Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated

Cator, Eric; Mieghem, P.F.A. Van

doi:10.1103/physreve.89.052802

Cited by 42 publications

(70 citation statements)

References 25 publications

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“…Clearly, if cov[X i (t),X k (t)] = 0 for each nodal pair (i,k), then the NIMFA equations (3) are equal to the exact SIS equations (2). Moreover, as shown in [27], cov[X i (t),X k (t)] 0 for a Markovian SIS and SIR process on any graph, so that βR i 0 and NIMFA always upper bounds the viral infection probability (v i w i ) and, thus, lower bounds the epidemic threshold τ c τ…”

Section: Mean-field Approximation For Markovian Sis Epidemics On mentioning

confidence: 99%

“…are the ordered real eigenvalues of an N × N symmetric matrix M. We remark that criterion (6) does not hold for non-Markovian SIS epidemics on networks [30], since then, as shown in [27], c ij = cov[X i ,X j ] can be negative. A consequence of the WielandtHoffman theorem [28, p. 252] for symmetric matrices A and C is that…”

Section: Accuracy Criterionmentioning

confidence: 99%

“…As demonstrated in [27], c ij 0, and, since a ij 0, we have R i 0 for each node i. Let R,V ,W be the vector with ith component R i ,v i (t), and w i (t), respectively.…”

Section: Accuracy Criterionmentioning

confidence: 99%

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Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks

Mieghem

Bovenkamp

2015

Phys. Rev. E

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Mean-field approximations (MFAs) are frequently used in physics. When a process (such as an epidemic or a synchronization) on a network is approximated by MFA, a major hurdle is the determination of those graphs for which MFA is reasonably accurate. Here, we present an accuracy criterion for Markovian susceptible-infectedsusceptible (SIS) epidemics on any network, based on the spectrum of the adjacency and SIS covariance matrix. We evaluate the MFA criterion for the complete and star graphs analytically, and numerically for connected Erdős-Rényi random graphs for small size N 14. The accuracy of MFA increases with average degree and with N . Precise simulations (up to network sizes N = 100) of the MFA accuracy criterion versus N for the complete graph, star, square lattice, and path graphs lead us to conjecture that the worst MFA accuracy decreases, for large N , proportionally to the inverse of the spectral radius of the adjacency matrix of the graph.

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Section: Mean-field Approximation For Markovian Sis Epidemics On mentioning

confidence: 99%

Section: Accuracy Criterionmentioning

confidence: 99%

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Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks

Mieghem

Bovenkamp

2015

Phys. Rev. E

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“…(C6) and by simulations. The N-intertwined mean-field approximation (NIMFA) epidemic threshold τ (1) c is defined as the reciprocal of the largest eigenvalue λ 1 of the adjacency matrix and is a lower bound of the real epidemic threshold [23,24], i.e., τ …”

Section: A the Complete Graphmentioning

confidence: 99%

“…7(b). For large N in K N , both the NIMFA epidemic threshold τ , which may seem surprising, since the NIMFA threshold is proved to be a lower bound for the exact epidemic threshold [24]. For small network sizes N , the epidemic threshold is not precisely defined, because there is a relatively broad transition region in τ .…”

Section: The Largermentioning

confidence: 99%

Survival time of the susceptible-infected-susceptible infection process on a graph

Bovenkamp

Mieghem

2015

Phys. Rev. E

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The survival time T is the longest time that a virus, a meme, or a failure can propagate in a network. Using the hitting time of the absorbing state in an uniformized embedded Markov chain of the continuous-time susceptible-infected-susceptible (SIS) Markov process, we derive an exact expression for the average survival time E[T ] of a virus in the complete graph K N and the star graph K 1,N−1 . By using the survival time, instead of the average fraction of infected nodes, we propose a new method to approximate the SIS epidemic threshold τ c that, at least for K N and K 1,N−1 , correctly scales with the number of nodes N and that is superior to the epidemic threshold τ . However, when the average fraction of infected nodes is used as a basis for comparison, the virus will survive in the star graph longer than in any other graph, making the star graph the worst-case graph instead of the complete graph. Finally, in non-Markovian SIS, the distribution of the spreading attempts over the infectious period of a node influences the survival time, even if the expected number of spreading attempts during an infectious period (the non-Markovian equivalent of the effective infection rate) is kept constant. Both early and late infection attempts lead to shorter survival times. Interestingly, just as in Markovian SIS, the survival times appear to be exponentially distributed, regardless of the infection and curing time distributions.

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Some aspects of the Markovian SIRS epidemic on networks and its mean‐field approximation

Ottaviano

Bonaccorsi

2020

Math Methods in App Sciences

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We study the spread of an SIRS‐type epidemic with vaccination on network. Starting from an exact Markov description of the model, we investigate the mean epidemic lifetime by providing a sufficient condition for fast extinction that depends on the model parameters and the topology of the network. Then, we pass to consider a first‐order mean‐field approximation of the exact model and its stability properties, by relying on the graph‐theoretical notion of equitable partition. In the case of graphs possessing this kind of partition, we prove that the endemic equilibrium can be computed by using a lower‐dimensional dynamical system. Finally, in the special case of regular graphs, we investigate the domain of attraction of the endemic equilibrium.

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Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated

Cited by 42 publications

References 25 publications

Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks

Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks

Survival time of the susceptible-infected-susceptible infection process on a graph

Some aspects of the Markovian SIRS epidemic on networks and its mean‐field approximation

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