Epidemic spreading with awareness and different timescales in multiplex networks

One of the major issues in theoretical modeling of epidemic spreading is the development of methods to control the transmission of an infectious agent. Human behavior plays a fundamental role in the spreading dynamics and can be used to stop a disease from spreading or to reduce its burden, as individuals aware of the presence of a disease can take measures to reduce their exposure to contagion. In this paper, we propose a mathematical model for the spread of diseases with awareness in complex networks. Unlike previous models, the information is propagated following a generalized Maki-Thompson rumor model. Flexibility on the timescale between information and disease spreading is also included. We verify that the velocity characterizing the diffusion of information awareness greatly influences the disease prevalence. We also show that a reduction in the fraction of unaware individuals does not always imply a decrease of the prevalence, as the relative timescale between disease and awareness spreading plays a crucial role in the systems' dynamics. This result is shown to be independent of the network topology. We finally calculate the epidemic threshold of our model, and show that it does not depend on the relative timescale. Our results provide a new view on how information influence disease spreading and can be used for the development of more efficient methods for disease control.

Figure 1

Schematic illustration showing the states of the nodes in the network and the associated transition probabilities between states, indicated by the Greek letters.

Figure 2

Stationary density of infected nodes ${\rho }_I^{*}$ (disease prevalence) as a function of the disease propagation probability ($\beta$), for different values of $\gamma , \mathrm{\Gamma }$, and $\pi$, using the baseline model. For $\pi =0.5$, the rumor spreading and the epidemic propagation have the same timescale, whereas for $\pi =0.1$ (0.9) rumor events are slower (faster) than the events of the epidemic process. The solid lines are Markov chain calculations (see Appendix), whereas symbols are results from Monte Carlo simulations. Other parameters of the model are set to $\mu =0.9, \alpha =0.6, \kappa =0.5$, and $\sigma =0.6$.

Figure 3

Stationary densities of nodes in states I (infected), A (aware), R (stifler), and U (unaware) normalized by their values when there is no self-awareness (at $\kappa =0$) as a function of $\kappa$, for (a) $\pi =0.1$ (slow rumor spreading) and (b) $\pi =0.9$ (fast rumor spreading). The baseline model was used for this figure. Squares are the results of Monte Carlo simulations, whereas the solid lines are Markov chain calculations (see the Appendix). The dotted lines are guides to the eyes. Other parameters are set to $\beta =1.0, \mu =0.9, \gamma =0.5, \alpha =0.6, \mathrm{\Gamma }=0.0$, and $\sigma =0.6$.

Figure 4

Normalized stationary densities vs. $\kappa$ using the modified model (see text), for (a) $\pi =0.1$ (fast epidemic spreading) and (b) $\pi =0.9$ (fast rumor propagation). Squares and solid lines correspond to MC simulations and the Markov chain approach, respectively. Parameter values are the same as those described in the caption of Fig. 3.

Figure 5

Stationary densities for the susceptible nodes as a function of the self-awareness probability $\kappa$, normalized by their values with $\kappa =0$, using the modified model, for different values of the timescale parameter $\pi$. The increasing on SU population with $\kappa$ for $\pi =0.9$ helps explaining the behavior in Fig. 4. Other parameters are set to $\beta =1.0, \mu =0.9, \gamma =0.5, \alpha =0.6, \mathrm{\Gamma }=0.0$, and $\sigma =0.6$.

Figure 6

Normalized disease prevalence ${\rho }_I^{*}\left(\kappa \right)/{\rho }_I^{*}(\kappa =0)$ vs $\kappa$ for the (a) baseline and (b) modified models. The values of the timescale parameter $\pi$ increase from the darker to the brighter color, showing how the curves change their behavior with $\kappa$ as $\pi$ increases. Other parameters are set to $\beta =1.0, \mu =0.9, \gamma =0.5, \alpha =0.6, \mathrm{\Gamma }=0.0, \sigma =0.6$.

Figure 7

Markov chain calculations of the normalized prevalence for the modified model, using different pairs of network models between the configurational scale-free (SF), Erdős-Rényi (ER), and Watts-Strogatz (WS) models. On the upper plot, the prevalence is shown as a function of the self-awareness parameter ($\kappa$) for a fixed value of $\pi =0.9$, and on the lower plot it is shown as a function of $\pi$ for a fixed value of $\kappa =0.8$. Other parameters are set to the same values as in Fig. 6.

Figure 8

Phase diagram of the model, showing the phase transition curve between the healthy and endemic phases for different values of $\mathrm{\Gamma }$. The inset shows the critical epidemic transmission probability $\beta$ as a function of the protection factor $\mathrm{\Gamma }$, for different values of the information transmission probability $\gamma$. The diagram is the same for the modified and baseline models, and does not depend on $\pi$ and $\kappa$. Other parameters are set to $\mu =0.9, \alpha =0.6, \sigma =0.6$.

Figure 9

Probability trees with all the possible transitions for each state. The informational group has an associated probability of $\pi$, whereas the epidemic group carries the complementary probability $1-\pi$. For the modified model, the $\text{IR}\to \text{IU}$ transition follows the factors in red (instead of the ones in black).