A Coinfection Model of Leptospirosis and Melioidosis With Optimal Control

Leptospirosis and melioidosis are emerging tropical diseases that are seriously affecting both human and animal populations worldwide. The actual incidence and fatal cases of the diseases are underreported due to a lack of awareness of the diseases, underuse of clinical microbiology laboratories test, and limitations of the model. In this paper, a new deterministic mathematical model for the coinfection of leptospirosis and melioidosis with optimal controls is presented. Based on the next-generation matrix approach, the basic reproduction numbers for the coinfection model as well as for submodels are computed to analyze their dynamics behavior. The disease-free equilibrium point of the melioidosis-only submodel is proven to be globally asymptotically stable when the basic reproduction number (R0m) is less than unity, whereas the existence of its unique positive endemic equilibrium is shown if R0m>1. Based on the center manifold theory, the endemic equilibrium point of the leptospirosis-only submodel is proven to be locally asymptotically stable when the basic reproduction number (R0l) is greater than unity. The disease-free equilibrium point of the full model is locally asymptotically stable whenever the basic reproduction number (R0ml) less than unity. Sensitivity analysis for the basic reproduction number of the model is performed to determine the most influencing parameters on the transmission dynamics of the model. Furthermore, the model was extended into an optimal control problem by incorporating four time-dependent control functions. Pontryagin’s maximum principle was used to derive the optimality system for the optimal control problem. The optimality system was simulated using the forward–backward sweep method to show the effectiveness and cost-effectiveness of different optimal control strategies in combating the burden of leptospirosis–melioidosis coinfection. The incremental cost-effectiveness ratio was applied to determine the most cost-effective strategy. The numerical results revealed that Strategy 6 which implements a combination of all optimal control measures is the most effective strategy for minimizing the spread of the coinfection of the epidemics, whereas Strategy 1 which implements rodenticide control measure is the most effective when available resources are limited.