We develop a class of models to represent the dynamics of a virus spreading in a cellphone network, employing a taxonomy that includes five key characteristics. Based on the resulting dynamics governing the spread, we present optimization models to rapidly detect the virus, subject to resource limitations. We consider two goals, maximizing the probability of detecting a virus by a time threshold and minimizing the expected time to detection, which can be applied to all spread models we consider. We establish a submodularity result for these two objective functions that ensures that a greedy heuristic yields a well-known constant-factor (63%) approximation. We relate the latter optimization problem, under a specific virus-spread mechanism from our class of models, to a classic facility-location model. Using data from a large carrier, we build several base cellphone contact networks of different scale. We then rescale these base networks using the so-called c-core decomposition that removes vertices of low degree in a recursive way. We further show that this down-sampling strategy preserves, in general, the topological properties of the base networks, based on testing several measures. For the objective that maximizes the probability that we detect a virus by a time threshold, we provide a sample-average optimization model that yields an asymptotically-optimal design for locating the detection devices, as the number of samples grows large. To choose a relevant time threshold, we perform a simulation for some spread models. We then test the performance of our proposed solution methods by solving the presented optimization models for some spread dynamics using some of the contact networks built after the c-core decomposition. The computational results show that the greedy algorithm is an efficient way to solve the corresponding sample-average approximation model, and the greedy solutions outperform some other simple solution approaches. / text

The following essay describes and compares Fisher's Randomization Test and Neyman's average treatment test, with the intention of concluding an easily understood blueprint for the comprehension of the practical execution of the tests and the conditions surrounding them. Focus will also be directed towards the tests' different implications on statistical inference and how the design of a study in relation to assumptions affects the external validity of the results. The essay is structured so that firstly the tests are presented and evaluated, then their different advantages and limitations are put against each other before they are applied to a data set as a practical example. Lastly the results obtained from the data set are compared in the Discussion section. The example used in this paper, which compares cigarette consumption after having treated one group with nicotine patches and another with fake nicotine patches, shows a decrease in cigarette consumption for both tests. The tests differ however, as the result from the Neyman test can be made valid for the population of interest. Fisher's test on the other hand only identifies the effect derived from the sample, consequently the test cannot draw conclusions about the population of heavy smokers. In short, the findings of this paper suggests that a combined use of the two tests would be the most appropriate way to test for treatment effect. Firstly one could use the Fisher test to check if any effect at all exist in the experiment, and then one could use the Neyman test to compensate the findings of the Fisher test, by estimating an average treatment effect for example.