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

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 R 0 < 1 and unstable otherwise. A malaria endemic equilibrium exists and is LAS when R 0 > 1. Sensitivity analysis of R 0 shows that the use of bed nets and fumigation can reduce 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 R 0 > 1, a greater value of noise intensity σ generates a solution that is different from a deterministic solution; when 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 R 0 should receive close attention. A potentially endemic (R 0 > 1) environment requires more frequent fumigation than an environment that is not potentially endemic (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 problem arises from illegal poaching of southern white rhino mathematical model

“…To implement the intervention directly in the field, it is easier to do it as a constant effort for a specific time interval. Several authors have been implementing this in epidemiological models [19,28]. Another limitation is that we do not consider the government's possibility to open up the privatization of rhino hunting.…”

“…Optimal control theory is usually applied to control the spread of a disease in an epidemiology model, as proposed in [18,19]. It can also be used in a predator-prey model with infections among prey or predator populations, known as an eco-epidemiology model.…”

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“…In this paper, a quadratic objective functional is implemented to measuring the control cost, since we assume that the cost for intervention are nonlinear. This quadratic cost function has been frequently used by many authors [8,10,19,20,21].…”

“…We update the control variables using (5) and iterate this scheme until the convergence criteria achieved. Please see [8,20,21,[24][25][26][27][28] for more example on the implementation of this scheme.…”

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