Mathematical Model of Effect of Yellow Virus on Tomato Plants Through Bemisia tabaci Insects Using Verticillium lecanii Fungus

-The Yellow virus is a virus that causes tomato plants to die. The insect vector Bemisia tabaci spreads this virus. The goal of this study is to identify the shape of a mathematical model of the influence of yellow virus on tomato plants via the insect Bemisia tabaci and the fungus Verticilliun lecanii , as well as to interpret the results of the mathematical model analysis. This is referred to as basic research. This study employs a descriptive method in which theories are analyzed in relation to the topics to be discussed, and these theories are based on a literature review. Stability analysis is carried out using Routh-Hurwitz criteria. It indicates that the disease-free equilibrium point is asymptotically stable when Λ t = μ t N and the endemic equilibrium point is asymptotically stable for 𝑑 1 > 𝑒 1 , 𝑑 2 > 𝑒 2 and a 1 > a 12 +a 32 a 0 a 3 a 2 . The model simulation shows that if the efficacy of Verticillium lecanii is high, the population of infected tomato plants, as well as the population of Bemisia tabaci , will go extinct.


Introduction
Tomatoes, which are members of the Solanaceae family, are the second most important fruit and vegetable crop after potatoes. Tomatoes are grown as fresh fruit and processed into processed items. Vitamins, carotenoids, and phenolic compounds are among the healthpromoting substances found in tomatoes. Tomatoes have become a model for the study of the growth of fleshy fruits, in addition to being economically and nutritionally important. Tomatoes are a fruit with obvious metabolic changes during fruit development (Quinet et al., 2019). According to 2020 horticultural statistics, the use of tomatoes plants is increasing year after year. The increased consumption of tomatoes necessitates an increase in tomato plant output, yet there are various obstacles that cause tomato plant production to be less than optimal. The yellow virus that affects tomato plants is one of the causes of poor tomato plant productivity. The yellow virus is a pathogenic virus that is difficult to control since it lives as an obligatory paresis in plant cells, therefore eradicating the virus requires killing the host's cells or tissues (MAHENDRA et al., 2017). The first Yellow Virus was discovered in Israel in the late 1930s, and tomato farming in the Middle East has been seriously impacted since the 1960s. The yellow virus is carried naturally by the whitefly Bemisia tabaci. The Bemisia tabaci has a diverse host range. Initially, the yellow virus only infects weed species before spreading to kill nearby cultivated plants, one of which is tomato plants (Moriones & Navas-Castillo, 2000). In 1989, the yellow virus was discovered destroying mangroves in Indonesia. The yellow virus was discovered to have affected chili plants in 2001. The yellow virus attack was so pervasive in 2004 that 984,6 ha were infected, resulting in financial losses of Rp.7.31 billion and a yield losses of 20-100% (Gunaeni et al., 2008). The entomopathogenic fungus Verticillium lecanii can inhibit the spread of this yellow virus. Verticillium lecanii, an entomopathogenic fungus, is an excellent fungus for eradicating Bemisia tabaci insects. The fungus Verticillium lecanii has the advantage of preventing the hatching of vector insect eggs. This fungus can also infect all stages of Bemisia tabaci, including nymphs and imago (detrivores), as well as natural enemies, including parasitoids and predators (Prayogo, 2014). This problem can be described mathematically to describe the real-world characteristics of the problem and also as a tool for policy planning and control.

Methods
The research is referred to as basic research by descriptive research methods. This study was carried out using a literature review, which entailed gathering books and references. The theories developed will be used to solve existing problems and draw conclusions. The following steps have been conducted to explain the model analysis: 1. Identifying the problem to be modeled. 2. Collecting and reviewing relevant theories about the problem. 3. Determine the variables, parameters, and assumptions that will be used in the formation of the model. 4. Forming a model of variables, parameters, and assumptions that have been determined. 5. Analyzing the mathematical model that has been formed. 6. Interpreting the results of the analysis 7. Draw a conclusion. Based on Figure 1, we get the mathematical model for yellow virus on tomato plants through the insect Bemisia tabaci using the fungus Verticillium lecanii as the following system:
The basic reproduction number ( 0 ) is the largest spectral radius or eigenvalue of the matrix ( −1 ), thus it determines as: To simulate the model, we use Maple Software to show the trajectory near the steady states. The parameters used to simulate the model are: From the parameter values in Table 1, 0 will be calculated so that the basic reproduction number is obtained 0 = 0,3346958083.. Since 0 < 1, the free-disease equilibrium is asymptotically stable. The parameter values in Table 1 then are substituted in the disease-free equilibrium 0 such that 0 = (50, 0, 3.8647,0). The trajectory is given in Figure 2: This indicates that 0 is asymptotically stable and the value of 0 < 1, which means that within a certain time the spread of the yellow virus will disappear. While the simulation for the endemic equilibrium point uses the following parameter values:  The numerical simulation shows 1 is asymptotically stable and 0 > 1, which means that the yellow virus will spread for a long time.
(c) Interpretation Based on the stability analysis, the disease-free equilibrium point is asymptotically stable if = which means the yellow virus can disappear within a certain time. Meanwhile, the endemic equilibrium point will be asymptotically stable if 1 > 1 , 2 > 2 , and 1 > 1 2 + 3 2 0 3 2 which means that the yellow virus will spread for a long time. The simulation results given indicated that if the parameter value of the effectiveness of the use of the fungus Verticillium lecanii is large, the population of infected tomato plants will decrease to extinction as well as the insect population of Bemisia tabaci. And conversely, if the parameter value of the effectiveness of using Verticillium lecanii is small, the yellow virus will spread for a long time.

Conclusion
The mathematical model of the spread of the yellow virus in tomato plants is stated in the system of differential equations. There are two equilibrium points, namely the disease-free equilibrium point and the endemic equilibrium point. The stability analysis for disease-free equilibrium point is asymptotically stable if Λ t = μ t N and for the endemic equilibrium point it is asymptotically stable if d 1 > e 1 , d 2 > e 2 , and a 1 > a 1 2 +a 3 2 a 0 a 3 a 2 . The numerical simulation model shows that if the effectiveness of the fungus Verticillium lecanii is large, the infected tomato plant population will decrease to extinction as well as the Bemisia tabaci insect population.