key: cord-0205252-vaoj7v30 authors: vCern'ak, Jozef title: Spatio-temporal dynamics of outbreak on a lattice with quenched mobility patterns date: 2021-12-21 journal: nan DOI: nan sha: 6df5bfec8e75f6fd0d7f8c2fdbb314511c0ee100 doc_id: 205252 cord_uid: vaoj7v30 We have designed a computational model of a virus spread near the outbreak threshold. Using computer simulation we studied the Susceptible - Infected - Recovered (SIR) process where in consequence of a force of habit that is manifested by the population mobility patterns, the recovered persons create the spatio-temporal patterns as the barriers to a virus transmission. The results show a spontaneous stopping of the virus spread without a need to infect the whole population, a non-trivial random noise of daily count of infected cases, and power laws of a cumulative count of infected cases. Outbreak evolution strongly depends on the initial conditions thus we concluded that the model has the features of chaotic systems that makes it difficult to predict its behaviors. The SARS-CoV-2 virus and their variants continue in the evolution in the global space. The destiny of virus extinction is unclear (1 ) and it depends on many evolutionary factors (2 ) . An increase of social distance is a measure to slow down the pandemic. It has been successfully demonstrated that either local or national non-pharmaceutical interventions led to a significant reduction of the virus rate transmission on large scales with beneficial and measurable health outcomes (3 , 4 ) . The mathematical models, susceptible-infected-recovered (SIR), susceptible -infected -susceptible (SIS), susceptible -infected -recovered -susceptible (SIRS) and susceptibleexposed -infected -recovered -susceptible (SEIRS) (5 , 6 ) , predict an exponential growth of a cumulative count of infected cases I(t) ∼ exp(t), where t is time. Similarly, the SI, SIS, and SIR models on the complex networks show an exponential growth of the number of links available for future transmission (7 ) . These theoretical predictions are in contrast to a cumulative count of infected cases I(t) in China during the beginning of the first SARS-CoV-2 outbreak wave. The first wave in China shows a power law, where α is an exponent. A deviation from the expected exponential growth models, we consider for a signature of a quite different mechanism of the contagion, than it is widely accepted for traditional SIR or SEIR models (9 ) . The authors Rhodes and Anderson (10 ) analyzed distribution of epidemic sizes and epidemic durations of measles outbreaks. They observed that dynamical structures of the measles returns reflect the existence of an underlying scaling mechanism. Random structures often exhibit self-similar geometry that is characterized by a power law (11 ) . So, it is useful to consider a virus contagion as a dynamical process on the fractal networks like a diffusion in the percolating networks (11 ) as well as the branching process (12 ) on these networks. Kumamoto and Kamihigashi (13 ) reviewed mathematical mechanisms that are known to generate the power laws. In particular, they focused on stochastic processes based on growth and preferential attachments including the Yule process, the Simon process, the Barabási -Albert Model, and stochastic models based on geometric Brownian motion. Stroud et al. (14 ) introduced the power laws of some variables in the traditional homogeneous models, SIR and SEIR, to better model the real outbreaks. In the stochastic version of the SIR process, the authors Ben-Naim and Krapivsky (15 ) found nontrivial scaling relations of a maximal size of outbreak M ∼ N 2 3 and duration of outbreak T ∼ T 1 2 on a population size N near the outbreak threshold. We observed that the time series of daily count of infected cases i(t) show a noise that is not possible to neither reproduce nor explain by the classical models of epidemic. Hurst investigated annual values of some phenomena such as river discharges, rainfall, temperatures. He observed that these values are approximately normally distributed if no account is taken of order of occurrence. So far as is known, there is no regularity in the occurrence or the length of these periods, and usually there is no significant correlation over one of them between a year and its successor. Hurst considered this phenomenon to be important in problems of storage (16 ) . Mandelbrot found that the Hurst exponent H must be 0 ≤ H ≤ 1. If H < 1 2 then a noise shows an anti-persistent fractional Brownian motion (17 ) . If H > 1 2 then the noise exhibits a long term persistence and nonperiodic cycles. The Hurst noise was observed in many natural phenomena (18 ) and it was demonstrated in the laboratory insect populations (19 ) , i.e. the Hurst phenomenon was reproduced in well controlled experimental conditions. Our motivation is to better understand the emergence of power laws in the outbreak evolution, the mechanisms of spontaneous stopping of the outbreak and the nature of the noise of the daily count of infected cases i(t). We were inspired by the cellular automata (20 ) , a few features of the self organized criticality models and forest fire models (21 -23 ) . However, we had to constrain mobility of individuals and to introduce a spatio-temporal memory effect. World Health Organization (WHO) provides the SARS-CoV-2 data (Supplementary Materials). We carefully selected countries: China, the Czech Republic, Belgium and Kenya ( Figure 1 ), that show general as well as country specific features of the virus spread. The daily count of infected cases i(t) and the cumulative number of infected cases I(t) Many species that are exposed to pathogens can alter their behaviors in ways to maximize benefits and minimize cost. Studies of social behaviors of nonhuman animals have the potential to provide important insights into ecological and evolutionary processes relevant to human health, including pathogen transmission dynamics and virulence evolution (24 ) . Our aim has been to implement the most important feature of animals to adapt to the pathogen enemy (24 ) The diffusion of individuals is defined on a two-dimensional (2D) lattice of size L×L, L is a size in one dimension. An individual is assigned to the lattice node n ij . The individuals create a metapopulation of the size N = L×L. The individual can be mobile or immobile. The mobile individual may escape the node n ij and diffuse, in contrast to the immobile individual that stands in the node n ij . We define the time t. If all mobile individuals diffuse m steps from the site n ij (Eq. 1) then the time t is increased by one unit t = t + 1. After m steps each mobile individual returns to the initial node n ij ( Figure S1 ) i k=0 = i, j k=0 = j, a 1 and a 2 are random integer numbers and ξ 1 and ξ 2 are uncorrelated random variables 0 ≤ ξ 1,2 < 1 that are evaluated for each k. In simulations, the periodic boundary conditions are used to conserve a number of individuals N on the lattice. We The computational model (Section S1) is implemented in the Python language ("m5.py" (1 )). We tested the program "m5.py" using the OS Ubuntu 18.04.6 LTS. Type in the terminal: python3 m5.py to run a computer simulation. The program prints the time series: the time t, the daily count of infected cases i(t) and the cumulative count of infected cases I(t). World Health Organization (WHO) collects the statistics of the COVID-19 pandemic, which are publicly available in the WHO archive (5 ) . Statistical data were downloaded from the link (6 ), as the comma-separated values (CSV file format). Data columns: "Date_reported", "Country_code", "New_cases", and "Cumulative_cases" were exported from the Microsoft Excel data sheet "WHO-COVID-19-global-data.xlsx". The file "WHO-COVID-19-global-data.xlsx" and exported data used to plot the graphs in Figure 1 (the main text) are enclosed in the Local Data Directories (1 ). The results of computer simulations are stored in data files "File.txt" (1 ). The probability of infection transmission p, the duration of infection window T i , and the maximal path length l m are the parameters that were used to generate the time series of i(t) and I(t), that are stored in data files "File.txt". Data files and the parameters p, T i and l m are arranged in Tables S1 and S2. These data were used to plot the graphs in Figures 2, 3 (the main text), S2 and S3. Data files and the parameters p, T i and l m are arranged in Tables S1. p T i l m File (.txt) Figure 2 0.130 14 20 v4 (B) v7 v70_1 v70_2 Table S1 : The parameters p, T i and l m , and data files ("File.txt") of the graphs in Figures 2 and 3 (the main text) . Table S2 : The probability of infection transmission p = 0.132, the duration of infection window T i = 5, and the maximal path lengths l m = 90, 100, 110 and 120 are parameters. Data files ("File.txt") were used to plot the graphs in Figures S2 and S3 and to determine the Hurst exponents. Simulations are initiated using only one infected case, that it is placed randomly on the lattice. In such case, a probability to start an outbreak may be low (7 ) . We had to run the initialization of simulation many times for certain parameters. We observed a spontaneous stopping of the outbreak after the time T , for all parameters p, T i and l m We observed that, the growth of the outbreak and its decay follow the power laws Table S1 ). The exponents, α and β, are of the power laws i(t) ∼ t β and I(t) ∼ t α . The fluctuations of daily count infected cases i(t), for parameters and data files that arranged in Table S2, Network Science Proceedings of the Royal Society of London. Series B: Biological Sciences how nature works, the science of self-organized criticality Self-Organised Criticality, Theory, Models and Characterisation Introduction to Percolation Theory Complex Media and Percolation Theory Local Data Directories Gnuplot 5.2: an interactive plotting program The Sage Developers, SageMath, the Sage Mathematics Software System Fractal market analysis : applying chaos theory to investment and economics World Health Organization, WHO Coronavirus (COVID-19) Dashboard-Information Note World Health Organization, WHO Coronavirus (COVID-19) Dashboard-Download link