Malaria and leptospirosis co-infection: A mathematical model analysis with optimal control and cost-effectiveness analysis

Malaria and leptospirosis are emerging vector-borne diseases that pose significant global health problems in tropical and subtropical regions. This study aimed to develop and analyze a mathematical model for the transmission dynamics of malaria-leptospirosis co-infection with optimal control measures. The model’s dynamics are examined through its two sub-models: one for malaria alone and the other for leptospirosis alone. We apply a next-generation matrix approach to derive the basic reproduction numbers for the sub-models. By using the reproduction number, we demonstrate the local and global asymptotic stability of both disease-free and endemic equilibria in these sub-models. We perform numerical experiments to validate the theoretical outcomes of the full co-infection model. The graphical results show that malaria-leptospirosis co-infection will be eradicated from the population through time if $R_{0ml}<1$. Conversely, if $R_{0ml}>1$, the co-infection will persist in the population. Furthermore, we investigate an optimal control model to demonstrate the impact of various time-dependent controls in reducing the spread of both diseases and their co-infection. We use the forward–backward sweep iterative method to perform numerical simulations of the optimal control problem. Our findings of the optimal control problem imply that strategy $D$, which incorporates all optimal controls, namely malaria prevention ${\omega }_1\left(t\right)$, leptospirosis prevention ${\omega }_2\left(t\right)$, insecticide control measure for malaria ${\omega }_3\left(t\right)$, control sanitation rate of the environment ${\omega }_4\left(t\right)$ is the most effective in minimizing our objective function. We also conduct a cost-effectiveness analysis to identify the predominant strategy in terms of cost among the optimal strategies.