Chaos: An Interdisciplinary Journal of Nonlinear Science

In this article a mathematical treatment of Covid-19 as a stochastic process is discussed. The chance of extinction and the consequences of introducing new Covid-19 infectives into the population are evaluated by using certain approximate arguments. It is shown, in general terms, that the stochastic formulation of a recurrent epidemic like Covid-19 leads to the prediction of a permanent succession of undamped outbreaks of disease. It is also shown that one is able to derive certain useful conclusions about Covid-19 without consideration of immune individuals in a population.

The novel SARS_CoV-2 virus, prone to variation when interacting with spatially extended ecosystems and within hosts1 can be considered a complex dynamic system2. Therefore, it behaves creating several space-time manifestations of its dynamics. However, these physical manifestations in nature have not yet been fully disclosed or understood. Here we show 4-3 and 2-D space-time patterns of rate of infected individuals on a global scale, giving quantitative measures of transitions between different dynamical behaviour. By slicing the spatio-temporal patterns, we found manifestations of the virus behaviour such as cluster formation and bifurcations. Furthermore, by analysing the morphogenesis processes by entropy, we have been able to detect the virus phase transitions, typical of adaptive biological systems3. Our results for the first time describe the virus patterning behaviour processes all over the world, giving for them quantitative measures. We know that the outcomes of this work are still partial and more advanced analyses of the virus behaviour in nature are necessary. However, we think that the set of methods implemented can provide significant advantages to better analyse the viral behaviour in the approach of system biology4, thus expanding knowledge and improving pandemic problem solving.

Virus cause multiple outbreaks, for which comprehensive tailored therapeutic strategies are still missing. Virus and host cell dynamics are strictly connected, and convey in virion assembly to ensure virus spread in the body. Study of the systemic behavior of virus-host interaction at the single-cell level is a scientific challenge, considering the difficulties of using experimental approaches and the limited knowledge of the behavior of emerging novel virus as a collectivity. This work focuses on positive-sense, single-stranded RNA viruses, like human coronaviruses, in their virus-individual host interaction, studying the changes induced in the host cell bioenergetics. A systems-thinking representation, based on stock-flow diagramming of virus-host interaction at the cellular level, is used here for the first time to simulate the system energy dynamics. We found that reducing the energy flow which fuels virion assembly is the most affordable strategy to limit the virus spread, but its efficacy is mitigated by the contemporary inhibition of other flows relevant for the system.Summary Positive-single-strand ribonucleic acid ((+)ssRNA) viruses can cause multiple outbreaks, for which comprehensive tailored therapeutic strategies are still missing. Virus and host cell dynamics are strictly connected, generating a complex dynamics that conveys in virion assembly to ensure virus spread in the body.This work focuses on (+)ssRNA viruses in their virus-individual host interaction, studying the changes induced in the host cell bioenergetics. A systems-thinking representation, based on stock-flow diagramming of virus-host interaction at the cellular level, is used here for the first time to simulate the energy dynamics of the system.By means of a computational simulator based on the systemic diagramming, we identifid host protein recycling and folded-protein synthesis as possible new leverage points. These also address different strategies depending on time setting of the therapeutic procedures. Reducing the energy flow which fuels virion assembly is addressed as the most affordable strategy to limit the virus spread, but its efficacy is mitigated by the contemporary inhibition of other flows relevant for the system. Counterintuitively, targeting RNA replication or virion budding does not give rise to relevant systemic effects, and can possibly contribute to further virus spread. The tested combinations of multiple systemic targets are less efficient in minimizing the stock of virions than targeting only the virion assembly process, due to the systemic configuration and its evolution overtime. Viral load and early addressing (in the first two days from infection) of leverage points are the most effective strategies on stock dynamics to minimize virion assembly and preserve host-cell bioenergetics.As a whole, our work points out the need for a systemic approach to design effective therapeutic strategies that should take in account the dynamic evolution of the system.Competing Interest StatementThe authors have declared no competing interest.View Full Text

Complex random networks provide a powerful mathematical framework to study high-dimensional physical and biological systems. Several features of network structure (e.g. degree correlation, average path length, clustering coefficient) are correlated with descriptors of network dynamics and function. However, it is not clear which features of network structure relate to the dynamics of biological neuronal networks (BNNs), characterized by non-linear nodes with high in- and out degrees, but being weakly connected and communicating in an event-driven manner, i.e. only when neurons spike. To better understand the structure-dynamics relationship in BNNs, we analysed the structure and dynamics of > 9, 000 BNNs with different sizes and topologies. In addition, we also studied the effect of network degeneration on neuronal network structure and dynamics. Surprisingly, we found that the topological class (random, small-world, scale-free) was not an indicator of the BNNs activity state as quantified by the firing rate, network synchrony and spiking regularity. In fact, we show that different network topologies could result in similar activity dynamics. Furthermore, in most cases, the network activity changes did not depend on the rules according to which neurons or synapses were pruned from the networks. The analysis of dynamics and structure of the networks we studied revealed that the effective synaptic weight (ESW) was the most crucial feature in predicting the statistics of spiking activity in BNNs. ESW also explained why different synapse and neuron pruning strategies resulted in almost identical effects on the network dynamics. Thus, our findings provide new insights into the structure-dynamics relationships in BNNs. Moreover, we argue that network topology and rules by which BNNs degenerate are irrelevant for BNN activity dynamics. Beyond neuroscience, our results suggest that in large networks with non-linear nodes, the effective interaction strength among the nodes, instead of the topological network class, may be a better predictor of the network dynamics and information flow.

There is growing evidence that many forms of neural computation may be implemented by low-dimensional dynamics unfolding at the population scale. However, neither the connectivity structure nor the general capabilities of these embedded dynamical processes are currently understood. In this work, the two most common formalisms of firing-rate models are evaluated using tools from analysis, topology and nonlinear dynamics in order to provide plausible explanations for these problems. It is shown that low-rank structured connectivity predicts the formation of invariant and globally attracting manifolds in both formalisms, which generalizes existing theories to different neural models. Regarding the dynamics arising in these manifolds, it is proved they are topologically equivalent across the considered formalisms. It is also stated that under the low-rank hypothesis, dynamics emerging in neural models are universal. These include input-driven systems, which broadens previous findings. It is then explored how low-dimensional orbits can bear the production of continuous sets of muscular trajectories, the implementation of central pattern generators and the storage of memory states. It is also proved these dynamics can robustly simulate any Turing machine over arbitrary bounded memory strings, virtually endowing rate models with the power of universal computation. In addition, it is shown how the low-rank hypothesis predicts the parsimonious correlation structure observed in cortical activity. Finally, it is discussed how this theory could provide a useful tool from which to study neuropsychological phenomena using mathematical methods.

The present work proposes a surrogate method for understanding and analyzing the replication of SARS CoV-2 through fractal and inertia moment (IM) analysis of cell culture images at different stages. The fractal analysis of images of cell culture, calculated by the box-counting and power spectral density methods, reflect the stages of virus infection, leading to the replication of the virus RNA and damaging the host cell. The linear increase of IM value reveals not only the proliferation of SARS CoV-2 by replication but also damage to the host cell with time. Thus, the work shows the possibility of fractal analysis and IM measurement for understanding the dynamics of the virus infection.

We study a statistical lattice model of a mitigated epidemic. The level of mitigation, defined by measures to slow down the spread of the infection, is characterized by the infection transmissivity. It is determined by peoples mobility, frequency of contacts, and probability to catch the virus from a contact. In the absence of testing the infected people are isolated for a finite period of time during which they are symptomatic. In the presence of testing, people become isolated a soon as they are tested positive. We compute time dependence of daily new infections as function of transmissivity, initial infection, and testing. The duration of the epidemic increases rapidly with the increased level of mitigation while the number of people falling sick daily decreases. Testing, regardless of the level, has little effect on the duration of the epidemic. The total number of people who contract the disease over the lifetime of the epidemic depends weakly on its duration. It does not change significantly for the homogeneous testing of the population at the level below 10% daily.

The dynamical activity of the human brain describes an extremely complex energy landscape changing over time and its characterisation is central unsolved problem in neuroscience. We propose a novel mathematical formalism for characterizing how the landscape of attractors sustained by a dynamical system evolves in time. This mathematical formalism is used to distinguish quantitatively and rigorously between the different human brain states of wakefulness and deep sleep. In particular, by using a whole-brain dynamical ansatz integrating the underlying anatomical structure with the local node dynamics based on a Lotka-Volterra description, we compute analytically the global attractors of this cooperative system and their associated directed graphs, here called the informational structures. The informational structure of the global attractor of a dynamical system describes precisely the past and future behaviour in terms of a directed graph composed of invariant sets (nodes) and their corresponding connections (links). We characterize a brain state by the time variability of these informational structures. This theoretical framework is potentially highly relevant for developing reliable biomarkers of patients with e.g. neuropsychiatric disorders or different levels of coma.

The brain criticality hypothesis has been a central research topic in theoretical neuroscience for two decades. This hypothesis suggests that the brain operates near the critical point at the boundary between order and disorder, where it acquires its information-processing capabilities. The mechanism that maintains this critical state has been proposed as a feedback system known as self-organized criticality (SOC); brain parameters, such as synaptic plasticity, are regulated internally without external adjustment. Therefore, clarifying how SOC occurs can help us to understand the mechanisms that maintain brain function and cause brain disorders. From the standpoint of neural network structures, the topology of neural circuits also plays a crucial role in information processing, with healthy neural networks exhibiting small-world, scale-free, and modular characteristics. However, how these network structures affect SOC remains poorly understood. In this study, we numerically investigated the possibility that the structure of neural networks contributes to the brains critical state and dysfunction using a mathematical model. Our results reveal that the time scales at which synaptic plasticity operates to achieve a critical state differ depending on the network structure. Additionally, we observed Dragon king phenomena associated with abnormal neural activity, depending on the network structure and synaptic plasticity time scales. Notably, Dragon king was observed over a wide range of synaptic plasticity time scales in scale-free networks with high-degree hub nodes. This study emphasizes the importance of neural network topology in neuroscience from the perspective of SOC.

In this work an author is building a theoretical model of a non-infectious disease, which shows that there is a connection between diseases frequencies and their causes. This connection allows to determine how many factors are causing a specific non-infectious disease if we know the disease rate in a population. The model shows that for a majority of non-infectious diseases there are at least two simultaneously acting factors which cause a disease and in many cases there are more factors simultaneously involved. This helps researchers to improve an understanding of specific disease causation and physiological mechanisms behind it and will lead to a research for additional, still missing causes to complete these mechanisms. This work determines a number of simultaneously acting factors causing diseases such as a breast cancer, coronary heart disease(CHD), multiple sclerosis, etc. and explains so called French Paradox for CHD. The work also deduces a formula and a method of determining that a specific risk factor is the one which really causes a disease or it is not. Applying a method developed in this work the author shows how three different simultaneously acting causes of atrial fibrillation are determined using an existing research data. This method should allow medical researchers to determine if a found risk factor for a disease is really a cause of the disease or not and covers a significant gap in current understanding of risk factors nature and its connection to the physiological parameters of the human body.. SummaryO_LIUsing statistical and experimental data a mathematical model of non-infectious disease is created which is applicable to any non-infectious disease (and to some degree to infectious diseases as well) C_LIO_LIThe model is considering that a cause of a disease is a change in a physiological parameter of the body approximately beyond 1-sigma interval of its measurements, slightly less. C_LIO_LIThe model predicts and the experimental data confirm its predictions that rate of non-infectious diseases is closely connected to the number of changes to physiological parameters of the human body which are causing the disease. Based on this connection the model allows to determine number of disease causes (as number of physiological parameters changed) for any non-infectious disease by only knowing the disease rate. For example, if the disease rate is 72 per 1000 people then the model determines that disease has 2 causes. Despite the differences in the rates of a specific disease in different countries or populations the model determines the same number of causes. The model introduces a formula to calculate a number of disease causes as below: C_LI O_FD O_INLINEFIG[Formula 1]C_INLINEFIGM_FD(1)C_FD where a rate of disease can be for example as 45 per 10000 people and number of causes(as physiological parameters changes) is an integer number like 1, 2, 3, etc. O_LIAccording to this non-infectious disease model in order to cause a disease all of specific for each disease physiological parameters should be changed by actions of the external environment so their measurements will be beyond 1-sigma interval. Actually, slightly less than this interval. C_LIO_LIThe model shows the more the rate of non-infectious disease the less number of simultaneous changes to the physiological parameters is required to cause it. The changes can be acquired over the lifetime of individual and have to be such so each physiological parameters measurement goes beyond 1-sigma interval*. C_LIO_LIThe work also allows to use vast volume of existing medical researchs data about diseases risk factors in order to determine real causes of non-infectious diseases using a simple criteria which mathematically derived from the model. C_LIO_LIThe work mathematically derives the criteria which allows to determine which risk factor is a cause of disease and which is not. It mathematically proves that in order to be a cause of the specific disease the found risk factors (calculated as [ risk_cases - normal_cases ] / normal_cases) should rich the value of 3.5 (or 350%) or 19.7 (1970%), etc. plus/minus a practical margin (recommendation is +/-50%) C_LIO_LIThe calculations based on the model has shown that a majority of non-infectious diseases are caused by at least 2 simultaneously occurring changes to physiological parameters of human body or more. For example it shows that Coronary Heart Disease (CHD) is caused by 4 physiological parameters changes taking place at the same time. C_LIO_LIAs the model derived in this work predicts that non-infectious diseases are caused at least by 2 changes (beyond 1-sigma) to physiological parameters of human body then it shows there is no reason and actually, often it is incorrect to search for a single cause of the non-infectious disease. The single cause of non-infectious disease does not exist for a majority of them, according to this mathematical model. C_LIO_LIBased on the smoking risk factor value, the developed in this work criteria determines that smoking is one of the causes of lung cancer and that is matching to already a well recognized medical fact, and supports the validity of the criteria used to make this prediction. C_LIO_LIThe model shows that Multiple Sclerosis in men is caused by changes to 5 physiological parameters (beyond 1 sigma interval *) while the same disease is caused by changes to 4 physiological parameters in women. This explains why a rate of the Multiple Sclerosis is higher in women than in men as the more physiological parameters needs to be changed to cause the disease the less the rate of disease. C_LIO_LIThe work shows why breast cancers and leukemias are caused by a change to 6 physiological parameters of the human body. C_LIO_LIThe model leads to a conclusion that the majority of non-infectious diseases cannot occur if only one physiological parameter of human body changes beyond 1-sigma interval. The multiple and simultaneously taking place changes to physiological parameters needs to be there for a non-infectious disease to occur but the changes can be acquired over the time. C_LIO_LIThe work leads to a conclusion that by controlling few physiological parameters of the human body so they are located within the 1-sigma of its measurements it is possible to prevent a non-infectious disease such cancers or strokes from occurring at all in the individual. C_LIO_LIThe work shows that the fact that few of physiological parameters changes needed to simultaneously occur in order to cause the non-infectious disease, allows to select some risks factors as real causes of the disease and that we have some room for an error in selecting these risk factors as disease causes because if few risk factors are correctly chosen they will compensate for an incorrectly chosen one. This way we can prevent many diseases from occurrence by keeping the right risk factors (physiological parameters) under control (meaning within 1-sigma interval). C_LIO_LIThe work mathematically derives a formula which connecting the numbers of disease causes (physiological parameters changes) determined in populations with a risk factor and without, to a risk factor value determined for the population impacted by this risk factor. The Risk Factor value here is 0 and more and defined as (cases_with_risk - cases_without_risk) / cases_without_risk. Here Risk Causes / No Risk Causes are numbers of causes (physiological parameters changes) determined for a disease in a population with a risk factor and without it using a formula presented above and are integer numbers (0,1,etc.). The formula is: C_LI O_FD O_INLINEFIG[Formula 2]C_INLINEFIGM_FD(2)C_FD The formula is a foundation of the criteria to determine if the risk factor is really a disease cause. It connects physiological parameters changes in the human body to the risk factor which creates them for an individual. Using this formula researchers can determine a number of physiological parameters changes the specific risk factor is causing in the human body.

We 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 hierarchys levels composing this barrier. As the most appropriate for the recent situation in Sweden, we consider linearly increasing (with respect to hierarchys 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 models 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. O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=126 SRC="FIGDIR/small/20146209v1_fig2.gif" ALT="Figure 2"> View larger version (8K): org.highwire.dtl.DTLVardef@1b3b3fcorg.highwire.dtl.DTLVardef@ed8c38org.highwire.dtl.DTLVardef@190b208org.highwire.dtl.DTLVardef@9848be_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOFigure 2:C_FLOATNO Asymptotic behavior of probability to become immune; increasing of the herd immunity (for fixed social temperature T, the upper graphs correspond to one-step barrier growth 10 and 100 times, respectively. C_FIG

A qualitative mathematical model of the notion of immunocompetence is developed, based on the formalism of Memory Evolutive Systems (MES), from which, immunocompetence is defined as an emergent structure of a higher order arising from the signal networks that are established between effector cells and molecules of the immune response in the presence of a given antigen. In addition, a possible mechanism of functorial nature is proposed, which may explain how immunocompetence is achieved in an organism endowed with innate and adaptive components of its immune system. Finally, a practical method to measure the immunocompetence status is established, using elements of the theory of small random graphs and taking into account the characteristics of the immune networks, established through transcriptional studies, of patients with severe COVID-19 and healthy patients, assuming that both types of patients were vaccinated with an effective biological against SARS-CoV-2.

By the growing number of available structural connectome data, the distributions of the synaptic weights can be determined which provides a hint at the learning mechanisms at play, both in the global and local level. In this work, we show a numerical analysis of this on the occasion of the latest large connectomes, the mouse visual cortex and the fruit-fly optical lobe, which, while evolutionarily distant share similar motion processing strategies. In general, it is found that the local node strengths can follow heavy-tailed distributions that decay faster than a power-law (PL), if we shuffle the weights among the nodes randomly. To understand this deviation from the global PL behavior, we propose a mechanism that can explain this difference which may resolve the ubiquitous contradicting observation of lognormal (LN) and PL tails related to critical behavior. We also show that synaptic weights of edges of fully proofread connectomes considered here, emanating from source and terminating at the sink nodes (broadcasters/integrators), respectively, exhibit PL tailed distributions, with exponents smaller than 3, so they are scale-free.

Withdrawal StatementThe authors have withdrawn their manuscript owing to data ownership concerns. Therefore, the authors do not wish this work to be cited as reference for the project. If you have any questions, please contact the corresponding author.

Since 2007, ZIKV outbreaks have been occurring around the world. While ZIKV is mainly spread by mosquito vectors, transmission via sex activities enables the virus to spread in regions without mosquito vectors. Modeling the patterns of ZIKV outbreak in these regions remain challenging. We consider age as an asymmetric factor in transmitting ZIKV, in addition to gender as seen in previous literature, and modify the graph structure for better modeling of such patterns. We derived our results by both solving the underlying differential equations and simulation on population graph. Based on a double asymmetric percolation process on sexual contact networks. we discovered a quadruple ZIKV epidemic transition. Moreover, we explored the double asymmetric percolation on scale-free networks. Our work provides more insight into the ZIKV transmission dynamics through sexual contact networks, which may potentially provide better public health control and prevention means in a ZIKV outbreak.

1Traveling waves of electrical activity are ubiquitous in biological neuronal networks. Traveling waves in the brain are associated with sensory processing, phase coding, and sleep. The neuron and network parameters that determine traveling waves evolution are synaptic space constant, synaptic conductance, membrane time constant, and synaptic decay time constant. We used an abstract neuron model to investigate the propagation characteristics of traveling wave activity. We formulated a set of evolution equations based on the network connectivity parameters. We numerically investigated the stability of the traveling wave propagation with a series of perturbations with biological relevance.

We present a novel mean-field theory for balanced neuronal networks with arbitrary levels of symmetry in the synaptic connectivity. The theory determines the fixed point of the network dynamics and the conditions for its stability. The fixed point becomes unstable by increasing the synaptic gain beyond a critical value that depends on the level of symmetry. Beyond this critical gain, for positive levels of symmetry, we find a previously unreported phase. In this phase, the dynamical landscape is dominated by a large number of marginally-stable fixed points. As a result, the network dynamics exhibit non-exponential relaxation and ergodicity is broken. We discuss the relevance of such a glassy phase for understanding dynamical and computational aspects of cortical operation.

Neurodegenerative diseases are frequently associated with aggregation and propagation of toxic proteins. In particular, it is well known that along with amyloid-beta, the tau protein is also driving Alzheimers disease. Multiscale reaction-diffusion models can assist in our better understanding of the evolution of the disease. We have modified the heterodimer model in such a way that it can now capture some of critical characteristics of this evolution such as the conversion time from healthy to toxic proteins. We have analyzed the modified model theoretically and validated the theoretical findings with numerical simulations.

1In animal species with fission-fusion dynamics, individuals can split from or follow others during collective movements. In spider monkeys (Ateles geoffroyi) this decision depends in part on the information they have about the location of available feeding trees. Foraging widely and continuously splitting and joining from others, individuals could be pooling their partial information such that the group as a whole has a more complete picture of a heterogeneous foraging environment. Here we use individual utilization areas over a realistic foraging landscape to infer the sets of potentially known trees by each individual. Then we measure the spatial entropy of these areas, considering tree species diversity and spatial distribution. We measure how complementary pairs of areas are, by decomposing the spatial entropy into redundant and unique components. We find that the areas uniquely known by each pair member still contain considerable amounts of information, but there is also a high redundancy in the information that a pair has about the foraging landscape. The networks joining individuals based on the unique information components seem to be structured efficiently for information transmission. Distributed foraging in fission-fusion dynamics would be an example of adaptive pooling of information and thus, collective intelligence.

Ensembles of phase-oscillators are known to exhibit a variety of collective regimes. Here, we show that a simple mean-field model involving two heterogenous populations of pulse-coupled oscillators, exhibits, in the strong-coupling limit, a robust irregular macroscopic dynamics. The resulting, strongly synchronized, regime is sustained by a homeostatic mechanism induced by the shape of the phase-response curve combined with adaptive coupling strength, included to account for energy dissipated by the pulse emission. The proposed setup mimicks a neural network composed of excitatory and inhibitory neurons.