Abstract
This paper is concerned with the spatial dynamics of a time-delayed reaction and diffusion malaria model. We first analyse the well-posedness of the initial-boundary value problem of the model. Then, we study the global stability of the disease-free or endemic steady state for the system by structuring two Lyapunov functionals. Moreover, by applying Schauder fixed-point theorem, we establish the existence of travelling wave solutions connecting the two steady states: the disease-free steady state and the endemic steady state if the basic reproduction ratio |$\mathcal {R}_0>1$|, and show the non-existence of travelling wave solutions connecting the disease-free steady state to itself if |$\mathcal {R}_0<1$|. Our analytic results show that |$\mathcal {R}_0$| is a threshold value for global dynamics and the existence of travelling wave solutions to the malaria model.
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