Nonperturbative heterogeneous mean-field approach to epidemic spreading in complex networks

Since roughly a decade ago, network science has focused among others on the problem of how the spreading of diseases depends on structural patterns. Here, we contribute to further advance our understanding of epidemic spreading processes by proposing a nonperturbative formulation of the heterogeneous mean-field approach that has been commonly used in the physics literature to deal with this kind of spreading phenomena. The nonperturbative equations we propose have no assumption about the proximity of the system to the epidemic threshold, nor any linear approximation of the dynamics. In particular, we first develop a probabilistic description at the node level of the epidemic propagation for the so-called susceptible-infected-susceptible family of models, and after we derive the corresponding heterogeneous mean-field approach. We propose to use the full extension of the approach instead of pruning the expansion to first order, which leads to a nonperturbative formulation that can be solved by fixed-point iteration, and used with reliability far away from the epidemic threshold to assess the prevalence of the epidemics. Our results are in close agreement with Monte Carlo simulations, thus enhancing the predictive power of the classical heterogeneous mean-field approach, while providing a more effective framework in terms of computational time.

Figure 1

Epidemic prevalence $\rho$ as a function of the infection rate $\beta$. The different curves correspond (as indicated) to Monte Carlo results and the numerical solutions of the fixed-point equations of the different approaches (MMCA, HMF, and npHMF) for a SF network made up of $N={10}^4$ nodes using the configuration model. The exponent of the degree distribution is $\gamma =2.7$. The contact probabilities used correspond to the case of a fully reactive process and $\mu$ has been set to $0.5$.

Figure 2

Fraction ${\rho }_k$ of infected individuals of degree $k$ from npHMF vs the infection probabilities $p_i$ (black) and their average ${\langle p_i\rangle }_k$ over nodes with the same degree (blue) obtained from MC. The epidemic model is a RP with reinfections with $\mu =0.5$ and $\beta =0.3$. For the sake of clarity, we have used in this case an SF network of 500 nodes with $\gamma =2.7$.

Figure 3

Epidemic prevalence $\rho$ as a function of the infection rate $\beta$ for increasing values of $\mu$ from left ($\mu =0.1$) to right ($\mu =1.0$). The top panel corresponds to the numerical solutions of the fixed-point equations of the MMCA formulation, while the bottom figure has been obtained using the npHMF equations. In both cases, the numerical solutions are compared with the results of MC simulations on top of a SF network made up of $N={10}^4$ nodes. The exponent of the degree distribution is $\gamma =2.7$. The contact probabilities used correspond to the case of a fully reactive process.