  In this work, two mathematical models for malaria under resistance are
presented. More precisely, the first model shows the interaction between humans
and mosquitoes inside a patch under infection of malaria when the human
population is resistant to antimalarial drug and mosquitoes population is
resistant to insecticides. For the second model, human-mosquitoes population
movements in two patches is analyzed under the same malaria transmission
dynamic established in one patch. For a single patch, existence and stability
conditions for the equilibrium solutions in terms of the local basic
reproductive number are developed. These results reveal the existence of a
forward bifurcation and the global stability of disease-free equilibrium. In
the case of two patches, a theoretical and numerical framework on sensitivity
analysis of parameters is presented. After that, the use of antimalarial drugs
and insecticides are incorporated as control strategies and an optimal control
problem is formulated. Numerical experiments are carried out in both models to
show the feasibility of our theoretical results.
