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

Abstract

Abstract 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 $\mathcal{R}_{0} &lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math> and unstable otherwise. A malaria endemic equilibrium exists and is LAS when $\mathcal{R}_{0} &gt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:math> . Sensitivity analysis of $\mathcal{R}_{0} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> shows that the use of bed nets and fumigation can reduce $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> . 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 $\mathcal{R}_{0} &gt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:math> , a greater value of noise intensity σ generates a solution that is different from a deterministic solution; when $\mathcal{R}_{0} &lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math> , 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 $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> should receive close attention. A potentially endemic ( $\mathcal{R}_{0}&gt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:math> ) environment requires more frequent fumigation than an environment that is not potentially endemic ( $\mathcal{R}_{0}&lt;1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math> ). A combination of the use of bed nets and fumigation can reduce the number of infected individuals at minimal cost.