Global stability of a diffusive Leishmaniasis model with direct and indirect infection rate

Abstract

Visceral leishmaniasis (VL), or black fever (kala-azar), is a fatal parasitic disease that infects a hosts internal organs. According to the World Health Organization (WHO), leishmaniasis is a major public health problem that has been neglected, and the control measures in various seriously infected areas have not been successful. Therefore, in order to analyze and control the spread of leishmaniasis, we establish a diffusive model with direct and indirect infection rate. Firstly, we prove the uniform bounds of solutions of the system, and analyze the sensitivity of the parameters. Secondly, sufficient conditions for the existence of the disease-free equilibrium and the endemic equilibrium are given, respectively. In addition, the stability of the model is studied in local and global sense by using the Routh Hurwitz criterion and Lyapunov theory, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R 0 1 and the endemic equilibrium is globally asymptotically stable when the basic reproduction number R 0 > 1. Finally, the theoretical results of the diffusive Leishmaniasis model with direct and indirect infection rate are verified by simulation. The results show that direct or indirect infection rates may affect the prevalence of the disease.