Various reports show how the level of human concern can affect the speed of the spread of disease. The high level of awareness of the dangers of disease, including vaccination, can make humans try to protect themselves from the possibility of infection. In this undergraduate thesis, the Susceptible-Vaccinated-Infected-Recovered (SVIR) model is used to understand how to control the disease with vaccination interventions and consider the level of awareness as an independent variable. The population of susceptible in the model is divided into groups of susceptible individuals aware of infections and susceptible individuals unaware of infections. It is assumed that only susceptible individuals aware of infections could get vaccinated. Vaccination is assumed couldn't be able to protect the individual completely from diseases. Analytical studies of disease-free equilibrium points, endemic equilibrium points, and basic reproduction number (R_0) are carried out in this undergraduate thesis to understand the long-term dynamics of the established model. It was found that disease-free equilibrium when the level of awareness of the diseases is constant would be stable if R_0 < 1, and otherwise. Some numerical simulations are given to support the results of analytic studies and provide interpretation. From all the analytical results that have been discussed, it could be said that vaccination is one of the effective ways of minimizing the spread of the disease. However, with the level of disease awareness, the intensity of vaccinations needed will not be as massive as when there is no people awareness.
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Pada tesis ini, dikonstruksi sebuah model matematika penyebaran TB yang melibatkan relapse, reinfeksi dan kegagalan treatment dan memperkenalkan pula efek dari vaksin jenis terbaru M72/AS01E untuk pencegahan terjadinya relapse. Model yang dibentuk menggunakan persamaan diferensial biasa orde satu. Proses nondimensi dilakukan terhadap model untuk menyederhanakan masalah tanpa kehilangan esensi utama dari tujuan tesis ini. Model yang telah dibentuk dilakukan kajian analitik. Analisa yang dilakukan antara lain adalah eksistensi dan kestabilan titik keseimbangan dan basic reproduction number. Adapun analisis kestabilan dari titik keseimbangan dilakukan menggunakan pendekatan Van den Driessche and Watmough untuk titik keseimbangan bebas penyakit serta Teori Center Manifold oleh Castilo Song disekitar R0=1 untuk titik keseimbangan endemik penyakit. Analisa kestabilan dengan Teorema Center Manifold juga menghasilkan bahwa model yang telah terbentuk mampu menghasilkan bifurkasi mundur, bifurkasi maju dan bifurkasi maju+hysteresis. Kajian yang dilakukan menghasilkan bahwa koefisien saturasi sangat berperan penting dalam terjadinya fenomena bifurkasi dalam model. Lebih jauh, fenomena relapse, reinfeksi dan kegagalan treatment memegang peran penting terhadap peningkatan nilai R0. Namun, hal ini dapat diminimalisir dengan keberadaan vaksin M72/AS01E.
In this thesis, a mathematical model of TB spread was constructed involving relapse, reinfection, and failure of treatment. It also introduces the effect of the latest vaccine type M72/AS01E to prevent the occurrence of relapse. The model was formed using firstorder ordinary differential equations. The non-dimensionalization process is carried out on the model to simplify the problem without losing the main essence of the purpose of this thesis. The model that has been formed is an analytical study. The analysis carried out includes the existence and stability of the balance point and the basic reproduction number. The stability analysis of the equilibrium point was carried out using the Van den Driessche and Watmough approach for the disease-free equilibrium point and Castilo Song’s Theory Center around R0=1 for the endemic balance point of the disease. Stability analysis with the Center Manifold Theorem also shows that the established model can produce backward bifurcation, forward bifurcation, and forward + hysteresis bifurcation. The study conducted resulted that the saturation coefficient plays an essential role in the occurrence of the bifurcation phenomenon in the model. Furthermore, the phenomenon of relapse, reinfection, and failure of treatment plays an essential role in increasing the value of R0. However, this can be minimized by the existence of this M72/AS01E vaccine.
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