Anomalous epidemic spreading in heterogeneous networks

Epidemic spreading in heterogeneous networks has attracted great interest in recent years. To capture the significant effect of residence of individuals on epidemic spreading, we consider herein a simple susceptible-infected-susceptible model with random waiting time in heterogeneous networks. We provide the analytical dynamical expressions for the time evolution for infected individuals and find a fractional memory effect of power-law waiting time on anomalous epidemic spreading. This work provides new quantitative insights in describing contagion processes and could help model other spreading phenomena in social and technological networks.

Figure 1

Epidemic SIS model with random waiting time in heterogeneous networks. An infected or susceptible individual waits at the node $i$ with degree $k_1$ for some time $\tau$ when it may react according to Eqs. (17) and (18), and then it moves away to one of neighbor node $j$ with degree $k_2$ along an edge, after which the process is renewed.

Figure 2

A random zipf sequence with $V=100$ nodes obeying the power-law degree distribution $P\left(k\right)=k^{-2.5}$.

Figure 3

A scale-free network respecting the degree sequence in Fig. 2.

Figure 4

The time evolution of the average numbers $I\left(t\right)$ [orange (light gray)] and $S\left(t\right)$ [blue (dark gray)] in the time interval $[0,1000]$ for anomalous epidemic spreading with power-law waiting time $\psi \left(t\right)=\frac{\alpha }{t+{\tau }_0}{\left(\frac{{\tau }_0}{t+{\tau }_0}\right)}^{\alpha }$ with $\alpha =1$ (a) and $\alpha =0.55$ (b) in heterogeneous network of size $V=100$. Here ${\tau }_0=0.22, \mu =0.02$, and $\beta =0.01$; the initial total number $\rho \left(0\right)=601$; and thus the average number $N\left(t\right)=6.01$. One can see that the stationary number for infectious individuals reduces with small anomalous exponent $\alpha$ of random waiting time, and the starting time when $I\left(t\right)$ becomes more than $S\left(t\right)$ in (b) is also retardant.

Figure 5

The stationary average numbers $I\left(t\right)$ (orange square) and $S\left(t\right)$ (blue circle) for anomalous epidemic spreading in heterogeneous network of size $V=100$ with power-law waiting time $\psi \left(t\right)=\frac{\alpha }{t+{\tau }_0}{\left(\frac{{\tau }_0}{t+{\tau }_0}\right)}^{\alpha }$ with $\alpha =1$ (a) and $\alpha =0.55$ (b) with respect to the total average number $N\left(t\right)$ for $t=1000, {\tau }_0=0.22, \mu =0.02, \beta =0.01$, and $\rho \left(0\right)=101,201,301,401,501,601,701,801,901$, and 1001. One can see that $I\left(t\right)$ is monotonous increasing, and in both cases $I\left(t\right)=0$ when $N\left(t\right)=1.01$ and $I\left(t\right)>0$ [$I\left(t\right)=0.0227$ for $\alpha =1$ and $I\left(t\right)=0.0074$ for $\alpha =0.55$] when $N\left(t\right)=2.01$. The simulation for $\alpha =1$ verifies the analytical solution for the epidemic threshold for type I that if $N\left(t\right)>\frac{\mu }{\beta }\frac{\langle k^2\rangle }{{\langle k\rangle }^2}\approx 1.55$, then $I\left(t\right)>0$. Notice also that the epidemic spreading is slowed down when $\alpha$ reduces, and the abscissa value for the intersection point of two discrete curves becomes larger with the decrease of exponent.

Figure 6

The stationary average numbers $N_k\left(t\right)$ (green triangle), $I_k\left(t\right)$ (orange square), and $S_k\left(t\right)$ (blue circle) with respect to degree $k$ for anomalous epidemic spreading in heterogeneous network of size $V=100$ with power-law waiting time for $\alpha =1$ (a) and $\alpha =0.55$ (b) when $t=1000$. Here $\rho \left(0\right)=801, {\tau }_0=0.22$, and the reaction rates $\mu =0.02$ and $\beta =0.01$. It is shown that they are all linear-like in $k$. Note that such linear relation has also been found in epidemic spreading in heterogeneous networks without considering the effect of random waiting time (see Ref. [4]). The linear dependence of the stationary average number $N_k\left(t\right)$ on the degree $k$ for $\alpha =1$ is in agreement with the analytical result presented in Eq. (48).