Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays

To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0 > 1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and centermanifold theorem, the explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solutions are derived. Some numerical simulations are given to confirm our analytic results.


Introduction
Plants are very important not only to man's survival but to every species on Earth; however, plants may contract a disease by many different ways.Tremendous crop losses and global threat to food security have been caused by plant diseases [1,2].In recent years, plant diseases have attracted the interest of many mathematical modeling researchers and epidemiologists [3][4][5][6][7].
Mathematical models provide powerful tools for investigating how an infection propagates within a population of plants.Shi et al. [8] have proposed an epidemic model which describes vector-borne plant diseases, and the global dynamics of the system have been analyzed in terms of the basic reproduction number.Luo et al. [9] studied a discrete plant virus disease model with roguing and replanting; they proved that the basic reproduction number serves as a threshold parameter in determining the global dynamics of the model.The plant diseases epidemic models have been extensively studied by many authors (see [10][11][12][13][14]).
In [15], a delay differential equations was proposed to model the interaction between plants, a plant virus, and the insect-vector that transfers the virus from one plant to another.Since insects can only bite a limited number of plants, then the interaction between vector and plant is of predator-prey Holling type II [16].In order to consider the time it takes for the virus to spread throughout the plant or insect-vector, a couple of delays were introduced to the model (see [15] for more details).They obtained the following model: where the state variables (), (), and () represent the number of susceptible, infected, and recovered plants at time , respectively.Because when a plant dies by the virus or natural death in farms, it is replaced with a new healthy plant.
The new plant shares the same characteristics of the plant it replaced, before it was infected.Then it is supposed that the total number of plants stabilizes at ,  =  +  + . is the natural death rate of plants;  is the infection rate of plants due to vectors;  is the saturation constant of plants due to vectors;  is the death rate of infected plants due to the disease;  is the recovery rate of plants.The insect-vectors are divided into two populations: susceptible and infective denoted by () and (), respectively.The total number of insects is denoted by (), and then () = () + ().Λ is the replenishing rate of vectors (birth and/or immigration);  1 is the infection rate of vectors due to plants;  1 is the saturation constant of vectors due to plants;  is the natural death rate of vectors. 1 is the time it takes a plant to become infected after contagion, and  2 is the time it takes a vector to become infected after contagion.Notice that and then () → Λ/ as  → ∞.Thus, one can consider the following reduced system: where  =  +  + .
For model (3), Jackson and Chen-Charpentier [15] gave the basic reproduction number and found the equilibria of the model, and then they studied the stability of equilibria only for particular values of the parameters using numerical methods.Therefore, in this paper, we reconsider the plant disease model (3) in theoretical aspects, and we establish the stability of equilibria, the existence of Hopf bifurcation, and the stability, direction, and other properties of bifurcating periodic solution will also be discussed.
This paper is organized as follows.In Section 2, we discuss the stability of the equilibria and the existence of the Hopf bifurcations occurring at the endemic equilibrium.In Section 3, the formulae determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold are obtained by using the normal form theory and the center manifold theorem by Hassard et al. [17].In Section 4, we perform numerical simulations to illustrate the analytical results.We conclude with a brief discussion in Section 5.
Let Ω be the following subset of R 3 + : Using a proof process similar to that in [19,20], we obtain the following lemma.
Lemma 1.The solutions of system (3) which satisfy the initial conditions (4) are positive.The set Ω is positively invariant.
We now consider the local stability of the coexistence equilibrium  * ( * ,  * ,  * ) and the existence of Hopf bifurcation at  * .The linearized system of (3) at  * is given by with Therefore, we obtain the following characteristic equation: ( Case 1 ( 1 =  2 = 0).Characteristic equation ( 13) becomes where Note that then we get and thus Since , it is easy to show that  11  12 − 10 > 0, and, thus, all roots of (15) have negative real parts.That is,  * ( * ,  * ,  * ) is locally asymptotically stable.Actually, by a similar proof as in [23], we can show that  * ( * ,  * ,  * ) is globally asymptotically stable for  1 =  2 = 0.
).We consider (3) with  2 in its stable interval and regard  1 as a parameter.Let  =  1 ( 1 > 0) be a root of (13), separating real and imaginary parts, and we have the following: where From (26), we have where We make the following assumption.
In addition to (41), we further assume that Therefore, by the Hopf bifurcation theorem for functional differential equations in Hale [18], the following result holds.

Discussion
In this paper, we have studied the dynamics of a plant virus propagation model with two delays (3) proposed by Jackson and Chen-Charpentier [15].The model describes the disease transmission dynamics between the insects and the plants.
Jackson and Chen-Charpentier [15] studied model (3) using numerical methods.However, the problem of the theoretical analysis of this model remained unsolved and was an open problem.
For this problem, first, by analyzing the characteristic equation, constructing a Lyapunov functional, and using LaSalle's invariance principle, we prove that the disease-free equilibrium  0 is globally asymptotically stable if  0 ≤ 1 (Theorem 2), regardless of the length of the time delays, the sufficient conditions for the stability of the endemic equilibrium, and existence of Hopf bifurcation if  0 > 1 have been given, respectively.Then, by using normal form theory and center manifold theorem introduced by Hassard et al. [17], regarding  1 as a parameter, we investigate the direction and stability of the Hopf bifurcation, and the explicit formulae which determine the direction and stability of the bifurcating periodic solutions are derived.Finally, the numerical simulation results in Figures 1-4 have verified the obtained analytic results.
Our simulation results show that, for the parameter values considered, the disease will persist and exhibit oscillatory bahavior, and this manifests that the densities of the plants and insect-vectors will remain in an oscillatory case, and then  agriculture workers must be alert to the virus even if they have noticed that fewer plants are becoming infected.