Stability Analysis of the Corona Virus (Covid-19) Dynamics SEIR Model in Indonesia

Covid-19 is a type of virus that infects the respiratory tract or is also known as severe acute respiratory syndrome Corona virus-2 (SARS-CoV-2). Researchers through this study were trying to build a mathematical model of the spread of the Covid-19 virus and analyzed the stability of its critical points. This virus dynamic model was built based on the interaction between the suspect group with the exposed group and the interaction with the infected group. The virus spread model in this study consists of two critical points, namely a disease-free critical point and a disease endemic critical point. The two critical points will be stable if they meet the requirements for certain parameter values. Numerical simulations of virus dynamics in Indonesia were carried out using data on the number of susceptible people, the number under monitoring, the number of the infected, the number of cures and the number of deaths due to the virus. These data were obtained from the Ministry of Health of the Republic of Indonesia. Based on the data obtained, it was found that the simulation model of stability reached a critical point in a very long time. Therefore, treatment or vaccines are needed as a step to reduce the dynamics of Covid-19.


Introduction
Corona virus or severe acute respiratory syndrome corona virus 2 (SARS-CoV-2) is a virus that attacks the respiratory system. The virus causes this disease is called Covid-19 (Corona Virus Disease 2019). The ICTV Corona virus disease Study Group stated that this virus is a species associated with severe acute respiratory syndrome. Covid-19 was first discovered in humans in December 2019. This outbreak was first detected in Wuhan City, Hubei Province, China in mid-December 2019. In the end, the outbreak due to SARS-CoV-2 was declared a global health emergency or pandemic by the Health Organization. World (WHO) on January 30, 2020. The Chinese government conducted quarantine in the city of Wuhan on January 23, 2020 as a step to control the pandemic [2].
Based on data from BPS (Statistics Indonesia), it is known that the total population in 2020 is projected to be 271,066,400 people [1]. The discovery of Covid-19 was first confirmed in Indonesia on March 02, 2020. At that time it was declared that there were 2 people who were positively infected, of which 0 recovered and 0 died [7]. The investigators suggest that the main reason for the high number of cases may be due to a lack of testing so that many cases go undetected. The Indonesian Ministry of Health officially changed the terms ODP, PDP, OTG and confirmed cases to be suspect cases, probable cases, close contacts and confirmed cases. This change is contained in the Decree of The assumptions in the modification of the SEIR model are: 1. The population size is constant. 2. The population is homogeneous where every individual has the same opportunity to make contact with other individuals. 3. The spread of the corona virus occurs from person to person. 4. Individuals who have recovered cannot be reinfected. Based on the modified SEIR Model scheme, a system of differential equations is obtained as follows: (1)

The Critical Points of SEIR Model for the Spread of Covid-19
The critical point of the system can be found if it satisfies . Thus, the model critical point is processed as follows: (2) (3) (4) (5) From (2) it is obtained (6) From (4) it is obtained (7) Furthermore, Equation (7) is substituted into Equation (6) as follows Equation (7) and Equation (8) is substituted into Equation (3) as follow: For E * = 0, the S * , I * , R * obtained are as follows: And for the S * , I * , R * obtained are as follows:

Stability Analysis of Disease-Free Critical Point
Based on the SEIR model, suppose the functions are determined as follows: so that the Jacobian matrix is obtained as follows: The Jacobian matrix at the free of virus critical point J(E 0 ) is as follows: can be written into a characteristic equation: r . Based on the theory of stability, the system will be stable if all eigen values are negative. According to the characteristic equation, the value of is positive, will be positive for , and will be positive value for Therefore it can be concluded that all characteristic roots are negative. Thus, the critical point is stable.

Stability Analysis of Disease-Endemic Critical Point
The point of (S, E, I, R) is The stability analysis of the endemic critical point was carried out by looking for the equation of the Jacobian matrixJ(E 1 ) as follows: Then the eigenvalues J(E 1 ) will be determined.
It can be written asa characteristic equation: The eigen value .From the solution above, all characteristic roots are negative so that the critical point of 1 E is stable.

Numerical Simulation
The SEIR dynamics model of the corona virus will be simulated using covid-19 data in May-September, 2020 in Indonesia. The determination of the parameters in this simulation was carried out A numerical simulation can be obtained with the initial values of and based on the data in the Table 3. and Table 4.and assume that 6 . 0   as follows:  Figure 2 presents that the number of individuals S, E, I and R will be stable over a long time. If it is viewed based on graphs, the end of the spread of Covid-19 is expected to occur at the 800th time unit (month). The graph of Suspect will initially decrease and begin to increase due to the parameter (rate of increase in the individual supect).The graph of Exposed is known to have increased even though in the end it also decreased. In this case, the increase that occurs is caused by the parameter (or the rate at which S moves to E)while the decrease is caused by the  parameter (rate at which E moves toI).
The simulation of Svalue based on the  parameter is presented by Figure 3 below, so that the influence of  on Scan be obtained.  Figure 3. S simulation based on  parameter Based on these Figure 3, it was found that if the parameters were higher, the graph of Swould also increase. Meanwhile, for the simulation of class Infectedvalue based on the parameter is as follows: The following Figure 4 is a graph of the number of infected based on the assumed value of the theta parameter. Of course, the higher the value of the direct contact rate of the suspect group with the infected group will increase the spread of the virus. So, the theta parameter greatly affects the virus dynamics. Disciplined quarantine is needed to decrease the balance point.

Conclusions
The model for the spread of the Covid-19 virus has two critical points, the free of virus critical point 0 E and an endemic critical point 1 E . The two critical points will be called asymptotically stable if they meet certain inequality of parameters. The dynamics of the corona virus in Indonesia will reach a stable point in the long term. The theta parameter, namely the rate of direct contact of the suspect groups to the exposed and infected groups plays an important role in the spread of the virus. Avoiding direct contact between individuals, disciplined quarantine of infected groups and vaccinations are needed to stop the spread of the virus.