  We study a simple case of the susceptible-weakened-infected-removed model in
regular random graphs in a situation where an epidemic starts from a finite
fraction of initially infected nodes (seeds). Previous studies have shown that,
assuming a single seed, this model exhibits a kind of discontinuous transition
at a certain value of infection rate. Performing Monte Carlo simulations and
evaluating approximate master equations, we find that the present model has two
critical infection rates for the case with a finite seed fraction. At the first
critical rate the system shows a percolation transition of clusters composed of
removed nodes, and at the second critical rate, which is larger than the first
one, a giant cluster suddenly grows and the order parameter jumps even though
it has been already rising. Numerical evaluation of the master equations shows
that such sudden epidemic spreading does occur if the degree of the underlying
network is large and the seed fraction is small.
