Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate

A time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number $ {R}_{0} $ is calculated. We focus on the global dynamics of the model with respect to $ {R}_{0} $. If and only if $ {R}_{0} > 1 $, the model exists a unique malaria-infected equilibrium $ E^{\ast} $, whereas it always possesses the malaria-free equilibrium $ E_{0} $. We first prove the local stability of the equilibria $ E_0 $ and $ E^{\ast} $ by using proof by contradiction and the properties of complex modulus. Secondly, by utilizing the Lyapunov functional method and the limiting system of the model with some novel details, we show that the equilibrium $ {E}_{0} $ is globally asymptotically stable (GAS) when $ {R}_{0} < 1 $, globally attractive (GA) when $ {R}_{0} = 1 $ and unstable when $ {R}_{0} > 1 $; the equilibrium $ E^{\ast} $ is GAS if and only if $ {R}_{0} > 1 $. In particular, in order to obtain global attractivity of the equilibrium $ E^{\ast} $, we demonstrate the weak persistence of the system for $ {R}_{0} > 1 $. Our results imply that malaria will gradually disappear if $ {R}_{0}\leq1 $ and persistently exist if $ {R}_{0} > 1 $.