THE DYNAMICS OF COVID-19: THE EFFECT OF LARGE-SCALE SOCIAL RESTRICTIONS

Due to the global pandemic of Covid-19, Indonesian government released a large-scale social restriction policy to reduce the spread of the disease. It still allowed people to have activities outside. The government also had another policy to divide covid-19 patient into three categories, such as People under Monitoring (ODP), Patients under Surveillance (PDP), and Confirmed Patients. This study aimed to explore the Covid-19 model in which large scale social restriction had been involved. It was not only to figure out the stability analysis of the model but also to predict the spread of the disease through numerical simulation. We constructed model based on the characteristic of Covid19. Human population have been divided into five sub population, such as Susceptible (0), Susceptible (1), Exposed, Infected, and Recovered. Three of them related to the Covid-19 patient’s category. A disease-free equilibrium and endemic equilibrium have been determined. By using Next Generation Matrix, the basic reproduction number also had been obtained. Stability analysis have been done to explore the existence of the disease. A large scale-social restriction had a significant effect to the spread of the disease. The effectiveness of policy ranges from 80%-100% and it will reduce the number of people under monitoring. 2 ARTIONO, PRAWOTO, HIDAYAT, YUNIANTI, ASTUTI


INTRODUCTION
Since the end of 2019, the disease caused by the corona virus has spread throughout the world.
The disease, known as Covid-19 and first appeared in Wuhan China, has infected 6,535,354 patients as of early June 2020. Meanwhile, data obtained from the WHO site on June 5th, 2020, showed that 213 countries had been infected by . Developed countries such as the United States, Brazil, Russia and Italy have been recorded as the countries with the most cases of Covid patients in the world [1][2][3]. Meanwhile, the number of patients who die in developing countries continues to increase every day. The number of death due to Covid-19 that occurred in Iran, Pakistan, India, Phillipine and Indonesia until the end of May 2020 were 7734, 1483, 5164, 950 and 1,573 people respectively [4][5].
Several countries issued new policies related to handling Covid-19. Countries such as Austria, Belgium, France, Ireland, Italy, Luxembourg, Portugal, Spain and the United Kingdom have implemented a full lockdown on their citizens to prevent the spread of the disease, while Korea and the United States had implemented a partial lockdown in several of its states [6][7].
Indonesia with an increasing number of corona patients had also issued new policies to reduce the movement of its citizens. Some big cities with a relatively large number of patients (known as red and black zones) were subject to large-scale social restrictions. This restriction did not apply to citizens who work in certain fields, such as: health, logistics, banking, and communications [8]. In addition to large-scale social restrictions, Indonesia had also implemented a naming system for cases of covid-19 patients, such as: People under Monitoring (ODP), Patients under Surveillance (PDP), and Confirmed Patients [9]. This form of labelling was done to classify the risk and appearance of symptoms from people who might or have been exposed to the corona virus.
In addition, based on the https://kawalcovid19.id site, the labelling system used by the Indonesian government was based on one of the characteristics of the Covid-19 or a person's travel is based on the labelling system used by the Indonesian Government.

MATERIALS AND METHOD
This modelling is constructed from the basic model of SIR disease spread which was first developed by Kermack-Mckendrik in 1927 [10]. We propose a mathematical model with the compartments shown in Figure 1. The human population is divided into five sub-populations, namely Susceptible ( 0 ( )), Susceptible ( 1 ( )), Exposed ( ( )), Infected ( ( )), and Recovered Regard to the compartment diagram, mathematical model is described as follows

RESULTS AND DISCUSSION
Furthermore, one of the most significant thresholds in studying disease transmission models is knowing the position where the disease will remain in a population or will extinct for a long time.
This threshold is known as the basic reproduction number. In this section we show the basic reproduction numbers generated by next generation matrix, stability analysis, sensitivity analysis, and numerical simulation.

a. Basic Reproduction Number
By using the next generation matrix approach, the determination of the basic reproduction number can be determined using a matrix and which is the Jacobian matrix associated with the incidence rate of new infections [11].
The basic reproduction number 0 is determined using a spectral radius . −1 so that the following form is obtained

b. Local Stability
From equations (1) -(5) and the disease-free equilibrium point, the Jacobian matrix is obtained as Furthermore, the stability analysis is carried out by analyzing the eigenvalues at the disease-free equilibrium [12]. From the Jacobian matrix above, five eigenvalues are obtained as follows: a. 1 = = − − . The value of lies between 0 and 1 so that the eigenvalue 1 is always negative. It means that the disease-free equilibrium will be stable when the value of 1 is satisfied.
b. 2 = − . All parameter are assumed to be positive so that the eigenvalue of 2 is always negative. It means that the disease-free equilibrium will be stable when the value of 2 is satisfied.
c. 3 = − − . All parameter are assumed to be positive so that the eigenvalue of 3 is always negative. It means that the disease-free equilibrium will be stable when the value of 3 is satisfied.
. The value of is always negative while the value of and are always positive so that the real part of this eigenvalue 4 is always positive.
It means that the disease-free equilibrium will not be stable when ( + ) 2 − (4 2 − 4 ) < 0. This condition will be satisfied when the of value 0 > 1. . The value of is always negative while the value of and are always positive so that the real part of this eigenvalue 4 is always positive. It means that the disease-free equilibrium will not be stable when ( + ) 2 − (4 2 − 4 ) < 0. This condition will be satisfied when the of value 0 > 1.
Regard to the eigenvalues (a) -(e), it can be concluded that the disease-free equilibrium of the equations (1) -(5) is locally asymptotically stable if 0 < 1 and unstable if 0 > 1.

c. Sensitivity Analysis
Sensitivity analysis on the predetermined threshold value, 0 , needs to be done to find out which parameter has the most influence on the spread of the disease. This analysis is not only for experimental design but also for data assimilation and simplification of complex models [13]. This analysis is used to determine the robustness of the model's predictions to parameter values so that the parameter which has the highest effect on the threshold value can be determined. The following is the definition for determining the sensitivity of a parameter value [14]. In this study, the sensitivity analysis of parameter values use the ± 10% rule which is the result can be seen in the change of exposed and infected populations. The elasticity of parameters can be seen in Figure 2.  Regard to the results of the sensitivity analysis, there are two parameter values that are interesting to explore, such as and parameters. Furthermore, by using numerical simulation, the behaviour of the developed model will be seen using several values and . The stability of the endemic equilibrium is also explored with respect to the threshold value of 0 .

d. Numerical Simulation
Numerical simulation is used to see the behavior of the model that has been developed with the correspond to the change of the natural death rate. This natural reduction in mortality can occur if the human immune system is enhanced [15]. It means that a good immune system in a population can reduce the spread of Covid-19, although for a long time period ( → ∞), the disease will remain in a population.

CONCLUSION
In this study, a mathematical model of the spread of the Covid-19 has been produced that takes into account large-scale social restriction policies and the labelling of people under monitoring (ODP), patients under surveillance (PDP) and patient confirmation (Confirmed patients). In addition, this research has also produced a disease-free equilibrium point obtained analytically, the threshold requirement, namely 0 , the critical point stability requirements through the threshold value, sensitivity analysis to determine the most influential parameters in the model, and numerical simulation in around the endemic equilibrium point.
From the results of this study, it can be concluded that large-scale social restriction policies will be successful if the effectiveness of their implementation ranges from 80% -100%. This policy will reduce the number of people under monitoring. In addition, this study can also conclude that the change of the death rate in a population will also reduce the spread of the Covid-19. This changing in mortality can be done by increasing the immune system in a population.