Susceptible-infected-susceptible Epidemics Research Articles

Probabilistic predictions of SIS epidemics on networks based on population-level observations

We predict the future course of ongoing susceptible–infected–susceptible (SIS) epidemics on regular, Erdős–Rényi and Barabási–Albert networks. It is known that the contact network influences the spread of an epidemic within a population. Therefore, observations of an epidemic, in this case at the population-level, contain information about the underlying network. This information, in turn, is useful for predicting the future course of an ongoing epidemic. To exploit this in a prediction framework, the exact high-dimensional stochastic model of an SIS epidemic on a network is approximated by a lower-dimensional surrogate model. The surrogate model is based on a birth-and-death process; the effect of the underlying network is described by a parametric model for the birth rates. We demonstrate empirically that the surrogate model captures the intrinsic stochasticity of the epidemic once it reaches a point from which it will not die out. Bayesian parameter inference allows for uncertainty about the model parameters and the class of the underlying network to be incorporated directly into probabilistic predictions. An evaluation of a number of scenarios shows that in most cases the resulting prediction intervals adequately quantify the prediction uncertainty. As long as the population-level data is available over a long-enough period, even if not sampled frequently, the model leads to excellent predictions where the underlying network is correctly identified and prediction uncertainty mainly reflects the intrinsic stochasticity of the spreading epidemic. For predictions inferred from shorter observational periods, uncertainty about parameters and network class dominate prediction uncertainty. The proposed method relies on minimal data at population-level, which is always likely to be available. This, combined with its numerical efficiency, makes the proposed method attractive to be used either as a standalone inference and prediction scheme or in conjunction with other inference and/or predictive models.

Multigroup SIS Epidemics With Simplicial and Higher Order Interactions

This article analyzes a susceptible–infected–susceptible (SIS) model of epidemic propagation over hypergraphs, and motivated by an important special case, we refer to the model as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simplicial SIS model</i> . Classically, the multigroup SIS model has assumed pairwise interactions of contagion across groups and, thus, has been vastly studied in the literature. It is only recently that renewed special attention has been drawn to the study of contagion dynamics over higher order interactions and over more general graph structures, such as simplexes. Previous work on mean-field approximation scalar models of the simplicial SIS model has indicated that a new dynamical behavior domain, compared to the classical SIS model, appears due to the newly introduced higher order interaction terms: both a disease-free equilibrium and an endemic equilibrium coexist and are both locally asymptotically stable. This article formally establishes that bistability (as a new epidemiological behavior) also appears in the multigroup simplicial SIS model. We give sufficient conditions over the model’s parameters for the appearance of this and the other behavioral domains present in the classical multigroup SIS model. We additionally provide an algorithm to compute the value of the endemic equilibrium and report numerical analysis of the transition from the disease-free domain to the bistable domain.

Semi‐Markov models of epidemics over networks with time delays

AbstractIn this article, we extend the Markov models describing the susceptible‐infected‐susceptible (SIS) epidemics over undirected networks to take into account the virus minimum incubation period and the minimum recovery period of an infected individual. We represent both periods as time delays in the states of the extended model. However, due to the addition of time delays, the process loses its Markovain property. We use the generalized semi‐Markov theory to introduce both the incubation and recovery delays to the probabilistic dynamical models. Hence, in this paper, we propose a time‐delay version of the two principal models of the SIS epidemics over undirected networks: the exactstate model and the approximatedintertwined model. Finally, using Lyapunov analysis, we give sufficient conditions that guarantee the global exponential stability of the time‐delayintertwined model.

A State Feedback Controller for Mitigation of Continuous-Time Networked SIS Epidemics

The paper considers continuous-time networked susceptible-infected-susceptible (SIS) diseases spreading over a population. Each agent represents a sub-population and has its own healing rate and infection rate; the state of the agent at a time instant denotes what fraction of the said sub-population is infected with the disease at the said time instant. By taking account of the changes in behaviors of the agents in response to the infection rates in real-time, our goal is to devise a feedback strategy such that the infection level for each agent strictly stays below a pre-specified value. Furthermore, we are also interested in ensuring that the closed-loop system converges either to the disease-free equilibrium or, when it exists, to the endemic equilibrium. The upshot of devising such a strategy is that it allows health administration officials to ensure that there is sufficient capacity in the healthcare system to treat the most severe cases. We demonstrate the effectiveness of our controller via numerical examples.

Suppressing the endemic equilibrium in SIS epidemics: A state dependent approach

Abstract This paper considers the susceptible-infected-susceptible (SIS) epidemic model with an underlying network structure and focuses on the effect of social distancing to regulate the epidemic level. We demonstrate that if each subpopulation is informed of its infection rate and reduces interactions accordingly, the fraction of the subpopulation infected stays below half for all time instants. To this end, we first modify the basic SIS model by introducing a state dependent parameter representing the frequency of interactions between subpopulations. Thereafter, we show that for this modified SIS model, the spectral radius of a suitably-defined matrix being not greater than one causes all the agents, regardless of their initial sickness levels, to converge to the healthy state; assuming non-trivial disease spread, the spectral radius being greater than one leads to the existence of a unique endemic equilibrium, which is also asymptotically stable. Finally, by leveraging the aforementioned results, we show that the fraction of (sub)populations infected never exceeds half.

Optimal Induced Spreading of SIS Epidemics in Networks

Induced spreading aims to maximize the infection probabilities of some target nodes by adjusting the nodal infection rates, which can be applied in biochemical and information spreading. We assume that the adjustment of the nodal infection rates has an associated cost and formulate the induced spreading for susceptible-infected-susceptible (SIS) epidemics in networks as an optimization problem under a constraint on total cost. We address and solve both a static model and a dynamic model for the optimization of the induced SIS spreading. By numerical results in some artificial and real networks, we investigate the effect of the network topology on the optimal induced strategy with a quadratic cost function. In the static method, the infection rate increment on each node is coupled to both the degree and the average hops to the targets. In the dynamic method, we show that the effective resistance could be a good metric to indicate the minimum total cost for targeting a single node. We also illustrate that the minimum total cost increases much more slowly with the increasing fraction of targets in the SIS model than in linear control systems.

Game-Theoretic Protection Against Networked SIS Epidemics by Human Decision-Makers

Abstract We study networks of human decision-makers who independently decide how to protect themselves against Susceptible-Infected-Susceptible (SIS) epidemics. Motivated by studies in behavioral economics showing that humans perceive probabilities in a nonlinear fashion, we examine the impacts of such misperceptions on the equilibrium protection strategies. In our setting, nodes choose their curing rates to minimize the infection probability under the degree-based mean-field approximation of the SIS epidemic plus the cost of their selected curing rate. We establish the existence of a degree based equilibrium under both true and nonlinear perceptions of infection probabilities (under suitable assumptions). When the per-unit cost of curing rate is sufficiently high, we show that true expectation minimizers choose the curing rate to be zero at the equilibrium, while curing rate is nonzero under nonlinear probability weighting.

The spreading time in SIS epidemics on networks

In a Susceptible–Infected–Susceptible (SIS) process, we investigate the spreading time Tm, which is the time when the number of infected nodes in the metastable state is first reached, starting from the outbreak of the epidemics. We observe that the spreading time Tm resembles a lognormal-like distribution, though with different deep tails, both for the Markovian and the non-Markovian infection process, which implies that the spreading time can be very long with a relatively high probability. In addition, we show that a stronger virus, with a higher effective infection rate τ or an earlier timing of the infection attempts, does not always lead to a shorter average spreading time E[Tm]. We numerically demonstrate that the average spreading time E[Tm] in the complete graph and the star graph scales logarithmically as a function of the network size N for a fixed fraction of infected nodes in the metastable state.

Assessing network robustness under SIS epidemics: The relationship between epidemic threshold and viral conductance

Telecommunication networks, as well as other network types, are critical infrastructures where any service disruption has a notable impact on individuals. Hence, studying network dynamics under failures or attacks is of paramount importance. In this paper, we assess the robustness of networks with respect to the spread of Susceptible-Infected-Susceptible (SIS) epidemics, using the N-Intertwined Mean-Field Approximation (NIMFA). A classical robustness metric is the NIMFA epidemic threshold, which is inversely proportional to the largest eigenvalue of the adjacency matrix, also called the spectral radius. Besides the NIMFA epidemic threshold, the viral conductance has been proposed as a measure incorporating the average fraction of infected nodes in the steady state for all possible effective infection rates. In general, the viral conductance provides more information about the network’s behavior with respect to virus spreading, however, the full picture is not always necessary. The aim of this paper is to understand when the spectral radius is adequate for reflecting robustness. By analyzing the relationship between spectral radius and viral conductance in several graph classes, we show that the two metrics are highly correlated. We thus conclude that the spectral radius is sufficient to compare the robustness of networks belonging to the same class.

Diffusion in Social Networks as SIS Epidemics: Beyond Full Mixing and Complete Graphs

Peer influence and interactions between agents in a population give rise to complex, nonlinear behaviors. This paper adopts the SIS (susceptible-infected-susceptible) framework from epidemiology to analytically study how network topology affects the diffusion of ideas/opinions/beliefs/innovations in social networks. We introduce the scaled SIS process, which models peer influence as neighbor-to-neighbor infections. We model the scaled SIS process as a continuous-time Markov process and derive for this process its closed form equilibrium distribution. The adjacency matrix that describes the underlying social network is explicitly reflected in this distribution. The paper shows that interesting population asymptotic behaviors occur for scenarios where the individual tendencies of each agent oppose peer influences. Specifically, we determine how the most probable configuration of agent states (i.e., the population configuration with maximum equilibrium distribution) depends on both model parameters and network topology. We show that, for certain regions of the parameter space, this and related issues reduce to standard graph questions like the maximum independent set problem.

How to run a campaign: Optimal control of SIS and SIR information epidemics

Information spreading in a population can be modeled as an epidemic. Campaigners (e.g., election campaign managers, companies marketing products or movies) are interested in spreading a message by a given deadline, using limited resources. In this paper, we formulate the above situation as an optimal control problem and the solution (using Pontryagin’s Maximum Principle) prescribes an optimal resource allocation over the time of the campaign. We consider two different scenarios—in the first, the campaigner can adjust a direct control (over time) which allows her to recruit individuals from the population (at some cost) to act as spreaders for the Susceptible-Infected-Susceptible (SIS) epidemic model. In the second case, we allow the campaigner to adjust the effective spreading rate by incentivizing the infected in the Susceptible-Infected-Recovered (SIR) model, in addition to the direct recruitment. We consider time varying information spreading rate in our formulation to model the changing interest level of individuals in the campaign, as the deadline is reached. In both the cases, we show the existence of a solution and its uniqueness for sufficiently small campaign deadlines. For the fixed spreading rate, we show the effectiveness of the optimal control strategy against the constant control strategy, a heuristic control strategy and no control. We show the sensitivity of the optimal control to the spreading rate profile when it is time varying.

Susceptible-infected-susceptible epidemics on networks with general infection and cure times

The classical, continuous-time susceptible-infected-susceptible (SIS) Markov epidemic model on an arbitrary network is extended to incorporate infection and curing or recovery times each characterized by a general distribution (rather than an exponential distribution as in Markov processes). This extension, called the generalized SIS (GSIS) model, is believed to have a much larger applicability to real-world epidemics (such as information spread in online social networks, real diseases, malware spread in computer networks, etc.) that likely do not feature exponential times. While the exact governing equations for the GSIS model are difficult to deduce due to their non-Markovian nature, accurate mean-field equations are derived that resemble our previous N-intertwined mean-field approximation (NIMFA) and so allow us to transfer the whole analytic machinery of the NIMFA to the GSIS model. In particular, we establish the criterion to compute the epidemic threshold in the GSIS model. Moreover, we show that the average number of infection attempts during a recovery time is the more natural key parameter, instead of the effective infection rate in the classical, continuous-time SIS Markov model. The relative simplicity of our mean-field results enables us to treat more general types of SIS epidemics, while offering an easier key parameter to measure the average activity of those general viral agents.