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.