scholarly journals An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 931 ◽  
Author(s):  
Andrei Khrennikov ◽  
Klaudia Oleschko
Keyword(s):  
Random Walk ◽  
Spin Glasses ◽  
Statistical Physics ◽  
Herd Immunity ◽  
Social Hierarchy ◽  
Disease Spread ◽  
Ultrametric Spaces ◽  
Social Barriers ◽  
Preventative Measures ◽  
One Step

We present a mathematical model of disease (say a virus) spread that takes into account the hierarchic structure of social clusters in a population. It describes the dependence of epidemic’s dynamics on the strength of barriers between clusters. These barriers are established by authorities as preventative measures; partially they are based on existing socio-economic conditions. We applied the theory of random walk on the energy landscapes represented by ultrametric spaces (having tree-like geometry). This is a part of statistical physics with applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, a virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy levels composing this barrier. Infection spreads rather easily inside a social cluster (say a working collective), but jumps to other clusters are constrained by social barriers. The model implies the power law, 1−t−a, for approaching herd immunity, where the parameter a is proportional to inverse of one-step barrier Δ. We consider linearly increasing barriers (with respect to hierarchy), i.e., the m-step barrier Δm=mΔ. We also introduce a quantity characterizing the process of infection distribution from one level of social hierarchy to the nearest lower levels, spreading entropy E. The parameter a is proportional to E.

scholarly journals Utrametric diffusion equation on energy landscape to model disease spread in hierarchic socially clustered population

2021 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Diffusion Equation ◽  
Mathematical Models ◽  
Statistical Physics ◽  
Herd Immunity ◽  
Number Fields ◽  
Disease Spread ◽  
Ultrametric Spaces ◽  
Social Barriers ◽  
Hierarchic Structure ◽  
Medical Specialities

We present a new mathematical model of disease spread reflecting some specialities of the covid-19 epidemic by elevating the role of hierarchic social clustering of population. The model can be used to explain slower approaching herd immunity, e.g., in Sweden, than it was predicted by a variety of other mathematical models and was expected by epidemiologists; see graphs Fig. \ref{fig:minipage1},\ref{fig:minipage2}. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider trees with the constant number $p>1$ of branches leaving each vertex. Such trees are endowed with an algebraic structure, these are $p$-adic number fields. We apply theory of the $p$-adic diffusion equation to describe a virus spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling {\it dynamics on energy landscapes.} To move from one social cluster (valley) to another, a virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. We consider {\it linearly increasing barriers.} A virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the covid-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model; in particular, by notion of the probability to be infected (at time $t$ in a social cluster $C).$ We present socio-medical specialities of the covid-19 epidemic supporting our model.

scholarly journals Ultrametric model for covid-19 dynamics: an attempt to explain slow approaching herd immunity in Sweden

2020 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Random Walk ◽  
Spin Glasses ◽  
Herd Immunity ◽  
Human Society ◽  
Ultrametric Spaces ◽  
Real Situation ◽  
Social Barriers ◽  
Infection Dynamics ◽  
Hierarchic Structure ◽  
Social Clusters

AbstractWe present a mathematical model of infection dynamics that might explain slower approaching the herd immunity during the covid-19 epidemy in Sweden than it was predicted by a variety of other models; see graphs Fig. 2. The new model takes into account the hierarchic structure of social clusters in the human society. We apply the well developed theory of random walk on the energy landscapes represented mathematically with ultrametric spaces. This theory was created for applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, the virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy’s levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider linearly increasing (with respect to hierarchy’s levels) barriers. This structure of barriers matches with a rather soft regulations imposed in Sweden in March 2020. In this model, the infection spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This model’s feature matches with the real situation during the covid-19 epidemy, with its cluster spreading structure. Clusters need not be determined solely geographically, they are based on a number of hierarchically ordered social coordinates. The model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. In particular, our model describes such a specialty of spread of covid-19 virus as the presence of “super-spreaders” who by performing a kind of random walk on a hierarchic landscape of social clusters spreads infection. In future, this model will be completed by adding the SIR-type counterpart. But, the latter is not a specialty of covid-19 spreading.

scholarly journals Utrametric diffusion model for spread of covid-19 in socially clustered population: Can herd immunity be approached in Sweden?

2020 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Mathematical Models ◽  
Statistical Physics ◽  
Herd Immunity ◽  
Number Fields ◽  
Disease Spread ◽  
Ultrametric Spaces ◽  
Medical Specialties ◽  
Social Barriers ◽  
Hierarchic Structure ◽  
Diffusional Model

We present a new mathematical model of disease spread reflecting specialties of covid-19 epidemic by elevating the role social clustering of population. The model can be used to explain slower approaching herd immunity in Sweden, than it was predicted by a variety of other mathematical models; see graphs Fig. 2. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider homogeneous trees with p-branches leaving each vertex. Such trees are endowed with algebraic structure, the p-adic number fields. We apply theory of the p-adic diffusion equation to describe coronavirus' spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling dynamics on energy landscapes. To move from one social cluster (valley) to another, the virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider linearly increasing barriers. This structure matches with mild regulations in Sweden. The virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the covid-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. We present socio-medical specialties of the covid-19 epidemic supporting our purely diffusional model.

scholarly journals P-adic (treelike) Diffusion Model for COVID-19 Epidemic

2020 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Mathematical Models ◽  
Statistical Physics ◽  
Herd Immunity ◽  
Number Fields ◽  
Disease Spread ◽  
Ultrametric Spaces ◽  
Medical Specialties ◽  
Social Barriers ◽  
Hierarchic Structure ◽  
Diffusional Model

We present a new mathematical model of disease spread reflecting specialties of covid-19 epidemic by elevating the role social clustering of population. The model can be used to explain slower approaching herd immunity in Sweden, than it was predicted by a variety of other mathematical models; see graphs Fig. \ref{GROWTH2}. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider homogeneous trees with $p$-branches leaving each vertex. Such trees are endowed with algebraic structure, the $p$-adic number fields. We apply theory of the $p$-adic diffusion equation to describe coronavirus' spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling {\it dynamics on energy landscapes.} To move from one social cluster (valley) to another, the virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider {\it linearly increasing barriers.} This structure matches with mild regulations in Sweden. The virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the covid-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. We present socio-medical specialties of the covid-19 epidemic supporting our purely diffusional model.

scholarly journals Ultrametric Model for COVID-19 Dynamics: An Attempt to Explain Slow Approaching Herd Immunity in Sweden

2020 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Random Walk ◽  
Spin Glasses ◽  
Herd Immunity ◽  
Human Society ◽  
Ultrametric Spaces ◽  
Social Barriers ◽  
Infection Dynamics ◽  
Hierarchic Structure ◽  
Standard Models ◽  
Social Clusters

We present a model of infection dynamics that might explain slower approaching the herd immunity during the covid-19 epidemy in Sweden than it was predicted by a variety of other models; see graphs Fig. \ref{GROWTH2}. The new model takes into account the hierarchic structure of social clusters in the human society. We apply the well developed theory of random walk on the energy landscapes represented mathematically with ultrametric spaces. This theory was created for applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, the virus should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy's levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider linearly increasing (with respect to hierarchy's levels) barriers. This structure of barriers matches with a rather soft regulations imposed in Sweden in March 2020. In this model, the infection spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This model's feature matches with the real situation during the covid-19 epidemy, with its cluster spreading structure. Clusters need not be determined solely geographically, they are based on a number of hierarchically ordered social coordinates. The model differs crucially from the standard models of spread of disease, such as the SIR-model. Our model describes such a specialty of spread of covid-19 virus as the presence of ``super-infectors'' who by performing a kind of random walk on a hierarchic landscape of social clusters spreads infection. In future, this model will be completed by adding the SIR-type counterpart. But, the latter is not a specialty of covid-19 spreading.

scholarly journals Covid-19: Why Herd Immunity was not Approached Anywhere? Ultrametric Diffusion Modeling of Virus Spread in Hierarchically Clustered Population

2021 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Herd Immunity ◽  
Disease Spread ◽  
The Novel ◽  
Virus Spread ◽  
Ultrametric Spaces ◽  
The Social ◽  
Standard Models ◽  
Models Of Disease ◽  
Basic Features ◽  
Novel Model

In spite of numerous predictions, the natural herd immunity for covid-19 visru had not been approahed anywhere in the world. Thus, the traditional mathematical models of disease spread demonstrated their inability to describe adequately the covid-19 pandemic. In author's works, the novel model of the disease spread was developed. This model reflects the basic features of the covid-19 pandemic: a) the social clustering character of virus spread, b) . Social clustering is mathematically modelled with ultrametric spaces having the treelike geometry encoding hierarchy of the regulation constraints. The virus spread is described by ultrametric diffusion or random walk on the hierarchic energy landscape. In contrast to the standard models which are characterized by the exponential decrease of the probability to become infected - at the stage of approaching of the herd immunity, the ultrametric model is characterized by the power law. Moreover, the model gives the possibility to quantify the influence of restriction measures up to the lockdown. Our main result is that the play with restrictions, including lockdowns, is counterproductive and leads to the essential slowdown of approaching the herd immunity or even makes this impossible.

One-step random-walk process of nanoparticles in cement-based materials

2021 ◽  
Vol 28 (6) ◽  
pp. 1679-1691
Author(s):  
Ali Bahari ◽  
Aref Sadeghi-Nik ◽  
Elena Cerro-Prada ◽  
Adel Sadeghi-Nik ◽  
Mandana Roodbari ◽  
...  
Keyword(s):  
Random Walk ◽  
Random Walk Process ◽  
Cement Based Materials ◽  
One Step

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

scholarly journals The Role of Risk Preferences in Responses to Messaging About COVID-19 Vaccine Take-Up

2021 ◽  
pp. 194855062199962
Author(s):  
Jennifer S. Trueblood ◽  
Abigail B. Sussman ◽  
Daniel O’Leary
Keyword(s):  
Large Scale ◽  
Risk Preferences ◽  
Herd Immunity ◽  
Disease Spread ◽  
Short Term ◽  
Policy Makers ◽  
Health Crisis ◽  
Flu Vaccine ◽  
Nationally Representative

Development of an effective COVID-19 vaccine is widely considered as one of the best paths to ending the current health crisis. While the ability to distribute a vaccine in the short-term remains uncertain, the availability of a vaccine alone will not be sufficient to stop disease spread. Instead, policy makers will need to overcome the additional hurdle of rapid widespread adoption. In a large-scale nationally representative survey ( N = 34,200), the current work identifies monetary risk preferences as a correlate of take-up of an anticipated COVID-19 vaccine. A complementary experiment ( N = 1,003) leverages this insight to create effective messaging encouraging vaccine take-up. Individual differences in risk preferences moderate responses to messaging that provides benchmarks for vaccine efficacy (by comparing it to the flu vaccine), while messaging that describes pro-social benefits of vaccination (specifically herd immunity) speeds vaccine take-up irrespective of risk preferences. Findings suggest that policy makers should consider risk preferences when targeting vaccine-related communications.

scholarly journals Scribes et enquêteurs Note sur le personnel judiciaire en Égypte aux quatre premiers siècles de l’hégire

2011 ◽  
Vol 54 (3) ◽  
pp. 370-404 ◽  
Author(s):  
Mathieu Tillier
Keyword(s):  
Social Hierarchy ◽  
Social Barriers ◽  
High Ranking ◽  
Long Time

AbstractThis article undertakes first a reconstruction of lists of legal scribes (kātibs) and investigators (ṣāḥibsal-masāʾil) active in Fusṭāṭ between the 1st/early 8th and the 4th/10th century. Identification of these people allows a better understanding of the recruitment of Egyptian judiciary staff. Their reputations as scholars, as well as their ethnical, geographical and tribal origins, show that legal careers were limited by social barriers for a long time. Up until the 3rd/9th century, the office of scribe was mostly held bymawālī—high-ranking clients could possibly aspire to the office of investigator—, whereasqāḍīs were recruited among Arabs. The partitioning of the judiciary reveals a complex social hierarchy beyond the mere distinction between Arabs and non-Arabs. The results of this study also allow a re-evaluation of the Abbasid revolution’s impact on Egyptian society.

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