Optimal control of the treatment frequency in a stochastic model of tuberculosis
AbstractThis paper presents a stochastic model of the Tuberculosis (TB) infection with treatment in a population composed of four individuals compartments: susceptible individuals, latent infected individuals, active infected individuals and recovered individuals after the therapy. The mathematical model of TB infection includes in addition to the deterministic term, a stochastic term that translates the random noise. The random nature of this model is due to the fact that the contraction of the Mycobacterium Tuberculosis, the vector agent of the TB infection and his transmission within the population are done in a random manner according to the variable efficiency of control of the immune system of the individuals. While supposing that only the active infected individuals transmit the infection, their survey in order to observe the rules of hygiene, to adopt a positive behavior with respect to the susceptible individuals (who must also take some precautions) to follow the treatment up to finish, constitute measures of adequate control. A preliminary survey of the model is made before approaching the crucial left of the topic. The main objective of this paper is to control the frequency of treatment in a stochastic model of the TB infection while minimizing the cost of the measures. We formulate an optimal control problem that consists in minimizing the relative cost to the dynamics of the model in order to reduce the prevalence and the mortality due to this infection. The optimal control problem is solved by applying the Projection Stochastic Gradient Method in order to find the optimal numerical solution. Finally, we provide some numerical simulations of the controlled model.
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