Immunization of networks with limited knowledge and temporary immunity

Modern view of network resilience and epidemic spreading has been shaped by percolation tools from statistical physics, where nodes and edges are removed or immunized randomly from a large-scale network. In this paper, we produce a theoretical framework for studying targeted immunization in networks, where only n nodes can be observed at a time with the most connected one among them being immunized and the immunity it has acquired may be lost subject to a decay probability ρ. We examine analytically the percolation properties as well as scaling laws, which uncover distinctive characters for Erdős–Rényi and power-law networks in the two dimensions of n and ρ. We study both the case of a fixed immunity loss rate as well as an asymptotic total loss scenario, paving the way to further understand temporary immunity in complex percolation processes with limited knowledge.

Percolation theory has been extensively employed to study network resilience and spread of infectious diseases. It has successfully explained important behaviors such as Achilles heels of scale-free networks and targeted immunization. It is recently reported that partial observation ability in the targeted immunization strategy can effectively influence the network robustness. In this paper, we add a further dimension to the picture by incorporating temporary immunity, where a node that acquires immunity at one step may lose it later. By accommodating limitations in space (knowledge of node) and time (immunity of node), we investigate percolation properties and scaling lows analytically for networks with arbitrary degree distributions. Distinctive characteristics for targeted immunization in Erdős-Rényi networks and power-law networks have been revealed, extending, for example, the well-known Achilles heels phenomenon under the two dimensions of limitation. We solve for both cases of a fixed fading rate of immunity and an asymptotic total loss of immunity. Our results suggest that increasing level of knowledge in targeted immunization may not be as effective as one would expect in fighting some epidemics such as COVID-19.