Modelling and stability analysis of the dynamics of measles with application to Ethiopian data

Hailay Weldegiorgis Berhe; Abadi Abay Gebremeskel; Habtu Alemayehu Atsbaha; Yohannes Yirga Kefela; Abadi Abraha Asgedom; Woldegebriel Assefa Woldegerima; Shaibu Osman; Lamin Kabareh

doi:10.1016/j.heliyon.2024.e33594

Modelling and stability analysis of the dynamics of measles with application to Ethiopian data

Heliyon. 2024 Jul 2;10(13):e33594. doi: 10.1016/j.heliyon.2024.e33594. eCollection 2024 Jul 15.

Authors

Hailay Weldegiorgis Berhe^{1

2}, Abadi Abay Gebremeskel³, Habtu Alemayehu Atsbaha¹, Yohannes Yirga Kefela¹, Abadi Abraha Asgedom¹, Woldegebriel Assefa Woldegerima⁴, Shaibu Osman⁵, Lamin Kabareh⁶

Affiliations

¹ Department of Mathematics, Mekelle University, Mekelle, Tigray, Ethiopia.
² Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology (BRICS), Helmholtz Centre for Infection Research, Rebenring 56, Braunschweig, 38106, Germany.
³ Department of Mathematics, Raya University, Maichew, Tigray, Ethiopia.
⁴ Department of Mathematics and Statistics, York University, Toronto, Canada.
⁵ Department of Basic Sciences, University of Health and Allied Sciences, Ghana.
⁶ Department of Statistics, University of the Gambia, Gambia.

Abstract

Measles, a highly contagious airborne disease, remains endemic in many developing countries with low vaccination coverage. In this paper, we present a deterministic mathematical compartmental model to analyze the dynamics of measles. We establish global stability conditions for both disease-free and endemic equilibria using the Lyapunov functional stability method. By using arbitrary parameters, we find that the proposed model exhibits forward bifurcation. To simulate the solution of the model for the forward problem, we perform numerical integration using MATLAB software. Moreover, we calibrate the model with real data from Ethiopia and estimate the parameters along with a 95 percent confidence interval (CI) by formulating an inverse problem. It is noteworthy that our model fits well with the actual data from Ethiopia. The estimated basic reproduction number ( $R_{0}$ ) is determined to be $R_{0} = 1.3973$ , demonstrating the endemic status of the disease. Additionally, our local sensitivity analysis indicates that reducing the transmission rate and increasing vaccination coverage can effectively minimize $R_{0}$ .

Keywords: Basic reproduction number; Forward bifurcation; Global stability; Measles; Model calibration; Vaccination.