# Optimal control for SIR Model with The Influence of Vaccination, Quarantine and Immigration factor

## Authors

• Susi Agustianingsih Universitas Jenderal Soedirman
• Rina Reorita Universitas Jenderal Soedirman
• Renny Renny Universitas Jenderal Soedirman

## Keywords:

Quarantine, optimal control, SIR model, Pontryagin maximum principle, vaccination

## Abstract

The SIR model is one of the mathematical model which describes the characteristic of the spread of infectious disease in differential equation form by dividing the human populations into three groups. There are individual susceptible group, individual infective group, and individual recovered group. This model involves vaccination, quarantine, and immigration factors. Vaccination and quarantine must be given as much as it needs, so a control is required to minimize infection of disease and the number of individual infective with a minimum costs. In this research, optimal control of SIR model with vaccination, quarantine, and immigration factor is solved by using Pontryagin maximum principle and numerically simulated by using Runge-Kutta method. Numerical simulation results show optimal control of treatment, citizen of vaccination, immigrant of vaccination, and quarantine will accelerate the decline of infected number with the minimum cost, compared with the optimal control of SIR model without quarantine factor.

## Author Biographies

### Susi Agustianingsih, Universitas Jenderal Soedirman

Jurusan Matematika

### Rina Reorita, Universitas Jenderal Soedirman

Jurusan Matematika

### Renny Renny, Universitas Jenderal Soedirman

Jurusan Matematika

## References

Anggriani, N., Supriatna, A., Subartini, B., dan Wulantini, R. 2015. Kontrol Optimum pada Model Epidemik SIR dengan Pengaruh Vaksinasi dan Faktor Imigrasi. Jurnal Matematika Integratif. 11(2): 111-118.

Murray, J. D. 2002. Mathematical Biology: I An Introduction, Third Edition. USA: Springer.

Salmani, M. 1995. A Model for Disease Transmission in a Patchy Environment. Thesis. Department of Mathematics and Statistics, University of Isfahan, Iran.

Sari, I., dan Tasman, H. 2014. Model Epidemik SIR untuk Penyakit yang Menular secara Horizontal dan Vertikal. Makalah. Dalam: Prosiding Konferensi Nasional Matematika XVII di ITS, 11-14 Juni.

2020-04-28

## How to Cite

Agustianingsih, S., Reorita, R., & Renny, R. (2020). Optimal control for SIR Model with The Influence of Vaccination, Quarantine and Immigration factor. Jurnal Matematika, Statistika Dan Komputasi, 16(3), 311-324. https://doi.org/10.20956/jmsk.v16i3.6942

## Section

Research Articles