the following set of differential equations: with ( ) ( ) ( ) as initial conditions. Therefore, system (1) has the domain *( ) +. Clearly, system (1) contains functions and, therefore, these functions are Lipschitzain. Hence, the solution of system (1) exists and is unique. Furthermore, the uniform boundedness of the solutions of system (1) is proved in the following theorem. Theorem (1). All the solutions of the system (1) are uniformly bounded. Proof: We define , then can be written as here, * + , then direct computation shows that, for to go to , we have .
Therefore, all the variables are bounded.

Existence of EPs and Their Local Stability Analysis
It is observed that system (1) has four nonnegative biologically reasonable equilibrium points (EPs). The existence conditions for each of these EPs are established as the following: The trivial EP, ( ), always exists. The axial EP, ( ), always exists as the prey population grows to carrying capacity in the absence of predation. The infected predator free EP, which exists under the condition .
(2b) The coexistence or positive EP, ( ), is given by where . While ( ) represents the positive intersection point of the following two isoclines Obviously, as then the isoclines becomes ( ) . / , ( ) . Therefore, ( ) intersects the axis at the positive point , Consequently, in the following, the stability near these EPs is studied locally using the linearization method. Note that it is easy to confirm that the Jacobian matrix (JM) of system (1) about an arbitrary . (5) Now, it is obtained that the JM of system (1) around trivial EP, ( ), has the following eigenvalues: . (6) Hence, the trivial EP is unstable (saddle point). The JM of the system (1) at the axial EP, ( ), can be written as: Therefore, the eigenvalues of ( ) are given by .
(8) Now, the JM of system (1) at the EP, ( ̅ ̅ ), can be written as: Clearly, one of the eigenvalue is ̅ and the other two eigenvalues are the roots of the equation: , . Clearly, Eq. (9b) has the following roots (9c) Hence, all these eigenvalues have negative real parts and hence is LAS if and only if the following condition holds: ̅ .
(10) Finally, the JM evaluated at the positive EP, , is given by / , Then the characteristic equation of ( ) can be written as: Accordingly, by using the Routh-Hawirtiz criterion, all roots of Eq. (11b) have negative real part roots and, hence, the EP, , is LAS if the following sufficient conditions are satisfied: Now, we study the global stability and the persistence of system (1). It is well known that any biological system persists if and only if all its species persist for all the time. This means, from the mathematical point of view, that there is no trajectory that approaches asymptotically to the boundary axis or planes. Now, according to system (1), if the infected individuals disappear, then the following subsystem is obtained Clearly, subsystem (13) is a 2D system that has a unique positive point given by ( ̅ ̅ ), which are given by Eq. (2a) and exist uniquely in the plane under the condition (2b). According to the wellknown Poincare Bendixon theorem, the solution of system (13) approaches either to EP ( ̅ ̅ ) or else to the periodic dynamics. Now, by using the continuous function ( ) , we obtain the following quantity . Then, from the Dulac criterion, there is no periodic dynamics in the interior of the positive quadrant of the plane. Therefore, according to the Poincare Bendixon theorem, the EP, ( ̅ ̅ ), is a globally asymptotically stable (GAS) whenever it exists. Hence, the 3D system (1) has no periodic dynamics in the boundary planes. Recall that the system (1) persists if and only if all the species coexist for all the future time. Hence, the following theorem contains the conditions that guarantee the persistence of the system. Theorem (2). System (1) is uniformly persistent provided that: , (14b) Proof: Suppose that is a point in the interior of and ( ) is the orbit through and let ( ) be the omega limit set of ( ). Furthermore, 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 [19] there is at least one other point such that ( ) ( ), where ( ) is the stable manifold of . Now, since the stable manifold of is given by plane and the entire orbit through , say ( ), is contained in ( ), hence, if is on either boundary axes of plane, then we obtain a contradiction to the boundedness of ( ), due to the containment of the unbounded positive axis in it. Now, let belongs to the interior of plane. Since there is no EP in the interior of plane, then the orbit through , which is contained in ( ), must be unbounded. This gives a contradiction and leads to ( ). Now, to proof that ( ), we assume the converse. Since is a saddle point under the condition (14a), then by Butler-McGhee lemma, there is another point, say , so that ( ) ( ). Now, since the stable manifold of is given by plane and the entire orbit through that denoted by ( ) is contained in ( ) hence if ( ) or ( ) , then a contradiction to the boundedness of ( ) is obtained and then ( ). Finally, since the point is a saddle point under the condition (14b), then by using similar argument as given in the first part of the proof, we obtain that ( ). Hence, the proof is complete.

Global Stability Analysis
In this section, the global stability of all the locally stable EPs is studied with the use of Lyapunov method, as shown in the following theorems. Theorem (3). Assume that the EP, ( ), of system (1) is LAS in , and the following conditions are satisfies: Then it is GAS in . Proof: Consider the following function: . / . Then is a function, which is a positive, definite, real valued function. Now, the function can be calculated as:

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Obviously, under the condition (15), we have that is negative definite. Also, since is radially unbounded function, then is GAS. Theorem (4). Assume that the EP, , of system (1) is LAS in , and the following conditions are satisfied: .
(16b) Then it is GAS in . Proof: Consider the following function: .
where are positive constants that shall be determined later on. Clearly, is that is positive definite real valued function. Then we have So, by choosing the positive constant as below: and , we obtain that: Obviously, under the conditions (16a)-(16b), we have that is negative semi definite. Therefore, the EP, , is a stable point. Now, since is the only invariant set that satisfies , then by using LaSalle'sinvarianceprinciple, it is attracting. Hence, is GAS. Theorem (5). Assume that the EP, ( ), of system (1) is LAS in , and the following conditions are satisfied: where all the symbol are defined in the proof. Then it is GAS in . Proof: Consider the following function: where are positive constants that shall be determined later on. Clearly, is a function that is a positive definite function. Then we have where . So, by choosing and , we get, after using the given condition (17), that: Obviously, we have that is negative definite. Also, since is radially unbounded function, then is GAS.

Local bifurcation
It is well known that the bifurcation occurs if and only if there is a qualitative change in the behavior of the solution of a system, as occurs by varying the control parameter. Therefore in this section, a study of the occurrence of local bifurcation (LB) near the EPs of system (1) is performed using Sotomayor'stheorem [20]. Also, it is well known that the non-hyperbolic property of an EP is a necessary but not sufficient condition for the occurrence of bifurcation in the neighborhood of that point. Therefore, the parameters, which change the EPs from hyperbolic to non-hyperbolic EPs, are considered as candidate bifurcation parameters of system (1), as shown in the next theorems. Now, for simplifying the notations, we rewrite system (1) in the vector form as follows ( ), with ( ) and ( ) .
So, according to the JM of system (1) at the point( ), it is easy to verify that for any vector ( ) , we have that where , On other hand, we have also In the following theorems, the occurrence of LB around the EPs, , is investigated, respectively. Theorem (6). Assume that the parameter satisfies that .
(20) Then system (1) near the EP, , has a transcritical bifurcation (TB). However, saddle-node bifurcation (SNB) and pitchfork bifurcation (PB) cannot occur. Proof: Note that, when , then the JM of system (1) at can be written as So, has the following eigenvalues: and and, hence, the necessary but not sufficient condition for bifurcation is satisfied and is a nonhyperbolic point. Let ( ) be the eigenvectors of corresponding to the eigenvalue . Then, simple computation gives that ( ) , where represents any nonzero real number and .
Also, let ( ) represents the eigenvectors of corresponding to the eigenvalue . Then again, simple calculation shows that ( ) , where is any nonzero real number and .
Since the partial derivative of vector field with respect to the parameter is given by ( ) , hence by substituting and in this derivative, we obtain that ( ) ( ) . Therefore , ( )-. Thus system (1)  .
(24b) Then, when the parameter passes through the following value system (1) near the coexistence EP, , has a SNB, provided that the following condition holds , (25) where all the symbols are given in the proof. Proof: Straightforward computation shows that under the conditions (12a), (12b), (24a), (24b) and (24c) the coefficients of the characteristic equation given by Eq.(11b) are , and . Hence, Eq. (11b) has three roots (eigenvalues of ( )) given by Clearly, the eigenvalues and have negative real parts. Hence the JM of system (1)  We have that ( ) , hence we obtain that ( ) ( ) .
Therefore, we obtain that , ( )- under the condition (25), and hence system (1) undergoes a SNB near the coexistence equilibrium.

Numerical Simulation
In this section, the global dynamics of system (1) is further investigated. To specify the control set of parameters, the system is solved numerically. System (1) is solved numerically using Runge-Kutta of order six, followed by a four-step Predictor-Corrector method. Then, all the obtained numerical results are drawn in the form of phase portrait 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. (26) It is observed, for this set of data, that the system (1) approaches asymptotically to the unique coexistence EP, ( ), starting from three different initial values, as shown in the following two figures ( Figures-1 and 2).

Alwan and Satar
Iraqi Journal of Science, 2021, Vol. 62, No. 3, pp: 981-996 990  According to these two figures, system (1) persists at the coexistence point in . Now, in order to discuss the effect of varying the parameters' values of system (1) on the dynamical behavior of the system, the system is solved numerically for the data given in Eq. (26) 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. (26), system (1) approaches asymptotically to infected predator-free EP in the interior of plane, otherwise it has a GAS coexistence EP, see Figure-   Obviously, as the carrying capacity increases, the population, especially the infected predator, increases too, but the system still persists at the coexistence EP. Now, varying in the range of leads to an extinction in the infected predator and the system (1) approaches to , as shown in the typical illustration given by Figure-5. While for the system approaches to periodic dynamics in 3D, as shown in the typical illustration given by Figure-6. Otherwise, system (1) still has a global coexistence EP.  Moreover, for the parameter in the range it is observed that system (1) approaches to periodic dynamics in 3D as shown in the typical illustration given by Figure-7. However, it approaches to otherwise. It is observed that varying the parameters and have similar effect as that shown with varying . Now, for the parameter in the range it is observed that system (1) approaches to periodic dynamics in 3D, as shown in the typical illustration given by Figure-8. However, it approaches to otherwise. On the other hand, varying the parameters of the infection rate of system (1) was also studied. It is observed that, for system (1) has a GAS at with quantitative changes in the sizes of populations. 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 , system (1) approaches gradually to , as shown in Figure-9 for the values respectively. Finally, for the parameter in the range with the rest of parameters being as in Eq. (26), system (1) approaches asymptoticaly to as shown in the typical illustration given by Figure-10. Otherwise system (1) still approaches to in the interior of . Similar effect was obtained, as that happened with varying , when we varied the parameter .