Optimum Control of SEIR Model on COVID-19 Spread with Delay Time and Vaccination Effect in South Sulawesi Province

Abstract

The increasing number of cases and the development of new variants of the Covid-19 virus globally including the territory of Indonesia, especially in the province of South Sulawesi are increasingly worrying and need to be prevented. Therefore, this study aims to develop a SEIR model on the spread of Covid-19 with vaccination control, optimal control analysis, stability analysis and numerical simulation of the SEIR model on the spread of Covid-19 in South Sulawesi. This study uses the SEIR epidemic model to predict the spread of Covid-19 in South Sulawesi Province with parameters such as birth rate, cure rate, mortality rate, interaction rate and vaccination. The SEIR model was chosen because it is one of the basic methods in the epidemiological model. The method used to build the model is a time delay model by considering the vaccination factor as a model parameter, model analysis using the next generation matrix method to determine the basic reproduction number and stability of the Covid-19 distribution model in South Sulawesi. Numerical model simulation using secondary data on the number of Covid-19 cases in South Sulawesi starting in 2021 which was obtained from the South Sulawesi Provincial Health Office. The results obtained are model analysis provides evidence of the existence of optimal control in the model. Based on the results obtained, it can also be seen that vaccination greatly influences the spread of Covid-19 in South Sulawesi, so that awareness is needed for the people of South Sulawesi to follow the government's recommendation to vaccinate to prevent or reduce the rate of transmission of Covid-19 in South Sulawesi.

Analisis dan Solusi Numerik Model Matematika pada Penyebaran Covid-19 di Provinsi Sulawesi Selatan dengan Metode Runge Kutta dan Adams-Bashforth-Moulton

“…Teorema 1 Titik kesetimbangan bebas penyakit model matematika penyebaran Covid-19 dikatakan stabil asimtotik jika 𝑅 0 < 1 dan tidak stabil ketika 𝑅 0 > 1 (Side, 2022). Bukti: Berdasarkan sistem persamaan (4.2) diperoleh matriks Jacobian (𝐽) berikut:…”

“…Analisis dan Solusi Numerik Model Matematika pada Penyebaran Covid-19 di Provinsi Sulawesi Selatan dengan Metode Runge Kutta dan Adams-Bashforth-Moulton (Side, dkk) 𝜇)(𝜆 + (𝜇 + 𝜇 𝑖 + 𝜎)) | 𝜆 + (𝑣 + 𝜇 + (𝛼 + 𝛿)𝑖) 0 𝜇)(𝜆 + (𝜇 + 𝜇 𝑖 + 𝜎)) (( 𝜆 + (𝑣 + 𝜇 + (𝛼 + 𝛿)𝑖)) ((𝜆 + (𝛽 + 𝜇))(𝜆 + (𝛾 + 𝜇 + 𝜇 𝑖 )) − 𝛼𝛽𝑠) + 𝛼𝑖(𝛽(𝛼 + 𝛿))𝑠) = 0 (4.1)Substitusi nilai titik kesetimbangan bebas penyakit (𝑠, 𝑒, 𝑖, 𝑎, 𝑟) yaitu 𝑠 = 𝑢(𝑢 + 𝑣) −1 dan 𝑖 = 0 ke persamaan (4.1) sehingga diperoleh:(𝜆 + 𝜇)(𝜆 + (𝜇 + 𝜇 𝑖 + 𝜎))( 𝜆 + 𝑣 + 𝜇) ((𝜆 + (𝛽 + 𝜇)) (𝜆 + (𝛾 + 𝜇 + 𝜇 𝑖 )) − 𝛼𝛽𝑢(𝑢 + 𝑣) −1 ) = 0 (4.2) Perhatikan bahwa: (𝜆 + (𝛽 + 𝜇)) (𝜆 + (𝛾 + 𝜇 + 𝜇 𝑖 )) − 𝛼𝛽𝑢(𝑢 + 𝑣) −1 = 0 (4.3) Berdasarkan aturan tanda Descartes , persamaan (4.3) akan memiliki akar-akar negatif jika semua tanda pada setiap suku-sukunya positif sehingga agar persamaan (4.2) memiliki akar-akar negatif maka haruslah 𝑅 0 ≤ 1. Oleh karena itu maka dapat disimpulkan bahwa titik kesetimbangan bebas penyakit akan stabil asimtotik jika 𝑅 0 ≤ 1 dan tidak stabil jika 𝑅 0 > 1.∎ Teorema 2 Titik kesetimbangan endemik penyakit model matematika penyebaran Covid-19 stabil asimtotik(Side, 2022). Bukti: Berdasarkan persamaan (4.1) subtitusi nilai titik kesetimbangan endemik (𝑠 * , 𝑒 * , 𝑖 * , 𝑎 * , 𝑟 * ) sehingga diperoleh: (𝜆 + 𝜇)(𝜆 + (𝜇 + 𝜇 𝑖 + 𝜎)) (( 𝜆 + 𝑣 + 𝜇 + (𝛼 + 𝛿)𝑖)(𝜆 2 + ((𝛽 + 𝜇) + (𝛾 + 𝜇 + 𝜇 𝑖 ))𝜆) + (𝛽 + 𝜇)(𝛾 + 𝜇 + 𝜇 𝑖 )(𝛼 + 𝛿)𝑖) = 0 (4.4) Perhatikan bahwa: (( 𝜆 + 𝑣 + 𝜇 + (𝛼 + 𝛿)𝑖)(𝜆 2 + ((𝛽 + 𝜇) + (𝛾 + 𝜇 + 𝜇 𝑖 ))𝜆) + (𝛽 + 𝜇)(𝛾 + 𝜇 + 𝜇 𝑖 )(𝛼 + 𝛿)𝑖) = 0 (4.5) Dari persamaan (4.5) diperoleh bahwa semua tanda pada setiap suku-sukunya positif sehingga berdasarkan aturan tanda Descartes(Martha, 2022), persamaan (4.4) memiliki akar-akar negatif.…”

Analysis of the Spread of Covid-19 Disease Considering the Vaccination Factor

“…[4] carried out an analysis of the COVID-19 model using the Susceptible-Infected-Recovered (SIR) approach by randomly adding vaccination factors to newborn individuals, thus becoming a Susceptible-Vaccinated-Infected-Recovered (SVIR) model. Furthermore, there is also a model built previously, namely the delay model, by considering vaccination factors, which develops the basic SEIR model [5]. The same basic model is also used in [6]; the model considers vaccination and isolation factors as model parameters.…”