Fixing

,

,

,

, and

. The three lines are curves of

as a function of

, keeping

constant. We select three different values of

which correspond to solid black, red, and dashed lines, respectively. The value

is a special case that leads to

. It divides the phase space in two different regions. All the values of

below are characterized by

. In this case for large values of

the model is reduced to an SIR with reproductive number

below

and the epidemic is halted. Interestingly, this behavior starts in an intermediate regime of

. There is a critical value

of

above which (i.e.,

) the epidemic size is zero. This transition happens with a jump, as shown by the solid black line. All the values of

above

are instead characterized by

. Also in this case the model is reduced to an SIR with reproductive number

for large values of

, but in this case this value is above

. This results in a epidemic size that is always non-zero. In this region of parameters no jumps are present (see the dashed line). The values shown in the plot are computed through numerical integration of the equations.

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