Introduction

Bananas are a significant source of income, a favourite fruit, and a staple crop in many parts of the world. However, the emergence of devastating plant diseases such as banana bunchy top disease (BBTD) prevents banana production from reaching its full potential. Hence, it affects people’s livelihood. BBTD caused by a banana bunchy top virus (BBTV) seriously threatens banana production in Africa1. BBTD causes significant stunting and a severe decrease in the production of 90 to \(100\%\) within two production seasons2

Banana bunchy top virus is transmitted from an infected banana plant to a healthy banana plant via an aphid vector (Pentalonia nigronervosa Coquerel)3. Aphid vectors transmit the BBTV in a circulative and nonpropagative manner4. Aphid vectors acquire BBTV after feeding on an infected banana plant and later feed on a susceptible banana plant. On the other hand, the BBTV can also be transmitted from one area to another through latently infected banana planting material3. Therefore, the disease spreads more quickly. The incubation period of BBTV ranges from \(25-85\) days after inoculation depending on the speed of leaf emergence, the type of banana cultivar, the genetic variation across BBTV strains and temperature5. Signs of BBTD infection include dark green streaks on the leaves; thinner, shrunk upright and bunched leaves; fruits produced by infected plants grow improperly and may not emerge entirely from the leaf cluster; and plants infected when they are young may not produce fruits6

The control strategies applied to control BBTD include farming education campaigns, aphid vector clearance, timely detection and removal of symptomatic infected banana plants, herbicides and restricting the transportation of planting material from infected areas7. Despite these efforts to control the disease, BBTD remains one of the most devastating viral diseases affecting banana production worldwide, causing severe yield losses and economic hardship for farmers. Niyongere et al.6 observed that no cultivar resists BBTD. Although no cultivar is resistant to the BBTV, differences in the levels of susceptibility and time to symptom expression among different cultivars have been reported8. Furthermore,6,9 Argued that the more sustainable way of managing the BBTD is to design plants resistant to the virus and the vector. Fortunately,10 reported the existence of Musa balbisiana, a natural genetic resource in the Philippines, with complete resistance to BBTV. The existence and application of banana planting material that resists BBTV affect the dynamics of BBTD9 .

Mathematical models are valuable and indispensable for organizing, implementing, and evaluating the effectiveness of disease diagnosis, management, and preventative strategies and comprehending the transmission dynamics of infectious diseases; mathematical models are indispensable11. The mathematical modelling of crop diseases in plant pathology is a rapidly advancing field12,13,14. Researchers have developed various models to study the transmission dynamics of BBTD, including15,16,17,18. Allen15 developed a stochastic spatiotemporal polyclinic mathematical model to study the dynamics of BBTD, specifically by determining the mean inoculation distance of BBTV spread by the aphid vector and the latent period from the day of infection. The study revealed that the mean distance of spread for the aphid vector was 15.2 M, and the latent period was 59.8 days. However, a significant limitation of this study was its omission of the potential resistance of various banana planting materials to the virus, which could influence the outcomes of the disease model.

Smith et al.16 developed a nonspatial and spatial model to examine the quantitative epidemiology of BBTD and its control. These findings demonstrated that, for disease incidence to increase exponentially over time, the rate of disease progression may be influenced by either the rate of growth under external inoculum pressure or the rates of internal transmission and roguing rate. Nevertheless, like Allen’s work, this study did not account for the impact of planting material resistance to BBTV. On the other hand, a linked model using STELLA software was developed to simulate epidemics of BBTD in a changing climate17. The effect of climate change on bunchy top epidemics was simulated by adding \({1}\,^{\circ }\)C and \({2}\,^{\circ }\)C to the average monthly temperature for 1998-2007 in Davao City, Philippines17. The results revealed that an increase in the monthly average temperature reduced the number of viruliferous aphids and disease incidence. However, this study did not consider the resistance planting materials to the banana bunchy top virus, this is factor may significantly alter disease outcomes.

Recently,18 formulated a novel stochastic model to study within-field disease dynamics of the banana bunchy top virus via approximate Bayesian computation. This study’s results show that temperature variations affect disease dynamics. However, the study did not account for the resistance of planting material to the banana bunchy top virus. Furthermore, In their study,19 developed a mathematical model to analyze the transmission dynamics of banana bunchy top disease (BBTD) in banana plants and to evaluate various control strategies. They found that increasing the removal rate of infected plants and effectively managing the transmission of the virus from infected plants to susceptible ones are crucial strategies for reducing the number of infected plants and controlling the disease. Nonetheless, this study also overlooked the resistance of planting material to the banana bunchy top virus.

Most of the mathematical models developed in the literature have yet to investigate the impact of replanting with BBTV-resistant planting material on the dynamics of BBTD. Hence, this underscores the need to develop a mathematical model that considers BBTV-resistant planting material in the dynamics of BBTD.

Therefore, this paper develops and analyses a mathematical model that integrates BBTV-resistant planting material into the dynamics of BBTD, underscoring the manuscript’s significant contribution to plant pathology specifically in understanding the transmission dynamics of the BBTD. The remaining parts of the article are organized as follows: The next section presents the model formulation and analysis results. The following section discusses numerical simulation results. Finally, we outline some concluding remarks which summarize the work.

Main results

Model formulation

This study applies the concepts from20,21 to formulate a mathematical model for analysing the dynamics of BBTD incorporating disease resistance components. Banana plants and aphid vector populations are considered in the development of the model system and are categorised into compartments according to their infection levels. The banana plant population is categorized into susceptible \((S_p)\), latently infected \((I_l)\), and symptomatic infected \((I_s)\) plants. Susceptible banana plants \((S_p)\) refer to healthy banana plants, encompassing both those vulnerable to BBTV infection and those resistant to BBTV infection. We subsequently categorised the vector population into susceptible vectors \((S_a)\) and contaminated vectors \((I_a)\), referred to as contaminated vectors in this paper. Since a limited number of banana plants must be allowed on the farm, we assume logistic population growth, as shown in22,23. The rate of \(S_p\) recruitment is denoted by \(\Lambda\). The recruitment process may involve the importation of planting material or the emergence of new suckers from banana plants on the farm. The recruitment of \(I_l\) on the farm depends on the probability of selecting a latently infected planting material (\(\omega\)). An \(S_p\) acquires a BBTV from an \(I_a\) at \(\beta _b\). The infection happens when an \(I_a\) feeds on a \(S_p\). A \(I_l\) can progress to \(I_s\) at the rate of \(\theta\) after showing symptoms. When an contaminated vector vector feeds on a BBTV-resistant banana plant, the resistant plants do not become infected and therefore do not contribute to disease transmission. A \(S_p\) resists BBTV infection at a rate of \(\alpha\). Early symptoms are not easily detectable, often leading to latent infections and a lack of clean seeds5. According to24, complete roguing could result in the undesirable removal of healthy plants. Therefore, the model includes only the removal of \(I_s\) at the rate \(\gamma\). Mature banana plants are harvested at the rate of \(\mu\). We assume that the vector population grows logistically at a rate of \(\Lambda _a\), either through birth or immigration from neighbouring farms. The probability of recruiting aphid vectors infected by the BBTV is given by \(\rho\). A susceptible vector acquires the BBTV after feeding on an \(I_s\) at a rate of \(\beta _v\); after being contaminated with BBTV, \(S_a\) progresses to the \(I_a\) compartment. The natural or artificial clearance of aphid vectors on a farm is given by \(\mu _a\). The following assumptions guide this study:

  1. i.

    The replanting rate of banana plants exceeds the harvesting rate of mature plants \((\Lambda > \mu )\). This assumption is intended to ensure that farming activities remain continuous. If it happen that the number of banana plants harvested exceed the rate of replanting, there would eventually be no banana plants left over time.

  2. ii.

    The recruitment of aphid vectors, including susceptible and infected individuals, exceeds the death rate. \((\Lambda _a > \mu _{a})\). This assumption is intended to ensure that there are aphid vectors in the plantation to complete the system. If the death rate of aphid vectors exceeds its recruitment rate, there will eventually be no aphid vectors left.

  3. iii.

    Due to limited resources, such as available planting space, we assume that the populations in this study will demonstrate logistic growth. This approach aims to adhere to scientific good farming practices by allowing only the number of banana plants that are scientifically advised in a specific plantation space.

  4. iv.

    There is a uniform transmission rate across all stages plant growth, regardless of growth stage. This assumption is based on the observation that Banana Bunchy Top Disease (BBTD) transmission primarily depends on aphid activity and host susceptibility rather than plant age. Since aphid vectors do not strongly prefer specific growth stages, the probability of infection remains relatively consistent across plants in a given area.

  5. v.

    BBTV-resistant banana plants are not infective; hence, after resisting BBTV, they remain in a group of healthy banana plants.

Under the assumption above, we establish the system of ordinary differential equations presented in (1). Table 1 details the parameters used in this model. Refer to Fig. 1 for the best representation of the model.

Table 1 Variables and parameters of the model.
Full size table
Fig. 1
figure 1

Compartmental diagram illustrating the dynamics of BBTD, considering banana plants’ resistance to BBTV infection. The green rectangles denote susceptible banana plants and aphid vectors, whereas red rectangles signify symptomatic infected banana plants and contaminated aphid vectors. The blue rectangle represents latently infected banana plants. The solid lines indicate transitions between compartments, recruitment rates, harvesting rates, and vector mortality rates. The dashed lines illustrate the typical interactions between several compartments in transmitting the disease.

Full size image

From the compartmental diagram in Figure 1 above, we formulate a system of differential equations as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{dS_p}{dt} & =\Lambda S_p\Big (1-\frac{N_p}{K_p}\Big )+\alpha I_l- \frac{\beta _bI_aS_p}{N_a}-\mu S_p,\\ \frac{dI_l}{dt} & = \Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big ) + \frac{\beta _bI_aS_p}{N_a} - \theta I_l - \mu I_l-\alpha I_l,\\ \frac{dI_s}{dt} & = \theta I_l-\mu I_s - \gamma I_s,\\ \frac{dS_a}{dt} & =\Lambda _a S_a\Big (1-\frac{N_a}{K_a}\Big )- \frac{\beta _vI_sS_a}{N_p}-\mu _aS_a,\\ \frac{dI_a}{dt} & =\Lambda _a \rho I_a\Big (1-\frac{N_a}{K_a}\Big )+ \frac{\beta _vI_sS_a}{N_p}-\mu _aI_a.\\ \end{array}\right. } \end{aligned}$$
(1)

with initial conditions, \(S_p(0)>0, I_l(0)>0, I_s(0)>0, S_a(0)>0, I_a(0)>0\)

where;

$$\begin{aligned} \lambda _1= \displaystyle \frac{\beta _bI_aS_p}{N_a}, \end{aligned}$$
(2)

and

$$\begin{aligned} \lambda _2= \displaystyle \frac{\beta _vI_sS_a}{N_p}. \end{aligned}$$
(3)

The equations for the total population of banana plants and vectors are derived from the model system in (1) and presented in (4).

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{dN_p}{dt} & =\Lambda S_p\Big (1-\frac{N_p}{K_p}\Big )+ \Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big ) -\mu N_p,\\ \frac{dN_a}{dt} & =\Lambda _a S_a\Big (1-\frac{N_a}{K_a}\Big )+\Lambda _a \rho I_a\Big (1-\frac{N_a}{K_a}\Big )-\mu _a N_a. \end{array}\right. } \end{aligned}$$
(4)

Positivity of solutions and boundedness of the system

We check the model system’s epidemiological and mathematical validity by ensuring that the invariant region is contained and that the model solution is positive. This ensures that the model variables do not take on negative values.

Positivity of solutions

We use the steps explained in23 to show that the model system’s solution (1) remains positive for all nonnegative initial conditions in the fixed region \(\Omega\). Consider the model system’s initial equation (1).

$$\begin{aligned} \begin{aligned} \frac{dS_p}{dt}&=\Lambda S_p\Big (1-\frac{N_p}{K_p}\Big )+\alpha I_l- \frac{\beta _bI_aS_p}{N_a}-\mu S_p, \end{aligned} \end{aligned}$$
(5)

Since \(S_p \ge 0\), then \(\Lambda S_p\Big (1-\frac{N_p}{K_p}\Big ) \ge 0\) applying this in (5). results to (6)

$$\begin{aligned} \begin{aligned} \frac{dS_p}{dt}&\ge \alpha I_l- \frac{\beta _bI_aS_p}{N_a}-\mu S_p. \end{aligned} \end{aligned}$$
(6)

separating the variables in (6), gives (7).

$$\begin{aligned}\begin{aligned} \frac{dS_p}{S_p}&\ge \Big (\frac{\beta _bI_a}{N_a}-\mu \Big )dt \end{aligned} \end{aligned}$$
(7)

Integrating (7) both sides and simplifying results to (10)

$$\begin{aligned} \begin{aligned} \int \frac{dS_p}{S_p}&\ge \int \Big (\frac{\beta _bI_a}{N_a}-\mu _b\Big )dt \end{aligned} \end{aligned}$$
(8)
$$\begin{aligned} \begin{aligned} \ln {S_p}&\ge -\Big (\frac{\beta _bI_a}{N_a}-\mu \Big )t+C \end{aligned} \end{aligned}$$
(9)
$$\begin{aligned} \begin{aligned} \ln {S_p(t)}&\ge B \exp \Big (\frac{\beta _bI_a}{N_a}-\mu \Big )t \end{aligned} \end{aligned}$$
(10)

for \(t=0\), \(S_p(0) \ge B\) consequently,

$$\begin{aligned} \begin{aligned} S_p(t)&\ge S_p(0) \exp \Big (\frac{\beta _bI_a}{N_a}-\mu \Big )t \ge 0 \forall t \ge 0. \end{aligned} \end{aligned}$$
(11)

After applying the same method to each of the other equations in the model system (1) \(\forall t \ge 0\), we conclude that all of the model system’s (1) solutions are positive \(\forall t \ge 0\).

Boundedness of the system

Lemma 1

Given the model system in (1) with initial conditions \(S_p(0)>0\), \(I_l(0)\ge 0\), \(I_s(0)\ge 0\), \(S_a(0)>0\), \(I_a(0)\ge 0\), the solutions of the model system (1) in \(\mathbb {R}^{5}_{+}\) enter the invariant region \(\Omega =\{S_p(t), I_l(t), I_s(t), S_a(t), I_a(t)\ge 0 \in \mathbb {R}^{5}_{+}, \forall {t\ge 0}\}.\)

Proof

We employed the box method, which was utilized by23 and25, to determine the boundness of the model system. We consider the solution of our dynamic system \(\frac{dX}{dt}=\vartheta (X,t)\), where \(X \in \mathbb {R}^{n}\), and assume its continuity and Lipschitz characteristics. The form in (12) is obtained by reducing the model system (1).

$$\begin{aligned} \frac{dX}{dt}=NX+M, \end{aligned}$$
(12)

where X is the column vector such that \(X=(S_p, I_l, I_s, S_a, I_a)^{T}\) and

$$\begin{aligned} \displaystyle M=\Big (\Lambda S_p\big (1-\frac{N_p}{K_p}\big )- \frac{\beta _bI_aS_p}{N_a},\Lambda \omega I_l\big (1-\frac{N_p}{K_p}\big ), 0, \Lambda _a S_a\big (1-\frac{N_a}{K_a}\big )- \frac{\beta _vI_pS_a}{N_p}, \Lambda _a \rho I_a\big (1-\frac{N_a}{K_a}\big )\Big )^{T}, \end{aligned}$$
(13)

and the Metzler matrix N for all X in \(\mathbb {R}^{5}_+\) is given as,

$$\begin{aligned} N= \begin{pmatrix} -\mu & \alpha & 0& 0& 0 \\ \displaystyle \frac{\beta _bI_a}{N_a}& -(\theta +\mu +\alpha ) & 0 & 0& \displaystyle \frac{\beta _bS_p}{N_a}\\ 0 & \theta & -(\mu +\gamma )& 0& 0 \\ 0 & 0 & 0& -\mu _a& 0 \\ 0 & 0 & \displaystyle \frac{\beta _vS_a}{N_b}& 0& -\mu _a \\ \end{pmatrix}. \end{aligned}$$
(14)

In Eq. (14), a reduced matrix N contains all negative values on its principal diagonal and all non-negative values on its off-diagonal. These results suggest that the invariant region \(\Omega\) is the boundary of the model system (1). The positivity and boundedness of the solutions of the model system in (1) indicate that the model possesses both mathematical and biological significance. Therefore, we will proceed with a more detailed analysis of the model. \(\square\)

Disease free equilibrium point

The disease-free equilibrium point \((X^{0})\) is where BBTV infection is absent among the involved populations. In this work, we derived \(X^{0}\) by setting the equations in the model system (1) to zero and solving the resulting system.

$$\begin{aligned} & \Lambda S_p\Big (1-\frac{N_p}{K_p}\Big )+\alpha I_l- \frac{\beta _bI_aS_p}{N_a}-\mu S_p=0,\\& \Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big ) + \frac{\beta _bI_aS_p}{N_a} - (\theta +\mu +\alpha )I_l= 0,\\ & \theta I_l-(\mu + \gamma )I_s= 0,\\& \Lambda _a S_a\Big (1-\frac{N_a}{K_a}\Big )- \frac{\beta _vI_sS_a}{N_p}-\mu _aS_a=0,\\& \Lambda _a \rho I_a\Big (1-\frac{N_a}{K_a}\Big )+ \frac{\beta _vI_sS_a}{N_p}-\mu _aI_a= 0. \end{aligned}$$
(15)

To simplify the subsystem (15) solving procedure, let \(\omega _1,\omega _2 >0\) such that;

$$\begin{aligned} \begin{aligned} \omega _{1}&=\theta +\mu +\alpha , \omega _{2}&=\mu +\gamma . \end{aligned} \end{aligned}$$
(16)

When solving the system (15) simultaneously, we obtain the solution \((S^{0}_p, I^{0}_l, I^{0}_s, S^{0}_a, I^{0}_a)\). When \(I_l=0\), \(I_s=0\), and \(I_a=0\), we obtain the value of \(X^{0}\) given by the Eq. (17).

$$\begin{aligned} X^{0}=\Big (\frac{K_b(\Lambda -\mu )}{\Lambda }, 0, 0, \frac{K_a(\Lambda _a-\mu _a)}{\Lambda _a}, 0\Big ). \end{aligned}$$
(17)

where from the model assumptions \(\Lambda _a>\mu _a\) and \(\Lambda >\mu\)

Endemic equilibrium point

This point occurs when the disease continues to persist in a specific area. Researchers obtained the endemic equilibrium point (\(X^{*}\)) for the model system (1) by setting the right-hand side of the model equations to zero and solving the resulting system. Considering the model system Eq. (15), let \(\eta = \Big (1-\frac{N_p}{K_p}\Big )\) and \(\phi = \Big (1-\frac{N_a}{K_a}\Big )\). Solving the resulting model system results to and endemic equilibrium point \(X^{*}= (S^{*}_p, I^{*}_l, I^{*}_s, S^{*}_a , I^{*}_a)\), where;

$$\begin{aligned} S^{*}_p= & \frac{\delta _{2} N_p(\Lambda _a-\mu _a)(\delta _{1}-\Lambda \omega \eta )}{\beta _{v} \theta (\Lambda \eta + \alpha -\mu )}, \end{aligned}$$
(18)
$$\begin{aligned} I^{*}_l= & \frac{\delta _{2}N_p(\Lambda _a \phi -\mu _{a})}{ \beta _v \theta },\end{aligned}$$
(19)
$$\begin{aligned} I^{*}_s= & \frac{N_p(\Lambda _{a} \phi - \mu _{a})}{\beta _v},\end{aligned}$$
(20)
$$\begin{aligned} S^{*}_a= & \frac{(\mu _{a}-\Lambda _{a} \phi \rho )(\Lambda \eta + \alpha -\mu )N_a}{\beta _{b}(\Lambda _a \phi -\mu _a)},\end{aligned}$$
(21)
$$\begin{aligned} I^{*}_a= & \frac{N_a(\Lambda \eta + \alpha -\mu )}{\beta _{b}}. \end{aligned}$$
(22)

From the model assumptions \(\Lambda _a>\mu _a\) and \(\Lambda >\mu\).

Effective reproduction number

Effective reproduction number (\(R_e\)) measures the actual spread of a disease in a population by considering various factors and control strategies. Hence, \(R_e\) takes into account the impact of control strategies and interventions on reducing transmission. According to26, the \(R_e\) aids in understanding the disease’s capacity to spread throughout the population. If \(R_e<1\), then the number of diseased individuals within a completely susceptible banana plant population will, on average, not replace themselves, leading to the cessation of disease transmission. However, if \(R_e>1\), each infected individual generates, on average, more than one new infection, facilitating the spread of the disease into the broader population. As part of this study, we used the next-generation method developed by26 and27 as applied by28 to find the model system’s \(R_e\). Consider the infected subsystem (23) from the model’s system of equations (1).

$$\begin{aligned} `{\left\{ \begin{array}{ll} \begin{aligned} \frac{dI_l}{dt} & = \Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big ) + \frac{\beta _bI_aS_p}{N_a} - \omega _1I_l,\\ \frac{dI_s}{dt} & = \theta I_l-\omega _2I_s,\\ \frac{dI_a}{dt} & =\Lambda _a \rho I_a\Big (1-\frac{N_a}{K_a}\Big )+ \frac{\beta _vI_sS_a}{N_p}-\mu _aI_a. \end{aligned} \end{array}\right. } \end{aligned}$$
(23)

Dividing the infected subsystem (23) into two parts results in (24) and (25). \(\mathscr {F}(x,y)\) in (24) represents the transmission part, which depicts the generation of new infections, and \(\mathscr {V}\)(x,y) in (25) represents the transition part, which involves changes in states.

$$\begin{aligned} \mathscr {F}(x,y)= \begin{pmatrix} \Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big ) + \frac{\beta _bI_aS_p}{N_a}\\ 0\\ \Lambda _a \rho I_a\Big (1-\frac{N_a}{K_a}\Big )+ \frac{\beta _vI_sS_a}{N_p}\\ \end{pmatrix}, \end{aligned}$$
(24)
$$\begin{aligned} \mathscr {V}(x,y)= \begin{pmatrix} \omega _1 I_l \\ -\theta I_l+\omega _2 I_s\\ \mu _aI_a \\ \end{pmatrix}. \end{aligned}$$
(25)

where \(\omega _1\) and \(\omega _2\) are defined in (16). By computing the partial derivatives of \(\mathscr {F}_j\) in (24) and \(\mathscr {V}_j\) in (25) with respect to \(I_l, I_s\) and \(I_a\) and evaluating at \(X^{0}\) results in (26) and (28),

$$\begin{aligned} F= \begin{pmatrix} M_1 & 0 & M_2\\ 0 & 0 & 0\\ 0 & M_3 & M_4\\ \end{pmatrix}, \end{aligned}$$
(26)

where,

$$\begin{aligned} M_1=\Lambda \omega \Big (1-\frac{S^0_p}{K_b}\Big ), M_2 = \frac{\beta _b\Lambda _a K_p(\Lambda - \mu )}{\Lambda K_a (\Lambda _a- \mu _{a})},\hspace{0.1cm} M_3 = \frac{\beta _v \Lambda K_a (\Lambda _a- \mu _{a})}{\beta _b \Lambda _a K_p(\Lambda - \mu )}, M_4 = \Lambda _a \rho \Big (1-\frac{S^0_a}{K_a}\Big ). \end{aligned}$$
(27)

and

$$\begin{aligned} V= \begin{pmatrix} \omega _1 & 0 & 0\\ -\theta & \omega _2 & 0\\ 0 & 0 & \mu _a\\ \end{pmatrix}, \end{aligned}$$
(28)

Then it follows that, the inverse of the matrix V in (28), is given by (29)

$$\begin{aligned} V^{-1}= \begin{pmatrix} \displaystyle \frac{1}{\omega _1} & 0 & 0 \\ \displaystyle \frac{\theta }{\omega _1 \omega _2} & \displaystyle \frac{1}{\omega _2} & 0 \\ 0 & 0 & \displaystyle \frac{1}{\mu _a}\\ \end{pmatrix}. \end{aligned}$$
(29)

Let,

$$\begin{aligned} A = \frac{1}{\omega _1} \hspace{0.2cm}\text {,}\hspace{0.2cm} B = \frac{\theta }{\omega _1 \omega _2} \hspace{0.2cm}\text {and}\hspace{0.2cm} C=\frac{1}{\omega _2}. \end{aligned}$$
(30)

Let K be the next generation matrix. Now, the Next generation Matrix is obtained by \(K=FV^{-1}\), Therefore,

$$\begin{aligned} K= \begin{pmatrix} M_{1}A & 0& \frac{M_{2}}{\mu _{a}}\\ 0& 0& 0\\ M_{3}B & M_{3}C & \frac{M_{4}}{\mu _{a}}\\ \end{pmatrix}, \end{aligned}$$
(31)

The effective reproduction number \(R_e\) of the model is a dominant eigenvalue of the Next Generation Matrix Q. Therefore,

$$\begin{aligned} R_e=\frac{\omega _{2}\mu _a(\omega \mu + \rho \omega _{1}) + \sqrt{\omega _{2}\mu _{a}\Big (\omega _{2}\mu _{a}\big (\rho \omega _{1}-\omega \mu \big )^2 +4\beta _{b}\beta _{v}\theta \omega _{1}\Big )}}{2\omega _{1}\omega _{2}\mu _{a}}, \end{aligned}$$
(32)

where \(\omega _{1}\) and \(\omega _{2}\) are defined in (16).

According to \(R_e\) in (32), \(\frac{1}{\omega _{1}}\) is the average duration that a banana plant remains in a latent infection stage prior to being harvested, transitioning to the symptomatic infected stage, or return to the susceptible stage after resistance to BBTV infection. The duration for which an infected banana plant remains in a symptomatic infected group before removal or harvest is represented by \(\frac{1}{\omega _{2}}\). Moreover, \(\frac{1}{\mu _a}\) represents the typical lifespan of an aphid vector prior to experiencing natural death or being exterminated by insecticides.

Local stability of the disease free equilibrium point

Theorem 2

The disease-free equilibrium point is locally asymptotically stable if \(R_e < 1\) and unstable if \(R_e > 1\)

Proof

To prove this theorem, consider the Jacobian matrix in (33) evaluated at the \(X^{0}\) point

$$\begin{aligned} J_{X^{0}}=\left[ \begin{array}{ccccc} -(\Lambda -\mu )& 0& 0& 0& -J_{1}\\ 0& \mu (\omega -1)-\theta -\alpha & 0& 0& J_{1}\\ 0& \theta & -(\mu +\gamma )& 0& 0\\ 0& 0& -J_{2}& -(\Lambda _a-\mu _a)& 0\\ 0& 0& J_{2}& 0& \mu _{a}(\rho -1)\end{array} \right] . \end{aligned}$$
(33)

where:

$$\begin{aligned} J_{1}= \frac{\beta _{b} S^{0}_p}{S^{0}_a} \hspace{0.1cm} \text {and} \hspace{0.1cm} J_{2}=\frac{\beta _{v}S^{0}_a}{S^{0}_p} \end{aligned}$$

Given the assumptions that \(-(\Lambda -\mu )<0\) and \(-(\Lambda _a-\mu _a)<0\), it follows that the diagonal entries \(-(\Lambda -\mu )\) and \(-(\Lambda _a-\mu _a)\) of the Jacobian matrix \(J_{X^{0}}\) constitute the first two eigenvalues. The removal of the rows and columns corresponding to the observed eigenvalues yields the matrix presented in (34).

$$\begin{aligned} J^{1}_{X^{0}}=\left[ \begin{array}{ccc}\mu (\omega -1)-\theta -\alpha & 0 & J_{1} \\ \theta & -(\mu +\gamma ) & 0 \\ 0 & J_{2} & \mu _{a}(\rho -1) \end{array} \right] . \end{aligned}$$
(34)

Additionally, based on assumptions, it can be noted that \((\rho -1)<0\) and \((\omega -1)<0\). Consequently, the expressions \(\mu (\omega -1)-\theta -\alpha\) and \(\mu _{a}(\rho -1)\) are negative. Furthermore, the entry \(-(\mu +\gamma )\) is negative as well. Define \(b_1 = -\mu (\omega - 1) + \theta + \alpha\), \(b_2 = \mu + \gamma\), and \(b_3 = \mu _{a}(1 - \rho )\). The remaining eigenvalues are thus the roots of the polynomial: The equation \(|J^{1}_{X^{0}}-\lambda I|=0\) is specified in the characteristic equation presented in (35).

$$\begin{aligned} d_0\lambda ^{3}+d_1\lambda ^{2}+d_2\lambda +d_3=0. \end{aligned}$$
(35)

where;

$$\begin{aligned} \begin{aligned} d_0&= 1,\\ d_1&= b_1 + b_2 + b_3,\\ d_2&= b_1 b_2 + b_1 b_3 + b_2 b_3,\\ d_3&= b_1 b_2 b_3 - \beta _{b} \beta _{v} \theta .\\ \end{aligned} \end{aligned}$$
(36)

The Routh stability criteria necessitate that all roots of a third-degree polynomial, as presented in equation (35), possess negative real parts. To show that all the roots of the cubic polynomial in equation (35) have a negative real part, we use Routh’s stability criteria which require that all values of the coefficients \(d_i\) for \(i = 0, 1, 2, 3\) must be positive. Additionally, the product \(d_1d_2\) must be greater than the product \(d_0d_3\). From (36), it is clear that \(d_0>0, d_1>0,\) and \(d_2>0\). Since \(b_1 b_2 b_3 > \beta _{b} \beta _{v} \theta\) the coefficient \(d_3\) is also positive. Similarly, from (36), it can be shown that \(d_1d_2 - d_0d_3 > 0\). Since all the Routh stability criteria for a cubic polynomial are met, it indicates that all eigenvalues of the Jacobian matrix in (33) are negative, implying that the disease-free equilibrium point (\(X^{0}\)) is locally asymptotically stable for \(R_e < 1\) and unstable for \(R_e> 1\). The local stability of the disease-free point means that if \(R_e<1\), the system will return to disease-free after minor changes. These results suggest that the disease will eventually die out under these conditions. However, if \(R_e>1\), the disease can spread and persist. In this case, introducing virus-resistant planting material would lower the \(R_e\) to less than a unit, potentially leading to a situation where the disease cannot establish itself in the long term. \(\square\)

Global stability of the disease-free equilibrium point

The global stability of the disease-free equilibrium point means that the solutions of system are attracted to the \(X^{0}\) point over an indefinite time.

Theorem 3

If \(X^{0}\) is a disease free equilibrium point of the model system (1), then \(X^{0}\) is globally asymptotically stable if \(R_e<1\) and unstable if \(R_e>1\).

Proof

By applying the comparison theorem in29, we can reformulate the rate of change of the variables representing the infected components of model system (23) in the following manner:

$$\begin{aligned} \begin{pmatrix} \frac{dI_l}{dt}\\ \frac{dI_s}{dt}\\ \frac{dI_a}{dt}\\ \end{pmatrix} = (F-V)\begin{pmatrix} I_l\\ I_s\\ I_a\\ \end{pmatrix} - \begin{pmatrix} M_1 & 0 & M_2\\ 0 & 0 & 0\\ 0 & M_3 & M_4\\ \end{pmatrix} \begin{pmatrix} I_l\\ I_s\\ I_a\\ \end{pmatrix}, \end{aligned}$$
(37)

implying that

$$\begin{aligned} \begin{pmatrix} \frac{dI_l}{dt}\\ \frac{dI_s}{dt}\\ \frac{dI_a}{dt}\\ \end{pmatrix} \le (F-V)\begin{pmatrix} I_l\\ I_s\\ I_a\\ \end{pmatrix} \end{aligned}$$
(38)

where F and V represent the Jacobian matrices as referenced in Eqs. (26) and (28) respectively. On the basis of the stability results of Theorem 1, the eigenvalues of the matrix \((F-V)\) possess negative real components. Consequently, system (23) is stable when \(R_{0}<1\). This stability ensures that \((S_p, I_l, I_s, S_a, I_a)\) will reliably converge to \((S^{0}_p, 0, 0, S^{0}_a, 0)\) as t approaches infinity, with \(S_p\) approaching \(S^0_p\) and \(S_a\) approaching \(S^0_a\). According to the comparison theorem (refer to29 for further details), \((S_p, I_l, I_s, S_a, I_a)\) will also converge to \(X^{0}\) as t approaches infinity. This reassures us that \(X^{0}\) is globally asymptotically stable if \(R_e<1\).

The epidemiological significance of Theorem 0.3 is based in its provision of essential insights for disease control and prevention strategies. The global asymptotic stability of the BBTD-free equilibrium point indicates that, regardless of initial conditions, the dynamics of the system will converge, ultimately leading to the eradication of the disease, with populations achieving and maintaining a state free of the disease. This property is crucial as it highlights the potential for completely eliminating the disease from a population and ensuring effective long-term disease management. However, if the control measures do not reduce the effective reproduction number to below one, BBTD will persist. \(\square\)

Global stability of the endemic equilibrium point

Theorem 4

The endemic equilibrium point of model (1) is globally asymptotically stable in \(\Omega\) if \(R_e>1\).

Proof

In this study, we determine the global stability of the endemic equilibrium point (Xee) via the Lyapunov function as described by30. The Lyapunov function is constructed via the formula in

$$\begin{aligned} L=\sum c_j(X_i-X^{*}_i \ln X_i). \end{aligned}$$
(39)

where L is the Lyapunov function, \(a_i\) are carefully selected nonnegative constants, and \(X^{*}_i\) is the endemic equilibrium point for \(i,j\in \mathbb {Z}\). In this study, the Lyapunov function is given by (40)

$$\begin{aligned} L= & c_1(S_p-S^{*}_p \ln S_p)+c_2(I_l-I^{*}_l \ln I_l)+c_3(I_s-I^{*}_s \ln I_s)+c_4(S_a-S^{*}_a \ln S_a)\nonumber \\ & +\>c_5(I_a-I^{*}_a \ln I_a). \end{aligned}$$
(40)

Differentiating (40) with respect to time gives

$$\begin{aligned} \frac{dL}{dt}= & c_1 \Big (1-\frac{S^{*}_p}{S_p}\Big )\frac{dS_p}{dt} +c_2 \Big (1-\frac{I^{*}_l}{I_l} \Big )\frac{dI_l}{dt} +c_3 \Big (1-\frac{I^{*}_s}{I_s}\Big )\frac{dI_s}{dt} +c_4 \Big (1-\frac{S^{*}_a}{S_a} \Big )\frac{dS_a}{dt} \nonumber \\ & +\> c_5 \Big (1-\frac{I^{*}_a}{I_a}\Big )\frac{dI_a}{dt}. \end{aligned}$$
(41)

Substituting values from the system (1) in (41) results to,

$$\begin{aligned} \frac{dL}{dt}= & c_1(1-\frac{S^{*}_p}{S_p})[\Lambda S_p\Big (1-\frac{N_p}{K_p}\Big )- \lambda _1S_p-\mu S_p] + c_2(1-\frac{I^{*}_l}{I_l})[\Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big )\nonumber \\ & \> + \lambda _1S_p - (\theta + \mu +\alpha ) I_l] + c_3(1-\frac{I^{*}_s}{I_s})[\theta I_l-(\mu + \gamma ) I_s]\nonumber \\ & \> + c_4(1-\frac{S^{*}_a}{S_a})[\Lambda _a S_a\Big (1-\frac{N_a}{K_a}\Big ) -\lambda _2 S_a-\mu _aS_a]\nonumber \\ & +\>c_5(1-\frac{I^{*}_a}{I_a})[\Lambda _a\rho I_a\Big (1-\frac{N_a}{K_a}\Big )+\lambda _2 S_a - \mu _aI_a]. \end{aligned}$$
(42)

At endemic equilibrium point,

$$\begin{aligned} \Lambda S_p\Big (1-\frac{N_p}{K_p}\Big )= & \lambda ^{*}_1S^{*}_p+\mu S^{*}_p, \end{aligned}$$
(43)
$$\begin{aligned} \Lambda \omega I_l\Big (1-\frac{N_p}{K_p}\Big )= & \lambda ^{*}_1S^{*}_p -\omega _1I^{*}_l, \end{aligned}$$
(44)
$$\begin{aligned} \mu _b +\gamma= & \frac{\theta I^{*}_l}{I_s}, \end{aligned}$$
(45)
$$\begin{aligned} \Lambda _a S_a\Big (1-\frac{N_a}{K_a}\Big )= & \lambda ^{*}_2 S^{*}_a+\mu _aS^{*}_a, \end{aligned}$$
(46)
$$\begin{aligned} \Lambda _a\rho I_a\Big (1-\frac{N_a}{K_a}\Big )= & \mu _aI^{*}_a - \lambda ^{*}_2 S^{*}_a. \end{aligned}$$
(47)

Therefore, substituting into (42) results to

$$\begin{aligned} \frac{dL}{dt}\le & c_1(1-\frac{S^{*}_p}{S_p})[\lambda ^{*}_1S^{*}_p+\mu S^{*}_p- \lambda _1 S_p-\mu S_p] + c_2(1-\frac{I^{*}_l}{I_l})[\lambda _1 S_p + \lambda ^{*}_1S^{*}_p -\omega _1I^{*}_l -\omega _1 I_l] \nonumber \\ & +\>c_3(1-\frac{I^{*}_s}{I_s})[\theta I_l-\frac{\theta I^{*}_l}{I_s} I_s] + c_4(1-\frac{S^{*}_a}{S_a})[\lambda ^{*}_2 S^{*}_a+\mu _aS^{*}_a -\lambda _2 S_a-\mu _aS_a]\nonumber \\ & +\> c_5(1-\frac{I^{*}_a}{I_a})[\lambda _2 S_a + \mu _aI^{*}_a - \lambda ^{*}_2 S^{*}_a- \mu _aI_a]. \end{aligned}$$
(48)

Further simplification

$$\begin{aligned} \frac{dL}{dt}= & -c_1\lambda _1 S_p\Big (1-\frac{S^{*}_p}{S_p}\Big )(1-\frac{\lambda ^{*}_1S^{*}_p}{\lambda _1S_p}\Big )-c_1\lambda _1 S_p\Big (1-\frac{S^{*}_p}{S_p}\Big )^{2}+c_2\lambda _1 S_p\Big (1-\frac{I^{*}_l}{I_l}\Big )(1-\frac{\lambda ^{*}_1S^{*}_p}{\lambda _1S_p}\Big )\nonumber \\ & -\>c_2\omega _1 I_l\Big (1-\frac{I^{*}_l}{I_l}\Big )^{2}+c_3\theta I_l\Big (1-\frac{I^{*}_s}{I_s}\Big )(1-\frac{I^{*}_l I_s}{I_l I^{*}_s}\Big )-c_4\lambda _2 S_a\Big (1-\frac{S^{*}_p}{S_p}\Big )(1-\frac{\lambda ^{*}_2 S^{*}_a}{\lambda _2 S_a}\Big )\nonumber \\ & \>-c_4\mu _a S_a\Big (1-\frac{S^{*}_a}{S_a}\Big )^{2}+c_5\lambda _2S_a\Big (1-\frac{I^{*}_a}{I_a}\Big )(1-\frac{\lambda ^{*}_2 S^{*}_a}{\lambda _2 S_a}\Big )-c_5\mu _a I_a\Big (1-\frac{I^{*}_a}{I_a}\Big )^{2}. \end{aligned}$$
(49)

Choosing

$$\begin{aligned} c_1=c_2=\frac{1}{\lambda _{1}S_{p}}\hspace{0.2cm} \text {,} \hspace{0.2cm}c_3=\frac{1}{\theta I_{l}}\hspace{0.2cm} \text {and} \hspace{0.2cm} c_4=c_5=\frac{1}{\lambda _{2} S_a} \end{aligned}$$

and collecting the like terms

$$\begin{aligned} \frac{dL}{dt}= & -\Big (1-\frac{S^{*}_p}{S_p}\Big )^{2}-\frac{\mu _a}{\lambda _{2}} \Big (1-\frac{S^{*}_a}{S_a}\Big )^{2}-\frac{\mu _a I_a}{\lambda _2 S_a}\Big (1-\frac{I^{*}_a}{I_a}\Big )^{2}-\frac{\omega _1 I_l}{\lambda _{1} S_p}\Big (1-\frac{I^{*}_l}{I_l}\Big )^{2}+P(X) \end{aligned}$$
(50)

Where

$$\begin{aligned} P(X)= & \Big (1-\frac{I^{*}_l}{I_l}\Big )\Big (1-\frac{\lambda ^{*}_1S^{*}_p}{\lambda _1S_p}\Big )+\Big (1-\frac{I^{*}_s}{I_s}\Big )\Big (1-\frac{I^{*}_l I_s}{I_l I^{*}_s}\Big )-\Big (1-\frac{S^{*}_p}{S_p}\Big )\Big (1-\frac{\lambda ^{*}_2 S^{*}_a}{\lambda _2 S_a}\Big )\nonumber \\ & +\Big (1-\frac{I^{*}_a}{I_a}\Big )\Big (1-\frac{\lambda ^{*}_2 S^{*}_a}{\lambda _2 S_a}\Big )-\Big (1-\frac{S^{*}_p}{S_p}\Big )\Big (1-\frac{\lambda ^{*}_1S^{*}_p}{\lambda _1S_p}\Big ). \end{aligned}$$
(51)

Where

$$\begin{aligned} X= {(S_p,I_l,I_s,S_a,I_a)>0}. \end{aligned}$$
(52)

From (51), \(P(X)\le 0\) for all the elements in X which implies that \(\frac{dL}{dt}\le 0\) in X. \(\frac{dL}{dt}=0\) in X only when \(X=X^{*}\), implying that the largest invariant set in X when \(\frac{dL}{dt}=0\) is singleton X which is the endemic equilibrium point. Therefore, by LaSalle’s invariance principle described in31, the endemic equilibrium point exists and is asymptotically stable in \(\Omega\) when \(R_e>1\) and unstable otherwise. The epidemiological significance of Theorem 0.4 The global asymptotic stability of the BBTD endemic equilibrium point indicates that the disease will persist relatively constantly within the population, neither diminishing nor decreasing when the effective reproduction number is greater than one. \(\square\)

Sensitivity analysis

Sensitivity analysis examines the impact of each model parameter on \(R_e\). The sensitivity analysis results can be used to identify and suggest the best control techniques for disease elimination. To obtain the sensitivity indices of the model parameters, we carefully analysed the sensitivity of the parameters in \(R_e\) by using the normalized forward sensitivity index as described in32. If \(R_e\) is differentiable with respect to its parameter \(\beta\), then the sensitivity index of \(\beta\) is given by (53).

$$\begin{aligned} \S ^{R_e}_\beta =\frac{\partial R_e}{\partial \beta } \times \frac{\beta }{R_e}. \end{aligned}$$
(53)

Given,

$$\begin{aligned} R_e=\frac{\omega \mu \omega _{2}\mu _a + \rho \mu _{a}\omega _{1}\omega _{2} + \sqrt{\omega _{2}\mu _{a}\Big (\omega _{2}\mu _{a}\big (\rho \omega _{1}-\omega \mu \big )^2 +4\beta _{b}\beta _{v}\theta \omega _{1}\Big )}}{2\omega _{1}\omega _{2}\mu _{a}}, \end{aligned}$$
(54)

where from (16),

$$\begin{aligned} \omega _{1} = \theta +\mu +\alpha \hspace{0.1cm} \text {and} \hspace{0.1cm} \omega _{2} = \mu +\gamma . \end{aligned}$$

Therefore, differentiating (54) with respect to \(\beta _{v}\) gives

$$\begin{aligned} \frac{\partial R_e}{\partial \beta _v}=\frac{\theta \beta _{b}}{\sqrt{\mu _{a} \left( \mu _{a} \left( \rho \omega _{1}-\omega \mu \right) ^{2} \omega _{2}+4 \theta \beta _{b} \beta _{v} \omega _{1}\right) \omega _{2}}} \end{aligned}$$
(55)

Substituting (55) in (53) and replacing \(\beta\) with the parameter \(\beta _v\) results in a sensitivity index in (56).

$$\begin{aligned} \S ^{R_e}_{\beta _v}=0.4702 \end{aligned}$$
(56)

Since \(R_e\) in (54) is differentiable to all its parameters, we now apply (53) to calculate the sensitivity indices of the model parameter via the values in Table 2. This results in sensitivity indices as indicated in Fig. 2.

Table 2 Values of the model parameters.
Full size table

The parameters with positive indices, as indicated by the sensitivity analysis results in Fig. 2, are \(\rho , \beta _b, \mu , \theta\), \(\omega\) and \(\beta _v\). The positivity of the sensitivity index implies that changing any of these parameters will directly increase the effective reproduction number (\(R_e\)) while maintaining the same values for the other parameters. For example, when the parameter \(\beta _b\) is increased (or decreased) by \(10\%\), the \(R_e\) is increased (or decreased) by \(10\%\). These findings suggest that when infection of banana plants from a contaminated vector increases, the number of secondary infections caused by one infected banana plant increases by \(10\%\), which enhances the further spread of BBTD disease. On the other hand, the parameters \(\gamma , \alpha\), and \(\mu _a\) have negative indices, indicating that altering any of these values will have an inverse relationship with \(R_e\). Optimizing disease control involves lowering the \(R_e\) value and increasing the negative indexed parameters. For example, based on Fig. 2, the sensitivity index for parameter \(\alpha\) is \(-0.2793\). These findings suggest that increasing replanting with BBTV-resistant plant material will decrease the number of secondary infections caused by one infected banana plant by \(10\%\), which will help limit the further spread of BBTD disease.

Since \(\mu\) has a sensitivity index of \(-0.0284\) closer to zero than the other parameters, it is said to be less sensitive to \(R_e\). These results suggest that harvesting operations have less of an effect on the propagation of BBTDs.

Fig. 2
figure 2

Sensitivity indices of the model parameters.

Full size image

Bifurcation analysis

Theorem 5

The system (1) exibits a forward bifurcation at \(R_e=1\)

Proof

Change the variables \(S_p=x_1\), \(I_l=x_2\), \(I_s=x_3\), \(S_a=x_4\), and \(I_a=x_5\). Hence \(X = (x_1, x_2, x_3, x_4, x_5)^{T}\) and \(\frac{dX}{dt}=F(t,x(t))\) where \(F(t,x(t))=(f_1, f_2, f_3, f_4, f_5)^{T}\). Applying the new names of the variables in the system (1) results to

$$\begin{aligned} \begin{aligned} f_1&= \Lambda x_1\Big (1-\frac{x_1+x_2+x_3}{k_p}\Big )- \frac{\beta _bx_5x_1}{x_4+x_5}-\mu x_1,\\ f_2&= \Lambda \omega x_2\Big (1-\frac{x_1+x_2+x_3}{k_p}\Big )+ \frac{\beta _bx_5x_1}{x_4+x_5} -\omega _1x_2,\\ f_3&= \theta x_2-\omega _2 x_3,\\ f_4&= \Lambda _a x_4\Big (1-\frac{x_4+x_5}{k_a}\Big ) -\frac{\beta _vx_3 x_4 }{x_1+x_2+x_3}-\mu _ax_4,\\ f_5&= \Lambda _a \rho x_5\Big (1-\frac{x_4+x_5}{k_a}\Big )+ \frac{\beta _vx_3 x_4}{x_1+x_2+x_3} - \mu _ax_5. \end{aligned} \end{aligned}$$
(57)

Choose \(\beta _b\) be the bifurcation parameter, evaluating \(\beta _b\) at \(R_e=1\) results to

$$\begin{aligned} \begin{aligned} \beta ^{*}_{b}&= \frac{\omega _1 \omega _2 \mu _{a}}{\theta \beta _v}. \end{aligned} \end{aligned}$$
(58)

find the Jacobian matrix for the model system at the disease-free equilibrium point (\(X^{0}\)) and \(\beta _b = \beta ^{*}_b\) results to

$$\begin{aligned} J_{X^{0}}=\left[ \begin{array}{ccccc} -(\Lambda -\mu )& 0& 0& 0& -J_{1}\\ 0& \mu (\omega -1)-\theta -\alpha & 0& 0& J_{1}\\ 0& \theta & -(\mu +\gamma )& 0& 0\\ 0& 0& -J_{2}& -(\Lambda _{a}-\mu _a)& 0\\ 0& 0& J_{2}& 0& \mu _{a}(\rho -1)\end{array} \right] . \end{aligned}$$
(59)

where,

$$\begin{aligned} J_{1}= \frac{\beta _{b} \Lambda _{a}k_b(\Lambda -\mu )}{\Lambda k_v(\Lambda _{a}-\mu _{a})} \hspace{0.1cm} \text {and} \hspace{0.1cm} J_{2}=\frac{\beta _{v} \Lambda k_v(\Lambda _{a}-\mu _{a})}{\Lambda _{a}k_b(\Lambda -\mu )} \end{aligned}$$

Also, recall from (28) that \(\omega _1 = \mu +\theta +\alpha\) and \(\omega _2 = \mu + \gamma\). Now, from the Jacobian matrix in (59) we compute the right eigenvector \(w = (w_1, w_2, w_3, w_4, w_5)^{T}\) that satisfy the condition \(J_{(X^{0}, \beta ^{*}_{b})}. w =0\) and the left eigenvector \(v = (v_1, v_2, v_3, v_4, v_5)\) that satisfy the condition \(v. J_{(X^{0}, \beta ^{*}_{b})}=0\). This computation results to

$$\begin{aligned} w_1 = \frac{-\beta _{b}x_1w_{5}}{\mu },\hspace{0.1cm} w_2 = \frac{\beta _{b}x_1w_{5}}{\omega _1}, \hspace{0.1cm} w_3 = \frac{\beta _{b}x_1\theta w_{5}}{\omega _{1}\omega _{2}}, \hspace{0.1cm} w_4 = \frac{-\beta _{b}x_4\omega _{5}}{\beta ^{*}_{b}},\hspace{0.2cm} \text {and} \hspace{0.2cm} w_5 = w_5 > 0 \end{aligned}$$
(60)

Computing the left eigenvector results to

$$\begin{aligned} v_1 = 0,\hspace{0.1cm} v_2 = \frac{\mu _{v}v_{5}}{\beta _{b}x_1}, \hspace{0.1cm} v_3=\frac{\beta _{v}x_{4}v_5}{\omega _{2} x_1}, \hspace{0.1cm} v_4 = 0, \hspace{0.1cm} \text {and} \hspace{0.1cm}v_5 = v_5>0, \end{aligned}$$
(61)

The non zero second order partial derivative of f(k) with respect to \(x_{i,j}\) for \(k, i, j=1,2,3,4,5\) are given by

$$\begin{aligned} \frac{\partial ^2 f_{2}}{\partial x_{5} \partial x_{1}} =\frac{\partial ^2 f_{2}}{\partial x_{1} \partial x_{5}}= \frac{ \beta _{b}}{x_4},\hspace{0.1cm} \hspace{0.1cm}\hspace{0.1cm}\frac{\partial ^2 f_{2}}{\partial x_{5} \partial x_{4}}=\frac{\partial ^2 f_{2}}{\partial x_{4} \partial x_{5}}= \frac{ \beta _{b}x_1}{x^{2}_4}, \end{aligned}$$
(62)
$$\begin{aligned} \frac{\partial ^2 f_{5}}{\partial x_{2} \partial x_{3}} = \frac{\partial ^2 f_{5}}{\partial x_{3} \partial x_{2}}=\frac{- \beta _{v}x_4}{x^{2}_1}, \frac{\partial ^2 f_{5}}{\partial x_{3} \partial x_{4}}=\frac{\partial ^2 f_{5}}{\partial x_{4} \partial x_{3}}=\frac{ \beta _{v}}{x_1} \end{aligned}$$
(63)

Again, Computing the partial derivative with respect to the bifurcation constant \(\beta ^{*}_{v}\) gives

$$\begin{aligned} \frac{\partial ^2 f_{2}}{\partial x_{i} \partial \beta ^{*}_{b}} = 0,\hspace{0.1cm}\text {for}\hspace{0.1cm} i=1,2,3,4 \hspace{0.1cm}\frac{\partial ^2 f_{2}}{\partial x_{5} \partial \beta ^{*}_{b}} = \frac{ x_1}{x_5}, \hspace{0.1cm} \frac{\partial ^2 f_{5}}{\partial x_{i} \partial \beta ^{*}_{b}} = 0, \hspace{0.1cm}\text {for}\hspace{0.1cm} i=1,2,3,4,5 \end{aligned}$$
(64)

Now, computing the value of the coefficient b

$$\begin{aligned} & b=\sum _{k,i=1}^{5} v_{k}w_{i}\frac{\partial ^2 f_{k}}{\partial x_{i} \partial \beta _{b}} \end{aligned}$$
(65)
$$\begin{aligned} & b= v_{2}w_{5}\frac{\partial ^2 f_{2}}{\partial x_{5} \partial \beta _{b}} \nonumber \\ & b = \frac{\mu _{v}v_{5} w_{5}}{\beta ^{*}_{b}x_1}>0 \end{aligned}$$
(66)

Furthermore, computing the value of the coefficient b

$$\begin{aligned} a=\sum _{k,i,j=1}^{5} v_{k}w_{i}w_{j}\frac{\partial ^2 f_{k}}{\partial x_{i} \partial x_{j}} \end{aligned}$$
(67)
$$\begin{aligned} a= & v_{2}\Big (2w_{5}w_{1}\frac{\partial ^2 f_{2}}{\partial x_{5} \partial x_{1}}+2w_{5}w_{4}\frac{\partial ^2 f_{2}}{\partial x_{5} \partial x_{4}}\Big )+v_{6}\Big (2w_{3}w_{2}\frac{\partial ^2 f_{2}}{\partial x_{3} \partial x_{2}}+ 2w_{3}w_{4}\frac{\partial ^2 f_{2}}{\partial x_{3} \partial x_{4}}\Big ) \end{aligned}$$
(68)
$$\begin{aligned} a= & v_{2}\Big (\frac{2w_{5}w_{1} \beta _{b}}{x_4}+\frac{2w_{5}w_{4} \beta _{b}x_1}{x^{2}_4}\Big )+v_{5}\Big (\frac{-2w_{3}w_{2} \beta _{v}x_4}{x^{2}_1}+\frac{2w_{3}w_{4} \beta _{v}}{x_1}\Big ) \end{aligned}$$
(69)

Since \(w_1, w_4 <0\) and \(v_2, v_5, w_2, w_3, w_5 >0\) it can be observed that form (69) and (66) \(a<0\) and \(b>0\) respectively. Therefore, from theorem 4.1 in34, the model (1) exhibits a forward bifurcation as \(a < 0\) and \(b>0\). This result means that a slight change in the rate of susceptible banana plants to acquire BBTV from an aphid vector (\(\beta ^{*}_{b}\)) can cause system stability to transition from an asymptotically stable disease-free equilibrium point to an asymptotically stable disease persistence equilibrium point. The bifurcation plot in Fig. 3 shows how varying key parameters affect the effective reproduction number and the endemic equilibrium. The shape and position of the bifurcation point are directly affected by transmission rates (\(\beta _b\) and \(\beta _v\)), the roguing rates (\(\gamma\)), and the effectiveness of resistant planting materials (\(\alpha\)). Similarly, improving the effectiveness of BBTV-resistant planting materials lowers the bifurcation threshold, increasing the chances of disease eradication. Figure 3 shows that with respect to the effective reproduction number \((R_e)\), this result means that when \(R_e<1\), the disease-free equilibrium point is asymptotically stable, and when \(R_e>1\), the global disease persistence is asymptotically stable. The epidemiological significance of the existence of forward bifurcation in Theorem 0.5 reveals that lowering the effective reproduction number (\(R_e\)) to 1 can effectively control BBTD. These results show how important it is to use a lot of virus-resistant planting material and get rid of infected banana plants regularly. \(\square\)

Fig. 3
figure 3

The forward bifurcation diagram for the banana bunchy top transmission dynamics model with resistant planting material. When \(R_e < 1\), the disease-free equilibrium is stable. However, when \(R_e > 1\), the disease-free equilibrium is unstable, while the endemic equilibrium is stable. The parameter values in Table 2 with the initial condition \(S_p(0)=400\) and \(S_a(0)=350\) were employed for the simulation.

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Numerical results and discussion

This study applied the Runge–Kutta fourth-order schemes to numerically solve model system (1). The Fourth-order Runge-Kutta method was selected because it balances accuracy and computational efficiency. The Fourth-order Runge-Kutta method is a fourth-order technique characterized by a truncation error of \(O(h^{5})\), where h denotes the step size; this indicates that the error at each step is proportional to \(h^{5}\), making the approach highly precise35. The Fourth-order Runge–Kutta method is engineered for stability across various situations and is widely regarded as highly effective35. Furthermore, the convergence rate of the Fourth-order Runge-Kutta method is \(O(h^{4})\), indicating that the inaccuracy in the numerical solution diminishes significantly when the step size is reduced35. MATLAB software was used to implement the Runge-Kutta fourth-order schemes. Plots of the numerical solutions demonstrated the impact of replanting with BBTV-resistant planting material.

We simulate the model system (1) using the parameter values as shown in Table 2 and \(S_p(0)=400, I_l(0)=50, I_s(0)=50, S_a(0)=350, I_a(0)=200\) as initial condition values of the model variables. The selected initial conditions represent a possible outbreak in a banana plantation affected by Banana Bunchy Top Disease (BBTD). In particular, \(S_p(0)=400\) means that there are a lot of healthy but susceptible plants in the plantation. \(I_l(0)=50\) means that there are latently infected banana plants and \(I_s(0)=50\) means that there are symptomatic infected banana plants in the plantation, showing that the disease is active. The susceptible aphid population \(S_a(0)=350\) suggests a considerable number of susceptible aphid vectors present that could contribute to further disease spread of the disease once contaminated with BBTV after feeding on an infected banana plant, and contaminated vectors \(I_a(0)=200\) indicate a high BBTV transmission potential in the plantation. These conditions help assess the impact of BBTV-resistance planting material on the transmission dynamics of BBTD under moderate infection levels and a substantial vector population.

Fig. 4
figure 4

The impact of resistance breed (\(\alpha\)) on the number of susceptible banana plants (\(S_p\)) on the farm when no other control strategy applied and \(\Lambda =0.9\), \(\gamma =0.105, \mu =0.0056\), \(\beta _{b}=0.3\).

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Figure 4 shows that when there are no control strategies, all the susceptible banana plants (400) will be infected with BBTD and there will be no healthy plants on the farm. However, increasing the number of resistant banana plants from \(10\%\) to \(90\%\) increased the number of healthy banana plants from 400 to 500 over two years.

Fig. 5
figure 5

The impact of resistance breed (\(\alpha\)) on the number of latent infected banana plants (\(I_l\)) in the farm when no other control strategy applied and \(\Lambda =0.9\), \(\gamma =0.105, \mu =0.0056\), \(\beta _{b}=0.3\).

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Figure 5 shows that when no other control strategy is applied and the percentage of healthy banana plants that resists BBTV is \(10\%\), the number of latent infected banana plants rapidly increases from 50 to 90 in the first 5 months and decreases to 80 in the next 5 months. However, when the percentage of banana plants resisting BBTV infection increases to \(90\%\) the number of latent infected banana plants decreases to 12 after 24 months.

Fig. 6
figure 6

The impact of resistance breed (\(\alpha\)) on the number of symptomatic infected banana plants (\(I_s\)) in the farm when no other control strategy applied and \(\Lambda =0.9\), \(\gamma =0.105, \mu =0.0056\), \(\beta _{b}=0.3\).

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Figure 6 illustrates that when only \(10\%\) of banana plants resist BBTV, and merely \(10\%\) of infected banana plants are removed from the farm, the number of latently infected banana plants rises from 50 to 90 over four months. This number then decreases to 80 after 10 months and remains stable at that level for the subsequent 14 months. In contrast, when the proportion of BBTV-resistant banana plants increases to \(90\%\) and \(90\%\) of the symptomatic infected plants are removed, the count of latent infected banana plants drops dramatically from 50 to 2 within 24 months.

Fig. 7
figure 7

The impact of resistance breed (\(\alpha\)) on the number of symptomatic infected banana plants (\(I_s\)) on the farm when no other control strategy was applied and \(\Lambda =0.9\), \(\gamma =0.105, \mu =0.0056\), \(\beta _{b}=0.3\).

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The impact of resistance breed (\(\alpha\)) on the number of symptomatic infected banana plants (\(I_s\)) is shown in Fig. 7. Figure 7 shows that increasing the resistance rate of susceptible banana plants from \(10\%\) to \(90\%\) reduces the number of symptomatic infected banna pants from 145 to 30 in two years when the rate of removing symptomatic infected banana plants is \(\gamma =0.105.\)

Fig. 8
figure 8

The impact of resistance breed (\(\alpha\)) on the number of symptomatic infected banana plants (\(I_s\)) on the farm when no other control strategy was applied and \(\Lambda =0.9\), \(\gamma =0.105, \mu =0.0056\), \(\beta _{b}=0.3\).

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Figure 8 illustrates the impact of banana plants’ resistance to BBTV. When only \(10\%\) of the banana plants are resistant to the virus and only \(10\%\) of the infected banana plants are removed from the farm, the number of symptomatic infected banana plants rises from 50 to 140 over 24 months. Conversely, if the percentage of banana plants resistant to BBTV increases to \(90\%\) and \(90\%\) of the symptomatic infected plants are removed, the number of symptomatic infected banana plants drops from 50 to 0 within the same 24 months.

Fig. 9
figure 9

Banana population dynamics when \(\alpha =0.9\) and \(\gamma =0.105\).

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Figure 9 illustrates the dynamics of the banana population when the initial conditions are set to \(S_p(0) = 100\), \(I_l(0) = 100\), and \(I_s(0) = 50\). In this scenario, \(90\%\) of the healthy banana plants are resistant to BBTV infection, and the rate of removing symptomatic infected plants is \(\gamma = 0.105\). Figure 9 shows that the number of symptomatic infected banana plants decreases slightly, contributing to disease persistence on the farms. On the other hand, Fig. 10 shows that within 10 months, there are fewer symptomatic and latently infected banana plants, leaving only healthy ones. This happens when the percentage of healthy banana plants resistant to BBTV infection reaches \(90\%\), and a \(90\%\) removal rate of symptomatic infected banana plants occurs.

Fig. 10
figure 10

Banana population dynamics when \(\alpha =0.9\) and \(\gamma =0.9\).

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This study examined the impact of replanting with BBTV-resistant planting material on the transmission dynamics of BBTD. The results from this study show that increasing the number of resistant banana plants to \(90\%\) reduces the number of latent infected banana plants by \(76\%\) and symptomatic infected plants by \(36\%\) in 25 months when the rate of removing symptomatic infected banana plants \((\gamma )\) is \(10\%\). Furthermore, when \(90\%\) of the banana plants are resistant to BBTV infection, and the rate of removing symptomatic infected plants increased to \(90\%\) after 15 months, the banana population consisted of only susceptible plants, and there were no latent or symptomatic infected plants. These results imply that increasing the number of healthy banana plants resistant to BBTV reduces latent and symptomatic infected plants.

The numerical simulation results suggest that while virus-resistant planting materials alone can reduce disease prevalence, they are most effective when combined with timely roguing and replanting strategies. The results indicate that increasing the number of resistant plants beyond a certain threshold can lead to disease elimination. Delays in adopting and implementing these control strategies may cause persistent BBTV infections. However, the success of this approach depends on the availability, affordability, and farmers adoption rate of resistant planting material. Furthermore, the rouging strategy’s feasibility depends on farmer compliance, monitoring efficiency, and economic constraints.

This study aligns with the findings in6,9, which suggested that introducing resistant cultivars is the most effective option for managing Banana Bunchy Top Disease (BBTD). Additionally, the results of this study support the observations made by36,37, which indicated that roguing is an effective method for preventing the further spread of the disease. Therefore, BBTD can be effectively eliminated by removing the infected banana plants and replanting with healthy, BBTV-resistant banana plant material. These findings directly affect farmers, agricultural policymakers, and disease management programs. The model provides insights for disease elimination by replanting with BBTV-resistant planting material and roguing strategies. The insights from this study can help inform banana breeding programs to ensure effectiveness of the BBTV-resistant planting material in fighting against BBTV. Furthermore, these results can assist decision-makers in allocating resources for BBTD control initiatives, ensuring the availability and effectiveness of the BBTV-resistant planting material, leading to disease elimination.

Conclusion

In this work, a deterministic mathematical model was developed and analysed to investigate the impacts of replanting with BBTV-resistant banana planting material on the transmission dynamics of BBTD. The effective reproduction number was derived via the next-generation matrix method, and the stability of the disease equilibrium points was determined. The results revealed that the disease-free equilibrium point is locally and globally asymptotically stable if \(R_e<1\) and unstable when \(R_e>1\). By using the Lyapunov function, it was further proved that the endemic equilibrium point is globally stable when \(R_e>1\) and unstable when \(R_e<1\). We further conducted a bifurcation analysis to support the stability results, which confirmed a forward bifurcation. The forward bifurcation implies that the disease-free equilibrium point is stable when the \(R_e<1\) and unstable when the \(R_e>1\). Using the Runge-Kutta fourth-order method, we conducted numerical simulations of the model system. The results showed that replanting with BBTV-resistant plant material while the rate of removing symptomatic infected plants is \(10\%\) reduces the number of latent and symptomatic infected banana plants by \(36\%\) and \(76\%\) respectively in the two years. Moreover, when the percentage of roguing was increased to \(90\%\) and the rate of replanting with BBTV-resistant planting material to \(90\%\), the diseased plants were cleared within 15 months. Hence, the disease can be eliminated. These results have significant practical implications for scientists, policymakers and farmers in controlling the further spread of BBTD. Every study has limitations of its own. One limitation of our research is the need for real data to support our findings because there is limited access to empirical data regarding the dynamics of Banana Bunchy Top Disease (BBTD), especially in areas where the disease is emerging. For this instance, We used simulated data from earlier studies to conduct numerical simulations in this study. From a computational perspective, we have noted the computational complexity of our model approach, especially when logistic recruitment is considered. Despite these limitations, this study builds on existing models in the literature concerning the transmission dynamics of BBTD when resistant planting materials are considered. Our model provides theoretical insights into the control of BBTD. We acknowledge that the feasibility of the proposed control strategies may be influenced by practical issues such as environmental unpredictability, farmer willingness to employ the proposed control, and economic drawbacks. This study may be extended by including additional control strategies like an education campaign, using pesticides to control the aphid vector population, and screening planting material for BBTV infections. Moreover, researchers can extend this study by including seasonal fluctuation parameters and other external elements such as soil quality and farming methods.Furthermore, future studies may consider incorporating optimal control, cost-effective analysis of control strategies, explore more complex dynamics such as backward bifurcations or oscillatory behavior, stage structured models and Stochastic model to assess the uncertainties.