Belief Contagion in a Small Population

Department of Psychohistory | Intermediate | 45 minutes | English draft; human subject and learner review pending

Objectives and project question

You will compute a reproduction number and its threshold, find the final size that a deterministic epidemic curve predicts, simulate the same model one event at a time, and read the distribution from which a single run is drawn. Project question: the department lists “belief state epidemiology”. If a belief spreads among 14 agents, how much does one run differ from the deterministic epidemic curve?

PSY-002 reads a shift in belief labels as a sign of regime change; this module asks what a model of spreading can forecast. PSY-004 studies 14 agents whose answers share a cause, and PSY-017 the calibration belief met in section 1.

1. The model, and what “infected” means here

Observed. The department record lists, as research area 7 of 10, “Belief state epidemiology — how beliefs spread across agents and repos”, and its key principle speaks of “14 personas across 4 repos”. extract.py snapshots what the repository records about beliefs at commit 4b03838 into observed-input.json: the 8 current belief files of state/beliefs/ and 13 archived ones. No proposition appears under two holders (0 of 21), but this count could not have shown sharing: the 12 files with content have one evaluator, the governance persona Demerzel, and the other 9 are empty scaffolds.

Two current files name a source outside their evaluator: confidence-calibration and remediation-effectiveness each rest on one evidence item, an ml-feedback-recommendation message whose rationale and confidence the applying script copies into Demerzel’s belief. Its schema says “from an ix ML pipeline to Demerzel”; the two messages are not in the repository, so the sender is known from that description and the script’s docstring only. Each is one hop, from a document to one holder.

Two record types are designed to hold an agent’s beliefs: a belief snapshot (one agent’s beliefs at a given time) and a knowledge state (a teacher, a learner, the learner’s belief before and after). Of the 722 JSON files under state/, course library excluded, none has the shape of either (the test recognises their three sample files) and none has a key naming a teacher, a learner or a message’s sender (origin_repo, origin_agent). Under these tests no file records a holder passing a belief on, a second hop, or an agent exposed without adopting. No rate can be estimated, so everything below is synthetic.

Analogy. In the SIR model of epidemiology, an agent is susceptible (S) until it adopts the belief, infected (I) while it holds the belief and passes it on, and recovered (R) once it no longer passes it on. “Infected” does not mean wrong or harmful; it names the state of transmitting. Among N agents, each holder passes the belief to agents picked at random at rate β and stops transmitting at rate γ:

adoption:            (S, I) -> (S-1, I+1)   at rate beta*S*I/N
stops transmitting:  (S, I) -> (S, I-1)     at rate gamma*I
R0 = beta/gamma
P(next event is an adoption) = R0*S / (R0*S + N)

The basic reproduction number R0 is the mean number of agents one holder converts in a large population when everybody else is susceptible. The last line is the embedded jump chain: it says which event comes next and ignores when. The final size, the number of agents that ever held the belief, depends only on that order, so analyze.py draws no waiting time.

Assumptions, in scenarios.json: N = 14 (the key principle’s count; PSY-004 finds 17 persona files at the pin) and N = 1,400; one initial holder; R0 = 0.8, 1.2 and 3.

Practice Exercise 1

With N = 14, R0 = 3 and one initial holder, what is the probability that the first event is an adoption? That the belief is never passed on? Recompute the second for N = 1,400.

Answer: With S = 13, 3×13 / (3×13 + 14) = 39/53 = 0.7358. Otherwise the only holder stops first and the belief ends with it: 14/53 = 0.2642. For N = 1,400: 1,400 / (3×1,399 + 1,400) = 1,400/5,597 = 0.2501.

2. The deterministic curve and its threshold

Replace counts by shares s, i, r of a large population and random events by their mean rates, with time in mean holding periods (1/γ):

ds/dt = -R0*s*i        di/dt = R0*s*i - i        dr/dt = i
holders increase while R0*s > 1
final size z:   1 - z = s0 * exp(-R0*z)

This is the mean-field curve: every agent feels the average of all the others. At its threshold, R0 × s = 1, each holder is replaced by exactly one. The final-size equation gives the share z that ever holds the belief.

Synthetic. For 14 agents (s0 = 13/14) the equation gives 3.13 agents at R0 = 0.8, 6.65 at R0 = 1.2 and 13.24 at R0 = 3. At R0 = 3 the integrated curve rises to 4.55 simultaneous holders after 1.59 holding periods and ends at the same 13.24 (the known-answer check). At R0 = 1.2 it peaks at 1.07 holders yet ends at 6.65.

Practice Exercise 2

For N = 14 and one initial holder, compute R0 × S/N at the start for the three values of R0. Where it exceeds 1, after how many adoptions do holders stop increasing?

Answer: 0.8 × 13/14 = 0.7429, 1.2 × 13/14 = 1.1143, 3 × 13/14 = 2.7857. At R0 = 0.8 the curve never rises. Holders increase while S exceeds N/R0: 14/1.2 = 11.67, so growth stops once S is 11, after 13 − 11 = 2 adoptions; 14/3 = 4.67, so once S is 4, after 13 − 4 = 9.

3. Fourteen agents: the distribution one run comes from

Synthetic. Runs at each final size, over 10,000 runs for each R0 (results.json, section cases):

final size     1     2    3    4    5    6    7    8    9   10   11   12    13    14
R0 = 0.8    5708  1553  818  520  364  273  231  176  133  101   65   41    11     6
R0 = 1.2    4706  1253  751  504  451  371  319  302  324  291  261  244   153    70
R0 = 3      2664   606  267  167  124  114  119  112  175  237  414  809  1464  2728

R0 = 3 has two humps. 2,664 runs never get past the first holder and 4,192 end at 13 or 14. The deterministic 13.24 lies inside the upper hump: 5,001 runs end at 12, 13 or 14, within 1.4 agents (a tenth of the population) of it. The curve describes half of the runs.
The mean falls in the valley. It is 8.38, and sizes 5 to 8 are the four rarest outcomes.
R0 = 1.2 has one hump, at 1, and a long shelf, like R0 = 0.8 on the other side of the threshold. Only 992 runs end at 6, 7 or 8, around the deterministic 6.65.

Computed check. Summing over every path of the same chain gives exact probabilities with no random number (fractions at N = 14, floating-point sums at N = 1,400). In all six cases the simulated counts stay within 2.65 standard errors of them, and the test rejects counts simulated with another R0.

Practice Exercise 3

From the R0 = 3 row: what share of the runs ends with at most 2 agents, and with at least 12? What share ends at 8 or 9, the whole numbers around the mean? Is “about 8 agents” a usable forecast?

Answer: 2,664 + 606 = 3,270 runs, 32.70%. 809 + 1,464 + 2,728 = 5,001 runs, 50.01%. 112 + 175 = 287 runs, 2.87%. No: the two ends hold 3,270 + 5,001 = 8,271 runs, 82.71%. For a two-humped outcome, state both humps and their shares, not the average.

4. A hundred times more agents, and the strongest objection

Synthetic. The same model with N = 1,400. “Near” means within a tenth of the population of the deterministic final size; an “early fade-out” ends at or below a tenth of the population (1 agent of 14, 140 of 1,400).

R0	Deterministic final size	Runs near it	Early fade-outs
0.8	4.96	9,995	9,995
1.2	443.03	1,097	8,455
3	1,316.74	6,634	3,366

Size does not remove the fade-out. At N = 1,400 and R0 = 3, 3,366 runs die out early; the exact probability is 0.3339. While nearly everybody is susceptible, each holder independently converts a random number of others, geometric with mean R0 because the holding time is exponential. That branching process dies out with probability 1/R0 when R0 exceeds 1: here 0.3333. With a fixed holding time the value would be 0.0595: the fade-out share rests on that assumption too. At R0 = 1.2 the count of fade-outs depends on the cut.

What size buys is separation. At N = 1,400 and R0 = 3 no run ended between 16 and 1,256 agents, and 90% of the 6,634 runs above the cut ended between 1,293 and 1,337: once a run has taken off, 1,316.74 is a sharp forecast. At N = 14 no gap separates the humps.

The fade-out risk falls with the number m of initial holders: the branching value is 1/R0 to the power m, and the exact probabilities at N = 1,400 and R0 = 3 are 0.3339, 0.1117, 0.0374 and 0.0042 for m = 1, 2, 3 and 5.

Practice Exercise 4

Compute the branching value for R0 = 3 and m = 1, 2, 3 and 5, and compare it with the exact probabilities above. What is the smallest m that brings the risk of an early fade-out below 5%?

Answer: 1/3 = 0.3333, 1/9 = 0.1111, 1/27 = 0.0370 and 1/243 = 0.0041, close to the exact values. Three holders: 0.0374 is below 0.05, while two leave 0.1117. Size alone is not enough; several initial holders are.

The strongest objection to this reading is that SIR may be the wrong family of models. It needs holders who convert other agents one by one. Personas that read the same governance files can instead acquire a belief together, from a common source: a broadcast, with no chain, no reproduction number and no fade-out. What is recorded leans that way: the two sample knowledge states name one teacher, the department’s head persona, and the two recorded adoptions both come from a document. This fixture cannot distinguish contagion from broadcast: two adoptions from a document and no chain are too few to tell. A transmission record would settle it, with six items per agent and proposition: who holds it, what it says, from whom it came, when it was adopted, when the agent stopped passing it on, and who else was exposed without adopting. The knowledge-state schema has keys for four and, through its assessment of one learner at a time, for part of the sixth; the belief snapshot has keys for three; neither records when a holder stops.

Practice Exercise 5

Classify each statement as observed, synthetic, analogy or unsupported, with the reason: (a) two current belief files rest on a message from another pipeline; (b) with R0 = 3, about a quarter of the runs among 14 agents never pass the belief on; (c) an agent that holds a belief is infected; (d) beliefs spread among the project’s personas with R0 above 1.

Answer: (a) Observed in the two files and in the script that writes them; the sender is read from the schema’s description, not from the messages. (b) Synthetic: 2,664 of 10,000 runs (exact probability 14/53), under assumed rates. (c) Analogy: a word borrowed from epidemiology for “holds it and passes it on”. (d) Unsupported: no file read here records a holder passing a belief to another agent, so R0 cannot be estimated; the six-item record would make it observable.

Sources, limits, and next check

Observed, pinned at Demerzel 4b03838800dcd4894f1e026461979bff1d985ca8: extract.py refuses to run if one of its 32 files, or either folder it lists (722 JSON files under state/, 11 contract schemas), differs from the pin. results.json lists the 42 self-checks; the README gives the commands and the controls.

Method references. Kermack, W. O., and McKendrick, A. G. (1927), “A Contribution to the Mathematical Theory of Epidemics”, Proceedings of the Royal Society of London, Series A 115, 700–721 (deterministic model, threshold, final size). Bailey, N. T. J. (1975), The Mathematical Theory of Infectious Diseases and its Applications, 2nd edition, Griffin. Whittle, P. (1955), “The Outcome of a Stochastic Epidemic: A Note on Bailey’s Paper”, Biometrika 42, 116–122 (minor outbreaks and 1/R0).

Not done. No rate was estimated from project data: the three values of R0 are assumed. Only the final size was simulated, and only uniform mixing: no network, no unequal agents, no second adoption. The shape and key tests cover JSON files under state/; beyond one YAML belief file (no evaluator; 10 of its 49 evidence items cite GPT-4o), no YAML, Markdown or other repository was read. No human has reviewed the subject matter of this lesson and no learner has tried it.

Next check. Keep the six-item record for a handful of propositions. If most adoptions name a document or a single teacher as their source, replace this model by a broadcast model. If they name other agents, estimate R0 from the adoptions per holder and compare the observed final sizes with section 3.