From Immensity to Eradication? Extending RAYOBETH Theory to Explain the Endgame of Human and Livestock Disease Outbreaks

Only two infectious diseases have ever been eradicated, smallpox in humans and rinderpest in cattle, while campaigns against polio, Guinea worm, malaria, measles and many livestock diseases have stalled within sight of the goal. The literature explains this with checklists of biological criteria and with the reproduction number, but it has no measure of what happens in the endgame, when cases are few. This article extends RAYOBETH Theory, a conditional theory of generative systems, to that endgame. It defines the endgame elasticity, the elasticity of next year’s cases with respect to this year’s when cases are low, and the control elasticity, the rate at which removal capacity falls as cases fall. A control elasticity rule proves that a disease without an external reservoir is eliminated in finite time if and only if the control elasticity is below the endgame elasticity, and that otherwise cases settle on a floor; mass control, whose effort is proportional to cases, therefore stalls whenever the endgame elasticity is below one. A reseeding theorem shows that any reservoir in animals, the environment or vaccines makes eradication impossible unless capacity is held permanently above the rate of reseeding, and a law of the weakest host dates human elimination by the slowest animal loop. Estimated across 5,349 country years of endgame data, the endgame elasticity is below one for smallpox (0.92), measles (0.83) and cholera (0.59) and close to one for Guinea worm and malaria. In two thirds of countries the last 1% of cases took longer to clear than the previous 9%. Global smallpox cases fell by 0.9% a year under mass vaccination from 1953 to 1966 and by 31% a year after targeted surveillance and containment began in 1967, when the net endgame elasticity rose from 0.90 to 1.10. Countries still infected with rinderpest fell by 1.5% a year before 1987 and 21% a year after, ending with Kenya in 2001. Guinea worm fell 34% a year to 22 human cases in 2015 and has since held a floor of 10 to 54 cases, sustained by 886 animal infections in 2023, as the reseeding theorem predicts. Notorious outbreaks, from Ebola in West Africa and cholera in Haiti to foot and mouth disease in the United Kingdom and avian influenza, fit the same rules.