  We consider a family of McKean-Vlasov equations arising as the large particle
limit of a system of interacting particles on the positive half-line with
common noise and feedback. Such systems are motivated by structural models for
systemic risk with contagion. This contagious interaction is such that when a
particle hits zero, the impact is to move all the others toward the origin
through a kernel which smooths the impact over time. We study a rescaling of
the impact kernel under which it converges to the Dirac delta function so that
the interaction happens instantaneously and the limiting singular
McKean--Vlasov equation can exhibit jumps. Our approach provides a novel method
to construct solutions to such singular problems that allows for more general
drift and diffusion coefficients and we establish weak convergence to relaxed
solutions in this setting. With more restrictions on the coefficients we can
establish an almost sure version showing convergence to strong solutions. Under
some regularity conditions on the contagion, we also show a rate of convergence
up to the time the regularity of the contagion breaks down. Lastly, we perform
some numerical experiments to investigate the sharpness of our bounds for the
rate of convergence.
