Propagation of interacting diseases on multilayer networks

The study on the dynamics of interacting diseases has attracted considerable attention in recent years. This paper proposes a model for two interacting epidemics spreading concurrently on a two-layer network, where both the epidemic thresholds and the dynamics of disease outbreaks are investigated. The analytical expression of the epidemic threshold shows strong coupling between the two interacting epidemics. Moreover, two metrics, the maximum prevalence and the highest spreading speed, are proposed to describe the outbreak process. Theoretical analysis together with extensive simulations illustrate the functions of various factors, including the network topological parameters, percentage of overlapped network links, vulnerable individuals, and the reciprocity of the two diseases. It is found that the seemingly important factor, i.e., the percentage of overlapped links, possesses no effect on the propagation, while the frequently overlooked factor, i.e., the percentage of vulnerable individuals, has significant effects. For the interaction of the two diseases, the recovery state of one disease is more influential than the other in both the mutually enhanced and the mutually impaired situations.

Figure 1

Temporal evolution of ${\lambda }_1$ under different parameter changes in a two-layer random graph when $P_I<1$ and $P_R<1$. In this interaction scenario, the propagation of disease 1 is impaired by disease 2. A common phenomenon illustrated by each panel is that ${\lambda }_1$ increases with the outbreak of disease 2. In addition to this, different dynamic parameters have influences on the temporal evolution. (a) ${\lambda }_1$ decreases at every propagation step of disease 2 with the increase of the average degree of layer 1. (b) When the number of initial infected individuals of disease 2 increase, ${\lambda }_1$ only varies at an earlier step. (c, d) In the case of strong influences ($P_I<0.4$ or $P_R<0.4$), the evolution of ${\lambda }_1$ shows the shape resembling the evolution of prevalence (c) and percentage of recovered individuals (d), which can be observed from the outbreak process illustrated in Fig. 3. (e) The increase in the percentage of vulnerable individuals results in the decrease of ${\lambda }_1$ at every propagation step of disease 2. Percentages of vulnerable individuals imposes similar effects as that of the average degree. (f) The evolution of ${\lambda }_1$ shows no difference with the increase of $\tau$, which implies that changes in the number of overlapped links have on effect on the propagation process.

Figure 2

Temporal evolution of ${\lambda }_1$ under different parameter changes for $P_I>1$ and $P_R>1$ in a two-layer random graph. In this interaction scenario, the propagation of disease 1 is enhanced by disease 2. Compared with the panels in Fig. 1, panels in this figure show that ${\lambda }_1$ decreases with the outbreak of disease 2, regardless of the changes of other dynamic parameters. The influences of the average degree (a), percentage of initial infected individuals of disease 2 (b), percentage of vulnerable individuals (e) overlapped links (f) are similar with those in Fig. 1. The only difference is that, for $P_I$ (c) and $P_R$ (d), the evolution of ${\lambda }_1$ illustrates the shape resembling the evolutions of prevalence and percentage of recovered individuals upside down for $P_I>1.6$ and $P_R>1.6$.

Figure 3

Dynamical processes of the initial parameter spaces. For different network types, i.e., (a) random graph, (b) scale-free network, (c) WS small world network, both diseases show similar dynamic propagation processes due to the identical dynamic parameters for the two diseases in the initial parameter spaces. As is illustrated in the figure, the outbreaks of both diseases consist of three evolution phases: the decrease in the percentage of susceptible individuals; the increase in the percentage of recovered individuals; the temporal evolution of infected individuals (i.e., increase and then decrease).

Figure 4

Influences of network topological parameters on MI and MS. (a) In a random graph, the increase of the average degree leads to remarkable growths in outbreak scale and speed for both diseases. (b) In a scale-free network, the power exponent of a scale-free network has no effects on the epidemic outbreak of disease 1. (c) In a WS small world network, the increase of reconnection probability leads to faster epidemic outbreak with a larger scale. Panels (d), (e), and (f) show how the changes of the network topological parameters of disease 1 influence the outbreak scale and speed of disease 2 under different interaction conditions. It can be observed that, when the two diseases are mutually impaired or enhanced, the outbreak of disease 2 is inevitably affected by the changes of the average degree (d) and reconnection probability (f). For the change of the power exponent, it has no effect on both outbreaks (e).

Figure 5

Influence of initial infected individuals on MI and MS in a scale-free network. (a) For disease 1, the increase of the percentage of initial infected individuals slightly enlarges the scale of the propagation and fastens the outbreak, since the increase in the percentage of initial infected individuals is equivalent to delaying the outbreak for a few steps. More obvious trend can be found in the mutual impaired scenario. (b) shows the influences of the initial infected individuals of disease 1 on the outbreak of disease 2 under different interaction scenarios, which seems inconspicuous due to its puny influence on the outbreak of disease 1.

Figure 6

Influence of infectiousness and recovery rates on MI and MS in a scale-free network. (a) The increase of infectiousness of disease 1 causes considerably faster epidemic outbreak with fairly large scale. (b) With the increase of infectiousness of disease 1, the outbreak scale and speed of disease 2 show different changing trends with different interaction scenarios. When the two contagions are mutually impaired, the outbreak speed and scale of disease 2 decrease. In contrast, when the two diseases are mutually enhanced, the outbreak speed and scale of disease 2 increase. (c) The increase of recovery rate of disease 1 causes considerably slower epidemic outbreak with smaller scale. (d) The influences of recovery rate of disease 1 on the outbreak of diseases 2 show an opposite trend with mutual enhancement and impairment compared to those of infectiousness, i.e., the outbreak speed and scale of disease 2 decrease when the two diseases are mutually enhanced, while increase when the two diseases are mutually impaired.

Figure 7

Influence of overlapped links on MI and MS in a scale-free network. For both disease 1 (a) and disease 2 (b), the percentage of overlapped links between their contact layers has no effect on the outbreak scale and speed of both diseases. As can be observed from the figure, the increase in percentage of overlapped links makes no difference in the outbreaks of both diseases.

Figure 8

Influence of vulnerable individuals on MI and MS in a scale-free network. For both disease 1 (a) and disease 2 (b), increase in the percentage of vulnerable individuals leads to nearly linear growths on the scale and speed of epidemic outbreaks. It implies that a host population with more vulnerable individuals is prone to epidemic outbreaks. This phenomenon is supposed to raise concern about the characteristics of host population in the study of epidemic dynamics.

Figure 9

Influences of interactions between diseases on MI and MS in a scale-free network. (a) The influences of parameter $P_I$ and $P_R$ on the outbreak scale. (b) The influences of parameter $P_I$ and $P_R$ on the outbreak speed. With the increase of $P_I$ and $P_R$ from 0.2 to 2, the outbreak of disease 1 shows an increase in both scale and speed. However, the outbreak scale and speed react relatively more sensitive to parameter $P_R$ than $P_I$, which implies the more significant effect of ‘R’ state in the interacting SIR diseases concurrently spreading framework.