Budget-Constrained Optimal Control and Cost-Effectiveness of Vaccination, Vector Control, and Movement Restriction for Lumpy Skin Disease

Lumpy skin disease (LSD) is a rapidly spreading viral disease of cattle that threatens livestock productivity and agricultural trade across Africa, the Middle East, and Asia. Despite effective vaccines being available, evidence-based guidance on allocating limited veterinary budgets across competing interventions remains lacking, and no LSD study has yet combined dynamic transmission modelling with a hard budget constraint and formal cost-effectiveness analysis. We develop a deterministic ShVhEhIhRh–SvIv (SVEIRSI) compartmental model for LSD, embedded within a budget-constrained optimal control framework with three time-dependent controls—vaccination (u1), vector control (u2), and movement restriction (u3)—evaluated across all eight strategy combinations. The basic reproduction number is R0=2.7899. Pontryagin’s Maximum Principle characterises the optimality conditions. The resulting two-point boundary value problem is solved via forward–backward sweep with budget bisection over a 180-day horizon at three budget levels (WL=NAD5000, WM=NAD15,000, WH=NAD40,000). Vaccination dominates the optimal allocation at every budget, reducing peak infectious prevalence by 54–57% and averting 15,645–17,970 animal-days of cumulative burden relative to the uncontrolled baseline (I0=97,313 animal-days). Shadow-price analysis reveals that the budget constraint is non-binding across all three policy budget levels (actual optimal spend ≈ NAD 572), indicating that logistical (rather than financial) barriers are the true bottleneck, with the shadow price dropping to zero at W≈NAD1862. Incremental cost-effectiveness analysis places vaccination alone (S1), vaccination plus vector control (S12), and the combined strategy (S123) on the efficiency frontier, with ICERs of NAD 0.012, 0.015, and 0.166 per animal-day averted, respectively. One-way sensitivity analysis identifies vaccine efficacy ε as the dominant driver of cases averted (swing ≈85,601 animal-days), while global sensitivity analysis via Latin Hypercube Sampling identifies the midge biting rate a as the strongest driver of cumulative infectious burden (PRCC=0.926, p<0.001). These results provide actionable, quantitative guidance for LSD outbreak response in Namibia and comparable resource-limited settings.