Numerical analysis of distributed order SVIR model by nonstandard finite difference method

Öz

Differential equations are used in mathematical modelling in many disciplines of science. In this study, distributed order differential equations which are more comprehensive are considered instead of the ordinary, partial and fractional order differential equations. SVIR (Susceptible, Vaccinated, Infectious, Recovered) model which is an epidemic model is defined with the help of distributed order differential equation and numerical solution is investigated by (NSFD). The model used in examination of contagious disease exhibits the effect and improvement of vaccination with the term V. The basic idea of the usage of distributed order differential equations is that these kinds of differential equations are the general case of ordinary and partial differential equations in some way and these equations leads to make interpretation with an only equation about different cases with the help of the intensity function included. Numerical solution and analysis of SVIR model are done in the study and later on, stability analysis of the discrete system is presented. In consequence of these studies, it is seen that the usage of the distributed order modelling is possible for this kind of epidemic modelling.

Anahtar Kelimeler

• Distributed order differential equations
• SVIR epidemic model
• vaccination

Proje Numarası

2211-E

Standart olmayan sonlu fark metodu ile dağılımlı mertebeden SVIR modelinin nümerik analizi

Öz

Çoğu bilim dalındaki matematiksel modellemelerde diferansiyel denklemler kullanılmaktadır. Ancak genelde kullanılan adi, kısmi ve kesirli mertebeden diferansiyel denklemlerin kullanımı yerine bu çalışmada daha kapsamlı bir diferansiyel denklem olan dağılımlı (distributed) mertebeden diferansiyel denklem ele alınmıştır. Bu çalışmada dağılımlı mertebeden diferansiyel denklem yardımı ile epidemik model olan SVIR (Susceptible, Vaccinated, Infectious, Recovered) modeli tanımlanmış ve nümerik çözümü standart olmayan sonlu fark metodu (NSFD) ile araştırılmıştır. Bulaşıcı hastalıkların incelenmesinde kullanılan bu model aynı zamanda içinde barındırdığı V terimi ile hastalık evresinde aşılamanın etkisini ve gelişimini ortaya koymaktadır. Dağılımlı mertebeden diferansiyel denklemlerin kullanılmasında ki temel düşünce hem bu tip denklemlerin bir nevi adi ve kesirli diferansiyel denklemlerin genel hali olması hem de içinde tanımlanan yoğunluk fonksiyonu ile farklı durumlar hakkında tek bir denklem ile yorum yapılabilmesindendir. SVIR modelinin nümerik çözümü ve analizi çalışma içerisinde yapılmış ve sonrasında ayrıklaştırılmış sisteme ait kararlılık analizi ifade edilmiştir. Bu çalışmalar neticesinde dağılımlı mertebeden modellemenin bu tip epidemik modellemelerde kullanımının mümkün olduğu görülmüştür.

Anahtar Kelimeler

• Dağılımlı mertebeden diferansiyel denklemler
• SVIR epidemik model
• aşılama

Destekleyen Kurum

Türkiye Bilimsel ve Teknolojik Araştırma Kurumuna (TÜBİTAK)

Proje Numarası

2211-E

Teşekkür

Yazarlardan Mehmet KOCABIYIK, 2211-E Programı ile maddi ve manevi destek veren Türkiye Bilimsel ve Teknolojik Araştırma Kurumuna (TÜBİTAK) teşekkür eder.

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