The susceptible-infected-susceptible (SIS) model is a canonical model for emerging disease outbreaks. Such outbreaks are naturally modeled as taking place on networks. A theoretical challenge in network epidemiology is the dynamic correlations coming from that if one node is infected, then its neighbors are likely to be infected. By combining two theoretical approaches-the heterogeneous mean-field theory and the effective degree method-we are able to include these correlations in an analytical solution of the SIS model. We derive accurate expressions for the average prevalence (fraction of infected) and epidemic threshold. We also discuss how to generalize the approach to a larger class of stochastic population models.

Recently, Gómez et al. proposed a microscopic Markov-chain approach (MMCA) [S. Gómez, J. Gómez-Gardeñes, Y. Moreno, and A. Arenas, Phys. Rev. E 84, 036105 (2011)PLEEE81539-375510.1103/PhysRevE.84.036105] to the discrete-time susceptible-infected-susceptible (SIS) epidemic process and found that the epidemic prevalence obtained by this approach agrees well with that by simulations. However, we found that the approach cannot be straightforwardly extended to a susceptible-infected-recovered (SIR) epidemic process (due to its irreversible property), and the epidemic prevalences obtained by MMCA and Monte Carlo simulations do not match well when the infection probability is just slightly above the epidemic threshold. In this contribution we extend the effective degree Markov-chain approach, proposed for analyzing continuous-time epidemic processes [J. Lindquist, J. Ma, P. Driessche, and F. Willeboordse, J. Math. Biol. 62, 143 (2011)JMBLAJ0303-681210.1007/s00285-010-0331-2], to address discrete-time binary-state (SIS) or three-state (SIR) epidemic processes on uncorrelated complex networks. It is shown that the final epidemic size as well as the time series of infected individuals obtained from this approach agree very well with those by Monte Carlo simulations. Our results are robust to the change of different parameters, including the total population size, the infection probability, the recovery probability, the average degree, and the degree distribution of the underlying networks.

a b s t r a c tThe transmission of infectious, yet vaccine-preventable, diseases is a typical complex social phenomenon, where the increasing level of vaccine update in the population helps to inhibit the epidemic spreading, which in turn, however, discourages more people to participate in vaccination campaigns, due to the ''externality effect'' raised by vaccination. We herein study the impact of vaccination strategies, pure, continuous (rather than adopt vaccination definitely, the individuals choose to taking vaccine with some probabilities), or continuous with randomly mutation, on the vaccination dynamics with a spatial susceptible-vaccinated-infected-recovered (SVIR) epidemiological model. By means of extensive Monte-Carlo simulations, we show that there is a crossover behavior of the final vaccine coverage between the pure-strategy case and the continuous-strategy case, and remarkably, both the final vaccination level and epidemic size in the continuous-strategy case are less than them in the pure-strategy case when vaccination is cheap. We explain this phenomenon by analyzing the organization process of the individuals in the continuous-strategy case in the equilibrium. Our results are robust to the SVIR dynamics defined on other spatial networks, like the Erd} os-Rényi and Barabási-Albert networks.

We study a susceptible-vaccinated-infected-recovered (SVIR) epidemic-spreading model with diversity of infection rate of the individuals. By means of analytical arguments as well as extensive computer simulations, we demonstrate that the heterogeneity in infection rate can either impede or accelerate the epidemic spreading, which depends on the amount of vaccinated individuals introduced in the population as well as the contact pattern among the individuals. Remarkably, as long as the individuals with different capability of acquiring the disease interact with unequal frequency, there always exist a cross point for the fraction of vaccinated, below which the diversity of infection rate hinders the epidemic spreading and above which expedites it. The overall results are robust to the SVIR dynamics defined on different population models; the possible applications of the results are discussed.

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