Qualitative behavior of a highly non-linear Cutaneous Leishmania epidemic model under convex incidence rate with real data

Manal Alqhtani; Khaled M. Saad; Rahat Zarin; Amir Khan; Waleed M. Hamanah; Manal Alqhtani; Khaled M. Saad; Rahat Zarin; Amir Khan; Waleed M. Hamanah

doi:10.3934/mbe.2024092

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2084-2120. doi: 10.3934/mbe.2024092

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Qualitative behavior of a highly non-linear Cutaneous Leishmania epidemic model under convex incidence rate with real data

1.
Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Kingdom of Saudi Arabia
2.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology, Thonburi (KMUTT), Bangkok 10140, Thailand
3.
Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhawa, Pakistan
4.
Applied Research Center for Metrology, Standards, and Testing, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
5.
Department of Electrical Engineering, College of Engineering and Physics, King Fahd University for Petroleum and Minerals, Dhahran, Saudi Arabia

Received: 21 October 2023 Revised: 14 December 2023 Accepted: 02 January 2024 Published: 08 January 2024

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.
- geometric approach,
- compound matrix,
- stability analysis,
- real data,
- bifurcation,
- sensitivity analysis
Citation: Manal Alqhtani, Khaled M. Saad, Rahat Zarin, Amir Khan, Waleed M. Hamanah. Qualitative behavior of a highly non-linear Cutaneous Leishmania epidemic model under convex incidence rate with real data[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2084-2120. doi: 10.3934/mbe.2024092

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Abstract

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

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