Farklı İnsidans Oranlarının Etkisi Altında Bir SEIR Epidemiyolojik Modelinin Optimal Kontrolü

Abstract

Bu çalışmada, farklı insidans hızı fonksiyonlarının etkisi altında bir kesirli SEIR salgın modeli için optimal kontrol problemi incelenmiştir. Bu fonksiyon, bir popülasyonun duyarlı ve enfekte bireyleri arasındaki etkileşimi ifade ederek bir salgın hastalığın en gerçekçi biçimde modellenmesinde dikkate değer bir göreve sahiptir. Bu etkileşim, hastalığın pandemiye dönüşüp dönüşmeyeceği konusunda oldukça belirleyicidir. Dolayısıyla bu fonksiyon, salgının durumuna göre farklı şekillerde tanımlanabilir. Bu çalışmada, bilineer ve doymuş insidans fonksiyonlarının etkileri tartışılmaktadır. İncelenen epidemiyolojik model, Caputo kesirli türevlidir. Temel amaç, enfekte hasta sayısını ve duyarlı önleyici bir tedbir mahiyetinde duyarlı bireylere verilen eğitim maaliyetini en aza indirmektir. Bu amaçla öncelikle ele alınan modelin optimallik koşullarına karşılık gelen Euler-Lagrange denklemleri hesaplanır. Daha sonra sağ ve sol kesirli Caputo türevli optimal sistem, kesirli Euler yöntemi ile birleştirilmiş ileri-geri süpürme yöntemi ile sayısal olarak çözülmüştür. Simülasyon sonuçlarına göre duyarlı bireylere verilen eğitimin salgını yavaşlatmada yadsınamaz derecede etkili olduğu görülmektedir.

Keywords

• Optimal kontrol
• Caputo kesirli türev
• SEIR modeli
• İnsidans oranı
• kesirli Euler yöntemi.

Optimal Control for A SEIR Epidemiological Model Under the Effect of Different Incidence Rates

Abstract

In this study, optimal control problem for a fractional SEIR epidemiological model under the effect of bilinear and saturate incidence rate functions is investigated. These rates play an important role in the realistic modeling of an epidemic by describing the interaction between susceptible and infected individuals of a population. This interaction is highly decisive in whether the disease will turn into a pandemic or not. Therefore, these functions can be defined in different forms depending on the course of the epidemic. The model discussed in this study is defined in terms of Caputo. Dimensional compatibility is guaranteed before posing the optimal control problem. The main objective of the proposed optimal control problem is to minimize the number of infected individuals and the cost of education given to susceptible individuals as a preventive measure. Euler-Lagrange equations corresponding to the optimality conditions of the considered model are first determined by Hamiltonian’s formalism. Afterward, the optimal system with right and left fractional Caputo derivatives are solved numerically by the forward-backward sweep method combined with the fractional Euler method. Optimal solutions are interpreted graphically for varying values of the incidence rate coefficients and the fractional parameter. According to the simulation results, it is seen that the education given to susceptible individuals is significantly effective in slowing down the epidemic.

Keywords

• Optimal control
• Hamiltonian formalism
• Caputo fractional derivative
• SEIR model
• Bilinear incidence rate
• Saturated incidence rate
• Forward-backward sweep method
• Fractional Euler method

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