key: cord-1031578-1loqavom authors: Annas, Suwardi; Isbar Pratama, Muh.; Rifandi, Muh.; Sanusi, Wahidah; Side, Syafruddin title: Stability Analysis and Numerical Simulation of SEIR Model for pandemic COVID-19 spread in Indonesia date: 2020-07-03 journal: Chaos Solitons Fractals DOI: 10.1016/j.chaos.2020.110072 sha: 346664e5a49f2647ff716ba7bba1b7e45b39411f doc_id: 1031578 cord_uid: 1loqavom The Aim of this research is construct the SEIR model for COVID-19, Stability Analysis and numerical simulation of the SEIR model on the spread of COVID-19. The method used to construct the model is the SEIR model by considering vaccination and isolation factors as model parameters, the analysis of the model uses the generation matrix method to obtain the basic reproduction numbers and the global stability of the COVID-19 distribution model. Numerical simulation models use secondary data on the number of COVID-19 cases in Indonesia. The results obtained are the SEIR model for COVID-19; model analysis yields global stability from the spread of COVID-19; The results of the analysis also provide information if no vaccine, Indonesia is endemic COVID-19. Then the simulation results provide a prediction picture of the number of COVID-19 in Indonesia in the following days, the simulation results also show that the vaccine can accelerate COVID-19 healing and maximum isolation can slow the spread of COVID-19. The results obtained can be used as a reference for early prevention of the spread of COVID-19 in Indonesia According to the World Health Organization (WHO), COVID-19 is transmitted through people who have been infected with the corona virus. The virus can easily spread through small droplets from the nose or mouth compilation of someone infected with this virus to sneeze or cough. The drops then land on objects or surfaces which are touched and the healthy person adjusts their eyes, nose or mouth. How to spread the corona virus compilation of small droplets inhaled by someone compiling switch with the one supported by corona [1] . It's important to spend 1 meter more distance than people who are sick. Until now there has been no research that states the COVID-19 corona virus can be transmitted through the air, "explained WHO as quoted from its website on March 23, 2020 [2] . parts of the world. The rate of increase, both for the number of cases of infection, death and cure, varies in each region. Each country also has its own policy to curb the spread of viruses that occur in its territory. According to data collected by John Hopkins University, as of March 23, 2020, the total number of Covid-19 cases worldwide had reached 331,273 cases, with 14,450 deaths, and 97,847 patients declared cured. The highest number of cases is still recorded in China, namely 81,397 cases, followed by Italy with 59,138 cases, and the United States as many as 33,073 cases. In terms of mortality, the largest number is in Italy, with 5,476 cases. The number exceeds the death rate that occurred in China, which is 3,265. Meanwhile, in terms of recovery, the largest number is in China, which is 72,362 patients. Both the progress of the number of infections, deaths, patients recovered until certain policies continue to be reported in various countries [2] . The number of COVID-19 cases in Indonesia continues to increase, until March 22, 2020 the number of positive cases of COVID-19 numbered 514 people with 29 people (5.64%) recovered and the number of deaths 48 people (9.34%) or the largest in Southeast Asia. The disease that has become a pandemic and causes a fairly high death in this world has not been found the cure [3] . Trend of the number of cases and spread map of COVID-19 in Indonesia is presented in Figure 1 and Source: [2] Negative Positive Gambar 2. spread map of COVID-19 in Indonesia on March, 2020 Source: [2] Mathematical modeling of SIR, SIRS, SEIR and SEIRS on transmission of diseases such as dengue fever, tuberculosis, diabetes, HIV-AIDS has been done by [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] , then mathematical modeling on the spread of COVID-19 has been carried out by [20] namely SEIRV mathematical modeling in the Wuhan, China taking into account environmental factors, while the analysis and simulation of the model used data of the number of COVID-19 cases in Wuhan. This SEIRV model does not consider vaccination and isolation factors as parameters in the model. Therefore, this research build, analyze using generation matrices method and simulate the SEIR model through isolation and vaccination on the spread of COVID-19 in Indonesia using Mathematica software. The SEIR mathematical modeling on the spread of COVID-19 is a theoretical study. The method used to construct the model is the SEIR model [12] by considering vaccination and isolation factors as model parameters, the model analysis uses the generation matrix method [9] The SEIR model on the spread of COVID-19 is divided into four compartments namely Suspected (S), Definition of variables and parameters of model SEIR for COVID-19 presented in Table 1 . Based on the population scheme in Figure 3 , the rate of change in the number of people Suspected, Exposed, Infected and Recovered over time in the SEIR mathematical model of the spread of Covid-19 can be interpreted as follows: Let and , simplified model becomes: Based on Equations (5) -(8), stability analysis is carried out to determine the disease free equilibrium point and endemic equilibrium point. To determine the two equilibrium points, each equation in equations (5) -(8), must be equal to zero, or , , , and , thus obtained: Then, we found the equilibrium point of and . Equilibrium points for disease-free are conditions where there is no spread of COVID-19 then, From Equation (9): From Equation (12): thus obtained: Then, the Equilibrium points of disease-free for COVID-19 are: Endemic equilibrium points are used to indicate the possibility of disease spread. Because in endemic conditions and disease spread, the population and . From Equation (9)-(12) obtained endemic equlibrium points from COVID-19 are: Then, the Equilibrium points of endemic for COVID-19 are: Base on the Equation (5) - (8), found the Jacobian matrices Then find the eigenvalue of the Jacobian matrix in equation (15) ( Subtitution the value ( ) and , obtained: If Equation (16) is resolved, then: Based on Descartes' rule [14] the number of negative roots of the characteristic equation (17) is equal to the number of variations in the change in the coefficient sign, so equation (17) has four negative values if it is in the form of: This concludes that if and , then: Because the characteristic values of the equation system in the COVID-19 model are negative, the equilibrium point is stable global asymptotic. The basic reproduction number ( ) is determined using the matrices generation method, Based on Equations (5) - (8) , to determine ( ) Then , we found: ) and then, Then, by the matrices generation method [9] , we found: Theorem 1 1. If then system in Equations (5) It means, ( ), - This proves that the system is global asymptotic stable. In contrast, if , then it follows from the continuity of the vector fields that in a neighborhood of the system in . Thus the system in Equation (5-(8) is unstable by the Lyapunov stability theory. The last part of the theorem can be proved by the persistent theory [22] which is similar to the proof of Theorem 2.5 in [23] . Model simulations are performed using Mathematica software. The initial values   0 and parameter values of the models used in this simulation are presented in Table 2, Table 3, and Table 4 , with the basic reproduction number values obtained based on equation (18), is [25] 0.62 x 10 -8 /person/day [11] 0.0006667 per day [11] 7.344 x 10 -7 [21] The value of the equilibrium points of the SEIR model is determined by substituting the parameter values (simulation 1) in Tables 3 and 4 in Equations (5) -(8) which are equated with zero, then the following equation (5) -(8) system is obtained: The equation system (19) provides the equilibrium points of the endemic SEIR model of the spread of COVID-19, namely: These equilibrium points explain that the number of COVID-19 suspected populations was 982 people, exposed to 1793 people, infected 1811 people and those recovered were 89008 people from the total 100,000 human population. The eigenvalues based on Equation (17) with the parameter values in Table 2 and Table 3 for the Covid-19 transmission SEIR model are: In the same way, the equilibrium values and eigenvalue for simulation 2 and simulation 3 are written in Table 5 . The eigenvalues obtained are real and negative, based on [17] , the type of stability at this equilibrium point is asymptotic stable . The stability phase of the system can be illustrated in Figure 4 and the fitting data of the SEIR model using Runge Kutta method versus the real data for covid 19 in Indonesia can be illustrated in Figure 5 . 1% vaccination is = 3.2094. This means that, if a person is infected with Covid-19 it will infect 3 other people. Whereas the value for simulation 2 and simulation 3 presented in Table 5 explains that 50% vaccine will reduce transmission of COVID-19 and 100% does not couse spreading of Covid-19 in Indonesia. Numerical simulation to determine the effect of variations in population isolation time on the dynamics of the number of Exposed and Infected Covid-19. The simulation uses the initial values in table 2 and the parameters in Table 3 with vaccination ν = 50%. 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