Domination-time dynamics in susceptible-infected-susceptible virus competition on networks

When two viruses compete for healthy nodes in a simple network and both spreading rates are above the epidemic threshold, only one virus will survive. However, if we prevent the viruses from dying out, rich dynamics emerge. When both viruses are identical, one virus always dominates the other, but the dominating and dominated virus alternate. We show in the complete graph that the domination time depends on the total number of infected nodes at the beginning of the domination period and, moreover, that the distribution of the domination time decays exponentially yet slowly. When the viruses differ moderately in strength and/or speed the weaker and/or slower virus can still dominate the other but for a short time. Interestingly, depending on the number of infected nodes at the start of a domination period, being quicker can be a disadvantage.

Figure 1

Competition between two viruses in a complete graph of 500 nodes. Each virus is prevented from dying out by reinfecting the last infected node of a virus at the moment it is cured.

Figure 2

Markov chain representation of the number of infected nodes. From each state there are four different transitions possible: North, South, East, and West. The transition probabilities of a direction can be zero at the edges of the “triangle”.

Figure 3

Structure of matrices $Y_1$, $Y_2$, and $T$. Gray areas consists of only zeros.

Figure 4

Probability that both viruses are alive as a function of the number of transitions. The solid line indicates the numerical results obtained from the transition matrix and the markers indicate the simulation results. The probability that both viruses are alive is the same as the probability that the epidemic process is in a transient state, i.e., the element wise sum of the entries of submatrix $T$. The inset shows the same plot on a logarithmic vertical axis and emphasizes the exponential decay of the transient state.

Figure 5

Probability density function of $(N_A-N_B)/N$ for different sizes of the complete graph (a) and different graph types of the same size (b).

Figure 6

Heatmap representation of the steady-state distribution of two equally strong viruses in a complete graph of 100 nodes, and effective spreading rate $\tau =0.025$.

Figure 7

Example state space for a network with seven nodes. States where virus $A$ dominates (the lower half) are green, states where virus $B$ dominates (the upper half) are blue, and states where the viruses are balanced are yellow. In this state diagram state 1 corresponds to $N_A=N_B=1$.

Figure 8

Measured and calculated domination time in a complete graph of 50 nodes as a function of the number of infected nodes at the start of the domination period for three different spreading rates $\beta$, and unit curing rate. The right-hand side vertical axis shows the measured probability density function of the number of infected nodes at the start of the domination period.

Figure 9

Probability distribution of the domination time $T$ in the complete graph for three different network sizes. For each network size $y_s=y=N/4$.

Figure 10

Measured probability density function of domination time $T$ for a domination period that starts with $y_s=y$ in five different graph types of 1000 nodes. The spreading rate is chosen so that the $y=N/4$.

Figure 11

Average duration of a domination period for two viruses with different effective spreading rates as a function of $y_s$. The green curves indicate the stronger of the two viruses, whereas the blue curves indicate the weaker of the two.

Figure 12

Ratio between the average duration of a domination period for the stronger virus and the duration of a domination period for the weaker virus for various differences in strength.

Figure 13

Probability density function of $(N_A-N_B)/N$ for a complete graph of 50 nodes where ${\tau }_B=\gamma {\tau }_A$, with $1\le \gamma \le 1.17$.

Figure 14

Average duration of a domination period for two viruses with different spreading speed as a function of the number of infected nodes at the start of the domination period. The green curves indicate the quicker of the two viruses, whereas the blue curves indicate the slower of the two.

Figure 15

Ratio between the average duration of a domination period for the slower virus and the duration of a domination period for the quicker virus for various differences in speed: ${\beta }_B=\alpha {\beta }_A$ for $1\le \alpha \le 15$.

Figure 16

Transition rate graph of the MSIS process. (a) Nodal transition rate graph: the three states are susceptible ($S$), infected with virus $A$ ($I_A$), and infected with virus $B$ ($I_B$). A node moves from state $I_A$ to $S$ with curing rate ${\delta }_A$ and from $I_B$ to $S$ with curing rate ${\delta }_B$. (b) Edge-based transition graph for type $A$: a node moves from state $S$ to state $I_A$ with spreading rate ${\beta }_A$ for each incident link from a node in $I_A$. (c) Edge-based transition graph for type $B$: a node moves from state $S$ to state $I_B$ with spreading rate ${\beta }_B$ for each incident link from a node in $I_B$. The contact networks for $I_A$ and $I_B$ are identical.

Figure 17

Comparison between the average number of infected nodes in the GEMF model simulations in a complete graph of 500 nodes. The simulation results are averaged over $100000$ runs.