Instability of disease-free equilibrium in a model of malaria with immune delay

A recent paper Ncube (2013) [11] considered the disease-free equilibrium in a mathematical model for intra-host dynamics of Plasmodium falciparum malaria with discrete immune time delay. The author showed that depending on system parameters, the disease-free steady state can be absolutely stable (i.e. asymptotically stable for arbitrary positive values of the time delay), or it can be asymptotically stable for sufficiently small values of the time delay and then undergo Hopf bifurcation once the time delay exceeds certain critical value. In this paper we show by direct calculation that the conclusions regarding stability and Hopf bifurcation of the disease-free equilibrium are incorrect, and, in fact, the disease-free equilibrium of the model is always unstable. Furthermore, we provide a general argument why the disease-free steady state of the model can never undergo Hopf bifurcation.