Critical Behavior Of The Contact Process Delayed By Infection And Immunization Periods

We analyze the absorbing state phase transition exhibited by two distinct unidimensional delayed contact process (CP). The first is characterized by the introduction of an infection period and the second by an immune period in the dynamics of the original model. We characterize these CP by the quantitiestd(infection or disease period) andti(immune period). The quantitytdcorresponds to the period interval an individual remains infected after being contaminated, while the periodtiis the time interval an individual remains immune after being cured. We used Monte Carlo simulations to compute the critical parameters associated with the absorbing state phase transition exhibited by these models. We find two distinct power-law scale relations for the critical infection rate$\lambda_{{\rm in}}^{*} \propto t_{{\rm d}}^{-\mu_{{\rm d}}}$and the critical cure rate$\lambda_{{\rm cu}_{\rm c}}^{*} \propto t_{{\rm i}}^{-\mu_{{\rm i}}}$. For the CP delayed by the minimum infection period we find μd= 0.98, while we obtained μi= 0.80 for the case of a delay due to immunity. In addition, we used a finite-size scaling analysis to estimate the critical exponents β/ν and ν, and found that these models belong to the universality class of directed percolation irrespective to the time delay.