Optimal Control and Cost-Effectiveness Analysis of Measles Transmission

Source abstract

This study develops and analyzes a deterministic compartmental model for measles transmission incorporating two-dose vaccination, imperfect dose-specific vaccine protection, breakthrough infection, and nonlinear saturated incidence. The population is divided into susceptible, first-dose vaccinated, second-dose vaccinated, exposed, infectious, and recovered classes. The fundamental mathematical properties of the model are established, including positivity and boundedness of solutions. The disease-free and endemic equilibria are characterized, and the basic reproduction number R0 is derived using the next-generation matrix approach. The disease-free equilibrium is globally asymptotically stable when R0<1, whereas for R0>1 the model admits a unique endemic equilibrium that is globally asymptotically stable relative to the interior of the biologically feasible region. Analytical sensitivity analysis identifies the local influence of the model parameters on R0. In particular, the saturation parameter does not enter R0, but numerical analysis shows that it can substantially affect the endemic infection level. The model is further extended to an optimal-control framework with three time-dependent, dimensionless controls representing intensified first-dose vaccination, intensified second-dose vaccination, and enhanced infectious-case removal. The existence of an optimal control is established, and Pontryagin’s Maximum Principle is used to derive the necessary optimality conditions. The resulting state–adjoint system is solved numerically using the Forward–Backward Sweep Method. Numerical simulations show that the combined optimal intervention reduces the exposed and infectious disease burdens over the finite control horizon, while sensitivity analysis demonstrates that the optimal-control profiles depend on the relative weights assigned to disease burden and intervention effort. Seven intervention combinations are subsequently compared using the model-based Average Cost-Effectiveness Ratio (ACER) and Incremental Cost-Effectiveness Ratio (ICER) measures. Under the baseline weighting scheme, first-dose vaccination alone has the smallest weighted intervention effort per case averted. After strong and extended dominance are accounted for, the incremental frontier consists of first-dose vaccination alone, first-dose vaccination combined with infectious-case removal, and the full three-control strategy. Increasing epidemiological benefit along this frontier is accompanied by increasing incremental weighted intervention effort. These results illustrate how dose-specific vaccination, nonlinear transmission, and infectious-case intervention can be integrated within a unified framework for comparing measles-control strategies while explicitly accounting for the assumptions underlying intervention effort.