  We study the changes of opinions about vaccination together with the
evolution of a disease. In our model we consider a multiplex network consisting
of two layers. One of the layers corresponds to a social network where people
share their opinions and influence others opinions. The social model that rules
the dynamic is the M-model, which takes into account two different processes
that occurs in a society: persuasion and compromise. This two processes are
related through a parameter $r$, $r<1$ describes a moderate and committed
society, for $r>1$ the society tends to have extremist opinions, while $r=1$
represents a neutral society. This social network may be of real or virtual
contacts. On the other hand, the second layer corresponds to a network of
physical contacts where the disease spreading is described by the SIR-Model. In
this model the individuals may be in one of the following four states:
Susceptible ($S$), Infected($I$), Recovered ($R$) or Vaccinated ($V$). A
Susceptible individual can: i) get vaccinated, if his opinion in the other
layer is totally in favor of the vaccine, ii) get infected, with probability
$\beta$ if he is in contact with an infected neighbor. Those $I$ individuals
recover after a certain period $t_r=6$. Vaccinated individuals have an
extremist positive opinion that does not change. We consider that the vaccine
has a certain effectiveness $\omega$ and as a consequence vaccinated nodes can
be infected with probability $\beta (1 - \omega)$ if they are in contact with
an infected neighbor. In this case, if the infection process is successful, the
new infected individual changes his opinion from extremist positive to totally
against the vaccine. We find that depending on the trend in the opinion of the
society, which depends on $r$, different behaviors in the spread of the
epidemic occurs. An epidemic threshold was found.
