The NADI Mathematical Model on the Danger Level of the Bili-Bili Dam

The research discusses the NADI mathematical model due to the overﬂow of the Bili-Bili dam, using secondary data obtained through online literature review by collecting various information related to the Bili-Bili Dam, starting from the Jeberang River Scheme, the chronology of ﬂoods, normal or dry conditions, and dam operation patterns. The aim of this study is to predict the level of danger of Bili-bili dam overﬂow over time, considering extreme weather factors and standard operating procedures performed by humans. The research uses analytical and computational methods. The study obtained the NADI mathematical model due to the overﬂow of the Bili-Bili dam, with two equilibrium points: (1) the equilibrium point free of disaster, (2) the disaster equilibrium point, and a basic disaster reproduction number of R0 = 1.219. This indicates that the water discharge from the dam is high and has an impact on the overﬂowing water for communities around the Jeneberang river. Therefore, it can be concluded that the NADI model can be used to simulate the Bili-bili dam process based on extreme weather and dam SOP, and predict the level of danger of Bili-bili dam overﬂow, which is also a novelty that has not been done in previous studies.


A. INTRODUCTION
In the historical development of civilization in the world, there are many innovative ideas to help solve human problems. One of the problems that humans often face is natural disasters (Suwaryo and Yuwono, 2017), which have a negative impact on all aspects of human life. One of the most frequent natural disasters in Indonesia is flooding (Darmawan et al., 2017;Niode et al., 2016). Floods are the most frequent natural disasters which are around 40 percent of other natural disasters (Risman et al., 2016). The flood itself is one of the natural phenomena that always occurs in the rainy season. Flooding occurs due to high rainfall intensity which causes excess water that exceeds the capacity limit in an area. The flood plain area is a lowland area on the side of the river which has a very gentle and relatively flat elevation. So, to reduce this, a reservoir is built and large excess runoff during the rainy season can be accommodated in the reservoir. South Sulawesi with the capital city of Makassar is famous for the Bili-Bili reservoir which is located in Gowa Regency, South Sulawesi.
The reservoir was built with a main building height of 73 m and a length of 750 m. The catchment area of the reservoir is 384.40 km 2 with a storage capacity of 375 million m 3 . The Bili-Bili reservoir is the confluence of the Jeneberang River and the Jenelata River. The Bili-Bili reservoir is multifunctional, namely as a flood control or reducing discharge (Achsan et al., 2015) where the discharge is 2200 m 3 /s to 1200 m 3 /s, the provision of raw water sources is 3300 liters/s, irrigation water services with an area of

Fluid
Fluid (Risman et al., 2016), in physics, is a substance or subsystem that will continuously deform when exposed to a shear force (tangential force), even though the force is small. When there is an overflow of water across the crest of a spillway, flow contraction occurs both on the side walls of the spillway and around the pillars built on top of the spillway, so that hydraulically the effective width of a spillway will be smaller than the actual overall width of the spillway. The flow of water that crosses the spillway in question is always based on its effective width, which is the result of subtracting the actual width from the total number of contractions that occur in the flow of water passing through the spillway.

SEIR Mathematical Model
SEIR mathematical model (Side et al., 2018b) is a system of differential equations to calculate the number of model distributions or non-linear incidence rates in epidemiology. The global stability of catastrophic equilibrium is proved by using the general criteria for orbital stability of the periodic orbits associated with higher-dimensional nonlinear autonomous systems as well as the theory of competitive systems of differential equations. As far as development goes, this mathematical model that we refer to deals with the extent of disaster spread.

System Equilibrium Point
The equilibrium point is a state of a system that does not change with time. If the dynamics system is described in a differential equation, then the equilibrium point can be obtained by taking the first derivative which is equal to zero. In the Definition 1.1

Stability Analysis of Equilibrium Point
The linear differential equation system of order n is given as follows: where i = 1, 2, . . . , n The first step to get a solution to equation (*) is to find the equilibrium point. Suppose we obtained the equilibrium point (x * 1 , x * 2 , . . . , x * n ), then the next step is to find the Jacobian matrix. Let G i (x 1 , x 2 , . . . , x n ) = Ax + f i (x 1 , x 2 , . . . , x n ), the Jacobian matrix is The next step is the substitution of the equilibrium point in the Jacobian matrix to obtain the linear system as follows: .,x * n ) u Determination of the stability of the equilibrium point is obtained by looking at the eigenvalues λ i with i = 1, 2, . . . , n as follows: In general, the stability of the equilibrium point has two behaviors, namely: 1. Stable if (a) Re(λ i ) < 0 for every i, or (b) There are Re(λ j ) = 0 for any j and Re(λ j ) < 0 for every ij. 2. Unstable if there is at least one i so that Re(λ i ) > 0

Basic Reproductive Number
The basic reproduction number can be found using the next-generation matrix method. This matrix is formed by taking into account the positive and negative parts of the population transmission rate, namely the exposed and infected populations. The formula for determining the number reproduction base is given in the equation K = F (V ) −1  In general, the basic reproduction number R 0 has three possibilities, namely: If R 0 < 1 then the probability of overflow at the Bili-Bili dam is low. If R 0 = 1, then the Bili-Bili dam will be stable. If R 0 > 1, then the probability of overflow of the Bili-Bili dam is high.

B. RESEARCH METHOD
This research is a theoretical and applied research study that is reviewing the literature on mathematical modeling related to the overflow of the Bili-Bili dam. The data used in this study is secondary data obtained from the literature that has been previously studied so that the resulting 4 stages in the dam are produced due to the volume of water in the dam, namely fluid sample N (0) is 2000 m 3 /s; score fluid sample Ad(0) is 2500 m 3 /s; score fluid sample Ds (0)   (4) f-class (I) namely Infection Recovery or Jenelata River flow rate i.e. 600 m 3 /s with a maximum of 987 m 3 /s. The maximum limit meant here is limit shelter from a variable, where if pass the limit, can be assumed the overflow happened and causes disaster.
There are several assumptions used in creating models, namely: 1. The population must have a system control rate. 2. The data rate of deep-water utilization at Bili-Bili dam is considered constant. 3. The data rate comply the fluid rule 4. The data entered into f-class (N ) are Jeneberang River, Malino River, and Bontomanai. River as well as raining around downstream of the Jeneberang River. 5. The data entered into f-class (AD) from f-class (N ) is the fluid flow of the Jeneberang River, Malino River, and Bontomanai.
River fluids join rain above as well as around downstream of the Jeneberang River going to the estuary Bili-Bili dam 6. The data entered into f-class (AD) are rain over the estuary of the Bili-Bili Dam and annual sedimentation due to landslides and other sedimentation factors. 7. The data entered into the f-class (Ds) from the f-class (AD) is the estuary fluid flow to the Spillway reservoir. 8. The data entered into the f-class (I) from the f-class (Ds) is the flow of the Spillway reservoir fluid to the Jenelata River when the Spillway door is opened. 9. The assumption of water utilization in the Bili-Bili dam is to make an artificial waterfall as a control gate to an artificial lake that functions as a water reservoir in Gowa Regency or create a new flow from the dam with a storage capacity of 1000 cubic meters per second to the Makassar Strait. The developed SEIR model schema becomes a NADI mathematical model can be seen in Figure 2 below. Figure 2. Schematic of the SEIR Model that has been developed become NADI model Figure 2 can also be interpreted in the mathematical model which is an equality nonlinear differential like the following: To determine point equilibrium free disaster and point equilibrium disaster, every equation in equation (5) Furthermore, using a simple substitution method, the values of S, Ad, Ds, and I will be determined for the disasterfree equilibrium point and the disaster equilibrium point. The disaster-free equilibrium point is the state where no occur deployment disaster overflow the Bili-Bili dam so that Ad = 0. By doing a little algebraic manipulation on equations (9)-(12) then we obtained equations (13)-(16) as follows: By substituting each equation (13)-(16) by first determining the value Ad = 0 then the disaster-free equilibrium point is obtained as follows: Next, in the same way, by substituting equation (13)-(16) then the disaster equilibrium point is obtained as follows:

Basic Reproduction Number
The basic reproduction number could be found using the next-generation matrix method. This matrix is formed by taking into account the positive and negative parts of speed fluid from along the waterway within the Bili-Bili dam area. The formula for determining the basic reproduction number is given in equation (19) Based on equation (6), then So that obtained So, we get the inverse matrix of equation (21) as follows: The eigenvalue of matrix R is determined based on equation (19) as follows: After obtaining matrix R in equation (23), then the Eigenvalue is found with the formula det(λI − R) = 0, where I is matrix identity. The basic reproduction number is determined based on the highest eigenvalue (λ).
So that we obtained the eigenvalue based on equation (24) as follows: and λ 2 We get the highest eigenvalue which is λ 1 = α(µ 1 + ϕ) The basic reproduction number based on equation (24) is given as follows:

Stability Analysis of Equilibrium Point
Based on equation (1) -(4), matrix Jacobian (J) can be formed as follows: Disaster-free equilibrium point on the spread of Covid-19 is said stable if R 0 ≤ 1 and not stable if R 0 > 1.

Proof.
Substitution of disaster-free equilibrium point to matrix J equation (26), so that we obtained new matrix as in equation (27).
Then the eigenvalue of the matrix in equation (27) is solved with a description as follows: Furthermore, substitute Ad in equation (28) so that we obtained equation (29).
Based on the rule of Descartes sign, equation (29) will have the root of all negative if all marks on each of the tribes are positive. So, it can be concluded that a disaster-free equilibrium-free point is stable if R0 ≤ 1 and not stable if R 0 > 1.

NADI Model Simulation in Disaster overflow from Bili-Bili dam
The simulation was conducted using Maple 17 software with a given score for each of the parameters. The parameter values are taken so that we obtained R 0 = 1.2195 > 1 which indicates that the event overflow and possibly collapse dam could happen. The score fluid sample N (0) is 2000m 3 /s; the score fluid sample Ad(0) is 2500m 3 /s; the score fluid sample Ds(0) is 1426m 3 /s; score fluid sample I(0) is 600m 3 /s; sample W is maximum model fluid.

Equilibrium Point of the NADI Model
The equilibrium point is determined using the NADI model set with the parameters for Bili-Bili that have already been defined. To determine the fixed point, the system of equations (1)-(4) is equated to zero and it is given in equations (9)-(12). By substituting the parameter values into equations (9)-(12), we get equations (30)-(33) as follows: (1)N + (0.5 + 1) − (1.82)Ad = 0 (1)Ad − (1.02)Ds = 0 (32) (1)Ds − (1)I = 0 (33) The system for the model in equations (30) These equilibrium points explain that the potential for the Jeneberang river flow and the Bili-Bili dam to overflow is 182%, the potential for the Bili-Bili dam spillway flow is 102% or close to normal conditions, the Jenelata river flow potential to overflow is 100% or in under normal conditions. The most potential solution to be given is the Jeneberang river which has the greatest potential to trigger the overflow of the Bili-Bili dam.

Stability of the NADI Model of Overflow from the Bili-Bili dam
By using equations (30)-(33) and the specified parameter values, the NADI model is converted into a Jacobian matrix (34)-(37) to find the eigenvalues λ.

Simulation Results of the NADI Model with Time Delay for the spread of the Bili-Bili Dam Overflow in South
Sulawesi The simulation of the NADI model without Stage II water level control can be seen in Figure 3. River, Plot (ADI) represents the water level in the dam crossing the elevation limit (standby stage), Plot (Ds) represents the capacity of the spillway gate that reaches the limit (the dam that makes the water elevation in the dam increases faster and causes catastrophic collapse and will cause a greater impact than the January 22, 2020 event. Plot (IR) represents the Jenelata River overflowing and submerging residential houses around the Bili-Bili Dam and along the Jenelata River.
The simulation of the NADI model with Stage II water level control can be seen in Figure 4.  Figure 4 shows the model plot for the non-overflowing of the Bili-Bili Dam. Plot (N F F ) represents the non-overflowing Jeneberang River, Plot (ADI) represents the water level in the dam that does not exceed the elevation limit (standby stage), Plot (Ds) represents the capacity of the spillway gate that does not reach the limit. Plot (IR) represents the Jenelata River which does not overflow and does not inundate residential houses around the Bili-Bili and along the Jenelata River.