Effects of distribution of infection rate on epidemic models

Self-driven criticality in a stochastic epidemic model

We present a generic epidemic model with stochastic parameters in which the dynamics self-organize to a critical state with suppressed exponential growth. More precisely, the dynamics evolve into a quasi-steady state, where the effective reproduction rate fluctuates close to the critical value 1 for a long period, as indeed observed for different epidemics. The main assumptions underlying the model are that the rate at which each individual becomes infected changes stochastically in time with a heavy-tailed steady state. The critical regime is characterized by an extremely long duration of the epidemic. Its stability is analyzed both numerically and analytically in different models.

Initial growth rates of malware epidemics fail to predict their reach

Empirical studies show that epidemiological models based on an epidemic’s initial spread rate often fail to predict the true scale of that epidemic. Most epidemics with a rapid early rise die out before affecting a significant fraction of the population, whereas the early pace of some pandemics is rather modest. Recent models suggest that this could be due to the heterogeneity of the target population’s susceptibility. We study a computer malware ecosystem exhibiting spread mechanisms resembling those of biological systems while offering details unavailable for human epidemics. Rather than comparing models, we directly estimate reach from a new and vastly more complete data from a parallel domain, that offers superior details and insight as concerns biological outbreaks. We find a highly heterogeneous distribution of computer susceptibilities, with nearly all outbreaks initially over-affecting the tail of the distribution, then collapsing quickly once this tail is depleted. This mechanism restricts the correlation between an epidemic’s initial growth rate and its total reach, thus preventing the majority of epidemics, including initially fast-growing outbreaks, from reaching a macroscopic fraction of the population. The few pervasive malwares distinguish themselves early on via the following key trait: they avoid infecting the tail, while preferentially targeting computers unaffected by typical malware.

Moment dynamics for gene regulation with rational rate laws

This aim of this paper is mainly to investigate the performance of two typical moment closure schemes in gene regulatory master equations of rational rate laws. When the reaction rate is polynomial, the error bounds between the authentic and approximate moments obtained by schemes of Gaussian moment closure and log-normal moment closure are explicitly given. When the reaction rate is not polynomial, it is shown that the two schemes both behave well in the absence of active-inactive state switch, but in the presence of active-inactive state switch the log-normal closure scheme is far superior to the Gaussian closure scheme in capturing the asymptotic ensemble statistics. Moreover, the accuracy of the log-normal closure method is further confirmed by steady-state analytic results and the conditional Gaussian closure method. It is also disclosed that optimal negative feedback exists in suppressing protein noise in the presence of the on-off switch control.

Fragility and robustness of self-sustained oscillations in an epidemiological model on small-world networks

We investigate the susceptible-infected-recovered-susceptible epidemic model, typical of mathematical epidemiology, with the diversity of the durations of infection and recovery of the individuals on small-world networks. Infection spreads from infected to healthy nodes, whose infection and recovery periods denoted by τ_I and τ_R, respectively, are either fixed or uniformly distributed around a specified mean. Whenever τ_I and τ_R are narrowly distributed around their mean values, the epidemic prevalence in the stationary state is found to reach its maximal level in the typical small-world region. This non-monotonic behavior of the final epidemic prevalence is thought to be similar to the efficient navigation in small worlds with cost minimization. Besides, pronounced oscillatory behavior of the fraction of infected nodes emerges when the number of shortcuts on the underlying network become sufficiently large. Remarkably, we find that the synchronized oscillation of infection incidences is quite fragile to the variability of the two characteristic time scales τ_I and τ_R. Specifically, even in the limit of a random network (where the amplest oscillations are expected to arise for fixed τ_I and τ_R), increasing the variability of the duration of the infectious period and/or that of the refractory period will push the system to change from a self-sustained oscillation to a fixed point with negligible fluctuations in the steady state. Interestingly, negative correlation between τ_I and τ_R can give rise to the robustness of the self-sustained oscillatory phenomenon. Our findings thus highlight the pivotal role of, apart from the external seasonal driving force and demographic stochasticity, the intrinsic characteristic of the system itself in understanding the cycle of outbreaks of recurrent epidemics.