The paper presents a detailed analytical and numerical study of epidemic outbreaks in complex networks with acquired immunity. The authors focus on scale-free networks, which exhibit diverging connectivity fluctuations and lack an epidemic threshold, leading to a finite fraction of infected individuals. They show that the large connectivity fluctuations in these networks significantly enhance the incidence of epidemic outbreaks. The study uses the susceptible-infected-removed (SIR) model on two prototype complex networks: the Watts-Strogatz (WS) model and the Barabási-Albert (BA) model. The analytical approach recovers the total size of the epidemics in an infinite population and confirms the absence of a finite epidemic threshold for connectivity distributions with diverging fluctuations. Numerical simulations on the WS and BA networks validate these findings, demonstrating that the interplay between network topology and epidemic modeling leads to a new theoretical framework with implications for understanding the spread of information and diseases in complex networks.The paper presents a detailed analytical and numerical study of epidemic outbreaks in complex networks with acquired immunity. The authors focus on scale-free networks, which exhibit diverging connectivity fluctuations and lack an epidemic threshold, leading to a finite fraction of infected individuals. They show that the large connectivity fluctuations in these networks significantly enhance the incidence of epidemic outbreaks. The study uses the susceptible-infected-removed (SIR) model on two prototype complex networks: the Watts-Strogatz (WS) model and the Barabási-Albert (BA) model. The analytical approach recovers the total size of the epidemics in an infinite population and confirms the absence of a finite epidemic threshold for connectivity distributions with diverging fluctuations. Numerical simulations on the WS and BA networks validate these findings, demonstrating that the interplay between network topology and epidemic modeling leads to a new theoretical framework with implications for understanding the spread of information and diseases in complex networks.