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Journals / Hacettepe Journal of Mathematics and Statistics / 2018 / Cilt: 47 - Sayı: 6

Numerical solution and stability analysis of a nonlinear vaccination model with historical effects

Davood ROSTAMY
Ehsan MOTTAGHİ
Pages
1478–1494
Publication Date
2018-12-12
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Our results show that historical effects play an important role on the disease spreading.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical effects by formulating it as a system of fractional-order differential equations (FDEs).
📊
Key Findings & Data:The basic reproduction number $R_0$ of the resulting fractional model is computed and it is shown that if $R_0$ is less than one, the disease-free equilibrium is locally asymptotically stable.
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Conclusion & Impact:Our results show that historical effects play an important role on the disease spreading.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:epidemic model · fractional-order differential equations · equilibrium points

Abstract

In this paper, we extend the classical vaccination epidemic model from a deterministic framework to a model with historical effects by formulating it as a system of fractional-order differential equations (FDEs). The basic reproduction number $R_0$ of the resulting fractional model is computed and it is shown that if $R_0$ is less than one, the disease-free equilibrium is locally asymptotically stable. Particularly, we analytically calculate a certain threshold-value for $R_0$ and present the existence conditions of endemic equilibrium. By using stability analysis, we prove stability and $\alpha$-stability of the endemic equilibrium points. The proposed model is applied on \emph{Pertussis} disease and the fractional nonlinear system of the model is solved by applying multi-step generalized differential transform method (MSGDTM). Our results show that historical effects play an important role on the disease spreading.

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Cite This Article

ROSTAMY, D., MOTTAGHİ, E. (2018). Numerical solution and stability analysis of a nonlinear vaccination model with historical effects. Hacettepe Journal of Mathematics and Statistics, Cilt: 47 - Sayı: 6, 1478–1494. https://ajindex.com/articles/669413

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