A REAL TURKEY DATA APPLICATION TO THE SEIR MODEL FOR TUBERCULOSIS DISEASE

Öz

The SEIR mathematical model in the literature is applied to analyze the tuberculosis transmission dynamics in Turkey. It consists of four classes: susceptible (S), exposed (E), infected (I) and recovered (R). The equilibrium points and the basic reproduction number of the SEIR model are given. The structural identifiability analysis of the model is conducted in the Maple program and also with the differential algebra method. As a result of the structural identifiability analysis, all parameters in the model are found as locally structurally identifiable, but aren’t found to be globally identifiable. In addition, if an additional parameter is known, all other parameters are obtained to be globally identifiable. The number of tuberculosis cases in Turkey from 2005 to 2016 is used to solve the parameter estimation problem. The practical identifiability of the model is examined using Monte Carlo simulations. Sensitivity analysis of the model is also studied. Matlab program is used to estimate the model parameters and perform practical identifiability analysis. It is concluded that three parameters μ,β and γ, except for the parameter ε, are all practically identifiable. As a result, the basic reproduction number calculated with the obtained values shows that tuberculosis disease will continue in Turkey.

Anahtar Kelimeler

• Parameter estimation
• Tuberculosis
• Epidemic model
• Structural identifiability
• Practical identifiability
• Sensitivity analysis

TÜBERKÜLOZ HASTALIĞI İÇİN SEIR MODELİNE GERÇEK TÜRKİYE VERİ UYGULAMASI

Öz

Türkiye'de tüberküloz iletim dinamiklerini belirlemek için literatürdeki SEIR matematiksel model uygulanmıştır. SEIR matematiksel model, duyarlı (S), maruz kalmış (E), enfekte (I) ve iyileşmiş (R) olmak üzere dört sınıftan oluşmaktadır. SEIR matematiksel modelin denge noktaları ve temel üreme sayısı verilmiştir. Modelin yapısal tanımlanabilirlik analizi Maple programında ve ayrıca diferansiyel cebir yöntemi ile belirlenmiştir. Yapısal tanımlanabilirlik analizi sonucunda, modeldeki tüm parametreler local yapısal tanımlanabilir olarak bulunmuş, fakat, global tanımlanabilir bulunmamıştır. Ek olarak, ek bir parametrenin bilinmesi durumunda tüm diğer parametreler global tanımlanabilir olarak elde edilmiştir. Parametre tahmin probleminin çözümü için Türkiye’de 2005’den 2016 yılına kadar tüberküloz vaka sayıları kullanılmıştır. Modelin pratik tanımlanabilirliği Monte Carlo simülasyonları kullanılarak incelenmiştir. Modelin duyarlılık analizi de incelenmiştir. Modelin parametre tahmini ve pratik tanımlanabilirlik analizi için Matlab programı kullanılmıştır. ε parametresi hariç μ, β ve γ parametrelerinin hepsi pratik tanımlanabilir olduğu sonucuna ulaşılmıştır. Sonuç olarak, elde edilen değerlerle hesaplanan temel üreme sayısı Türkiye'de tüberküloz hastalığının devam edeceğini göstermektedir.

Anahtar Kelimeler

• Parametre belirleme
• Tüberküloz
• Epidemik model
• Yapısal tanımlanabilirlik
• Pratik tanımlanabilirlik
• Duyarlılık analizi

Kaynakça

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