Analysis of continuous-time Markovian $\varepsilon$-SIS epidemics on networks

We analyze continuous-time Markovian $\varepsilon \text{−}\mathrm{SIS}$ epidemics with self-infections on the complete graph. The majority of the graphs are analytically intractable, but many physical features of the $\varepsilon \text{−}\mathrm{SIS}$ process observed in the complete graph can occur in any other graph. In this work, we illustrate that the timescales of the $\varepsilon \text{−}\mathrm{SIS}$ process are related to the eigenvalues of the tridiagonal matrix of the SIS Markov chain. We provide a detailed analysis of all eigenvalues and illustrate that the eigenvalues show staircases, which are caused by the nearly degenerate (but strictly distinct) pairs of eigenvalues. We also illustrate that the ratio between the second-largest and third-largest eigenvalue is a good indicator of metastability in the $\varepsilon \text{−}\mathrm{SIS}$ process. Additionally, we show that the epidemic threshold of the Markovian $\varepsilon \text{−}\mathrm{SIS}$ process can be accurately approximated by the effective infection rate for which the third-largest eigenvalue of the transition matrix is the smallest. Finally, we derive the exact mean-field solution for the $\varepsilon \text{−}\mathrm{SIS}$ process on the complete graph, and we show that the mean-field approximation does not correctly represent the metastable behavior of Markovian $\varepsilon \text{−}\mathrm{SIS}$ epidemics.

Figure 1

The Markov chain $\mathcal{M}$ of the $\varepsilon \text{−}\mathrm{SIS}$ process on the complete graph is a finite birth and death process with birth rate ${\lambda }_k=(\beta k+\varepsilon )(N-k)$ and death rate ${\mu }_k=\delta k$. The state $M=0,\cdots ,N$ of the Markov chain denotes the number of infected nodes in the graph. Furthermore, $\beta$ is the infection rate, $\delta$ is the curing rate, and $\varepsilon$ is the self-infection rate.

Figure 2

Plateau-behavior in the Markovian $\varepsilon \text{−}\mathrm{SIS}$ process on the complete graph. Each curve is computed using the exact solution (5), and the parameters are $N=30$ nodes, effective infection rate $\tau =2.5{\tau }_c^{\left(1\right)}=2.5/(N-1)$, the process starts with a single infected node, and the effective self-infection rate ${\varepsilon }^{*}=\{{10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5},{10}^{-6},{10}^{-7},0\}$. For ${\varepsilon }^{*}\le {10}^{-5}$, the background colors indicate the current phase of the $\varepsilon \text{−}\mathrm{SIS}$ process: (I) initial phase, (II) metastable behavior, and (III) convergence to the steady state.

Figure 3

The exact Markovian solution (solid lines), the mean-field approximation without self-loops (dashed lines), and exact mean-field solution with self-loops from Theorem IV.1 (dotted lines) of the $\varepsilon \text{−}\mathrm{SIS}$ process on the complete graph with $N=30$ nodes for various effective self-infection rates ${\varepsilon }^{*}$ for (a) effective infection rate $\tau =2.5/(N-1)$ and (b) effective infection rate $\tau =1/(N-1)$. The mean-field approximation is generally a loose upper bound for the Markovian $\varepsilon \text{−}\mathrm{SIS}$ dynamics.

Figure 4

All eigenvalues ${\xi }_k$ of the transition matrix $P$ on the complete graph with effective infection rate $\tau =2/(N-1)$, and effective self-infection rate ${\varepsilon }^{*}=0.01/N$ on (a) linear-linear scale and (b) log-log scale for $k=1,...,N+1$. The eigenvalues have been normalized with respect to the number of nodes $N$. Normalized index 0 corresponds to the largest eigenvalue ${\xi }_1=0$, and normalized index 1 corresponds to the smallest eigenvalue ${\xi }_{N+1}$. The horizontal dash-dotted line indicates the lower bound for ${\xi }_{N+1}$ from Theorem V.1. The largest eigenvalue ${\xi }_1=0$ is not shown in (b) because of the logarithmic axis. The inset in (a) zooms in for $0\le (k-1)/N\le 0.05$ and shows the critical index $k_2$ as a vertical dash-dotted line. In this case, the critical index $k_1=0$.

Figure 5

The critical indices $k_1$ (a) and $k_2$ (b) for different values of the effective infection rate $\tau$ and the effective self-infection rate ${\varepsilon }^{*}$ on the complete graph $K_N$ with $N=20$ nodes.

Figure 6

The normalized eigenvalues ${\xi }_k/N$ on the complete graph $K_N$ with $N=20$ nodes and $\tau =10$ for different effective self-infection rates between ${\varepsilon }^{*}=10$ and ${\varepsilon }^{*}=20$. By varying the effective self-infection rate ${\varepsilon }^{*}$, one eigenvalue of the nearly degenerate pair changes significantly, whereas the other remains largely unchanged.

Figure 7

The eigenvalues of the transition matrix $P$ for the complete graph with $N=30$ nodes and effective infection rate $\tau =2.5{\tau }_c^{\left(1\right)}$ for varying effective self-infection rates ${\varepsilon }^{*}$ on a log-log scale. Subfigure (a) shows the second-to-fifth-largest eigenvalues ${\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5$ and smallest eigenvalue ${\xi }_{N+1}$ as a function of the effective self-infection rate ${\varepsilon }^{*}$. The vertical line indicates the relation ${\varepsilon }^{*}=\tau$. Subfigure (b) is equivalent to (a), but here the eigenvalue ratio $\rho$ is colored green (light) and the area with the other eigenvalues is colored blue (dark). Finally, (c) shows a plot of the eigenvalue ratio $\rho$ vs the effective self-infection rate ${\varepsilon }^{*}$.

Figure 8

Illustration of the explosive phase transition for effective self-infection rates ${\varepsilon }^{*}>0$ and no phase transition for ${\varepsilon }^{*}=0$ for the second-largest eigenvalue ${\xi }_2$. The remaining eigenvalues ${\xi }_3,...,{\xi }_{N+1}$ are nearly indistinguishable for different ${\varepsilon }^{*}$. The network size equals $N=30$.

Figure 9

The eigenvalues of the transition matrix $P$ for the complete graph with $N=30$ nodes and effective self-infection rate ${\varepsilon }^{*}={10}^{-5}$ for varying effective infection rates $\tau$. Subfigure (a) shows the second-to-fifth-largest eigenvalues ${\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5$ and smallest eigenvalue ${\xi }_{N+1}$ as a function of the effective infection rate $\tau$. The vertical line indicates the mean-field epidemic threshold ${\tau }_c^{\left(1\right)}$, which is slightly smaller than the true epidemic threshold ${\tau }_c$. Subfigure (b) is equivalent to (a), but here the eigenvalue ratio $\rho$ is colored green (light) and the area with the other eigenvalues is colored blue (dark). Finally, (c) shows a plot of the eigenvalue ratio $\rho$ vs the effective infection rate $\tau$. Finally, (a) illustrates the lower bound for the smallest eigenvalue ${\xi }_{N+1}$ from Theorem V.1 by a dashed line.

Figure 10

Several formulas for the suggested epidemic threshold ${\tau }_c$ for various network sizes $N$.

Figure 11

The steady-state prevalence $y_{\infty }$, the mean-field steady-state prevalence $y_{\infty }^{\left(1\right)}$, and the eigenvalue ratio $\rho$ with $N=30$ nodes and effective self-infection rate ${\varepsilon }^{*}={10}^{-5}$ for varying effective infection rate $\tau$. The vertical line indicates the mean-field epidemic threshold ${\tau }_c^{\left(1\right)}=\frac{1}{N-1}$, which is slightly smaller than the true epidemic threshold ${\tau }_c$.

Figure 12

The normalized effective infection rate $x=\tau /{\tau }_c^{\left(1\right)}$ vs the steady-state prevalence $y_{\infty }$ for varying network sizes $N$ and the mean-field steady-state prevalence $y_{\infty }^{\left(1\right)}=1-\frac{1}{x}$ and ${\varepsilon }^{*}=0.01/N$.

Figure 13

The eigenvalues of the transition matrix $P$ for the complete graph with effective self-infection rate ${\varepsilon }^{*}=0.01/N$. Both subfigures show the second-to-fifth-largest eigenvalues ${\xi }_2,{\xi }_3,{\xi }_4,{\xi }_5$ and smallest eigenvalue ${\xi }_{N+1}$ as a function of the network size $N$. Subfigure (a) corresponds to effective infection rate $\tau =1.5{\tau }_c^{\left(1\right)}=1.5/(N-1)$ and (b) to $\tau =1.2{\tau }_c^{\left(1\right)}=1.2/(N-1)$.

Figure 14

The critical network size $N_c$ as a function of the normalized effective infection rate $x=\tau /{\tau }_c^{\left(1\right)}$. The analysis is conducted on the complete graph with effective self-infection rate ${\varepsilon }^{*}=0.01/N$. The critical network size $N_c$, which indicates when the $\varepsilon \text{−}\mathrm{SIS}$ process can be safely approximated using a mean-field method, increases if the normalized effective infection rate $x$ approaches the epidemic threshold $x_c=1$.

Figure 15

Illustration of the effective self-infection rate ${\varepsilon }^{*}$ for which the metastable prevalence $y$ and the steady-state prevalence $y_{\infty }$ are equal. The power-law tail is observed for $\tau >0.3\approx 9{\tau }_c^{\left(1\right)}$ and the results are obtained on a graph with $N=30$ nodes.

Figure 16

The time-varying prevalence $y\left(t\right)$ for the complete graph with $N=30$ nodes, effective infection rate $\tau =2.5{\tau }_c^{\left(1\right)}$, and effective self-infection rate ${\varepsilon }^{*}={10}^{-6}$. The solid line indicates the solution (5), and the dashed lines show the approximation (D1) for various values of $m$.

Figure 17

The relation between the critical number of eigenvalues $m^{*}$ and the number of nodes $N$ in the graph. The critical number $m^{*}$ is defined as the smallest integer $m$ for which the initial prevalence $\tilde{y}\left(0\right)$ of the approximated solution (D1) differs at most ${10}^{-3}$ from the initial prevalence $y\left(0\right)$ of the exact solution (1).