Stability Analysis and Optimal Control of Endemic Malaria Disease Transmission Model with Cost-Effective Strategies

Abstract

In this study, a non-linear system of ordinary differential equation model that describes the dynamics of malaria disease transmission is formulated and analyzed. Conditions are derived from the existence of disease-free and endemic equilibria. The basic reproduction number R0 of the model is obtained, and we investigated that it is the threshold parameter between the extinction and persistence of the disease. If R0 is less than unity, then the disease-free equilibrium point is both locally and globally asymptotically stable resulting in the disease removing out of the host populations. The disease can persist whenever R0 is greater than unity and the conditions for the existence of both forward and backward bifurcation at R0 is equal to unity are derived. Sensitivity analysis is also performed and the important parameter that derive the disease dynamics is identified. Furthermore, optimal combinations of time dependent control measures are incorporated to the model, and we derived the necessary conditions of optimal control using Pontryagins’s maximum principal theory. Numerical simulations were conducted using MATLAB to confirm our analytical results. Our findings were that, malaria may be controlled by reducing the requirement rate of mosquito populations and the use of a combination of vaccination, insecticide treated net ITN, indoor residual spray IRS and active treatment or strategy d can also help to reduce the number of populations with malaria symptoms to zero. We also find that the same strategy that is, strategy d proves to be efficacious and cost-effective.