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. AbstrakDemam 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