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Register a new userSolvable delay model for epidemic spreading: the case of Covid-19 in Italy
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Authors
Luca DellAnna
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We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.
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In late December 2019, a novel strand of Coronavirus (SARS-CoV-2) causing a severe, potentially fatal respiratory syndrome (COVID-19) was identified in Wuhan, Hubei Province, China and is causing outbreaks in multiple world countries, soon becoming a pandemic. Italy has now become the most hit country outside of Asia: on March 16, 2020, the Italian Civil Protection documented a total of 27980 confirmed cases and 2158 deaths of people tested positive for SARS-CoV-2. In the context of an emerging infectious disease outbreak, it is of paramount importance to predict the trend of the epidemic in order to plan an effective control strategy and to determine its impact. This paper proposes a new epidemic model that discriminates between infected individuals depending on whether they have been diagnosed and on the severity of their symptoms. The distinction between diagnosed and non-diagnosed is important because non-diagnosed individuals are more likely to spread the infection than diagnosed ones, since the latter are typically isolated, and can explain misperceptions of the case fatality rate and of the seriousness of the epidemic phenomenon. Being able to predict the amount of patients that will develop life-threatening symptoms is important since the disease frequently requires hospitalisation (and even Intensive Care Unit admission) and challenges the healthcare system capacity. We show how the basic reproduction number can be redefined in the new framework, thus capturing the potential for epidemic containment. Simulation results are compared with real data on the COVID-19 epidemic in Italy, to show the validity of the model and compare different possible predicted scenarios depending on the adopted countermeasures.
The control of Covid 19 epidemics by public health policy in Italy during the first and the second epidemic waves has been driven by using reproductive number Rt(t) to identify the supercritical (percolative), the subcritical (arrested), separated by the critical regime. Here we show that to quantify the Covid-19 spreading rate with containment measures (CSRwCM) there is a need of a 3D expanded parameter space phase diagram built by the combination of Rt(t) and doubling time Td(t). In this space we identify the dynamics of the Covid-19 dynamics Italy and its administrative Regions. The supercritical regime is mathematically characterized by i) the power law of Td vs. [Rt(t)-1] and ii) the exponential behaviour of Td vs. time, either in the first and in the second wave. The novel 3D phase diagram shows clearly metastable states appearing before and after the second wave critical regime. for loosening quarantine and tracing of actives cases. The metastable states are precursors of the abrupt onset of a next nascent wave supercritical regime. This dynamic description allows epidemics predictions needed by policymakers to activate non-pharmaceutical interventions (NPIs), a key issue for avoiding economical losses, reduce fatalities and avoid new virus variant during vaccination campaign
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.
The outbreak of the novel coronavirus, COVID-19, has been declared a pandemic by the WHO. The structures of social contact critically determine the spread of the infection and, in the absence of vaccines, the control of these structures through large-scale social distancing measures appears to be the most effective means of mitigation. Here we use an age-structured SIR model with social contact matrices obtained from surveys and Bayesian imputation to study the progress of the COVID-19 epidemic in India. The basic reproductive ratio R0 and its time-dependent generalization are computed based on case data, age distribution and social contact structure. The impact of social distancing measures - workplace non-attendance, school closure, lockdown - and their efficacy with durations are then investigated. A three-week lockdown is found insufficient to prevent a resurgence and, instead, protocols of sustained lockdown with periodic relaxation are suggested. Forecasts are provided for the reduction in age-structured morbidity and mortality as a result of these measures. Our study underlines the importance of age and social contact structures in assessing the country-specific impact of mitigatory social distancing.
We revisit well-established concepts of epidemiology, the Ising-model, and percolation theory. Also, we employ a spin $S$ = 1/2 Ising-like model and a (logistic) Fermi-Dirac-like function to describe the spread of Covid-19. Our analysis reinforces well-established literature results, namely: emph{i}) that the epidemic curves can be described by a Gaussian-type function; emph{ii}) that the temporal evolution of the accumulative number of infections and fatalities follow a logistic function, which has some resemblance with a distorted Fermi-Dirac-like function; emph{iii}) the key role played by the quarantine to block the spread of Covid-19 in terms of an emph{interacting} parameter, which emulates the contact between infected and non-infected people. Furthermore, in the frame of elementary percolation theory, we show that: emph{i}) the percolation probability can be associated with the probability of a person being infected with Covid-19; emph{ii}) the concepts of blocked and non-blocked connections can be associated, respectively, with a person respecting or not the social distancing, impacting thus in the probability of an infected person to infect other people. Increasing the number of infected people leads to an increase in the number of net connections, giving rise thus to a higher probability of new infections (percolation). We demonstrate the importance of social distancing in preventing the spread of Covid-19 in a pedagogical way. Given the impossibility of making a precise forecast of the disease spread, we highlight the importance of taking into account additional factors, such as climate changes and urbanization, in the mathematical description of epidemics. Yet, we make a connection between the standard mathematical models employed in epidemics and well-established concepts in condensed matter Physics, such as the Fermi gas and the Landau Fermi-liquid picture.
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