Modeling environmental-born melioidosis dynamics with recurrence: An application of optimal control

Melioidosis is a significant health problem in tropical and subtropical regions, especially in Southeast Asia and Northern Australia. Recurrent melioidosis is a major obstacle to eliminating the disease from the community in these nations. This work aims to propose and analyze a human melioidosis model with recurrent phenomena and an optimal control model by incorporating time-dependent control functions. The basic reproduction number ($R_0$) of the uncontrolled model is derived using the method of the next-generation matrix. Using the construction of a Lyapunov functional, we present the global asymptotic dynamics of the autonomous model in the presence of recurrent for both disease-free and endemic equilibria. The global asymptotic stability of the model’s equilibria shows the absence of a backward bifurcation for the model in both cases, whether in the absence or presence of relapse. The sensitivity analysis aims to identify the parameters that have the most significant impact on the model’s dynamics. Furthermore, qualitative analysis of the model’s global dynamics and the changing effect of the most influential parameters on $R_0$ are supported by numerical experiments, with the results being illustrated graphically. The model with time-dependent controls is analyzed using optimal control theory to assess the impact of various intervention strategies on the spread of the epidemic. The numerical results of the optimality system are carried out using the Forward–Backward Sweep method in Matlab. We also conducted a cost-effectiveness analysis using two approaches: the average cost-effectiveness ratio and the incremental cost-effectiveness ratio.