Dynamics of the discrete-time man-environment-man epidemic model with a free boundary

In this paper, we investigate the discrete-time man-environment-man epidemic model with a free boundary, which can be viewed as a time-discrete version of the free boundary model studied by Ahn, Beak, and Lin. First, applying the properties of the principal eigenvalue of the corresponding eigenvalue problem, we obtain the global dynamics of the corresponding fixed boundary problem. Then, we solve the problem step by step and establish the well-posedness of the solution. Moreover, we provide some sufficient conditions for the diseases spreading and vanishing by using the modified comparison principle. Finally, we give the long-time behavior of the solution by making use of the above results about the corresponding fixed boundary problem.