Abstract:Network epidemics is a ubiquitous model that can represent different phenomena and finds applications in various domains. Among its various characteristics, a fundamental question concerns the time when an epidemic stops propagating. We investigate this characteristic on a SIS epidemic induced by agents that move according to independent continuous time random walks on a finite graph: Agents can either be infected (I) or susceptible (S), and infection occurs when two agents with different epidemic states meet … Show more
“…Epidemics with motion were extensively studied in the random graph setting (without considering geometry). The basic model for SIS on graphs is that where agents perform a random walk on the random graph and where agents meeting at a give node of the graph may infect each other -see [5] and the references therein.…”
This paper is focused on SIS epidemic dynamics (also known as the contact process) on stationary Poisson point processes of the Euclidean plane, when the infection rate of a susceptible point is proportional to the number of infected points in a ball around it. Two models are discussed, the first with a static point process, and the second where points are subject to some random motion. For both models, we use conservation equations for moment measures to analyze the stationary point processes of infected and susceptible points. A heuristic factorization of the third moment measure is then proposed to derive simple polynomial equations allowing one to derive closed form approximations for the fraction of infected nodes and the steady state. These polynomial equations also lead to a phase diagram which tentatively delineates the regions of the space of parameters (population density, infection radius, infection and recovery rate, and motion rate) where the epidemic survives and those where there is extinction. According to this phase diagram, the survival of the epidemic is not always an increasing function of the motion rate. These results are substantiated by simulations on large two dimensional tori. These simulations show that the polynomial equations accurately predict the fraction of infected nodes when the epidemic survives. The phase diagram is also partly substantiated by the simulation of the mean survival time of the epidemic on large tori. The phase diagram accurately predicts the parameter regions where the mean survival time increases or decreases with the motion rate.
“…Epidemics with motion were extensively studied in the random graph setting (without considering geometry). The basic model for SIS on graphs is that where agents perform a random walk on the random graph and where agents meeting at a give node of the graph may infect each other -see [5] and the references therein.…”
This paper is focused on SIS epidemic dynamics (also known as the contact process) on stationary Poisson point processes of the Euclidean plane, when the infection rate of a susceptible point is proportional to the number of infected points in a ball around it. Two models are discussed, the first with a static point process, and the second where points are subject to some random motion. For both models, we use conservation equations for moment measures to analyze the stationary point processes of infected and susceptible points. A heuristic factorization of the third moment measure is then proposed to derive simple polynomial equations allowing one to derive closed form approximations for the fraction of infected nodes and the steady state. These polynomial equations also lead to a phase diagram which tentatively delineates the regions of the space of parameters (population density, infection radius, infection and recovery rate, and motion rate) where the epidemic survives and those where there is extinction. According to this phase diagram, the survival of the epidemic is not always an increasing function of the motion rate. These results are substantiated by simulations on large two dimensional tori. These simulations show that the polynomial equations accurately predict the fraction of infected nodes when the epidemic survives. The phase diagram is also partly substantiated by the simulation of the mean survival time of the epidemic on large tori. The phase diagram accurately predicts the parameter regions where the mean survival time increases or decreases with the motion rate.
“…The analysis of the case with mobility is more recent. The situation where agents perform a random walk on a finite graph and agents meeting at a given point of the graph may infect each other was studied in [6]. The situation where agents form a Poisson point process and migrate in the Euclidean plane was studied in [3], which proposed a computational framework based on moment closure techniques for evaluating the role of mobility in the propagation of epidemics.…”
Section: Introductionmentioning
confidence: 99%
“…The queueing model studied here may be seen as a discrete version of the model in [3], or as a thermodynamic limit of that of [6]. Figure 1. On the left, the SIS reactor.…”
Consider the following migration process based on a closed network of N queues with
$K_N$
customers. Each station is a
$\cdot$
/M/
$\infty$
queue with service (or migration) rate
$\mu$
. Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate
$\alpha Y$
if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate
$\beta$
. We let N tend to infinity and assume that
$\lim_{N\to \infty} K_N/N= \eta $
, where
$\eta$
is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to
$\eta$
. The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.
“…The analysis of the case with mobility is more recent. The situation where agents perform a random walk on a finite graph and agents meeting at a given point of the graph may infect each other was studied in [5]. The situation where agents form a Poisson point processes and migrate in the Euclidean plane was studied in [2], were a computational framework based onmoment closure techniques was proposed for evaluating the role of mobility on the propagation of epidemics.…”
Section: Introductionmentioning
confidence: 99%
“…The queuing model studied here may be seen as a discrete version of the model in of [2], or as a thermodynamic limit of that of [5].…”
Consider the following migration process based on a closed network of N queues with K N customers. Each station is a •/M/∞ queue with service (or migration) rate µ. Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to an SIS (Susceptible, Infected, Susceptible) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate αY if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate β. We let N tend to infinity and assume that lim N →∞ K N /N = η, where η is a positive constant representing the customer density. The main problem of interest is about the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We establish several structural properties (monotonicity and convexity) of the system, which allow us to give the structure of the phase transition diagram of this thermodynamic limit w.r.t. η. The analysis of this SIS model reduces to that of a wave-type PDE for which we found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the DOCS (Departure On Change of State) and the AIR (Averaged Infection Rate), which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed systems and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest.
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