Mathematics in Applied Sciences and Engineering https://doi.org/10.5206/mase/9451 Volume 1, Number 1, March 2020 , pp.27-38 https://ojs.lib.uwo.ca/mase GLOBAL ASYMPTOTIC STABILITY OF A DELAYED PLANT DISEASE MODEL YUMING CHEN AND CHONGWU ZHENG Abstract. In this paper, we consider the following system of delayed differential equations,{ S′(t) = σϕ − βS(t)I(t − τ) − ηS(t), I′(t) = σ(1 − ϕ) + βS(t)I(t − τ) − (η + ω)I(t), which can be used to model plant diseases. Here ϕ ∈ (0, 1], τ ≥ 0, and all other parameters are positive. The case where ϕ = 1 is well studied and there is a threshold dynamics. The system always has a disease-free equilibrium, which is globally asymptotically stable if the basic reproduction number R0 ≜ βσ/η(η + ω) ≤ 1 and is unstable if R0 > 1; when R0 > 1, the system also has a unique endemic equilibrium, which is globally asymptotically stable. In this paper, we study the case where ϕ ∈ (0, 1). It turns out that the system only has an endemic equilibrium, which is globally asymptotically stable. The local stability is established by the linearization method while the global attractivity is obtained by the Lyapunov functional approach. The theoretical results are illustrated with numerical simulations. 1. Introduction Plant diseases can decrease the economic, aesthetic, and biological value of many types of plants. Examples of plant diseases caused by plant viruses can be found in the journal, Plant Disease. As a result, integrated management concepts have been developed to combat various plagues suffered by crops. Integrated management strategies combine available host resistance with cultural, chemical and biological control measures. For example, a cultural control strategy including replanting, and/or removing (roguing) diseased plants is a widely accepted treatment for plant epidemics. Mathematical models of plant-virus disease epidemics are often built to provide a detailed exposition on how to describe, analyze, and predict epidemics of plant diseases. To name a few, see [7, 14, 16, 18, 19] and the references therein. Based on the biological background in [2, 19], a simple model for plant diseases with a continuous cultural strategy can be built as follows. The plant population is divided into three compartments, susceptible plants (S), infected plants (I), and removed plants (R). Removal occurs by death or by sanitation. The removed plants are assumed not to be infected again, so we need to focus only on S and I. We make the following assumptions. a) New plants enter the system at the rate σ with a proportion ϕ being susceptible and (1 − ϕ) being infected. b) The incidence rate is proportional to SI and the transmission rate is β. c) Plants are removed from the system at the rate η, in other words, 1/η is either the harvest time or the end of their reproductive lifetime. Received by the editors 23 January 2020; revised 22 February 2020; accepted 22 February, 2020; published online 24 February 2020. 2000 Mathematics Subject Classification. Primary 34K20; Secondary 92D30. Key words and phrases. Delay, equilibrium, stability, plant disease. The first author was supported in part by NSERC of Canada. 27 28 Y. CHEN AND C. ZHENG d) The rouging (removing infected plants from the system) rate is ω. Figure 1 schematically sketches the transmission of a plant disease. The above assumptions lead to the IS βSI ηS ωI ηI σ σϕ σ(1 − ϕ) Figure 1. Schematic diagram for the transmission of a plant disease following SI model for plant diseases,  dS(t) dt = σϕ − βS(t)I(t) − ηS(t), dI(t) dt = σ(1 − ϕ) + βS(t)I(t) − (η + ω)I(t). (1.1) Here the removal is continual, which is common for plant diseases. For human epidemics, the removal can be impulsive at some certain time instants. For example, see De la Sen et al. [4]. However, in disease transmission models, time delay is an important quantity for many epidemiolog- ical mechanisms, but this is not reflected in (1.1). We refer to van den Driessche [20] for a brief review of delay differential equations arising from disease modeling. As in Cooke [3], we incorporate a constant delay into (1.1) to obtain the following system of delayed differential equations,  dS(t) dt = σϕ − βS(t)I(t − τ) − ηS(t), dI(t) dt = σ(1 − ϕ) + βS(t)I(t − τ) − (η + ω)I(t). (1.2) Here τ is the average latent period of the plant disease, that is, the average time for a susceptible plant from getting infected to becoming infectious. An alternative approach, perhaps more realistic, is to incorporate a distributed delay as follows,  dS(t) dt = σϕ − βS(t) ∫ τ 0 I(t − s)dk(s) − ηS(t), dI(t) dt = σ(1 − ϕ) + βS(t) ∫ τ 0 I(t − s)dk(s) − (η + ω)I(t), (1.3) where k : [0, τ] → R is nondecreasing and has bounded variation such that ∫ τ 0 dk(s) = k(τ) − k(0) = 1. The discussion for (1.3) is quite similar to that for (1.2). We deal with (1.2) in the sequel and in Section 4 we will mention the necessary modification for a general system including (1.3). Let C = C([−τ, 0], R2) be the Banach space of continuous functions from [−τ, 0] to R2 equipped with the usual supremum norm ∥ · ∥. The initial condition of (1.2) is given as S(t) = φ1(θ) and I(t) = φ2(θ) for θ ∈ [−τ, 0], (1.4) A DELAYED PLANT DISEASE MODEL 29 where φ = (φ1, φ2) ∈ C such that φi(θ) ≥ 0 for θ ∈ [−τ, 0] and i = 1, 2. If (S(t), I(t)) is a solution of (1.2), then we have S(t) = S(0)e− ∫ t 0 [βI(s−τ)+η]ds + ϕσ ∫ t 0 e− ∫ t u [βI(s−τ)+η]dsdu (1.5) and I(t) = e−(η+ω)tI(0) + ∫ t 0 e−(η+ω)(t−u)[(1 − ϕ)σ + βS(u)I(u − τ)]du (1.6) for t ≥ 0. It follows that given an initial condition (1.4), we can use the step-by-step method to find S(t) first by (1.5) and then I(t) by (1.6). One can also easily see that any solution of (1.2) with initial condition (1.4) satisfies S(t) > 0 and I(t) > 0 for t > 0. Without loss of generality, we need to consider only positive solutions of (1.2). Sometimes, the solution of (1.2) with the initial condition (1.4) is also denoted by (S(t; φ), I(t; φ)). As usual, for a solution (S(t), I(t)) of (1.2) and t > 0, we define (St, It) ∈ C by St(θ) = S(t + θ) and It(θ) = I(t + θ) for θ ∈ [−τ, 0]. For more information on delay differential equations, we refer to Hale and Verduyn Lunel [8] or Diekmann et al. [5]. The case where ϕ = 1 of (1.2) is the standard delayed SI or SIR model. These standard models with discrete delays and distributed delays have been well studied. To name a few, see [1, 11, 12, 17]. The dynamics of (1.2) with ϕ = 1 is quite simple, which is a threshold dynamics. Precisely, the system always has a disease free equilibrium E0 = ( σ η , 0), which is globally asymptotically stable if R0 ≜ βσ/η(η + ω) ≤ 1 and is unstable if R0 > 1; when R0 > 1, it also has an endemic equilibrium E∗ = ( η + ω β , βσ − η(η + ω) β(η + ω) ), which is globally asymptotically stable. R0 is called the basic reproduction number of (1.2). Though (1.2) with ϕ = 1 and its modifications have been extensively studied, system (1.2) with ϕ ∈ (0, 1) has not been studied yet. One reason is that the epidemics considered in the literature are mainly human diseases. In such situations, in order to lower the economic cost, infected persons usually will be quarantined or hospitalized. Therefore, it is reasonable to assume that ϕ = 1. But, for plant diseases, it is necessary and important to study the dynamics of (1.2) with ϕ ∈ (0, 1), which is the goal of this paper. For example, African small holders replant cassava and sweet potato with cuttings only from previous crop while commercial sweet potato growers in China use disease-free material from in vitro propagation programmes [6]. The first corresponds to ϕ ∈ (0, 1) and the latter corresponds to ϕ = 1 if we use (1.2) to model them, respectively. The remainder of the paper is organized as follows. We first study the existence of equilibria and their local stability in Section 2. We then show that the system is globally asymptotically stable in Section 3 by the Lyapunov functional approach. The paper concludes with general remarks. It is mentioned that the approach here can be applied to deal with generalized versions of (1.2) with nonlinear incidence functions and distributed delays. 2. Existence of equilibria and their local stability In this section, we study the local dynamics of (1.2) with ϕ ∈ (0, 1). We first consider the existence of equilibria. 30 Y. CHEN AND C. ZHENG Proposition 2.1. Suppose ϕ ∈ (0, 1). Then (1.2) only has an endemic equilibrium E∗ϕ = (S ∗ ϕ, I ∗ ϕ), where S∗ϕ = βσ + η(η + ω) − √ ∆ϕ 2βη , I∗ϕ = σ − ηS∗ϕ η + ω , (2.1) ∆ϕ = [η(η + ω) + βσ] 2 − 4ϕβση(η + ω). Proof. Let (S, I) be an equilibrium of (1.2). Then we have σϕ − βSI − ηS = 0 (2.2) and σ(1 − ϕ) + βSI − (η + ω)I = 0. (2.3) Adding (2.2) and (2.3) yields σ − ηS − (η + ω)I = 0, or I = σ−ηS η+ω . Then substituting it into (2.2), we obtain G(S) = βη η + ω S2 − ( βσ η + ω + η ) S + ϕσ = 0. Note that G(0) = ϕσ > 0 and G(σ η ) = −(1 − ϕ)σ < 0. It follows that G(S) = 0 has two positive solutions, one less than σ η and the other larger than σ η . From I = (σ − ηS)/(η + ω) ≥ 0, we know that S ≤ σ/η. Hence (1.2) has only one equilibrium, which is positive and is given by (2.1). This completes the proof. □ Proposition 2.1 tells us that (1.2) has no disease-free equilibrium, which is due to the recruitment of infected plants. Therefore, the basic reproduction number cannot be defined. In the following, we investigate the local stability of E∗ϕ. E ∗ ϕ is locally stable if for any ε > 0 there exists δ > 0 such that ∥(St(·; φ), It(·; φ)) − E∗ϕ∥ ≤ ε for t ≥ 0 and ∥φ − E∗ϕ∥ ≤ δ; otherwise, it is unstable. The local stability of E∗ϕ is obtained by the technique of linearization. Precisely, if all roots of the characteristic equation associated with the linearized system about E∗ϕ have negative real parts then E∗ϕ is locally exponentially stable and hence stable; if at least one of the roots has positive real part then E∗ϕ is unstable. E ∗ ϕ is locally exponentially stable if there exist positive constants m, α, and δ such that ∥(St(·; φ), It(·; φ)) − E∗ϕ∥ ≤ me −αt∥φ − E∗ϕ∥ for all t ≥ 0 and ∥φ − E∗ϕ∥ ≤ δ. To prove the local stability of E∗ϕ, it is worthy to note that ∆ϕ > [η(η + ω) + βσ] 2 − 4βση(η + ω) = [βσ − η(η + ω)]2 as ϕ ∈ (0, 1). Theorem 2.1. Suppose ϕ ∈ (0, 1). Then the equilibrium E∗ϕ of (1.2) is locally exponentially stable. Proof. Linearize (1.2) around E∗ϕ to obtain  dS(t) dt = −(βI∗ϕ + η)S(t) − βS ∗ ϕI(t − τ), dI(t) dt = βI∗ϕS(t) + βS ∗ ϕI(t − τ) − (η + ω)I(t). The corresponding characteristic equation is det ( λ + βI∗ϕ + η βS ∗ ϕe −λτ −βI∗ϕ λ + η + ω − βS ∗ ϕe −λτ ) = 0, A DELAYED PLANT DISEASE MODEL 31 or λ2 + λ(2η + ω + βI∗ϕ) − βS ∗ ϕe −λτ (λ + η) + (βI∗ϕ + η)(η + ω) = 0. (2.4) To finish the proof, we only need to show that all roots of (2.4) have negative real parts. First, suppose that τ = 0. Then (2.4) reduces to λ2 + λ(2η + ω + βI∗ϕ − βS ∗ ϕ) + [(βI ∗ ϕ + η)(η + ω) − βS ∗ ϕη] = 0. With (2.1), we get λ2 + Pϕλ + √ ∆ϕ = 0, where Pϕ = 2η3 + 3η2ω + ηω2 − βσω + (2η + ω) √ ∆ϕ 2η(η + ω) . If βσ ≥ η(η + ω), then √ ∆ϕ > βσ − η(η + ω) and hence Pϕ ≥ 2η3 + 3η3ω + ηω2 − βσω + (2η + ω)[βσ − η(η + ω)] 2η(η + ω) = βσ η + ω > 0; if βσ < η(η + ω), then Pϕ > 2η3 + 3η2ω + ηω2 − ωη(η + ω) + (2η + ω) √ ∆ϕ 2η(η + ω) = 2η3 + 2η2ω + (2η + ω) √ ∆ϕ 2η(η + ω) > 0. Therefore, if τ = 0 then all roots of (2.4) have negative real parts. Next, we claim that (2.4) has no roots on the imaginary axis. By way of contradiction, suppose that (2.4) has a root iξ with ξ ∈ R for some τ > 0. Then substitute it into (2.4) and separate the real and imaginary parts to obtain (βI∗ϕ + η)(η + ω) − ξ 2 = βS∗ϕ(ξ sin ξτ + η cos ξτ) and (2η + ω + βI∗ϕ)ξ = βS ∗ ϕ(ξ cos ξτ − η sin ξτ). It follows that ξ4 + [(2η + ω + βI∗ϕ) 2 − 2(βI∗ϕ + η)(η + ω) − (βS ∗ ϕ) 2]ξ2 + [(βI∗ϕ + η) 2(η + ω)2 − (βS∗ϕη) 2] = 0. (2.5) Note that (βI∗ϕ + η) 2(η + ω)2 − (βS∗ϕη) 2 = [(βI∗ϕ + η)(η + ω) + βS ∗ ϕη][(βI ∗ ϕ + η)(η + ω) − βS ∗ ϕη] = [(βI∗ϕ + η)(η + ω) + βS ∗ ϕη] √ ∆ϕ > 0 32 Y. CHEN AND C. ZHENG and (2η + ω + βI∗ϕ) 2 − 2(βI∗ϕ + η)(η + ω) − (βS ∗ ϕ) 2 = [(η + ω) + (η + βI∗ϕ)] 2 − 2(βI∗ϕ + η)(η + ω) − (βS ∗ ϕ) 2 = (η + ω)2 + (η + βI∗ϕ) 2 − (βS∗ϕ) 2 = (η + βI∗ϕ) 2 + (η + ω + βS∗ϕ)(η + ω − βS ∗ ϕ) = (η + βI∗ϕ) 2 + (η + ω + βS∗ϕ) η(η + ω) − βσ + √ ∆ϕ 2η > 0 (as √ ∆ϕ > |η(η + ω) − βσ|). It follows that (2.5) cannot hold and this proves the claim. Note that the roots of (2.4) depend continuously on τ and that all roots of (2.4) with non-negative real parts are uniformly bounded (which is easy to see). This, combined with the claim and the result that all roots of (2.4) have negative real parts for the case where τ = 0, tells us that all roots of (2.4) have negative real parts for any τ ≥ 0. Therefore, we have completed the proof. □ 3. Global asymptotic stability of the endemic equilibrium In the previous section, we have shown that (1.2) has a unique endemic equilibrium E∗ϕ which is locally stable. Indeed, E∗ϕ is also globally asymptotically stable, that is, E ∗ ϕ is locally stable and globally attractive. E∗ϕ is globally attractive if lim t→∞ (St, It) = E ∗ ϕ for every solution (S(t), I(t)) of (1.2). To prove the global attractivity, we first establish the permanence of (1.2). Definition 3.1 ([9]). System (1.2) is said to be permanent if there are positive constants νi and Mi (i = 1, 2) such that ν1 ≤ lim inf t→∞ S(t) ≤ lim sup t→∞ S(t) ≤ M1 and ν2 ≤ lim inf t→∞ I(t) ≤ lim sup t→∞ I(t) ≤ M2 hold for any solution of (1.2) with the initial condition (1.4). Here νi and Mi (i = 1, 2) are independent of (1.4). Proposition 3.1. Suppose ϕ ∈ (0, 1). Then (1.2) is permanent. In fact, for every solution (S(t), I(t)), we have σϕη βσ + η2 ≤ lim inft→∞ S(t) ≤ lim sup t→∞ S(t) ≤ σ η , (3.1) σ(1 − ϕ) η + ω ≤ lim inft→∞ I(t) ≤ lim sup t→∞ I(t) ≤ σ η . (3.2) Proof. Let (S(t), I(t)) be any solution of (1.2). Then d(S(t) + I(t)) dt = σ − ηS(t) − (η + ω)I(t) ≤ σ − η(S(t) + I(t)) and hence by the comparison principle (see, for example, [10]) we get S(t) + I(t) ≤ [ σ(eηt − 1) η + S(0) + I(0) ] e−ηt = σ η + η(S(0) + I(0)) − σ η e−ηt. A DELAYED PLANT DISEASE MODEL 33 It follows that lim sup t→∞ (S(t) + I(t)) ≤ σ η , which implies that lim sup t→∞ S(t) ≤ σ η and lim sup t→∞ I(t) ≤ σ η . (3.3) In particular, for any ε > 0, there exists a T > 0 such that I(t) ≤ σ η + ε for all t ≥ T. Then, for t ≥ T + τ, dS(t) dt ≥ ϕσ − [ β ( σ η + ε ) + η ] S(t), which immediately yields from the comparison principle that lim inf t→∞ S(t) ≥ ϕσ β(σ η + ε) + η . As ε is arbitrary, we have lim inf t→∞ S(t) ≥ ϕση βσ + η2 . (3.4) Finally, note that dI(t) dt ≥ (1 − ϕ)σ − (η + ω)I(t) for t ≥ 0. Again, by the comparison principle we get lim inf t→∞ I(t) ≥ (1 − ϕ)σ η + ω . (3.5) It follows from (3.3)–(3.5) immediately that (1.2) is permanent and both (3.1) and (3.2) hold. This completes the proof. □ Now, we are ready to prove the main result of this section. Theorem 3.2. Suppose ϕ ∈ (0, 1). Then the equilibrium E∗ϕ of (1.2) is globally asymptotically stable. Proof. We have shown in Theorem 2.1 that E∗ϕ is locally stable and hence it suffices to show that E ∗ ϕ is globally attractive. The proof is completed by using the Lyapunov functional approach and is parallel to that of Theorem 4.1 of McCluskey [13] (but we use the notation in Hale and Verduyn Lunel [8]). To construct the Lyapunov functional, we need the function g(x) = x − 1 − ln x for x > 0. It is well-known that g(x) ≥ 0 and g(x) = 0 if and only if x = 1. Let D = { φ = (φ1, φ2) ∈ C ∣∣∣∣∣ φ1(θ) ∈ [ σϕη βσ+η2 , σ η ] and φ2(θ) ∈ [ (1−ϕ)σ η+ω , σ η ] for θ ∈ [−τ, 0] } . It is not difficult to see that D is an invariant set of (1.2). Moreover, by Proposition 3.1, D is attractive. Therefore, to study the global attractivity of E∗ϕ, we only need to consider solutions starting in D. We define V : D → R by V (φ) = 1 βI∗ ϕ VS(φ1) + 1 βS∗ ϕ VI(φ2) + V+(φ2) for φ = (φ1, φ2) ∈ D, where VS(φ1) = g( φ1(0) S∗ϕ ), VI(φ2) = g( φ2(0) I∗ϕ ) and V+(φ2) = ∫ τ 0 g( φ2(−s) I∗ϕ )ds. 34 Y. CHEN AND C. ZHENG To find the derivative of V along solutions of (1.2), the following relationships between S∗ϕ and I ∗ ϕ will be helpful, ϕσ = βS∗ϕI ∗ ϕ + ηS ∗ ϕ, (3.6) (η + ω)I∗ϕ = (1 − ϕ)σ + βS ∗ ϕI ∗ ϕ. (3.7) For clarity, we first calculate the derivatives of VS, VI, and V+ along solutions of (1.2) one by one. Firstly, V̇S(φ1) = lim sup h→0+ 1 h [VS(Sh) − VS(φ1)] = dVS(Sh) dh |h=0 = 1 S∗ϕ ( 1 − S∗ϕ S(0) ) dS(0) dt = 1 S∗ϕ ( 1 − S∗ϕ S(0) ) (ϕσ − βS(0)I(−τ) − ηS(0)) = 1 S∗ϕ ( 1 − S∗ϕ φ1(0) ) (ϕσ − βφ1(0)φ2(−τ) − ηφ1(0)). Using (3.6) to replace ϕσ gives us V̇S(φ1) = 1 S∗ϕ ( 1 − S∗ϕ φ1(0) ) {η[S∗ϕ − φ1(0)] + β[S ∗ ϕI ∗ ϕ − φ1(0)φ2(−τ)]} = −η (φ1(0) − S∗ϕ) 2 φ1(0)S ∗ ϕ + βI∗ϕ ( 1 − S∗ϕ φ1(0) )( 1 − φ1(0) S∗ϕ φ2(−τ) I∗ϕ ) . For simplicity of notation, let x = φ1(0) S∗ϕ , y = φ2(0) I∗ϕ , and z = φ2(−τ) I∗ϕ . Then we can get V̇S(φ1) = −η (φ1(0) − S∗ϕ) 2 φ1(0)S ∗ ϕ + βI∗ϕ ( 1 − xz − 1 x + z ) . (3.8) Secondly, V̇I(φ2) = 1 I∗ϕ ( 1 − I∗ϕ φ2(0) ) [(1 − ϕ)σ + βφ1(0)φ2(−τ) − (η + ω)φ2(0)] = 1 I∗ϕ ( 1 − I∗ϕ φ2(0) )[ (1 − ϕ)σ + βφ1(0)φ2(−τ) − (η + ω)I∗ϕ φ2(0) I∗ϕ ] . With the help of (3.7), we get V̇I(φ2) = 1 I∗ϕ ( 1 − I∗ϕ φ2(0) )[ (1 − ϕ)σ ( 1 − φ2(0) I∗ϕ ) (3.9) +βS∗ϕI ∗ ϕ ( φ1(0) S∗ϕ φ2(−τ) I∗ϕ − φ2(0) I∗ϕ )] = −(1 − ϕ)σ (φ2(0) − I∗ϕ) 2 φ2(0)(I ∗ ϕ) 2 + βS∗ϕ ( 1 − y − xz y + xz ) . (3.10) A DELAYED PLANT DISEASE MODEL 35 Finally, V̇+(φ2) = d dt ∫ τ 0 g ( I(t − s) I∗ϕ ) ds|t=0 = ∫ τ 0 d dt g ( I(t − s) I∗ϕ ) |t=0ds = ∫ τ 0 − d ds g ( I(−s) I∗ϕ ) ds = g ( I(0) I∗ϕ ) − g ( I(−τ) I∗ϕ ) = g(y) − g(z) = y − z + ln z − ln y. (3.11) Now, we obtain from (3.8), (3.10), and (3.11) that V̇ (φ) = − η βI∗ϕS ∗ ϕ (φ1(0) − S∗ϕ) 2 φ1(0) − (1 − ϕ)σ βS∗ϕ(I ∗ ϕ) 2 (φ2(0) − I∗ϕ) 2 φ2(0) +2 − 1 x − xz y + ln z − ln y = − η βI∗ϕS ∗ ϕ (φ1(0) − S∗ϕ) 2 φ1(0) − (1 − ϕ)σ βS∗ϕ(I ∗ ϕ) 2 (φ2(0) − I∗ϕ) 2 φ2(0) + ( 1 − 1 x − ln x ) + ( 1 − xz y + ln xz y ) = − η βI∗ϕS ∗ ϕ (φ1(0) − S∗ϕ) 2 φ1(0) − (1 − ϕ)σ βS∗ϕ(I ∗ ϕ) 2 (φ2(0) − I∗ϕ) 2 φ2(0) −g ( 1 x ) − g ( xz y ) . Then V̇ (φ) ≤ 0 and K = {φ ∈ D|V̇ (φ) = 0} = {φ = (φ1, φ2) ∈ C|φ1(0) = S∗ϕ, φ2(0) = I ∗ ϕ}. Obviously, the largest set in K that is invariant with respect to (1.2) is the singleton {E∗ϕ}. By Theorem 3.1 in Chapter 5 of Hale and Verduyn Lunel [8], every solution converges to E∗ϕ. This completes the proof. □ 4. Concluding remarks In this paper, we considered a plant disease model (1.2) with ϕ ∈ (0, 1). Unlike the case where ϕ = 1, there is no threshold dynamics. The model has a unique endemic equilibrium E∗ϕ, which is always globally asymptotically stable. Figure 2 and Figure 3 illustrate this result for the cases βσ < η(η + ω) and βσ > η(η + ω), respectively. In both cases, τ = 4, β = 0.0064, η = 0.002, ϕ = 0.75. But for Figure 2, σ = 0.0015 and ω = 0.003; while for Figure 3, σ = 0.015 and ω = 0.03. These parameter values are reasonable for cassava (Figure 2) and sweet potatoes (Figure 3), respectively. We refer to van den Bosch et al [19] for more detail. The global asymptotic stability implies that the disease cannot be eradicated. This is realistic since, in practice, it is almost impossible to guarantee that all replanted plants are healthy. This result looks quite different from that for the case where ϕ = 1. However, when ϕ is very close to 1, the result is 36 Y. CHEN AND C. ZHENG 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 S I Figure 2. When σ = 0.0015, ϕ = 0.75, β = 0.0064, η = 0.002, ω = 0.003 and τ = 4, the equi- librium E∗ϕ ≈ (0.3826, 0.1470) of (1.2) is globally asymptotically stable. 1.5 2 2.5 3 3.5 4 4.5 5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 S I Figure 3. When σ = 0.015, ϕ = 0.75, β = 0.0064, η = 0.002, ω = 0.03 and τ = 4, the equilibrium E∗ϕ ≈ (2.9428, 0.2848) of (1.2) is globally asymptotically stable. consistent with that for the case where ϕ = 1. In fact, I∗ϕ = βσ − η(η + ω) + √ ∆ϕ 2β(η + ω) . If βσ = η(η + ω) then I∗ϕ = √ (1 − ϕ)βση(η + ω) β(η + ω) = η √ 1 − ϕ β . Suppose that βσ ̸= η(η + ω). If ϕ ≈ 1, then√ ∆ϕ ≈ |βσ − η(η + ω)|[1 + 2(1 − ϕ)βση(η + ω) [βσ − η(η + ω)]2 ] = |βσ − η(η + ω)| + 2(1 − ϕ)βση(η + ω) |βσ − η(η + ω)| and hence I∗ϕ ≈ (1 − ϕ)ση η(η + ω) − βσ if βσ < η(η + ω) while I∗ϕ ≈ (βσ − η(η + ω))2 + (1 − ϕ)ση(η + ω) (η + ω)[βσ − η(η + ω)] if βσ > η(η + ω), which are quite close to the disease levels in the case where ϕ = 1. The discussion also tells us that in order to develop a good control strategy, we should make βσ < η(η + ω). In this case, though the disease cannot be eradicated, the disease level can be very small and hence it cannot reach the economic injury level [15] (the lowest population density that will cause economic damage or the amount of injury which will justify the cost of using controls). We mention that we can use the same Lyapunov functional to show the global attractivity of the endemic equilibrium of the following generalization of (1.2),  dS(t) dt = ϕσ − βS(t)f(I(t − τ)) − ηS(t), dI(t) dt = σ(1 − ϕ) + βS(t)f(I(t − τ)) − (η + ω)I(t), (4.1) provided that an equilibrium (which is of course an endemic equilibrium) is guaranteed by the incidence function f. For example, f(x) = 1 1+αx in McCluskey [13]. Here, we considered the case where f(x) = x for the simple reason that we could obtain the local stability of the endemic equilibrium. For a general A DELAYED PLANT DISEASE MODEL 37 incidence function f, it may not be easy to analyze the local stability. Moreover, consider modifications of (4.1) with a distributed delay,  dS(t) dt = ϕσ − βS(t) ∫ τ 0 f(I(t − s))dk(s) − ηS(t), dI(t) dt = σ(1 − ϕ) + βS(t) ∫ τ 0 f(I(t − s))dk(s) − (η + ω)I(t), where k : [0, τ] → R is nondecreasing and has bounded variation such that ∫ τ 0 dk(s) = k(τ) − k(0) = 1. Same results can be obtained by replacing V+(φ2) with V̂+(φ2), where V̂+(φ2) = ∫ τ 0 α(s)g ( φ2(−s) I∗ϕ ) ds and α(h) = ∫ τ h dk(s) for h ∈ [0, τ]. We refer to McCluskey [12] for some guidance. References [1] E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with times, J. Math. Biol. 33(1995) 250–260. [2] M.S. Chan and M.J. Jeger, An analytical model of plant virus disease dynamics with roguing and replanting, J. Appl. Ecol. 31(1994) 413–427. [3] K.L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math. 7(1979) 253–263. [4] M. De la Sen, Ravi P. Agarwal, A. Ibeas and S. Alonso-Quesada, On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls, Advances in Differential Equations 2010 (2010), Article ID 281612, 42 pages. [5] O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. [6] G. Feng, G. Yifu and Z. Pinbo, Production and development of virus-free sweet potato in China, Crop Prot. 19(2000) 105–111. [7] S. Fishman and R. Marcus, A model for spread of plant disease with periodic removals, J. Math. Biol. 21(1984) 149–158. [8] J. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993. [10] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Vol. I: Ordinary Differential Equations, Academic Press, New York-London, 1969. [11] W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54(2002) 581–591. [12] C.C. McCluskey, Complete global stability for an SIR epidemic model with delay–Distributed or discrete, Nonlinear Anal. Real World Appl. 11(2010) 55–59. [13] C.C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl. 11(2010) 3106–3109. [14] S. Soubeyrand, L. Held, M. Höhle and I. Sache, Modelling the spread in space and time of an airborne plant disease, J. Roy. Statist. Soc. Ser. C 57(2008) 253–272. [15] V.M. Stern, R.F. Smith, R. van den Bosch and K.S. Hagen, The integrated control concept, Hilgardia, 29(1959) 81–101. [16] N. Stollenwerk and K.M. Briggs, Master equation solution of a plant disease model, Phys. Lett. A 274(2000) 84–91. [17] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42(2000) 931–947. [18] H.R. Thieme and J.A.P. Heesterbeek, How to estimate the efficacy of periodic control of an infectious plant disease, Math. Biosci. 93(1989) 15–29. [19] F. van den Bosch, M.J. Jeger and C.A. Gilligan, Disease control and its selection for damaging plant virus strains in vegetatively propogated staple food crops; a theoretical assessment, Proc. R. Soc. B 274(2007) 11–18. [20] P. van den Driessche, Some epidemiological models with delays, in Differential Equations and Applications to Biology and to Industry, pp. 507–520, World Sci. Publ., River Edge, NJ, 1996. 38 Y. CHEN AND C. ZHENG Corresponding author. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada E-mail address: ychen@wlu.ca Department of Applied Mathematics, Yuncheng University, Yuncheng, Shanxi 044000, P.R. China E-mail address: chongyang1894@163.com