Endemicity and Periodicity
In a toy model of a measles-like disease, where beta (proxy for R0 or infectivity), birth rate, and population are such that we are above the Critical Community Size (CCS) threshold, we might see a prevalence plot like the red curve below:
(Generated at https://deepnote.com/app/idm/jonathanhhbs-Untitled-project-fda259dd-514c-4710-90d1-53849d90a543?POP=1000000&CBR=42&BETA=2).
For a CBR of 42 (which is relatively high), beta of 2, and a population of 1 million, we see a period of about 2 years.
But in that DeepNote dashboard one can adjust cbr and see that the period changes. Lowering just the CBR to, say, a more late-20th century Europe of say 17 produces a period of about 2000 days, though not constant:
Raising beta to 5 from 2 produces a period of about 900.
Adjusting population (assuming you stay above CCS threshold), doesn't seem to modify the period.
Theoretical work to explore periodicities of resonant SIR models are here (TBD).
- Should our models be tested in single node to confirm they exhibit periodicities consistent with theory or reference models?
- Is periodicity sensitive to other inputs? Which?
In full models of the real world, the "natural frequency" of the model is dominated by seasonal factors producing annual, biennial, or "multi-ennial" outbreaks. What if a model exhibits annual or biennial periodicity without seasonal forcing?
Note that the question of amplitude damping as a quantitative and measurable effect is omitted here, though I would note that this is something I've observed occasionally in my model version.