# Dynamics Analysis of a Stochastic Plant Disease Model with Continuous Cultural Control Strategy

Authors: Haisu Zhang; Tongqian Zhang
Journal: Complexity
 (2021)

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## Abstract

In this paper, a new stochastic plant disease model with continuous control strategy is proposed and analyzed. The dynamics of the system are explored under white noise disturbance. We prove that ifR1>1, then the disease is persistent; moreover, if R2>1, then the solutions of the system have a stationary distribution. For the special case, we prove that if R1<1, then the disease will eventually disappear. Finally, some numerical simulations were implemented to illustrate the theoretical results.


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## Body

## 1. Introduction

Plant disease is an important constraint on global crop production and severely damages the yield and quality of crops. The main sources of infections that cause crop disease are viruses, bacteria, fungi, nematodes, and plant disease caused by parasitic plants. Plant viruses are a common source of infection that cause plant disease. Plant virologists have selected 10 plant viruses that harm plant growth [1]. Plant viruses have caused a large reduction in crop production. For example, in the late 1940s, tobacco mosaic virus caused an average loss of approximately 40 million pounds of tobacco in the United States each year, accounting for 2%-3% of tobacco production [2]. For tomato growers in Florida, the cost of producing an acre of tomatoes is about 6000–7000 US dollars, and nearly 25% of the expenses are used to control the tomato yellow leaf curl virus [3]. In the past ten years, cassava in East Africa and Central Africa has lost 47% of its production due to cassava brown streak virus and cassava mosaic virus [4, 5].Plant infectious diseases are mainly infectious diseases caused by pathogens, which are generally passively spread by external forces, which can cause the occurrence and prevalence of disease. The main modes of transmission include air current transmission, rain transmission, insect and other biological transmission, and human factor transmission [6, 7]. Farmers have taken a variety of measures to control the harm of plant viruses, such as developing new plants with stronger resistance, cultural, physical, chemical, and biological control measures, and so on. However, a single control measure cannot control the disease well because it generally only plays an important role in the early stage of disease development. Facts have showed that the comprehensive utilization of several different control measures can achieve a good effect of disease control. This combined control strategy is often called integrated disease management (IDM) [8]. Researchers have established many mathematical models to explain the implementation process of IDM [9–14]. Recently, considering continuous replanting and roguing or removing diseased plants, a simple plant-virus disease model by ordinary differential equations has been proposed:(1)dStdt=σϕ−βStIt−ηSt,dItdt=σ1−ϕ+βStIt−η+ωIt,where S,I are the densities of susceptible and infected plants, respectively, σ is the total rate of the plants, 1−ϕ0<ϕ≤1 is the proportion of infected plants in the replanted plants, β represents the transmission rate of diseased plants, η is the death rate of the plant, and ω represents the roguing (or removal) rate for the infected plants. For detailed parameter meaning and model explanation, we refer the readers to [11, 15]. Obviously, model (1) is different from the general infectious disease model because it considers the characteristics of plant disease transmission, that is, the disease can be transmitted by infecting seedlings or seeds, and it can also be transmitted by infecting mature seedlings. For system (1), if 0<ϕ<1, simple mathematical analysis shows that system (1) always has a globally asymptotically stable disease equilibrium ES∗,I∗ and no disease-free equilibrium; this means that plant disease will eventually persist. If ϕ=1, system (1) always has a disease-free equilibrium ES0,0, and it is globally asymptotically stable for R<1 and unstable for R>1; if R>1, system (1) also has a globally asymptotically stable disease equilibrium E1S1,I1, where R=σβ/ηη+ω.As is known, biological systems are inevitably disturbed by environmental noise [16–21]. The spread of disease is also easily disturbed by environmental noise [22–25]; many researchers have thoroughly studied the impact of environmental noise on the spread of disease by establishing epidemic models by stochastic differential equations [26–29]. In the real world, because the occurrence, development, and spread of plant diseases is a very complex process, it will inevitably be disturbed by various unpredictable random factors, such as environmental conditions (temperature, moisture, and light), soil texture, pH value, and so on. For example, under the influence of high temperature, certain viruses (bacteria) may die, leading to the extinction of diseases; due to the scarcity of water, the growth cycle of certain viruses (bacteria) becomes longer, and the spread of diseases slows down; poor soil will cause the activity of some viruses (bacteria) to weaken, and disease infection is reduced [30, 31]. In this paper, on the basis of model (1), we assume that the environmental white noise is proportional to the variables S and I, respectively, and propose a simple plant disease model under stochastic perspective as follows:(2)dSt=σϕ−βStIt−ηStdt+δ1StdB1t,dIt=σ1−ϕ+βStIt−η+ωItdt+δ2ItdB2t,where Biti=1,2 are independent standard Brownian motions with Bi0=0i=1,2 and δi>0i=1,2 represent the intensities of the white noise on the susceptible and infected plants, respectively. In the whole paper, we let Ω,ℱ,ℙ be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions. The function Biti=1,2 is defined on Ω,ℱ,ℙ, and we denote ℝ+2=x∈ℝ2:xi>0,1≤i≤2.Our purpose is to investigate the impact of environmental noise on plant disease. The structure of the paper is as follows. The well-posedness of solutions of the system is discussed in Section2. The persistence of the plant disease is discussed in Section 3. In Section 4, we explore the existence of the stationary distribution of the solution of the system. In Section 5, numerical simulations are implemented to illustrate the theoretical results. A brief conclusion is given in Section 6.


## 2. The Existence and Uniqueness of Global Positive Solution of System (2)

Theorem 1.
The solution of system (2) that satisfies the initial value S0,I0∈ℝ+2, is unique and positive; moreover, the solution will remain in ℝ+2 with probability one.Proof.
Firstly, letℏe be the explosion time, and we claim that for any initial value S0,I0∈ℝ+2 there exists a unique local solution St,It∈ℝ+2 on t∈0,ℏe a.s. In fact, it is easy to get from the local Lipschitz property of the coefficients of system (2).
Secondly, we proveℏe=∞ a.s. This means that the solution is global. Let n0≥1 be sufficiently large such that S0,I0 all lie within the interval 1/n0,n0. For each integer n≥n0, define the stopping time(3)ℏn=inft∈0,ℏe:St∉1n,norIt∉1n,n.
In the whole paper, we denoteinf∅=∞ (the empty set). Obviously, ℏn is increasing as n⟶∞. Let ℏ∞=limn⟶∞ℏn, whence ℏ∞≤ℏe a.s. If ℏ∞=∞ a.s. is true, then ℏe=∞ and St,It∈ℝ+2 a.s. for all t≥0. Then, in order to achieve our purpose, we only need to show ℏ∞=∞ a.s. If this affirmation is not true, then there exist two constants T>0 and ε∈0,1 such that(4)ℙℏ∞≤T>ε.
Thus, for an integern1≥n0, we have(5)ℙℏn≤T>ε,∀n≥n1.
Define aC2-function V: ℝ+2⟶ℝ+ as follows:(6)V=S−a1−a1lnSa1+I−1−lnI,where a1 is a positive constant to be determined later.
From Itô’s formula, one gets(7)dVS,I=ℒVS,Idt+1−a1Sδ1SdB1t+1−1Iδ2IdB2t,where ℒV:ℝ+2⟶ℝ+ is defined by(8)ℒV=1−a1Sσϕ−βSI−ηS+1−1Iσ1−ϕ+βSI−η+ωI+a1δ122+δ222=σϕ−βSI−ηS−a1σϕS+a1βI+a1η+σ1−ϕ+βSI−η+ωI−σ1−ϕI−βS+η+ω+a1δ122+δ222≤σϕ+a1η+σ1−ϕ+η+ω+a1β−η+ωI+a1δ122+δ222.
Choosea1=η+ω/β. Then, we obtain(9)ℒV≤σϕ+ηη+ωβ+σ1−ϕ+η+ω+η+ωδ122β+δ222K1,where K1 is a positive constant. So, we obtain(10)dVS,I≤K1dt+1−η+ωβSδ1SdB1t+1−1Iδ1IdB2t.
Integrating the above formula from 0 toℏn∧T = minℏn,T and then taking the expectation on both sides, we have(11)EVSℏn∧T,Iℏn∧T≤VS0,I0+K1Eℏn∧T.
Hence,(12)EVSℏn∧T,Iℏn∧T≤VS0,I0+K1T.
LetΩn=ω∈Ω:ℏn=ℏnω≤T for n≥n1 and in view of (5), we obtain ℙΩn≥ε. Note that for every ω∈Ωn, there exists Sℏn,ω or Iℏn,ω equals either n or 1/n. Therefore, VSℏn,ω,Iℏn,ω is no less than either(13)n−1−lnnor1n−1−ln1n.
Therefore, we have(14)VSℏn,ω,Iℏn,ω≥n−1−lnn∧1n−1+lnn.
In view of (12), we obtain(15)VS0,I0+K1T≥E1ΩnωVSℏn,ω,Iℏn,ω≥εn−1−lnn∧1n−1+lnn,where 1Ωn denotes the indicator function of Ωn. Letting n⟶∞ leads to the contradiction(16)∞>VS0,I0+K1T=∞.
So, we must haveℏ∞=∞ a.s. This completes the proof.


## 3. Persistence in Mean

System (2) is said to be persistent in the mean if liminft⟶∞1/t∫0tIsds>0 a.s. For convenience, we define St=1/t∫0tSsds, It=1/t∫0tIsds.The following lemma is essentially the same as that in [32], so we omit it.Lemma 1.
Assumeη>δ12∨δ22/2. Let St,It be the solution of system (2) with any initial value S0,I0∈ℝ+2; then,(17)limt⟶∞∫0tSrdB1rt=0,limt⟶∞∫0tIrdB2rt=0a.s.Lemma 2.
LetSt,It be the solution of system (2) with initial value S0,I0∈ℝ+2. Then,(18)limt⟶∞Stt=0,limt⟶∞Itt=0a.s.Proof.
Letwt=St+It. Define(19)Vw=1+wζ,where ζ is a positive constant to be determined later. Then,(20)dVw=LVdt+ζ1+wζ−1δ1StdB1t+δ2ItdB2t,where(21)LVw=ζ1+wζ−1σ−ηS−ηI−ωI+ζζ−121+wζ−2δ12S2+δ22I2=ζ1+wζ−21+wσ−ηS−ηI−ωI+ζ−12δ12S2+δ22I2≤ζ1+wζ−21+wσ−ηw+ζ−12δ12S2+δ22I2≤ζ1+wζ−21+wσ−ηw+ζ−12δ12∨δ22w2=ζ1+wζ−2−η−ζ−12δ12∨δ22w2+σ−ηw+σ.
Chooseζ>0 such that(22)η−ζ−12∨0δ12∨δ22≔χ>0,so(23)LVw≤ζ1+wζ−2−χw2+σ−ηw+σ,(24)dVw≤ζ1+wζ−2−χw2+σ−ηw+σ+ζ1+wζ−1δ1StdB1t+δ2ItdB2t.
For0<k<ζχ, we have(25)dektVwt=LektVwtdt+ektζ1+wtζ−1δ1StdB1t+δ2ItdB2t,so(26)EektVwt=Vw0+E∫0tLeksVwsds,where(27)LektVw=kektVw+ektLVw≤ζekt1+wζ−2kζ1+w2−χW2+σ−ηw+σ=ζekt1+wζ−2−χ−kζw2+σ−η+2kζw+σ+kζ≤ζektQ,and(28)Q≔supw∈ℝ+1+wζ−2−χ−kζw2+σ−η+2kζw+σ+kζ+1.
Then, from (26), we get(29)Eekt1+wtζ≤1+w0ζ+ζQkekt.
Consequently, we have(30)limsupt⟶∞E1+wtζ≤ζQk≔Q0a.s.
By the continuity of functionwt, there exists a constant M>0 such that(31)E1+wtζ≤M,t≥0.
From (24), for sufficiently small ε>0,k=1,2,…, we have(32)Esupkε≤t≤k+1ε1+wtζ≤E1+wkεζ+I1+I2≤M+I1+I2,where(33)I1=Esupkε≤t≤k+1ε∫kεtζ1+wrζ−2−χw2r+σ−ηwr+σdr≤c1Esupkε≤t≤k+1ε∫kεt1+wrζdr≤c1E∫kεk+1ε1+wrζdr≤c1εEsupkε≤t≤k+1ε1+wtζ,(34)I2=Esupkε≤t≤k+1ε∫kεtζ1+wrζ−1δ1SrdB1r+δ2IrdB2r≤32E∫kεk+1εζ21+wr2ζ−1δ12S2r+δ22I2rdr1/2≤32Eζδ12∨δ221/2ε1/2Esupkε≤t≤k+1ε1+wr2ζ1/2≤32Eζδ12∨δ221/2ε1/2Esupkε≤t≤k+1ε1+wrζ.
By Burkholder–Davis–Gundy inequality [33], we obtain(35)Esupkε≤t≤k+1ε1+wtζ≤E1+wkεζ+c1ε+32ζδ12∨δ221/2ε1/2Esupkε≤t≤k+1ε1+wtζ.
In particular, letε>0 such that c1ε+32ζδ12∨δ221/2ε1/2≤1/2, and then(36)Esupkε≤t≤k+1ε1+wtζ≤2E1+wkεζ≤2M.
For sufficiently smallεw>0 and using Chebyshev’s inequality, we have(37)Psupkε≤t≤k+1ε1+wtζ>kε1+εw≤Esupkε≤t≤k+1ε1+wtζkε1+εw≤2Mkε1+εw,k=1,2,….
According to Borel–Cantelli lemma [33], we know that for almost all ω∈Ω,(38)supkε≤t≤k+1ε1+wtζ≤kε1+εwholds for all but finitely many k. Hence, there exists a k0ω, for almost all ω∈Ω, for which (38) holds whenever k≥k0. Therefore, for almost all ω∈Ω, if k≥k0 and kε≤t≤k+1ε,(39)log1+wtζlogt≤1+εwlogkεlogkε=1+εw.
Taking the limit superior, we have(40)limsupt⟶∞log1+wtζlogt=1+εwa.s.
Letεw⟶0, and we get(41)limsupt⟶∞log1+wtζlogt≤1a.s.
For1<ζ<1+2η/δ12∨δ22, i.e., η>ζ−1δ12δ22/2, we have(42)limsupt⟶∞logwtlogt≤limsupt⟶∞log1+wtζlogt≤1ζa.s.
In other words, for arbitrarily small0<ρ<1−1/ζ, there exist constant T=Tω and a set Ωρ such that PΩρ≥1−ρ, and for t≥T, ω∈Ωρ,(43)logwt≤1ζ+ρlogt,(44)limsupt⟶∞wtt≤limsupt⟶∞t1/ζ+ρt=0.
According to Theorem1, we get(45)limt⟶∞wtt=limSt+Itt=0a.s.,(46)limStt=0,limItt=0a.s.
The proof of Lemma2 is completed.
Denote(47)R1=σβηη+ω+δ22/2>1.Theorem 2.
IfR1>1, for the solution St,It of system (2) with any initial value S0,I0∈ℝ+2, we have(48)limsupt⟶∞It≥ηη+ω+δ22/2R1−1η+ωβ>0a.s.,i.e., the disease is persistent in mean.Proof.
Integrate and add the two equations of system (2) on both sides to get(49)St−S0t+It−I0t=σ−ηSt−η+ωIt+δ1∫0tSrdB1rt+δ2∫0tIrdB2rt.
Further calculation yields(50)St=ση−η+ωηIt+ω1+ω2−St−S0ηt−It−I0ηt,where(51)ω1=δ1∫0tSrdB1rηt,ω2=δ2∫0tIrdB2rηt.
By using Itô’s formula, one gets(52)dlnI=σ1−ϕI+βS−η+ω−δ222dt+δ2dB2t,(53)lnIt−lnI0t=σ1−ϕt∫0t1Irdr+βSt−η+ω+δ222+δ2B2tt≥βSt−η+ω+δ222+δ2B2tt=βση−η+ωηIt+ω1+ω2−St−S0ηt−It−I0ηt−η+ω+δ222+δ2B2tt=σβη−η+ωβηIt+βω1+βω2−βSt−S0ηt−βIt−I0ηt−η+ω+δ222+δ2B2tt.
Solving the above inequality, we have(54)It≥ηη+ωβlnI0−lnItt+ση+ω+ηη+ωω1+ηη+ωω2−St−S0η+ωt−It−I0η+ωt−ηη+ω+δ22/2η+ωβ+ηδ2B2tη+ωβt=ηη+ωβlnI0−lnItt+ηη+ωω1+ηη+ωω2−St−S0η+ωt−It−I0η+ωt+σβ−ηη+ω+δ22/2η+ωβ+ηδ2B2tη+ωβt=ηη+ω+δ22/2R1−1η+ωβ+ηη+ωω1+ηη+ωω2−St−S0η+ωt−It−I0η+ωt+ηη+ωβlnI0−lnItt+ηδ2B2tη+ωβt.
By taking the limit inferior of both sides of (54) and using Lemmas 1 and 2, we can obtain(55)limsupt⟶∞It≥ηη+ω+δ22/2R1−1η+ωβ.
This finishes the proof of Theorem2.


## 4. Existence of a Stationary Distribution

In this section, we explore existence of a stationary distribution for the solutions of system (2) under the condition R1>1.LetYt be a homogenous Markov process in l-dimensional Euclidean space El, and Yt satisfies(56)dYt=bYdt+∑r=1khrYdBrt.The diffusion matrix is defined as follows:(57)Ax=aijy,aijx=∑r=1khriyhrjy.Lemma 3 (see [34]).
The Markov processYt has a unique ergodic stationary distribution π⋅ if there exists a bounded domain U⊂El with regular boundary Γ and(A.1) There is a positive numberM such that(58)∑i,j=1naijyζiζj≥Mζ2,y∈U,ζ∈El.(A.2) There exists a nonnegativeC2-function V such that ℒV is negative for any El∖U. Then,(59)ℙylimτ⟶∞1τ∫0τgYtdt=∫Engypdy=1,for all y∈El, where g⋅ is a function integrable with respect to the measure p.Theorem 3.
For any given initial valueS0,I0∈ℝ+2, if(60)R2=σϕβδ12/2+ηη+ω+δ22/2>1holds, then system (2) has a unique ergodic stationary distribution.Proof.
According to Lemma3, we need to verify the conditions of the lemma one by one. Firstly, we get the diffusion matrix of system (2) with the form(61)A˜=δ12S200δ22I2.
Then, letM=minS,I∈U¯δ12S2,δ22I2, and we have(62)∑i,j=12aijξiξj=δ12S2ξ12+δ22I2ξ22≥Mξ2,S,I∈U¯,ξ∈ℝ2.
Thus, condition (A.1) in Lemma3 is satisfied.
In order to verify (A.2), we will construct a functionV∈C2ℝ+2;ℝ and a compact set Uϵ⊂ℝ+2 such that(63)ℒVS,I≤−1onS,I∈ℝ+2∖Uε.
In the following, let us construct aC2-function V: ℝ+2⟶ℝ with the form(64)V¯S,I=M−alnS−lnI−lnS+1θ+1S+Iθ+1,where θ and c are positive constants satisfying 0<θ<2η/δ12∨δ22,a=σϕβ/δ12/2+η2. Let us choose M>0 large enough such that(65)−Mλ2+D≤−2,where(66)λ2≔η+ω+δ222R2−1.
Obviously,(67)liminfw⟶S,I∈ℝ+2\UwV¯S,I=∞,where Uw=1/w,w×1/w,w. Because of the continuity of function V¯S,I, there exists a unique point S∗¯,I∗¯ in ℝ+2 which is the minimum point of V¯S,I. Thus, consider a positive-definite C2-function V: ℝ+2⟶ℝ+ with the form(68)VS,I,C=V¯S,I−V¯S∗¯,I∗¯=M−alnS−lnI−lnS+1θ+1S+Iθ+1−V¯S∗¯,I∗¯=MV1+V2+V3,where V1=−alnS−lnI, V2=−lnS, V3=1/θ+1S+Iθ+1−V¯S∗¯,I∗¯. From Itô’s formula, we have(69)ℒV1=−aσϕS+aβI+aη+aδ122−σ1−ϕI−βS+η+ω+δ222≤−aσϕS−βS+aβI+η+ω+aη+aδ122+δ222≤−2aσϕβ+aβI+η+ω+aδ122+η+δ222≤−σϕβδ12/2+η+η+ω+δ222+aβI≤−η+ω+δ222R2−1+aβI≔−λ2+aβI.
By using Itô’s formula, we obtain(70)ℒV2=−σϕS+βI+η+δ122.
Similarly,(71)ℒV3=S+Iθσ−ηS−η+ωI+θ2S+Iθ−1δ12S2+δ22I2≤σS+Iθ−ηS+Iθ+1+θ2δ12∨δ22S+Iθ+1=σS+Iθ−η−θ2δ12∨δ22S+Iθ+1≤B−12η−θ2δ12∨δ22S+Iθ+1,where(72)B=supS,I∈ℝ+2σS+Iθ−12η−θ2δ12∨δ22S+Iθ+1<∞.
Therefore,(73)ℒV≤−Mλ2+aMβI+βI−σϕS+η+δ122+B−12η−θ2δ12∨δ22S+Iθ+1≤−Mλ2+β1+aMI−σϕS−12η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122.
Consider a compact subsetU:(74)U=ε≤S≤1ε,ε≤I≤1ε,where ε is a constant small enough that(75)−σϕε+C≤−1,(76)−Mλ2+β1+aMε+D≤−1,(77)−14η−θ2δ12∨δ221εθ+1+E≤−1,(78)−14η−θ2δ12∨δ221εθ+1+F≤−1,where C, D, E, and F are positive constants with the form in equations (82), (84), (86), and (88), respectively. Then,(79)ℝ+2\U=U1∪U2∪U3∪U4,with(80)U1=S,I∈ℝ+2|0<S<ε,U2=S,I∈ℝ+2|0<I<ε,U3=S,I∈ℝ+2|S>1ε,U4=S,I∈ℝ+2|I>1ε.
Next, we discuss the negative ofℒV for any S,I∈ℝ+2\U in different regions.Case I. If S,I∈U1, we get(81)ℒV≤−σϕS+β1+aMI−12η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122≤−σϕε+C,where(82)C=supS,I∈ℝ+2β1+aMI−12η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122<∞.According to (75), we can have ℒV≤−1 for all S,I∈U1.Case II. If S,I∈U2, we can obtain that(83)ℒV≤−Mλ2+β1+aMI−12η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122≤−Mλ2+β1+aMε+D,where(84)D=supS,I∈ℝ+2−12η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122<∞.According to (76), we can have ℒV≤−1 for all S,I∈U2.Case III. If S,I∈U3, we have(85)ℒV≤−14η−θ2δ12∨δ22Sθ+1+β1+aMI−14η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122≤−14η−θ2δ12∨δ22Sθ+1+E≤−14η−θ2δ12∨δ221εθ+1+E,where(86)E=supS,I∈ℝ+2β1+aMI−14η−θ2δ12∨δ22Sθ+1−12η−θ2δ12∨δ22Iθ+1+B+η+δ122<∞.According to (77), we can get that ℒV≤−1 for all S,I∈U3.Case IV. If S,I∈U4, it follows that(87)ℒV≤−14η−θ2δ12∨δ22Iθ+1+β1+aMI−12η−θ2δ12∨δ22Sθ+1−14η−θ2δ12∨δ22Iθ+1+B+η+δ122≤−14η−θ2δ12∨δ22Iθ+1+F≤−14η−θ2δ12∨δ221εθ+1+F,where(88)F=supS,I∈ℝ+2β1+aMI−12η−θ2δ12∨δ22Sθ+1−14η−θ2δ12∨δ22Iθ+1+B+η+δ122<∞.According to (78), we can get that ℒV≤−1 for all S,I∈U4.
Thus, from the discussion of the above four cases, for a sufficiently smallε, we can get(89)ℒV≤−1forallS,I∈ℝ+2\U,which means condition (A.2) in Lemma 3 is satisfied. Hence, all the conditions in Lemma 3 are satisfied; then, according to Lemma 3, system (2) has a unique ergodic stationary distribution π⋅. This completes the proof of Theorem 3.


## 5. Numerical Simulation

In this section, we employ numerical simulation to verify the main results.For the deterministic system, we first consider the caseϕ=1, and the parameters for system (1) are chosen as shown in Table 1. If we let σ=0.0005, direct calculations show that system (1) has a disease-free equilibrium E00.25,0 which is globally asymptotically stable (see Figure 1), where R=0.8929<1.Table 1
Model parameters, their interpretation, and default value.
ParameterDescriptionDefault valueRef.ηHarvest rate0.002Holt and Chancellor [35]σPlanting rate0.0015Gibson and Aritua [36]ϕFraction planted from in vitro propagated, virus free, material1.0–0.0Feng et al. [37]ωRoguing rate parameter0.005Fondong et al. [38]βTransmission rate parameter0.025Gibson et al. [39]Figure 1
Simulation of solutions for system (1) with initial value S0,I0=1,0.6, where R=0.8929<1. (a) The series graph of S and I for system (1). (b) The phase diagrams of S and I.
(a)(b)If we increase planting rateσ to 0.0015, direct calculations show that system (1) has a disease equilibrium E1S1,I1=0.28,0.1343 which is globally asymptotically stable (see Figure 2), where R=2.6786>1.Figure 2
Simulation of solutions for system (1) with initial value S0,I0=1,0.6, where R=2.6786>1. (a) The series graph of S and I for system (1). (b) The phase diagrams of S and I.
(a)(b)For the caseϕ<1, the parameters are chosen as ϕ=0.8,β=0.025,η=0.002,andω=0.005, and simple calculations show that system (1) has a disease equilibrium ES∗,I∗=0.2032,0.1562 which is globally asymptotically stable (see the blue curve in Figures 3(a), 3(c), and 3(e)).Figure 3
Comparisons of solutions of system (1) and system (2), where δ1=0.04,δ2=0.03. (a) The sample paths of S. (b) The density functions of S in system (2). (c) The sample paths of I. (d) The density functions of I in system (2). (e) The phase diagrams of S and I.
(a)(b)(c)(d)(e)For the stochastic system, we get the following discrete system:(90)Sk+1=Sk+σϕ−βSkIk−ηSkΔt+δ1SkΔB1,k,Ik+1=Ik+σ1−ϕ+βSkIk−η+ωIkΔt+δ2IkΔB2,k,where ΔBi,k≜Btk+1−Btki=1,2 obeys the Gaussian distribution N0,Δt.Letδ1=0.04,δ2=0.03, and direct calculation shows R1=2.5168>1,R2=1.4382>1. According to Theorems 2 and 3, the disease will persist eventually, and moreover system (2) has a unique stationary distribution. The red curves in Figures 3(a) and 3(c) and the blue curves in Figures 3(e) show that the solutions of the stochastic system will fluctuate around the solution of the deterministic system. Figures 3(b) and 3(d) show that system (2) has a unique stationary distribution.From Theorems2 and 3, we get the key values(91)R1=σβηη+ω+δ22/2,R2=σϕβδ12/2+ηη+ω+δ22/2easily, and we can find that R1 is monotonically decreasing with respect to the intensity of white noise δ2 (see Figure 4(a)), and R2 is monotonically decreasing with respect to the intensity of white noises δ1 and δ2 (see Figure 4(b)).Figure 4
The relationship of the key values and intensity of white noise for stochastic system (2), where σ=0.0015,β=0.025,ϕ=0.8,η=0.002,andω=0.005. (a) The relationship of R1 and δ2 for stochastic system (2). (b) The relationship of R2 and δ1,δ2 for stochastic system (2).
(a)(b)


## 6. Discussion and Conclusion

In this paper, a simple stochastic model has been developed to model the spread of plant diseases. By using the theory and methods of stochastic differential equations, the dynamics of the system, such as the stochastic persistence of the disease and the existence of a stationary distribution, are discussed, and relevant sufficient conditions are obtained. Our results show that the system is deeply affected by stochastic interference.However, in the real world, the spread of plant diseases is very complicated. Plant diseases can be spread through air, rain, soil, insects and other organisms, grafting, sowing and transplanting, fertilizing, and irrigation [40, 41]. Considering two common transmission modes of plant disease transmission: insect and plant cuttings, Van den Bosch et al. [12] established the following model on the basis of model (1) as follows:(92)dStdt=σϕ+σ1−ϕr1−pIt+St1−pIt+St−ηSt−βStIt,dItdt=σ1−ϕ1−r1−pIt1−pIt+St−ηIt−ωIt+βStIt.In the model,σ represents the replanting rate, which is divided into two parts in the replanted plants, ϕ is the proportion of disease-free plants propagated in vitro, and 1−ϕ is the proportion of plants cultivated from cuttings of infected plants. r represents the probability that some infected plants may become disease-free plants due to reversal. Vision or other diagnostic methods are used to choose whether the plant is retained. If it is identified as an infected plant, it will be discarded with a probability of p. 1−r indicates the probability that the infected plant has not been reversed and is still an infected plant. 1−p indicates the probability that the infected plant is not detected by the diagnostic method and is retained. σ1−ϕ1−r1−pIt/1−pIt+St indicates the number of infected plants cultivated from cuttings of infected plants among all replanted plants. σ1−ϕr1−pIt+St/1−pIt+St represents the number of disease-free plants cultivated from cuttings of infected plants among all replanted plants. It is an interesting question to consider the model (92) from a stochastic perspective to explore the extinction or persistence of plant diseases, which helps us understand the impact of randomness on the spread of plant diseases. We will leave this question for future research.



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