The phase transition lines in pair approximation for the basic reinfection model SIRI

Abstract

For a spatial stochastic epidemic model we investigate in the pair approximation scheme the differential equations for the moments. The basic reinfection model of susceptible-infected-recovered-reinfected or SIRI type is analysed, its phase transition lines calculated analytically in this pair approximation.

A scaling analysis in the SIRI epidemiological model

“…In pair approximation, these critical points have different values for the SIS system [7] and the SIR system [6], as opposed to the mean field values which are the same for SIS and SIR. In the present article, we prove the analytical formula of the transition line presented in [10] between no-growth and annular growth. The proof of this analytical formula of the transition line has a difficulty far beyond the calculation of the other transition lines.…”

“…The existence of these two thresholds in systems with the possibility of reinfection with partial immunity, as known from, e.g., influenza, malaria and tuberculosis (see [3], and personal communication G. Gomes, Oeiras), has consequences for disease control and vaccination policies. The accurate matching of the models used in biology with those used in physics is vital for the mentioned applications and is given in [10] and the present study, in which the most interesting phase transition line is rigorously proven (the second phase transition having trivial form of a straight line in parameter space).…”

“…We compare the especially tricky phase transition line between no-growth and annular growth with simulations [10]. In pair approximation, these critical points have different values for the SIS system [7] and the SIR system [6], as opposed to the mean field values which are the same for SIS and SIR.…”

“…In [10], we presented the phase transition lines between no-growth and compact growth, between compact growth and annular growth and between no-growth and annular growth in pair approximation for a reinfection model, called SIRI. It describes a susceptible, infected, recovered epidemic process, SIR, with additional partial reinfection, a transition from partially immune recovered R to the infected class I , hence the name SIRI.…”

“…The geometric interpretation in the spatial set-up as given in [5] for their partial immunity models has helped us to understand the two thresholds, found in the biological mean field models to correspond to the transitions from no-growth to annular growth and from annular growth to compact growth [10]. The existence of these two thresholds in systems with the possibility of reinfection with partial immunity, as known from, e.g., influenza, malaria and tuberculosis (see [3], and personal communication G. Gomes, Oeiras), has consequences for disease control and vaccination policies.…”