Tinjauan Matematis Waktu Tundaan pada Model Covid-19 dengan Vaksinansi

Fitriana Yuli Saptaningtyas; Ahmadi Ahmadi

doi:10.21831/pythagoras.v17i1.49372

Tinjauan Matematis Waktu Tundaan pada Model Covid-19 dengan Vaksinansi

Fitriana Yuli Saptaningtyas, Program Studi Matematika, Universitas Negeri Yogyakarta, Indonesia
Ahmadi Ahmadi, Program Studi Matematika, Universitas Negeri Yogyakarta, Indonesia

10.21831/pythagoras.v17i1.49372

Abstract

Artikel ini membahas pemodelan matematika penyebaran Covid-19 dengan vaksinasi yang melibatkan waktu tundaan. Waktu tundaan merepresentasikan waktu individu mengalami penurunan kekebalan tubuh sehingga kembali rentan terhadap Covid-19 setelah sembuh. Kita tahu bahwa individu yang dinyatakan sembuh dari Covid-19 dapat terinfeksi kembali. Penelitian ini menganalisa titik ekuilibirum beserta kestabilannya, menentukan bilangan reproduksi dasar untuk melihat penyebaran penyakit, menentukan jenis bifurkasi yang muncul yang diakibatkan oleh waktu tundaan, dan melakukan simulasi numerik untuk melihat perilaku penyebaran penyakit. Di samping itu juga dilakukan kajian analitik untuk menentukan bilangan reproduksi dasar dan analisa perbandingan kestabilan lokal untuk model tanpa waktu tundaan dan dengan waktu tundaan. Hasil dari analisis terhadap model didapat dua titik ekuilibrium, yakni satu bebas penyakit dan satu endemik. Pada model dengan waktu tundaan diperoleh bahwa waktu tundaan tertentu dapat menyebabkan munculnya solusi periodik artinya akan terjadi fluktuasi banyaknya individu yang terinveksi pada periode waktu tertentu. Simulasi numerik dengan mengubah ubah parameter waktu tundaan dan tingkat vaksinasi menunjukkan pada kondisi endemik model dengan waktu tundaan akan menyebabkan lebih banyak individu yang terinveksi dari pada model tanpa waktu tundaan.

Mathematical Overview of Time Delay on Covid 19 Models with Vaccination

Abstract

This article discusses the mathematical modeling of the spread of Covid-19 with vaccination which involves a time delay. The time delay is represented when an individual experiences a decreased immune system so that he is declared susceptible to Covid-19 after recovering. Because we know that individuals who are declared cured of Covid-19 can be reinfected, this study analyzes the equilibrium point of the model and its stability, determines the primary reproduction number to see the spread of the disease, determines the type of bifurcation that appears due to the time delay, and performs numerical simulations. To see the behavior of the spread of the disease. In addition, analytical studies were carried out to determine the primary reproduction number and local stability comparison analysis for models without time delay and with time delay. The model analysis results obtained two equilibrium points, one free of disease and one endemic. In the time-delayed model, the value of the time-delay parameter is obtained, which causes the emergence of a periodic solution, meaning that there will be fluctuations in the number of individuals infected in a certain period. Numerical simulations by changing the time delay parameters and vaccination rates show that in endemic conditions, models with time delays will cause more individuals to be infected than models that do not use time delays.

Keywords

Covid-19; vaksinasi; waktu tundaan

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DOI: https://doi.org/10.21831/pythagoras.v17i1.49372