Abstract
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 < mu < 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.
Highlights
According to [1], emergence of microorganisms resistant to multidrug has become a global concern
We extend our understanding of antimicrobial resistance (AMR) by analyzing further the delay differential equation model of drug resistance proposed in [6] for its stability and Hopf bifurcation development
The model is modified to take into account the impulsive antibacterial drug treatment, which periodically reduces the level of sensitive strain which is susceptible to the drugs, as well as the sudden increases in the resistant strain as the susceptible population being killed by the antibiotics leaves the resistant strain to grow, on the limiting nutrients, and multiply in periodic bursts
Summary
According to [1], emergence of microorganisms resistant to multidrug has become a global concern. We extend our understanding of AMR by analyzing further the delay differential equation model of drug resistance proposed in [6] for its stability and Hopf bifurcation development. The model is modified to take into account the impulsive antibacterial drug treatment, which periodically reduces the level of sensitive strain which is susceptible to the drugs, as well as the sudden increases in the resistant strain as the susceptible population being killed by the antibiotics leaves the resistant strain to grow, on the limiting nutrients, and multiply in periodic bursts. Nutrients are removed by natural means at the rate ω3z, utilized for growth of sensitive strain and the resistant strain according to the second and last terms of (3), respectively.
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