The spread of infectious diseases from a physics perspective
Authors: Brabers J.H.V.J. 
Published: June 2022

This paper presents a theoretical investigation of the spread of infectious diseases (including Covid-19) in a population network. The central idea is that a population can actually be considered as a network of interlinked nodes. The nodes represent the members of the population, the edges between the nodes the social contacts linking 2 population members. Infections spread throughout the population along these network edges. The actual spread of infections is described within the framework of the SIR compartmental model. Special emphasis is laid on understanding and on the interpretation of phenomena in terms of concepts borrowed from condensed-matter and statistical physics. To obtain a mathematical framework that deals with the influence of the network structure and topology, the original SIR model by Kermack and McKendrick was augmented, leading to a system of differential equations that is in principle exact, but the solution of which appears to be intractable. Therefore, combined algebraic/numerical solutions are presented for simplified (approximative) cases that nevertheless capture the essentials of the effect of the network details on the spread of an infection. Solutions of this kind were successfully tested against the results of direct statistical simulations based on Monte-Carlo methods, indicating the appropriateness of the model. Expressions for the (basic) reproduction numbers in terms of the model parameters are presented, and justify some mild criticisms on the widely spread interpretation of reproduction numbers as being the number of secondary infections due to a single active infection. Throughout the entire paper, special attention is paid to the concept of herd-immunity, its nature and its definition. The model allows for obtaining an exact (algebraic) criterion for the most relevant form of herd-immunity to occur in unvaccinated populations. Analysis of the effects of vaccination leads to an even more general version of this criterion in terms of not only the model parameters but also the effectiveness of the vaccine(s) and the vaccination rate(s). This general criterion is also exact within the context of the SIR model. Furthermore it is shown that the onset of herd-immunity can be considered as a 2nd-order phase transition of the kind that is known from thermodynamics and statistical physics, thus offering a fundamentally new viewpoint on the phenomenon. The role of percolation is highlighted and extensively investigated. It is shown that the herd-immunity transition is actually related to a percolation transition, and marks therewith the transition from a regime where the cumulative infections grow into a large macroscopic cluster that spans a major part of the population, towards a regime were the cumulative infections only occur in smaller secondary clusters of limited size. It appears that percolation phenomena become particularly important in the case of (strict) lock-downs. It is also demonstrated how a system of differential equations can be obtained that accounts for the presence of such percolation phenomena. The analyses presented in this paper also provide insight in how various measures to prevent an epidemic spread of an infection work, how they can be optimised and what potentially deceptive issues have to be considered when such measures are either implemented or scaled down. Herd-immunity appears to be a particularly tricky concept in this respect. Phenomena such as a saturation of the cumulative infection number or a fade-out of the number of active infections may easily be mistaken for a stable case of herd-immunity setting in, whereas in reality such phenomena may be no more than an artefact of protective or contact-reducing measures taken, without any meaning for the vulnerability of a population at large under normal (social) conditions. On the other hand, the paper also highlights and explains the theoretical possibility of “smothering” an epidemic via very restrictive measures that prevent it from developing out of a limited number of initial seed-infections.