  We study the nonequilibrium phase transitions from the absorbing phase to the
active phase for the model of disease spreading
(Susceptible-Infected-Refractory-Susceptible (SIRS)) on a regular one
dimensional lattice. In this model, particles of three species (S, I and R) on
a lattice react as follows: $S+I\rightarrow 2I$ with probability $\lambda$,
$I\rightarrow R$ after infection time $\tau_I$ and $R\rightarrow I$ after
recovery time $\tau_R$. In the case of $\tau_R>\tau_I$, this model has been
found to has two critical thresholds separate the active phase from absorbing
phases \cite{ali1}. The first critical threshold $\lambda_{c1}$ is
corresponding to a low infection probability and second critical threshold
$\lambda_{c2}$ is corresponding to a high infection probability. At the first
critical threshold $\lambda_{c1}$, our Monte Carlo simulations of this model
suggest the phase transition to be of directed percolation class (DP). However,
at the second critical threshold $\lambda_{c2}$ we observe that, the system
becomes so sensitive to initial values conditions which suggests the phase
transition to be discontinuous transition. We confirm this result using order
parameter quasistationary probability distribution and finite-size analysis for
this model at $\lambda_{c2}$. Additionally, the typical space-time evolution of
this model at $\lambda_{c2}$ shows that, the spreading of active particles are
compact in a behavior which remind us the spreading behavior in the compact
directed percolation.14
