Hamiltonian structure and integrability of the susceptible-cleric-zombie-recovered epidemic model

The susceptible-cleric-zombie-recovered (SCZR) model is a compelling generalization of classical epidemic frameworks, introducing a cleric subclass that can cure infectives through direct intervention. This work uncovers the profound mathematical structure underlying this model. We demonstrate that the SCZR dynamics admit a noncanonical bi-Hamiltonian formulation, a definitive signature of complete integrability. By identifying three independent invariants of motion, we construct two distinct, compatible Poisson brackets and their corresponding Hamiltonian functions. Leveraging this integrable structure, we reduce the four-dimensional dynamics to a single first-order autonomous equation that is solvable by quadrature, yielding the formal analytical solution. Furthermore, we derive an explicit closed-form solution for the special case where the cleric and susceptible infection rates are identical (α=β). Analysis of the solution, supported by numerical illustrations, reveals a rich bifurcation structure. We demonstrate that the transition from a susceptible-infected-like (zombie-dominated) to a susceptible-infected-recovered-like (human survival) outcome not only depends on the critical rate ratio γ/α (where γ is the cleric-induced healing rate), but is also critically controlled by the initial cleric fraction c_{0} and the susceptible infection rate β. Our analytical framework provides a complete characterization of the SCZR model's mean-field behavior, establishing its integrability and offering a powerful baseline for studying more complex, nonintegrable variations.