Optimal Control and Cost‐Effectiveness in a Dynamic Vaccination Model for COVID‐19

We develop a deterministic compartmental model of COVID‐19 transmission structured by vaccination status, incorporating primary vaccination (two doses), booster vaccination and independently waning protection at each of the three vaccination stages. The model comprises nine living compartments and 10 state variables. We derive the basic reproduction number via the next‐generation matrix method and show that the disease‐free equilibrium is locally stable for , with global stability established on the region of state space corresponding to an established vaccination programme. Bifurcation analysis reveals a backward, rather than forward, bifurcation at : Numerical continuation locates a fold at , below which no alternative equilibrium was located and the disease‐free state governs long‐run behaviour, but between the fold and , a stable high‐burden equilibrium coexists with it, so alone does not guarantee elimination from an arbitrary initial condition. A global sensitivity analysis shows that vaccine efficacy at each dose, not merely coverage, ranks among the strongest determinants of , extending a local analysis that omits the vaccination pathway entirely. Using optimal control theory and a forward‐backward sweep, either sustained personal protection or rapid first‐dose vaccination alone suppresses transmission almost completely under idealised control bounds; under operationally realistic administration‐rate limits, this advantage largely disappears, and a combined strategy becomes preferable both epidemiologically and economically once cost‐index units are translated into pounds sterling using UK unit‐cost data, at a marginal cost within conventional UK willingness‐to‐pay thresholds per disability‐adjusted life year averted. A face‐validity comparison shows the model’s realistic‐bound hospital‐occupancy outcomes falling within the range actually observed during comparable UK epidemic waves. These results indicate that the reproduction number alone is an insufficient guide to elimination policy in vaccination‐structured models of this type and offer a template for jointly evaluating the epidemiological, dynamical and economic performance of adaptive vaccination strategies.