Gradually Declining Immunity Retains the Exponential Duration of Immunity-Free Diffusion

TL;DR

This paper introduces the cSIRS model, where immunity wanes gradually so that the expected immunity matches that of SIRS, and rigorously analyzes its survival time on stars and expanders. Using coupling with SIS and standard tools like gambler's ruin and Gamma-function bounds, the authors show that on star and expander graphs, the survival time of cSIRS remains essentially as large as in SIS, implying that gradual immunity is nearly as ineffective as having no immunity at all for sustaining infections. In contrast, the classical SIRS process yields polynomial survival on stars and endemic behavior on expanders; thus, full temporary immunity is necessary to significantly curb endemic spread. The results indicate that gradually declining immunity offers limited practical benefit, guiding future work toward upper bounds for cSIRS, alternative immunity decay models, and analyses on broader graph families.

Abstract

Diffusion processes pervade numerous areas of AI, abstractly modeling the dynamics of exchanging, oftentimes volatile, information in networks. A central question is how long the information remains in the network, known as survival time. For the commonly studied SIS process, the expected survival time is at least super-polynomial in the network size already on star graphs, for a wide range of parameters. In contrast, the expected survival time of the SIRS process, which introduces temporary immunity, is always at most polynomial on stars and only known to be super-polynomial for far denser networks, such as expanders. However, this result relies on featuring full temporary immunity, which is not always present in actual processes. We introduce the cSIRS process, which incorporates gradually declining immunity such that the expected immunity at each point in time is identical to that of the SIRS process. We study the survival time of the cSIRS process rigorously on star graphs and expanders and show that its expected survival time is very similar to that of the SIS process, which features no immunity. This suggests that featuring gradually declining immunity is almost as having none at all.

Figures (1)

• Figure 1: State transitions of a vertex in the shown processes. Vertices are susceptible (S), infected (I), or recovered (R). Edges represent the existence of a Poisson clock that triggers a transition attempt with a rate dependent on the arrow type. Note that edges to infected vertices represent one clock for each infected neighbor. The numbers on the arrows represent the probability of a successful attempt. We use $t_h$ to denote the time passing since the vertex last healed, and $t_i$ to denote the time passing since the last infection attempt after the vertex healed (or the last time the vertex healed, whichever is smaller).

Theorems & Definitions (31)

• Theorem 1: Gambler's ruin feller1957introduction
• Theorem 2: Ratio of Gamma Functions tricomi1951asymptotic
• Corollary 1
• proof
• Theorem 3
• proof
• Definition 1: $\textrm{labeled cSIRS}$
• proof
• Theorem 4
• proof