Since the Morris worm was released in 1988, Internet worms continue to be one of top security threats. For example, the Conficker worm infected 9 to 15 million machines in early 2009 and shut down the service of some critical government and medical networks. Moreover, it constructed a massive peer-to-peer (P2P) botnet. Botnets are zombie networks controlled by attackers setting out coordinated attacks. In recent years, botnets have become the number one threat to the Internet. The objective of this research is to characterize spatial-temporal infection structures of Internet worms, and apply the observations to study P2P-based botnets formed by worm infection. First, we infer temporal characteristics of the Internet worm infection structure, i.e., the host infection time and the worm infection sequence, and thus pinpoint patient zero or initially infected hosts. Specifically, we apply statistical estimation techniques on Darknet observations. We show analytically and empirically that our proposed estimators can significantly improve the inference accuracy. Second, we reveal two key spatial characteristics of the Internet worm infection structure, i.e., the number of children and the generation of the underlying tree topology formed by worm infection. Specifically, we apply probabilistic modeling methods and a sequential growth model. We show analytically and empirically that the number of children has asymptotically a geometric distribution with parameter 0.5, and the generation follows closely a Poisson distribution. Finally, we evaluate bot detection strategies and effects of user defenses in P2P-based botnets formed by worm infection. Specifically, we apply the observations of the number of children and demonstrate analytically and empirically that targeted detection that focuses on the nodes with the largest number of children is an efficient way to expose bots. However, we also point out that future botnets may self-stop scanning to weaken targeted detection, without greatly slowing down the speed of worm infection. We then extend the worm spatial infection structure and show empirically that user defenses, e.g., patching or cleaning, can significantly mitigate the robustness and the effectiveness of P2P-based botnets. To counterattack, we evaluate a simple measure by future botnets that enhances topology robustness through worm re-infection.