MATHEMATICAL MODEL FOR SOYBEAN MOSAIC DISEASE TRANSMISSION WITH ENTOMOPATHOGEN INTERVENTIONS AND PHOTOPERIODICITY

: Mosaic is a severe disease of soybeans that has the potential to reduce the quality and quantity of soybean production. The Mosaic disease can infect soybean plants by the Aphis vector carrying Soybean Mosaic Virus (SMV). In this study, a mathematical model of the spread of Mosaic disease was built by considering two interventions, namely the application of entomopathogen and the regulation of photosynthesis intensity. This research focuses on knowing the effect of intervention in controlling Mosaic disease and increasing the population of susceptible generative plants. Using dynamical system theory, non-endemic and endemic equilibrium points and their stability are obtained. Then, the basic reproduction ratio (ℜ 0 ) is obtained for this model. Sensitivity analysis was carried out to determine the most influential parameters in the model. Optimal control theory was used to determine the optimal conditions of the model by considering the cost of entomopathogen application and photoperiodicity. The results of numerical simulations show that the application of entomopathogen and photosynthetic intensity can suppress the population of plant and vector infections and increase the population of susceptible plants in the generative phase at the same time


INTRODUCTION
Soybean is an important food containing protein and oil and is commonly used in the program of diets for both humans and animals [1]. Soybean is often called a marvel plant because it has a high protein of about 39-44% and an oil content of 21% [2]. It is the best source of protein and oil that may be an alternative to meat. Soybean content is used in the manufacturing industry, such as oil, cakes, flour, herbal cheese, and some additional products in the food industry [3]. These facts show that soybean is a crop needed for daily consumption and industrial needs. However, the high demand for soybean is not followed by the ability of the soybean farms to produce the grains. The main factors include climate change, inappropriate growth time, planting space, weeds, and disease [4].
Soybean Mosaic Disease (SMD) is a significant issue in soybean agriculture. The dis-ease begins when the plants is eaten by Aphid [5], a vector transmission that brings the Soybean Mosaic Virus (SMV). SMV occurs in almost all the soybean agriculture areas of the world and potentially infects other economic crops [5]. SMD causes the yield of agriculture to reduce between 35-50% of what it should be [6]. As SMV is an aphid-transmitted and seed-transmitted virus, it has three ways to infect the plants: 1) mechanical transmission, 2) insect transmission, and 3) grafting transmission [5]. The primary method of SMV transmission is transmission by insects such as various Aphids [7].
The presence of Mosaic transmitting vectors, namely Aphid insects, which are possible to be present in agricultural ecosystems increases the possibility of spreading the virus. In the concept of Plant Pest Control (IPM), the control goal is not to completely eradicate the pest/disease population, but it is dominant to manage the population below the threshold [8]. The use of insecticide against vectors showed a slight decrease in Mosaic cases [5]. The concept of IPM considers this step, but in terms of the use of insecticides, it is necessary to pay more attention because it causes damage to plants. The author in [9] explains that Lecanicillium Lecanii (L. Lecanii) has the potential to be a natural bio-insecticide or entomopathogen for Aphid.
The mathematical model can be used to study the phenomena such as infectious disease transmission both on human [10,11] and plant population [12,13]. The model for this study was usually formed as a compartmental-based model, which divided the population into subpopulations with a unique description. Many researchers have developed a model to study the long-term behavior of various health problem. Disease such as COVID-19 [11], dengue [14], and 3 MATHEMATICAL MODEL FOR SOYBEAN MOSAIC DISEASE hepatitis [15] were studied with some interventions as an effort to control the disease. The problem of plant disease transmission is studied by several researchers using the same analogy of the transmission of human disease.
A mathematical model for plant disease transmission is included as a vector-borne disease model. The model indirectly represents the process of spreading disease from infected plants to susceptible plants by involving the Aphis as a vector transmission. This relationship is interpreted through one of the mathematical studies, namely differential equations system. By knowing the spreading patterns, the disease transmission can be studied through analysis and simulation with some scenarios to predict disease behavior and the impact of interventions involved in the model.
Many researchers conduct to develop a vector-borne model for describing the plant disease and studying its behavior. Jeger [16] built a mathematical model for plant disease considering the latent period of infection in the plant population. The local stability [17] and global stability [18] of the plant disease model is investigated to learn the disease behavior for the long term. Luo et al. [19] and Al-Basir et al. [20] developed a plant disease model that considered the roguing and replanting of plants as an effort to eradicate the disease. However, the plant disease epidemic may be prevented through curative fungicide application [21] and protect the plants with applied some methods, such as roguing and insecticide spraying [20]. In order to control the disease by IPM concept, the insecticide or bio-insecticide can reduce the vector population, which interprets as a parameter con-trol in a mathematical model [22]. Suryaningrat et al. [23] built a vector-borne model for Tungro disease with considered the existence of biological agents as predators of vector transmission.
Mathematical studies of optimal control theory also applied to some vector-borne models for plant disease. A parameter interprets numerous interventions in a mathematical model set to be a control parameter in an optimal control problem, see [24][25][26] for examples. Chowdhury et al. [24] considered a model for pest management in plant dis-ease models through pesticides. Dynamical analysis and optimal control theory are applied to study the behavior of diseases with pesticide effort. In [25], Anggriani et al. developed an optimal control model to study the plant disease model by determining the curative treatment as a parameter control in the optimal control problem. Bokil et al. [26] built a mathematical model to describe the plant disease transmission, especially African Cassava Mosaic Disease (ACMD), and study the disease's behavior through mathematical analysis and optimal control theory. The roguing and insecticide effort was set to be the control parameter 4 SANUBARI TANSAH TRESNA, NURSANTI ANGGRIANI, ASEP K. SUPRIATNA in the optimal control problem.
In this article, we construct a mathematical model considering both insecticides to reduce the vector population and photoperiodicity to optimize plant growth. Photoperiodicity is the ratio between day and night lengths that may impact the process of plant growth and development [27].
This research focuses on comparing the advantages of interventions, including 1) entomopathogen without photoperiodicity, 2) photoperiodicity without entomopathogen, and 3) combining entomopathogen and photoperiodicity to the model. We look for the equilibrium points of the model and study their stability. A sensitivity analysis through Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) was carried out to determine parameters that have the dominant influence on the model. Then, we set the insecticide and photoperiodicity as two control parameters in the optimal control problem. We formulate the optimal control model and solve it to determine the optimal condition of both controls and minimize the infected vector population and maximize the susceptible generative population. However, we first satisfy the necessary and sufficient condition of the Pontryagin Maximum Principle [28] before solving the optimal control problem. Numerical simulations were conducted to confirm the analytical result.
Finally, we discussed the results comprehensively and presented some insights for future work.  Table 1). Finally, we can figure out a schematic diagram to describe the insect transmission process of the Mosaic virus in both plant and vector populations (see Figure 1).  Based on Figure 1, the differential equations system for the spread of the disease is written as equation (1).

MODEL FORMULATION
With the initial condition of each compartment as equation (2).

MATHEMATICAL ANALYSIS
In this section, we carried out the equilibrium points, both non-endemic and endemic, and analyzed their stability. The basic reproduction ratio (ℜ 0 ) is obtained through the next-generation matrices method.

Non-Endemic Equilibrium Point
The non-endemic equilibrium point is a state which represents that there is no disease infection in the system with ℜ 0 < 1. Based on the model in equation (1), the non-endemic equilibrium point is obtained as follows:

Endemic Equilibrium Point
The endemic equilibrium point is a state which represents that there is disease infection in the system with ℜ 0 > 1 . Based on the model in equation (1), the endemic equilibrium point is obtained as follows:

Basic Reproduction Ratio
In epidemiology, the basic reproduction ratio is essential to know, which shows the potential emergence of the spread of disease. Biologically, this ratio indicates the number of subsequent infections from one infective host or vector to susceptible hosts or vectors. Mathematic looks at the ratio as a parameter in studying the disease transmission through a compartmental model. The next-generation matrices method in [30] was used to obtain this parameter. We set as the new infection matrix and as the matrix of changes in the infection compartment. Based on equation (1), the and matrices are obtained as follows: We carried out and as the Jacobian from f and v matrices, then the spectral radius (dominant eigenvalue) of the −1 matrix is determined at the non-endemic equilibrium point (see equation 3). This process can be written in equation (5-8): 8 SANUBARI TANSAH TRESNA, NURSANTI ANGGRIANI, ASEP K. SUPRIATNA Finally, the parameter of the basic reproduction ratio is obtained radius spectral of F −1 as follows:

Stability Analysis
Analysis at the equilibrium points is carried out to determine the behavior of the system in the long term. The investigation began with formulating the Jacobian matrix for the model in equation (1). We write this matrix in equation (10).  (1) is locally asymptotically stable if ℜ 0 < 1.

Proof.
Through the method in [31], the local stability of the non-endemic equilibrium point in (3) can be determined by substituting 0 into the Jacobian matrix in (10). Then we get the characteristic polynomial, as follows: From the equation (11), we get the eigenvalues of the evaluated Jacobian matrix at the non- 2 ) )/(1 + )} and determine the character of polynomial 1 2 + 2 + 3 through Routh-Hurwitz criterion. All coefficients 1 , 2 , and 3 are known and can be written as: Since with = 1,2,3 are negative and 1 , 2 , 3 > 0 indicates that 4 and 5 have negative values -also ℜ 0 < 1. This completes the proof.

Endemic Point
The endemic equilibrium point of the model in equation (1) is locally asymptotically stable if ℜ 0 > 1.
Proof. Through the method in [31], the local stability of the endemic equilibrium point in (4) can be determined by substituting * into the Jacobian matrix in (10). Then we get the characteristic polynomial, as follows: From the equation (12)

NUMERICAL SENSITIVITY ANALYSIS
The sensitivity analysis of the plant disease epidemic model is presented in this section. We   increases. Therefore, it means that decreasing the value of Λ, , 1 and 2 can reduce the risk of losing crop yields.

OPTIMAL CONTROL PROBLEM
In order to control the spread of Mosaic disease with respect to the cost of interventions.
Globally, our goal is to minimize the number of vector populations and maximize the number of susceptible generative subpopulations. But, keep in mind the cost of interventions, both entomopathogen ( 1 ) and photoperiodicity ( 2 ) , remains low. We elaborated the optimal control problem to be three scenarios and explained it in the following subsection.

Scenario 1
The goal of optimal control in this case is to minimize the number of vector populations, both susceptible and infected, with respect to the cost of intervention through applied entomopathogen.
The objective function for this case is written in the equation (11).
Note that 2 1 1 2 = 2 1 > 0 satisfies the minimum criterion of optimal control theory with 1 * being the optimal control of the system.

Scenario 2
The goal of optimal control in this case is to maximize the number of susceptible generative subpopulations with respect to the cost of intervention through controlling the photoperiodicity.
The objective function for this case is written in the equation (17). The stationary condition of the system for scenario 2 is Note that 2 2 2 2 = −2 2 < 0 satisfies the maximum criterion of optimal control theory with 2 * being the optimal control of the system.

Scenario 3
The goal of optimal control, in this case, is to minimize the number of vector populations and maximize the number of susceptible generative subpopulations. We are concerned about the cost of interventions, both entomopathogen and photoperiodicity. The objective function for this case is written in the equation (21). The stationary condition of the system for scenario 3 is theory with 1 * and 2 * being the optimal control of the system.

NUMERICAL SIMULATION
In this section, we evaluate the model to confirm the system behavior and all scenarios of optimal control problems through a numerical approach. The value parameters used are described in Table 1. Note that scenario 1 and 2 in the optimal control problem has a different use of parameter, including 1) scenario 1 ( = 0) and 2) scenario 2 ( = 0). It represents there are no use interventions in the scenario, respectively. We obtained some graphical simulation that represents the population dynamics in the phenomenon of soybean Mosaic disease spreads.

Population Dynamics without Optimal Control Theory
In this subsection, the population dynamics without intervention and with interventions are presented to compare the spread of Mosaic disease in soybean plants. We also confirm the stability of each equilibrium point through the simulation for the long term.   Figure 3(b) shows that the number of infected populations will go to zero. It is caused by the number of vectors controlled; moreover the subpopulation of the infected plant will go to zero.
The explanation can be concluded that the risk of Mosaic disease spreading can be reduced.

Effect of Entomopathogen
In  In scenario 1, a simulation is carried out by considering the control optimal of entomopathogen, but without photoperiodicity intervention. Therefore, parameter value 1 ( ) = 1 * ( ) is applied to the simulation. The result is obtained in Figure 5. In scenario 2, a simulation is carried out by considering the control optimal of photoperiodicity but without entomopathogen intervention. Therefore, parameter value 2 ( ) = 2 * ( ) is applied to the simulation. The result is obtained in Figure 6.
(a) Population dynamics (b) Control function of photoperiodicity Based on Figure 6, it is presented that the disease is still spreading in the system. The control in this scenario is indirect to impact the vector populations, but optimize the growth and development of plants. So the number of vectors is uncontrolled and causes the infected plant to increase. We conclude that scenario 2 is not successful in reducing the risk of disease spreading.

Population Dynamics with Optimal Control Scenario 3
A simulation is carried out in scenario 3 by considering the control optimal of entomopathogen and photoperiodicity intervention. Therefore, parameter value 1 ( ) = 1 * ( ) and 2 ( ) = 2 * ( ) is applied to the simulation. The result is obtained in Figure 7. The control function graph shows that the control values for each 1 ( ) and 2 ( ) are not different from those obtained in scenarios 1 and 2. Globally, we see that the number of infected plants and vectors can be reduced. It represents that the disease is not spreading in the system and scenario 3 can be said to control the disease successfully. But, we have to compare the population dynamics in each scenario, including no control, 1, 2, and 3, to see the difference specifically.

Comparison of Population Dynamics
This subsection shows the population dynamics in the susceptible generative plants, susceptible vectors, and infected vectors as the main subpopulations of optimal control targets through four scenarios. The first scenario is to show population dynamics without intervention in the system. While the second, third, and fourth scenarios are scenarios 1, 2, and 3 in the optimal control problem. The results obtained are shown in Figure 8.