Effects of measures on phase transitions in two cooperative susceptible-infectious-recovered dynamics

TL;DR

This paper studies how immunization measures, like vaccination, influence phase transitions in cooperative co-infection models, showing that high rates can convert abrupt transitions into gradual ones and increase thresholds.

Contribution

It introduces a mean-field framework to analyze the impact of vaccination on phase transitions in coupled infectious dynamics, including heterogeneity effects.

Findings

High vaccination rates can change discontinuous transitions to continuous.

Measures proportional to infection size increase transition thresholds.

Analytical and numerical solutions confirm the effects across parameters.

Abstract

In recent studies, it has been shown that a cooperative interaction in a co-infection spread can lead to a discontinuous transition at a decreased threshold. Here, we investigate effects of immunization with a rate proportional to the extent of the infection on phase transitions of a cooperative co-infection. We use the mean-field approximation to illustrate how measures that remove a portion of the susceptible compartment, like vaccination, with high enough rates can change discontinuous transitions in two coupled susceptible-infectious-recovered dynamics into continuous ones while increasing the threshold of transitions. First, we introduce vaccination with a fixed rate into a symmetric spread of two diseases and investigate the numerical results. Second, we set the rate of measures proportional to the size of the infectious compartment and scrutinize the dynamics. We solve the equations numerically and analytically and probe the transitions for a wide range of parameters. We also determine transition points from the analytical solutions. Third, we adopt a heterogeneous mean-field approach to include heterogeneity and asymmetry in the dynamics and see if the results corresponding to homogeneous symmetric case stand.