Chemotactic cells known as microglia are involved in the inflammation associated with pathology in Alzheimer's disease (AD). We investigate conditions that lead to aggregation of microglia and formation of local accumulations of chemicals observed in AD senile plaques. We develop a model for chemotaxis in response to a combination of chemoattractant and chemorepellent signaling chemicals. Linear stability analysis and numerical simulations of the model predict that periodic patterns in cell and chemical distributions can evolve under local attraction, long-ranged repulsion, and other constraints on concentrations and diffusion coefficients of the chemotactic signals. Using biological parameters from the literature, we compare and discuss the applicability of this model to actual processes in AD.