Finally On A Clockwork Contagion

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Over the course of the year my fellow students and I have spent our free time building mathematical models of the spread of disease, initially assuming that upon contracting the infection a person would immediately and forever be infectious[1], then adding periods of incubation and recovery[2] before finally introducing the concept of location whereby the proximate are significantly more likely to interact than the distant[3] and examining the consequences for a population distributed between several disparate villages.
Whilst it is most certainly the case that this was more reasonable than assuming entirely random encounters it failed to take into account the fact that folk should have a much greater proclivity to meet with their friends, family and colleagues than with their neighbours and it is upon this deficiency that we have concentrated our most recent efforts.

A Fifth Model Of Infections

To model close relationships, as opposed to vague acquaintances, my fellow students and I set explicit connections between members of a population with some random chance of both contacting and infecting or becoming infected by them. With deck 1 we gave each person forty eight compatriots of varying intimacy, out of a population of ten thousand, and tracked the total number who became infected over the course of one hundred days.

Deck 1: Total Infected By Day

We next constructed deck 2 to chart the number of newly infected people upon each day given the same assumptions.

Deck 2: Newly Infected By Day

To gain some insight into this model's behaviour we then took the familiar path of simplifying it.

Ring A Ring O' Roses

To that end we decided to commit the \(i^\mathrm{th}\) member of the population to approaching the \((i+1)^\mathrm{th}\) through to the \((i+k+1)^\mathrm{th}\), modulo the population size, thus forming a ring of associations and additionally transmitting or receiving the pathogen with a constant probability of \(p\). We put together deck 3 to figure the total number of consequent infections

Deck 3: Total Infections Upon The Ring

and thereafter deck 4 to figure the number of novel infections per day.

Deck 4: Novel Infections Upon The Ring

Whilst it is most evident that the progress of the disease under these assumptions is quite unlike that that results from random interactions and transmissions, we were nevertheless hopeful that we might find some aspects of it that we might consequently quantify. In particular we were interested in the effect of varying numbers of individually initiated contacts from one randomly chosen member of the population to another upon the contagion and therefore assembled deck 5 to track the total number of those who had contracted the infection upon there no longer being any infectious members of the population.

Deck 5: Final Totals Upon The Ring

Remarkably, we find a sudden and dramatic transition from there being but a few infections to an overwhelming proportion of the population succumbing!

In order to create a deterministic model we attempted to propagate the expected spread of disease by admitting fractional infections, effectively replacing each individual with an infinite population representing the infinitude of possibilities. Specifically, when an individual having an infectious proportion of \(i\) interacts with another having a susceptible proportion of \(u\) then the latter should accrue an infected proportion of
\[ i \times p \times u \]
Deck 6 exercises this averaging, recording each individual's historical fraction of infected and infectious in their associated inf arrays.

Deck 6: The Expected Infection Rates

Satisfied that this was at least similar to our random model, in that it yields a temporal series of peaks in the rates of infection centred upon approximately the same dates, we put together deck 7 to reckon the effect of the number of contacts upon its implied total number of infections.

Deck 7: The Expected Final Totals

Clearly something is amiss! Here we have a gradual, rather than a sudden, rise in the eventual number of infected.

It's A Small World

The disparity is a consequence of the fact that infection will occasionally leap forward to more distant acquaintances, creating pools of infectiousness from which disease might spread both fore and aft, rapidly accelerating its transmission.

This is fundamentally related to the concept of small-world networks[4] in which a relatively small number of friends leads to every member of a population being associated with almost every other by a chain of friends of friends of friends. Deck 8 illustrates the rate at which a set of isolated individuals coalesce into but a few such mutually associated clans, concluding with folk having just three friends upon the average.

Deck 8: Counting Clans

Similarly, deck 9 figures the size of the largest of the clans.

Deck 9: The Maximal Clan Size

Here we gain some inkling as to why the rate of infection increases so very dramatically once a threshold of individual acquaintances has been reached; the number of members of the most populous clan exhibits very much the same behaviour, giving the disease the opportunity to spread throughout the population at large!

Now a truly remarkable property of small-world networks is how short these chains of association typically are. Defining the degree of separation between two people as the length of such a chain, so that an immediate acquaintance is at one degree of separation, a friend of a friend is at two, a friend of a friend of a friend is at three and so on and so forth, deck 10 charts the approximate average degree of separation between members of a population of five thousand as their average number of friends grows from one to five.

Deck 10: Average Degrees Of Separation

At an average of five friends apiece we find that the mean degree of separation between two members of the population is just four! Deck 11, in contrast, samples the greatest degrees of separation between members of the population as their circles of friends grow.

Deck 11: Maximal Degrees Of Separation

Here we find that they are not so much very greater than five in length should the commonest number of acquaintances be five.
This then is the second piece of the puzzle! Once each individual's circle of friends grows past a relatively small threshold, the vast majority of the population join a single tribe whilst simultaneously decreasing the degrees of separation between each other so that the disease may be propagated along relatively short chains of infection.
To confirm that our intuition was sound my fellow students and I put together deck 12 to figure the number of people who had at some point become infected at the cessation of the outbreak for an increasing number of individual contacts.

Deck 12: Final Totals

Several Small Worlds

Another interesting aspect of small-world networks is that they can take many forms. For example, if we divide the population into a set of villages and give each individual a much greater chance of associating with members of their own village than with those of another, we still observe surprisingly few degrees of separation between members of the population at large, as demonstrated by deck 13 which plots their maxima for a sample of the population as their average number of associates increase.

Deck 13: Maximal Villager Degrees Of Separation

Note that the initially small values are a consequence of our ignoring the infinite degrees of separation between members of different tribal affiliation and so reflect those of fellow villagers. Nevertheless, after an initial growth as folk begin to interact with inhabitants of other villages we observe a rapid decrease, just as we did with a homogeneous population.
Furthermore, the final total number of cases exhibits the same rapid growth after an initial stagnation, as shown by deck 14.

Deck 14: Final Totals In The Villages

To observe the effect of friendships between members of different villages, we compiled deck 15 to record the ultimate number of infections as their likelihood increases.

Deck 15: Increasing Global Acquaintances

Unsurprisingly, we find that for small numbers of such acquaintances the disease is unlikely to spread beyond one or two villages, whereas for large numbers there is a good chance that much of the population will be afflicted.
We were also curious as to what effect this village-like network might have upon daily infection rates and so we put together deck 16 to track them.

Deck 16: Newly Infected Villagers By Day

Much like our previous model of the spread of infection between villages, the rate has multiple peaks reflecting the eventual transmission of the disease from the first village to the others and its consequent propagation within them.

My fellow students and I have been quite unable to figure simple formulae that capture the behaviour of this, our final, model of contagion and it is with some regret that it is upon this failure that we must conclude our investigations.
\(\Box\)

References

[1] On A Clockwork Contagion, www.thusspakeak.com, 2021.

[2] Further On A Clockwork Contagion, www.thusspakeak.com, 2021.

[3] Further Still On A Clockwork Contagion, www.thusspakeak.com, 2021.

[4] Watts, D., Small Worlds, Princeton University Press, 1999.

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