## 1 Introduction  
The mathematical modeling of plant virus diseases has been increasing in recent decades since it constitutes one of the most relevant tools for understanding virus propagation and implementing strategies of control, which include the development of appropriate technologies [1; 2; 3; 4; 5; 6]. The interaction between plants and viruses is complex; consequently, mathematical models are built to provide information about the most relevant variables in the environment. Additionally, there is a range of techniques used to construct these mathematical models. These include regression analysis, mixed model and generalized linear model analyses, multivariate and time series analyses, differential equations, and stochastic models [7; 8; 9; 10; 11; 6]. Specifically, in this paper, we focus on applying ordinary differential equations to model the dynamics of plant virus disease epidemics that affect plant populations and are transmitted by vector populations. For a more detailed review of plant virus disease epidemics we refer to [12; 13; 14; 15]. Periodicity is a common characteristic in biologically relevant phenomena. Typically, these systems exhibit a range of scenarios, from disease-free states to mutual extinction, often including stable periodic orbits. These orbits can be influenced by seasonal variations, leading to fluctuating periods of high and low virus transmission rates. Understanding these patterns is crucial for predicting virus outbreaks and devising effective control measures. For instance, studies have highlighted the significance of seasonal variations in disease dynamics, underscoring the need for the strategic implementation of control measures during critical periods, such as the rainy season [16]. This concept of periodic environments is well-documented [17], particularly in plant virus contexts where roguing infected plants is a known control strategy [18]. For the agricultural industry, which plays a pivotal role in food security, understanding these dynamics is vital. Viral outbreaks in plants can severely impact agriculture. Periodic forcing studies are, therefore, instrumental for farmers and agricultural policymakers in strategizing crop rotations, planting schedules, and timing treatments effectively. The distinction between continuous and impulsive roguing control is a pertinent example [19]. A deeper comprehension of the periodic nature of plant virus interactions paves the way for more sophisticated control strategies. This study examines a mathematical model that incorporates periodic forcing, offering more precise predictions of virus spread and outbreak intensity. Such models are indispensable for preparing and responding to agricultural crises, ensuring a proactive approach to safeguarding crop health and food security. Periodic forcing is especially important in scenarios where the plants are cassava plantations, the vectors are Bemisia tabaci whiteflies, and the virus is the cassava mosaic virus: Whiteflies inflict damage on a variety of plants--including cassava, tomatoes, sweet potatoes, roses, and households--from their nymph stage, which lasts about 3 days, through to their adult phase, spanning 5 to 28 days. While mature adults typically fly short distances, usually no more than 2 meters during daylight, air currents play a crucial role in dispersing them across the cultivation area. Immediately upon reaching adulthood, these insects mate, with each female laying between 80 and 300 eggs [20; 21]. The species Trialeurodes vapor-arriorum is more suited to temperate climates, whereas Bemisia tabasi thrives in warmer or tropical environments. Consequently, the emergence of these vector insects, as well as their interactions with healthy plants, occurs during times of the year when temperatures favor the presence of whiteflies [22]. This underscores the importance of investigating periodic forcing in the plant virus dynamic. Identifying the specific times of the year when whiteflies are most likely to appear, according to models, is crucial.