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 τ_c = 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.

Spreading processes are ubiquitous in nature and societies, e.g. spreading of diseases and computer virus, propagation of messages, and activation of neurons. Computer viruses cause an enormous economic loss. Moreover, many illnesses/diseases still causing a serious threat to public health. For example, the outbreaks of circulating influenza strains cause millions of illness and deaths worldwide every year. Pronounced outbreaks of flu usually occur during winter. This recognized timing allows public health agencies to organize their flu-related mitigation and response activities to prepare for the winter flu season. Although the general wintertime peak of influenza incidence in temperate regions can be easily forecast, the specific intensity, duration, and time of individual local outbreaks are quite changeable. Even after an outbreak has begun, it is still difficult to predict the future characteristics of the epidemic curve. If the diseases/viruses outbreak characteristics could be reliably predicted, the public health response will be better coordinated.The goal is to develop a fast and accurate epidemic model to estimate, fit and forecast the spreading of an epidemic on a defined network. The aim is to conduct a study over viruses spreading phenomena both theoretically and numerically, then create a general model/algorithm that can be easily applied to different diseases and computer viruses. In this master thesis, we propose a new approach which can be used on real illness/viruses data (such as influenza) to estimate and forecast the epidemic more accurately. The approach is to use a model-inference system combining the network science, susceptible-infected-recovered-susceptible (SIRS) model, statistical filtering techniques and gradient descent. We are able to fit and estimate with a relatively low error compared to other algorithms. Moreover, we forecast the out-breaker with a high accuracy, four weeks before the true out-breaker on synthetic epidemic data. The model is evaluated on a regular graph, Erdös-Rényi graph, Watts-Strogatz small-world graph, & Barabási-Albert graph. Furthermore, the model is carried out on real-world epidemic data (influenza data) for four countries (the Netherlands, Germany, Belgium and the United Kingdom), from the years 2012 to 2017.

Epidemic models are applied to describe epidemic processes such as the spreading of infectious viruses, opinions and fake news on real-life or online
social networks, and to analyse the epidemic processes mathmatically. The viral state evolution is closely related to the underlying network topology. Therefore, the network topology is of vital importance to describing the viral state of each individual in a network. This master thesis focuses on the network reconstruction problem of the NIMFA approximation of the Susceptible-Infected-Susceptible (SIS) epidemic process. Given the viral state series generated by the NIMFA epidemic process, we aim to estimate the adjacency matrix A of the underlying network given that the spreading parameters are known. In this thesis, we estimate the adjacency matrix of the network from the viral states by a constrained linear least-squares formulation. Our algorithm gives an accurate estimate of the adjacency matrix provided that suciently many epidemic outbreaks are observed.