Modeling the Evolution of Xenophobic Attitudes: A Fractional-Order Approach and Optimal Strategies

Xenophobia persists as a significant societal challenge with profound impacts on community cohesion, yet conventional policy approaches often prove insufficient. This research presents a Caputo fractional-order model to examine xenophobia dynamics, capturing memory effects. The model classifies populations into five distinct groups: Protected (P), Susceptible (S), Exposed (E), Affected (A), and Recovered (R). We rigorously establish solution positivity and boundedness, identify both xenophobia-free and endemic equilibrium states, and conduct comprehensive stability analysis to characterize long-term behavioral patterns. The study derives a threshold reproduction number governing xenophobia spread dynamics and formulates an associated optimal control problem. This framework evaluates two complementary intervention strategies: socio-economic initiatives (w₁(t)) addressing poverty and inequality, and legal-judicial measures (w₂(t)) strengthening law enforcement. Numerical simulations reveal that coordinated implementation (w₁=w₂=0.7) significantly reduces xenophobia prevalence, while lower fractional orders (κ→0.5) demonstrate enhanced recovery rates due to memory effects. These results strongly support integrated policy interventions combining socio-economic and legal approaches to effectively mitigate xenophobic transmission and foster inclusive communities.