The United States has a long-standing issue with measles, even with high levels of vaccination.  In this study, we develop and analyze a deterministic mathematical model to investigate measles transmission dynamics in a heterogeneous population.  To capture both the burden of disease and its transmission, we include some of the most important epidemiological and demographic processes, such as recruitment, transitions between age groups, transitions between risk classes, and disease-induced mortality.  We define the basic analytical properties of the model, such as positivity and boundedness of solutions, and calculate the basic reproduction number to describe the potential of transmission.  We examine both local and global stability of the disease-free equilibrium (DFE) and establish the conditions under which measles can be eliminated, which includes a clear herd immunity level.  We supplement the analytical results with the numerical simulation and the global sensitivity analysis to study the impact of the important parameters, determine the effect of the vaccination coverage, and the impact of the intervention strategies.  According to this study, to eliminate measles from the United States, at least 94 % individuals should be vaccinated (herd immunity threshold), consistent with existing literature.  Our results give a quantitative understanding of the role of age and risk heterogeneity in determining the processes of measuring the transmission and control of measles, and offer an informative framework to guide the processes of vaccination planning and decision-making in health on the part of the public in a heterogeneous population.  Simulations also show that incorporating a single supplementary immunization activity (SIA) achieving 80% coverage of the susceptible population, alongside routine vaccination, reduces cumulative measles cases by approximately 72% over the 48-week outbreak period.  We further analyze an optimal-control extension incorporating two intervention strategies: an awareness-based transmission-reduction program and a continuous vaccination program.  Numerical simulations show that both strategies reduce infection burden, while the cost-effectiveness analysis identifies vaccination as the most cost-effective intervention under the assumed cost structure.