Stability, Hopf bifurcation and effects of impulsive antibiotic treatments in a model of drug resistance with conversion delay

Abstract

© 2019, The Author(s). Drug resistance has become a problem of grave concern for members of the medical profession for many decades. More and more bacterial infections cannot be contained. Therefore it is of imperative importance for us to be able to keep drug resistance under control. Mathematical models can help us to discover possible treatment strategies that could alleviate the problem. In this paper, the dynamic behavior of drug resistance is investigated by studying a model system of differential equations incorporating a delay in the process whereby the sensitive bacteria (bacteria that antibiotics can still attack) is converted into the resistant bacterial strain through plasmid transfers. We give the conditions under which a Hopf bifurcation occurs, leading to a periodic solution. The result indicates that the conversion rate and the delay play a significant role in the development of drug resistance. Also, the impact of periodic antibiotic intakes is taken into account, making the model an impulsive one. Each time a patient takes antibiotics, a fraction μ (0 < μ< 1) of sensitive bacteria dies, but resistant bacteria are left to grow and multiply in periodic bursts. Analysis is carried out on the impulsive system to find the stability criteria for the steady-state solution where bacterial strains are washed out. Numerical simulation is carried out to support our theoretical predictions.