  In recent studies, it has been shown that a cooperative interaction in a
co-infection spread can lead to a discontinuous transition at a decreased
threshold. Here, we investigate effects of immunization with a rate
proportional to the extent of the infection on phase transitions of a
cooperative co-infection. We use the mean-field approximation to illustrate how
measures that remove a portion of the susceptible compartment, like
vaccination, with high enough rates can change discontinuous transitions in two
coupled susceptible-infectious-recovered dynamics into continuous ones while
increasing the threshold of transitions. First, we introduce vaccination with a
fixed rate into a symmetric spread of two diseases and investigate the
numerical results. Second, we set the rate of measures proportional to the size
of the infectious compartment and scrutinize the dynamics. We solve the
equations numerically and analytically and probe the transitions for a wide
range of parameters. We also determine transition points from the analytical
solutions. Third, we adopt a heterogeneous mean-field approach to include
heterogeneity and asymmetry in the dynamics and see if the results
corresponding to homogeneous symmetric case stand.
