Approximate formula and bounds for the time-varying susceptible-infected-susceptible prevalence in networks

Based on a recent exact differential equation, the time dependence of the SIS prevalence, the average fraction of infected nodes, in any graph is first studied and then upper and lower bounded by an explicit analytic function of time. That new approximate “tanh formula” obeys a Riccati differential equation and bears resemblance to the classical expression in epidemiology of Kermack and McKendrick [Proc. R. Soc. London A 115, 700 (1927)] but enhanced with graph specific properties, such as the algebraic connectivity, the second smallest eigenvalue of the Laplacian of the graph. We further revisit the challenge of finding tight upper bounds for the SIS (and SIR) epidemic threshold for all graphs. We propose two new upper bounds and show the importance of the variance of the number of infected nodes. Finally, a formula for the epidemic threshold in the cycle (or ring graph) is presented.

Figure 1

The prevalence as a function of time $t^{*}=t$ (for $\delta =1$) in one realization of an Erdős-Rényi random graph $G_p\left(N\right)$ with $N=50$ nodes and link density $p=0.4$, spectral radius ${\lambda }_1\left(A\right)=20.85$, and algebraic connectivity ${\mu }_{N-1}=10.11$. The normalized infection strength $x=\frac{\tau }{{\tau }_c^{\left(1\right)}}={\lambda }_1\left(A\right)\tau$ was chosen to operate below the epidemic threshold ${\tau }_c$. Both NIMFA and the tanh formula in (13) for $\tilde{y}\left(t^{*}\right)$ with $c=0$ are compared with simulations. Initially, only one node is infected, and ${10}^6$ realizations of the Markovian SIS process are simulated to produce an accurate average fraction of infected nodes $y\left(t^{*}\right)$.

Figure 2

The prevalence as a function of time $t^{*}=t$ (for $\delta =1$) for precisely the same graph as in Fig. 1, with normalized infection strength $x=\frac{\tau }{{\tau }_c^{\left(1\right)}}={\lambda }_1\left(A\right)\tau$ well above the epidemic threshold ${\tau }_c$.