Global stability of a time-delayed malaria model with standard incidence rate

TL;DR

This paper analyzes a delay differential equations model of malaria, establishing conditions for the global stability of disease-free and endemic states based on the basic reproduction number, using Lyapunov methods.

Contribution

It introduces a four-dimensional delay differential equations model for malaria and proves its global stability properties with respect to the basic reproduction number R0.

Findings

Disease-free equilibrium is globally stable if R0<1.

Endemic equilibrium is globally stable if R0>1.

The model's weak persistence is established for R0>1.

Abstract

A four-dimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number ${R}_{0}$. Specifically, it shows that the disease-free equilibrium ${E}^{0}$ is globally asymptotically stable (GAS) for ${R}_{0}<1$, and globally attractive (GA) for ${R}_{0}=1$, while the endemic equilibrium $E^{\ast}$ is GAS and ${E}^{0}$ is unstable for ${R}_{0}>1$. Especially, to obtain the global stability of the equilibrium $E^{\ast}$ for $R_{0}>1$, the weak persistence of the model is proved by some analysis techniques.