## 1 Introduction
\begin{array}{ll}\dot{\epsilon},&\text{abundance- replanting model},\\ \frac{eI}{S+\bar{c}I},&\text{frequency-replanting model}.\end{array} \right. \tag{12}\]  
The parameters \(b,\theta,\beta,h,g,\alpha,a,\gamma,\mu,c\), and \(\epsilon\) are positive constants denoting some rates as described in Table 1. We remark that the parameters \(\epsilon/\bar{\epsilon}\) used in both models have different meanings: In the case of the frequency-replanting model, the selection is based on the composition of susceptible and infected plants, and in the case of the abundance-replanting model, the selection is based on the abundance of infected plants. Moreover, some important remarks about the assumptions used to deduce the system (7)-(12) are the following: (i) plants are propagated vegetatively, by continuous replanting, using plant cuttings, within the crop field; (ii) it is not possible to identify infected plants prior to replanting, so \"frequency-of-selection\" or \"replanting-by-abundance\" replanting includes healthy plants and some infected plants; (iii) the growth of both populations is limited by the carrying capacity of the crop field; (iv) the rate of replanting, in any form, is density dependent; (v) there is no latency period in the plant population; and (vi) vectors are infectious throughout their lives, so there is no latency period. In this paper, we are interested in the analysis of periodic solutions of the plant virus system (7)-(12). We then propose a generalization by assuming that some parameters are positive, continuous, and periodic functions. To be more precise, we present the mathematical model as follows:  
\[\frac{dS}{dt} =b(t)\ r_{S}(S,I)S\bigg{(}1-\frac{S+I}{\theta}\bigg{)}-\beta(t)SW-h (t)S+g(t)I, \tag{13}\] \[\frac{dI}{dt} =b(t)\ r_{I}(S,I)I\bigg{(}1-\frac{S+I}{\theta}\bigg{)}+\beta(t)SW- (\alpha+h+g)(t)I,\] (14) \[\frac{dU}{dt} =a(t)(U+W)\bigg{(}1-\frac{U+W}{K(S+I)}\bigg{)}-\gamma(t)IU-(\mu+c) (t)U,\] (15) \[\frac{dW}{dt} =\gamma(t)IU-(\mu+c)(t)W. \tag{16}\]  
Moreover, we consider that the system is supplemented with the following initial condition:  
\[(S,I,U,W)(0)=(S_{0},I_{0},U_{0},W_{0}). \tag{17}\]  
Figure 1 shows the relationship between the compartments of the model equations. **Theorem 1**. : _Assume that \(\dot{e},\epsilon,\theta,\) and \(K\) are positive constants with \(\dot{e}<1\); the initial condition \((S_{0},I_{0},U_{0},W_{0})\in\mathbb{R}^{4}_{+}\); and the functions \(b\), \(\beta\), \(h\), \(g\), \(\alpha\), \(\gamma\), \(\mu\), and \(c\) are positive, continuous, \(\omega\)-periodic on \([0,\omega]\). Then, system (13)-(17) has at least one positive \(\omega\)-periodic solution. Study of a Preliminary Model shown in Appendix A._  
A previous result related to Theorem 1 for frequency-replanting models and abundance-replanting models when the coefficients are constant functions was deduced in [35], where the authors achieved the result of existence for non-negative solutions by considering the assumption that \(h+\alpha\leq\mu+c\) (see [35], theorem A.1). Thus, Theorem 1 is an improvement on the result from [35] since we do not consider the restrictions on \(h+\alpha\) and \(\mu+c\). In this paper, we apply topological degree theory, and the organization is as follows. In Section 2, for a self-contained presentation, we recall some notation, concepts, and results related to the statement of [PERSON]'s continuation theorem [36]. Moreover, in Section 2 we recall a useful result on the lower estimates for a linear differential equation. In Section 3, we define some particular Banach spaces \(X\), \(Y\) and operators \(L\) and \(N\), and prove that these particular operators satisfy the assumptions of [PERSON]'s continuation theorem. In