Bifurkasi Mundur dalam Model Matematika Penyebaran Penyakit Tuberkulosis dengan Mempertimbangkan Laju Deteksi dan Pengobatan

Tuberculosis is a disease caused by the bacteria Mycobacterium tuberculosis. This disease can spread bacteria from one individual to another. In this article, we analyzed the spread of Tuberculosis using the SEIR model. The mathematical model is presented in a system of first-order nonlinear ordinary differential equations. This mathematical model also observes the rate of case detection and treatment. This article also discusses the analysis of the equilibrium point, the stability of the equilibrium points of the model that has been formed, and the basic reproduction number (R0). This model shows a backward bifurcation, that is the appearance of an endemic equilibrium point when R0<1, which means that the disease will not necessarily disappear even though R0<1. The numerical solution for this model is obtained using the fifth order Runge-Kutta method

Model Matematika COVID-19 dengan Vaksinasi Dua Tahap, Karantina, dan Pengobatan Mandiri

Penelitian ini mengembangkan model SEIR untuk memodelkan penyebaran COVID-19 dengan menambahkan vaksinasi dua tahap, isolasi mandiri, karantina di rumah sakit, dan pengobatan mandiri. Pembentukan model diawali dengan membuat asumsi dan diagram transfer penyebaran COVID-19 dengan populasi dibagi menjadi sembilan subpopulasi yaitu subpopulasi rentan, subpopulasi vaksinasi dosis 1, subpopulasi vaksinasi dosis 2, subpopulasi laten, subpopulasi terinfeksi, subpopulasi isolasi mandiri, subpopulasi karantina di rumah sakit, subpopulasi pengobatan mandiri, dan subpopulasi removed, kemudian dibentuk sistem persamaan diferensial nonlinear. Dari analisis model diperoleh titik ekuilibrium bebas penyakit, titik ekuilibrium endemik penyakit, dan bilangan reproduksi dasar (R0). Titik ekuilibrium bebas penyakit stabil asimtotik lokal ketika R0<1. Eksistensi titik ekuilbirum endemik terdapat satu atau tiga akar positif jika R0>1 dan terdapat nol atau dua akar positif jika R0<1. Bifurkasi mundur terjadi pada kondisi R0<1 sehingga diperoleh persamaan bifurkasi mundur R0c<R0<1. Simulasi numerik untuk model yang dibuat sesuai dengan analisis yang telah dilakukan. Analisis sensitivitas diperoleh parameter yang berpengaruh signifikan pada penyebaran COVID-19 adalah tingkat kontak dengan individu terinfeksi dan tingkat perpindahan vaksinasi dosis satu

Analisis Bifurkasi Mundur Dan Solusi Numerik Pada Model Penyebaran Penyakit Menular Dengan Kekebalan Parsial

Pada tugas akhir ini dibahas model penyebaran penyakit menular tipe SIS (Susceptible Infected Susceptible) yang terdiri dua tahap. Model penyebaran penyakit ini dianalisis berdasarkan kestabilan titik kesetimbangan, dan bifurkasi dengan satu parameter. Dalam hal ini parameternya adalah bilangan reproduksi dasar atau biasa disebut R0 yang digunakan untuk mengetahui tingkat penyebaran suatu penyakit. Analisa bifurkasi diperlukan untuk mengetahui perubahan stabilitas dan perubahan banyaknya titik tetap akibat perubahan nilai parameter. Selanjutnya dilakukan penyelesaian numerik untuk model dengan menggunakan metode numerik Runge-Kutta orde empat yang disimulasikan dengan menggunakan Matlab. Hasil analisa yang diperoleh yaitu fenomena bifurkasi mundur muncul bergantung pada cakupan vaksinasi dan keefektifan vaksin dan simulasi numerik dari model menunjukkan bahwa diperlukan keefektifan vaksin yang cukup tinggi untuk pemberantasan penyakit secara efektif. ======================================================================================================================== This �nal project discusses SIS type of infectious disease transmission model which consist of two stages. This disease transmission model be analyzed based on the stability of equilibrium point and bifurcation with one parameter. This parameter is basic reproduction number or R0. R0 is used to determine the rate of transmission disease. Bifurcation analysis is needed to know the change of stability and number of �xed point due to value of parameter. Then, we �nd numerical solution for Runge-Kutta numerical method model. This phenomenon of backward bifurcation does not arise depending on vaccination coverage and e�cacy of vaccine. Numerical simulations of the model show that, the use of an imperfect vaccine can lead to e�ective control of the disease if the vaccination coverage and the e�cacy of vaccine are high enoug

Backward bifurcation analysis of epidemiological model with partial immunity

This paper presents a two stage SIS epidemic model in animal population with bovine tuberculosis (BTB) in African buffalo as a guiding example. The proposed model is rigorously analyzed. The analysis reveals that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) coexists with a stable endemic equilibrium (EE) when the associated reproduction number (Rv) is less than unity. It is shown under two special cases of the presented model, that this phenomenon of backward bifurcation does not arise depending on vaccination coverage and efficacy of vaccine. Numerical simulations of the model show that, the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough.http://www.elsevier.com/locate/camwahb201

Backward bifurcation and reinfection in mathematical models of tuberculosis

Mathematical models are widely used for understanding the transmission mechanisms and control of infectious diseases. Numerous infectious diseases such as those caused by bacterial and viral infections do not confer life long immunity after recovering from the first episode. Consequently, they are characterized by partial or complete loss of immunity and subsequent reinfection. This thesis explores the epidemiological implications of loss of immunity using simple and complex mathematical models. First, a simple basic model mimicking transmission mechanisms of tuberculosis (TB) is proposed with the aim of correcting problems that are often repeated by mathematical modellers when determining underlying bifurcation structures. Specifically, the model makes transparent the problems that may arise if one aggregates all the bifurcation parameters when computing backward bifurcation thresholds and structures. The backward bifurcation phenomenon is an important concept for public health and disease management. This is because backward bifurcation signals that disease will not be eliminated even when the basic reproduction number R0 is decreased below unity; rather, for the disease to be eliminated, R0 has to be reduced below another critical threshold. I provide conditions to find the threshold correctly. Secondly, the simple basic TB model is extended to incorporate epidemiological and biological aspects pertinent to TB transmission such as recurrent TB, which is defined as a second episode of TB following successful recovery from a previous episode. I study the conditions for backward bifurcation in this extended model that features recurrent TB. Mathematical techniques based on the center manifold approach, are used to derive an exact backward bifurcation threshold. Furthermore, both analytical and numerical findings reveal that recurrent TB is capable of inducing a new and rare hysteresis effect where TB will persist when the basic reproduction number is below unity even though there is no backward bifurcation. Moreover, when the reinfection pathway among latently infected individuals is switched off, leaving only recurrent TB, the model analysis indicates that recurrent TB can independently induce a backward bifurcation. However, this will only occur if recurrent TB transmission exceeds a certain threshold. Although this threshold seems to be relatively high when realistic parameters are used, it falls within the recent range estimated in the relevant literature. The second TB model is extended by dividing the latent compartment into two: fast (early latent) and slow (late latent) latent compartments, to enhance realism. Individuals in both early and late compartments are subjected to treatment. The proposed TB model is used to investigate how heterogeneity in host susceptibility influences the effectiveness of treatment. It is found that making the assumption that individuals treated with preventive therapy and recovered individuals (previously treated for active TB) acquire equal levels of protection after initial infection, and are therefore reinfected at the same rate, may obscure dynamics that are imperative when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated from active TB provides important epidemiological insights. That is, the reinfection parameter accounting for the relative rate of reinfection of the cohort treated with preventive therapy is the one that plays the key role in generating qualitative changes in TB dynamics. In contrast, the parameter accounting for the risk of reinfection among recovered individuals (previously treated for active TB) does not play a significant role. The study shows that preventive treatment during early latency is always beneficial regardless of the level of susceptibility to reinfection. And if patients have greater immunity following treatment for late latent infection, then treatment is again beneficial. However, if susceptibility increases following treatment for late latent infection, the effect of treatment depends on the epidemiological setting: (a) for (very) low burden settings, the effect on reactivation predominates and burden declines; (b) for high burden settings, the effect on reinfection predominates and burden increases. This is mostly observed between the two reinfection thresholds, RT2 and RT1, respectively associated with individuals being treated with preventive therapy and individuals with untreated late latent TB infection. Finally, a mathematical model that examines how heroin addiction spreads in society is formulated. The model has many commonalities with the TB model. The global stability properties of the proposed model are analysed using both the Lyapunov direct method and the geometric approach by Li and Muldowney. It is shown that even for a four dimensional model, the use of two well known nonlinear stability techniques becomes nontrivial. When all the parameters of the model are accounted for, it is difficult if not impossible, to design a Lyapunov function. Here I apply the geometric approach to establish a global condition that accounts for all model parameters. If the condition is satisfied, then heroin persistence within the community is globally stable. However, if the global condition is not satisfied heroin users can oscillate periodically in number. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users, are the key mechanisms fuelling heroin epidemic proliferation. However, in the long term, relapse of heroin users undergoing treatment back to a heroin using career, has the most significant impact