Epidemic threshold for the SIRS model on the networks

Saif, Mehrdad

doi:10.1016/j.physa.2019.122251

Cited by 14 publications

(14 citation statements)

References 33 publications

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“…In addition to that, we can confirm the importance of this parameter in the occurrence of the synchronization state from Ref. [7], which emphasizes the need for the loops in the networks in order to the disease to spread frequently throughout the nodes of the networks. Whereas, clusters tend to spread infection among close-knit neighborhoods [14].…”

Section: Numerical Resultssupporting

confidence: 78%

“…In Ref. [7] we have proved that, for the SIRS model on the networks and when the recovery time τ R is larger than the infection time τ I , the infection will flow directionally from the ancestors to descendants however, the descendants will be unable to reinfect their ancestors during the time of their illness. That behavior leads him to deduce that, in order to the disease spreads frequently throughout the nodes of the networks, the loops on the network are necessary, which means the clustering coefficient will play an important role in this model.…”

Section: Introductionmentioning

confidence: 99%

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Emergence of Self-Sustained Oscillations for SIRS Model on Random Networks

Saif¹

2019

Preprint

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We study the phase transition from the persistence phase to the extinction phase for the SIRS (susceptible/ infected/ refractory/ susceptible) model of diseases spreading on random networks. By studying temporal evolution and synchronization parameter of this model on random networks, we find that, this model on random networks, shows a synchronization phase in a narrow range of very small values of clustering coefficient. This finding corroborates the conclusion reached in Ref.[4] that, the clustering coefficient is responsible for the emergence of the synchronization phase in the small world networks.

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Section: Numerical Resultssupporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Emergence of Self-Sustained Oscillations for SIRS Model on Random Networks

Saif¹

2019

Preprint

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“…When all individuals are susceptible, the infection dies out, i.e. all susceptibles is an absorbing state of the system 29 . Lastly note a significant difference from certain earlier models here.…”

Section: Modelmentioning

confidence: 99%

Localized spatial distributions of disease phases yield long-term persistence of infection

Moitra

Sinha

2019

Sci Rep

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We explore the emergence of persistent infection in two patches where the phases of disease progression of the individuals is given by the well known SIRS cycle modelling non-fatal communicable diseases. We find that a population structured into two patches with significantly different initial states, yields persistent infection, though interestingly, the infection does not persist in a homogeneous population having the same average initial composition as the average of the initial states of the two patches. This holds true for inter-patch links ranging from a single connection to connections across the entire inter-patch boundary. So a population with spatially uniform distribution of disease phases leads to disease extinction, while a population spatially separated into distinct patches aids the long-term persistence of disease. After transience, even very dissimilar patches settle down to the same average infected sub-population size. However the patterns of disease spreading in the patches remain discernibly dissimilar, with the evolution of the total number of infecteds in the two patches displaying distinct periodic wave forms, having markedly different amplitudes, though identical frequencies. We quantify the persistent infection through the size of the asymptotic infected set. We find that the number of inter-patch links does not affect the persistence in any significant manner. The most important feature determining persistence of infection is the disparity in the initial states of the patches, and it is clearly evident that persistence increases with increasing difference in the constitution of the patches. So we conclude that populations with very non-uniform distributions, where the individuals in different phases of disease are strongly compartmentalized spatially, lead to sustained persistence of disease in the entire population.

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“…(5.6) 1+𝜀 𝑛 (4.10) WCA+17], consist of mean-eld analysis of the model [BP10], or consider deterministic variants of the model [Sai19] 2 . Although the similarities of the SIRS model with its SIS counterpart suggest that understanding the behavior of SIRS on simple structures could be used to yield results on real-world graph models, this fundamental rst step has not been done yet.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the survival time of the SIRS process via expansion

Friedrich¹,

Göbel²,

Klodt³

et al. 2022

Preprint

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We study two continuous-time Markov chains modeling the spread of infections on graphs, namely the SIS and the SIRS model. In the SIS model, vertices are either susceptible or infected; each infected vertex becomes susceptible at rate 1 and infects each of its neighbors independently at rate 𝜆. In the SIRS model, vertices are either susceptible, infected, or recovered; each infected vertex becomes recovered at rate 1 and infects each of its susceptible neighbors independently at rate 𝜆; each recovered vertex becomes susceptible at a rate 𝜚 , which we assume to be independent of the size of the graph. The survival time of the SIS process, i.e., the time until no vertex of the host graph is infected, is fairly well understood for a variety of graph classes. Stars are an important graph class for the SIS model, as the survival time of SIS on stars has been used to show that the process survives on real-world graphs for a long time. For the SIRS model, however, to the best of our knowledge, there are no rigorous results, even for simple graphs such as stars.We analyze the survival time of the SIS and the SIRS process on stars and cliques. We determine three threshold values for 𝜆 such that when 𝜆 < 𝜆 ℓ , the expected survival time of the process is at most logarithmic, when 𝜆 < 𝜆 𝑝 , it is at most polynomial, and when 𝜆 > 𝜆 𝑠 , it is at least super-polynomial in the number of vertices. Our results show that the survival time of the two processes behaves fundamentally di erent on stars, while it behaves fairly similar on cliques. Our analyses bound the drift of potential functions with globally stable equilibrium points. On the SIRS process, our two-state potential functions are inspired by Lyapunov functions used in mean-eld theory.

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Epidemic threshold for the SIRS model on the networks

Cited by 14 publications

References 33 publications

Emergence of Self-Sustained Oscillations for SIRS Model on Random Networks

Emergence of Self-Sustained Oscillations for SIRS Model on Random Networks

Localized spatial distributions of disease phases yield long-term persistence of infection

Analysis of the survival time of the SIRS process via expansion

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