Mathematical Modeling of Measles Outbreak with Two Patches

Abstract

In this study, single-patch and two-patch mathematical models are developed to investigate the progression of measles. First, the parameters of the single-patch model are estimated based on available data. The simulation results for the single-patch model show good agreement with real data. Additionally, the parameter values are used to construct a two-patch model that incorporates mutual migration or travel of both susceptible and infectious individuals between two countries. Then, the disease-free equilibrium points and the basic reproduction numbers for both models are obtained, and stability theorems are proven. The stability theorems are exemplified with simulations showing that the results align well with the theoretical findings.

Keywords

• Mathematical modeling
• measles
• stability analysis
• two-patch model

Supporting Institution

The authors have no received any financial support for the research, authorship, or publication of this study.

Ethical Statement

The work does not require ethics committee approval and any private permission.

Thanks

This paper is produced from Master Thesis of Turgay Küçük during his studies at Gazi University Thesis number (Council of Higher Education, Thesis Center): 684539

Kızamık Salgınının İki Yamalı Matematiksel Modellemesi

Abstract

Bu çalışmada, kızamık hastalığının seyrini araştırmak için tek yamalı ve iki yamalı matematiksel modeller geliştirilmiştir. İlk olarak, tek yamalı modelin parametreleri mevcut verilere dayanarak tahmin edilmiştir. Tek yamalı modelin simülasyon sonuçlarının, gerçek verilerle iyi bir uyum gösterdiği görülmüştür. Ek olarak, parametre değerleri, hem duyarlı hem de bulaşıcı bireylerin iki ülke arasında karşılıklı göçünü veya seyahatini içeren iki parçalı bir model oluşturmak için kullanılmıştır. Ardından, her iki model için hastalıksız denge noktaları ve temel üreme sayıları elde edilerek kararlılık teoremleri kanıtlanmıştır. Kararlılık teoremleri, sayısal sonuçların teorik bulgularla iyi uyum sağladığını gösteren simülasyonlarla örneklenmiştir.

Keywords

• Matematiksel modelleme
• kızamık
• kararlılık analizi
• iki yamalı model.

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