Stability and persistence of delayed resistant and sensitive bacterial strains interaction under impulsive drug treatment

Abstract

Drug resistance occurs when living organisms such as bacteria, viruses, fungi and parasites change in ways that render the medications used incapable of curing the disease or medical conditions they cause. The microorganisms which have become resistant to most antimicrobials are commonly referred to as "superbugs". This development is a major concern to patients and physicians because a resistant infection may be vital, being able to spread to others, and impose huge costs to individuals and society. Here, we consider a model of the dynamic interaction between sensitive and resistant strains of pathogens in a nutrient limiting environment of the gastrointestinal tract. A delay τ in the process of conversion from a sensitive strain to a resistant strain is incorporated by addition of a delay in the rate equation of the resistant strain with an exponential factor to account for the probability that a resistant bacteria survives from the time t - τ to the time t. The system is analyzed for the stability of its various equilibrium solutions. The model is then expanded to take into account the effect of periodic drug treatment leading us to a system of delayed impulsive differential equations. Conditions are discovered under which the system is persistent, and stability of the susceptible strain free equilibrium and the bacterial free equilibrium can be expected.