Investment is the wealth of one or more assets in the hope of future benefits. Things to consider in investing are profit and risk. So investors need to diversify their investments, which means investors need to form a portfolio through the selection of several assets so that risk can be minimized without reducing expected profits. The COVID-19 pandemic period had a big impact on the economy, especially for investors in making optimal portfolio formation. This study aims to determine the optimal portfolio formation during the COVID-19 pandemic using the Single Index Model. In this study a Single Index Model was be studied systematically and then translated into a programming. The data used are data of consistent shares included in the Jakarta Islamic Index (JII) shares over the past two years. Furthermore, these stocks are chosen which have an average return that is higher than the profits obtained if investors save their money in the bank. The results showed six JII companies included in the candidate for optimal portfolio formation. After the analysis, two shares were produced, namely BRPT with a proportion of 63.8043% and EXCL 36.1957%. The proportion is expected to provide a profit of 1.57% per week and a risk of 6.06% per week. With the proportions obtained, an investment simulation was then carried out during the COVID-19 pandemic. The results of the simulation obtained a gain of 0.0771504% every week. These results are below the risk-free return of assets (SBIS) during the COVID-19-19 pandemic with an average profit of 0.087445% per week. It was concluded that optimal portfolio formation with the Single Index Model did not provide optimal benefits during the COVID-19 pandemic.

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ABSTRAKMakalah ini membahas dinamika transmisi COVID-19 dengan melibatkan intervensi karantina. Model dikonstruksi dengan melibatkan tiga kelas penyebab infeksi, yaitu kelas manusia terpapar, kelas manusia terinfeksi tanpa gejala klinis, dan kelas manusia terinfeksi disertai gejala klinis. Variabel yang merepresentasikan intervensi karantina untuk menekan pertumbuhan infeksi juga dipertimbangkan pada model. Selanjutnya, analisis model difokuskan pada eksistensi titik kesetimbangan dan simulasi numerik untuk menunjukkan dinamika populasi secara visual. Model yang dikonstruksi membentuk model SEAQIR yang memiliki dua titik kesetimbangan, yaitu titik kesetimbangan bebas penyakit dan titik kesetimbangan endemik. Analisis kestabilan menunjukkan bahwa titik kesetimbangan bebas penyakit bersifat stabil asimtotik lokal pada saat R01 dan tidak stabil pada saat R01. Simulasi numerik menunjukkan bahwa peningkatan intervensi berupa karantina dapat berkontribusi memperlambat transmisi COVID-19 sehingga diharapkan dapat mencegah terjadinya wabah pada populasi.ABSTRACTThis paper discusses the dynamics of COVID-19 transmission by involving quarantine interventions. The model was constructed by involving three classes of infectious causes, namely the exposed human class, asymptotically infected human class, and symptomatic infected human class. Variables were representing quarantine interventions to suppress infection growth were also considered in the model. Furthermore, model analysis is focused on the existence of equilibrium points and numerical simulations to visually showed population dynamics. The constructed model forms the SEAQIR model which has two equilibrium points, namely a disease-free equilibrium point and an endemic equilibrium point. The stability analysis showed that the disease-free equilibrium point was locally asymptotically stable at R01 and unstable at R01. Numerical simulations showed that increasing interventions in the form of quarantine could contribute to slowing the transmission of COVID-19 so that it is hoped that it can prevent outbreaks in the population.

This paper discusses the dynamics of COVID-19 transmission by involving quarantine interventions. The model was constructed by involving three classes of infectious causes, namely the exposed human class, asymptotically infected human class, and symptomatic infected human class. Variables were representing quarantine interventions to suppress infection growth were also considered in the model. Furthermore, model analysis is focused on the existence of equilibrium points and numerical simulations to visually showed population dynamics. The constructed model forms the SEAQIR model which has two equilibrium points, namely a disease-free equilibrium point and an endemic equilibrium point. The stability analysis showed that the disease-free equilibrium point was locally asymptotically stable at R0<1 and unstable at R0>1. Numerical simulations showed that increasing interventions in the form of quarantine could contribute to slowing the transmission of COVID-19 so that it is hoped that it can prevent outbreaks in the population.

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ABSTRAKMakalah ini membahas dinamika transmisi COVID-19 dengan melibatkan intervensi karantina. Model dikonstruksi dengan melibatkan tiga kelas penyebab infeksi, yaitu kelas manusia terpapar, kelas manusia terinfeksi tanpa gejala klinis, dan kelas manusia terinfeksi disertai gejala klinis. Variabel yang merepresentasikan intervensi karantina untuk menekan pertumbuhan infeksi juga dipertimbangkan pada model. Selanjutnya, analisis model difokuskan pada eksistensi titik kesetimbangan dan simulasi numerik untuk menunjukkan dinamika populasi secara visual. Model yang dikonstruksi membentuk model SEAQIR yang memiliki dua titik kesetimbangan, yaitu titik kesetimbangan bebas penyakit dan titik kesetimbangan endemik. Analisis kestabilan menunjukkan bahwa titik kesetimbangan bebas penyakit bersifat stabil asimtotik lokal pada saat R01 dan tidak stabil pada saat R01. Simulasi numerik menunjukkan bahwa peningkatan intervensi berupa karantina dapat berkontribusi memperlambat transmisi COVID-19 sehingga diharapkan dapat mencegah terjadinya wabah pada populasi.ABSTRACTThis paper discusses the dynamics of COVID-19 transmission by involving quarantine interventions. The model was constructed by involving three classes of infectious causes, namely the exposed human class, asymptotically infected human class, and symptomatic infected human class. Variables were representing quarantine interventions to suppress infection growth were also considered in the model. Furthermore, model analysis is focused on the existence of equilibrium points and numerical simulations to visually showed population dynamics. The constructed model forms the SEAQIR model which has two equilibrium points, namely a disease-free equilibrium point and an endemic equilibrium point. The stability analysis showed that the disease-free equilibrium point was locally asymptotically stable at R01 and unstable at R01. Numerical simulations showed that increasing interventions in the form of quarantine could contribute to slowing the transmission of COVID-19 so that it is hoped that it can prevent outbreaks in the population.

Coronavirus Disease-2019 (COVID-19) merupakan suatu virus yang dapat menyebabkan penyakit menular dengan gejala batuk, demam, hilangnya indra perasa maupun penciuman, hingga sesak napas. Proses penyebaran penyakit terjadi ketika adanya kontak dari individu terinfeksi dengan individu rentan, baik secara langsung maupun tidak langsung. Penelitian ini dilakukan untuk merumuskan model matematika penyebaran COVID-19, melakukan analisis kestabilan model, dan simulasi numerik. Berdasarkan asumsi serta pertimbangan dalam membangun model diperoleh suatu model matematika penyebaran Covid-19 tipe SVIR yang terdiri atas empat kelas populasi, yaitu kelas populasi; individu rentan (S), individu tervaksin (V), individu terinfeksi (I), dan individu sembuh dari COVID-19 (R). Sifat kestabilan titik kesetimbangan model bergantung pada perubahan nilai bilangan reproduksi dasar (<0). Titik kesetimbangan bebas penyakit E0 bersifat stabil asimtotik lokal jika <0 < 1, serta tidak stabil jika <0 > 1. Titik kesetimbangan bebas penyakit E0 akan selalu ada, serta titik kesetimbangan endemik E∗ unik dan positif jika dan hanya jika <0 > 1. Simulasi numerik menggambarkan bahwa sistem berada pada kondisi stabil disekitar titik kesetimbangan endemik. Peningkatan laju vaksinasi dan laju efektivitas pengobatan masing-masing dapat menekan jumlah kasus infeksi COVID-19. Sedangkan peningkatan laju penyusutan vaksin dan laju penurunan efektivitas vaksin dapat mengakibatkan jumlah kasus infeksi COVID-19 terus meningkat.Kata Kunci: COVID-19, Model SVIR, Titik Kesetimbangan

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