Modeling dual-colony Nosema transmission in honeybees: The role of distributed delays and antiviral treatment

In this paper, we develop a comprehensive mathematical model to investigate the transmission dynamics of dual Nosema infections (Nosema apis and Nosema ceranae) in two interacting honeybee colonies. The model incorporates distributed time delays to capture biological realism in latency, incubation, and parasite maturation periods, and includes an environmental pathogen compartment to account for indirect, environment-mediated transmission. First, we analyze a simplified ordinary differential equation (ODE) version of the model, thereby deriving the basic reproduction number ${\mathcal{R}}_0$ and establishing the global asymptotic stability of both disease-free and endemic equilibria using Lyapunov functions. Then, the analysis is extended to the full distributed-delay system, where we derive the delayed basic reproduction number ${\mathcal{R}}_0^d$ and prove the global stability of its equilibria via carefully constructed Lyapunov functionals. A sensitivity analysis identifies key parameters—most notably transmission rates, spore shedding rates, and natural mortality—that dominate the infection dynamics. Furthermore, we introduce an antiviral treatment term to quantify the efficacy required to drive ${\mathcal{R}}_0^d$ below unity and achieve disease eradication. Numerical simulations validate the analytical results and illustrate how distributed delays and treatment interventions critically influence the long-term disease outcomes. The study provides a robust theoretical framework to understand Nosema spread in multi-colony settings. Its key contributions are as follows: (1) The derivation of an additive basic reproduction number reveals the necessity of apiary-wide management; (2) provides rigorous global stability proofs for the delayed system; and (3) provides actionable quantitative insights to design effective apiary management, identify critical intervention targets, and establish treatment efficacy thresholds for disease eradication.