The increase in antibiotic resistance continues to pose a public health risk as very few new antibiotics are being produced, and bacteria resistant to currently prescribed antibiotics is growing. Within a typical hospital setting, one may find patients colonized with bacteria resistant to a single antibiotic, or, of a more emergent threat, patients may be colonized with bacteria resistant to multiple antibiotics. Precautions have been implemented to try to prevent the growth and spread of antimicrobial resistance such as a reduction in the distribution of antibiotics and increased hand washing and barrier preventions; however, the rise of this resistance is still evident. As a result, there is a new movement to try to re-examine the need for the development of new antibiotics. In this paper, we use mathematical models to study the possible benefits of implementing a new antibiotic in this setting; through these models, we examine the use of a new antibiotic that is distributed in various ways and how this could reduce total resistance in the hospital. We compare several different models in which patients colonized with both single and dual-resistant bacteria are present, including a model with no additional treatment protocols for the population colonized with dual-resistant bacteria as well as models including isolation and/or treatment with a new antibiotic. We examine the benefits and limitations of each scenario in the simulations presented.

In this paper, we investigate three particular algorithms: a stochastic simulation algorithm (SSA), and explicit and implicit tau-leaping algorithms. To compare these methods, we used them to analyze two infection models: a Vancomycin-resistant enterococcus (VRE) infection model at the population level, and a Human Immunodeficiency Virus (HIV) within host infection model. While the first has a low species count and few transitions, the second is more complex with a comparable number of species involved. The relative efficiency of each algorithm is determined based on computational time and degree of precision required. The numerical results suggest that all three algorithms have the similar computational efficiency for the simpler VRE model, and the SSA is the best choice due to its simplicity and accuracy. In addition, we have found that with the larger and more complex HIV model, implementation and modification of tau-Leaping methods are preferred.