This paper presents an analytical and numerical study of the SIR model on two complex networks: the Watts-Strogatz (WS) model and the Barabási-Albert (BA) model. The study shows that the large connectivity fluctuations in complex networks significantly enhance the incidence of epidemic outbreaks. Scale-free networks, characterized by diverging connectivity fluctuations, lack an epidemic threshold and always show a finite fraction of infected individuals. This is due to the highly heterogeneous response of the system to the introduction of infected individuals with different connectivity. The understanding of epidemics in complex networks provides insights into the spread of information and diseases in biological and technological networks. The SIR model on complex networks reveals that the epidemic threshold depends on the connectivity distribution, with scale-free networks having no finite threshold. The study confirms that the interplay of complex network topology and epidemic modeling leads to a new theoretical framework. The results show that the epidemic incidence in scale-free networks is non-zero for any non-zero spreading rate, and that the threshold is inversely proportional to the connectivity fluctuations. The study also highlights the importance of network heterogeneity in epidemic spreading and the effectiveness of targeted immunization strategies. The findings have implications for understanding the spread of diseases in real-world networks and for developing better protection methods.This paper presents an analytical and numerical study of the SIR model on two complex networks: the Watts-Strogatz (WS) model and the Barabási-Albert (BA) model. The study shows that the large connectivity fluctuations in complex networks significantly enhance the incidence of epidemic outbreaks. Scale-free networks, characterized by diverging connectivity fluctuations, lack an epidemic threshold and always show a finite fraction of infected individuals. This is due to the highly heterogeneous response of the system to the introduction of infected individuals with different connectivity. The understanding of epidemics in complex networks provides insights into the spread of information and diseases in biological and technological networks. The SIR model on complex networks reveals that the epidemic threshold depends on the connectivity distribution, with scale-free networks having no finite threshold. The study confirms that the interplay of complex network topology and epidemic modeling leads to a new theoretical framework. The results show that the epidemic incidence in scale-free networks is non-zero for any non-zero spreading rate, and that the threshold is inversely proportional to the connectivity fluctuations. The study also highlights the importance of network heterogeneity in epidemic spreading and the effectiveness of targeted immunization strategies. The findings have implications for understanding the spread of diseases in real-world networks and for developing better protection methods.