Spreading dynamics of forget-remember mechanism

We study extensively the forget-remember mechanism (FRM) for message spreading, originally introduced in Eur. Phys. J. B 62, 247 (2008). The freedom of specifying forget-remember functions governing the FRM can enrich the spreading dynamics to a very large extent. The master equation is derived for describing the FRM dynamics. By applying the mean field techniques, we have shown how the steady states can be reached under certain conditions, which agrees well with the Monte Carlo simulations. The distributions of forget and remember times can be explicitly given when the forget-remember functions take linear or exponential forms, which might shed some light on understanding the temporal nature of diseases like flu. For time-dependent FRM there is an epidemic threshold related to the FRM parameters. We have proven that the mean field critical transmissibility for the SIS model and the critical transmissibility for the SIR model are the lower and the the upper bounds of the critical transmissibility for the FRM model, respectively.

Figure 1

Time evolution of infection ratio $n_A$ [(a),(b)] and time duration distribution [(c),(d)]. Each of the simulation results is averaged over 1000 different realizations of networks of $10000$ vertices with $z=4, \nu =0.002$, and ${\nu }_0=0.005$. (a) Time evolution of infection ratios with different forget and remember functions (from top to bottom): (1) $F\left(t\right)=0, R\left(t\right)=0$; (2) $F\left(t\right)=\theta (0.5+0.5t), R\left(t\right)=\theta (0.05+e^{-0.05t})$; (3) $F\left(t\right)=\theta (0.05-e^{-0.05t}), R\left(t\right)=\theta (0.05-0.05t)$; (4) $F\left(t\right)=0.5, R\left(t\right)=0.05$; (5) $F\left(t\right)=0.05, R\left(t\right)=0$. (b) Comparison of numerical solutions of Eq. (8) with simulations: the solid lines are simulations and the dashed lines are numerical solutions. (c),(d) Distributions of forget time duration and remember time duration, with $F\left(t\right)=0.1$ and $R\left(t\right)=\theta (0.6-0.05t)$, where in these two panels the results taken from Table 1 have also been drawn.

Figure 2

Left panel: simulation result for stationary infection ratio $n_{A,s}$ vs constant, $F\left(t\right)$ and $R\left(t\right)$, taken from paper [30]. Right panel: the difference $\mathrm{\Delta }{n}_{A,s}=n_{A,s}-n_{A,s-}$ vs constant $F\left(t\right)$ and $R\left(t\right)$.

Figure 3

Impacts of linear remember mechanism on the spread of epidemics. Simulations and calculations were performed on networks of $N=10000$ vertices with fixed forget function $F\left(t\right)=0.1$, the spreading rate $\nu =0.002$, and the initial infection ratio ${\nu }_0=0.005$. (a) Stationary infection ratio $n_{A,s}$ with respect to the remember parameters ${\alpha }_r$ and ${\beta }_r$. (b) Stationary infection ratio $n_{A,s-}$ as given by Eq. (13). (c) The size of the giant component consisting of infected individuals with respect to remember parameters. The red dashed line for critical remember parameters $({\alpha }_{rc},{\beta }_{rc})$ is estimated via Eq. (27) at ${\mathcal{T}}_c(0.1,0,{\alpha }_{rc},{\beta }_{rc})=0.039$. (d) The dependence of transmissibility $\mathcal{T}$ on linear remember parameters.