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Malaria is one of the world’s most serious health problems because of the increasing number of cases every year. First, we discuss a deterministic model of epidemic SIR-SI spread of malaria with the intervention of bed nets and fumigation. We found that the malaria-free equilibrium is locally asymptotically stable (LAS) when $\mathcal{R}_{0} <1$ and unstable otherwise. A malaria endemic equilibrium exists and is LAS when $\mathcal{R}_{0} >1$. Sensitivity analysis of $\mathcal{R}_{0} $ shows that the use of bed nets and fumigation can reduce $\mathcal{R}_{0}$. We modify the previous model into a stochastic differential equation model to understand the effect of random environmental factors on the spread of malaria. Numerical simulations show that when $\mathcal{R}_{0} >1$, a greater value of noise intensity σ generates a solution that is different from a deterministic solution; when $\mathcal{R}_{0} <1$, regardless of the σ value, the solution approaches a deterministic solution. Then the deterministic model was modified into an optimal control model to determine the best strategy in controlling the spread of malaria by using fumigation as the control variable. Numerical simulations show that periodic fumigations cost less than constant intervention and can reduce the number of infected humans. Priority is given to the endemic prevention strategy rather than to the endemic reduction strategy. For more effective intervention, the value of $\mathcal{R}_{0}$ should receive close attention. A potentially endemic ($\mathcal{R}_{0}>1$) environment requires more frequent fumigation than an environment that is not potentially endemic ($\mathcal{R}_{0}<1$). A combination of the use of bed nets and fumigation can reduce the number of infected individuals at minimal cost.

Optimal control in a malaria model: intervention of fumigation and bed nets