Mathematical Analysis of an Epidemic-Species Hybrid Dynamical System

We consider an epidemic-species hybrid dynamical system. The disease is spread among the prey only and the infected prey can reproduce virus.The predator only eats the infected prey.Mathematical analyses are given for the systemwith regard to the existence of equilibria, local stability, Hopf bifurcation, and the orbital stability of the Hopf bifurcating limit cycle. We further analyse the system under impulsive releasing of virus and predator.


Introduction
Epidemic models and species models have received much attention from scientists, respectively.There are many literatures about them.Now we consider an epidemic-species hybrid dynamical system which is motivated by the integrated pest management (IPM).As we know, IPM is becoming more and more popular among farmers, researchers, and policy makers.IPM seeks to minimize reliance on pesticides by emphasizing the contribution of other control methods, including biological control, host-plant resistance breeding, and cultural techniques.Potentially, the use of viruses, fungi, and bacteria is one of the most effective biological methods for controlling pests.One reason for this optimism is that the generation time lapse of microbes is much shorter than that of an insect pest [1].The role of microbial pesticides in the integrated management of insect pests has been recently reviewed for agriculture [2][3][4][5], forestry [6,7], and public health [8].In most cases no single microbial control agent will provide sustainable control of an insect pest or complex of pests.As components of an integrated approach, entomopathogens can provide significant and selective insect control.In the not too distant future we can envision a broader appreciation for the attributes of entomopathogens and expect to see synergistic combinations of microbial control agents with other technologies (in combination with semichemical technology, soft chemical pesticides, other natural enemies, resistant plants, chemigation, remote sensing, etc.) that will enhance the effectiveness and sustainability of integrated control strategies.
There is a vast amount of literature on the applications of microbial disease to suppress pests [9,10].But there are only a few papers on mathematical models of the dynamics of microbial disease in pest control [11,12].On the other hand, there are many mathematical models of infectious diseases of human beings [13][14][15][16].All the earlier pest control models are normally two-dimensional consisting of susceptible and infected pests, and infectious disease models are at most three-dimensional with a susceptible, infected, and removable class; the dynamics of the diseasescausing organism is not considered in the model.However, presently the insect pathogens are used in two ways [17].In the first method, a small amount of pathogens is introduced in the pest population with the expectation that it will generate an epidemic and will be subsequently endemic.The success of the approach depends on the survival of the microbes which in turn depends on environmental factors, such as temperature, humidity, and crop conditions.In the second method, an insect pathogen is used like a nonresidual chemical insecticide.In this case, it is applied whenever a pest population is at an economically extreme level for pest damage and there is no expectation that the pathogen will survive for an appreciable period.In this paper, we examine the use of pathogens in a four-dimensional prey-predator model for the agricultural or forestry ecosystem with the assumption that viral disease spreads only among the insect pests.In nature, the infection by baculovirus begins when an insect eats virus particles on a plant, perhaps from a sprayed treatment.Virus infection causes the cell lysis in the host body and produces more virus particles (virus replication), until the cell and ultimately the insect die.Most baculoviruses cause the host insect to die in a way that maximizes the chance that other insects come in contact with the virus and become infected by eating the foliage that has been contaminated by virus-killed larvae.We further assume that the natural enemy (predator) in the system survives on the infected prey.This is due to the fact that the viral infection makes some behavioral changes and sublethal effects on host, which make them more vulnerable to predation by natural enemies than healthy hosts before death [18].
The dynamical behavior of the considered system is investigated from the point of view of stability and persistence.The model shows that infection can be sustained only above a threshold force of infection resulting from virus replication parameter .Still, on increasing the value of , the endemic equilibrium bifurcates towards a periodic solution.The stability of the limit cycles arising from Hopf bifurcation is analysed using Poore's condition [19].Whereas, in consideration of the practice in pest control, the natural enemy and the pathogens are introduced discontinuously at some fixed moment, the impulsive releasing of the virus and natural enemy is added in the system.We prove that the pest extinction periodic solution exists and is globally asymptotically stable when the impulsive period  is less than the critical value; otherwise, the system can be permanent.The arrangement of the paper is as follows.In Section 2 we describe the model formulation.Section 3 shows its equilibria.Section 4 discusses the local stability and Hopf bifurcation.In Section 5 we give the details about the orbital stability of the periodic orbit arising from Hopf bifurcation by Poore's condition.In Section 6 we consider the corresponding impulsive system by releasing virus and natural enemy at some fixed moment.Lastly in Section 7 we end the paper with a concluding discussion.

The Model
The model contains three species, namely, pest, predator, and virus.The total pest population  is divided into two subpopulation classes: the susceptible pest, denoted by , and the virus-infected pest, denoted by .Therefore, at any time , () = () + ().We assume that only susceptible pest  is capable of reproducing with logistic law; that is, the infected class of pest is removed by lysis before having the possibility of reproducing.However, they still contribute with  to population growth towards the carrying capacity . is the intrinsic birth rate.Then the growth equation of susceptible pest  is given by A susceptible pest becomes infected as the two mentioned methods ahead.The incidence is assumed to be the simple mass action incidence ;  represents the effective per pest contact rate with viruses.Hence, the evolution equation of the susceptible pest is The infected pest  has a latent period, the period between the instant of infection and that of lysis, during which the virus is reproduced inside the larval tissue.The lysis death rate  gives a measure of such latency period , given by  = 1/.The equation of infected class  takes the form The lysis largely produces virus polyhedra or polyhedral inclusion bodies (PIB) on average  PIB per insect;  ( > 1) is called the virus replication parameter.The virus particles  have natural mortality denoted by  V due to temperature changes, enzymatic attack, pH dependence, and so forth.The equation of the virus is given by We assume the natural enemy or predator  consumes only the infected prey , the consuming means is the standard bilinear expression , and  is the predation coefficient.We assume the natural enemies in the system remain unharmed from the virus.Then the equations of the infected pest and the natural enemy are where  is the death rate of the predator,   is the density restriction coefficient of predator, and  is the conversion factor for the predator.Keeping these in view, the dynamics of the epidemic-species may be governed by the following autonomous system of differential equations: In the above model, all of the coefficients, , , , , , ,   , ,  V , and , are positive real numbers.If using the dimensionless time  =  and the transformations  = /,  = /,  = /, and V = /, we have the dimensionless form of model (6).For convenience, we still write  in place of ; the corresponding dimensionless form of model ( 6) is given as follows: where we denote  = /(),  = /(),  =   /(),  = ()/, and  =  V /().(7) Equilibria of model ( 7) are obtained by solving / = / = / = V/ = 0.It can be checked that model (7) has the following three boundary equilibria: the vanishing equilibrium  0 (0, 0, 0, 0); the disease-free equilibrium  1 (1, 0, 0, 0); and the predator-free equilibrium  2 (, , 0, V), where  = /,  = ( − )/( + ), and V = ( − )/( + ); when  > , the equilibrium  2 is nonnegative.The positive equilibrium  * ( * ,  * ,  * , V * ) is obtained by solving the following equations:

Equilibria of System
From the fourth equation of the above equations, we have If  < , then the predator-free equilibrium  2 and the positive equilibrium  * do not exist, which implies the virus replication parameter  is too small to support the virus invasion in insect pest.When the virus replication parameter  is larger than  and very close to , then the predator-free equilibrium  2 will collapse to the disease-free equilibrium  1 .Thus it can be seen that the virus replication parameter  plays an important role for the dynamics of system (7).In summary, we have the following theorem about the equilibria of system (7).

Local Stability and Hopf Bifurcation
The dynamical behaviour of equilibria can be studied by computing variational matrices corresponding to each equilibrium.The variational matrix of ( 7) is For the vanishing equilibrium  0 , correspondingly, variational matrix is Obviously,  0 is unstable saddle point with threedimensional stable manifold and one-dimensional unstable manifold.
For the disease-free equilibrium  1 , the corresponding variational matrix is The characteristic equation is When 1 <  < , there are four real negative eigenvalues and consequently the disease-free equilibrium  1 is locally asymptotically stable; when  = , corresponding to the disease-free equilibrium  1 , there are a simple eigenvalue 0 and another three negative eigenvalues.The system (7) enters into a saddle-node bifurcation at  1 ; when  > , there are three real negative eigenvalues and one positive eigenvalue so the disease-free equilibrium  1 is unstable.
For the predator-free equilibrium  2 , the corresponding variational matrix is The characteristic equation is where  1 =  +  + ,  2 = ( +  + V), and The characteristic equation has clearly one negative real root, namely,  − , and another three roots are given by the equation and  3 can be looked as the function of ; since  = / and the predator-free equilibrium  2 exists if  > ,  ∈ (0, 1).
Obviously  is positive definite.The derivative of  along the solution of the equation Ẋ() =   * (), where () = ((), (), (), V())  , is as follows: The symmetric matrix corresponding to V is the following: The positive equilibrium  * is locally asymptotically stable if V is negative definite, which in turn follows if the symmetric matrix  is negative definite; that is, the odd rank principal minor in order is negative and the even rank principal minor in order is positive, which in turn follows if We choose  2 V * −  1  * = 0,  3  * − ( 2 /) * = 0, and  4  −  2  * = 0; then from the above expression, we know if that is,  < (/ * )( +  * /), for the symmetric matrix , the odd rank principal minor in order is negative and the even rank principal minor in order is positive.That is to say, if  < (/ * )( +  * /), the positive equilibrium  * is locally asymptotically stable.Furthermore, to find out the basin of attraction, we choose the Lyapunov function as follows: Differentiating  along the solution of the model and choosing  1 = /V * , we can obtain the stable manifold The proof is completed.Now, we will find out the conditions for which the equilibrium  * enters into Hopf bifurcation.The characteristic equation for  * is where  The existence of  * can be obtained by solving Ψ() = 0.
In the next section we will consider the stability of bifurcating periodic orbits applying Poore's condition [19].

Stability of the Bifurcating Periodic Solution
We apply Poore's condition for verification of the orbital stability of the Hopf bifurcating limit cycle.For convenience, we list the preliminaries firstly.Let a real, -dimensional ( ≥ 2), first-order system of autonomous differential equations be of the form where ] = (] 1 , ] 2 , . . ., ]  )  denotes a vector of -real parameters.There must firstly exist a combination of the parameters, say, ] 0 ; a critical point  0 , such that the variational matrix F ( 0 , ] 0 ) has exactly two, nonzero, purely imaginary eigenvalues, say, ± 0 with  0 > 0; and other −2 eigenvalues with nonzero real parts.To vary one, some, or all of the  parameters, an -dimensional vector function () is introduced with the property that (0) = 0, and hence we confine our analysis to the following system of ODE: It follows from the definition of (, ) that ( 0 , 0) = 0 and eigenvalues of   ( 0 , 0) = 0 are the same as those of F( 0 , ] 0 ).
It is assumed that (, ) ∈   [×(− 0 ,  0 )], where  ≥ 3 and  is a domain in R  containing  0 and  0 > 0. As there is no nonzero eigenvalue of the variational matrix,   ( 0 , 0) ̸ = 0 ∈ R  , and hence the implicit function theorem guarantees the existence of a critical point   which is -times continuously differentiable in  and satisfies (  , ) = 0 for  in a small neighborhood of  = 0. Using this definition of   , a change of variables is introduced: This reduces the differential (25) to the following form: where  0 =   ( 0 , 0) and  is defined as given in [19].Thus, the problem of periodic solutions of ( 25) is reduced to a perturbation problem in the small parameter of   .Now, the stability information of the bifurcating periodic orbits is contained in the following theorem of Poore, coupled with an algebraic expression, which completely reduces the determination of stability to an algebraic problem.By the assumptions of Theorem 2.3 [19], the differential equation in ( 27) is continuously differentiable in   , where  = (  ) and  = (  ), and in the function  in a neighborhood of the periodic orbit.Thus, the existing periodic orbit will be asymptotically orbitally stable with asymptotic phase if  − 1 of the characteristic multipliers of the variational equation have moduli less than one.The following theorem developed by Poore about the modulus of each of the characteristic multipliers reduces the condition of orbital stability of the Hopf bifurcating limit cycle to an algebraic expression as follows.
Written out in component form, the above expression reduces to where the repeated indices within each term imply a sum from 1 to  and all the derivatives of  are evaluated at the equilibrium  0 .The sign of the real and imaginary parts of the right-hand side of expression in Theorem 4 is independent of the choice of (  ) in ⋃ (,   ) = Φ()(  ) and the eigenvectors  and V so long as (0) ⋅ (0) > 0 and V = 1.So positivity of the real part of the above expression in parenthesis really indicates the orbital stability of the periodic solution arising out of Hopf bifurcation.
For model (7) we considered, let the right four equations be  1 = (1 −  − ) − V,  2 = V − (1/) − ,  3 = (−−+), and  4 = −V+, respectively.To verify that the conditions of Theorem 4 are satisfied, we will calculate the second-and the third-order derivatives of   ,  = 1, 2, 3, 4, as follows: Other second-order and third-order derivatives are all zero.The inverse matrix  of   * − 2 0 and the left and right eigenvectors  and V, respectively, in the sense that V = 1, of the variational matrix   * can be calculated.Based on the preliminary work, we can calculate expression (29) in parenthesis as follows: where Putting the values of   ,   , , and V and components of the matrix  in terms of the parameters of the model, the positivity of the real part of expression (29) in parenthesis can be deduced.This in turn indicates the orbital stability of the limit cycle arising out of Hopf bifurcation.

Extinction and Permanence with Impulsive Effect
In the sections ahead, we focus our attention on model ( 6) or (7), in which the growth of the predator and the virus is continuous.But in fact, it is needed to throw in the predator or virus according to the pest control at some moments for better effects.So in this section, we develop model ( 6) into an impulsive system as follows and the presence of impulses gives the system a mixed nature, both continuous and discrete: where the predator and virus are thrown at ,  = 0, 1, 2, . ..,  is the period, and ,  are the amount of impulsive stocking on the predator and the virus; ,  are positive constants.
For the system we have the following lemma.
The pest-eradication solution (0, 0,  * (),  * ()) is globally asymptotically stable when  < / V ; that is to say, the pest population is eradicated totally.But in practice, we only need to control the pest population under the economic threshold level (ETL), instead of eradicating it totally, and hope the pest population, the predator (natural enemy), and the virus population can coexist when the pests do not bring about immense economic losses.As for this, we have the uniform permanence of system (33); see the next theorem.Theorem 9.If  > / V , then system (33) is uniformly permanent.
The proofs of the lemmas and the theorems listed in this section are all omitted here since they are similar to Hui and Zhu [21].
Figure 5 is the bifurcation graph of system (7).The virus replication factor  is chosen as the bifurcation parameter.When 1 <  < 8.866, there is a stable positive equilibrium  * ; when  > 8.866, the positive equilibrium  * is unstable and the Hopf bifurcation appears at  = 8.86; the stable periodic solution appears.Figure 6 illustrates the bifurcation phenomenon of the impulsive system (33).When 1 <  < 14, there exists a stable periodic solution; when  > 14, the system (33) enters into the chaos state.

Conclusions
In this paper, from Sections 1 to 5 we focus our attention on an epidemic-species hybrid system without impulse influence.From our analyses, we found the virus replication factor  plays a very important role for dynamics of system.The system always has two equilibria, namely, the vanishing equilibrium  0 (0, 0, 0, 0) and the disease-free equilibrium  1 (1, 0, 0, 0).When the virus replication factor  is larger than its natural death rate , that is,  > , the predatorfree equilibrium appears; furthermore, the virus replication factor  satisfies  >  and / < (( − ) +  2 )/( +  2  2  +  2 ) < /( + ); the positive equilibrium  * exists.The vanishing equilibrium  0 (0, 0, 0, 0) is a saddle point with three-dimensional stable and onedimensional unstable manifold.Biologically, it means that in the absence of prey, that is, the insect pest, the predator and virus population becomes extinct.When 1 <  < , the disease-free equilibrium  1 is locally asymptotically stable; it means that if the virus replication parameter or virulence of the applied virus on the system is less than its natural death rate , then only  viruses obtained from lysis of the infected pest are not enough to maintain the viral infection in the pest population.When  = , the system passes through saddle-node bifurcation and for value of  higher than , the system gives two branches of equilibria, namely, the diseasefree equilibrium  1 and the predator-free equilibrium  2 , one hyperbolic equilibrium with three-dimensional stable manifold and one nonhyperbolic equilibrium, respectively; that is, for  > ,  1 becomes unstable and  2 becomes feasible.Biologically, for spreading of infection into the system, the virus replication factor  must be greater than .When  > , the predator-free equilibrium  2 is feasible and stable.For higher value of , there exists a  0 which Horizontal axis represents the bifurcation parameter  and vertical axis represents the positive equilibrium state of the susceptible pest; the full line represents stable equilibrium state and the dotted line represents unstable equilibrium state; the small dots represent the maximum and the minimum amplitude of limit cycle bifurcating from the positive equilibrium and the Hopf bifurcation appears at  = 8.86.