Traveling waves for a nonlocal delayed reaction-diffusion SIR epidemic model with demography effects.

Considering the importance of time delay phenomenon in disease transmission and the interdependence of time delay and spatial location, a reaction-diffusion SIR epidemic model with nonlocal time delay (or time-space time delay) is proposed and the travelling wave solutions are discussed. Specifically, we define the basic reproduction number $ \mathcal {R}_0 $ R0 and the critical wave speed $ c^* $ c∗. For every wave speed $ c\geq c^* $ c≥c∗, the existence of travelling wave solutions is studied by using the upper–lower solutions, the fixed-point theorem and some limit techniques when $ \mathcal {R}_0 \gt 1 $ R0>1. The nonexistence of travelling waves when $ \mathcal {R}_0 \gt 1 $ R0>1 for any $ 0 \lt c \lt c^* $ 0<c<c∗ or $ \mathcal {R}_0 \lt 1 $ R0<1 for any <italic>c</italic>>0 is also proved. Finally, the influence of time-delay on disease propagation, especially the critical wave speed, is discussed through analysis and simulations. [ABSTRACT FROM AUTHOR]