Stability Analysis and Optimal Control of Mathematical Model of Thypoid Fever Spread
Abstract
Typhoid fever is an endemic disease caused by infection with Salmonella Typhi. The transmission of typhoid fever is through food and drink contaminated with Salmonella Typhi bacteria, which is excreted through the feces or urine of an infected person. The problem of typhoid fever is increasingly complex because of the increase in carrier cases, making it difficult for treatment and prevention efforts. This study develops a mathematical model for the control of typhoid fever, which consists of two equilibrium points, namely endemic and non-endemic equilibrium points. The endemic and non-endemic equilibrium point is asymptotically stable if it satisfies the condition given by the Routh-Hurwitz criterion. Optimal control theory is applied to the mathematical model by providing control through health campaigns, screening, and treatment to minimize the number of asymptomatic individuals, symptomatic individuals, and chronic carriers. The Pontryagin Minimum principle is used to determine the optimal control form. Numerical simulations are performed using the Forward-Backward Sweep Runge-Kutta method of order 4. The simulation results indicate a decrease in each infected subpopulation after applying optimal control for ten months. It is found that control in health campaigns has a more significant impact than control in screening and treatment in decreasing the number of asymptomatic and symptomatic individuals. The control of treatment effectively reduces infected individuals with symptoms of becoming chronic carriers. In conclusion, the most effective strategy in controlling the spread of typhoid fever is to simultaneously apply controls in the form of health campaigns, screening, and treatment.
Keywords: health campaign; screening; treatment; optimal control; Pontryagin minimum principle; forward-backward sweep.
Abstrak
Demam tifoid merupakan penyakit endemik yang disebabkan oleh infeksi bakteri Salmonella Typhi. Proses penularan demam tifoid melalui makanan dan minuman yang telah terkontaminasi bakteri Salmonella Typhi yang dikeluarkan melalui tinja maupun urin dari orang yang telah terinfeksi. Permasalahan tentang demam tifoid semakin kompleks karena meningkatnya kasus - kasus carrier, sehingga menyulitkan upaya pengobatan dan pencegahan. Model matematika yang dikembangkan memiliki dua titik kesetimbangan yaitu titik setimbang nonendemik dan titik setimbang endemik. Titik setimbang nonendemik dan endemik akan stabil asimtotik jika memenuhi kondisi yang diberikan oleh aturan Routh-Hurwitz. Teori kontrol optimal diterapkan pada model matematika dengan pemberian kontrol berupa kampanye kesehatan, screening dan pengobatan untuk meminimumkan jumlah individu asymptomatic, individu symptomatic dan carrier chronic. Penentuan bentuk kontrol optimal menggunakan prinsip Minimum Pontryagin. Simulasi numerik dilakukan dengan menggunakan metode Forward-Backward Sweep Runge-Kutta orde 4. Berdasarkan hasil simulasi, terjadi penurunan disetiap subpopulasi terinfeksi setelah penerapan kontrol optimal selama 10 bulan. Kontrol berupa kampanye kesehatan memiliki pengaruh yang besar dibandingkan kontrol berupa screening dan pengobatan dalam menekan meningkatnya individu asymptomatic dan individu symptomatic. Penerapan kontrol berupa pengobatan sangat efektif dalam menekan individu terinfeksi dengan gejala menjadi individu carrier chronic.
Kata Kunci: kampanye kesehatan; screening; pengobatan; kontrol optimal; prinsip minimum Pontryagin; forward-backward sweep.
2020MSC: 00A71, 92B05
Keywords
References
OMS, "Données épidémiologiques sur la typhoïde," pp. 153–172, 2018.
WHO, "Typhoid and other invasive salmonellosis," 2018. [Online]. Available: https://www.who.int/publications/m/item/vaccine-preventable-diseases-surveillance-standards-typhoid. [Accessed 03 May 2022].
K. A. A. Prayoga and N. N. D. Fatmawati, "Identifikasi Salmonella spp pada feses penjamah makanan di rumah potong ayam RJ dengan metode kultur," Intisari Sains Medis, vol. 9, no. 3, pp. 1-5, 2018.
W. Mulu et al., "Seroconfirmed Typhoid Fever and Knowledge, Attitude, and Practices among Febrile Patients Attending at Injibara General Hospital, Northwest Ethiopia," BioMed Res. Int., vol. 2, no. Article ID 8887266, pp. 1-8, 2021. https://doi.org/10.1155/2021/8887266.
J. S. Gauld, H. Hu, D. J. Klein and M. M. Levine, "Typhoid fever in Santiago, Chile: Insights from a mathematical model utilizing venerable archived data from a successful disease control program," PLoS Negl. Trop. Dis., vol. 12, no. 9. p. e0006759, 2018. doi: 10.1371/journal.pntd.0006759.
S. T. Tresna, Subiyanto and S. Supian, "Mathematical Models for Typhoid Disease Transmission: A Systematic Literature Review," Mathematics, vol. 10, no. 14, p. 2506, 2022, doi: 10.3390/math10142506.
J. S. Gunn, J. M. Marshall, S. Baker, S. Dongol, R. C. Charles and E. T. Ryan, "Salmonella chronic carriage: epidemiology, diagnosis, and gallbladder persistence," Trends Microbiol, vol. 22, no. 11, p. 648–655, 2014.
V. Gupta, M. Kaur, P. Datta and J. Chander, "Chronic urinary carrier state due to Salmonella Typhi causing urinary tract infection in an immunocompetent healthy woman," Trop. Doct., vol. 48, no. 3, p. 236–238, 2018.
S. C. Olsen, P. Boggiatto, J. Wilson-Welder, P. Nol, J. Rhyan and N. Srirangathan, "Immunogenicity and Efficacy of a Rough Brucella suis Vaccine Delivered Orally or Parenterally to Feral Swine," International Journal of Vaccine Research, vol. 2, no. 1, p. 6, 2017.
C. H. Watson and W. J. Edmunds, "A review of typhoid fever transmission dynamic models and economic evaluations of vaccination," Vaccine, vol. 33, p. C42–C54, 2015, doi: 10.1016/j.vaccine.2015.04.013.
S. Toaha, Pemodelan Matematika dalam Dinamika Populasi, Makassar: Dua Satu Pers, 2013.
K. Kasbawati, "Kontrol Optimal Upaya Pencegahan Infeksi Virus Flu Burung H5N1 dalam Populasi Burung dan Manusia," J. Mat. Stat. Dan Komputasi, vol. 8, no. 1, pp. 12-24, 2011.
O. J. Peter, A. F. Adebisi, M. O. Ajisope, F. O. Ajibade, A. I. Abioye and F. A. Oguntolu, "Global Stability Analysis of Typhoid Fever Model," Adv. Syst. Sci. Appl., vol. 20, no. 2, p. Art. no 2, 2020, doi: 10.25728/assa.2020.20.2.792.
H. O. Nyaberi and J. S. Musaili, "Mathematical modeling of the impact of treatment on the dynamics of typhoid," J. Egypt. Math. Soc., vol. 29, no. 1, pp. 1-11, 2021.
G. T. Tilahun, O. D. Makinde and D. Malonza, "Modelling and optimal control of typhoid fever disease with cost-effective strategies," Computational and Mathematical Methods in Medicine, vol. 1, pp. 1-16, 2017.
J. Mushanyu, F. Nyabadza, G. Muchatibaya, P. Mafuta and G. Nhawu, "Assessing the potential impact of limited public health resources on the spread and control of typhoid," J. Math. Biol., vol. 77, no. 3, p. 647–670, 2018.
S. Mushayabasa, C. P. Bhunu and E. T. Ngarakana-Gwasira, "Assessing the impact of drug resistance on the transmission dynamics of Typhoid fever," Comput. Biol. J., vol. 2013, 2013.
C. B. Acosta-Alonzo, I. V. Erovenko, A. Lancaster, H. Oh, J. Rychtář and D. Taylor, "High endemic levels of typhoid fever in rural areas of Ghana may stem from optimal voluntary vaccination behaviour," Proc. R. Soc. A, vol. 476, no. 2241, p. p. 20200354, 2020.
A. S. Shaikh and K. S. Nisar, "Transmission dynamics of fractional order Typhoid fever model using Caputo–Fabrizio operator," Chaos Solitons Fractals, vol. 128, p. 355–365, Nov. 2019, doi: 10.1016/j.chaos.2019.08.012.
H. Abboubakar and R. Racke, "Mathematical modeling, forecasting, and optimal control of typhoid fever transmission dynamics," Chaos Solitons Fractals, vol. 149, p. 111074, 2021.
T. A. Ayoola, H. O. Edogbanya, O. J. Peter, F. A. Oguntolu, K. Oshinubi and M. L. Olaosebikan, "Modelling and optimal control analysis of typhoid fever," J. Math. Comput. Sci., vol. 11, no. 6, pp. 6666-6682, 2021.
J. T. Workman and S. Lenhart, Optimal Control Applied to Biological Models, in Optimal Control Applied to Biological Models, London: Taylor & Francis Group, 2007.
DOI: 10.15408/inprime.v5i1.27205
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