A Mathematical Modeling Approach of Drug-Sensitive Tuberculosis (DS-TB) and Drug-Resistant Tuberculosis (DR-TB) Transmission with Vaccination and Isolation

Tuberkulosis (TB) masih menjadi masalah kesehatan yang serius. Berdasarkan tingkat kepekaan bakteri terhadap obat anti-tuberkulosis, TB dibedakan menjadi TB sensitif obat (TB-SO) dan TB resisten obat (TB- RO), di mana keberadaan TB-RO menjadi tantangan pengendalian karena penanganannya lebih kompleks serta berpotensi mempertahankan penularan dalam suatu populasi. Model matematika dapat digunakan untuk mengetahui dinamika penyebaran TB-SO dan TB-RO. Penelitian ini menganalisis model 𝑆𝑉𝐸_𝑠𝐸_𝑟𝐼_𝑠𝐼_𝑟𝑄𝑅 yang di mana kebaruan penelitian ini terletak pada pengembangan model dua strain TB yang secara simultan memuat pemisahan fase laten spesifik strain, perbedaan laju penularan antar strain, dan intervensi isolasi khusus TB-RO dalam satu kerangka. Analisis yang dilakukan di antaranya menentukan titik kesetimbangan, bilangan reproduksi dasar dengan next generation matrix, analisis kestabilan, dan simulasi numerik. Diperoleh tiga titik kesetimbangan: bebas penyakit, monoeksistensi TB-RO, dan koeksistensi kedua strain. Analisis kestabilan di tiap titik kesetimbangan dilakukan dengan bantuan nilai 𝑅₀. Analisa kestabilan dilakukan melalui syarat nilai parameter dari laju penularan TB-SO (𝛽𝑠) dan laju penularan TB-RO (𝛽_𝑟). Analisis sensitivitas menunjukkan bahwa penurunan 𝛽_𝑠 dan 𝛽_𝑟 berdampak besar menekan penyebaran TB-SO dan TB-RO, peningkatan 𝜎 juga berdampak besar dalam menekan penularan TB-RO melalui isolasi kasus TB-RO.

Kata Kunci: model matematika; analisis kestabilan lokal; tuberkulosis sensitif obat; tuberkulosis resisten obat.

Tuberculosis (TB) remains a serious public health problem. Based on bacterial susceptibility to anti- tuberculosis drugs, TB is classified into drug-susceptible TB (DS-TB) and drug-resistant TB (DR-TB), where the presence of DR-TB poses a challenge for control because its management is more complex and it has the potential to sustain transmission within a population. Mathematical models can be used to understand the transmission dynamics of DS-TB and DR-TB. This study analyzes the 𝑆𝑉𝐸_𝑠𝐸_𝑟𝐼_𝑠𝐼_𝑟𝑄𝑅 model The novelty of this research lies in the development of a two-strain TB model that simultaneously incorporates strain-specific latent phase separation, differences in transmission rates between strains, and TB-RO-specific isolation interventions within a single framework. The analysis include determining equilibrium points, the reproduction number using next generation matrix, stability analysis, and numerical simulations. Three equilibrium points are obtained: disease-free, TB-RO mono-existence, and coexistence of both strains. Stability analysis at each equilibrium point is carried out using the value of 𝑅₀ . The analysis is performed through parameter conditions related to the transmission rate of DS-TB (𝛽_𝑠) and the transmission rate of DR-TB (𝛽_𝑟). Sensitivity analysis shows that reducing 𝛽_𝑠 and 𝛽_𝑟 has a major impact on suppressing the transmission of DS-TB and DR-TB. In addition, increasing 𝜎 substantially reduces DR-TB transmission by enhancing the isolation of DR-TB cases.

Keywords: mathematical model; local stability analysis; drug-sensitive tuberculosis; drug-resistant tuberculosis.