  We consider the qualitative behavior of a mathematical model for transmission
dynamics with two nonlinear stages of contagion. The proposed model is inspired
by phenomena occurring in epidemiology (spread of infectious diseases) or
social dynamics (spread of opinions, behaviors, ideas), and described by a
compartmental approach. Upon contact with a promoter (contagious individual), a
naive (susceptible) person can either become promoter himself or become
$\textit{weakened}$, hence more vulnerable. Weakened individuals become
contagious when they experience a second contact with members of the promoter
group. After a certain time in the contagious compartment, individuals become
inactive (are insusceptible and cannot spread) and are removed from the chain
of transmission. We combine this two-stage contagion process with renewal of
the naive population, modeled by means of transitions from the weakened or the
inactive status to the susceptible compartment. This leads to rich dynamics,
showing for instance coexistence and bistability of equilibria and periodic
orbits. Properties of (nontrivial) equilibria are studied analytically. In
addition, a numerical investigation of the parameter space reveals numerous
bifurcations, showing that the dynamics of such a system can be more complex
than those of classical epidemiological ODE models.
