Optimal Control of Mathematical Model of Diphtheria Spreading

Putri Amalia(1*), Syamsuddin Toaha(2),

(1) Department Mathematics, FMIPA Universitas Hasanuddin, Indonesia
(2) Department Mathematics, FMIPA Universitas Hasanuddin, Indonesia
(*) Corresponding Author




DOI: https://doi.org/10.26858/jdm.v10i2.35776

Abstract


This article examines optimal control model for the spread of diphtheria disease. Diphtheria is an infectious disease caused by the bacterium Corynebacterium diphtheriae. This model is divided into six compartments, namely population of susceptibles (𝑆), population of latent (L), population of infected with symptoms (Is), population of infected without symptoms (Ia), population of recovered with full immunity (Rf) and population of recovered with partial immunity (Rp). Two optimal controls are applied in the model, namely vaccination and treatment. The problem of optimal control is solved by using Pontryagin's minimal principle, which consists in solving a set of necessary conditions that must be satisfied by the optimal control and its associated state. The numerical method used to solve the optimal control problem is the forward-backward sweep method. Based on the results of numerical simulations, both controls should be administered in large numbers and continuously since the beginning of observation in order to reduce the number of diphtheria infected population and to control the spread of diphtheria.

Keywords


Mathematical model of diphtheria spread, optimal control problem, state, forward-backward sweep.

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References


Cahyono E., 2013. Pemodelan Matematika, Edisi Pertama. Graham Ilmu., Yogyakarta.

WHO, 2020. Immunization Diphtheria. https://www.who.int/immunization/monitoringsurveillance/burden/diphtheria/en/ [Accessed 22 Februari 2021].

Zakikhany, K. and Efstratiou A., 2012. Diphtheria in Europe: Current Problems and Challenges. Future Microbiology, Vol 7 No. 5, pages: 595–607. Doi: 10.2217/fmb.12.24.

U.S. Department of Health and Human Services, 2020. Diphtheria Vaccination. United State of America. https://www.cdc.gov/vaccines/vpd/diphtheria/index.html [Accessed 10 Juni 2022].

Gower, C., Lacy, J., Oeser, C., O’Boyle, S., Amirthalingam, G., Fry, N. et al., 2022. Public Health Control and Management of Diphtheria in England 2022 Guidelines. UK Health Security Agency. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1083946/Diphtheria-guidelines-2022-v16.1.pdf [Accessed 4 Juli 2022].

Kementerian Kesehatan RI. 2017. Buku Pedoman Pencegahan dan Pengendalian Difteri Edisi Pertama. Jakarta, Direktorat Surveilans dan Karantina Kesehatan;Direktorat Pencegahan dan Pengendalian Penyakit. https://sehatnegeriku.kemkes.go.id/wp-content/uploads/2018/01/buku-pedoman-pencegahan-dan-penanggulangan-difteri.pdf [Accessed 22 Februari 2021].

Kementerian Kesehatan RI, 2018. Pemerintah Optimis KLB Difteri Bisa Teratasi. https://www.kemkes.go.id/article/view/18011500004/pemerintah-optimis-klb-difteri-bisa-teratasi.html#:~:text=Situasi%20KLB%20Difteri,dengan%20kematian%20sebanyak%2044%20kasus. [Accessed 22 Februari 2021 ]

Kementerian Kesehatan RI, 2020. Buletin Surveilains PD3I dan Imunisasi Edisi 2: Tetap Terlindungi di Masa Pandemi Covid-19. Jakarta. Kementerian Kesehatan Republik Indonesia. https://www.who.int/docs/default-source/searo/indonesia/sit-rep/buletin-surveilans-pd3i-dan-imunisasi-edisi-3---okt-2020.pdf?Status=Master&sfvrsn=8a2c9dea_5 [Accessed 22 Februari 2021]

Islam, Z., Ahmed, S., Rahman, M.M., Karim, M.F. and Amin, M.R., 2022. Global Stability Analysis and Parameter Estimation for a Diphtheria Model: A Case Study of an Epidemic in Rohingya Refugee Camp in Bangladesh. Hindawi, Computational and Mathematical Methods in Medicine, Volume 2022. Doi: 10.1155/2022/6545179.

Sornbundit, K., Triampo, W. dan Modchang, C., 2017. Mathematical Modeling of Diphtheria Transmission in Thailand.Bangkok. Elsevier, Computers in Biology and Medicine, Volume 87, Pages 162-168. Doi: 10.1016/j.compbiomed.2017.05.031.

Puspita, G., Kharis, M. dan Supriyono. (2017). Pemodelan Matematika pada Penyebaran Penyakit Difteri dengan Pengaruh Karantina dan Vaksinasi. Unnes Journal ofMathematics, Vol 6 No 1 (2017). doi.org/10.15294/ujm.v6i1.11867.

Hethcote, H., Ma, Z. and Liao, S.,2002 . Effects of Quarrantine in Six Endemic Models for Infectious Disease. Elsevier, Mathematical Biosciences, Volume 180, Issues 1–2, pages 141-160. Doi: 10.1016/S0025-5564(02)00111-6.

Suryani, I. and Yuenita, M.E., 2016. Analisis Kestabilan Model MSEIR Penyebaran Penyakit Difteri dengan Saturated Incidence Rate. Jurnal Sains Matematika dan Statistika, Vol 2, No 1 (2016). Doi: 10.24014/jsms.v2i1.3104.

Cheuvart, B., Burgess, M., Zepp, F., Mertsola, J., Wolter, J. and Schuerman, L., 2004. Anti-Diphtheria Antibody Seroprotection Rates are Similar 10 Years After Vaccination with Dtpa or Dtpa Using A Mathematical Model; Vaccine 23 (2004), pages:336–342. Doi: 10.1016/j.vaccine.2004.06.012.

Aryani, I. and Widyaningsih, P., 2020. Model Susceptible Vaccinated Infected Recovered (SVIR) dan Penerapannya pada Penyakit Difteri di Indonesia. PRISMA 3 (2020), pages:173-178.

Liu X., Takeuchi Y., Iwami S. (2008). SVIR Epidemic Models with Vaccination Strategies. Elsevier, Journal Theoretical Biology, Volume 253, Issue 1, pages 1-11.

Lenhart, S. and Workman, J.T., 2007. Optimal Control Applied to Biological Models. Edisi pertama. Chapman and Hall/CRC, London.


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