We consider the constrained-degree percolation model in a random environment (CDPRE) on the square lattice. In this model, each vertex v has an independent random constraint $\kappa_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $\rho_j$. The dynamics is as follows: at time $t=0$ all edges are closed; each edge e attempts to open at a random time $U(e)\sim \mathrm{U}(0,1]$, independently of all the other edges. It succeeds if at time U(e) both its end vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, this result is meaningful only if the supremum of all values of t for which the expected size of the open cluster of the origin is finite is larger than $\frac12$. We prove this last fact by showing a sharp phase transition for an intermediate model.

We introduce a new 1-dependent percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are independent and identically distributed, but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but, conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above by the corresponding quantities for independent bond percolation with a certain density; this generalizes a result of Kuulasmaa (1982). Many models in the literature are special cases of our general model.