A scaling analysis in the SIRI epidemiological model

Abstract

For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I , then recover and remain only partial immune against reinfection R, we determine the phase transition lines using pair approximation for the moments derived from the master equation. We introduce a scaling argument that allows us to determine analytically an explicit formula for these phase transition lines and prove rigorously the heuristic results obtained previously.

Dynamics of Epidemiological Models

“…hSi þ hIi þ hRi ¼ N, one can reduce the ODE system for total expectation values and for pair expectation values to five independent variables hIi, hRi, hSIi, hRIi and hSRi (see Martins et al 2009;Stollenwerk et al 2007): Considering the special cases for reinfection rate equal to first infection rate (the SIS limit of the SIRI model), vanishing the reinfection rate (the SIR limit of the SIRI model) and the limit of vanishing transition from recovered to susceptible a, the above system can be solved analytically. In these cases, we can give the stationary values hIi à etc.…”

“…In Martins et al (2009) and Stollenwerk et al (2007), it is investigated the Eqs. (1.18, 1.19) further, using the information that when hIi à goes to zero, so does hRi à , but the quotient stays finite This completes the expression for the critical curve bðbÞ for the general a and c case.…”

“…To calculate the dynamics of the moments we apply the master equation into the time derivative of the moments obtaining for the mean total number of susceptible, infected and recovered hosts the following ODE's (see Martins et al 2009;Stollenwerk et al 2007)…”

“…For the spatial stochastic epidemic model SIRI we investigate, in the pair approximation scheme, its phase transition lines. We present the analytic expression for the phase transition line between no-growth and nontrivial stationary equilibria (see Martins et al 2009;Stollenwerk et al 2007). …”

Stochastic epidemic models revisited: analysis of some continuous performance measures

“…the papers by Coolen-Schrijner and van Doorn [11] and Artalejo et al [4]) and also to more complicated variants of the SIS and SIR epidemic models (see, e.g. some recent publications in this journal such as the papers by McCormack and Allen [24], Clémençon et al [9] and Martins et al [23]). …”