Optimal control of a malaria model with long-lasting insecticide-treated nets

A deterministic multi-stage malaria model with a non-therapeutic control measure and the effect of loss of immunity due to the use of the Long-Lasting bednets with a control perspective is formulated and analyzed both theoretically and numerically. The model basic reproduction number is derived, and analytical results show that the model's equilibria are locally and globally asymptotically stable when certain threshold conditions are satisfied. Pontryagin's Maximum Principle with respect to a time dependent constant is used to derive the necessary conditions for the optimal usage of the Long-Lasting Insecticide-treated bednets (LLINs) to mitigate the malaria transmission dynamics. This is accomplished by introducing biologically admissible controls and $ϵ\mathrm{\%}$-approximate sub-optimal controls. Forward-backward fourth-order Runge-Kutta method is used to numerically solve the optimal control problem. We observe that the disadvantage (loss of immunity, even at its maximum) in the use of bednets is compensated by the benefit of the number of susceptible/infected individuals excluded from the malaria disease dynamics, the only danger being the poor use of the long-lasting bednets. Moreover, it is possible to get closer to the optimal results with a realistic strategy. The results from this study could help public health planners and policy decision-makers to design reachable and more practical malaria prevention programs "close" to the optimal strategy.