The Dynamical Analysis For The Resistance Of Bacteria With Resistant Plasmids

The infection caused by resistant bacteria leads to the failure of the antibiotic treat-ment. The evolution of the resistance of bacteria against antibiotics is not a new problem, but would be needed to be solved. Horizontal gene transfer by resistant plasmid plays an enormous role in the dissemination of antibiotics resistant gene. Because of bacteri-a’s increasing resistance and slow development of new antibiotics, the research for the evolution of resistance may be urgent. In fact, the spread of resistant plasmid among different subspecies may not be limited in the laboratory, but turned into a mathematical problem, particularly a dynamical system problem. Thus, in this paper, we would like to investigate the long term behaviors of the population of bacteria with resistant fitness cost under different antibiotics pressure and its influence on the bacteria’s resistance level.The first chapter, Introduction, we review the history that the deterministic model is employed to describe the selection, evolution and dissemination of resistant bacteria and introduce four categories of classical mathematical models. In the preview research, the research focuses on the dissemination of resistant bacteria among human hosts and the dissemination dynamics with health care workers as carrier of resistant bacteria. Thus, we discuss the importance to employ the deterministic model to describe the long term behaviors of populations of different subspecies and its long term effect on bacteria’s resistance level.The second chapter is the generalized research for bacteria with wild-type bacteri-a and resistant plasmid-bearing bacteria. By establishing a generalized model for the population of wild-type bacteria and resistant plasmid-bearing bacteria, the existence of equilibria, the locally asymptotic stability of equilibria are provided. By providing the sufficient condition of existence of positive equilibria, we arrived two important mathe-matical results.·If positive equilibrium/equilibria exist, no closed orbits exist;·If positive equilibrium/equilibria and unique stable equilibria exist, the stable equi-libria is globally asymptotically stable.For the biological perspective, the resistance level of bacteria is nonzero and bounded and may not change periodically with respect to time t.The third chapter investigates dynamical behaviors on three two-dimensional invari-ant subspaces. We revise an existed four-dimensional system and partition it into three two-dimensional subspaces. The well-posedness, existence of equilibria and their glob-al asymptotic stability of dynamical systems on three invariant subspaces are provided. Thus, the wild-type bacteria and resistant bacteria may not only compete but also co-operate to survive under different antibiotic concentrations. The existence of subspecies heavily depends on the concentration of antibiotic and its fitness cost. This may provide a reasonable explanation for the contradiction the resistance gene may be dropped, but bacteria retain resistance gene.The forth chapter discusses the dynamical behavior of the evolution model of resis-tant plasmid-bearing bacteria. By employing qualitative analysis of ordinary differential equations, the well-posedness of the resistance is proved. Also, the existence of bound-ary equilibria and their locally asymptotic stability is provided. Moreover, we give the possibility of existence of positive equilibria. Moreover, we point out that the rate of plasmid segregation may be small, but it directly leads to the existence of the positive equilibrium. Biologically, the rate of plasmid segregation plays an enormous role in the evolution of resistance plasmid-bearing bacteria.The fifth chapter conducts the numerical simulation for the evolution model of re-sistant plasmid-bearing bacteria. By using three numerical simulations, we prove the mathematical theoretical results and provide the interval of existence and stability of e-quilibria. Furthermore, the three numerical simulations indicate that the existence of the positive equilibrium is coincided with bistable phenomenon and it may be common. Also, we provide the corresponding biological interpretation. Moreover, in the specific interval, the growth of one bacteria may not be decreased by using antibiotic, but increased.The sixth chapter concludes achieved results and innovations and points out the future research problem.The rational dosing of antibiotics is an inevitable problem that clinical doctors are facing everyday. In this manuscript, by detailedly studying the long term behaviors of the population of bacteria with resistant fitness cost under different antibiotics pressure and its influence on the bacteria’s resistance level, the theoretical results for the rational and efficient dosing may be a possible answer.