## 6 Conclusions and Future Work  
In this paper, we have presented a unified mathematical model for propagation dynamics of a pathogen living in a vector population that affects a plant population. The model considers the density distribution of both plants and vectors. Specifically, the plant population is categorized into two groups, healthy plants and diseased plants, with their time-dependent density populations denoted by \(S\) and \(I\), respectively. Similarly, the vector population is categorized into healthy and diseased vectors, with densities denoted by \(U\) and \(W\), respectively. These categorizations are part of the approach to model the dynamics of plant virus disease epidemics that affect plant populations and are transmitted by vector populations.  
We have successfully demonstrated the existence of positive periodic solutions, assuming that the coefficients are positive functions and the initial condition is positive. Significantly, our main result constitutes an advancement over the findings previously reported in [35]. The methodology employed to prove this result is based on the coincidence degree theory.  
Additionally, we have illustrated the model's implications through five numerical examples. These examples effectively demonstrate the periodic behavior of the system's solutions under the conditions stipulated by our main theorem. This practical application highlights the relevance and applicability of our theoretical findings in understanding the dynamics of pathogen propagation in vector-plant interactions.  
The real world applications of model predictions supports a predictive agriculture: With better models, farmers can predict when their crops are most at risk and take preventive measures, which leads to more stable food supplies and economic stability for farming communities. Complementarily, such predictions allow for resource allocation, e.g., of pesticides and labor, and biodiversity conservation by predicting and preventing outbreaks. Agricultural disease management and ecosystem health, an understanding of the periodic solutions of plant virus systems, helps in predicting disease outbreaks, optimizing control strategies, and enhancing crop protection.  
In our future work, we plan to conduct research on the epidemiology of plant-insect systems in at least three ways. First, we plan to study the comparison of the model with experimental data via the analysis of the parameter identification problem by using steepest descent methods with the adjoint and sensitivity methodologies [43; 44]. The second idea is to improve the model by including the effects of diffusion in the vector displacement and using the ideas of the SIS diffusion-reaction model [45]. Moreover, we expect to develop the model in the context of fractal-fractional differential theory [8].  
Conceptualization, A.C. and F.H.; methodology, A.C.; software, S.B.; validation, A.C., F.H. and S.B.; formal analysis, A.C., F.H. and S.B.; investigation, A.C., F.H. and S.B.; resources, A.C.; data curation, S.B.; writing--original draft preparation, F.H.; writing--review and editing, F.H. and S.B.; visualization, A.C., F.H. and S.B.; supervision, A.C.; project administration, F.H.; funding acquisition, A.C. All authors have read and agreed to the published version of the manuscript.  
The research was partially supported by Universidad del Bio-Bio (Chile) through the project INES I+D 22-14; Universidad Tecnologica Metropolitana through the project supported by the Competition for Research Regular Projects, year 2020, Code LPR20-06; and ANID (Chile) through projects Fondecyt project #1230560 and FONDEF project ID23110026.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflicts of interest.