A computational framework for evaluating the role of mobility on the propagation of epidemics on point processes

Baccelli, François; Ramesan, Nithin S.

doi:10.1007/s00285-021-01692-1

Cited by 4 publications

(9 citation statements)

References 14 publications

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“…This evidence suggests that the phase diagram in µ may be more complicated than in the case of other parameters. This is in line with what was observed in [2]. It is a subject of our future research plans.…”

Section: Phase Diagram In µsupporting

confidence: 92%

“…Note that, due to monotonicity and convexity of the function p o = g(p), we get p all t ≥ T (1) , the input rate of infected customers in the auxiliary upper process is bigger than that in the thermodynamic limit. Then it follows from simple sample-path arguments that Y (t) ≤ Y (2) (t) a.s. for all t ≥ T (1) and, in particular, E(Y (t)) ≤ E Y (2) (t), where the RHS is smaller than p (3) 1 , for all t ≥ T (3) . The induction argument implies that, for any n = 1, 2, .…”

Section: Proof Of Theorem 20mentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…Consider now the time interval [T (1) , ∞) and compare the processes ( X (2) (t), Y (2) (t)) and (X(t), Y (t)) within this interval. By construction, at the initial time T (1) of this time period, Y (2) (T (1) ) ≥ Y (T (1) ) a.s. and, for…”

mentioning

confidence: 99%

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Migration-Contagion Processes

Baccelli¹,

Foss²,

Shneer³

2022

Preprint

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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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Section: Phase Diagram In µsupporting

confidence: 92%