[en] Two stochastic epidemic lattice models, the susceptible-infected-recovered and the susceptible-exposed-infected models, are studied on a Cayley tree of coordination number k. The spreading of the disease in the former is found to occur when the infection probability b is larger than b_c = k/2(k - 1). In the latter, which is equivalent to a dynamic site percolation model, the spreading occurs when the infection probability p is greater than p_c = 1/(k - 1). We set up and solve the time evolution equations for both models and determine the final and time-dependent properties, including the epidemic curve. We show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p = b/[k - (k - 1)b]. These include the cluster size distribution and the density of individuals of each type, quantities that have been determined in closed forms.