The Impact of Caputo Fractional Derivative on Mathematical Modeling of Tuberculosis Disease

Öz

Memory effect, one of the most important advantages of fractional derivative and integral operators, is a prominent characteristic of a disease. Mycobacterium tuberculosis is one of such memory-dependent precarious disease that affects the lungs. Therefore, we investigate this serious public health infection by means of the Caputo operator having memory. Firstly, some fundamental features of the proposed model such as the positiveness of solutions, the disease-free equilibrium, and endemic equilibrium points are presented. Then, the basic reproduction number to characterize the disease model under consideration is calculated. In addition, stability analysis is performed by using these basic calculations, and comments are made on the course of the tuberculosis infection. On the other hand, we numerical simulations in order to show the effect of non-integer order on the mycobacterium tuberculosis transmission are carried out. Lastly, comparison analysis between the classical and fractional mathematical models is supported by various graphs. The results show that fractional order can greatly affect the course of the tuberculosis transmission

Anahtar Kelimeler

• Mathematical biology
• Caputo operator
• Tuberculosis transmission
• Fractional derivative
• Mycobacterium tuberculosis.

Tüberküloz Hastalığının Matematiksel Modellemesine Caputo Kesirli Türevinin Etkisi

Öz

Kesirli türev ve integral operatörlerinin en önemli avantajlarından biri olan hafıza etkisi, bir hastalığın belirgin bir özelliğidir. Mikobakteriyum tüberküloz, akciğerleri etkileyen hafızaya bağlı bu tür tehlikeli hastalıklardan biridir. Bu nedenle, bu ciddi halk sağlığı enfeksiyonu hafıza etkisine sahip Caputo operatörü aracılığıyla araştırılmıştır. İlk olarak, çözümlerin pozitifliği, hastalıksız denge ve endemik denge noktaları gibi önerilen modelin bazı temel özellikleri sunulmuştur. Daha sonra, söz konusu hastalık modelini karakterize etmek için temel üreme sayısı hesaplanmıştır. Ayrıca, bu temel hesaplamalar kullanılarak kararlılık analizi gerçekleştirilmiştir ve tüberküloz enfeksiyonunun seyri hakkında yorumlar yapılmıştır. Öte yandan, tam sayı olmayan mertebenin mikobakteriyum tüberküloz yayılımı üzerindeki etkisini göstermek için sayısal simülasyonlar gerçekleştirilmiştir. Son olarak, klasik ve kesirli matematiksel modeller arasındaki karşılaştırma analizi çeşitli grafiklerle desteklenmiştir. Sonuçlar, kesirli sıranın tüberküloz yayılmasını seyrini büyük ölçüde etkileyebileceğini ve bu küresel sağlık sorunu hakkında daha spesifik tahminlere izin verdiğini göstermektedir.

Anahtar Kelimeler

• Matematiksel biyoloji
• Caputo operatörü
• Tüberküloz iletimi
• Kesirli türev
• mikobakteriyum tüberküloz

Kaynakça

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