This is a quick script to try to understand the effect of quarantine using random graphs with a toy degree distribution. It is runnable interactively in any browser via Binder.
The toy degree distribution allows exactly 3 types of individuals. A fraction 1-p_a-ε of the population infect 0 new individuals. A fraction p_a of them infect a new individuals. Finally, a fraction ε of the population infect a large number b of new individuals (they are "superspreaders"). We assume that the basic reproduction number R0 is known and fixed, and look at different proportions p_a and activities a of "normal" individuals and different activity b of the superspreaders (ε is adjusted to match these parameters).
For a completely mixed population with no immigration, basic theoretical models prove that if we manage to get R0 even slightly below 1 (by social distancing, hand and face hygiene, closures of places where a lot of people gather closely together, etc.), an epidemic ends infecting approximately 0% of the population altogether. If R0 remains above one, unless stopped while there are just a few cases, an epidemic explodes and infects a significant fraction of the population.
When R0 is fixed (and R0 > 1), if most individuals observe quarantine, the same growth can only be sustained by those who don't or can't, the superspreaders in our toy model. If this is the case (most individuals do not pass the infection on, but the initial growth is still fast), the final size of an epidemic (the largest component) may be smaller than the one predicted by the simple G(n,p) model corresponding to a completely homogeneous population (initial growth is the same but everyone equally ignores the quarantine, the green line in the plot).
For G(n,p) the blue line is the "herd immunity threshold" 1 - 1/R0. If herd immunity is attained via "natural infection" as opposed to vaccination, the epidemic only begins to die out once this point is reached. The total infected fraction (the green line), or the fraction of vertices in the giant component, is quite a bit larger, it is the unique solution ρ of 1-ρ = exp(-R0 ρ) and 0 < ρ < 1.
In the terminology of graph theory the first chart shows the giant component size as a function of the largest degree b while other parameters, including R0, which is expectation of the size-biased degree distribution minus one, E D^2/E D - 1, are kept fixed.
My intuition of the "superspreader effect" is that the superspreaders are more likely to get infected first so the effective growth rate Rt is largest at the beginning. This is indeed the case in our toy configuration graph model, corresponding to b=10 in the above chart (66.88% infect 0, 30.00% infect 1, 3.12% infect 10). The analytical results from [1] with these parameters give:
Here the time is (monotonically) transformed by the proof of [1], it does not match the real time. y axis in the first two subplots shows the proportion of the population. It can be seen that an active epidemic does not magically stop when the herd immunity threshold Rt=1 is reached, a considerable additional fraction of the population is infected if no measure is taken. Conversely, if a stricter quarantine is enforced to stop the epidemic close to this threshold, the growth will not resume.
In this toy experiment I was interested in the simplified quarantine situation. A similar effect should occur more generally due to heterogeneity of a population, see [2], [3], [4] and recent papers citing them.
[1] Svante Janson and Malwina J. Luczak, A new approach to the giant component problem, Random Structures & Algorithms 34 (2009), 197-216.
[2] Håkan Andersson, Limit Theorems for a Random Graph Epidemic Model, The Annals of Applied Probability 8 (1998), 1331-1349.
[3] Tom Britton, Svante Janson and Anders Martin-Löf, Graphs with specified degree distributions, simple epidemics, and local vaccination strategies, Advances in Applied Probability 39 (2007), 922-948.
[4] Mark E. Newman, Spread of epidemic disease on networks, Physical Review E (2002) 66, 016128.