COVID-19 is a highly infectious disease spreading through human droplets and contact. To investigate the effect of human behavior changes on the spread of COVID-19, a reaction-diffusion model that contains contact rate functions related to human behavior is studied. The basic reproduction number $\mathcal{R}{0}$ for this system is derived and a threshold-type result on its global dynamics in terms of $\mathcal{R}{0}$ is established in this paper. More precisely, we show that the disease-free equilibrium is globally asymptotically stableif $\mathcal{R}{0}\leq1,$ and the system admits a positive solution and the disease is uniformly persistent if $\mathcal{R}{0}>1$. By the numerical simulations of the analytic results, we find that human behavior changes may lower infection levels and reduce the number of exposed and infected humans.