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 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 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 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 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.

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 Disease Detectives: Using Mathematics to Forecast the Spread of Infectious Diseases

2021 ◽  
Vol 8 ◽  
Author(s):  
Heather Z. Brooks ◽  
Unchitta Kanjanasaratool ◽  
Yacoub H. Kureh ◽  
Mason A. Porter
Keyword(s):  
Mathematical Model ◽  
Infectious Diseases ◽  
Mathematical Models ◽  
Human Behavior ◽  
Government Policies ◽  
Disease Spread

The COVID-19 pandemic has led to significant changes in how people are currently living their lives. To determine how to best reduce the effects of the pandemic and start reopening communities, governments have used mathematical models of the spread of infectious diseases. In this article, we introduce a popular type of mathematical model of disease spread. We discuss how the results of analyzing mathematical models can influence government policies and human behavior, such as encouraging mask wearing and physical distancing to help slow the spread of a disease.

scholarly journals Monitoring and forecasting the COVID-19 epidemic in Moscow: model selection by balanced identification technology - version: September 2021

2021 ◽  
Author(s):  
Alexander Sokolov ◽  
Lyubov Sokolova
Keyword(s):  
Mathematical Model ◽  
Model Selection ◽  
Mathematical Models ◽  
Statistical Data ◽  
Herd Immunity ◽  
Real Object ◽  
List Type ◽  
The Third ◽  
New Knowledge ◽  
Monitoring And Forecasting

AbstractA mathematical model is a reflection of knowledge on the real object studied. The paper shows how the accumulation of data (statistical data and knowledge) about the COVID-19 pandemic lead to gradual refinement of mathematical models, to the expansion of the scope of their use. The resulting model satisfactorily describes the dynamics of COVID-19 in Moscow from 19.03.2020 to 01.09.2021 and can be used for forecasting with a horizon of several months. The dynamics of the model is mainly determined by herd immunity. Monitoring the situation in Moscow has not yet (as of 01.09.2021) revealed noticeable seasonality of the disease nor an increase in infectivity (due to the Delta strain). The results of using balanced identification technology to monitor the COVID-19 pandemic are:models corresponding to the data available at different points in time (from March 2020 to August 2021);new knowledge (dependencies) acquired;forecasts for the third and fourth waves in Moscow.Discrepancies that manifested after 01.09.2021 and possible further modifications of the model are discussed

scholarly journals A scaling investigation of pattern in the spread of COVID-19: universality in real data and a predictive analytical description

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200689
Author(s):  
Subir K. Das
Keyword(s):  
Statistical Physics ◽  
Analytical Description ◽  
Real Data ◽  
Climatic Conditions ◽  
Disease Spread ◽  
Predictive Tool ◽  
General Relevance ◽  
Medical Facilities ◽  
Universality Classes ◽  
Novel Coronavirus

We analyse the spread of COVID-19, a disease caused by a novel coronavirus, in various countries by proposing a model that exploits the scaling and other important concepts of statistical physics. Quite expectedly, for each of the considered countries, we observe that the spread at early times occurs exponentially fast. We show how the countries can be classified into groups, like universality classes in the literature of phase transitions, based on the rates of infections during late times. This method brings a new angle to the understanding of disease spread and is useful in obtaining a country-wise comparative picture of the effectiveness of lockdown-like social measures. Strong similarity, during both natural and lockdown periods, emerges in the spreads within countries having varying geographical locations, climatic conditions, population densities and economic parameters. We derive accurate mathematical forms for the corresponding scaling functions and show how the model can be used as a predictive tool, with instruction even for future waves, and, thus, as a guide for optimizing social measures and medical facilities. The model is expected to be of general relevance in the studies of epidemics.

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