Economic and social factors in designing disease control strategies for epidemics on networks

Models for control of epidemics on local, global and small-world networks are considered, with only partial information accessible about the status of individuals and their connections. The main goal of an effective control measure is to stop the epidemic at a lowest possible cost, including treatment and cost necessary to track the disease spread. We show that delay in detection of infectious individuals and presence of long-range links are the most important factors determining the cost. However, the details of long-range links are usually the least-known element of the social interactions due to their occasional character and potentially short life-span. We show that under some conditions on the probability of disease spread, it is advisable to attempt to track those links. Thus, collecting some additional knowledge about the network structure might be beneficial to ensure a successful and cost-effective control.Comment: To be published in Acta Phys. Pol.

Levy--Brownian motion on finite intervals: Mean first passage time analysis

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by L\'evy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulae when the stability index $\alpha$ approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.Comment: 9 pages, 13 figure

Transport in a Levy ratchet: Group velocity and distribution spread

We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric L\'evy noise, being a minimal setup for a ``L\'evy ratchet.'' Due to the non-thermal character of the L\'evy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the L\'evy ratchet has to be based on the characteristics of directionality which are different from typically used measures like mean current and the dispersion of particles' positions, since these get inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport like the position of the median of the particles displacements' distribution characterizing the group velocity, and the interquantile distance giving the measure of the distributions' width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length unveiling qualitative differences between the noises with L\'evy indices below and above unity. Finally, we inspect the problem of the first escape from an interval of given length revealing independence of exit times on the structure of the potential.Comment: 9 pages, 12 figure

Anomalous diffusion and generalized Sparre-Andersen scaling

We are discussing long-time, scaling limit for the anomalous diffusion composed of the subordinated L\'evy-Wiener process. The limiting anomalous diffusion is in general non-Markov, even in the regime, where ensemble averages of a mean-square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion. To discriminate between truly memory-less process and the non-Markov one, we are analyzing deviation of the survival probability from the (standard) Sparre-Andersen scaling.Comment: 5 pages, 3 figure

Stationary states in Langevin dynamics under asymmetric L\'evy noises

Properties of systems driven by white non-Gaussian noises can be very different from these systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by $\alpha$-stable L\'evy type noises, which provide natural extension to the Gaussian noise having however a new property mainly a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic and in generic double well potential models subjected to the action of $\alpha$-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.Comment: 9 pages, 9 figures, 3 table

Controlling disease spread on networks with incomplete knowledge

Models for control of highly infectious diseases on local, small-world, and scale-free networks are considered, with only partial information accessible about the status of individuals and their connections. We consider a case when individuals can be infectious before showing symptoms and thus before detection. For small to moderately severe incidence of infection with a small number of nonlocal links, it is possible to control disease spread by using purely local methods applied in a neighborhood centered around a detected infectious individual. There exists an optimal radius for such a control neighborhood leading to the lowest severity of the epidemic in terms of economic costs associated with disease and treatment. The efficiency of a local control strategy is very sensitive to the choice of the radius. Below the optimal radius, the local strategy is unsuccessful; the disease spreads throughout the system, necessitating treatment of the whole population. At the other extreme, a strategy involving a neighborhood that is too large controls the disease but is wasteful of resources. It is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated

Bimodality and hysteresis in systems driven by confined L\'evy flights

We demonstrate occurrence of bimodality and dynamical hysteresis in a system describing an overdamped quartic oscillator perturbed by additive white and asymmetric L\'evy noise. Investigated estimators of the stationary probability density profiles display not only a turnover from unimodal to bimodal character but also a change in a relative stability of stationary states that depends on the asymmetry parameter of the underlying noise term. When varying the asymmetry parameter cyclically, the system exhibits a hysteresis in the occupation of a chosen stationary state.Comment: 4 pages, 5 figures, 30 reference

Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the abovementioned properties of "Gaussianity" and "whiteness" of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian L\'evy walks, so called L\'evy flights correspond to the class of Markov processes which still can be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. L\'evy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed to understand features of stochastic dynamics under the influence of L\'evy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by non-Gaussian, heavy-tailed fluctuations with infinite variance.Comment: 7 pages, 8 figure

Improving epidemic control strategies by extended detection

The majority of epidemics eradication programs work in a preventive responsive way. The lack of exact information about the epidemiological status of individuals makes responsive actions less efficient. Here, we demonstrate that additional tests can significantly increase the efficiency of “blind” treatment (vaccination or culling). Eradication strategy consisting of “blind” treatment in very limited local neighbourhood supplemented by extra tests in a little bit larger neighbourhood is able to prevent invasion of even highly infectious diseases and to achieve this at a cost lower than for the “blind” strategy. The effectiveness of the extended strategy depends on such parameters as the test efficiency and test cost