V3I7P33

A model of football-fan reaction to a refereeing decision: application to Argentina–Egypt, FIFA World Cup 2026 Round of 16

Felistus Shitakha^1*, Eunice Mueni Musyoki², Nancy Matendechere Imbusi³, Consolata Achieng Muganda⁴

Abstract

On 7 July 2026, in the Round of 16 of the FIFA World Cup at Atlanta Stadium, Argentina recovered from a two-goal deficit to beat Egypt 3–2 after a Video Assistant Referee (VAR) intervention had disallowed what would have been Egypt’s second goal. The Egyptian Football Association alleged biased officiating; FIFA’s refereeing chief replied two days later. We model the resulting diffusion of grievance through the global audience with a deterministic seven-compartment system –––––– (susceptible viewers, exposed deliberators, discerning evaluators, aggrieved, justifiers, propagators, refractory). Match-specific drivers—the disallowed goal, the memory of previous officiating decisions, the presence of Lionel Messi and the audience inflation it causes, the sporting-power asymmetry between the two nations, and the decisive late goal—enter as explicitly identified parameters rather than as ad hoc forcing. We establish well-posedness on a bounded, positively invariant region; obtain the outrage reproduction number  in closed form by the next-generation method; prove local and, in the absence of re-ignition, global asymptotic stability of the outrage-free equilibrium; and reduce endemic equilibrium existence to a quadratic whose constant coefficient changes sign exactly at . A Castillo-Chavez–Song centre-manifold analysis shows that the bifurcation at  is backward whenever the re-ignition amplification factor  exceeds , producing a saddle-node at  and a bistable window in which entrenched outrage coexists with the outrage-free state. Sensitivity indices and a Latin-hypercube/PRCC analysis identify the transmission rate , the propagator amplification weight  and the propagator exit rate  as dominant. An optimal three-control policy, characterised by Pontryagin’s Maximum Principle, reduces the cumulative aggrieved and propagator burdens by  and . Finally, for a fixed response with  there is a critical delay  days beyond which the identical policy fails outright; FIFA’s actual two-day pause raised the cumulative aggrieved burden by . Driving  below unity is neither sufficient nor, if it is done late, of any use at all.

Keywords:

Compartmental model; outrage dynamics; video assistant referee; backward bifurcation; centre manifold; optimal control; Pontryagin Maximum Principle; critical delay; sensitivity analysis.