Stability analysis of leishmania epidemic model with harmonic mean type incidence rate

Amir Khan; Rahat Zarin; Mustafa Inc; Gul Zaman; Bandar Almohsen

doi:10.1140/epjp/s13360-020-00535-0

Stability analysis of leishmania epidemic model with harmonic mean type incidence rate

Eur Phys J Plus. 2020;135(6):528. doi: 10.1140/epjp/s13360-020-00535-0. Epub 2020 Jun 26.

Authors

Amir Khan¹, Rahat Zarin¹, Mustafa Inc^{2

3}, Gul Zaman⁴, Bandar Almohsen⁵

Affiliations

¹ Department of Mathematics and Statistics, University of Swat, Swat, Khyber Pakhtunkhawa Pakistan.
² Department of Mathematics, Science Faculty, Firat University, Elazig, Turkey.
³ Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan.
⁴ Department of Mathematics, University of Malakand, Chakdara Dir, Khyber Pakhtunkhawa Pakistan.
⁵ Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451 Saudi Arabia.

Abstract

We discussed anthroponotic cutaneous leishmania transmission in this article, due to its large effect on the community in the recent years. The mathematical model is developed for anthroponotic cutaneous leishmania transmission, and its qualitative behavior is taken under consideration. The threshold number $R_{0}$ of the model is derived using the next-generation method. In the disease-free case, local and global stability is carried out with the condition that $R_{0}$ will be less than one. The global stability at the disease-free equilibrium point has been derived by utilizing the Castillo-Chavez method. On the other hand, at the endemic equilibrium point, the local and global stability holds with some conditions, and $R_{0}$ is greater than unity. The global stability at the endemic equilibrium point is established with the help of a geometrical approach which is the generalization of Lyapunov theory, by using the third additive compound matrix. The sensitivity analysis of the threshold number with other parameters is also taken into account. Several graphs of important parameters are discussed in the last section.