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Register a new userHow macroscopic laws describe complex dynamics: asymptomatic population and CoviD-19 spreading
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Macroscopic growth laws, solutions of mean field equations, describe in an effective way an underlying complex dynamics. They are applied to study the spreading of infections, as in the case of CoviD-19, where the counting of the cumulated number $N(t)$ of detected infected individuals is a generally accepted, coarse-grain, variable to understand the epidemic phase. However $N(t)$ does not take into account the unknown number of asymptomatic, not detected, cases $A(t)$. Therefore, the question arises if the observed time series of data of $N(t)$ is a reliable tool for monitoring the evolution of the infectious disease. We study a system of coupled differential equations which includes the dynamics of the spreading among symptomatic and asymptomatic individuals and the strong containment effects due to the social isolation. The solution is therefore compared with a macroscopic law for the population $N(t)$ coming from a single, non-linear, differential equation with no explicit reference to $A(t)$, showing the equivalence of the two methods. Indeed, $N(t)$ takes into account a more complex and detailed population dynamics which permits the evaluation of the number of asymptomatic individuals also. The model is then applied to Covid-19 spreading in Italy where a transition from an exponential behavior to a Gompertz growth for $N(t)$ has been observed in more recent data. Then the information contained in the data analysis of $N(t)$ is reliable to understand the epidemic phase, although it does not describe the total infected population. The asymptomatic population is larger than the symptomatic one in the fast growth phase of the spreading.
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We propose a mathematical model to analyze the time evolution of the total number of infected population with Covid-19 disease at a region in the ongoing pandemic. Using the available data of Covid-19 infected population on various countries we formulate a model which can successfully track the time evolution from early days to the saturation period in a given wave of this infectious disease. It involves a set of effective parameters which can be extracted from the available data. Using those parameters the future trajectories of the disease spread can also be projected. A set of differential equations is also proposed whose solutions are these time evolution trajectories. Using such a formalism we project the future time evolution trajectories of infection spread for a number of countries where the Covid-19 infection is still rapidly rising.
We investigated daily COVID-19 cases and deaths in the 337 lower tier local authority regions in England and Wales to better understand how the disease propagated over a 15-month period. Population density scaling models revealed residual variance and skewness to be sensitive indicators of the dynamics of propagation. Lockdowns and schools reopening triggered increased variance indicative of outbreaks with local impact and country scale heterogeneity. University reopening and December holidays triggered reduced variance indicative of country scale homogenisation which reached a minimum in mid-January 2021. Homogeneous propagation was associated with better correspondence with normally distributed residuals while heterogeneous propagation was more consistent with skewed models. Skewness varied from strongly negative to strongly positive revealing an unappreciated feature of community propagation. Hot spots and super-spreading events are well understood descriptors of regional disease dynamics that would be expected to be associated with positively skewed distributions. Positively skewed behaviour was observed; however, negative skewness indicative of cold-spots and super-isolation dominated for approximately 8 months during the period of study. In contrast, death metrics showed near constant behaviour in scaling, variance, and skewness metrics over the full period with rural regions preferentially affected, an observation consistent with regional age demographics in England and Wales. Regional positions relative to density scaling laws were remarkably persistent after the first 5-9 days of the available data set. The determinants of this persistent behaviour probably precede the pandemic and remain unchanged.
With the unfolding of the COVID-19 pandemic, mathematical modeling of epidemics has been perceived and used as a central element in understanding, predicting, and governing the pandemic event. However, soon it became clear that long term predictions were extremely challenging to address. Moreover, it is still unclear which metric shall be used for a global description of the evolution of the outbreaks. Yet a robust modeling of pandemic dynamics and a consistent choice of the transmission metric is crucial for an in-depth understanding of the macroscopic phenomenology and better-informed mitigation strategies. In this study, we propose a Markovian stochastic framework designed to describe the evolution of entropy during the COVID-19 pandemic and the instantaneous reproductive ratio. We then introduce and use entropy-based metrics of global transmission to measure the impact and temporal evolution of a pandemic event. In the formulation of the model, the temporal evolution of the outbreak is modeled by the master equation of a nonlinear Markov process for a statistically averaged individual, leading to a clear physical interpretation. We also provide a full Bayesian inversion scheme for calibration. The time evolution of the entropy rate, the absolute change in the system entropy, and the instantaneous reproductive ratio are natural and transparent outputs of this framework. The framework has the appealing property of being applicable to any compartmental epidemic model. As an illustration, we apply the proposed approach to a simple modification of the Susceptible-Exposed-Infected-Removed (SEIR) model. Applying the model to the Hubei region, South Korean, Italian, Spanish, German, and French COVID-19 data-sets, we discover a significant difference in the absolute change of entropy but highly regular trends for both the entropy evolution and the instantaneous reproductive ratio.
The study carries out predictive modeling based on publicly available COVID-19 data for the duration 01 April to 20 June 2020 pertaining to India and five of its most infected states: Maharashtra, Tamil Nadu, Delhi, Gujarat, and Rajasthan using susceptible, infected, recovered, and dead (SIRD) model. The basic reproduction number R0 is derived by exponential growth method using RStudio package R0. The differential equations reflecting SIRD model have been solved using Python 3.7.4 on Jupyter Notebook platform. For visualization, Python Matplotlib 3.2.1 package is used. The study offers insights on peak-date, peak number of COVID-19 infections, and end-date pertaining to India and five of its states. The results could be leveraged by political leadership, health authorities, and industry doyens for policy planning and execution.
By characterising the time evolution of COVID-19 in term of its velocity (log of the new cases per day) and its rate of variation, or acceleration, we show that in many countries there has been a deceleration even before lockdowns were issued. This feature, possibly due to the increase of social awareness, can be rationalised by a susceptible-hidden-infected-recovered (SHIR) model introduced by Barnes, in which a hidden (isolated from the virus) compartment $H$ is gradually populated by susceptible people, thus reducing the effectiveness of the virus spreading. By introducing a partial hiding mechanism, for instance due to the impossibility for a fraction of the population to enter the hidden state, we obtain a model that, although still sufficiently simple, faithfully reproduces the different deceleration trends observed in several major countries.
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