Stochastic cellular automata model for stock market dynamics

Stochastic formulation of multiwave pandemic: decomposition of growth into inherent susceptibility and external infectivity distributions

Many known and unknown factors play significant roles in the persistence of an infectious disease, but two that are often ignored in theoretical modelling are the distributions of (i) inherent susceptibility ( σ inh ) and (ii) external infectivity ( ι ext ), in a population. While the former is determined by the immunity of an individual towards a disease, the latter depends on the exposure of a susceptible person to the infection. We model the spatio-temporal propagation of a pandemic as a chemical reaction kinetics on a network using a modified SAIR (Susceptible-Asymptomatic-Infected-Removed) model to include these two distributions. The resulting integro-differential equations are solved using Kinetic Monte Carlo Cellular Automata (KMC-CA) simulations. Coupling between σ inh and ι ext are combined into a new parameter Ω, defined as Ω = σ inh × ι ext ; infection occurs only if the value of Ω is greater than a Pandemic Infection Parameter (PIP), Ω 0 . Not only does this parameter provide a microscopic viewpoint of the reproduction number R0 advocated by the conventional SIR model, but it also takes into consideration the viral load experienced by a susceptible person. We find that the neglect of this coupling could compromise quantitative predictions and lead to incorrect estimates of the infections required to achieve the herd immunity threshold. The figure represents the network model for spread of infectious diseases considered in this work. It also shows the resultant multiwave infection graph by inclusion of inherent susceptibility and external infectivity distributions and migration of infected individuals.

A computational model of liver tissue damage and repair

Drug induced liver injury (DILI) and cell death can result from oxidative stress in hepatocytes. An initial pattern of centrilobular damage in the APAP model of DILI is amplified by communication from stressed cells and immune system activation. While hepatocyte proliferation counters cell loss, high doses are still lethal to the tissue. To understand the progression of disease from the initial damage to tissue recovery or death, we computationally model the competing biological processes of hepatocyte proliferation, necrosis and injury propagation. We parametrize timescales of proliferation (α), conversion of healthy to stressed cells (β) and further sensitization of stressed cells towards necrotic pathways (γ) and model them on a Cellular Automaton (CA) based grid of lattice sites. 1D simulations show that a small α/β (fast proliferation), combined with a large γ/β (slow death) have the lowest probabilities of tissue survival. At large α/β, tissue fate can be described by a critical γ/β* ratio alone; this value is dependent on the initial amount of damage and proportional to the tissue size N. Additionally, the 1D model predicts a minimum healthy population size below which damage is irreversible. Finally, we compare 1D and 2D phase spaces and discuss outcomes of bistability where either survival or death is possible, and of coexistence where simulated tissue never completely recovers or dies but persists as a mixture of healthy, stressed and necrotic cells. In conclusion, our model sheds light on the evolution of tissue damage or recovery and predicts potential for divergent fates given different rates of proliferation, necrosis, and injury propagation.

Mathematical modeling and cellular automata simulation of infectious disease dynamics: Applications to the understanding of herd immunity

The complexity associated with an epidemic defies any quantitatively reliable predictive theoretical scheme. Here, we pursue a generalized mathematical model and cellular automata simulations to study the dynamics of infectious diseases and apply it in the context of the COVID-19 spread. Our model is inspired by the theory of coupled chemical reactions to treat multiple parallel reaction pathways. We essentially ask the question: how hard could the time evolution toward the desired herd immunity (HI) be on the lives of people? We demonstrate that the answer to this question requires the study of two implicit functions, which are determined by several rate constants, which are time-dependent themselves. Implementation of different strategies to counter the spread of the disease requires a certain degree of a quantitative understanding of the time-dependence of the outcome. Here, we compartmentalize the susceptible population into two categories, (i) vulnerables and (ii) resilients (including asymptomatic carriers), and study the dynamical evolution of the disease progression. We obtain the relative fatality of these two sub-categories as a function of the percentages of the vulnerable and resilient population and the complex dependence on the rate of attainment of herd immunity. We attempt to study and quantify possible adverse effects of the progression rate of the epidemic on the recovery rates of vulnerables, in the course of attaining HI. We find the important result that slower attainment of the HI is relatively less fatal. However, slower progress toward HI could be complicated by many intervening factors.

Understanding Financial Market States Using an Artificial Double Auction Market

The ultimate value of theories describing the fundamental mechanisms behind asset prices in financial systems is reflected in the capacity of such theories to understand these systems. Although the models that explain the various states of financial markets offer substantial evidence from the fields of finance, mathematics, and even physics, previous theories that attempt to address the complexities of financial markets in full have been inadequate. We propose an artificial double auction market as an agent-based model to study the origin of complex states in financial markets by characterizing important parameters with an investment strategy that can cover the dynamics of the financial market. The investment strategies of chartist traders in response to new market information should reduce market stability based on the price fluctuations of risky assets. However, fundamentalist traders strategically submit orders based on fundamental value and, thereby stabilize the market. We construct a continuous double auction market and find that the market is controlled by the proportion of chartists, P_c. We show that mimicking the real state of financial markets, which emerges in real financial systems, is given within the range P_c = 0.40 to P_c = 0.85; however, we show that mimicking the efficient market hypothesis state can be generated with values less than P_c = 0.40. In particular, we observe that mimicking a market collapse state is created with values greater than P_c = 0.85, at which point a liquidity shortage occurs, and the phase transition behavior is described at P_c = 0.85.

Emerging properties of financial time series in the "Game of Life"

We explore the spatial complexity of Conway's "Game of Life," a prototypical cellular automaton by means of a geometrical procedure generating a two-dimensional random walk from a bidimensional lattice with periodical boundaries. The one-dimensional projection of this process is analyzed and it turns out that some of its statistical properties resemble the so-called stylized facts observed in financial time series. The scope and meaning of this result are discussed from the viewpoint of complex systems. In particular, we stress how the supposed peculiarities of financial time series are, often, overrated in their importance.

Agent-based spin model for financial markets on complex networks: emergence of two-phase phenomena

We study a microscopic model for financial markets on complex networks, motivated by the dynamics of agents and their structure of interaction. The model consists of interacting agents (spins) with local ferromagnetic coupling and global antiferromagnetic coupling. In order to incorporate more realistic situations, we also introduce an external field which changes in time. From numerical simulations, we find that the model shows two-phase phenomena. When the local ferromagnetic interaction is balanced with the global antiferromagnetic interaction, the resulting return distribution satisfies a power law having a single peak at zero values of return, which corresponds to the market equilibrium phase. On the other hand, if local ferromagnetic interaction is dominant, then the return distribution becomes double peaked at nonzero values of return, which characterizes the out-of-equilibrium phase. On random networks, the crossover between two phases comes from the competition between two different interactions. However, on scale-free networks, not only the competition between the different interactions but also the heterogeneity of underlying topology causes the two-phase phenomena. Possible relationships between the critical phenomena of spin system and the two-phase phenomena are discussed.

Understanding the complex dynamics of stock markets through cellular automata

We present a cellular automaton (CA) model for simulating the complex dynamics of stock markets. Within this model, a stock market is represented by a two-dimensional lattice, of which each vertex stands for a trader. According to typical trading behavior in real stock markets, agents of only two types are adopted: fundamentalists and imitators. Our CA model is based on local interactions, adopting simple rules for representing the behavior of traders and a simple rule for price updating. This model can reproduce, in a simple and robust manner, the main characteristics observed in empirical financial time series. Heavy-tailed return distributions due to large price variations can be generated through the imitating behavior of agents. In contrast to other microscopic simulation (MS) models, our results suggest that it is not necessary to assume a certain network topology in which agents group together, e.g., a random graph or a percolation network. That is, long-range interactions can emerge from local interactions. Volatility clustering, which also leads to heavy tails, seems to be related to the combined effect of a fast and a slow process: the evolution of the influence of news and the evolution of agents' activity, respectively. In a general sense, these causes of heavy tails and volatility clustering appear to be common among some notable MS models that can confirm the main characteristics of financial markets.

Stochastic opinion formation in scale-free networks

The dynamics of opinion formation in large groups of people is a complex nonlinear phenomenon whose investigation is just beginning. Both collective behavior and personal views play an important role in this mechanism. In the present work we mimic the dynamics of opinion formation of a group of agents, represented by two states +/-1, as a stochastic response of each agent to the opinion of his/her neighbors in the social network and to feedback from the average opinion of the whole. In the light of recent studies, a scale-free Barabási-Albert network has been selected to simulate the topology of the interactions. A turbulent-like dynamics, characterized by an intermittent behavior, is observed for a certain range of the model parameters. The problem of uncertainty in decision taking is also addressed both from a topological point of view, using random and targeted removal of agents from the network, and by implementing a three-state model, where the third state, zero, is related to the information available to each agent. Finally, the results of the model are tested against the best known network of social interactions: the stock market. A time series of daily closures of the Dow-Jones index has been used as an indicator of the possible applicability of our model in the financial context. Good qualitative agreement is found.