Time dependence of susceptible-infected-susceptible epidemics on networks with nodal self-infections

The average fraction of infected nodes, in short the prevalence, of the Markovian $\varepsilon$-SIS (susceptible-infected-susceptible) process with small self-infection rate $\varepsilon >0$ exhibits, as a function of time, a typical “two-plateau” behavior, which was first discovered in the complete graph $K_N$. Although the complete graph is often dismissed as an unacceptably simplistic approximation, its analytic tractability allows to unravel deeper details, that are surprisingly also observed in other graphs as demonstrated by simulations. The time-dependent mean-field approximation for $K_N$ performs only reasonably well for relatively large self-infection rates, but completely fails to mimic the typical Markovian $\varepsilon$-SIS process with small self-infection rates. While self-infections, particularly when their rate is small, are usually ignored, the interplay of nodal self-infection and spread over links may explain why absorbing processes are hardly observed in reality, even over long time intervals.

Figure 1

The prevalence $y_N\left(\tau ,{\varepsilon }^{*};t\right)$ of the $\varepsilon$-SIS process on the complete graph $K_N$ on $N=30$ nodes with effective infection rate $\tau =2{\tau }_c^{\left(1\right)}=\frac{2}{N-1}$ versus time on a logarithmic scale for various values of the self-infection rate $\varepsilon$. The epidemic started with one infected node. Both simulations and numerical solutions of the differential Eqs. (A7) and (A8) (smooth lines) are presented for each self-infection rate $\varepsilon$.

Figure 2

The steady-state prevalence $y_{\infty ;N}\left(\tau ,{\varepsilon }^{*}\right)$ in a complete graph $K_N$ on $N=30$ nodes versus the effective infection rate $\tau$, for various self-infection rates varying from ${\varepsilon }^{*}={10}^{-1}$ down to ${\varepsilon }^{*}={10}^{-10}$. The dotted line at $\tau =2{\tau }_c^{\left(1\right)}=\frac{2}{N-1}$ corresponds to the setting in Fig. 1. The bold curve is the NIMFA steady-state prevalence (1).

Figure 3

The prevalence $y_N\left(\tau ,{\varepsilon }^{*};t\right)$ in the complete graph $N=30$ versus time for self-infection rate ${\varepsilon }^{*}={10}^{-5}$ and effective infection rates ${\tau }_c^{\left(1\right)}\le \tau \le 3{\tau }_c^{\left(1\right)}$ and each curves differs $0.2{\tau }_c^{\left(1\right)}$ from its neighboring curves. The curves are numerical solution of the differential Eqs. (A7) and (A8).

Figure 4

The prevalence on $K_{30}$ over time for a high effective infection rate $\tau =3{\tau }_c^{\left(1\right)}=\frac{3}{29}$ and several self-infection rates $\varepsilon$. The curves are numerical solutions of the differential equations.

Figure 5

The prevalence $y_N\left(\tau ,{\varepsilon }^{*};t\right)$ in the complete graph $N=30$ versus time for self-infection rate ${\varepsilon }^{*}={10}^{-1}$ with the same values of effective infection rates $\tau$ as in Fig. 3. The mean-field approximation (6) (dotted lines) lies above $\varepsilon$-SIS Markovian prevalence, computed via the numerical solution of the differential Eqs. (A7) and (A8).

Figure 6

The prevalence $y_N\left(\tau ,{\varepsilon }^{*};t\right)$ of the $\varepsilon$-SIS with same parameters as in Fig. 1 to which the the mean-field approximation (6) is added. Same colors have same self-infection rate $\varepsilon$. In contrast to Fig. 5 for a large self-infection rate, the mean-field approximation (dotted line) deviates considerably for smaller self-infection rates.

Figure 7

The prevalence $y_N\left(\tau ,{\varepsilon }^{*};t\right)$ with self-infection rate ${\varepsilon }^{*}={10}^{-5}$ versus time for four different types of graphs: the square lattice with $N=36$, the ring graph with $N=30$, the Erdős-Rényi random graph with $N=30$ and link density $p=0.098$ and the Barabasi-Albert power law graph with $N=500$. Initially, one node is infected and the plots are based upon (i.e., averaged over) 100 realizations of each $\varepsilon$-SIS process. The prevalence curves shown increase with a step of $\tau =0.2{\tau }_c^{\left(1\right)}$. The three curves in color illustrate the three different behaviors: blue shows two plateaus and upward bending, red the almost constant plateau, and green depicts the downward bending.

Figure 8

The “two-plateau” $\varepsilon$-SIS prevalence curve versus time for an Erdős-Rényi (ER) graph with $N=100$ nodes and link density $p=2p_c$ and $\varepsilon ={10}^{-5}$. The downward bending of the green curve only occurs for considerably larger times.

Figure 9

The $\varepsilon$-SIS prevalence $y_N\left(\tau ,{\varepsilon }^{*};t\right)$ in the star graph on $N=30$ node as a function of time (in units of the average curing time). Initially, one (leave) node is infected and the prevalence is averaged over ${10}^4$ realizations. The curves depicted have an effective infection rate, starting with the lowest curve at $\tau =2{\tau }_c^{\left(1\right)}=\frac{2}{\sqrt{N-1}}$ up to the highest curve $\tau =6.4{\tau }_c^{\left(1\right)}$ with a step of $\tau =0.2{\tau }_c^{\left(1\right)}$.