Random perturbations in a mathematical model of bacterial resistance: Analysis and optimal control

Hermann Mena; Lena-Maria Pfurtscheller; Jhoana P. Romero-Leiton; Hermann Mena; Lena-Maria Pfurtscheller; Jhoana P. Romero-Leiton

doi:10.3934/mbe.2020247

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 4477-4499. doi: 10.3934/mbe.2020247

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Random perturbations in a mathematical model of bacterial resistance: Analysis and optimal control

1.
Escuela de Ciencias Matemáticas y Computacionales, Universidad de Investigación de Tecnología Experimental Yachay, Ecuador
2.
Institut fur Mathematik, Universität Innsbruck, Austria

Received: 13 February 2020 Accepted: 17 June 2020 Published: 23 June 2020

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.
- sensitive bacteria,
- resistant bacteria,
- antibiotics,
- deterministic model,
- stochastic model,
- equilibrium solutions,
- stability,
- optimal control problem
Citation: Hermann Mena, Lena-Maria Pfurtscheller, Jhoana P. Romero-Leiton. Random perturbations in a mathematical model of bacterial resistance: Analysis and optimal control[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4477-4499. doi: 10.3934/mbe.2020247

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Abstract

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.

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