## 1 Introduction
This knowledge will facilitate the implementation of targeted mitigation and control strategies, reducing the need for indiscriminate use of insecticides and pesticides, which are detrimental to both the environment and the health of those applying them. We begin by considering the basic dynamics of a population of plants and vectors, organisms that can transmit a pathogen [23]. We assume that the plant population is categorized into two groups: healthy plants and diseased plants, whose time-dependent density populations are given by \(S\) and \(I\), respectively. Then, following the standard compartmental modeling [24; 25; 26; 27], we deduce that the basic dynamics are described by the following system:  
\[\frac{dS}{dt} = \bar{r}S-\bar{k}_{1}SI, \tag{1}\] \[\frac{dI}{dt} = -\bar{a}I+\bar{k}_{1}SI, \tag{2}\]  
where \(\bar{r}\), \(\bar{k}_{1}\), and \(\bar{a}\) are the positive constants modeling the rates of replanting of healthy plants, the \"contact\" between healthy and diseased plants, and the death of diseased (or removal of) plants, respectively. However, the \"contact\" between healthy and diseased plants is produced by a vector population, and the term \(\bar{k}_{1}SI\) should be improved by considering the vector population. We consider that the vector population is divided into non-infective and infective populations, with densities \(U\) and \(W\), respectively. Then, if we consider that \(\bar{k}_{1}\) is the contact between healthy plants and infective vectors, the term \(\bar{k}_{1}SI\) in (1) and (2) must be replaced by \(\bar{k}_{1}SW\). Moreover, we consider two kinds of behaviors in the dynamics of the vector population: logistic population growth and the acquisition of infectivity by the vectors through contact of non-infective individuals with diseased plants. Thus, the mathematical model is given by the following system:  
\[\frac{dS}{dt} = rS-\bar{k}_{1}SW, \tag{3}\]\[\frac{dI}{dt} = -\bar{a}I+\bar{k}_{1}SW, \tag{4}\] \[\frac{dU}{dt} = \bar{b}(U+W)\bigg{[}1-\frac{U+W}{\bar{m}(S+I)}\bigg{]}-\bar{k}_{2} IU-\bar{c}U,\] (5) \[\frac{dW}{dt} = \bar{k}_{2}IU-\bar{c}W, \tag{6}\]  
where \(\bar{b},\bar{c},\bar{k}_{2}\), and \(\bar{m}\) are positive constants denoting the rates of birth, death, and contact between infected plants and non-infective vectors, and the maximum vector abundance, respectively. Nowadays, there are more advanced mathematical models for plant virus diseases (see, for instance, [18; 23; 28; 29; 30; 31; 32; 33; 34; 35]). Between them, we are interested in two models analyzed in [35]. We remark that the models given in [35] are focused on _ cassava_ plantations infested by _begomoviruses_ carried by the whitfely _Bemisia tabaci_, commonly known as _cassava mosaic_ virus. However, the model can be applied to other types of crops where a cutting is a living part (a stem, a branch, or a shoot) that has been taken from a plant in order to graft it onto another plant or into a container for further development. This multiplication method, using cuttings, is a common reproductive process for plants like fruit trees (blueberries, grapes, citrus, kiwi, cherry, etc. ), various trees (ficus, olive trees, elms, willow, laurel, etc. ), and ornamentals (roses, magnolias, laverder, lilacs, etc.). To be precise, the two models considered in [35] can be written as follows:  
\[\frac{dS}{dt} = b\ r_{S}(S,I)S\bigg{(}1-\frac{S+I}{\theta}\bigg{)}-\beta SW-hS+gI, \tag{7}\] \[\frac{dI}{dt} = b\ r_{I}(S,I)I\bigg{(}1-\frac{S+I}{\theta}\bigg{)}+\beta SW-( \alpha+h+g)I,\] (8) \[\frac{dU}{dt} = a(U+W)\bigg{(}1-\frac{U+W}{K(S+I)}\bigg{)}-\gamma IU-(\mu+c)U,\] (9) \[\frac{dW}{dt} = \gamma IU-(\mu+c)W, \tag{10}\]  
with \(r_{S}(S,I)\) and \(r_{I}(S,I)\) considered to unify the models, referred to as the \"frequency-replanting model\" and \"abundance-replanting model\", and defined as  
\[r_{S}(S,I) = \left\{ \begin{array}{ll}1,&\text{abundance-replanting model},\\ \frac{S}{S+\bar{c}I},&\text{frequency-replanting model},\end{array} \right. \tag{11}\] \[r_{I}(S,I) = \left\{