Effective degree Markov-chain approach for discrete-time epidemic processes on uncorrelated networks

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)] 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)], 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.

Figure 1

(a) The final epidemic size as a function of the infection probability and (b) the fraction of infected individuals as a function of the time step. Network parameters: Erdős-Rényi network with total nodes $N={10}^4$, average degree $\langle k\rangle =4$. Lines (a) and thick red line (b) are the numerical solutions of the microscopic Markov-chain approach, while the scatters (a) and thin gray lines (b) are simulation results. The epidemic process is the reactive process with reinfections with the number trials $\lambda \to \infty$ [30, 31].

Figure 2

Schematic illustration of the transition of the whole classes for (a) SIS and (b) SIR epidemic processes at one time step. $X$ and $Y$ denote individual states, where $X\in \{S,I,R\}$ and $Y\in \{S,I\}$; $s$ and $i$ denote a $X\left(Y\right)$ individual having $s$ susceptible and $i$ infected neighbors, respectively.

Figure 3

Comparison of the results obtained by Monte Carlo simulations and by the effective degree Markov-chain approach and microscopic Markov-chain approach for the SIS epidemic process on networks with different degree distributions. The final epidemic size as a function of the infection probability $r$ (left column panels) and the fraction of infected individuals as a function of the time step (middle and right column panels) are explicitly shown for comparison. Network parameters: the top three panels are for random regular graphs with $N={10}^5$, the middle panels are for ER networks with $N={10}^5$, and the bottom panels are for uncorrelated random scale-free networks with $N={10}^4$. The parameter $z$ is the minimal degree of the scale-free networks. Solid lines (left column panels) and black lines (middle and right column panels) are the numerical solutions of EDMA [Eqs. (5) and (6)]; dashed lines (left column panels) and red (gray) lines (middle and right column panels) are the numerical solutions of MMCA; scatters (left column panels) and light gray lines (middle and right column panels) are simulation results, which are averages over 100 independent runs.

Figure 4

Comparison of the results obtained by Monte Carlo simulations and by the effective degree Markov-chain approach for the SIR epidemic process on networks with different degree distributions. The final epidemic size as a function of the infection probability $r$ (left column panels) and the fraction of infected individuals as a function of the time step (middle and right column panels) are explicitly shown for comparison. Network parameters are the same as those in Fig. 3. Solid lines (left column panels) and black lines (middle and right column panels) are the numerical solutions of the fixed-point equations (8) and (9), while scatters (left column panels) and light gray lines (middle and right column panels) are simulation results, which are averages over 100 independent runs.

Figure 5

(a) The epidemic threshold $r_c$ as a function of the recovery probability $g$ for the discrete-time SIS epidemic process on different networks. The ER random network consists of $N={10}^5$ nodes with $\langle k\rangle =4$, while the RRG consists of $N={10}^5$ nodes with $k_0=4$. (b) Comparison of the epidemic threshold $r_c$ on RRG obtained by Eq. (15), which is derived from the approximate analysis, with those by directly integrating Eqs. (5) and (6).

Figure 6

(a) The epidemic threshold $r_c$ as a function of the recovery probability $g$ for the discrete-time SIR epidemic process on different networks. The parameters are the same as in Fig. 5. (b) Comparison of the epidemic threshold $r_c$ on RRG obtained by Eq. (19), which is derived from the approximate analysis, with those by directly integrating Eqs. (8) and (9).