Effects of vector preferential settling in the dynamics of a plant virus model with latent periods

Abstract

Insect vectors transmit many plant viruses of agricultural importance. In most cases, these viruses manipulate their vectors' behavior and movement leading to vector settling and feeding preferences that influence virus spread. The latent period within the insect vector is also crucial in virus transmission during vector feeding, however is assumed to be negligible in previous studies. This paper proposes a plant-virus model with a focus on vector settling preference for virus-infected plants and considers the latent periods in both the plant and insect vector populations by introducing them as time delays. We analyze the existence and local stability of the equilibrium solutions using the basic reproduction number of the proposed model which is inversely related to the saturation parameter due to vector preference. We investigate the effects of the time delays on the local stability of equilibria and the possible emergence of stable limit-cycle solutions. Moreover, our theoretical results are corroborated using numerical continuation and bifurcation analysis to provide more insights into how the latent periods and vector preference affect the system's dynamics. Particularly, we show numerically that increasing the saturation parameter due to preferential settling reduces the number of new infections, if not eradicates the infection in the system.

Graphical abstract

Highlights

• •The study proposed a novel plant-virus model focused on vector settling preference for virus-infected plants.
• •The latent periods in both the plant and insect vector populations are considered.
• •Analyzed the existence and local stability of the equilibria in terms of the basic reproduction number of the proposed model.
• •Investigated the effects of the time delays on the local stability of equilibria.
• •Examined possible emergence of stable limit-cycle solutions.

1. Introduction

Plant viral diseases have always attained a high socio-economic impact around the world. Almost all plants and crops that humans grow for food are affected by at least one virus which significantly reduce yield and production. The most destructive plant viruses are those affecting major crops in the developing world mainly because the greatest number of people would be affected in these regions. For example, the African cassava mosaic disease and maize streak disease caused major damage to two of the most common staple food crops in Africa resulting to hunger for years, while rice tungro disease and banana bunchy top disease continue to cause severe losses to rice and banana production in many developing countries in Asia [1].

Majority of plant viruses utilize a living organism, called a vector, to ensure their ability to move from one plant to another. About three-quarters of the known plant pathogenic viruses are naturally transmitted by plant-feeding insects [2]. Insect transmission of plant viruses has assumed enormous significance in plant health as the severity of a viral disease in plant correlates with the proportion of the insect population that acts as a vector for the virus [3]. Many insect-borne plant viruses of agricultural importance are persistently transmitted by insects including aphids, whiteflies, leafhoppers, planthoppers and thrips [4], [1]. Such viruses require vectors to feed on an infected host plant for a sustained period before the virus can potentially be transmitted to healthy plants during insect feeding, and the inoculativity by the vector is retained for days, weeks, or months, and often throughout the vector's lifespan [2].

Plant viruses can have direct or plant-mediated effects on their insect vectors which includes alterations in their vector behavior and movement, performance, and life-cycle [5], [6]. Preferential settling and feeding behavior on virus-infected plants by insect vectors have been observed and identified for some persistently transmitted viruses [7], [8], [9]. A plant virus modifying plant quality sufficiently to improve it as a feeding resource can directly influence the transmission of the virus. For instance, for non-persistently transmitted viruses, inhibition of settling while allowing probing would encourage transmission, whereas prolonged settling would retard transmission [10]. These suggest that plant viruses manipulate their vectors in a way that could enhance or diminish virus transmission efficiency and spread.

Time lags also play important roles in plant-virus-vector systems. The infectiousness of an infective plant or viruliferous vector does not occur instantly after exposure to a pathogen. The acquisition of a persistently transmitted virus by vectors requires longer periods and subsequently a latent period, which is the time it takes for the ingested virus to be incorporated into the salivary secretions of an insect, that may take several hours or days. Similarly, it takes some incubation period, which is from the inoculation of a healthy plant with virus by a feeding viruliferous insect vector to symptom onset, and correspondingly a latent period, which is the time interval from inoculation of healthy plant to becoming infectious.

Vector-borne plant diseases have become of interest to many mathematical modeling researchers. Several mathematical models have been proposed taking into account the different transmission characteristics of plant viruses, plant-virus control strategies, vector-population dynamics, and incubation and latent periods [11], [12], [13], [14]. In such models, bilinear incidence assumptions have been continuously modified to capture essential biological mechanisms and processes. For example, Basir et al. in [15] proposed a plant disease model with incubation period and Beddington-DeAngelis type disease transmission

$$f(x,v)=\frac{\beta xv}{1+\alpha v+{\alpha }_{2}x}\text{with}\alpha ,{\alpha }_2,\beta \ge 0$$ (1)

where β is the maximum contact rate of an infected vector v to a susceptible plant x and the saturation constants α and ${\alpha }_2$ represent the measures of the inhibition effect resulting from the crowding of vectors and plant resistance rate, respectively. The incubation period introduced in this model is incorporated as a time delay in the system. Such delay processes may give rise to oscillations in solutions as well as induce stability switches and Hopf bifurcations [15], [16].

Chen-Charpentier [17] recently presented two models of plant–virus transmission by a vector, Model A with constant population and Model B with non-constant population and saturation (1) in the disease transmission in plant population following [15]. They then accounted for the latent period in the plant host in two ways: (i) incorporating time delay in the system of differential equations and (ii) adding an exposed compartment in the system. The models' equilibrium points and stability were analyzed but only for particular values of parameters, specifically the natural death rate m of vectors, using numerical methods. In contrast to [15], Basir et al. took a more theoretical approach in analyzing the stability of the equilibria of their delayed plant disease model and proving the existence of a Hopf bifurcation for some critical incubation period. However, their model did not explicitly capture the infection between virus-infected plant hosts and non-viruliferous vectors.

In [15], the acquisition of virus by the insect vector is not considered while in [17] a simple bilinear incidence is assumed in the virus transmission within the insect vector population. In reality, the forces of infection from the inoculation of virus in healthy plant hosts and acquisition of virus by vectors may increase more gradually than linear from saturation effects brought by preferential settling as a consequence of virus manipulation of the host and vector in plant virus transmission. Moreover, the latent period of the virus in insect vectors was assumed to be zero in both studies which is not the case in many plant-virus-vector systems.

We address the gap discussed above and consider a more general plant-virus model with focus on vector settling preference and latent periods in both the plant host and vector populations. This model follows the modeling assumptions in [14] with the additional assumption that the insect vectors have a settling preference for virus-infected plants. That is, the incidence rate in the plant population is now saturated with the susceptible plants too, hence the use of the disease transmission in equation (1), and the introduction of the corresponding saturation constant ${\alpha }_2$ due to vector preference. We examine the effects of the time delays on the dynamical behavior of our model by determining the regions in the parameter space where the equilibrium of the system is locally asymptotically stable. By varying the saturation parameter ${\alpha }_2$ when both the time delays are nonzero, we show numerically the creation of the so-called island of stability, i.e. an isolated region on the parameter space where the endemic equilibrium is asymptotically stable.

The rest of the paper is organized as follows. In Section 2, we present and analyze the proposed model system using the basic reproduction number. Next, we establish the local stability of the disease-free (DFE) and endemic equilibria (EE) both in the absence and presence of the time delays in Section 3. We then give numerical simulations in Section 4 to illustrate our theoretical results. Furthermore, we perform numerical continuation and numerical bifurcation analyses to discover other features of the system. Finally, we summarize our discussions.

2. The proposed model

We expound more on the basic properties of our proposed model. Particularly, we show that solutions of the corresponding system are non-negative and bounded. We also compute the basic reproduction number of the model and provide the conditions for the existence of equilibria.

2.1. Plant-virus model with vector preference and latent periods

Our proposed model of plant virus propagation by a vector with nonlinear incidence rates in the plant and insect populations and latent periods is given by the following system of delay differential equations

(2)

Here, the plant population is partitioned into three classes, namely, the susceptible class $x(t)$, the infective class $y(t)$ and the recovered class $z(t)$. We assume that the virus does not kill the vector nor the vector defend against the virus. Also, an infected vector keeps the virus throughout its life and does not recover. In other words, we only partition the vector population into two classes, namely, the susceptible vectors $u(t)$ and infective vector $v(t)$. The disease incidence in the vector population is saturated only with the infected plants for reasons such as vector limitations on virus acquisition and/or interference in infected plants such as preventive and protective measures facilitated by humans. Contrary to plants, the force of infection in vectors does not saturate at large number of non-viruliferous vectors. This is due to the assumption that insect vectors, regardless of their infectivity status, will tend to settle and feed more on the virus-infected plants, which can result to higher number of infections as vector population is increased.

Table 1 lists the model parameters together with their corresponding description and range of values. The parameter values shown in Table 1 are not specific to a particular virus and plant. Some of the range of values are taken from [14] while other parameter values are assumed for purposes of comparison and illustration. Note that if the saturation constant ${\alpha }_2=0$, then the system given in equation (2) reduces to the system studied in [14]. That is, the model given in the system (2) is a generalization of the plant virus propagation model introduced in [14]. In the later simulations, we will also look at the effects of ${\alpha }_2$ in the system dynamics.

Table 1.

Summary of the parameters of the system models (2) and (5).

Symbol	Description	Value
K	total plant host population	50 − 1000
μ	natural death rate of plant hosts	0 − 0.1
d	disease-related death rate of plant hosts	0.05 − 0.1
β	infection rate of plants due to infected vectors	0.01 − 0.033
α	saturation constant due to infected vectors	0.01
α2	saturation constant due to healthy plant host	0 − 0.1
γ	recovery rate of infected plant hosts	0 − 0.25
β1	infection rate of vectors due to infected plants	0.01 − 0.025
α1	saturation constant due to infected plant hosts	0.01 − 0.02
Λ	replenishing rate of vectors	8
m	death rate of vector population	0 − 0.5
τ1	latent period of the disease in plants
τ2	latent period of the virus in insect vector

In the absence of the virus, the total plant population is comprised of healthy susceptible plants which stabilizes to the fixed constant K and only non-viruliferous insect vectors thrive in the system with their population closing in on the capacity $\mathrm{\Lambda }/m$. With the introduction of virus in system, the total number of plants at any time t remains constant at K, i.e.

$$K=x(t)+y(t)+z(t)$$ (3)

because any plant that dies naturally or due to the virus is immediately replaced with a new healthy plant. Following [14], we get $u(t)+v(t)\to \mathrm{\Lambda }/m$ as $t\to \infty$ implying the total number of insect vectors in the system do not exceed the value $\mathrm{\Lambda }/m$ at any time t. For simpler analysis, we assume

$$u(t)=\frac{\mathrm{\Lambda }}{m}-v(t).$$ (4)

Thus, using the equations (3) and (4), we obtain the following reduced model corresponding to the system in equation (2)

(5)

where $\eta =d+\mu +\gamma$. All the parameters are assumed to be positive. The time delays ${\tau }_1$ and ${\tau }_2$ represent the latent period of viral disease in plants and the latent period of the virus in insect vectors, respectively, and are independent of each other.

Let $\tau =\max \{{\tau }_1,{\tau }_2\}$. The initial conditions for the system given in equation (5) take the form

$$x(\theta )={\varphi }_1(\theta ),y(\theta )={\varphi }_2(\theta ),v(\theta )={\varphi }_3(\theta ),0\le ({\varphi }_1+{\varphi }_2)\le K,0\le {\varphi }_3\le \mathrm{\Lambda }/m$$ (6)

where $({\varphi }_1(\theta ),{\varphi }_2(\theta ),{\varphi }_3(\theta ))\in C([-\tau ,0],{\mathbb{R}}_{+}^3)$. By the fundamental theory of functional differential equations, for each initial condition given in equation (6), the system in equation (5) has a unique solution $(x(t),y(t),v(t))$ for $t\ge 0$.

2.2. Non-negativity and boundedness of solutions

For biological reasons, we find relevant conditions so that the solution of system given in equation (5) has non-negative components for all $t\ge 0$. Suppose $x(t)=0$ or $y(t)=0$ or $v(t)=0$ for some time t and consider the non-negative initial history conditions in equation (6). Clearly, $y^{\prime }(t)\ge 0$ when $y(t)=0$ and $v^{\prime }(t)\ge 0$ when $v(t)=0$. If $x(t)=0$ and $y(t)\ge 0$, then $x^{\prime }(t)\ge 0$ provided that ${\varphi }_1$ and ${\varphi }_3$ satisfy

$$\frac{\beta {\varphi }_3{\varphi }_1}{1+\alpha {\varphi }_1+{\alpha }_2{\varphi }_3}\le \mu K.$$ (7)

Thus, the solution of system (5) with initial conditions in equations (6) and (7) lies in ${\mathbb{R}}_{+}^3\cup \{\mathbf{\text{0}}\}$ for all $t\ge 0$. In addition, the solution is always bounded from the assumptions that total number of plants is given by the fixed constant K and total number of insect vectors equals $\mathrm{\Lambda }/m$ for all $t\ge 0$.

Lemma 1

All solutions of system (5) are nonnegative on $[0,+\infty )$ given the initial conditions in equation (6) satisfying (7) . Moreover, the solutions of system (5) are bounded.

2.3. Equilibria of the system (5)

We now derive the biologically relevant equilibrium solutions of the system (5), and provide conditions for these equilibria to exist. An equilibrium $E_{⁎}=(x_{⁎},y_{⁎},v_{⁎})$ of the system (5) satisfies the following system of equations

$$\mu K+d{y}_{⁎}-\mu x_{⁎}-\frac{\beta v_{⁎}{x}_{⁎}}{1+\alpha v_{⁎}+{\alpha }_2{x}_{⁎}}=0,$$ (8)

$$\frac{\beta v_{⁎}{x}_{⁎}}{1+\alpha v_{⁎}+{\alpha }_2{x}_{⁎}}-\eta y_{⁎}=0,$$ (9)

$$\frac{{\beta }_1{y}_{⁎}}{1+{\alpha }_1{y}_{⁎}}(\frac{\mathrm{\Lambda }}{m}-v_{⁎})-m{v}_{⁎}=0.$$ (10)

Adding the equations (8) and (9) to eliminate $v_{⁎}$, and then solving for $y_{⁎}$ yields

$$y_{⁎}=\frac{\mu (K-x_{⁎})}{\mu +\gamma },$$ (11)

while solving for $v_{⁎}$ in the equation (10) gives

$$v_{⁎}=\frac{\mathrm{\Lambda }{\beta }_1{y}_{⁎}}{m^2+m^2{\alpha }_1{y}_{⁎}+m{\beta }_1{y}_{⁎}}.$$ (12)

Now, substituting the above expressions for $y_{⁎}$ and $v_{⁎}$ to the equation (9), we obtain the following cubic equation in $x_{⁎}$

$$\mu (x_{⁎}-K)(A{x}_{⁎}^2+B{x}_{⁎}+C)=0,$$ (13)

where the coefficients A, B, and C are as follows

$$A=-{\alpha }_2\mu \eta m({\alpha }_{1}m+{\beta }_1)<0,$$ (14)

$$B={\alpha }_2\mu \eta mK({\alpha }_{1}m+{\beta }_1)-\mu \eta (\alpha {\beta }_1\mathrm{\Lambda }+{\alpha }_1{m}^2+{\beta }_{1}m)-(\mu +\gamma )(\mathrm{\Lambda }\beta {\beta }_1-{\alpha }_2\eta m^2),$$ (15)

$$C=\mu \eta K(\alpha {\beta }_1\mathrm{\Lambda }+{\alpha }_1{m}^2+{\beta }_{1}m)+\eta m^2(\mu +\gamma )>0.$$ (16)

Clearly, $x_{⁎}=K$ is a root of the equation (13). This yields $y_{⁎}=0$ and $v_{⁎}=0$ using the equations (11) and (12), and the disease-free equilibrium (DFE)

$$E_0:=(K,0,0).$$ (17)

Next, we define the threshold value

$$\tilde{R}=\frac{\mathrm{\Lambda }\beta {\beta }_{1}K}{m^2\eta (1+{\alpha }_{2}K)}.$$ (18)

We show that if $\tilde{R}>1$, then the system (5) has a unique endemic equilibrium (EE) given by

$$E_1:=(x_1,y_1,v_1)$$ (19)

with $x_1,y_1,v_1>0$, and where the component $x_1$ is the unique positive root of the quadratic equation

$$p(x):=A{x}^2+Bx+C=0,$$ (20)

with coefficients $A,B$, and C in equations (14), (15), and (16), respectively. Since $A<0$ and $C>0$, the graph of $p(x)$ is a parabola that opens downward and intersects the vertical axis above the x-axis. Consequently, the equation (20) has a unique positive root $x_1$. Indeed, $x_1\in (0,K)$. To see this, note that

$$p(0)=C>0.$$

Moreover, if $\tilde{R}>1$, then

$$p(K)=-(\mu +\gamma )[\mathrm{\Lambda }\beta {\beta }_{1}K-m^2\eta (1+{\alpha }_{2}K)]<0$$

using the definition of $\tilde{R}$ in the equation (18). Thus, the existence of the positive root $x_1\in (0,K)$ is guaranteed by the continuity of $p(x)$ on $[0,K]$ and using the intermediate value theorem. Since $x_1<K$, we get a positive value for the corresponding $y_1$ using the equation (11), which then gives a positive value for the corresponding $v_1$ from equation (12). That is, the unique endemic equilibrium $E_1$ exists whenever $\tilde{R}>1$. However, observe that if $\tilde{R}\le 1$, then $p(K)\ge 0$. Thus, the quadratic equation (20) does not have a positive root in the open interval $(0,K)$, and consequently, the endemic equilibrium $E_1$ does not exist in this case. We summarize the above discussions in the following result.

Theorem 2

The disease-free equilibrium $E_0$ of the system (5) given in the equation (17) always exists, while the unique endemic equilibrium $E_1$ given in the equation (19) exists if and only if $\tilde{R}>1$ .

Remark 1

The threshold value $\tilde{R}$ given in equation (18) defines the basic reproduction number $R_0$ for the model (5). The basic reproduction number represents the average number of secondary infections caused by an infected individual in a completely susceptible population [18]. This number is significant since it provides us with a way of predicting the spread of an infection in the population. More precisely, if $R_0<1$, then the infection eventually dies out, while if $R_0>1$, then the infection persists. In the case of vector-transmitted diseases, the basic reproductive number is more often reported as the square root of the threshold parameter, taking into consideration the cyclically alternating generations that have taken place, one from the infected vector to the susceptible hosts and the second from the infected host to the vectors [19], [20]. Hence, for the plant virus model (5), the basic reproduction number is given by

$$R_0:=\sqrt{\tilde{R}}=\sqrt{\frac{\mathrm{\Lambda }\beta {\beta }_{1}K}{m^2\eta (1+{\alpha }_{2}K)}}.$$ (21)

Consequently, Theorem 2 can be restated in terms of the basic reproduction number $R_0$ as follows.

Theorem 3 Restatement of Theorem 2 —

The disease-free equilibrium $E_0$ of the system (5) given in the equation (17) always exists, while the unique endemic equilibrium $E_1$ given in the equation (19) exists if and only if $R_0>1$ .

Furthermore, notice that the expression for $R_0$ given in the equation (21) is independent of the saturation constants α and ${\alpha }_1$, but is dependent of the saturation constant ${\alpha }_2$. Moreover, observe that the parameter ${\alpha }_2$ that was introduced from incorporating insect preference on virus-infected plants has an inverse relationship to the basic reproduction number. This means that ${\alpha }_2$ can have a significant role on the persistence or eradication of the infection in the system. The effects of varying ${\alpha }_2$ will be explored later on in the numerical simulations.

3. Main results

We now derive the conditions needed for the equilibrium solutions of the system (5) to be locally asymptotically stable. To do this, we need to examine the linearized system about a given equilibrium and the distribution of the roots of its corresponding characteristic equation on the complex plane. The reader may refer to the texts [21], [22], [23] for further background on the theory of delay differential equations.

3.1. Characteristic equation

If we denote by $\mathbf{X}(t)={[x(t),y(t),v(t)]}^{\top }$, then the linearized system corresponding to the system (5) at an equilibrium $E_{⁎}=(x_{⁎},y_{⁎},v_{⁎})$ is given by

$$\frac{d}{dt}\mathbf{X}\left(t\right)={\mathbf{\text{M}}}_0\mathbf{X}\left(t\right)+{\mathbf{\text{M}}}_1\mathbf{X}(t-{\tau }_1)+{\mathbf{\text{M}}}_2\mathbf{X}(t-{\tau }_2)$$ (22)

where the $3\times 3$ matrices ${\mathbf{\text{M}}}_0$, ${\mathbf{\text{M}}}_1$, and ${\mathbf{\text{M}}}_2$ are given as follows

with

$$A_1=\frac{\beta v_{⁎}(1+\alpha v_{⁎})}{{(1+\alpha v_{⁎}+{\alpha }_2{x}_{⁎})}^2},$$ (23)

$$A_2=\frac{\beta x_{⁎}(1+{\alpha }_2{x}_{⁎})}{{(1+\alpha v_{⁎}+{\alpha }_2{x}_{⁎})}^2},$$ (24)

$$A_3=\frac{{\beta }_1}{{(1+{\alpha }_1{y}_{⁎})}^2}(\frac{\mathrm{\Lambda }}{m}-v_{⁎}),$$ (25)

$$A_4=\frac{{\beta }_1{y}_{⁎}}{1+{\alpha }_1{y}_{⁎}}.$$ (26)

Remark 2

For the disease-free equilibrium $E_0=(K,0,0)$, we obtain $A_1=0$, $A_2=\beta K/(1+{\alpha }_{2}K)>0$, $A_3={\beta }_1\mathrm{\Lambda }/m>0$ and $A_4=0$ after substituting $(x_{⁎},y_{⁎},v_{⁎})=(K,0,0)$ in the equations (23), (24), (25), and (26). Meanwhile, for the endemic equilibrium $E_1=(x_1,y_1,v_1)$ where the components $x_1>0$, $y_1>0$ and $v_1>0$, the corresponding quantities $A_i(i=1,2,3,4)$ are all positive.

The characteristic equation corresponding to the linearized system in equation (22) is

$$\det (\lambda {\mathbf{I}}_3-{\mathbf{M}}_0-{\mathbf{M}}_1{e}^{-\lambda {\tau }_1}-{\mathbf{M}}_2{e}^{-\lambda {\tau }_2})=0,$$ (27)

where ${\mathbf{I}}_3$ is the identity matrix of dimension 3. If all roots of the characteristic equation (27) lie in the open left-half of the complex plane, i.e. if the $\text{Re}(\lambda )<0$ for all roots λ of the equation (27), then the equilibrium $E_{⁎}=(x_{⁎},y_{⁎},v_{⁎})$ of the system (5) is locally asymptotically stable (LAS).

3.2. Local stability of the disease-free equilibrium

At the DFE $E_0=(K,0,0)$, the characteristic equation (27) takes the following form

$$(\lambda +\mu )[{\lambda }^2+(m+\eta )\lambda +m\eta -\frac{\mathrm{\Lambda }\beta {\beta }_{1}K}{m(1+{\alpha }_{2}K)}{e}^{-\lambda \tau }]=0$$ (28)

where $\tau ={\tau }_1+{\tau }_2$. Since $\lambda =-\mu <0$ is a root of the equation (28), the local stability of the DFE now only depends on the distribution of the roots of the following transcendental equation on the complex plane

$${\lambda }^2+(m+\eta )\lambda +m\eta -\frac{\mathrm{\Lambda }\beta {\beta }_{1}K}{m(1+{\alpha }_{2}K)}{e}^{-\lambda \tau }=0.$$ (29)

First, observe that if $\tau =0$, i.e. when both the time delays ${\tau }_1$ and ${\tau }_2$ are zero, then the equation (29) can be written as

$${\lambda }^2+(m+\eta )\lambda +m\eta (1-R_0^2)=0$$

using the expression for the basic reproduction number $R_0$ as given in the equation (21). Immediately, we see that for the case where both time delays are zero, the DFE is LAS if and only if $R_0<1$.

Let us now consider the case where $\tau >0$, i.e. the case where one or both the time delays are positive, and suppose further that $R_0<1$ so that the DFE is LAS when $\tau =0$. According to [24], as τ is increased, the stability of an equilibrium can change if a zero appears on the imaginary axis and cross it. So we first need to check if the equation (29) can have a root or roots along the imaginary axis, i.e. check if $\lambda =0$ or $\lambda =\pm i\omega$ with $\omega >0$ are roots of the equation (29). We show that both the former and the latter are not possible under the assumption that $R_0<1$, and hence the DFE remains LAS for all $\tau >0$ when $R_0<1$.

If $\lambda =0$ is a root of the equation (29), then we obtain $R_0=1$. Since we assume that $R_0<1$, we see that $\lambda =0$ is not a root of the equation (29). If $\lambda =i\omega$ with $\omega >0$ is a root of the equation (29), then we get the following equations which are obtained by substituting $\lambda =i\omega$ in the equation (29) and then separating the real and imaginary parts

$${\omega }^2-m\eta =-\frac{\mathrm{\Lambda }\beta {\beta }_{1}K}{m(1+{\alpha }_{2}K)}\cos \left(\omega \tau \right)$$

and

$$(m+\eta )\omega =-\frac{\mathrm{\Lambda }\beta {\beta }_{1}K}{m(1+{\alpha }_{2}K)}\sin \left(\omega \tau \right).$$

Eliminating τ in the above equations yields the following degree 4 even polynomial equation in ω

$${\omega }^4+(m^2+{\eta }^2){\omega }^2+m^2{\eta }^2(1-R_0^4)=0.$$ (30)

Since we assumed that $R_0<1$, we have $(1-R_0^4)>0$ and thus the equation (30) does not have any positive roots. Consequently, the equation (28) cannot have purely imaginary roots.

Theorem 4

The disease-free equilibrium $E_0$ of the system (5) is locally asymptotically stable for all ${\tau }_1\ge 0$ and ${\tau }_2\ge 0$ if and only if $R_0<1$ .

If an equilibrium of a system of delay differential equations is asymptotically stable for all time delays, then it is said to be absolutely stable [25]. Theorem 4 then tells us that a necessary and sufficient condition for the absolute stability of the DFE is that $R_0<1$. In other words, the latent periods do not affect the eradication of the infection on a particular plant population as long as the number of secondary cases is kept low, i.e. the basic reproduction number $R_0<1$. In the subsection that follows, we consider the case when $R_0>1$ and show that, in contrast to the case for the DFE discussed in this subsection, the latent periods may affect the local stability of the endemic equilibrium.

3.3. Local stability of the endemic equilibrium

Here, we assume that $R_0>1$ so that endemic equilibrium $E_1=(x_1,y_1,v_1)$ exists. That is, the components $x_1$, $y_1$ and $v_1$ are all positive. The characteristic equation (27) corresponding to the linearized system about $E_1$ can be written as follows

$$P_0\left(\lambda \right)+P_1\left(\lambda \right)e^{-\lambda {\tau }_1}+P_2\left(\lambda \right)e^{-\lambda {\tau }_2}+P_3\left(\lambda \right)e^{-\lambda ({\tau }_1+{\tau }_2)}=0$$ (31)

where

$$P_0\left(\lambda \right)=(\lambda +\eta )(\lambda +\mu )(\lambda +m),P_1\left(\lambda \right)=(\lambda +\gamma +\mu )(\lambda +m)A_1,P_2\left(\lambda \right)=(\lambda +\eta )(\lambda +\mu )A_4,P_3\left(\lambda \right)=(A_1{A}_4-A_2{A}_3)\lambda +(\gamma +\mu )A_1{A}_4-\mu A_2{A}_3.$$

To organize the rest of this subsection, we consider three cases: (i) when both the time delays are zero; (ii) when exactly one of the time delays is positive; and (iii) when both the time delays are positive. For sake of brevity, we only consider for (ii), the case where ${\tau }_1=0$ and ${\tau }_2>0$, and then (iii) builds on this case, fixing the value of ${\tau }_2$ where $E_1$ is LAS, and then varying the value of ${\tau }_1$.

Case 1: ${\tau }_1=0$ and ${\tau }_2=0$

We first show that the endemic equilibrium is LAS when the time-delay parameters in the system (5) are both zero. When ${\tau }_1=0$ and ${\tau }_2=0$ in the characteristic equation (31), we obtain the following cubic equation

$${\lambda }^3+a_2{\lambda }^2+a_1\lambda +a_0=0$$ (32)

where

$$a_0=(m+A_4)(\gamma +\mu )A_1+\eta \mu A_4+\mu (m\eta -A_2{A}_3),$$ (33)

$$a_1=\eta \mu +m\mu +(m+\gamma +\mu )A_1+(\eta +\mu )A_4+A_1{A}_4+(m\eta -A_2{A}_3),$$ (34)

$$a_2=\eta +\mu +m+A_1+A_4.$$ (35)

Lemma 5

The coefficients $a_0$ , $a_1$ and $a_2$ in the equation (32) are all positive. Moreover, $a_2{a}_1>a_0$ .

Proof

Recall that all system parameters appearing on the right-hand side of the equations (33), (34), and (35) are all positive. Moreover, since $R_0>1$, the quantities $A_i(i=1,2,3,4)$ are all positive as mentioned in Remark 2. Immediately, we see that $a_2>0$, while the coefficients $a_0$ and $a_1$ are positive if $(m\eta -A_2{A}_3)>0$. We now show that $A_2{A}_3<m\eta$. Using the expressions for $A_2$ and $A_3$ given in the equations (24) and (25) and the relations given in equations (9) and (10), we get

$$A_2{A}_3=\frac{\beta x_1(1+{\alpha }_2{x}_1)}{{(1+\alpha v_1+{\alpha }_2{x}_1)}^2}\cdot \frac{{\beta }_1}{{(1+{\alpha }_1{y}_1)}^2}(\frac{\mathrm{\Lambda }}{m}-v_1)=m\eta \left(\frac{1+{\alpha }_2{x}_1}{1+\alpha v_1+{\alpha }_2{x}_1}\right)\left(\frac{1}{1+{\alpha }_1{y}_1}\right).$$

Since the quantities inside the parenthesis are both less than one, we see that $A_2{A}_3<m\eta$. Consequently, $(m\eta -A_2{A}_3)>0$ and thus $a_0>0$ and $a_1>0$ proving the first assertion.

We next show the second assertion that $(a_2{a}_1-a_0)>0$. From the equations (34) and (35), we obtain the relations $a_1>\eta \mu +(\gamma +\mu )A_1$ and $a_2>m+A_4$. Hence, we have

$$(a_2{a}_1-a_0)>(m+A_4)[\eta \mu +(\gamma +\mu )A_1]-[(m+A_4)(\gamma +\mu )A_1+\eta \mu A_4+\mu (m\eta -A_2{A}_3)]=\mu A_2{A}_3.$$

Since $\mu A_2{A}_3>0$, the second assertion follows and the proof is complete.

Lemma 5 and the Routh-Hurwitz criterion tell us that all roots of the cubic equation (32) have negative real part. Thus, we have the following result.

Theorem 6

The endemic equilibrium $E_1$ of the system (5) with ${\tau }_1=0$ and ${\tau }_2=0$ , when it exists, i.e. when $R_0>1$ , is locally asymptotically stable.

Remark 3

The conditions listed in Lemma 5 disallow codimension-one bifurcations to occur in the system (5) with ${\tau }_1=0$ and ${\tau }_2=0$.

Case 2: ${\tau }_1=0$ and ${\tau }_2>0$

When ${\tau }_1=0$ in the characteristic equation (31), we obtain

$$({\lambda }^3+b_2{\lambda }^2+b_1\lambda +b_0)+(c_2{\lambda }^2+c_1\lambda +c_0)e^{-\lambda {\tau }_2}=0$$ (36)

where the coefficients are as follows

$$b_0=\eta \mu m+(\gamma +\mu )m{A}_1,$$ (37)

$$b_1=\eta \mu +m\mu +m\eta +(m+\gamma +\mu )A_1,$$ (38)

$$b_2=\eta +\mu +m+A_1,$$ (39)

$$c_0=\eta \mu A_4+(\gamma +\mu )A_1{A}_4-\mu A_2{A}_3,$$ (40)

$$c_1=(\eta +\mu )A_4+A_1{A}_4-A_2{A}_3,$$ (41)

$$c_2=A_4.$$ (42)

Using Lemma 5 and the expression for $a_0$ from the equation (33), we see that $b_0+c_0=a_0>0$. Thus, $\lambda =0$ is not a root of the equation (36). Suppose now that the equation (36) has a root $\lambda =i\omega$ with $\omega >0$. Then,

$$(-i{\omega }^3-b_2{\omega }^2+i{b}_1\omega +b_0)+(-c_2{\omega }^2+i{c}_1\omega +c_0)e^{-i\omega {\tau }_2}=0,$$ (43)

which yields the following equations after separating the real and imaginary parts of equation (43)

$$b_2{\omega }^2-b_0=(-c_2{\omega }^2+c_0)\cdot \cos (\omega {\tau }_2)+c_1\omega \cdot \sin (\omega {\tau }_2),$$ (44)

$${\omega }^3-b_1\omega =(c_2{\omega }^2-c_0)\cdot \sin (\omega {\tau }_2)+c_1\omega \cdot \cos (\omega {\tau }_2).$$ (45)

Eliminating ${\tau }_2$ in equations (44) and (45), we get

$${(b_2{\omega }^2-b_0)}^2+{({\omega }^3-b_1\omega )}^2={(c_2{\omega }^2-c_0)}^2+{\left(c_1\omega \right)}^2$$ (46)

or equivalently, equation (46) simplifies to the following cubic equation in ν

$$F(\nu ):={\nu }^3+d_2{\nu }^2+d_1\nu +d_0=0$$ (47)

with $\nu ={\omega }^2$ and where the coefficients are as follows

$$d_0=b_0^2-c_0^2,$$ (48)

$$d_1=b_1^2-c_1^2-2b_0{b}_2+2c_0{c}_2,$$ (49)

$$d_2=b_2^2-c_2^2-2b_1.$$ (50)

The cubic polynomial $F(\nu )$ defined in the equation (47) plays an important role in the local stability of the endemic equilibrium. Specifically, we look at the cases where the cubic equation (47) has positive roots or none at all. The latter case results to absolute stability of the endemic equilibrium, while the former case allows the occurrence of stability switches.

Theorem 7

If the cubic equation (47) does not have positive roots, then the endemic equilibrium $E_1$ of the system (5) with ${\tau }_1=0$ is LAS for all ${\tau }_2>0$ .

A sufficient condition is given in the following corollary.

Corollary 8

If the coefficients $d_i$ given in the equations (48) , (49) , and (50) are all positive, then the endemic equilibrium $E_1$ of the system (5) with ${\tau }_1=0$ is LAS for all ${\tau }_2>0$ .

We now turn our attention to the case where the cubic equation (47) has positive roots. The following lemma provides a sufficient condition for such case to occur.

Lemma 9

If the coefficient $d_0<0$ , then the cubic equation (47) has at least one positive root.

To provide a more general discussion, let us assume that the cubic equation (47) has exactly three positive simple roots denoted by ${\nu }_j$ with $j=1,2,3$. Corresponding to these positive roots of the equation (47) are the purely imaginary roots $\lambda =\pm i{\omega }_j$ of the characteristic equation (36), where ${\omega }_j=\sqrt{{\nu }_j}$ for $j=1,2,3$, occurring respectively at the time-delay values ${\tau }_2={\tau }_k^{(j)}(k=0,1,2,\dots )$ where

$${\tau }_k^{(j)}:=\frac{1}{{\omega }_j}\cdot \{{\cos }^{-1}\left(\frac{(c_0-c_2{\omega }_j^2)(b_2{\omega }_j^2-b_0)-c_1{\omega }_j(b_1{\omega }_j-{\omega }_j^3)}{{\left(c_1{\omega }_j\right)}^2+{(c_0-c_2{\omega }_j^2)}^2}\right)+2\pi \cdot k\}$$ (51)

obtained using the equations (44) and (45). The following lemma, whose proof can be found in [26, see e.g. pp.84-85], tells us how the roots of the equation (36) that are on the imaginary axis at ${\tau }_2={\tau }_k^{(j)}$ will traverse the imaginary axis.

Lemma 10

Let$\lambda \left({\tau }_2\right)$be a simple root of the characteristic equation(36)satisfying$\lambda \left({\tau }_k^{(j)}\right)=\pm i{\omega }_j$, where${\tau }_k^{(j)}$is given in equation(51). Then,

$$\mathrm{sign}\left\{{\frac{\mathrm{d}}{\mathrm{d}{\tau }_2}\left\{\mathrm{Re}\lambda \left({\tau }_2\right)\right\}|}_{{\tau }_2={\tau }_k^{(j)}}\right\}=\mathrm{sign}\left\{F^{\prime }\left({\nu }_j\right)\right\}.$$

Apparently, the movement of the roots of the equation (36) that are on the imaginary axis at ${\tau }_2={\tau }_k^{(j)}$ depends on whether the graph of the cubic polynomial $F\left(\nu \right)$ is increasing or decreasing at $\nu ={\nu }_j$ corresponding to the time-delay value ${\tau }_k^{(j)}$.

Theorem 11

Suppose $d_0<0$ and let

$${\tau }_2^{⁎}:=\min \{{\tau }_k^{(j)}>0|j=1,2,3\mathit{\text{and}}k=0,1,2,\dots \}.$$

If $F^{\prime }\left({\nu }^{⁎}\right)>0$ where ${\nu }^{⁎}$ corresponds to critical time-delay value ${\tau }_2^{⁎}$ , then the endemic equilibrium $E_1$ of the system (5) with ${\tau }_1=0$ is LAS whenever ${\tau }_2\in (0,{\tau }_2^{⁎})$ . Moreover, at ${\tau }_2={\tau }_2^{⁎}$ , the system undergoes a Hopf bifurcation at $E_1$ .

Example 1

Consider the system (5) with parameters $K=70$, $\mathrm{\Lambda }=8$, $m=0.25$, $d=0.05$, $\mu =0.02$, $\gamma =0.05$, $\alpha =0.01$, ${\alpha }_1=0.01$, ${\alpha }_2=0.10$, $\beta =0.033$, ${\beta }_1=0.025$, and ${\tau }_1=0$. Then, we get $R_0\approx 2.774887>1$ from the equation (21). Thus, the endemic equilibrium exists and is given by

$$E_1\approx (6.193191,18.230517,19.411224).$$ (52)

The coefficients of the cubic polynomial $F(\nu )$ defined in the equation (47) are $d_2\approx 0.015367$, $d_1\approx -0.002741$, and $d_0\approx -0.000024$, which are computed using the values in equations (37), (38), (39), (40), (41), and (42). Since $d_0<0$, the cubic equation (47) has at least one positive root using Lemma 9. In fact, this cubic equation has exactly one positive simple root ${\nu }^{⁎}\approx 0.049578$ as shown in the graph of $F(\nu )$ in Fig. 1.

The characteristic equation (36) has a pair of purely imaginary roots $\lambda =\pm i\sqrt{v^{⁎}}$ which occurs at the critical time-delay values ${\tau }_2={\tau }_k^{(1)}(k=0,1,2,\dots )$ as given in the equation (51). Since $\left\{{\tau }_k^{(1)}\right\}$ is an increasing sequence, the minimum occurs when $k=0$, so

$${\tau }_2^{⁎}={\tau }_0^{(1)}\approx 11.172956$$

as in Theorem 11. Moreover, the graph of $F(\nu )$, as shown in Fig. 1, is increasing at $\nu ={\nu }^{⁎}$, i.e. $F^{\prime }({\nu }^{⁎})>0$. Therefore, by Theorem 11, the endemic equilibrium $E_1$ given in equation (52) is LAS for ${\tau }_2\in (0,{\tau }_2^{⁎})$, as seen in Fig. 2(a) while for values of ${\tau }_2$ immediately beyond threshold ${\tau }_2^{⁎}$ we expect small-amplitude limit cycles. Fig. 2(b) illustrates this switch towards instability as well as the occurrence of Hopf bifurcation at ${\tau }_2={\tau }_2^{⁎}$.

Figure 1.

Graph of the cubic polynomial F(ν) showing the lone positive root ν^⁎ ≈ 0.049578.

Figure 2.

Time-series plots of x(t), y(t), and v(t) for the cases (a) ${\tau }_2=10.50<{\tau }_2^{⁎}$ and (b) ${\tau }_2=11.20>{\tau }_2^{⁎}$. Hopf Bifurcation occurs at ${\tau }_2^{⁎}\approx 11.1729556$.

Remark 4

The stability of the endemic equilibrium $E_1$ in the previous Example 1 can only switch towards instability. This is because the cubic equation $F(\nu )=0$ has exactly one positive root ${\nu }^{⁎}$ and $F^{\prime }({\nu }^{⁎})>0$. Lemma 10 tells us that the roots of the equation (36) that are on the imaginary axis at ${\tau }_2={\tau }_k^{(j)}$ can only move towards the open right-half of the complex plane. That is, once $E_1$ becomes unstable it can no longer regain its stability.

One can make analogous discussion as above for Case 2 but instead considering the case where ${\tau }_2=0$ and ${\tau }_1>0$. For sake of brevity, we do not show the derivations here but we will eventually obtain results similar to Theorem 11. That is, the endemic equilibrium $E_1$ is LAS for ${\tau }_1\in (0,{\tau }_1^{⁎})$, and the switch towards instability at ${\tau }_1={\tau }_1^{⁎}$ is due to a Hopf bifurcation. Figs. 3(a) and 3(b) show an example, using the same set of parameters as in Example 1 but with ${\tau }_2=0$, where a LAS endemic equilibrium $E_1$ becomes unstable at ${\tau }_1={\tau }_1^{⁎}\approx 10.731545$ where small-amplitude limit cycles are created by the Hopf bifurcation occurring at ${\tau }_1={\tau }_1^{⁎}$. This means that varying the value the latent period ${\tau }_1$ may also cause the endemic equilibrium to switch stability similar to the earlier discussions in Case 2 where ${\tau }_2$ is varying.

Figure 3.

Time-series plots of x(t), y(t), and v(t) for the cases (a) ${\tau }_1=10.50<{\tau }_1^{⁎}$ and (b) ${\tau }_1=10.75>{\tau }_1^{⁎}$. Hopf Bifurcation occurs at ${\tau }_1^{⁎}\approx 10.73154$.

Case 3: ${\tau }_1>0$ and ${\tau }_2>0$

We now consider the case where both time-delay parameters are positive. Specifically, we fix the value of ${\tau }_2\in (0,{\tau }_2^{⁎})$ where ${\tau }_2^{⁎}$ is the critical value of ${\tau }_2$ as described in Theorem 11. This assumption guarantees that the endemic equilibrium is LAS when ${\tau }_1=0$. We want to know what will happen as we increase the value of ${\tau }_1$ from zero.

If the value of ${\tau }_2$ is fixed, then the characteristic equation (31) can be written in the following form

$$P\left(\lambda \right)+Q\left(\lambda \right)\cdot e^{-\lambda {\tau }_1}=0$$ (53)

where $P\left(\lambda \right)=P_0\left(\lambda \right)+P_2\left(\lambda \right)\cdot e^{-\lambda {\tau }_2}$, $Q\left(\lambda \right)=P_1\left(\lambda \right)+P_3\left(\lambda \right)\cdot e^{-\lambda {\tau }_2}$, and $P_i(\lambda )$, $i=1,2,3,4$, are the same quantities as in the characteristic equation (31). Like in Case 2, we are interested when the characteristic equation (53) has a pair of purely imaginary simple roots, that is, when the endemic equilibrium becomes unstable due to the occurrence a Hopf bifurcation. If the equation (53) has a root $\lambda =i\omega$ with $\omega >0$, then $P\left(i\omega \right)+Q\left(i\omega \right)\cdot e^{-i\omega {\tau }_1}=0$. Using the notations

$$P\left(i\omega \right)=P_R\left(\omega \right)+i{P}_I\left(\omega \right)\text{and}Q\left(i\omega \right)=Q_R\left(\omega \right)+i{Q}_I\left(\omega \right),$$

where the real and imaginary parts of P and Q are as follows

$$P_R\left(\omega \right)=(-{\omega }^2+\eta \mu )\cdot [m+A_4\cos \left(\omega {\tau }_2\right)]-(\eta +\mu )\omega \cdot [\omega -A_4\sin \left(\omega {\tau }_2\right)],P_I\left(\omega \right)=(-{\omega }^2+\eta \mu )\cdot [\omega -A_4\sin \left(\omega {\tau }_2\right)]+(\eta +\mu )\omega \cdot [m+A_4\cos \left(\omega {\tau }_2\right)],$$

and

$$Q_R\left(\omega \right)=-A_1{\omega }^2+(\gamma +\mu )m{A}_1+[(\gamma +\mu )A_1{A}_4-\mu A_2{A}_3]\cos \left(\omega {\tau }_2\right)+(A_1{A}_4-A_2{A}_3)\omega \sin \left(\omega {\tau }_2\right),Q_I\left(\omega \right)=(\gamma +\mu +m)A_1\omega +(A_1{A}_4-A_2{A}_3)\omega \cos \left(\omega {\tau }_2\right)-[(\gamma +\mu )A_1{A}_4-\mu A_2{A}_3]\sin \left(\omega {\tau }_2\right),$$

we then get

$$[P_R\left(\omega \right)+i{P}_I\left(\omega \right)]+[Q_R\left(\omega \right)+i{Q}_I\left(\omega \right)]\cdot [\cos \left(\omega {\tau }_1\right)-i\sin \left(\omega {\tau }_1\right)]=0.$$ (54)

The following system of equations was obtained by separating the real and imaginary parts in the equation (54)

(55)

Eliminating ${\tau }_1$ in equation (55), we get

$$P_R^2\left(\omega \right)+P_I^2\left(\omega \right)=Q_R^2\left(\omega \right)+Q_I^2\left(\omega \right).$$

This means that if $\lambda =i\omega$ with $\omega >0$ is a root of the characteristic equation (53), then

$$G\left(\omega \right):=P_R^2\left(\omega \right)+P_I^2\left(\omega \right)-Q_R^2\left(\omega \right)-Q_I^2\left(\omega \right)=0$$ (56)

has a positive root. The contraposition of this statement yields a condition for absolute stability of the endemic equilibrium when the value of ${\tau }_2$ is fixed.

Theorem 12

Let ${\tau }_2\in (0,{\tau }_2^{⁎})$ be fixed. If the equation (56) has no positive roots, then the endemic equilibrium $E_1$ is LAS for all ${\tau }_1>0$ .

Switches in the stability of $E_1$ may occur if the equation $G(\omega )=0$ has positive roots. We explore this possibility for the rest of this section. Note that the function $G(\omega )$, given in the equation (56), is a combination of polynomials in ω, and the sine and cosine functions. Moreover, the function value $G(\omega )$ increases without bound as $|\omega |$ increases without bound. That is, the equation (56) cannot have infinitely many roots. Suppose now that the equation (56) has exactly n positive roots, say ${\omega }_1$, ${\omega }_2$, …, ${\omega }_k$, …, ${\omega }_n$. Corresponding to these ω values are the following respective sequences of time-delay values $\left\{{\tau }_j^{(1)}\right\}$, $\left\{{\tau }_j^{(2)}\right\}$, …, $\left\{{\tau }_j^{(k)}\right\}$, …, $\left\{{\tau }_j^{(n)}\right\}$ where $j=0,1,2,\dots$ such that $\lambda =i{\omega }_k$ is a root of the characteristic equation (53) when ${\tau }_1={\tau }_j^{(k)}$ for $j=0,1,2,\dots$. Using equation (55), we get

$${\tau }_j^{(k)}=\frac{1}{{\omega }_k}\cdot \{{\cos }^{-1}(-\frac{P_R\left({\omega }_k\right)Q_R\left({\omega }_k\right)+P_I\left({\omega }_k\right)Q_I\left({\omega }_k\right)}{Q_R^2\left({\omega }_k\right)+Q_I^2\left({\omega }_k\right)})+2\pi j\}.$$ (57)

The following lemma tells us how the roots of the characteristic equation (53) that are on the imaginary axis at ${\tau }_1={\tau }_j^{(k)}(j=0,1,2,\dots )$ will traverse the imaginary axis. Similar to Lemma 10, its proof can be found in [26].

Lemma 13

Let$\lambda \left({\tau }_1\right)$be a simple root of the characteristic equation(53)satisfying$\lambda \left({\tau }_j^{(k)}\right)=\pm i{\omega }_k$with values of${\tau }_j^{(k)}(j=0,1,2,\dots )$given in the equation(57). Then,

$$\mathrm{sign}\left\{{\frac{\mathrm{d}}{\mathrm{d}{\tau }_1}\left\{\mathrm{Re}\lambda \left({\tau }_1\right)\right\}|}_{{\tau }_1={\tau }_j^{(k)}}\right\}=\mathrm{sign}\left\{G^{\prime }\left({\omega }_k\right)\right\}.$$

In other words, the monotonicity of the function $G(\omega )$ given in the equation (56) at its zero ${\omega }_k$ determines the movement of the roots of the characteristic equation (53) that are on the imaginary axis at ${\tau }_1={\tau }_j^{(k)}(j=0,1,2,\dots )$.

The following theorem gives a scenario where a LAS endemic equilibrium becomes unstable when ${\tau }_1$ is varied and ${\tau }_2$ is fixed. This switch towards instability occurs at a Hopf bifurcation for some critical value of ${\tau }_1$. Here, the complex conjugate roots of the characteristic equation (53) that are on the imaginary axis at this critical ${\tau }_1$ value move towards the open right-half of the complex plane.

Theorem 14

Suppose that the equation $G(\omega )=0$ has at least one positive root and all roots are simple. Denote by

$${\tau }^{⁎}=\min \{{\tau }_j^{(k)}>0|k=1,2,\dots ,n\mathit{\text{and}}j=0,1,2,\dots \}.$$

If $G^{\prime }\left({\omega }^{⁎}\right)>0$ , where ${\omega }^{⁎}$ corresponds to critical time-delay value ${\tau }^{⁎}$ , then the endemic equilibrium $E_1$ of the system (5) with fixed ${\tau }_2\in (0,{\tau }_2^{⁎})$ is LAS whenever ${\tau }_1\in (0,{\tau }^{⁎})$ . Moreover, at ${\tau }_1={\tau }^{⁎}$ , the system undergoes a Hopf bifurcation at $E_1$ .

Example 2

We use the same parameter values as in Example 1. In addition, we fixed ${\tau }_2=6$ so that it is inside the interval $(0,{\tau }_2^{⁎})$ where ${\tau }_2^{⁎}\approx 11.172956$ as obtained in Example 1. The equation (56) has exactly 3 positive simple roots as shown by the graph of the function $G(\omega )$ in Fig. 4. We denote these 3 positive roots by ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$, and their approximate values are as follows

$${\omega }_1\approx 0.169202,{\omega }_2\approx 0.332718,\text{and}{\omega }_3\approx 0.388820.$$

Corresponding to the ω-values ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ are the sequences of time-delay values $\{{\tau }_j^{(1)}\}$, $\{{\tau }_j^{(2)}\}$, and $\{{\tau }_j^{(3)}\}$, respectively, which are computed using the formula given in the equation (57). The approximate values of the first few terms in these sequences are listed in Table 2.

Figure 4.

Graph of the function G(ω) showing the three positive roots ω₁ ≈ 0.169202, ω₂ ≈ 0.332718, and ω₃ ≈ 0.388820.

Table 2.

Approximate values of ${\tau }_j^{(k)}$ for k = 1,2,3 and j = 0,1,2.

j	${\tau }_j^{(1)}$	${\tau }_j^{(2)}$	${\tau }_j^{(3)}$
0	11.836393	07.602306	03.556582
1	48.970594	26.486748	19.716190
2	86.104796	38.824587	35.875799

The sequences $\{{\tau }_j^{(1)}\}$, $\{{\tau }_j^{(2)}\}$, and $\{{\tau }_j^{(3)}\}$ are all increasing. Hence, the critical value of ${\tau }_1$ as described in Theorem 14 is given by

$${\tau }_{⁎}={\tau }_0^{(3)}\approx 3.556582$$

which corresponds to the positive root ${\omega }_3$ of $G(\omega )=0$. From the graph of $G(\omega )$ in Fig. 4, we see that $G(\omega )$ is increasing at $\omega ={\omega }_3$, i.e. $G^{\prime }({\omega }_3)>0$. Hence, by Theorem 14, the endemic equilibrium $E_1$ of the system (5) with fixed ${\tau }_2\in (0,{\tau }_2^{⁎})$ is LAS whenever ${\tau }_1\in (0,{\tau }^{⁎})$, as observed in Fig. 5(a). Moreover, since the system undergoes a Hopf bifurcation at ${\tau }_1={\tau }^{⁎}$, we obtain small-amplitude limit-cycle solutions when ${\tau }_1$ is slightly beyond the threshold ${\tau }^{⁎}$. Fig. 5(b) illustrates the switch in the stability of $E_1$ due to the occurrence of Hopf bifurcation at ${\tau }_1={\tau }^{⁎}$, as well as the periodic solution obtained when ${\tau }_1=3.58>{\tau }^{⁎}$.

Figure 5.

Time-series plots of x(t), y(t), and v(t) for the cases (a) τ₁ = 3.25 < τ^⁎ and (b) τ₁ = 3.58 > τ^⁎. We fixed τ₂ = 6 in both cases, and the Hopf bifurcation occurs at τ^⁎ ≈ 3.556582.

Remark 5

From the graph of $G(\omega )$ in Fig. 4, we see that $G^{\prime }({\omega }_1)>0$, $G^{\prime }({\omega }_2)<0$, and $G^{\prime }({\omega }_3)>0$. By Lemma 13, this means that the roots of the characteristic equation (53) that are on the imaginary axis at ${\tau }_1={\tau }_j^{(k)}$ can move either towards the open half-left or towards the open right-half of the complex plane. In other words, multiple stability switches may occur in Example 2, unlike in Example 1 where there is only a one-time switch towards instability. We further explore these multiple stability switches in the next section using numerical continuation.

4. Numerical simulations

We now illustrate some of the main results from the previous section utilizing a numerical continuation and bifurcation analysis tool. In particular, we use the Matlab package DDE-Biftool that is suitable for analyzing systems of delay differential equations with several fixed discrete delays. DDE-Biftool was originally created by Koen Engelborghs at KU Leuven in 2001 [27], while the current version DDE-Biftool v3.1.1 is being maintained by Jan Sieber [28]. It is capable of computing, continuing, and analyzing the stability of the equilibrium solutions, as well as their bifurcations. We refer the readers to the following papers on DDE-Biftool [29], [30], and to the following manual and tutorials [16], [28]. We start by illustrating the case of multiple stability switches as mentioned in the Remark 5.

4.1. Multiple stability switches

In Example 2, by fixing the value of ${\tau }_2=6$ and then increasing value of ${\tau }_1$ from zero, a LAS $E_1$ becomes unstable at ${\tau }_1={\tau }_0^{(3)}$. This switch towards instability is due to a Hopf bifurcation, where a conjugate pair of characteristic roots that are on the imaginary axis at ${\tau }_1={\tau }_0^{(3)}$ move towards the open right-half of the complex plane since $G^{\prime }({\omega }_3)>0$. We want to know what happens to the stability of $E_1$ when we further increase the value of ${\tau }_1$. Notice that in Table 2, the next Hopf bifurcation occurs when ${\tau }_1={\tau }_0^{(2)}$. Since $G^{\prime }({\omega }_2)<0$, the conjugate pair of characteristic roots that are on the imaginary axis at ${\tau }_1={\tau }_0^{(2)}$ move towards the open left-half of the complex plane. Apparently, in this particular example, this conjugate pair of characteristic roots is the same pair that moved to the right-half of the complex plane when ${\tau }_1>{\tau }_0^{(3)}$ and is now moving to the left-half of the complex plane when ${\tau }_1>{\tau }_0^{(2)}$ and thus $E_1$ regains its stability. Figs. 6(a) and 6(b) illustrate the movement of this conjugate pair of characteristic roots as ${\tau }_1$ is increased. This shows the switch towards instability at ${\tau }_1={\tau }_0^{(3)}$, and another switch at ${\tau }_1={\tau }_0^{(2)}$ but this time towards stability.

Figure 6.

(a) Characteristic roots crossing the imaginary axis as τ₁ is varied. (b) Magnification of the same image showing the location of the roots before and after the threshold values ${\tau }_0^{(3)}\approx 3.556582$ and ${\tau }_0^{(2)}\approx 7.602306$.

The abovementioned dynamical behavior can be neatly illustrated using DDE-Biftool. In Fig. 7, the branch of the endemic equilibria is shown as ${\tau }_1$ is varied. Stable and unstable parts of this branch are shown in green and magenta, respectively. The Hopf bifurcations are marked with asterisk (⁎), and the values of ${\tau }_1$ where these Hopf bifurcations occur agree with the values shown in Table 2, which are obtained theoretically. The endemic equilibrium $E_1$ switches stability 3 times at ${\tau }_1={\tau }_0^{(3)}$, ${\tau }_1={\tau }_0^{(2)}$, and ${\tau }_1={\tau }_0^{(1)}$. For ${\tau }_1>{\tau }_0^{(1)}$, $E_1$ remains unstable.

Figure 7.

Branch of the endemic equilibrium E₁ obtained by fixing τ₂ = 6 and varying the time-delay parameter τ₁.

4.2. Hopf bifurcation curves

To obtain the Hopf bifurcation curves, we perform a 2-parameter continuation in DDE-Biftool using the time-delay parameters ${\tau }_1$ and ${\tau }_2$. Fig. 8 shows the Hopf bifurcation curves in this 2-parameter space. The value ${\tau }_2^{⁎}$ in the vertical axis is the threshold value identified in Case 2 when ${\tau }_1=0$ in the system (5). As computed in Example 1, ${\tau }_2^{⁎}\approx 11.172956$ and at this value of ${\tau }_2$ the system undergoes a Hopf bifurcation as guaranteed by Theorem 11. We can continue this Hopf bifurcation into a branch of Hopf bifurcations in DDE-Biftool varying the time delay ${\tau }_1$ to obtain the red curve shown in Fig. 8. Similarly, the value ${\tau }_1^{⁎}$ in the horizontal axis is the threshold value identified in the case when ${\tau }_2=0$ in the system (5). This was briefly discussed in the paragraph after Remark 4, where the threshold value ${\tau }_1^{⁎}\approx 10.731545$ was given. Since in this case, the system (5) also undergoes a Hopf bifurcation at ${\tau }_1={\tau }_1^{⁎}$, we can continue this Hopf bifurcation into a branch of Hopf bifurcations in DDE-Biftool this time varying the time delay ${\tau }_2$ to obtain the blue curve in Fig. 8.

Figure 8.

Hopf bifurcation curves in (τ₁,τ₂)−plane when α₂ = 0.10. The green regions are the stability regions of E₁.

The Hopf bifurcation curves in Fig. 8 partition the 2-parameter space into several regions. Specifically, the endemic equilibrium $E_1$ is LAS if values of the time-delay parameters ${\tau }_1$ and ${\tau }_2$ are chosen such that the point $({\tau }_1,{\tau }_2)$ is inside the green-shaded regions. Case 1 in the previous section, where $E_1$ is LAS, is shown here at the point $({\tau }_1,{\tau }_2)=(0,0)$. Also, Case 2 in the previous section where ${\tau }_1=0$, can be seen here by looking at the vertical axis which shows that $E_1$ is LAS when ${\tau }_2\in (0,{\tau }_2^{⁎})$. Moreover, Case 3 is shown here using the thin black horizontal line where ${\tau }_2=6$. This black line passed through the green-shaded region twice, which is actually the stability switches described in the previous subsection. We remark here that some choices for the time-delay parameters $({\tau }_1,{\tau }_2)$ in the stability regions may not be observable in the real world as vector latent periods are typically shorter than plant latent periods. However, we do not reject entirely the possibility that plant latent periods may be negligible for some plant-virus-vector systems since the independence of the two latent periods was assumed for the model.

Fig. 8 offers a more comprehensive viewpoint of our previous results, and more. One interesting case obtained in the 2-parameter continuation in Fig. 8 is the isolated region created by the loop of the blue Hopf bifurcation curve where the endemic equilibrium $E_1$ is LAS. In the succeeding subsection, we explore how this island of stability is created and the effects of varying the values of the parameter ${\alpha }_2$ to the dynamical behavior of the system.

4.3. Creation of the ‘island of stability’ and the effects of varying ${\alpha }_2$

We end this section by showing that the so-called island of stability, as shown in Fig. 8, is in fact originally a part of the larger stability region and was created when the new saturation parameter ${\alpha }_2$ of the model system (5) is increased. Fig. 9 shows the Hopf bifurcation curves and the stability regions for different values of ${\alpha }_2$. When ${\alpha }_2=0.0970$ and ${\alpha }_2=0.0975$, the stability region of $E_1$ consists of a single region, as seen in Figs. 9(a) and 9(b). Meanwhile, when ${\alpha }_2=0.0980$, a portion of the stability region becomes separated forming two stability regions, one of which is the island of stability. Increasing ${\tau }_2$ further, e.g. when ${\tau }_2=0.0985$, we observe that the stability regions become smaller. These islands of stability formed in the latter two cases are illustrated in Figs. 9(c) and 9(d), respectively. This shrinking of the stability regions for $E_1$ continues as ${\alpha }_2$ is increased, and eventually the regions will vanish.

Figure 9.

Creation of the ‘island of stability’ by the Hopf bifurcation curves as the saturation parameter α₂ is varied: (a) α₂ = 0.0970, (b) α₂ = 0.0975, (c) α₂ = 0.0980, and (d) α₂ = 0.0985.

5. Summary and conclusions

We considered a more general plant virus propagation model with vector preference, and latent periods in both the plant host and insect vector populations. In particular, we assumed that the virus affects the behavior and movement of the insect vectors, and additionally there are time lags to become infectious that occur both within the plant host and insect vector after exposure to viruses. These give rise to our proposed model, which is a system of delay differential equations with Beddington-DeAngelis type incidence rate to reflect the insect vector preference, and two discrete time delays to depict the latent periods. We provided conditions for the existence of the disease-free equilibrium and the endemic equilibrium in terms of the basic reproduction number of the system. Our results show that only the disease-free equilibrium exists when the reproduction number is less than one, while the endemic equilibrium only exists when the reproduction number is greater than one.

Using the time delays as main parameters, in the former case, the disease-free equilibrium is shown to be absolutely stable. That is, it is locally asymptotically stable for any positive values of the time delays. In other words, the latent periods do not affect the eradication of the infection in the system as long as the number of secondary cases is kept low. On the contrary, in the latter case, the latent periods may cause a locally asymptotically stable endemic equilibrium to become unstable. This occurs when the system undergoes a Hopf bifurcation as the value of one time-delay parameter reaches a threshold. Consequently, small-amplitude limit-cycle solutions are observed when a time-delay parameter crosses its threshold value.

Numerical continuation and bifurcation analysis allowed us to show scenarios where the endemic equilibrium can switch stability, possibly multiple times due to a series of Hopf bifurcations occurring as one of time-delay parameters is increased. These same numerical techniques gave us the Hopf bifurcation curves which partition the two-parameter space into regions where the endemic equilibrium is locally asymptotically stable and where there are limit-cycle solutions. Thus, we gained more insights on how the latent periods affect the dynamics of the system.

Lastly, we examined the effects of one of the saturation-rate parameter, which is a new parameter not considered in previous works. This saturation constant due to healthy plant hosts and vector settling preference is inversely related to the reproduction number, and so the intuitive approach is to increase the value of this parameter in order to lower the reproduction number. Specifically, we gave a scenario where increasing the value of this saturation parameter causes the region of stability of the endemic equilibrium in the 2-parameter space to decrease. Our analysis in the two-dimensional parameter space showed a better perspective on the dynamical behavior of the system when we vary the value of this saturation constant but the reproduction number remains larger than one.

This study primarily examined the effects of vector preference and latent periods on the dynamical behavior of the proposed plant virus propagation model, focusing on the local stability and bifurcations of equilibria. The parameter values used were derived from previous models to facilitate comparison with existing frameworks. However, a key limitation of this work is the reliance on theoretical parameters rather than empirical data. Future studies would greatly benefit from the integration of experimental data, which would allow for a more accurate parameter calibration, enhancing the predictive power and applicability of the model to real-world scenarios.

CRediT authorship contribution statement

Kimben M. Gonzales: Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization, Validation. Juancho A. Collera: Writing – review & editing, Writing – original draft, Visualization, Supervision, Software, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No new data was generated for the research described in the article.