Dynamical Analysis of Delayed Plant Disease Models with Continuous or Impulsive Cultural Control Strategies

Abstract

Delayed plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are presented and investigated. Firstly, we consider continuous cultural control strategy in which continuous replanting of healthy plants is taken. The existence and local stability of disease-free equilibrium and positive equilibrium are studied by analyzing the associated characteristic transcendental equation. And then, plant disease model with impulsive replanting of healthy plants is also considered; the sufficient condition under which the infected plant-free periodic solution is globally attritive is obtained. Moreover, permanence of the system is studied. Some numerical simulations are also given to illustrate our results.

1. Introduction

Plant viruses or pathogens are an important constraint to crop production worldwide and cause major production and economic losses in agriculture and forestry. For example, soybean rust (a fungal disease in soybeans) has caused a significant economic loss, and just by removing 20% of the infection, the farmers may benefit with an approximately 11 million-dollar profit [1]. Several plant diseases caused by plant viruses in cassava (Manihot esculenta) and sweet potato (Ipomoea batatas) are among the main constraints to sustainable production of these vegetatively propagated staple food crops in lesser-developed countries [2–4]. A strain of the virus causing cassava mosaic disease gives rise to losses in Africa [5]. Therefore, people have turned more attention to plant diseases. Several conferences have been held to discuss how to control or prevent plant virus. Therefore, farmers have been evolving practices for controlling plant diseases, which involves a number of dynamic processes such as the growth of plants and the spread of diseases. Recently, the integrated disease management (IDM) which combines biological, cultural, and chemical tactics and so on to reduce the numbers of infected individuals to a tolerable level and aims to minimize losses and maximize returns [6, 7] has been developed gradually. IDM includes four main control strategies for vegetatively propagated plant diseases, which are containing transmission vectors, improving the production of planting material, controlling the crop sanitation through removal of infected plants, and breeding plants for resistance to the virus. Breeding plants for resistance to the virus as an cultural strategy has been widely used into practice [8–11]. In the system of IDM, mathematical modeling has shown its unique value on describing, analyzing, and predicting plant epidemics [12–16]. Meng and Li have investigated vegetatively propagated plant diseases and developed a mathematical model with continuous control strategies and impulsive cultural control strategies [17], which leads to𝑆^′(𝑡)(𝑡)𝐼(𝑡)=𝜌−𝛽𝑆(𝑡)(𝑡)(𝑡),𝐼1+𝛼𝑆−𝜇𝑆+𝜔𝐼_′(𝑡)=𝛽𝑆(𝑡)𝐼(𝑡)(𝑡)−()𝐼(𝑡),1+𝛼𝑆𝑑+𝑟+𝜔 (1.1) where 𝑆(𝑡)and𝐼(𝑡) denote the number of susceptible and infected plants, respectively. 𝛽 is the transmission rate, 𝛼 denotes potentially density dependent, 𝜇 either denotes harvest time or the end of reproductive life time of plants, 𝜌 represents the total rate at which the susceptible plants enter the system, 𝑟 is the removal rate for the infected plants, 𝜔 is the recovery rate of the cured diseased plants, and the infected plants suffer an extra disease-related death with constant rate 𝑑. In system (1.1), the authors refer to two-control strategy: one is continuous control and the other is impulsive control by implementing periodic replanting of healthy plants or removing infected plants at a critical time. A model for the spread of an infectious disease (involving only susceptibles and infective individuals) transmitted by a vector after an incubation time was proposed by Cooke [18]. This is called the phenomena of time delay. Many authors have directly incorporated time delays in modeling equations, and, as a result, the models take the form of delay differential equations [19–23]. Motivated by Meng, we get the following reasonable plant disease models by introducing time delay:𝑆^′(𝑡)=𝜌−𝛽𝑒^−𝜇𝜏𝑆(𝑡)𝐼(𝑡−𝜏)(𝑡)(𝑡)(𝑡),𝐼1+𝛼𝑆−𝜇𝑆+𝜔𝐼_′(𝑡)=𝛽𝑒^−𝜇𝜏𝑆(𝑡)𝐼(𝑡−𝜏)(𝑡)−()𝐼(𝑡),𝑆1+𝛼𝑆𝑑+𝑟+𝜔_′(𝑡)=−𝛽𝑒^−𝜇𝜏𝑆(𝑡)𝐼(𝑡−𝜏)(𝑡)(𝑡)(𝑡),𝐼1+𝛼𝑆−𝜇𝑆+𝜔𝐼_′(𝑡)=𝛽𝑒^−𝜇𝜏𝑆(𝑡)𝐼(𝑡−𝜏)(𝑡)−()𝐼(𝑡),𝑆𝑡1+𝛼𝑆𝑑+𝑟+𝜔𝑡≠𝑛𝑇,₊(𝑡)𝐼𝑡=𝑆+𝜌,₊=𝐼(𝑡),𝑡=𝑛𝑇. (1.2)(1.3)

From the point of biology, we only consider system (1.2) and (1.3) in the biological meaning region: 𝐷={(𝑆,𝐼)∣𝑆,𝐼≥0}. Let 𝐶₊=𝜙=𝜙₁(𝑠),𝜙₂(𝑠)∈𝐶∶𝜙_𝑖(0)>0(𝑖=1,2), (1.4) where 𝜙_𝑖(𝑠) is positive, bounded, and continuous function for 𝑠∈[−𝜏,0]. Motivated by the application of systems (1.2) and (1.3) to population dynamics (refer to [24]), we assume that solutions of systems (1.1) satisfy the following initial conditions: 𝜙∈𝐶₊,𝜙(0)>0. (1.5)

2. Plant Disease Continuous Control for System (1.2)

In this section, we consider system (1.2) with continuous replanting and removing and without impulsive effect. By Smith [25, Theorem??5.2.1] or Zhao and Zou [26], for any 𝜙∈𝐶₊, there is a unique solution (𝑆(𝑡,𝜙),𝐼(𝑡,𝜙)) of system (1.2) with (𝑆(𝜁,𝜙),𝐼(𝜁,𝜙))=𝜙(𝜁), for any 𝜁∈[−𝜏,0] and 𝑆(𝜁,𝜙)≥0, 𝐼(𝜁,𝜙)≥0 for all 𝑡≥0 in its maximal interval of existence.

Define 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡); then we have 𝑑𝑁(𝑡)(𝑡)𝑆(𝑡)+()𝐼(𝑡).𝑑𝑡+𝐿𝑁=𝜌+𝐿−𝜇𝐿−𝑑−𝑟 (2.1) Let 𝐿=min{𝜇,𝑑+𝑟}; we have 𝑑𝑁(𝑡)(𝑡)𝑑𝑡+𝐿𝑁≤𝜌. (2.2) Then 𝑁(𝑡)≤𝑁(0)𝑒^−𝐿𝑡+𝜌𝐿1−𝑒^−𝐿𝑡⟶𝜌𝐿=𝑀 (2.3) as 𝑡→+∞. Hence, system (1.2) is uniformly bounded.

Since 𝑆(𝑡) and 𝐼(𝑡) denote the number of susceptible and infected plants, respectively, it is easy to observe that system (1.2) has a disease-free of the form 𝐸₁(𝜌/𝜇,0), and a unique infection equilibrium 𝐸₂(𝑆^∗,𝐼^∗) provided that we have the following condition:(H)𝛽𝑒^−𝜇𝜏>𝜇𝛼+𝜌(𝑑+𝑟+𝜔), (2.4) where𝑆^∗=𝑑+𝑟+𝜔𝛽𝑒^−𝜇𝜏−𝛼(𝑑+𝑟+𝜔),𝐼^∗=𝜌−𝜇𝑆^∗.𝑑+𝑟 (2.5)

2.1. The Stability of the Disease-Free Equilibrium 𝐸₁(𝜌/𝜇,0)

We may firstly consider the stability of the disease-free equilibria 𝐸₁(𝜌/𝜇,0). Let 𝑥(𝑡)=𝑆(𝑡)−𝜌/𝜇, 𝑦(𝑡)=𝐼(𝑡); then system (1.2) can be rewritten as the following equivalent system:𝑥^′(𝑡)=𝜌−𝛽𝑒^−𝜇𝜏𝑥(𝑡)+𝜌/𝜇𝑦(𝑡−𝜏)𝑥(𝑡)𝑥(𝑡)1+𝛼+𝜌/𝜇−𝜇+𝑆^∗(𝑡),𝑦+𝜔𝑦_′(𝑡)=𝛽𝑒^−𝜇𝜏𝑥(𝑡)+𝜌/𝜇𝑦(𝑡−𝜏)𝑥(𝑡)−()𝑦(𝑡).1+𝛼+𝜌/𝜇𝑑+𝑟+𝜔 (2.6) Thus, the disease-free equilibrium 𝐸₁(𝜌/𝜇,0) of system (1.2) is transformed into zero equilibrium of system (2.6). Linearizing system (2.6) about the equilibrium (0, 0) yields the following linear system:𝑥^′(𝑡)=−𝜇𝑥(𝑡)−𝛽𝜌𝑒^−𝜇𝜏𝑦()(𝑡),𝑦𝜇+𝛼𝜌𝑡−𝜏+𝜔𝑦_′(𝑡)=𝛽𝜌𝑒^−𝜇𝜏𝑦()−()𝑦(𝑡),𝜇+𝛼𝜌𝑡−𝜏𝑑+𝑟+𝜔 (2.7) with characteristic equation:det−𝜇−𝜆𝜔−𝛽𝜌𝑒^−𝜇𝜏𝑒𝜇+𝛼𝜌^−𝜆𝜏0𝛽𝜌𝑒^−𝜇𝜏𝑒𝜇+𝛼𝜌^−𝜆𝜏−(𝑑+𝑟+𝜔)−𝜆=0, (2.8) that is, −𝜇−𝜆𝛽𝜌𝑒^−𝜇𝜏𝑒𝜇+𝛼𝜌^−𝜆𝜏−(𝑑+𝑟+𝜔)−𝜆=0. (2.9)

The stability of trivial solution of system (1.2) depends on the locations of roots of characteristic equation (2.9). When all roots of (2.9) locate in the left half-plane of complex plane, the trivial solution of system (1.2) is stable; otherwise, it is unstable. In the following, we will investigate the distribution of roots of (2.9). Obviously, 𝜆₁=−𝜇<0. Let𝛽𝜌𝑒^−𝜇𝜏𝑒𝜇+𝛼𝜌^−𝜆𝜏−(𝑑+𝑟+𝜔)−𝜆=0. (2.10) For (2.10), the root of (2.10) with 𝜏=0 always has negative real part provided that 𝛽𝜌𝑒^−𝜇𝜏/(𝜇+𝛼𝜌)<𝑑+𝑟+𝜔.

In addition, 𝑖𝜛(𝜛>0) is a root of (2.10) if and only if 𝜛 satisfies the following equation:𝑖𝜛=𝛽𝜌𝑒^−𝜇𝜏()−().𝜇+𝛼𝜌cos𝜛𝜏−𝑖sin𝜛𝜏𝑑+𝑟+𝜔 (2.11)

Separating the real and imaginary parts of (2.11) gives the following equations:𝛽𝜌𝑒^−𝜇𝜏𝜇+𝛼𝜌cos𝜛𝜏=𝑑+𝑟+𝜔,𝛽𝜌𝑒^−𝜇𝜏𝜇+𝛼𝜌sin𝜛𝜏=−𝜛. (2.12)

Thus we can have𝜛²=𝛽𝜌𝑒^−𝜇𝜏𝜇+𝛼𝜌²−(𝑑+𝑟+𝜔)². (2.13) Then if 𝛽𝜌𝑒^−𝜇𝜏/(𝜇+𝛼𝜌)<𝑑+𝑟+𝜔, (2.13) has not positive real root, which leads to (2.10) that has not purely imaginary root. By the Rouche Theory, we know that all the roots of (2.9) have always negative real parts. So the equilibrium 𝐸₁(𝜌/𝜇,0) of system (1.2) is locally asymptotically stable.

Define 𝜏₁=1𝜇𝛽ln().𝛼+𝜇/𝜌𝑑+𝑟+𝜔 (2.14) For system (1.2), we have the following result on stability of the disease-free equilibrium 𝐸₁(𝜌/𝜇,0).

Theorem 2.1. For system (1.2), the following statements are true.(i)If 𝜏∈[0,𝜏₁), then the disease-free equilibrium 𝐸₁(𝜌/𝜇,0) of system (1.2) is unstable.(ii)If 𝜏∈(𝜏₁,+∞), then the disease-free equilibrium 𝐸₁(𝜌/𝜇,0) of system (1.2) is locally asymptotically stable.

2.2. The Stability of the Positive Equilibrium 𝐸₂(𝑆^∗,𝐼^∗) of System (1.2)

In this section, we show that the disease equilibrium is asymptotically stable in the case that time delay 𝜏 is less than the unity(𝜏<𝜏₁); then we have the following theorem.

Theorem 2.2. For system (1.2), if 𝜏∈[0,𝜏₁), then the positive equilibrium 𝐸₂(𝑆^∗,𝐼^∗) of system (1.2) is asymptotically stable.

Proof. Under the hypothesis (H), let 𝑥(𝑡)=𝑆(𝑡)−𝑆^∗,𝑦(𝑡)=𝐼(t)−𝐼^∗, then system (1.2) can be rewritten as the following equivalent system: 𝑥^′(𝑡)=𝜌−𝛽𝑒^−𝜇𝜏𝑥(𝑡)+𝑆^∗𝑦(𝑡−𝜏)+𝐼^∗1+𝛼(𝑥(𝑡)+𝑆^∗)−𝜇𝑥(𝑡)+𝑆^∗+𝜔𝑦(𝑡)+𝐼^∗,𝑦^′(𝑡)=𝛽𝑒^−𝜇𝜏𝑥(𝑡)+𝑆^∗𝑦(𝑡−𝜏)+𝐼^∗1+𝛼(𝑥(𝑡)+𝑆^∗)−(𝑑+𝑟+𝜔)𝑦(𝑡)+𝐼^∗. (2.15)
Thus, the positive equilibrium 𝐸^∗(𝑆^∗,𝐼^∗) of system (1.2) is transformed into zero equilibrium of system (2.15). Linearizing system (2.15) about the equilibrium (0, 0) yields the following linear system:𝑥^′(𝑡)=−𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇𝑥(𝑡)−𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗𝑦()(𝑡),𝑦𝑡−𝜏+𝜔𝑦_′(𝑡)=𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²𝑥(𝑡)+𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗𝑦(𝑡−𝜏)−(𝑑+𝑟+𝜔)𝑦(𝑡), (2.16) with characteristic equation: −det𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇−𝜆𝜔−𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗𝑒^−𝜆𝜏𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗𝑒^−𝜆𝜏−(𝑑+𝑟+𝜔)−𝜆=0. (2.17) The stability of trivial solution of system (1.2) depends on the locations of roots of characteristic equation (2.17). For the sake of simplicity, let 𝑝=𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇+𝑑+𝑟+𝜔>0,𝑞=−𝜇𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗=−𝜇(𝑑+𝑟+𝜔)<0,𝑔=−𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗()()=−𝑑+𝑟+𝜔<0,𝑠=𝑑+𝑟𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇+𝜇𝜔>0. (2.18) Then (2.17) can be briefly denoted as the following equation: 𝜆²+𝑝𝜆+𝑞𝑒^−𝜆𝜏+𝑔𝜆𝑒^−𝜆𝜏+𝑠=0. (2.19) For (2.19), we can claim that the two roots of (2.19) have always negative real parts. We will prove it in the following two steps.Step 1. If 𝜏=0, (2.19) can be simplified as 𝜆²+𝑝+𝑔𝜆+𝑞+𝑠=0. (2.20) Note that 𝑝+𝑔=𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇+𝑑+𝜔−(𝑑+𝜔)=𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²()+𝜇>0,𝑞+𝑠=−𝜇𝑑+𝜔+𝑑𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇+𝜇𝜔=𝑑𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²>0. (2.21) Therefore, the two roots of (2.19) with 𝜏=0 have always negative real parts.Step 2. If 𝜏>0,𝑖𝜔(𝜔>0) is a root of (2.15) if and only if 𝜔 satisfies the following equation: −𝜔²+𝑝𝜔𝑖+𝑞(cos𝜔𝜏−𝑖sin𝜔𝜏)+𝑟𝜔𝑖(cos𝜔𝜏−𝑖sin𝜔𝜏)+𝑠=0. (2.22)
Separating the real and imaginary parts of (2.22) gives the following equations: 𝑠−𝜔²+𝑞cos𝜔𝜏+𝑟𝜔sin𝜔𝜏=0,𝑝𝜔−𝑞sin𝜔𝜏+𝑟𝜔cos𝜔𝜏=0. (2.23) One can obtain 𝜔⁴+𝑝²−𝑟²−2𝑠𝜔²+𝑠²−𝑞²=0. (2.24) We can easily see that 𝑝²−𝑟²−2𝑠=𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇+𝑑+𝜔²−(𝑑+𝜔)²−2𝑑𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²=+𝜇+𝜇𝜔2𝜔𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²𝑠>0,₂−𝑞²=𝑑𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+𝜇+𝜇𝜔²−𝜇(𝑑+𝜔)²=𝑑𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²𝑑𝛽𝑒^−𝜇𝜏𝐼^∗(1+𝛼𝑆^∗)²+2𝜇(𝑑+𝜔)>0. (2.25) So (2.24) has not positive real root, which leads to (2.19) that has not purely imaginary root. By the Rouche Theory, we know that all the roots of (2.19) have always negative real parts. So the equilibrium 𝐸₂(𝑆^∗,𝐼^∗) of system (1.2) is asymptotically stable. The proof is complete.

3. Plant Disease Impulsive Control for System (1.3)

3.1. Boundedness

Let the initial data be 𝑆(0)>0,𝐼(0)>0. Then, one can easily prove that the solutions (𝑆(𝑡),𝐼(𝑡)) of system (1.3) are positive for all 𝑡>0. Now, let 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡). We calculate the upper right derivative of 𝑁(𝑡) along with a solution of system (1.3) with 𝑡≠𝑛𝑇:𝑑𝑁(𝑡)(𝑡)−()𝐼(𝑡).𝑑𝑡=−𝜇𝑆𝑑+𝑟 (3.1) Since 𝜇,𝑑,𝑟>0, one can deduce that𝑑𝑁(𝑡)(𝑡),𝑑𝑡≤−𝐿𝑁 (3.2) where 𝐿=min{𝜇,𝑑+𝑟}. We consider the following impulse differential inequalities:𝑑𝑁(𝑡)(𝑡)𝑁𝑑𝑡≤−𝐿𝑁,𝑡≠𝑛𝑇,𝑛𝑇⁺=𝑁(𝑛𝑇)+𝜌,𝑡=𝑛𝑇. (3.3) According to impulse differential inequalities theory, we get𝑁(𝑡)≤𝑁(0)𝑒^−𝐿𝑡+_{0<𝑛𝑇<𝑡}𝜌𝑒^{−𝐿(𝑡−𝑛𝑇)}⟶𝜌𝑒^𝐿𝑇𝑒^𝐿𝑇−1 (3.4) as 𝑡→+∞.

So 𝑁(𝑡) is uniformly ultimately bounded. Hence, by the definition of 𝑁(𝑡), for any 𝜀>0, there exists a constant 𝑀′=𝜌𝑒^𝐿𝑇/(𝑒^𝐿𝑇−1)+𝜀 such that 𝑆(𝑡)<𝑀^′ and 𝐼(𝑡)<𝑀′ for each solution of (1.3) with 𝑡 being large enough.

3.2. Global Attractivity of the Disease-Free Periodic Solution of System (1.3)

In the following, we shall prove that the disease-free periodic is stable if it exists. For this purpose, we give firstly some basic properties of the following subsystem:𝑆^′(𝑡)(𝑡)𝑆𝑡=−𝜇𝑆,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝜌,𝑡=𝑛𝑇. (3.5) We can find a unique positive periodic solution 𝑆^∗(𝑡)=𝜌𝑒^{−𝜇(𝑡−𝑛𝑇)}/(1−𝑒^−𝜇𝑇),𝑛𝑇<𝑡≤(𝑛+1)𝑇, which is globally asymptotically stable by using stroboscopic map. As a consequence, system (3.5) always has a disease-free periodic solution (𝑆^∗(𝑡),0). Now, we give the conditions which assure the global attractivity of disease-free periodic solution of the system (1.3).

Denote 𝐴=𝜔𝑀′𝜇+𝜌1−𝑒^−𝜇𝑇,ℜ₁=𝐴𝛽𝑒^−𝜇𝜏(𝑑+𝑟+𝜔)(1+𝛼𝐴). (3.6)

Theorem 3.1. The disease-free periodic solution (𝑆^∗(𝑡),0) of system (1.3) is globally attractive provided that ℜ₁<1. (3.7)

Proof. Let (𝑆(𝑡),𝐼(𝑡)) be any solution of system (1.3). From the first equation of system (1.3), it follows that 𝑆(̇𝑡)<−𝜇𝑆(𝑡)+𝜔𝑀′,𝑆(𝑡⁺)=𝑆(𝑡)+𝜌, for 𝑛𝑇<𝑡≤(𝑛+1)𝑇; then we consider the following impulse differential system: 𝑧̇(𝑡)=−𝜇𝑧(𝑡)+𝜔𝑀^′𝑧𝑡,𝑡≠𝑛𝑇,₊=𝑧(𝑡)+𝜌,𝑡=𝑛𝑇. (3.8) Obviously, system (3.8) has a globally asymptotically stable positive periodic solution: 𝑧^∗(𝑡)=𝜔𝑀^′𝜇+𝜌𝑒^{−𝜇(𝑡−𝑛𝑇)}1−𝑒^−𝜇𝑇,𝑡∈(𝑛𝑇,(𝑛+1)𝑇]. (3.9) By the comparison theorem in impulsive differential equation, for any sufficiently small positive 𝜀, there exists an integer 𝑛₁ such that 𝑆(𝑡)<𝑧^∗(𝑡)+𝜀<𝜔𝑀′𝜇+𝜌1−𝑒^−𝜇𝑇+𝜀=𝐴+𝜀,𝑛>𝑛₁. (3.10) Therefore, from the second equation of system (1.3), we have 𝐼̇(𝑡)≤𝛽𝑒^−𝜇𝜏(𝐴+𝜀)()𝐼()−()𝐼(𝑡)1+𝛼𝐴+𝜀𝑡−𝜏𝑑+𝑟+𝜔,𝑡>𝑛₁𝑇+𝜏. (3.11) Now we consider the following comparison equation: 𝑅̇(𝑡)=𝛽𝑒^−𝜇𝜏(𝐴+𝜀)()𝑅()−()𝑅(𝑡)1+𝛼𝐴+𝜀𝑡−𝜏𝑑+𝑟+𝜔,𝑡>𝑛₁𝑇+𝜏. (3.12) Since ℜ₁<1, we have 𝛽𝑒^−𝜇𝜏𝐴1+𝛼𝐴<𝑑+𝑟+𝜔. (3.13) We may choose three sufficiently small positive constants 𝜀 such that 𝛽𝑒^−𝜇𝜏(𝐴+𝜀)1+𝛼(𝐴+𝜀)<𝑑+𝑟+𝜔. (3.14) According to the theory of delay differential equation [24], we obtain that lim_𝑡→+∞𝑅(𝑡)=0. By impulsive comparison theorem, we have 𝐼(𝑡)<𝑅(𝑡) with 𝑡 being large enough. Therefore, we obtain that lim_𝑡→+∞𝐼(𝑡)=0.
Then for a sufficiently small 𝜀₁>0 and all 𝑡 being large enough, we have 0<𝐼(𝑡)<𝜀₁. Without loss of generality, we may assume 0<𝐼(𝑡)<𝜀₁ as 𝑡≥0. From the first equation of system (1.3), we have𝑆̇(𝑡)>−𝛽𝑒^−𝜇𝜏𝜀₁𝑆(𝑡)𝑆𝑡+𝜇,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝜌,𝑡=𝑛𝑇. (3.15) Consider the following comparison system: 𝑦̇(𝑡)=−𝛽𝑒^−𝜇𝜏𝜀₁𝑦(𝑡)𝑦𝑡+𝜇,𝑡≠𝑛𝑇,₊=𝑦(𝑡)+𝜌,𝑡=𝑛𝑇. (3.16) Then, system (3.16) has a positive periodic solution: 𝑦^∗(𝑡)=𝜌𝑒^{−(𝛽𝑒−𝜇𝜏𝜀1+𝜇)(𝑡−𝑛𝑇)}1−𝑒^{−(𝛽𝑒−𝜇𝜏𝜀1+𝜇)𝑇}, (3.17) which is globally asymptotically stable. Thus, for a sufficiently small 𝜀>0, when 𝑡 is large enough, we have 𝑆(𝑡)>𝑦(𝑡)>𝑦^∗(𝑡)−𝜀. (3.18) From the first equation of system (1.3), we also have 𝑆̇(𝑡)<−𝜇𝑆(𝑡)+𝜀₁𝑆𝑡𝜔,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝜌,𝑡=𝑛𝑇. (3.19) Consider the following comparison system: 𝑥̇(𝑡)=−𝜇𝑥(𝑡)+𝜀₁𝑥𝑡𝜔,𝑡≠𝑛𝑇,₊=𝑥(𝑡)+𝜌,𝑡=𝑛𝑇. (3.20) System (3.20) has a globally asymptotically stable positive periodic solution: 𝑥^∗(𝑡)=𝜀₁𝜔𝜇+𝜌𝑒^{−𝜇(𝑡−𝑛𝑇)}1−𝑒^−𝜇𝑇,𝑡∈(𝑛𝑇,(𝑛+1)𝑇]. (3.21) Thus, for a sufficiently small 𝜀>0, when 𝑡 is large enough, we have 𝑆(𝑡)<𝑥(𝑡)<𝑥^∗(𝑡)+𝜀. (3.22) Combining (3.18) with (3.22), we obtain 𝑦^∗(𝑡)−𝜀<𝑆(𝑡)<𝑥^∗(𝑡)+𝜀. (3.23) Let 𝜀₁→0; we have 𝑆^∗(𝑡)−𝜀<𝑆(𝑡)<𝑆^∗(𝑡)+𝜀, which implies lim_𝑡→+∞𝑆(𝑡)=𝑆^∗(𝑡). The proof is completed.

3.3. Permanence of the System (1.3)

Definition 3.2. System (1.3) is said to be permanent if there exist constants 𝑘,𝐾>0 (independent of initial value) and a finite time 𝑇₀ such that for every positive solution (𝑆(𝑡),𝐼(𝑡)) with initial conditions (1.3) satisfies 𝑘<𝑆(𝑡)<𝐾,𝑘<𝐼(𝑡)<𝐾 for all 𝑡>𝑇₀. Here 𝑇₀ may depend on the initial condition.

Denote ℜ₂=𝜌𝛽𝑒^−𝜇𝜏−𝛼(𝑑+𝑟+𝜔)(𝑑+𝑟+𝜔)𝑒^𝜇𝑇,𝑆−1_∗=𝑑+𝑟+𝜔𝛽𝑒^−𝜇𝜏(),𝐼−𝛼𝑑+𝑟+𝜔_∗=(1/𝑇)ln𝑒^𝜇𝑇−1ℜ₂+2−𝜇𝛽𝑒^−𝜇𝜏. (3.24)

Lemma 3.3. If ℜ₂>1, then there exists a positive constant 𝑚₂ such that lim_𝑡→∞inf𝐼(𝑡)≥𝑚₂.

Proof. Define 𝑈(𝑡)=𝐼(𝑡)+𝛽𝑒^−𝜇𝜏𝑆^∗1+𝛼𝑆^∗𝑡_𝑡−𝜏𝐼𝜚𝑑𝜚. (3.25) Calculating the derivative of 𝑈(𝑡) along with the solution of (1.3), we can get 𝑈̇(𝑡)=𝐼̇+𝛽𝑒^−𝜇𝜏𝑆^∗1+𝑆^∗(𝐼(𝑡)−𝐼(𝑡−𝜏))=𝛽𝑒^−𝜇𝜏𝑆(𝑡)(𝑡)−𝑆1+𝛼𝑆1+𝛼^∗𝑆^∗𝐼(𝑡−𝜏) (3.26) for 𝑡≥0.
Since ℜ₂>1, then 𝐼^∗=((1/𝑇)ln((𝑒^𝜇𝑇−1)ℜ₂+2)−𝜇)/𝛽𝑒^−𝜇𝜏>((1/𝑇)ln(𝑒^𝜇𝑇+1)−𝜇)/𝛽𝑒^−𝜇𝜏>((1/𝑇)ln(𝑒^𝜇𝑇)−𝜇)/𝛽𝑒^−𝜇𝜏=0. Note that ℜ₂=𝜌/𝑆^∗(𝑒^𝜇𝑇−1) and 𝐼^∗=(1/𝑇)(ln((𝑒^𝜇𝑇−1)ℜ₂+2)−𝜇)/𝛽𝑒^−𝜇𝜏>((1/𝑇)ln((𝑒^𝜇𝑇−1)ℜ₂+1)−𝜇)/𝛽𝑒^−𝜇𝜏=((1/𝑇)ln(𝜌/𝑆^∗+1)−𝜇)/𝛽𝑒^−𝜇𝜏.
Solving the aforementioned inequality, we can have that 𝑆^∗<𝜌𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}1−𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}. (3.27)
We can choose 𝜀>0 being small enough such that𝑆^∗<𝜌𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}1−𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}−𝜀=𝑆_Δ. (3.28) For any positive constant 𝑡₀, we claim that the inequality 𝐼(𝑡)<𝐼^∗ cannot hold for all 𝑡≥𝑡₀. Otherwise, there exists a positive constant 𝑡₀ such that 𝐼(𝑡)<𝐼^∗ for all 𝑡≥𝑡₀. From the first equation of (1.3), we have 𝑆̇(𝑡)>−𝛽𝑒^−𝜇𝜏𝐼^∗𝑆(𝑡)𝑆𝑡+𝜇,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝜌,𝑡=𝑛𝑇. (3.29) Similarly, we know that there exists such 𝑇₁>𝑡₀+𝜏, for 𝑡>𝑇₁ that 𝑆(𝑡)>𝜌𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}1−𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}−𝜀=𝑆_Δ. (3.30) Then, by (3.30), we have that, for 𝑡>𝑇₁, 𝑈̇(𝑡)=𝐼̇+𝛽𝑒^−𝜇𝜏𝑆^∗1+𝑆^∗(𝐼(𝑡)−𝐼(𝑡−𝜏))>𝛽𝑒^−𝜇𝜏𝑆_Δ1+𝛼𝑆_Δ−𝑆^∗1+𝛼𝑆^∗𝐼(𝑡−𝜏). (3.31) Let 𝐼^𝑙=min_{𝑇𝑡∈1,𝑇1+𝜏}𝐼(𝑡). (3.32) We show that 𝐼(𝑡)≥𝐼^𝑙 for all 𝑡≥𝑇₁. Otherwise, there exists a nonnegative constant 𝑇₂ such that 𝐼(𝑡)≥𝐼^𝑙 for 𝑡∈[𝑇₁,𝑇₁+𝜏+𝑇₂],𝐼(𝑇₁+𝜏+𝑇₂)=𝐼^𝑙 and 𝐼(̇𝑇₁+𝜏+𝑇₂)≤0. Thus from the second equation of (1.3) and (3.30), we easily see that 𝐼̇𝑇₁+𝜏+𝑇₂>𝛽𝑒^−𝜇𝜏𝑆(𝑡)(𝑡)−()𝐼1+𝛼𝑆𝑑+𝑟+𝜔^𝑙=(𝑑+𝑟+𝜔)𝛽𝑒^−𝜇𝜏𝑆(𝑡)𝑑+𝑟+𝜔(𝑡)𝐼1+𝛼𝑆−1^𝑙>()𝑆𝑑+𝑟+𝜔^Δ/1+𝛼𝑆_Δ𝑆^∗/(1+𝛼𝑆^∗)−1𝐼^𝑙>0, (3.33) which is a contradiction. Hence we get that 𝐼(𝑡)≥𝐼^𝑙>0 for all 𝑡≥𝑇₁. Therefore, for all 𝑡>𝑇₁+𝜏, we have 𝑈̇(𝑡)>𝛽𝑒^−𝜇𝜏𝑆_Δ1+𝛼𝑆_Δ−𝑆^∗1+𝛼𝑆^∗𝐼_𝑙>0, (3.34) which implies 𝑈(𝑡)→+∞ as 𝑡→+∞. This is a contradiction to 𝑈(𝑡)≤𝑀(1+𝛽𝜏𝑒^−𝜇𝜏(𝑆^∗/1+𝛼𝑆^∗)) for being 𝑡 large enough. Therefore, for any positive constant 𝑡₀, the inequality 𝐼(𝑡)<𝐼^∗ cannot hold for all 𝑡≥𝑡₀.
On the one hand, if 𝐼(𝑡)≥𝐼^∗ holds true for all 𝑡 being large enough, then our aim is obtained. Otherwise, 𝐼(𝑡) is oscillatory about 𝐼^∗.
Let 𝑚₂𝐼=min^∗2,𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏}. (3.35) In the following, we will show that 𝐼(𝑡)≥𝑚₂ for 𝑡 being large enough. There exist two positive constants 𝑡,𝜓 such that 𝐼𝑡=𝐼𝑡+𝜓=𝐼^∗,𝐼(𝑡)<𝐼^∗,for𝑡<𝑡<𝑡+𝜓. (3.36)(3.37) Since 𝐼(𝑡) is continuous and bounded and is not effected by impulses, we conclude that 𝐼(𝑡) is uniformly continuous. Hence there exists a constant 𝑇₃ (with 0<𝑇₃<𝜏 and 𝑇₃ is independent of the choice of 𝑡) such that 𝐼(𝑡)>𝐼^∗/2 for all 𝑡≤𝑡≤𝑡+𝑇₃.
If 𝜓≤𝑇₃, our aim is obtained.
If 𝑇₃<𝜓≤𝜏, from the second equation of (1.3), we have that 𝐼(̇𝑡)≥−(𝑑+𝑟+𝜔)𝐼(𝑡) for 𝑡<𝑡≤𝑡+𝜓. Then we have 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏} for 𝑡<𝑡≤𝑡+𝜓≤𝑡+𝜏 since 𝐼(𝑡)=𝐼^∗. It is clear that 𝐼(𝑡)≥𝑚₂ for 𝑡<𝑡≤𝑡+𝜓.
If 𝜓≥𝜏, then we have that 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏} for 𝑡<𝑡≤𝑡+𝜏. Next, we will show that 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏} for 𝑡+𝜏<𝑡≤𝑡+𝜓. In fact, if not, there exists a 𝑇₄>0 such that 𝐼(𝑡)≥𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏} for 𝑡<𝑡≤𝑡+𝜏+𝑇₄.𝐼(𝑡+𝜏+𝑇₄)=𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏} and 𝐼(̇𝑡+𝜏+𝑇₄)≤0. When 𝑡 is large enough, from (3.37) and the first equation of (1.3), we have𝑆̇(𝑡)>−𝛽𝑒^−𝜇𝜏𝐼^∗𝑆(𝑡)𝑆𝑡+𝜇,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝜌,𝑡=𝑛𝑇. (3.38) Similarly, we know that there exists 𝑇₅>𝑡+𝜏, for 𝑡>𝑇₅ that 𝑆(𝑡)>𝜌𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}1−𝑒^{−(𝛽𝑒−𝜇𝜏𝐼∗+𝜇)𝑇}−𝜀=𝑆_Δ. (3.39) Then the inequality 𝑆(𝑡)>𝑆_Δ holds true for 𝑡+𝜏<𝑡<𝑡+𝜓. On the other hand, we have from the second equation of (1.3) that 𝐼̇𝑡+𝜏+𝑇₄>𝛽𝑒^−𝜇𝜏𝑆(𝑡)(𝑡)−()𝐼1+𝛼𝑆𝑑+𝑟+𝜔^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏}=(𝑑+𝑟+𝜔)𝛽𝑒^−𝜇𝜏𝑆(𝑡)𝑑+𝑟+𝜔(𝑡)𝐼1+𝛼𝑆−1^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏}>()𝑆𝑑+𝑟+𝜔^Δ/1+𝛼𝑆_Δ𝑆^∗/(1+𝛼𝑆^∗)−1𝐼^∗𝑒^{−(𝑑+𝑟+𝜔)𝜏}>0. (3.40) This is a contradiction to 𝐼(̇𝑡+𝜏+𝑇₄)≤0. Therefore, 𝐼(𝑡)≥𝑚₂ for 𝑡∈[𝑡,𝑡+𝜓].
Since this kind of interval [𝑡,𝑡+𝜓] is arbitrarily chosen, we get that 𝐼(𝑡)≥𝑚₂ for 𝑡 being large enough. In view of our arguments previously, the choice of 𝑚₂ is independent of the positive solution of (1.3) which satisfies that 𝐼(𝑡)≥𝑚₂ for sufficiently large 𝑡. This completes the proof.

Theorem 3.4. If ℜ₂>1, the system (1.3) is permanent; that is, there exist two positive constants 𝑚₁,𝑚₂ such that 𝑆(𝑡)≥𝑚₁,𝐼(𝑡)≥𝑚₂ for 𝑡 being large enough.

Proof. Suppose that 𝑋(𝑡)=(𝑆(𝑡),𝐼(𝑡)) is any positive solution of system (1.3). From the first and third equations of (1.3), we have that 𝑆̇(𝑡)>−𝛽𝑒^−𝜇𝜏𝑀^′𝑆(𝑡)𝑆𝑡+𝜇,𝑡≠𝑛𝑇,₊=𝑆(𝑡)+𝜌,𝑡=𝑛𝑇. (3.41) Similarly, we can get such 𝑡 large enough and 𝜀>0 small enough that 𝑆(𝑡)≥𝜌𝑒^{−(𝛽𝑒−𝜇𝜏𝑀′+𝜇)𝑇}1−𝑒^{−(𝛽𝑒−𝜇𝜏𝑀′+𝜇)𝑇}−𝜀=𝑚₁. (3.42)
Set 𝐷=(𝑆,𝐼)∈𝑅²∣𝑚₁≤𝑆(𝑡)≤𝑀^′,𝑚₂≤𝐼(𝑡)≤𝑀^′. (3.43)
Then 𝐷 is a bounded compact region which has positive distance from coordinate axes. By Lemma 3.3, one obtains that every solution of system (1.3) eventually enters and remains in the region 𝐷. The proof of Theorem 3.4 is completed.

4. Numerical Simulation and Conclusion

To verify the theoretical results obtained in this paper, we will give some numerical simulations.

Under the continuous control strategy, we consider the hypothetical set of parameter values as 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,and𝜔=0.1, with 𝑆(0)=1,and𝐼(0)=1. Through calculation, we know 𝜏₁=0.5889 and 𝐸₁(1.5,0).(i)If 𝜏=1>𝜏₁, then according to Theorem 2.1, we know the disease-free equilibrium 𝐸₁ of system (1.2) is local stable for this case (see Figures 1, 2, and 3). (ii)If 𝜏=0.5<𝜏₁, through calculation, we know 𝐸₂(1.4356,0.0644). Then according to Theorem 2.2, the positive equilibrium 𝐸₂ of system (1.2) is local stable for this case (see Figures 4, 5, and 6).

Figure 1 

Time series of 𝑆(𝑡) with different initial values and parameters 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,𝜔=0.1,and𝜏=1.

Figure 2 

Time series of 𝐼(𝑡) with different initial values and parameters 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,𝜔=0.1,and𝜏=1.

Figure 3 

Phase diagram of 𝑆(𝑡) with different initial values and parameters 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,𝜔=0.1,and𝜏=1.

Figure 4 

Time series of 𝑆(𝑡) with different initial values and parameters 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,𝜔=0.1,and𝜏=0.5.

Figure 5 

Time series of 𝐼(𝑡) with different initial values and parameters 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,𝜔=0.1,and𝜏=0.5.

Figure 6 

Phase diagram of 𝑆(𝑡) with different initial values and parameters 𝜌=0.3,𝜇=0.2,𝛽=1.5,𝛼=1,𝑟=0.5,𝑑=0.2,𝜔=0.1,and𝜏=0.5.

Its epidemiological implication is that if time delay 𝜏 is greater than some key value 𝜏₁, then diseased plants will disappear in local scope. In contrast, if time delay 𝜏 is less than some key value 𝜏₁, then susceptible plants and diseased plants will coexist in local scope.

Under the impulsive control strategy, We consider the hypothetical set of parameter values as 𝜇=0.2,𝛽=1.2,𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0 with 𝑆(0)=1,and𝐼(0)=1.(i)We consider the susceptible plants rate 𝜌=0.8. Through calculation, we have ℜ₂=1.2044. Then according to Theorem 3.4, we know that system (1.3) is permanence, for this case (see Figures 10, 11, and 12).(ii) If we decrease the susceptible plants rate 𝜌 to 0.3, through calculation, we know ℜ₁=0.9505. Then according to Theorem 3.1, the disease-free periodic solution of system is globally attractive, for this case (see Figures 7, 8, and 9).

Figure 7 

Time series of 𝑆(𝑡) with parameters 𝜌=0.8,𝜇=0.2,𝛽=1.2, 𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0.

Figure 8 

Time series of 𝐼(𝑡) with parameters 𝜌=0.8,𝜇=0.2,𝛽=1.2, 𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0.

Figure 9 

Phase diagram of 𝑆(𝑡) and 𝐼(𝑡) with parameters 𝜌=0.8,𝜇=0.2,𝛽=1.2, 𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0.

Figure 10 

Time series of 𝑆(𝑡) with parameters 𝜌=0.3,𝜇=0.2,𝛽=1.2, 𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0.

Figure 11 

Time series of 𝐼(𝑡) with parameters 𝜌=0.3,𝜇=0.2,𝛽=1.2, 𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0.

Figure 12 

Phase diagram of 𝑆(𝑡) and 𝐼(𝑡) with parameters 𝜌=0.3,𝜇=0.2,𝛽=1.2, 𝛼=1,𝑟=0.6,𝑑=0.2,𝜔=0.1,𝑇=1,and𝜏=0.

We have other hypothetical parameter values under the impulsive control strategy as 𝜇=0.1,𝛽=1,𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2. with 𝑆(0)=0.5,and𝐼(0)=0.1(i)We consider the susceptible plants rate 𝜌=5. Through calculation, we have ℜ₂=1.2044. Then according to Theorem 3.4, we know that system (1.3) is permanence, for this case (see Figures 16, 17, and 18 ).(ii) If we decrease the susceptible plants rate 𝜌 to 0.4, through calculation, we know ℜ₁=0.9505. Then according to Theorem 3.1, the disease-free periodic solution of system is globally attractive, for this case (see Figures 13, 14, and 15).

Figure 13 

Time series of 𝑆(𝑡) with parameters 𝜌=0.4,𝜇=0.1,𝛽=1, 𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2.

Figure 14 

Time series of 𝐼(𝑡) with parameters 𝜌=0.4,𝜇=0.1,𝛽=1, 𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2.

Figure 15 

Phase diagram of 𝑆(𝑡) and 𝐼(𝑡) with parameters 𝜌=0.4,𝜇=0.1,𝛽=1, 𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2.

Figure 16 

Time series of 𝑆(𝑡) with parameters 𝜌=5,𝜇=0.1,𝛽=1, 𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2.

Figure 17 

Time series of 𝐼(𝑡) with parameters 𝜌=5,𝜇=0.1,𝛽=1, 𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2.

Figure 18 

Phase diagram of 𝑆(𝑡) and 𝐼(𝑡)with parameters 𝜌=5,𝜇=0.1,𝛽=1, 𝛼=1,𝑟=0.3,𝑑=0.2,𝜔=0.2,𝑇=0.8,and𝜏=0.2.

Its epidemiological implication is that we took such a strategy by improving planting susceptible plants in practice; as a result, if the susceptible plants rate is greater than some key value 𝜌^∗, both susceptible plants and diseased plants will coexist. In contrast, if we decrease the susceptible plants rate 𝜌 and make it less than some key value 𝜌_∗, then diseased plants will die out at length. In a word, we find that the impulse plants rate has played a very important role in the actual plant epidemic prevention.

In this paper, delay SIS plant epidemic model is constructed and investigated. We proposed two different control strategies in the model. Our primary results are to compare the difference between the two control methods. Firstly, we consider continuous cultural control strategy by continuous replanting of healthy plants. We come to the conclusion that if 𝜌<𝜌_Δ, then diseased plants will disappear in local scope where 𝜌_Δ=𝜇(𝑑+𝑟+𝜔)/(𝛽𝑒^−𝜇𝜏−𝛼(𝑑+𝑟+𝜔)). And if 𝜌>𝜌_Δ, then diseased plants will exist for a long time in local scope. Secondly, impulsive control strategy of plant disease model is considered; in this case, we get that if 𝜌<𝜌_∗, then diseased plants will disappear finally where 𝜌_∗=((1−𝑒^−𝐿𝑇)/(𝜇+𝜔))(𝜇(𝑑+𝑟+𝜔)/(𝛽𝑒^−𝜇𝜏−𝛼(𝑑+𝑟+𝜔))). And if 𝜌>𝜌^∗, then diseased plants will exist for a long time where 𝜌^∗=(𝑒^𝜇𝑇−1)(𝜇(𝑑+𝑟+𝜔)/(𝛽𝑒^−𝜇𝜏−𝛼(𝑑+𝑟+𝜔))). We think that our results will offer help to the actual plant infectious disease management.