Modelling Markovian epidemic and information diffusion over adaptive networks

Modelling the spread of contagious diseases among people has been a research topic for over a hundred years. However, the increase in computation power in the recent years allows for more advanced scientific models. The well-known susceptible-infected-susceptible model is used to describe the spreading of a disease among a group of people. This is modelled as a network, where persons are represented by nodes and their connections are links in the network. In this thesis, instead of a typical static network, the network itself changes structure based on the disease states of the nodes. In other words, there are two independent processes; the spreading of the disease over the network (state of the nodes) and the adaption of the network to the disease (state of the links). As a spreading model, the Markovian adaptive susceptible-infected-susceptible model (ASIS model for short) is introduced. It is shown that the model has one steady state, named the trivial steady state, in which all individuals are healthy. When the infection rate is sufficiently high, the system undergoes a phase transition from the disease-free state to an endemic state, where most nodes are infected. The state where most nodes are infected is named the metastable state. After being in the metastable state for a long time, the system collapses to the trivial steady state. The spreading of contagious processes is not just limited to disease spreading. Other relevant examples, such as opinion, gossip, fake news, neuron transmittance in the brain, etc. can be modelled using adaptive models as well. In this thesis, the ASIS model is extended by allowing different rules for the link-breaking and link-creation processes in what we call the Generalised ASIS framework. In total 36 models have been analysed simultaneously. Out of the 36 models, 9 showed a partially unstable metastable state. This resulted in rapid oscillations of the number of infected nodes just above the epidemic threshold. The relation between the epidemic threshold and the effective link-breaking rate was also determined. For 5 cases, the epidemic threshold is independent of the effective link-breaking rate. In 18 cases, the epidemic threshold scales linearly in the link-breaking rate. The remaining 13 cases are bounded between a constant and a linear link-breaking rate, however, its exact dependence remains unclear. In the G-ASIS framework, it was conjectured that the metastable state of the Markov process can be accurately approximated by the steady state of the mean-field approximation. It was shown this is not true for every model. However in the ASIS model, the mean-field approximation showed close resemblance to the averaged stochastic results. This may be caused by a positive correlation between the nodes. For the other models, the deviation from the mean field approximation can probably be contributed to the fact that nodes are not positively correlated. Other higher order mean field approximations should be examined to approximate the averaged behaviour of the Markov process.Applied Mathematic