Competing epidemics on complex networks

Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.

Figure 1

A two-dimensional slice through the phase diagram of the system, with $T_b$ held constant while $T_r$ and $\alpha$ vary. The colors of the regions represent the dominant disease in each phase, while the symbols give the leading-order scaling of the expected number of individuals infected by the blue (top) and red (bottom) diseases respectively.

Figure 2

Simulation results for the number of vertices infected with each disease as a function of network size for various values of the parameters. Red circles indicate values for the red disease, blue squares for the blue disease. Each point represents an average over 1000 networks. In the top panel $\alpha =0.3$, $T_b=1$, and $T_r$ is 0.37, 0.42, 0.50, 0.55, and 0.75 for the various data sets shown. These parameter values are all outside the coexistence region, in the regime where one disease or the other always dominates. In the lower panel $\alpha =0.7$, $T_b=1$, and $T_r$ is 0.4, 0.5, 0.6, 0.8, and 1. These parameters include parts of the coexistence region, where both diseases can infect an extensive number of vertices. The dotted lines represent the scaling expected from the arguments given in the text.

Figure 3

Simulation results for the average fraction of vertices infected with red (circles) and blue (squares) diseases as a function of $\alpha$ for the case $T_r=0.6$, $T_b=0.8$. The red (solid) and blue (dashed) lines represent the analytic predictions for the same parameter values. The parameters fall outside the coexistence region for all values of $\alpha$, so one disease or the other should always dominate, at least in the limit of large $n$, and there should be a single discontinuous transition as we cross the growth-rate boundary. The shades of the data points indicate network sizes of $n={10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (lightest points to darkest). Each point is an average over at least 100 networks.

Figure 4

Top: simulation results for the average fraction of vertices infected with red (circles) and blue (squares) as a function of $T_r$ for the case $\alpha =0.1$, $T_b=1$. Bottom: corresponding plot as a function of $T_b$ for $T_r=0.4$, $\alpha =0.2$. The red (solid) and blue (dashed) lines represent the analytic predictions. The parameter ranges in each case were chosen to overlap the coexistence region so that two transitions are visible in each panel, the discontinuous transition at the growth-rate boundary and the continuous transition at the coexistence threshold. In the central region of each panel the two diseases coexist. As in Fig. 3, the shades of the data points indicate network sizes of $n={10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ (lightest points to darkest). Each point is an average over at least 100 networks.

Figure 5

Simulation results comparing the fractions of vertices infected with the red and blue diseases for four different sets of parameter values close to the growth-rate boundary (the four rows in the figure). Left column: the fraction infected with red versus blue for networks with $n={10}^6$. All points lie on the curves defined by Eq. (29), which are shown as the solid black lines. Middle column: box plot of the fraction infected with each disease for various network sizes $n$. Right column: the fraction infected by the red (circles) and blue (squares) diseases at the end of the simulation as a function of the ratio of the numbers infected by the faster- and slower-growing diseases measured at the first time that either disease reaches $\frac{1}{2}\sqrt{n}$. First row: $T_b=0.8$, $T_r=0.6$, $\alpha =0.5$; second row: $T_b=0.8$, $T_r=0.6$, $\alpha =0.8$; third row: $T_b=1$, $T_r=0.45$, $\alpha =0.25$; fourth row: $T_b=1$, $T_r=0.45$, $\alpha =0.4$.