Burst of virus infection and a possibly largest epidemic threshold of non-Markovian susceptible-infected-susceptible processes on networks

Since a real epidemic process is not necessarily Markovian, the epidemic threshold obtained under the Markovian assumption may be not realistic. To understand general non-Markovian epidemic processes on networks, we study the Weibullian susceptible-infected-susceptible (SIS) process in which the infection process is a renewal process with a Weibull time distribution. We find that, if the infection rate exceeds $1/ln({\lambda }_1+1)$, where ${\lambda }_1$ is the largest eigenvalue of the network's adjacency matrix, then the infection will persist on the network under the mean-field approximation. Thus, $1/ln({\lambda }_1+1)$ is possibly the largest epidemic threshold for a general non-Markovian SIS process with a Poisson curing process under the mean-field approximation. Furthermore, non-Markovian SIS processes may result in a multimodal prevalence. As a byproduct, we show that a limiting Weibullian SIS process has the potential to model bursts of a synchronized infection.

Figure 1

The prevalence of the Weibullian SIS process on an Erdős-Rényi (ER) network $G_{0.15}\left(50\right)$ obtained by averaging over ${10}^5$ realizations. The effective infection rate $\tau =1$ is around 8.5 times the NIMFA threshold ($1/{\lambda }_1=0.1173$). The minimum prevalence decreases with the shape parameter $\alpha$. For $\alpha <1$, the prevalence is a trivial single-peak function with time. For small $\alpha >1$, the prevalence oscillates but eventually becomes approximately constant. When $\alpha \to \infty$, the metastable state prevalence is no longer constant.

Figure 2

Steady maximum prevalence and minimum prevalence of three different networks under the mean-field approximation and simulation. The prevalence is obtained by averaging over ${10}^5$ realizations of simulation with all nodes infected initially to prevent the inaccuracy caused by the early die-out [18]. The simulation runs for a long enough time (50 time units with $\delta =1$), and the maximum and minimum prevalence are plotted, which are selected from the last complete time period. The networks are (a) the ER network $G_{0.15}\left(50\right)$ corresponding to Fig. 1; (b) a Barabási-Albert scale-free network with size $N=500$, and number of links $L=1491$; and (c) a rectangular grid with size $N=484, L=924$.

Figure 3

The epidemic threshold vs the Weibull shape parameter $\alpha$. The thresholds are obtained by simulation of ${10}^5$ realizations. The simulation setup is the same as that in Fig. 2. The threshold is chosen as the value of the effective infection rate $\tau$ which leads to the maximum prevalence being around 0.001 at the last period.