Datasets:

---

LLMDH

/

OpenScience

like 0

Follow

LLM - Digital Humanities 16

ERROR: type should be string, got "https://doi.org/10.30598/barekengvol17iss3pp1325-1340 \nSeptember 2023 Volume 17 Issue 3 Page 1325–1340 \nP-ISSN: 1978-7227 \n E-ISSN: 2615-3017 \n \nBAREKENG: Journal of Mathematics and Its Applications https://doi.org/10.30598/barekengvol17iss3pp1325-1340 \nSeptember 2023 Volume 17 Issue 3 Page 1325–1340 \nP-ISSN: 1978-7227 \n E-ISSN: 2615-3017 \n \nBAREKENG: Journal of Mathematics and Its Applications September 2023 Volume 17 Issue 3 Page 1325–1340 \nP-ISSN: 1978-7227 \n E-ISSN: 2615-3017 \nBAREKENG: Journal of Mathematics and Its Applications September 2023 Volume 17 Issue 3 Page 1325–1340 \nP-ISSN: 1978-7227 \n E-ISSN: 2615-3017 \nBAREKENG: Journal of Mathematics and Its Applications THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE \nSUSCEPTIBLE, INFECTED AND RECOVERED (SIR) MODEL WITH \nVACCINATION Dorrah Azis 1*, La Zakaria2, Tiryono Ruby3, Muhammad Is’ad Arifaldi4 Dorrah Azis 1*, La Zakaria2, Tiryono Ruby3, Muhammad Is’ad Arifaldi4 1,2,3,4Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampung \nSumantri Brojonegoro Street, Bandar Lampung, 35141, Indonesia \nCorresponding author’s e mail: * dorrah azis@fmipa unila ac id 4Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampung \nSumantri Brojonegoro Street, Bandar Lampung, 35141, Indonesia , , , Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampun\nSumantri Brojonegoro Street, Bandar Lampung, 35141, Indonesia \nCorresponding author’s e-mail: * dorrah.azis@fmipa.unila.ac.id Corresponding author’s e-mail: * dorrah.azis@fmipa.unila.ac.id ABSTRACT The COVID-19 pandemic in 2020 has caused severe problems in Indonesia. The COVID-19 \nvirus epidemic can be modeled using the Susceptible, Infected, and Recovered (SIR) model. This \nmodeling aims to look at the dynamics of COVID-19 to predict when disease-free and endemic \ndisease occurs and to find the basic reproduction number (𝑅0) for policy making in suppressing \nthe spread of COVID-19. In this article, we describe and solve a research result on the SIR \nmodel with an assumption. The assumption in the model is that there is vaccination for the \npopulation. There are live stages of research conducted. The first is creating the SIR model and \ndetermining the equilibrium points on disease-free and disease-endemic. The Second is getting \nthe basic reproduction number. The third is determining the stability around the equilibrium \npoints using the Routh-Hurwitz criteria. Fourth, create a diagram for the subpopulations state \nat a specific time using Wolfram Mathematica software. As an implementation of the model \ncreated, COVID-19 data at the Batanghari Community Health Center Inpatient UPTD was \nused. Finally, determine the model error percentage with MAPE. The SIR COVID-19 model \nwas made using eight parameters, namely 𝑁, 𝛼, 𝛽, 𝜏, µ, 𝜎, 𝛿, 𝛾, which are all positive. The results \nshowed that the disease-free and disease-endemic equilibrium points were locally \nasymptotically stable after being analyzed using the Routh-Hurwitz stability criteria. The model \ntrial using data from UPTD Puskesmas Batanghari obtained a stable condition for up to 100 \nmonths with a MAPE of 2.8%. From this study, obtained an 𝑅0 =\n𝛽𝜎\n𝛼+µ. This means that if you \nwant to reduce the rate of spread, then reduce the number of people who are easily infected (𝜎) \nand reduce contacts (𝛽), and increase the healing rate (𝛼). Article History: \nReceived: 06th January 2023 \nRevised: 9th July 2023 \nAccepted: 19th July 2023 Article History: \nReceived: 06th January 2023 \nRevised: 9th July 2023 \nAccepted: 19th July 2023 y\nSIR model; \nCOVID-19; \nBasic reproduction \nnumber; \nRouth-Hurwitz Criterion This article is an open access article distributed under the terms and conditions of the \nCreative Commons Attribution-ShareAlike 4.0 International License. How to cite this article: \nD. Azis, L. Zakaria, T. Ruby and M. I. Arifaldi, “THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, \nAND RECOVERED (SIR) MODEL WITH VACCINATION,” BAREKENG: J. Math. &App., vol. 17, iss. 3, pp. 1325-1340, September, 2023. 1. INTRODUCTION For years, vaccines have been \nproven to reduce the incidence of infectious diseases through the mechanism of the human body's immunity \n[15]. The COVID-19 vaccine was developed to help the formation of individual body immunity so that the \nadministration of the COVID-19 vaccine is expected to accelerate the formation of group immunity (herd \nimmunity), which will have an impact on reducing the number of infected cases [16]. The vaccination policy \nhas an impact on reducing the number of COVID-19 cases that are still not under control in Indonesia [17]. This is reinforced by [18] which concluded that the severity of disease in individuals who had received the \nvaccine decreased, so it can be supposed that the vaccine effectively protects individuals from the SARS-\nCoV-2 variant. A disease can be modeled mathematically into an epidemiological model. One of the many types of \nmodeling available is the SIR model. The SIR model was first introduced in 1927 by Kermack and \nMcKendrick [19]. The SIR model determines the behavior of a pandemic and prediction [20]. This SIR \nmodel groups individuals in a population into three subpopulations, namely susceptible (groups of individuals \nwho are susceptible to being infected with a disease), infected (groups of individuals infected with disease), \nrecovered (groups of individuals who have recovered from the disease). SIR modeling is a model that is \nprepared with assumptions about a disease starting from the stage before being infected with a disease, being \ninfected, and until the individual is cured. Mathematical models of infectious diseases based on the classical \nSIR model are widely used to study the spread of a disease. These models show exciting results, especially \nin the early period of the pandemic [1]–[4], [21]–[28].The results of this model can be used as an illustration \nof how to suppress the spread of disease by looking at the effect of vaccination, ideal number of individuals \nwho must be vaccinated, and this model can predict within a certain time, the disease will become endemic. This requires field data to analyze the dynamics of the development of a disease. This modeling aims to \ndetermine the basic reproduction number (𝑅0), which plays a role in decision or policy-making for the \nauthorities in dealing with problems caused by the COVID-19 virus. 1. INTRODUCTION Mathematical models have played an essential role in explaining the dynamics of the disease. A \nmathematical model categorizes individuals into Susceptible, Infectious, and Recovered. In several very \nrecent research applied to the COVID-19 epidemic, researchers have developed and used SIR and SEIR-\nbased models with vaccination to overcome the limitations of the conventional SIR model. Mathematical \nmodeling specifically constructed to determine the development of disease outbreaks caused by COVID-19 \nhas not been widely carried out [1]–[9]. Handling the COVID-19 outbreak in Lampung Province involves various groups/communities of \nsociety, including academics. Even for Lampung province, dynamic models have not yet been found to solve \nthe problem of the COVID-19 outbreak in terms of the theory of the SIR model, which involves vaccination \nparameters. The research that has been carried out is to construct the SIR model to determine the development \nof the COVID-19 outbreak in Lampung Province, especially East Lampung Regency, by involving the \nparameters of vaccine administration to the community. Using data at the UPTD Puskesmas Batanghari \nDistrict, East Lampung Regency, the SIR model obtained was implemented to determine the trend of the \ngrowth of the COVID-19 outbreak in that place after the people were given the vaccine. In March 2020, WHO announced that the world was facing a pandemic called Corona Virus Infectious \nDisease 2019 or COVID-19 [10]. Until October 2020, the number of positive cases of. COVID-19 worldwide \nhas reached 37 million, with deaths reaching 1 million people [11]. The main medium of transmission of the \nSARS-Cov-2 virus is droplets that can be easily spread when humans interact directly with a certain distance. At the beginning of its spread, the average transmission power of the virus was still quite low, around 2.2 \n[12]. However, in its development, the SARS-Cov-2 virus underwent mutations so that several new virus \nvariants emerged with higher transmission capabilities, such as in England, South Africa, Brazil, and India \n[13]. The COVID-19 pandemic is developing so fast that many countries are not ready to adapt since the \nbeginning. WHO has advised focusing the handling of the pandemic on the health aspect by implementing \nregional isolation and banning activities that involve crowds. However, for some countries, this is not done \nbecause they doubt that the COVID-19 pandemic will last quite a long time [14]. One of the other efforts to \ndeal with the spread of COVID-19 is by implementing mass vaccination. How to cite this article: \nD. Azis, L. Zakaria, T. Ruby and M. I. Arifaldi, “THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, \nAND RECOVERED (SIR) MODEL WITH VACCINATION,” BAREKENG: J. Math. &App., vol. 17, iss. 3, pp. 1325-1340, September, 2023. How to cite this article: \nD. Azis, L. Zakaria, T. Ruby and M. I. Arifaldi, “THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, \nAND RECOVERED (SIR) MODEL WITH VACCINATION,” BAREKENG: J. Math. &App., vol. 17, iss. 3, pp. 1325-1340, September, 2023. Copyright © 2023 \nJournal homepage: https://ojs3.unpatti.ac.id/index.php/barekeng/ \nJournal e-mail: barekeng.math@yahoo.com; barekeng.journal@mail.unpatti.ac.id \nResearch Article • Open Access How to cite this article: \nD. Azis, L. Zakaria, T. Ruby and M. I. Arifaldi, “THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, \nAND RECOVERED (SIR) MODEL WITH VACCINATION,” BAREKENG: J. Math. &App., vol. 17, iss. 3, pp. 1325-1340, September, 2023. 1325 326 \n \nAziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1326 1. INTRODUCTION In addition, this modeling is also to find \nout whether this disease can disappear or will remain (endemic) in an area and predict when this disease will \ndisappear or remain (endemic) in an area. BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1327 2. RESEARCH METHODS This research is applied research conducted using secondary data obtained from the UPTD of the \nBatanghari District Inpatient Health Center related to cases of the spread of the COVID-19 disease that \noccurred throughout 2020-2021. The steps in this research are given in Figure 1: This research is applied research conducted using secondary data obtained from the UPTD of the \nBatanghari District Inpatient Health Center related to cases of the spread of the COVID-19 disease that \noccurred throughout 2020-2021. The steps in this research are given in Figure 1: Figure 1. 𝐒𝐭𝐞𝐩𝐬 of research diagram Figure 1. 𝐒𝐭𝐞𝐩𝐬 of research diagram 2.3 Next Generation Matrix Suppose there are 𝑛 infected classes and m uninfected classes. Furthermore, suppose that 𝑥 is an \ninfected sub-population and 𝑦 represents an uninfected (susceptible or cured) subpopulation, so that 𝑥̇ =\n𝜑𝑖(𝑥, 𝑦) −𝜔𝑖(𝑥, 𝑦), and 𝑦̇ = 𝑔𝑗(𝑥, 𝑦), where 𝑖= 1,2, … , 𝑛 and 𝑗= 1,2, … , 𝑚. 𝜑𝑖is the rate of secondary \ninfection in the infected class, and 𝜔𝑖 is the rate of disease progression, death, and recovery resulting in a \nreduced population of the infected class [32]. Next, Diekmann explain the next generation matrix 𝑲 is defined, which has the form 𝑲= 𝑭𝑽−1 \n(4) 𝑲= 𝑭𝑽−1 (4) where 𝑭 and 𝑽 are matrix of size 𝑛× 𝑛, which can also be written as follows: 𝑭= [\n𝜕𝜑𝑖\n𝜕𝑦𝑗] and 𝑽= [\n𝜕𝜔𝑖\n𝜕𝑦𝑗] \n(5) (5) (5) 2.1 SIR Model The SIR model divides the population into susceptible (S), infected (I), and recovered (R) \nsubpopulations. The number of susceptible, infected, and cured individuals at time t in a row can be written \nin the form of functions S(t), I(t), and R(t). The SIR model can be used to predict how a disease spreads, the total number of infected, the duration \nof the epidemic, and estimate various epidemiological parameters such as the number of reproductions. This \nmodel can determine how different public health interventions can affect epidemic outcomes [29]. Figure 2. 𝑺𝑰𝑹 model diagram \nS \n𝒓𝑺𝑰 \n𝜶𝑰 \nI \nR Figure 2. 𝑺𝑰𝑹 model diagram \nS \n𝒓𝑺𝑰 \n𝜶𝑰 \nI \nR Figure 2. 𝑺𝑰𝑹 model diagram 𝑆 = number of susceptible individuals in the population at the time \n𝐼 = number of infected individuals in the population at the time \n𝑅 = number of recovered individuals in the population at the time \n𝛼 = healing rate from infected to recovered \n𝑟 = rate of disease transmission from susceptible to infected \nThe SIR epidemic model is assumed as follows: 𝑆 = number of susceptible individuals in the population at the time 𝑟 = rate of disease transmission from susceptible to infected The SIR epidemic model is assumed as follows: 𝑑𝑆\n𝑑𝑡= −𝑟𝑆𝐼 \n(1) \n𝑑𝐼\n𝑑𝑡= 𝑟𝑆𝐼−𝛼𝐼 \n(2) \n𝑑𝑅\n𝑑𝑡= 𝛼𝐼 \n(3) 𝑑𝑆\n𝑑𝑡= −𝑟𝑆𝐼 \n(1) \n𝑑𝐼\n𝑑𝑡= 𝑟𝑆𝐼−𝛼𝐼 \n(2) \n𝑑𝑅\n𝑑𝑡= 𝛼𝐼 \n(3) 𝑑𝑆\n𝑑𝑡= −𝑟𝑆𝐼 \n𝑑𝐼\n𝑑𝑡= 𝑟𝑆𝐼−𝛼𝐼 \n𝑑𝑅\n𝑑𝑡= 𝛼𝐼 𝑑𝑆\n𝑑𝑡= −𝑟𝑆𝐼 \n𝑑𝐼\n𝑑𝑡= 𝑟𝑆𝐼−𝛼𝐼 \n𝑑𝑅\n𝑑𝑡= 𝛼𝐼 Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1328 Theorem Equilibrium Point [31]: 1. If all the real parts of the eigenvalues of the Jacobian matrix of a system of differential equations are \nnegative, then the equilibrium point of the system is stable. 1. If all the real parts of the eigenvalues of the Jacobian matrix of a system of differential equations are \nnegative, then the equilibrium point of the system is stable. 2. If one eigenvalue of the Jacobian matrix of a system of differential equations is positive, then the \nequilibrium point of the system is unstable. 2. If one eigenvalue of the Jacobian matrix of a system of differential equations is positive, then the \nequilibrium point of the system is unstable. 2.2 Equilibrium Point The equilibrium point is used to analyze the model. According to [19], the equilibrium point is the \nsolution of the system of differential equations which is independent of time. In [30], Meyer said the same \nthing too, the equilibrium condition is a condition where the system does not change over time. The equilibrium point is divided into 2, as follows: The equilibrium point is divided into 2, as follows: 1. The disease-free equilibrium point is the condition in which no individual is infected with the disease \ndiscussed in the population, so 𝐼 = 0. 1. The disease-free equilibrium point is the condition in which no individual is infected with the disease \ndiscussed in the population, so 𝐼 = 0. 2. The endemic equilibrium point is a condition where there are infected individuals in the population, so the \ncompartment at the endemic equilibrium point is 𝐼 ≠ 0. 2.4 Basic Reproduction Number (𝑹𝟎) The basic reproduction number 𝑅0 can be defined as the average number of infected individuals \ninfected by other infected individuals in a population. A basic reproductive number is a number that shows \nthe number of susceptible individuals who can suffer from diseases caused by one infected individual [33]. According [32], the basic reproduction number, which can be formulated as follows 𝑅0 = 𝜌(𝑲) = 𝜌(𝑭𝑽−𝟏) \n(6) (6) (6) The same thing regarding the basic reproduction number is denoted by 𝑅0 and is expressed by the \nfollowing equation [34]: 𝑅0 = 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑡ℎ𝑎𝑡 𝑐𝑎𝑢𝑠𝑒 𝑑𝑖𝑠𝑒𝑎𝑠𝑒\n𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑑𝑢𝑐𝑒 𝑑𝑖𝑠𝑒𝑎𝑠𝑒 \n(7) 𝑅0 = 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑡ℎ𝑎𝑡 𝑐𝑎𝑢𝑠𝑒 𝑑𝑖𝑠𝑒𝑎𝑠𝑒\n𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑡ℎ𝑎𝑡 𝑟𝑒𝑑𝑢𝑐𝑒 𝑑𝑖𝑠𝑒𝑎𝑠𝑒 \n(7) (7) BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1329 If 𝑅0 > 1, the each infected individual produces, on average, more than one new infection, and the disease \ncan invade the population. The disease will become an epidemic [35]. If 𝑅0 > 1, the each infected individual produces, on average, more than one new infection, and the disease \ncan invade the population. The disease will become an epidemic [35]. 2.5 Routh-Hurwitz Stability Criterion The necessary condition for the system to be said to be stable is if the coefficient of the characteristic \nequation is positive, while the sufficient condition is that each term of the first column of the matrix (6) is \npositive. 5. The necessary condition for the system to be said to be stable is if the coefficient of the characteristic \nequation is positive, while the sufficient condition is that each term of the first column of the matrix (6) is \npositive. Some of the conditions that will arise, namely: Some of the conditions that will arise, namely: the conditions that will arise, namely: If 𝑅0 < 1, then on average an infected individual produces \nless than one new infected individual over the course of its infectious period, and the infection \ncannot grow. The disease will disappear [35]. If 𝑅0 = 1, then the disease will persist [34]. BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 2.5 Routh-Hurwitz Stability Criterion The Routh-Hurwitz stability criterion is used to show a system's stability by taking into account the \ncoefficients of the characteristic equation without calculating the roots directly. If a polynomial equation is a \ncharacteristic equation, then this method can determine a system's stability. Thus, the procedures in the Routh-\nHurwitz criterion are [31]: 1. The 𝑛𝑡ℎ order polynomial equation is written in the form: 1. The 𝑛𝑡ℎ order polynomial equation is written in the form: 1. The 𝑛𝑡ℎ order polynomial equation is written in the form: det(𝜆𝐼−𝐴) = 𝑎𝑛𝜆𝑛+ 𝑎𝑛−1𝜆𝑛−1 + ⋯+ 𝑎1𝜆+ 𝑎0 \n(8) \n \nwhere the coefficients are real numbers and 𝑎𝑛≠0 det(𝜆𝐼−𝐴) = 𝑎𝑛𝜆𝑛+ 𝑎𝑛−1𝜆𝑛−1 + ⋯+ 𝑎1𝜆+ 𝑎0 \n(8) (8) where the coefficients are real numbers and 𝑎𝑛≠0. where the coefficients are real numbers and 𝑎𝑛≠0. 2. If there is a coefficient of 0 or negative, then there is one root or imaginary roots or has a positive real part \nwhich means the system is unstable. 2. If there is a coefficient of 0 or negative, then there is one root or imaginary roots or has a positive real part \nwhich means the system is unstable. 2. If there is a coefficient of 0 or negative, then there is one root or imaginary roots or has a positive real part \nwhich means the system is unstable. 3. If all coefficients are positive, then a matrix which is often called a Routh array, can be formed as follows |\n|\n𝜆𝑛\n𝑎𝑛\n𝑎𝑛−2\n𝑎𝑛−4\n⋯\n𝜆𝑛−1\n𝑎𝑛−1\n𝑎𝑛−3\n𝑎𝑛−5\n⋯\n𝜆𝑛−2\n𝑏1\n𝑏2\n𝑏3\n⋯\n⋮\n𝑐1\n𝑐2\n𝑐3\n⋯\n𝜆0\n⋮\n⋮\n⋮\n⋱\n|\n| \n(9) (9) The coefficients 𝑏1, 𝑏2, … , 𝑏𝑘 and 𝑐1, 𝑐2, … , 𝑐𝑘can be determined by the following formulas: 𝑏1 = −\n1\n𝑎𝑛−1 | 𝑎𝑛\n𝑎𝑛−2\n𝑎𝑛−1\n𝑎𝑛−3|, 𝑏2 = −\n1\n𝑎𝑛−3 |𝑎𝑛−2\n𝑎𝑛−4\n𝑎𝑛−3\n𝑎𝑛−5|,⋯ \n \n𝑐1 = −\n1\n𝑏1 |𝑎𝑛−1\n𝑎𝑛−3\n𝑏1\n𝑏2 |,𝑐2 = −\n1\n𝑏2 |𝑎𝑛−3\n𝑎𝑛−5\n𝑏2\n𝑏3 |,⋯ The scheme is continued until only zeroes appear (both to the right and down wards) [36]. The scheme is continued until only zeroes appear (both to the right and down wards) [36]. The scheme is continued until only zeroes appear (both to the right and down wards) [36]. 4. The number of unstable roots can be seen in the number of sign changes in the first matrix c 5. 2.6 Mean Absolute Percentage Error (MAPE) Mean Absolute Percentage Error is a method of finding the average absolute error value in the form of \na percentage in a comparison between actual data and existing projection or forecasting data. MAPE is \nformulated as follows: 𝑀𝐴𝑃𝐸= ∑|𝐴−𝑃|\n𝑛\n× 100% (10) 𝑀𝐴𝑃𝐸= ∑|𝐴−𝑃|\n𝑛\n× 100% (10) With With 𝐴: Actual data, 𝐴: Actual data, 𝑃: Forecasting data, 𝑃: Forecasting data, 𝑛: Total data. 𝑛: Total data. 𝑛: Total data. The percentage of MAPE is divided into four interpretation as follows [37]: Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1330 Table 1. MAPE Interpretation \nInterpretation of Typical MAPE Value’s \nMAPE \nInterpretation \n< 10 \nHighly Accurate Forecasting \n10 −20 \nGood Forecasting \n20 −50 \nReasonable Forecasting \n> 50 \nInaccurate Forecasting Table 1. MAPE Interpretation \nInterpretation of Typical MAPE Value’s \nMAPE \nInterpretation \n< 10 \nHighly Accurate Forecasting \n10 −20 \nGood Forecasting \n20 −50 \nReasonable Forecasting \n> 50 \nInaccurate Forecasting 3.1 SIR Model The assumptions in the SIR model for the COVID-19 disease are as follows: 1. Factors of birth and death are considered. Individuals born into the Susceptible (S) class because the \nindividual is assumed to be healthy but susceptible to COVID-19 disease. 1. Factors of birth and death are considered. Individuals born into the Susceptible (S) class because the \nindividual is assumed to be healthy but susceptible to COVID-19 disease. 2. The population birth and death rates in each compartment are assumed to be the same so that the total \npopulation is constant 2. The population birth and death rates in each compartment are assumed to be the same so that the total \npopulation is constant 3. Migration occurs in the population. No immigrants enter every S, I, and R class. 3. Migration occurs in the population. No immigrants enter every S, I, and R class. 4. COVID-19 disease can cause death (fatal). 5. Vaccinated individuals fall into class S. Vaccination efficacy is assumed as a percentage. WHO explained \nthat the performance of vaccines could be seen from three measurements, namely through the efficacy, \neffectiveness, and impact of the vaccine. One type of COVID-19 vaccine is Sinovac. The efficacy of the \nSinovac vaccine reaches 65.3% [38]. 5. Vaccinated individuals fall into class S. Vaccination efficacy is assumed as a percentage. WHO explained \nthat the performance of vaccines could be seen from three measurements, namely through the efficacy, \neffectiveness, and impact of the vaccine. One type of COVID-19 vaccine is Sinovac. The efficacy of the \nSinovac vaccine reaches 65.3% [38]. 6. COVID-19 disease can result in re-infection of individuals who have been infected before. However, some \nindividuals have recovered and formed antibodies to the COVID-19 virus. 6. COVID-19 disease can result in re-infection of individuals who have been infected before. However, some \nindividuals have recovered and formed antibodies to the COVID-19 virus. ased on the assumptions that have been made, the model parameters can be defined as follows: Figure 3. SIR model of COVID-19 disease with vaccination \n𝝉𝑰𝑹 \nS \n𝜷𝝈𝑰𝑺 \n𝜶𝑰 \nµ𝑰 \nµ𝑺 \nµ𝑹 \nµ𝑵 \nI \nR Figure 3. BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1331 value is not far from 1 or very close. However, birth and death rates are equated with making this model easy \nto analyze. value is not far from 1 or very close. However, birth and death rates are equated with making this model easy \nto analyze. Then the mathematical model of the spread of the COVID-19 disease with vaccination was obtained \nin the form of a system of differential equations as follows: 𝑑𝑆\n𝑑𝑡= µ −𝛽𝜎𝐼𝑆−µ𝑆 \n(12) \n𝑑𝐼\n𝑑𝑡= 𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−𝛼𝐼−µ𝐼 \n(13) \n𝑑𝑅\n𝑑𝑡= 𝛼𝐼−µ𝑅−𝜏𝐼𝑅 \n(14) 𝑑𝑆\n𝑑𝑡= µ −𝛽𝜎𝐼𝑆−µ𝑆 \n(12) \n𝑑𝐼\n𝑑𝑡= 𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−𝛼𝐼−µ𝐼 \n(13) \n𝑑𝑅\n𝑑𝑡= 𝛼𝐼−µ𝑅−𝜏𝐼𝑅 \n(14) (12) (13) (14) where 𝑆(0) > 0, 𝐼(0) ≥0, 𝑅(0) > 0; \n \nµ, 𝛽, 𝜎, 𝛿, 𝛾, 𝜏, 𝛼> 0. where 𝑆(0) > 0, 𝐼(0) ≥0, 𝑅(0) > 0; \n \nµ, 𝛽, 𝜎, 𝛿, 𝛾, 𝜏, 𝛼> 0. where 𝑆(0) > 0, 𝐼(0) ≥0, 𝑅(0) > 0; \n \nµ, 𝛽, 𝜎, 𝛿, 𝛾, 𝜏, 𝛼> 0. 3.2.1 Disease-Free Equilibrium Point Disease-free population means that in the population, no one is sick, 𝐼= 0. This means that no \nindividual has been exposed to COVID-19 or no individual has been infected with COVID-19. Because no \nindividual has been exposed to COVID-19, the transmission rate (𝛽)and the infected group (𝐼)is worth 0. From the Equation (𝟏𝟐), substitute each equation containing the variables 𝛽and 𝐼 with a value of 0, it is \nobtained: quation (𝟏𝟐), substitute each equation containing the variables 𝛽and 𝐼 with a value of 0, it is µ −(0)𝜎(0)𝑆0 −µ𝑆0 = 0 \nµ −µ𝑆0 = 0 \n−µ𝑆0 = −µ \n𝑆0 = 1 (15) (15) Substituting the value 𝐼0 = 0 into Equation (14), obtained: 𝛼(0) −µ𝑅0 −𝜏(0)𝑅0 = 0 \n−µ𝑅0 = 0 \n𝑅0 = 0 𝛼(0) −µ𝑅0 −𝜏(0)𝑅0 = 0 \n−µ𝑅0 = 0 \n𝑅0 = 0 (16) (16) Then the disease-free equilibrium point is obtained as follows: Then the disease-free equilibrium point is obtained as follows: (𝑆0, 𝐼0, 𝑅0) = (1,0,0) \n(17) (17) 3.2 Equilibrium Point The equilibrium condition is a condition where the system does not change over time [30]. 3.2.1 Disease-Free Equilibrium Point The equilibrium condition is a condition where the system does not change over time [30]. The equilibrium condition is a condition where the system does not change over time [30]. 3.2.1 Disease-Free Equilibrium Point 3.1 SIR Model SIR model of COVID-19 disease with vaccination Information: 𝑁 : Total number of individuals in the population (%) µ : Expresses the birth rate in the S compartment and the death rate in each compartment (% 𝛽 : Expresses the infection rate in compartment S (%) 𝛽 : Expresses the infection rate in compartment S (%) \n𝛼 : Expresses the healing rate in compartment I (%) σ : Stating the total number of susceptible individuals (total individuals who are not vaccinated and already \n vaccinated but infected) (%) 𝜏: Expresses the rate of reinfection in compartment R (%) \nwith \n(\n) with \n𝜎= 1 −𝛿𝛾 (11) \n𝛿 : Vaccine efficiency (%) \n𝛾 : Number of individuals who have been vaccinated (%) 𝜎= 1 −𝛿𝛾 (11) \nbeen vaccinated (%) (11) 𝛾 : Number of individuals who have been vaccinated (%) 𝛾 : Number of individuals who have been vaccinated (%) Since the birth rate is considered equal to the number of deaths, the total population will be constant, \nso 𝑆+ 𝐼+ 𝑅= 1. If the number of births is not equal to the number of deaths, then N is not constant, but the BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 3.2.2. Disease Endemic Equilibrium Point Finally, \nwe have the COVID-19 endemic equilibrium point for is as follows: them into Equation (20), Equation (19) and Equation (3), the equilibrium point () will be obtained. Finally, \nwe have the COVID-19 endemic equilibrium point for is as follows: (𝑆, 𝐼, 𝑅) = (𝑆∗, 𝐼∗, 𝑅∗) \n(21) (21) 3.2.2. Disease Endemic Equilibrium Point Endemic disease means that in the population, there are always individuals who are infected with the \ndisease, meaning that there are individuals who are prone to be exposed to COVID-19, and there are \nindividuals who are infected with COVID-19. Because there are individuals who are exposed to COVID-19, \nthe transmission rate (𝛽) and the infected group (𝐼) are worth 𝛽> 0, 𝐼≠0. Let (𝑆∗, 𝐼∗, 𝑅∗) be an endemic \nequilibrium point. From the Equation (𝟏𝟐) for 𝜎> 0 and µ > 0, we can rewrite the equation as: 𝑆∗=\nµ\n𝐼∗ 𝛽𝜎+ µ \n(18) (18) Substituting Equation (18) into Equation (𝟏𝟑), and then solving it for 𝐼∗, we have: Substituting Equation (18) into Equation (𝟏𝟑), and then solving it for 𝐼∗, we have: 𝐼∗= 1\n2 ( −µ(𝛼−𝛽𝜎+ µ)\n𝛽𝜎(𝛼+ µ)\n+ √µ(4 𝑅∗ 𝛽𝜎𝜏(𝛼+ µ) + µ(𝛼−𝛽𝜎+ µ)2)\n𝛽2𝜎2(𝛼+ µ)2\n) \n(19) (19) Substituting Equation (19) into Equation (𝟏𝟒), and then solving it for 𝑅∗, we have: Substituting Equation (19) into Equation (𝟏𝟒), and then solving it for 𝑅∗, we have: 1332 \n \nAziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1332 \n \nAziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 𝑅∗= 𝑎−𝑏 \n(20) \nWhere: \n𝑎=\n1\n2𝜏3 (𝛽𝜎𝜏µ + 𝛽𝜎µ2 −𝜏µ2 + 𝛼(2𝜏2 + 𝛽𝜎µ −𝜏µ) \n𝑏= 𝜏3√\n(𝛼2(2𝛽𝜎𝜏(2𝜏−µ)+𝛽2𝜎2µ+𝜏2µ)+µ(𝜏µ−𝛽𝜎(𝜏+µ))\n2+2𝛼µ(𝛽𝜎𝜏(𝜏−2µ)+𝜏2µ+𝛽2𝜎2(𝜏+µ)))\n(𝜏6µ) (20) (𝜏6µ) By selecting and setting parameter values that satisfy 𝜎> 0  µ > 0  ( (𝛼≥µ  ((𝛽= 𝛼+µ\n𝜎  𝜏>\n0) (𝜏>\nµ(𝛼−𝛽𝜎+µ)\n𝛼\n  0 < 𝛽<\n𝛼+µ\n𝜎)))  (0 < 𝛼< µ  0 < 𝛽≤\n𝛼+µ\n𝜎  𝜏>\nµ(𝛼−𝛽𝜎+µ)\n𝛼\n)) and substituting By selecting and setting parameter values that satisfy 𝜎> 0  µ > 0  ( (𝛼≥µ  ((𝛽= 𝛼+µ\n𝜎  𝜏>\n0) (𝜏>\nµ(𝛼−𝛽𝜎+µ)\n𝛼\n  0 < 𝛽<\n𝛼+µ\n𝜎)))  (0 < 𝛼< µ  0 < 𝛽≤\n𝛼+µ\n𝜎  𝜏>\nµ(𝛼−𝛽𝜎+µ)\n𝛼\n)) and substituting By selecting and setting parameter values that satisfy 𝜎> 0  µ > 0  ( (𝛼≥µ  ((𝛽= 𝛼+µ\n𝜎  𝜏> 0) (𝜏>\nµ(𝛼−𝛽𝜎+µ)\n𝛼\n  0 < 𝛽<\n𝛼+µ\n𝜎)))  (0 < 𝛼< µ  0 < 𝛽≤\n𝛼+µ\n𝜎  𝜏>\nµ(𝛼−𝛽𝜎+µ)\n𝛼\n)) an them into Equation (20), Equation (19) and Equation (3), the equilibrium point () will be obtained. 3.3 Basic Reproduction Number 1333 𝑅0 = 𝜌(\n𝛽𝜎\n(µ + 𝛼)\n0\n𝛼\n(µ + 𝛼)\n0\n) \n(26) (26) To get 𝑅0, the next step is to find the biggest eigenvalue of 𝐄𝐪𝐮𝐚𝐭𝐢𝐨𝐧 (𝟐𝟔) as follows: |𝑭∙𝑽−1 −𝜆𝑰| = ||(\n𝛽𝜎\n(µ + 𝛼)\n0\n𝛼\n(µ + 𝛼)\n0\n) −(𝜆\n0\n0\n𝜆)|| \n→|𝑭∙𝑽−1 −𝜆𝑰| = ||\n𝛽𝜎\n(µ + 𝛼) −𝜆\n0\n𝛼\n(µ + 𝛼)\n−𝜆\n|| \n→𝜆(𝜆−\n𝛽𝜎\n(µ + 𝛼)) = 0 \n𝜆1 = 0 and 𝜆2 =\n𝛽𝑂\nµ+𝛼 |𝑭∙𝑽−1 −𝜆𝑰| = ||(\n𝛽𝜎\n(µ + 𝛼)\n0\n𝛼\n(µ + 𝛼)\n0\n) −(𝜆\n0\n0\n𝜆)|| \n→|𝑭∙𝑽−1 −𝜆𝑰| = ||\n𝛽𝜎\n(µ + 𝛼) −𝜆\n0\n𝛼\n(µ + 𝛼)\n−𝜆\n|| \n→𝜆(𝜆−\n𝛽𝜎\n(µ + 𝛼)) = 0 \n𝜆1 = 0 and 𝜆2 =\n𝛽𝑂\nµ+𝛼 \n(27) (27) Because all variables are positive, so 𝜆2 > 𝜆1. Then obtained 𝑅0 as follows: Because all variables are positive, so 𝜆2 > 𝜆1. Then obtained 𝑅0 as follows: 𝑅0 =\n𝛽𝜎\nµ + 𝛼 , µ + 𝛼≠0 \n(28) 𝑅0 =\n𝛽𝜎\nµ + 𝛼 , µ + 𝛼≠0 (28) 3.3 Basic Reproduction Number The first step is to form a Jacobian matrix of the compartments containing infected individuals, namely \nI and R as follows: 𝑱(𝐼, 𝑅) = (\n𝑑(𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−µ𝐼−𝛼𝐼)\n𝑑𝐼\n𝑑(𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−µ𝐼−𝛼𝐼)\n𝑑𝑅\n𝑑(𝛼𝐼−𝜏𝐼𝑅−µ𝑅)\n𝑑𝐼\n𝑑(𝛼𝐼−𝜏𝐼𝑅−µ𝑅)\n𝑑𝑅\n) 𝑱(𝐼, 𝑅) = (𝛽𝜎𝑆+ 𝜏𝑅−µ −𝛼\n𝜏𝐼\n𝛼−𝜏𝑅\n−𝜏𝐼−µ) (22) (22) ext, substituting the disease-free equilibrium point (17) into the Jacobian matrix (22), we get: 𝑱(1,0,0) = (𝛽𝜎(1) + 𝜏(0) −µ −𝛼\n𝜏(0)\n𝛼−𝜏(0)\n−𝜏(0) −µ) \n𝑱(1,0,0) = (𝛽𝜎−µ −𝛼\n0\n𝛼\n−µ) \n(23) 𝑱(1,0,0) = (𝛽𝜎(1) + 𝜏(0) −µ −𝛼\n𝜏(0)\n𝛼−𝜏(0)\n−𝜏(0) −µ) \n𝑱(1,0,0) = (𝛽𝜎−µ −𝛼\n0\n𝛼\n−µ) \n(23) 𝑱(1,0,0) = (𝛽𝜎(1) + 𝜏(0) −µ −𝛼\n𝜏(0)\n𝛼−𝜏(0)\n−𝜏(0) −µ) \n𝑱(1,0,0) = (𝛽𝜎−µ −𝛼\n0\n𝛼\n−µ) 𝑱(1,0,0) = (𝛽𝜎−µ −𝛼\n0\n𝛼\n−µ) \n(23) (23) µ\nSince 𝑱= 𝑭−𝑽, by using manipulation, the Jacobian matrix (23) can be formed as follows: 𝑱(1,0,0) = (𝛽𝜎\n0\n𝛼\n0) −(µ + 𝛼\n0\n0\nµ) \n𝑭= (𝛽𝜎\n0\n𝛼\n0) and 𝑽= (µ + 𝛼\n0\n0\nµ) \n(24) (24) 𝛼\n0\nNext, looking for the inverse of the matrix V, we get: 𝛼\n0\n0\nµ\nNext, looking for the inverse of the matrix V, we get: ext, looking for the inverse of the matrix V, we get: 𝑽−1 =\n1\nµ(µ + 𝛼) (µ\n0\n0\nµ + 𝛼) \n𝑽−1 =\n(\n \n1\n(µ + 𝛼)\n0\n0\n1\nµ) 𝑽−1 =\n1\nµ(µ + 𝛼) (µ\n0\n0\nµ + 𝛼) \n𝑽−1 =\n(\n \n1\n(µ + 𝛼)\n0\n0\n1\nµ)\n \n(25) (25) (\nµ)\nThe next step is to find 𝑅0 = 𝜌(𝑭∙𝑽−1). Then obtained: (\nµ)\nThe next step is to find 𝑅0 = 𝜌(𝑭∙𝑽−1). Then obtained: (\nµ)\nThe next step is to find 𝑅0 = 𝜌(𝑭∙𝑽−1). Then obtained: (\nµ)\nThe next step is to find 𝑅0 = 𝜌(𝑭∙𝑽−1). Then obtained: 𝑅0 = 𝜌(𝐹∙𝑉−1) \n𝑅0 = 𝜌\n(\n \n (𝛽𝜎\n0\n𝛼\n0) ∙\n(\n \n1\n(µ + 𝛼)\n0\n0\n1\nµ)\n \n) BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1333 BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 3.4.1 \nDisease-Free Equilibrium Point Stability Analysis By substituting the disease-free equilibrium point value in Equation (17) into the Jacobian matrix (32),\nwe get: \n𝛽\n0 we get: 𝐽0 = [\n−µ\n−𝛽𝜎\n0\n0\n𝛽𝜎−𝛼−µ\n0\n0\n𝛼\n−µ\n] \n(33) (33) Next, look for the eigenvalues of the matrix (33). So, the characteristic equation for the Jacobian matrix \n(33), which is analyzed by substituting the disease-free equilibrium point, can be written as follows: Next, look for the eigenvalues of the matrix (33). So, the characteristic equation for the Jacobian matrix \n(33), which is analyzed by substituting the disease-free equilibrium point, can be written as follows: 𝑑𝑒𝑡[\n𝜆+ µ\n−𝛽𝜎\n0\n0\n𝜆−𝛽𝜎+ 𝛼+ µ\n0\n0\n𝛼\n𝜆+ µ\n] = 0 \n (𝜆+ µ)2(𝛼−βσ + 𝜆+ µ) = 0 (34) 𝑑𝑒𝑡[\n𝜆+ µ\n−𝛽𝜎\n0\n0\n𝜆−𝛽𝜎+ 𝛼+ µ\n0\n0\n𝛼\n𝜆+ µ\n] = 0 (𝜆+ µ)2(𝛼−βσ + 𝜆+ µ) = 0 (34) (34) Then the eigenvalues obtained are 𝜆1 = −µ and 𝜆2 = βσ −𝛼−µ. It can be seen that the value of 𝜆1 \nis clearly negative, and 𝜆2 is not necessarily negative. For that, it will be proved that 𝜆2 is negative. It is \nknown that the condition for the disease-free equilibrium point is said to be stable if 𝑅0 < 1. From Equation \n(28) known value 𝑅0 = 𝛽𝜎\n𝛼+µ. Then obtained: Then the eigenvalues obtained are 𝜆1 = −µ and 𝜆2 = βσ −𝛼−µ. It can be seen that the value of 𝜆1 \nis clearly negative, and 𝜆2 is not necessarily negative. For that, it will be proved that 𝜆2 is negative. It is \nknown that the condition for the disease-free equilibrium point is said to be stable if 𝑅0 < 1. From Equation \n𝛽 (28) known value 𝑅0 = 𝛽𝜎\n𝛼+µ. Then obtained: 𝛽𝜎\n𝛼+ µ < 1 \n𝛽𝜎< 𝛼+ µ \n𝛽𝜎−𝛼−µ < 0 Because 𝜆2 = βσ −𝛼−µ . Then obtained: Because 𝜆2 = βσ −𝛼−µ . Then obtained: 𝜆2 < 0 𝜆2 < 0 Based on the Routh-Hurwitz criteria, every eigenvalue that exists is the same, namely negative. As a \nresult, there is no change in sign. It can be concluded that the disease-free equilibrium point is locally \nasymptotically stable. 3.4.1 \nDisease-Free Equilibrium Point Stability Analysis 3.4.1 \nDisease-Free Equilibrium Point Stability Analysis 3.4 Stability Analysis After obtaining the model equilibrium point, the next step is to perform stability analysis for each \nequilibrium point (𝑆0, 𝐼0, 𝑅0) and (𝑆1, 𝐼1, 𝑅1). As the first step in analyzing the equilibrium point, it is \nnecessary to linearize the system of differential Equation (12), Equation (13), and Equation (14). For \nexample: 𝑓(𝑆, 𝐼, 𝑅) = µ −𝛽𝜎𝐼𝑆−µ𝑆 \n(29) \n𝑔(𝑆, 𝐼, 𝑅) = 𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−𝛼𝐼−µ𝐼 \n(30) \nℎ(𝑆, 𝐼, 𝑅) = 𝛼𝐼−µ𝑅−𝜏𝐼𝑅 \n(31) (29) \n(30) \n(31) (31) By linearizing Equation (𝟐𝟗), we get: 𝑑𝑓\n𝑑𝑆= 𝑑(µ −𝛽𝜎𝐼𝑆−µ𝑆)\n𝑑𝑆\n= −𝛽𝜎𝐼−µ \n𝑑𝑓\n𝑑𝐼= 𝑑(µ −𝛽𝜎𝐼𝑆−µ𝑆)\n𝑑𝐼\n= −𝐵𝜎𝑆 \n𝑑𝑓\n𝑑𝑅= 𝑑(µ −𝛽𝜎𝐼𝑆−µ𝑆)\n𝑑𝑅\n= 0 Then linearize Equation (𝟑𝟎). Then obtained: Then linearize Equation (𝟑𝟎). Then obtained: 𝑑𝑔\n𝑑𝑆= 𝑑(𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−𝛼𝐼−µ𝐼)\n𝑑𝑆\n= 𝛽𝜎𝐼 \n𝑑𝑔\n𝑑𝐼= 𝑑(𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−𝛼𝐼−µ𝐼)\n𝑑𝐼\n= 𝛽𝜎𝑆+ 𝜏𝑅−𝛼−µ \n𝑑𝑔\n𝑑𝑅= 𝑑(𝛽𝜎𝐼𝑆+ 𝜏𝐼𝑅−𝛼𝐼−µ𝐼)\n𝑑𝑅\n= 𝜏𝐼 Then linearize Equation (𝟑𝟏). Then obtained: Then linearize Equation (𝟑𝟏). Then obtained: 𝑑ℎ\n𝑑𝑆= 𝑑(𝛼𝐼−µ𝑅−𝜏𝐼𝑅)\n𝑑𝑆\n= 0 Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1334 𝑑ℎ\n𝑑𝐼= 𝑑(𝛼𝐼−µ𝑅−𝜏𝐼𝑅)\n𝑑𝐼\n= 𝛼−𝜏𝑅 \n𝑑ℎ\n𝑑𝑅= 𝑑(𝛼𝐼−µ𝑅−𝜏𝐼𝑅)\n𝑑𝑅\n= −µ −𝜏𝐼 The result of linearization will be the elements of the Jacobian matrix, which has the following general \nJacobian: The result of linearization will be the elements of the Jacobian matrix, which has the following general \nJacobian: Jacobian: 𝐽= [\n−𝛽𝜎𝐼−µ\n−𝛽𝜎𝑆\n0\n𝛽𝜎𝐼\n𝛽𝜎𝑆+ 𝜏𝑅−𝛼−µ\n𝜏𝐼\n0\n𝛼−𝜏𝑅\n−µ −𝜏𝐼\n] \n(32) (32) BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1335 1335 𝑑𝑒𝑡[\n−𝛽𝜎𝐼1 −µ\n−𝛽𝜎𝑆1\n0\n𝛽𝜎𝐼1\n𝛽𝜎𝑆1 + 𝜏𝑅1 −𝛼−µ\n𝜏𝐼1\n0\n𝛼−𝜏𝑅1\n−µ −𝜏𝐼1\n] = 0 𝐼1𝜏(−𝛼+ 𝜏𝑅1)(𝜆+ 𝐼1𝛽𝜎+ µ) + (𝜆+ 𝜏𝐼1 + µ)(𝑆1𝐼1𝛽2𝜎2 + (𝜆+ 𝐼1𝛽𝜎+ µ)(𝛼+ 𝜆−𝑆1𝛽𝜎+\n0 𝐼1𝜏(−𝛼+ 𝜏𝑅1)(𝜆+ 𝐼1𝛽𝜎+ µ) + (𝜆+ 𝜏𝐼1 + µ)(𝑆1𝐼1𝛽2𝜎2 + (𝜆+ 𝐼1𝛽𝜎+ µ)(𝛼+ 𝜆−𝑆1𝛽𝜎+ 𝑅1𝜏+ µ)) \n= 0 𝜆3 + 𝜆2(𝛼−𝑆1𝛽𝜎+ 𝐼1𝛽𝜎+ 𝐼1𝜏+ 𝑅1𝜏+ 3µ) + 𝜆(2𝐼1𝛽𝜎µ \n−2𝑆1𝛽𝜎µ + 2𝐼1𝜏µ + 2𝑅1𝜏µ + 3µ2 + 𝐼1𝛼𝛽𝜎−𝑆1𝐼1𝛽𝜎𝜏 \n+𝐼1\n2𝛽𝜎𝜏+ 𝐼1𝑅1𝛽𝜎𝜏+ 2𝐼1𝑅1𝜏2 + 2𝛼µ) + 2𝐼1\n2𝑅1𝛽𝜎𝜏2 \n+𝐼1𝛼𝛽𝜎µ −𝑆1𝐼1𝛽𝜎𝜏µ + 𝐼1\n2𝛽𝜎𝜏µ + 𝐼1𝑅1𝛽𝜎𝜏µ \n+2𝐼1𝑅1𝜏2µ + 𝛼µ2 −𝑆1𝛽𝜎µ2 + 𝐼1𝛽𝜎µ2 + 𝐼1𝜏µ2 \n+𝑅1𝜏µ2 + µ3 = 0 Then obtained: Then obtained: 𝑎0 = 1 \n𝑎1 = 𝛼−𝑆1𝛽𝜎+ 𝐼1𝛽𝜎+ 𝐼1𝜏+ 𝑅1𝜏+ 3µ \n𝑎2 = 2𝐼1𝛽𝜎µ −2𝑆1𝛽𝜎µ + 2𝐼1𝜏µ + 2𝑅1𝜏µ + 3µ2 + 𝐼1𝛼𝛽𝜎 \n−𝑆1𝐼1𝛽𝜎𝜏+ 𝐼1\n2𝛽𝜎𝜏+ 𝐼1𝑅1𝛽𝜎𝜏+ 2𝐼1𝑅𝜏2 + 2𝛼µ \n𝑎3 = 2𝐼1\n2𝑅1𝛽𝜎𝜏2 + 𝐼1𝛼𝛽𝜎µ −𝑆1𝐼1𝛽𝜎𝜏µ + 𝐼1\n2𝛽𝜎𝜏µ \n+𝐼1𝑅1𝛽𝜎𝜏µ + 2𝐼1𝑅𝜏2µ + 𝛼µ2 −𝑆1𝛽𝜎µ2 \n+𝐼1𝛽𝜎µ2 + 𝐼1𝜏µ2 + 𝑅1𝜏µ2 + µ3 Forming an Array-Routh based on the above equation, we get: Forming an Array-Routh based on the above equation, we get: ||\n𝜆3\n𝛼0\n𝛼2\n𝜆2\n𝛼1\n𝛼3\n𝜆\n(𝛼1𝛼2 −𝛼0𝛼3\n𝛼1\n)\n0\n|| ||\n𝜆3\n𝛼0\n𝛼2\n𝜆2\n𝛼1\n𝛼3\n𝜆\n(𝛼1𝛼2 −𝛼0𝛼3\n𝛼1\n)\n0\n|| The next step is to analyze 𝛼0, 𝛼1, and (\n𝛼1𝛼2−𝛼0𝛼3\n𝛼1\n)There is no change in signs of a stable condition. The condition for a disease to be endemic is 𝑆, 𝐼, 𝑅> 0. Each case will create a different compartment value, \nmeaning that the difference between the compartments is absolute. Then obtained: Then obtained: \n𝛼1𝛼2 −𝛼0𝛼3 = 𝐼1𝛼2𝛽𝜎+ |𝐼1 −𝑆1|𝐼1𝛼𝛽2𝜎2 + |𝐼1 −𝑆1|𝐼1𝛼𝛽𝜎𝜏+ 2𝑅1𝛼𝛽𝜎𝜏+ 𝑆1\n2𝐼1𝛽2𝜎2𝜏\n+ |𝐼1 −2𝑆1|𝐼1\n2𝛽2𝜎2𝜏+ |𝐼1 −𝑆1|𝐼1𝑅1𝛽2𝜎2𝜏+ 2𝐼1𝑅1𝛼𝜏2 + |𝐼1 −𝑆1|𝐼1\n2𝛽𝜎𝜏2\n+ |𝐼1 −3𝑆1|𝑆1𝐼1𝑅1𝛽𝜎𝜏2 + 𝐼1𝑅1\n2𝛽𝜎𝜏2 + 2𝐼2𝑅1𝜏3 + 2𝐼1𝑅1\n2𝜏3 + 2𝛼2µ\n+ |6𝐼1 −4𝑆1|𝛼𝛽𝜎µ + 2𝑆1\n2𝛽2𝜎2µ + |2𝐼1 −4𝑆1|𝑆1𝐼1𝛽2𝜎2µ + 4𝐼1𝛼𝜏µ + 4𝑅1𝛼𝜏µ\n+ |6𝐼1 −6𝑆1|𝐼1𝛽𝜎𝜏µ + |6𝐼1 −4𝑆1|𝑅1𝛽𝜎𝜏µ + 2𝐼1\n2𝜏2µ + 8𝑅1𝐼1\n2𝑅1µ + 2𝑅1\n2𝜏2µ + 8𝛼µ2\n+ |8𝐼1 −8𝑆1|𝛽𝜎µ2 + 8𝐼1𝜏µ2 + 8𝑅1𝜏µ2 + 8µ3 + |\n𝛼1𝛼2 −𝛼0𝛼3 > 0 + |\n𝛼1𝛼2 −𝛼0𝛼3 > 0 𝛼1𝛼2 −𝛼0𝛼3 > 0 𝛼1 = 𝛼+ |𝐼1 −𝑆1|𝛽𝜎+ 𝐼1𝜏+ 𝑅1𝜏+ 3µ \n𝛼1 > 0 In this case, 𝛼1𝛼2 −𝛼0𝛼3 and 𝛼1 are clearly positive, so it can be concluded that 𝛼1𝛼2−𝛼0𝛼3\n𝛼1\n is positive. BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. Then all column one in Array Routh is positive. So, this equation is asymptotically stable. This means that \nthe disease will remain in a population. 3.4.2 \nDisease Endemic Equilibrium Point Stability Analysis Consider an endemic equilibrium Equation (21). Let the point (𝑆1, 𝐼1, 𝑅1) = (𝑆∗, 𝐼∗, 𝑅∗) = (𝑆, 𝐼, 𝑅). By substituting the disease-free equilibrium point value in Equation (21) into the Jacobian matrix (32), we \nget: 𝐽= [\n−𝛽𝜎𝐼1 −µ\n−𝛽𝜎𝑆1\n0\n𝛽𝜎𝐼1\n𝛽𝜎𝑆1 + 𝜏𝑅1 −𝛼−µ\n𝜏𝐼1\n0\n𝛼−𝜏𝑅1\n−µ −𝜏𝐼1\n] \n(35) (35) Next, look for the eigenvalues of the matrix (35). So the characteristic equation for the Jacobian matrix \n(35), which is analyzed by substituting the disease-free equilibrium point, can be written as follows: Next, look for the eigenvalues of the matrix (35). So the characteristic equation for the Jacobian matrix \n(35), which is analyzed by substituting the disease-free equilibrium point, can be written as follows: BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. Table 2. Parameter values in COVID-19 data at Health Center UPTD Batangh Table 2. Parameter values in COVID-19 data at Health Center UPTD Batanghari Table 2. Parameter values in COVID 19 data at Health Center UPTD Batanghari \nParameter \nValue Parameter \nN(Total Population) \n100% (24.977) \nµ(Death and Birth Rate) \n2% (Assumption) \n𝛽 (Transmission or \nInfection Rate) \n100% (Assumption) \nα(Recovery Rate) \n66% (138 of 206) \nγ(Total Vaccinated) \n65% (16.361 of 24.977) \nδ(Vaccine Efficacy) \n65% (Sinovac Standard) \n𝜏(Reinfection Rate) \n50% (Assumption) \nI (Infected Individual) \n0,82% (206 of 24.977) Parameter \nValue Parameter \nN(Total Population) \n100% (24.977) \nµ(Death and Birth Rate) \n2% (Assumption) \n𝛽 (Transmission or \nInfection Rate) \n100% (Assumption) \nα(Recovery Rate) \n66% (138 of 206) \nγ(Total Vaccinated) \n65% (16.361 of 24.977) \nδ(Vaccine Efficacy) \n65% (Sinovac Standard) \n𝜏(Reinfection Rate) \n50% (Assumption) \nI (Infected Individual) \n0,82% (206 of 24.977) With initial values 𝑆1: 0.9868, 𝐼1: 0.0082, and 𝑅1: 0.005. Subtitute Equation (36), Equation (37), and \nEquation (38) with existing parameter values. Then obtained: 𝑆2 = 𝑆1 + µ −𝛽(1 −𝛿𝛾)𝐼1𝑆1 −µ𝑆1 \n𝑆2 = 0.9868 + 2% −100% × (1 −65% × 70%) × 0.0082 × 0.9868 −2% × \n0.9868 = 𝑆1 + µ −𝛽(1 −𝛿𝛾)𝐼1𝑆1 −µ𝑆1 \n𝑆2 = 0.9868 + 2% −100% × (1 −65% × 70%) × 0.0082 × 0.9868 −2% × \n0.9868 3.5 Model Application This simulation aims to test the balance points that have been formed following the Routh-Hurwitz \ncriteria. This simulation uses the following equation: This simulation aims to test the balance points that have been formed following the Routh-Hurwitz \ncriteria. This simulation uses the following equation: Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1336 𝑆𝑛+1 = 𝑆𝑛+ µ𝑁−𝛽𝜎𝐼𝑛𝑆𝑛−µ𝑆𝑛 \n(36) \n𝐼𝑛+1 = 𝐼𝑛+ 𝛽𝜎𝐼𝑛𝑆𝑛+ 𝜏𝐼𝑛𝑅𝑛−𝛼𝐼𝑛−µ𝐼𝑛 \n(37) \n𝑅𝑛+1 = 𝑅𝑛+ 𝛼𝐼𝑛−µ𝑅𝑛−𝜏𝐼𝑛𝑅𝑛 \n(38) (36) \n(37) \n(38) f this model uses data obtained from the UPTD Puskesmas Batanghari, as follows: lication of this model uses data obtained from the UPTD Puskesmas Batanghari, as follows: ACKNOWLEDGEMENT All authors would like to thank the academic community at the University of Lampung and LPPM at \nthe University of Lampung for supporting and funding the research and publication of this article. 4. CONCLUSIONS 4. CONCLUSIONS Our research results can be explained as follow: h results can be explained as follow: Our research results can be explained as follow: 1. The COVID-19 SIR Model, made locally asymptotically stable at the balance point of disease-free \nand endemic disease, means that this disease can disappear and remain in an area. 1. The COVID-19 SIR Model, made locally asymptotically stable at the balance point of disease-free \nand endemic disease, means that this disease can disappear and remain in an area. 2. The effect of the vaccination given is to inhibit the spike in infection. We can pay attention to the \nbasic reproduction number A in handling this case. 2. The effect of the vaccination given is to inhibit the spike in infection. We can pay attention to the \nbasic reproduction number A in handling this case. 3. Based on the application of the model to the data at the Batanghari Health Center UPTD, we obtained \nthat the dynamics of the COVID-19 development monitored to be conducive, and the condition was \nstable for the next 100 months, with a Mean Absolute Percentage Error (MAPE) percentage of 2.8% \nwhich it's Highly Accurate Forecasting 3.6 MAPE To find out the errors that exist in the forecasting process, the data in the Batanghari Health Center \nUPTD in months 1, 2, 3, 4, and 5 in 2022 with forecasting results with the SIR COVID-19 model in months \n1, 2, 3, 4, and 5 in 2022. Table 3. Comparison of Data for Months 1, 2, 3, 4, and 5 in 2022 Table 3. Comparison of Data for Months 1, 2, 3, 4, and 5 in 2022 \nMonths \nModel SIR \nCOVID-19 \n(𝑨) \nHealth \nCenter \n(𝑷) \n𝜟|𝑨−𝑷| \n1 \n12.239 \n10 \n2.239 \n2 \n10.965 \n11 \n0.035 \n3 \n9.796 \n7 \n2.796 \n4 \n8.761 \n8 \n0.761 \n5 \n7.837 \n9 \n1.163 \nTotal \n49.598 \n45 \n6.994 \n \nThen it can be obtained that the average absolute error or Mean Absolute Error (MAPE) is Then it can be obtained that the average absolute error or Mean Absolute Error (MAPE) is 𝑀𝐴𝐸= 𝛥|𝐴−𝑃|\n𝑛\n= 6,994\n5\n= 1.3988 \n𝑀𝐴𝑃𝐸= 1.3988\n49.598 = 0.028202 = 2.8% 𝑀𝐴𝐸= 𝛥|𝐴−𝑃|\n𝑛\n= 6,994\n5\n= 1.3988 \n𝑀𝐴𝑃𝐸= 1.3988\n49.598 = 0.028202 = 2.8% BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. decisions or policies implemented by the government and Health Center are correct. However, they must \ncontinue to apply health protocols properly so that there is no spike in unwanted infections. decisions or policies implemented by the government and Health Center are correct. However, they must \ncontinue to apply health protocols properly so that there is no spike in unwanted infections. [5] \nA. Mortellaro and P. Ricciardi-Castagnoli, “From vaccine practice to vaccine science: The contribution of human immunology \nto the prevention of infectious disease,” Immunol. Cell Biol., vol. 89, no. 3, pp. 332–339, 2011, doi: 10.1038/icb.2010.152. [2] \nA. Ajbar, R. T. Alqahtani, and B. Mourad, “Dynamics of an SIR-Based COVID-19 Model With Linear Incidence Rate, \nNonlinear Removal Rate, and Public Awareness,” Front. Phys., vol. 9, no. 634251, pp. 1–13, 2021, doi: \nhttps://doi.org/10.3389/fphy.2021.634251. ,\n,\n, pp\n,\n,\ng\nP. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.,” Sci. Rep., vol\n, 2020, doi: https://doi.org/10.1038/s41598-020-80007-8. [1] \nE. Callaway, “The race for coronavirus vaccines: a graphical guide,” Nature, vol. 580, no. 7805, pp. 576–577, 2020, doi: \ndoi:10.1038/d41586-020-01221-y. [4] \nZ. Liao, P. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.\n10, no. 1, 2020, doi: https://doi.org/10.1038/s41598-020-80007-8. 𝑺𝟐= 𝟎, 𝟗𝟕𝟗𝟒𝟗𝟖𝟐𝟎𝟒 𝑺𝟐= 𝟎, 𝟗𝟕𝟗𝟒𝟗𝟖𝟐𝟎𝟒 𝐼2 = 𝐼1 + 𝛽(1 −𝛿𝛾)𝐼1𝑆1 + 𝜏𝐼1𝑅1 −𝛼𝐼1 −µ𝐼1 \n𝐼2 = 0.000132495 + 100% × (1 −65% × 70%) × 0.000132495 × 0.999847505 + 10% × \n0.000132495 × 0.00002 −20% × 0.000132495 \n− 2% × 0.000132495 𝐼2 = 𝐼1 + 𝛽(1 −𝛿𝛾)𝐼1𝑆1 + 𝜏𝐼1𝑅1 −𝛼𝐼1 −µ𝐼1 \n𝐼2 = 0.000132495 + 100% × (1 −65% × 70%) × 0.000132495 × 0.999847505 + 10% × \n0.000132495 × 0.00002 −20% × 0.000132495 \n− 2% × 0.000132495 𝑅2 = 𝑅1 + 𝛼𝐼1 −µ𝑅1 −𝜏𝐼1𝑅1 \n𝑅2 = 0.00002 + 20% × 0.000132495 −2% × 0.00002 −10% × 0.000132495 × 0.00002 \n𝑹𝟐= 𝟎. 𝟎𝟎𝟎𝟎𝟒𝟔 2\n𝑅1 + 𝛼𝐼1\nµ𝑅1\n𝜏𝐼1𝑅1 \n2 = 0.00002 + 20% × 0.000132495 −2% × 0.00002 −10% × 0.000132495 × 0.00002 \n𝟐= 𝟎. 𝟎𝟎𝟎𝟎𝟒𝟔 Continued until the 100𝑡ℎ iteration, so that the graph is obtained as follows: a) \n(b) \n(c) \n \nFigure 4. Graph of the SIR Model for Covid 19 Data at Batanghari Health Center. (a) Population conditions \nfor individuals who are healthy but susceptible to COVID-19, (b) Population conditions for individuals infected \nwith COVID-19, and (c) Population conditions for individuals who have been given the COVID-19 vaccine. Based on the resulting graph, it can be seen that the dynamics of the development of COVID-19 are \nmonitored to be conducive, and the condition is stable for the next 100 months. It can be concluded that the Figure 4. Graph of the SIR Model for Covid 19 Data at Batanghari Health Center. (a) Population conditions \nfor individuals who are healthy but susceptible to COVID-19, (b) Population conditions for individuals infected \nwith COVID-19, and (c) Population conditions for individuals who have been given the COVID-19 vaccine. Based on the resulting graph, it can be seen that the dynamics of the development of COVID-19 are \nmonitored to be conducive, and the condition is stable for the next 100 months. It can be concluded that the Based on the resulting graph, it can be seen that the dynamics of the development of COVID-19 are \nmonitored to be conducive, and the condition is stable for the next 100 months. It can be concluded that the BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1337 [3] \nB. Wacker and J. Schlüter, “Time-continuous and time-discrete SIR models revisited: theory and applications,” Adv. Differ. \nEquations, vol. 2020, no. 556, pp. 1–44, 2020, doi: doi.org/10.1186/s13662-020-02995-1. \n[4] \nZ. Liao, P. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.,” Sci. Rep., vol. \n10, no. 1, 2020, doi: https://doi.org/10.1038/s41598-020-80007-8. q\n,\n,\n, pp\n,\n,\ng\nZ. Liao, P. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.,” Sci. R\n10\n1 2020 d i htt\n//d i\n/10 1038/ 41598 020 80007 8 [3] \nB. Wacker and J. Schlüter, Time continuous and time discrete SIR models revisited: theory and applicatio\nEquations, vol. 2020, no. 556, pp. 1–44, 2020, doi: doi.org/10.1186/s13662-020-02995-1. \n[4] \nZ. Liao, P. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.\n10 no 1 2020 doi: https://doi org/10 1038/s41598-020-80007-8 p\ng\np y\n[3] \nB. Wacker and J. Schlüter, “Time-continuous and time-discrete SIR models revisited: theory and applications,” Adv. Differ. \nEquations, vol. 2020, no. 556, pp. 1–44, 2020, doi: doi.org/10.1186/s13662-020-02995-1. \n[4]\nZ Liao P Lan Z Liao Y Zhang and S Liu “TW SIR: time window based SIR for COVID 19 forecasts ” Sci Rep vol REFERENCES [1] \nE. Callaway, “The race for coronavirus vaccines: a graphical guide,” Nature, vol. 580, no. 7805, pp. 576–577, 2020, doi: \ndoi:10.1038/d41586-020-01221-y. y\n[2] \nA. Ajbar, R. T. Alqahtani, and B. Mourad, “Dynamics of an SIR-Based COVID-19 Model With Linear Incidence Rate, \nNonlinear Removal Rate, and Public Awareness,” Front. Phys., vol. 9, no. 634251, pp. 1–13, 2021, doi: \nhttps://doi.org/10.3389/fphy.2021.634251. [3] \nB. Wacker and J. Schlüter, “Time-continuous and time-discrete SIR models revisited: theory and applications,” Adv. Differ. Equations, vol. 2020, no. 556, pp. 1–44, 2020, doi: doi.org/10.1186/s13662-020-02995-1. [4] \nZ. Liao, P. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.,” Sci. Rep., vol. 10, no. 1, 2020, doi: https://doi.org/10.1038/s41598-020-80007-8. [5] \nA. Mortellaro and P. Ricciardi-Castagnoli, “From vaccine practice to vaccine science: The contribution of human immunology \nto the prevention of infectious disease,” Immunol. Cell Biol., vol. 89, no. 3, pp. 332–339, 2011, doi: 10.1038/icb.2010.152. Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1338 [6] \nP. Thapa, “Predicating COVID19 Epidemic in Nepal Using the SIR Model,” Stud. Syst. Decis. Control, vol. 358, no. September, \npp. 229–237, 2021, doi: 10.1007/978-3-030-69744-0_14. pp\n[7] \nO. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio \nR0 in models for infectious diseases in heterogeneous populations,” J. Math. Biol., vol. 28, no. 4, pp. 365–382, 1990, doi: \n10.1007/BF00178324. [8] \nW. F. Putra, “Analisis Efikasi dan Efektivitas Vaksin COVID-19 terhadap Varian SARS-CoV-2: Sebuah Tinjauan Literatur,” \nJ. Kedokt. Meditek, vol. 28, no. 1, pp. 107–119, 2022, doi: 10.36452/jkdoktmeditek.v28i1.2243. [9] \nC. van Oosterhout, N. Hall, H. Ly, and K. M. Tyler, “COVID-19 evolution during the pandemic–Implications of new SARS-\nCoV-2 variants on disease control and public health policies,” Virulence, vol. 12, no. 1, pp. 507–508, 2021, doi: \n10.1080/21505594.2021.1877066. [10] J. Li et al., “Epidemiology of COVID-19 : A Systematic Review and Meta-analysis of Clinical Epidemiology of COVID-19 : \nA systematic review and meta-analysis of clinical characteristics, risk factors , and outcomes,” Med. Virol., pp. 1–10, 2020, \ndoi: 10.1002/jmv.26424. [11] WHO, \n“Coronavirus \ndisease \n( \nCOVID-19 \n),” \n2020. [Online]. Available: \nhttps://www.who.int/docs/default-\nsource/coronaviruse/situation-reports/20201012-weekly-epi-update-9.pdf p\ny p\np\np\n2] J. Sun et al., “COVID-19 : Epidemiology , Evolution , and Cross-Disciplinary Perspectives,” no. January, 2020. [\n]\n,\np\ngy ,\n,\np\ny\np\n,\ny,\n[13] C. Van Oosterhout, N. Hall, H. Ly, and K. M. REFERENCES Tyler, “COVID-19 evolution during the pandemic – Implications of new SARS-\nCoV-2 variants on disease control and public health policies,” Virulence, vol. 12, no. 1, pp. 507–508, 2021, doi: \n10.1080/21505594.2021.1877066. [14] S. Setiati and M. K. Azwar, “Dilemma of Prioritising Health and the Economy During COVID-19 Pandemic in Indonesia,” \nActa Med Indones, vol. 52, no. 3, pp. 196–198, 2020. [15] A. Mortellaro and P. Ricciardi-Castagnoli, “From vaccine practice to vaccine science : the contribution of human immunology \nto the prevention of infectious disease,” Immunol. Cell Biol., vol. 89, pp. 332–339, 2011, doi: 10.1038/icb.2010.152. [16] WHO “T\nJ\nb L\nkd\nd\nh d i\nit ” 2021\nh i t/i d\ni /\n/\nl\ni\n/\n/\nl\nkd\nd [15] A. Mortellaro and P. Ricciardi-Castagnoli, “From vaccine practice to vaccine science : the contribution of human immunology \nto the prevention of infectious disease,” Immunol. Cell Biol., vol. 89, pp. 332–339, 2011, doi: 10.1038/icb.2010.152. [16] WHO, “Tanya Jawab : Lockdown dan herd immunity,” 2021. who.int/indonesia/news/novel-coronavirus/qa/qa-lockdown-and-\nherd-immunity (accessed Jun. 10, 2020). y (\n,\n)\n[17] N. M. Nasir, I. S. Joyosemito, B. Boerman, and I. Ismaniah, “Kebijakan Vaksinasi COVID-19 : Pendekatan Pemodelan \nMatematika Dinamis Pada Efektivitas Dan Dampak Vaksin Di Indonesia,” J. Abdimas UBJ, vol. 4, no. 2, pp. 191–204, 2021. [18] W. F. Putra, “Analisis Efikasi dan Efektivitas Vaksin COVID-19 terhadap Varian SARS-CoV-2 : Sebuah Tinjauan Literatur \nAnalysis the Efficacy and Effectivity of COVID-19 Vaccines to the SARS-CoV-2 Variants : A Literature Review,” Meditek, \nvol. 28, no. 1, pp. 107–119, 2022, doi: https://doi.org/10.36452/jkdoktmeditek.v28i1.2243. pp\np\ng\nj\n[19] W. O. Kermarck and A. G. McKendrick, “A Contribution to the Mathematical Theory o f Epidemics.,” in The Royal Society \nLondon A, Royal Society, 1927, pp. 700–721. doi: 10.1098/rspa.1927.0118. [20] P. Thapa, “Predicating COVID19 epidemic in Nepal using the SIR model,” Artif. Intell. COVID-19., 2021, doi: 10.1007/978-\n3-030-69744-0_14. [21] M. A. Shereen, S. Khan, A. Kazmi, N. Bashir, and R. Siddique, “COVID-19 infection: origin, transmission, an\nof human coronaviruses.,” J. Adv. Res., vol. 24, no. 2020, pp. 91–98, 2020, doi: 10.1016/j.jare.2020.03.005. [22] Y. F. Lin et al., “Spread and Impact of COVID-19 in China: A Systematic Review and Synthesis of Predictions From \nTransmission-Dynamic Models. Frontiers in Medicine, 7. doi:10.3389/fmed.2020.00321,” Front. Med., vol. 7, no. 321, pp. 1–\n11, 2020, doi: doi:10.3389/fmed.2020.00321. [23] Q. Griette and P. REFERENCES Magal, “Clarifying predictions for COVID-19 from testing data: The example of New York State.,” Infect. Dis. Model., vol. 6, pp. 273–283, 2021, doi: https://doi.org/10.1016/j.idm.2020.12.011. pp\np\ng\nj\n[24] Z. Liu, P. Magal, and G. Webb, “Predicting the number of reported and unreported cases for the COVID- 19 epidemics in \nChina, South Korea, Italy, France, Germany and United Kingdom,” J. Theor. Biol. 509, 110501., vol. 509, no. 110501, 2021, \ndoi: doi:10.1016/j.jtbi.2020.110501. j j\n[25] A. J. Kucharski, T. W. Russell, J. Diamond, C., Liu, Y., Edmunds, S. Funk, and R. M. Eggo, “Early dynamics of transmission \nand control of COVID-19: a mathematical modelling study.,” Lancet Infect. Dis., vol. 20, pp. 1–7, 2020, doi: \nhttps://doi.org/10.1016/S1473-3099(20)30144-4. p\ng\n(\n)\n[26] E. S. Kurkina and E. M. Koltsova, “Mathematical Modeling of the Propagation of COVID-19 Pandemic Waves in the World.,” \nComput. Math. Model., vol. 32, no. 2, pp. 147–170, 2021, doi: doi:10.1007/s10598-021-09523-0. pp\n[27] A. Lobo et al., “COVID-19 epidemic in Brazil: where we at?,” Int. J. Infect. Dis., vol. 97, pp. 382–385, 2020, doi: \ndoi:10.1016/j.ijid.2020.06.044. [28] K. Roosa et al., “Real-time forecasts of the COVID-19 epidemic in China from February 5th to February 24th, 2020.,” Infect. Dis. Model., vol. 5, no. 2020, pp. 256–263, 2020, doi: 10.1016/j.idm.2020.02.002. [29] H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Rev., vol. 42, no. 4, pp. 599–653, 2000, [On\nhttp://www.siam.org/journals/sirev/42-4/37190.html p\ng j\n[30] K. R. Meyer, “Normal forms for the general equilibrium,” Funkc. Ekvacioj, vol. 27, pp. 261–271, 1984. p\ng j\n] K. R. Meyer, “Normal forms for the general equilibrium [31] G. J. Olsder and J. W. van der Woude, Mathematical Systems Theory, Second. Delft, The Netherland: Delft \n1997. [32] O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio \nRo in models for infectious diseases in heterogeneous populations,” J. Math. Biol, vol. 28, pp. 365–382, 1990. [33] A. Faruk, “Model Epidemik Tuberkulosis Seir dengan Terapi pada Individu Terinfeksi,” J. Penelit. Sains, vol. 18, no. 3, pp. 99–104, 2016, doi: 10.56064/jps.v18i3.16. jp\n[34] J. Giesecke, Modern Infectious Disease Epidemiology, Third. CRC Press Taylor & Francis Group, 2017. [35] P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental \nmodels \nof \ndisease \ntransmission,” \nMath. Biosci. 180, \nvol. 180, \npp. 29–48, \n2002, \n[Online]. Available: \nhttp://www.math.unb.ca/?watmough. p\ng\n[36] F. R. Gantmacher, The theory of matrices. REFERENCES Chelsea Publishing Company, 1959. p\ng\n[36] F. R. Gantmacher, The theory of matrices. Chelsea Publishing Company, 1959. ] F. R. Gantmacher, The theory of matrices. Chelsea Publi ewis, Demand Forecasting and Inventory Control: A computer aided learning approach, vol. 2. Woodhead Publishing\n7. [37] C. D. Lewis, Demand Forecasting and Inventory Control: A computer aided learning approach, vol. 2. Woodhead Publishing \nLtd, 1997. [38] WHO “Evaluation of COVID-19 vaccine effectiveness ” 2021 https://www who int/publications/i/item/WHO-2019-nCoV- Ltd, 1997. [38] WHO “Evaluation of COVID-19 vaccine effectiveness ” 2021 https://www who int/publications/i/item/WHO-2019-nCoV- Ltd, 1997. [38] WHO, “Evaluation of COVID-19 vaccine effectiveness,” 2021. https://www.who.int/publications/i/item/WHO-2019-nCoV- Evaluation of COVID-19 vaccine effectiveness,” 2021. https://www.who.int/publications/i/item/WHO-2019-nCoV [38] WHO, “Evaluation of COVID-19 vaccine effectiveness,” 2021. https://www.who.int/publications/i/item/WH BAREKENG: J. Math. & App., vol. 17(3), pp. 1325- 1340, September, 2023. 1339 \n \nvaccine_effectiveness- measurement-2021 1339 Aziz, et. al. THE DEVELOPMENT OF COVID-19 USING OUTBREAK …. 1340"