In this thesis, we investigate the optimal experimental design of some common biological experiments. The theory of optimal experimental design is a statistical tool that allows us to determine the optimal experimental protocol to gain the most information about a particular process, given constraints on resources. We focus on determining the optimal design for experiments where the underlying model is a Markov chain - a particularly useful stochastic model. Markov chains are commonly used to represent a range of biological systems, for example: the evolution and spread of populations and disease, competition between species, and evolutionary genetics. There has been little research into the optimal experimental design of systems where the underlying process is modelled as a Markov chain, which is surprising given their suitability for representing the random behaviour of many natural processes. While the first paper to consider the optimal experimental design of a system where the underlying process was modelled as a Markov chain was published in the mid 1980's, this research area has only recently started to receive significant attention. Current methods of evaluating the optimal experimental design within a Bayesian framework can be computationally inefficient, or infeasible. This is due to the need for many evaluations of the posterior distribution, and thus, the model likelihood - which is computationally intensive for most non-linear stochastic processes. We implement an existing method for determining the optimal Bayesian experimental design to a common epidemic model, which has not been considered in a Bayesian framework previously. This method avoids computationally costly likelihood evaluations by implementing a likelihood-free approach to obtain the posterior distribution, known as Approximate Bayesian Computation (ABC). ABC is a class of methods which uses model simulations to estimate the posterior distribution. While this approach to optimal Bayesian experimental design has some advantages, we also note some disadvantages in its implementation. Having noted some drawbacks associated with the current approach to optimal Bayesian experimental design, we propose a new method - called ABCdE - which is more efficient, and easier to implement. ABCdE uses ABC methods to calculate the utility of all designs in a specified region of the design space. For problems with a low-dimensional design space, it evaluates the optimal design in significantly less computation time than the existing methods. We apply ABCdE to some common epidemic models, and compare the optimal Bayesian experimental designs to those published in the literature using existing methods. We present a comparison of how well the designs - obtained from each of the different methods - performs when used for statistical inference. In each case, the optimal designs obtained via ABCdE are similar to those obtained via existing methods, and the statistical performance is indistinguishable. The main applications we consider are concerned with group dose-response challenge experiments. A group dose-response challenge experiment is an experiment in which we expose subjects to a range of doses of an infectious agent or bacteria (or drug), and measure the number that are infected (or, the response) at each dose. These experiments are routinely used to quantify the infectivity or harmful (or safe) levels of an infectious agent or bacteria (e.g., minimum dose required to infect 50% of the population), or the efficacy of a drug. We focus particularly on the introduction of the bacteria Campylobacter jejuni to chickens. C. jejuni can be spread from animals to humans, and is the species most commonly associated with enteric (intestinal) disease in humans. By quantifying the dose-response relationship of the bacteria in chickens - via group dose-response challenge experiments - we can determine the safe levels of bacteria in chickens with the aim to minimise, or eradicate, the risk of transmission amongst the flock, and thus, to humans. Thus, accurate estimates of the dose-response relationship are crucial - and can be obtained efficiently by considering the optimal experimental design. However, the statistical analysis of most dose-response experiments assume that the subjects are independent. Chickens engage in copraphagic activity (oral ingestion of faecal matter), and are social animals meaning they must be housed in groups. Thus, oral-faecal transmission of the bacteria may be present in these experiments, violating the independence assumption and altering the measured dose-response relationship. We use a Markov chain model to represent the dynamics of these experiments, accounting for the latency period of the bacteria, and the transmission between chickens. We determine the optimal experimental design for a range of models, and describe the relationship between different model aspects and the resulting designs.