Analysing the reinfection effect for transmitted diseases and social adoptions through stochastical SIRI model

The reinfection effect is relevant for many transmitted diseases such as Tuberculosis and Viral Hepatitis. In such cases, the second infection is less probable than the primary one. On the contrary, in social context, in general, the primary adoption of a product, innovation or idea is less probable than the second one [1]. For both cases, an stochastic version of the SIRI model is able to translate the spreading idea, based on the following compartments: Susceptible (Ignorant), Infected (Spreader) and Recovered (Inactive). The dynamical rules between compartments are given by: $S + I \rightarrow 2I$ with infection rate $\beta$; $I \rightarrow R$ with recovery rate $\gamma$; $R + I \rightarrow 2I$ with reinfection rate $\sigma \beta$. When $\sigma=0$, the stochastical SIR model is recovered. We set up the master equation for stochastical SIRI model. According to one-site and pair mean-field approximations of the master equation of SIRI model on a lattice, for $\sigma < 1$, the relevant scenario for transmited diseases, there is a phase transition between the endemic and epidemic states. For $\sigma<1$ , Monte Carlo simulations also exhibit a second-order phase transition between endemic and epidemic phases, revealing that SIRI model is in the same universality class of directed percolation. However, for $\sigma > 1$, the relevant scenario for social adoptions, the phase transition becomes abrupt in a network. We are studying analitically the stochastical SIRI model in homogeneous and heterogeneous networks in order to investigate the effect of the connectivity distribution on the phase transition as it was previously done for the stochastical SIS model [2].

[1] J. G?mes-Garde?es, A. S. De Barros, S. T. R. Pinho, R. F. S. Andrade. Abrupt transitions from reinfections in social contagions. EPL 110, 58006 (2015).

[2] R.S. Sander. Transi??es de Fase para Estados Absorventes: um estudo em redes regulares e complexas. Disserta??o de Mestrado: Universidade Federal de Vi?osa, Brasil (2011).