Analysis of a double age-structured anthrax model with saturating recruitment

Anthrax is a severe zoonotic disease of significant ecological and epidemiological concern. To better understand its transmission dynamics, we formulate and analyze a double age-structured anthrax model with a general saturating recruitment, in which physiological and infection age are incorporated to capture demographic and epidemiological heterogeneity. The model is rewritten as an abstract Cauchy problem, and well-posedness is established via the theory of non-densely defined operators. The local stability of its steady states is analyzed via its characteristic equation and Lumer–Phillips theorem. To characterize threshold dynamics, we construct the next-generation operator through two approaches: Volterra integral formulation and an operator-theoretic C–B decomposition. In addition, a demographic reproduction number K₀ is introduced to describe population persistence. Although both approaches yield the same explicit expression for the basic reproduction number R₀, they provide distinct analytical interpretations. The C–B framework further links the spectral bound of the linearized operator directly to R₀, allowing stability analysis without Lyapunov functionals. Moreover, we show that when R₀<1 all infected components vanish and, provided K₀>1, the disease-free steady state is globally attractive. Numerical simulations support the theoretical results, reveal non-monotonic effects of age-dependent transmission parameters on R₀, and suggest that interventions targeting environmental contamination may be more effective than directly reducing animal contact with contaminated environments.