The spread of epidemic under voluntary vaccination with heterogeneous infection rates

Epidemics usually spread widely and can cause a great deal of loss to humans. In the real world, vaccination is the principal method for suppressing the spread of infectious diseases. The Susceptible-Infected-Susceptible (SIS) model suggests that voluntary vaccination may affect the spread of an epidemic. Most studies to date have argued that the infection rates of nodes in the SIS model are not heterogeneous. However, in reality, there exist differences in the neighbor network structure and the number of contacts, which may affect the spread of infectious diseases in society. As a consequence, it can be reasonably assumed that the infection rate of the nodes is heterogeneous because of the amount of contact among people. Here, we propose an improved SIS model with heterogeneity in infection rates, proportional to the degree of nodes. By conducting simulations, we illustrate that almost all vaccinated nodes have high degrees when the infection rate is positively correlated with the degree of a node. These vaccinated nodes can divide the whole network into many connected sub-graphs, which significantly slows down the propagation of an epidemic; the heterogeneity of infection rates has a strong inhibitory effect on epidemic transmission. On the other hand, when the infection rate is negatively related to the degrees of the infection rate nodes, it is difficult for most nodes to meet the inoculation conditions, and the number of inoculations is close to zero.