This paper studies the dynamics of epidemic spreading on complex networks, focusing on the difference between exponential and scale-free (SF) networks. In exponential networks, an epidemic threshold exists, separating an infected phase from a healthy phase. The prevalence of infections decreases to zero at the threshold with linear behavior typical of a mean-field critical point. In contrast, SF networks, such as the Barabási-Albert model, do not exhibit an epidemic threshold. This implies that infections can spread and persist regardless of the infection rate. The absence of a threshold is due to the presence of highly connected nodes, which dominate the network's connectivity distribution. The study uses analytical methods and large-scale simulations to analyze the susceptible-infected-susceptible (SIS) model on these networks. For SF networks with 0 < γ ≤ 1, the model shows no threshold and infections can spread throughout the network. For 1 < γ ≤ 2, a threshold appears but is approached with a vanishing slope, indicating no critical behavior. For γ > 2, the usual critical behavior is recovered. The results have implications for understanding computer virus epidemics and other spreading phenomena on communication and social networks. The absence of an epidemic threshold in SF networks changes many standard conclusions about epidemic spreading. However, this is balanced by the exponentially small prevalence at low spreading rates. The findings suggest that SF networks, despite their lack of a threshold, can still be controlled due to their low prevalence at low infection rates. The study also highlights the importance of network topology in epidemic modeling and the potential for applying these results to real-world systems like the Internet and the World Wide Web.This paper studies the dynamics of epidemic spreading on complex networks, focusing on the difference between exponential and scale-free (SF) networks. In exponential networks, an epidemic threshold exists, separating an infected phase from a healthy phase. The prevalence of infections decreases to zero at the threshold with linear behavior typical of a mean-field critical point. In contrast, SF networks, such as the Barabási-Albert model, do not exhibit an epidemic threshold. This implies that infections can spread and persist regardless of the infection rate. The absence of a threshold is due to the presence of highly connected nodes, which dominate the network's connectivity distribution. The study uses analytical methods and large-scale simulations to analyze the susceptible-infected-susceptible (SIS) model on these networks. For SF networks with 0 < γ ≤ 1, the model shows no threshold and infections can spread throughout the network. For 1 < γ ≤ 2, a threshold appears but is approached with a vanishing slope, indicating no critical behavior. For γ > 2, the usual critical behavior is recovered. The results have implications for understanding computer virus epidemics and other spreading phenomena on communication and social networks. The absence of an epidemic threshold in SF networks changes many standard conclusions about epidemic spreading. However, this is balanced by the exponentially small prevalence at low spreading rates. The findings suggest that SF networks, despite their lack of a threshold, can still be controlled due to their low prevalence at low infection rates. The study also highlights the importance of network topology in epidemic modeling and the potential for applying these results to real-world systems like the Internet and the World Wide Web.