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Highlights • A new compartmental model for the co-dynamics of measles and dysentery diarrhea is proposed. • The system exhibits a backward bifurcation. • Optimal control and cost-effectiveness analysis are addressed. • Numerical simulations illustrate theoretical results and indicate suggestions for disease elimination. • The most cost-effective strategy for the control of the diseases is recommended. Abstract In this paper we propose a co-dynamics deterministic system for measles and dysentery diarrhea diseases in a single host population. Using center manifold theory, we show that the co-dynamics model may exhibit a backward bifurcation for some parameter values. Numerical simulations of the system show that the two diseases always coexist if R 0 > 1. The system is extended to include time-dependent control-variables: vaccination, treatment and sanitation of the environment, to minimize the number of infected humans and the cost of implementation of the controls. The Pontryagin Maximum Principle was employed to find the necessary conditions for the existence of the optimal controls. The numerical simulations show that the effective controls could reduce the diseases in the community. The incremental cost-effectiveness ratio was used to quantify the cost-effectiveness analysis. It is found that the control strategy which implements vaccination, treatment of dysentery diarrhea and sanitation of the environment is the most cost-effective. [ABSTRACT FROM AUTHOR]