  The term radicalization refers to the process of developing extremist
religious political or social beliefs and ideologies. Radicalization becomes a
threat to national security when it leads to violence. Prevention and
de-radicalization initiatives are part of a set of strategies used to combat
violent extremism, which taken together are known as Countering Violent
Extremism (CVE). Prevention programs aim to stop the radicalization process
before it starts. De-radicalization programs attempt to reform convicted
extremists with the ultimate goal of social reintegration. We describe
prevention and de-radicalization programs mathematically using a compartmental
model. The prevention initiatives are modeled by including a vaccination
compartment, while the de-radicalization process is modeled by including a
treatment compartment. The model exhibits a threshold dynamics characterized by
the basic reproduction number $ R _0 $. When $ R _0< 1 $ the system has a
unique equilibrium that is asymptotically stable. When $ R _0 >1 $ the system
has another equilibrium called "endemic equilibrium", which is globally
asymptotically stable. These results are established by using Lyapunov
functions and LaSalle's invariance principle. We perform numerical simulations
to confirm our theoretical results.

Physics and Society;Dynamical Systems