In this work, we study a mathematical model for the interaction of sensitive–resistant bacteria to antibiotics and analyse the effects of introducing random perturbations to this model. We compare the results of existence and stability of equilibrium solutions between the deterministic and stochastic formulations, and show that the conditions for the bacteria to die out are weaker in the stochastic model. Moreover, a corresponding optimal control problem is formulated for the unperturbed and the perturbed system, where the control variable is prophylaxis. The results of the optimal control problem reveal that, depending on the antibiotics, the costs of the prophylaxis, such as implementation, ordering and distribution, have to be much lower than the social costs, to achieve a bacterial resistance effective control.

中文翻译：

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细菌抗性数学模型中的随机扰动：分析和最佳控制

在这项工作中，我们研究了敏感性细菌与抗生素相互作用的数学模型，并分析了对该模型引入随机扰动的影响。我们比较了确定性公式和随机公式之间平衡解的存在性和稳定性的结果，并表明在随机模型中细菌死亡的条件较弱。此外，针对未受扰动和受扰动的系统，提出了相应的最优控制问题，其中控制变量是预防性的。最佳控制问题的结果表明，要实现细菌耐药性的有效控制，取决于抗生素的预防成本（例如实施，订购和分配）必须远低于社会成本。