Abstract
<p>Nosocomial infections caused by antibiotic resistant bacteria are a major threat to global public health today. In order to understand the diverse factors contributing to hospital acquired antibiotic resistant infections, we develop some mathematical models and address the theoretical, numerical and stochastic aspects of such models. In Chapter 2, both deterministic and stochastic mathematical models are developed to explore the roles that antibiotic exposure and environmental contamination play in the transmission dynamics of nosocomial infections in hospitals. Uncolonized patients without or with antibiotic exposure, colonized patients without or with antibiotic exposure, uncontaminated and contaminated health-care workers, and free-living Methicillin-resistant {\it Staphylococcus aureus} (MRSA) are included in the models. Under the assumption that there is no admission of the colonized patients, the basic reproduction number $R_0$ is calculated. We prove that when $R_0<1$, the infection-free equilibrium is globally asymptotically stable; when $R_0>1$, the infection is uniformly persistent. Numerical simulations show that environmental cleaning is the most important intervention. Increasing the stay of colonized patients with antibiotic exposure in hospitals will increase the prevalence of MRSA, which implies to treat patients with antibiotic exposure as efficiently and quickly as possible. Screening and isolating colonized patients at admission, and improving compliance with hand hygiene are also important control strategies. In Chapter 3, we extend the deterministic model developed in chapter 2. The extended model with periodic antibiotic prescribing rate is constructed to study the seasonality of Methicillin-resistant {\it Staphylococcus aureus} (MRSA) infections taking antibiotic exposure and environmental contamination into consideration. The basic reproduction number $R_0$ for the periodic model is also calculated under the assumption that there are only uncolonized patients with antibiotic exposure at admission. Sensitivity analysis of $R_0$ with respect to some essential parameters is performed. It is also shown that the infection would go to extinction if the basic reproduction number is less than unity and would persist if it is greater than unity. Numerical simulations indicate that environmental cleaning is the most important intervention to control the infection, which emphasizes the effect of environmental contamination in MRSA infections. It is also important to highlight the importance of effective antimicrobial stewardship programs, to increase active screening at admission and subsequent isolation of positive cases, and to treat patients quickly and efficiently. In Chapter 4, based on the results obtained from previous chapters, we apply the optimal control theory to the seven-compartment system of ordinary differential equations to minimize the numbers of colonized patients and bacteria in the environment while minimizing the cost associated with environmental cleaning rate and antibiotic use in a particular time period. Characterizations of optimal control strategies are formulated, and how hospitals should adjust their strategies when different hospital scenarios happen is discussed. Numerical simulations strongly suggest that environmental cleaning rate is key in the control of MRSA infections and hospital should use antibiotics as properly and little as possible. Meanwhile, how to treat colonized patients especially with antibiotic exposure as quickly and efficiently as possible is a big challenge in controlling MRSA infections. Screening and subsequent isolation can be an effective intervention supplement. In the last chapter, we summarized the results of the thesis and discuss some future studies.</p>