Susceptible-Infected-Susceptible (SIS) model is commonly used to describe the spreading of virus on networks. However, a real-life epidemic process is not necessarily Markovian. The spreading of diseases, behaviors and information in real systems are sometimes dependent on the characteristics and current status of individuals. Thus it is far from enough to just consider Markovian processes. We need to consider a more general model with non-Markovian processes. Although some recent works focus on the SIS model with a non-Markovian infection process, systematic research on the non-Markovian curing process is still lacking. Therefore, this thesis project is to study the influence of the non-Markovian curing process on the performance of SIS viral spreading on networks. Through continuous-time SIS epidemics simulator, we find some dramatic effects of a non-exponential curing time (while still assuming an exponential infection time) on the prevalence and critical point of effective infection rate by considering Weibullean curing times with same mean, but different shape parameter α. For α ∈ [0.2, 10], the epidemic threshold satisfies τc = 1/λ1, which is the same as the NIMFA conclusions of Markovian SIS process. Relatively, when α is too small, a large number of curing events synchronously happened at the beginning of the simulation, which will lead to collective deaths on finite network. The effect on initial condition of nodes further cause a decline on prevalence and an slow phase transition between healthy state and the meta-stable state. Furthermore, the heavy-tailed distribution of curing time leads to a small percent of nodes still surviving at the meta-stable state, even under a very low effective infection rate. The heavy-tailed distribution gives some nodes an extreme long curing time and thus can infect other nodes with a pretty small probability, thereby maintaining the virus' long-term spread in a small group of nodes. This spreading mode seems can explain some virus spreading phenomenon, like the spreading mode of hepatitis B virus (HBV). Additionally, when the shape parameter α of Weibull distribution is pretty large, the distribution of curing time is like a pulse or a Dirac delta function ( δ function), thus a huge amount of nodes can get synchronously recovered. We find when we control the successful curing probability p =1-1/e ≈0.632, the prevalence of pulse curing at the meta-stable state is equivalent to a Poisson curing process. Therefore, the pulse curing strategy can suppress the spreading of viruses and further save medical resources.