Virus spread on a scale-free network reproduces the Gompertz Growth observed in isolated COVID-19 Outbreaks - Supplementary Movies

ABSTRACT

The counts of confirmed cases and deaths in isolated SARS-CoV-2 outbreaks follow the Gompertz growth function for locations of very different sizes.  This lack of dependence on region size leads us to hypothesize that virus spread depends on the universal properties of the network of social interactions.  We test this hypothesis by simulating the propagation of a virus on networks of different topologies or connectivities.  Our main finding is that we can reproduce the Gompertz growth observed for many early outbreaks with a simple virus spread model on a scale-free network, in which nodes with many more neighbors than average are common.  Nodes that have very many neighbors are infected early in the outbreak and then spread the infection very rapidly.  When these nodes are no longer infectious, the remaining nodes that have most neighbors take over and continue to spread the infection.  In this way, the rate of spread is fastest at the very start and slows down immediately.  Geometrically we see that the "surface" of the epidemic, the number of susceptible nodes in contact with the infected nodes, starts to rapidly decrease very early in the epidemic and as soon as the larger nodes have been infected.  In our simulation, the speed and impact of an outbreak depend on three parameters: the average number of contacts each node makes, the probability of being infected by a neighbor, and the probability of recovery.  Intelligent interventions to reduce the impact of future outbreaks need to focus on these critical parameters in order to minimize economic and social collateral damage.

Supplementary Movies 1 and 2 represent two typical simulations of virus spread in a scale-free network with P_R/P_I = 1 and P_R/P_I =3, respectively.  Nodes are arranged so that larger nodes occupy the center of the graph, and smaller nodes are peripheral.  The size of each dot is proportional to the natural logarithm of its degree.  During the dynamics, infected nodes are represented in red, recovered nodes in yellow and susceptible nodes in blue.  The graphs on the right, plot the same quantities as in Figure 5.  Infection is seeded in a random node and spread slowly in the network until one the central (larger) nodes is hit.  From this time on the infection growth follows the Gompertz law, as evident from the linearity of the function J(t).  As larger nodes are well connected, this happens relatively early in the outbreak.