Frontiers in Physics

Due to the current COVID-19 epidemic plague hitting the worldwide population it is of utmost medical, economical and societal interest to gain reliable predictions on the temporal evolution of the spreading of the infectious diseases in human populations. Of particular interest are the daily rates and cumulative number of new infections, as they are monitored in infected societies, and the influence of non-pharmaceutical interventions due to different lockdown measures as well as their subsequent lifting on these infections. Estimating quantitatively the influence of a later lifting of the interventions on the resulting increase in the case numbers is important to discriminate this increase from the onset of a second wave. The recently discovered new analytical solutions of Susceptible-Infectious-Recovered (SIR) model allow for such forecast and the testing of lockdown and lifting interventions as they hold for arbitrary time dependence of the infection rate. Here we present simple analytical approximations for the rate and cumulative number of new infections.

The simplest approximation for the first stages of the infection spread is considered. The specific feature of the COVID-19 characterized by its long latent period is taken into account. Exponential increase of numbers of infected people is determined by the half period of the maximal latent time for the COVID-19. The averaging over latent period leads to additional increase of the infected numbers. PACS number(s)02.50.-r, 05.60.-k, 82.39.-k, 87.19.Xx

The effective reproduction ratio r(t) of an epidemic, defined as the average number of secondary infected cases per infectious case in a population in the current state, including both susceptible and non-susceptible hosts, controls the transition between a subcritical threshold regime (r(t) < 1) and a supercritical threshold regime (r(t) > 1). While in subcritical regimes, an index infected case will cause an outbreak that will die out sooner or later, with large fluctuations observed when approaching the epidemic threshold, the supercritical regimes leads to an exponential growths of infection. The super- or subcritical regime of an outbreak is often not distinguished when close to the epidemic threshold, but its behaviour is of major importance to understand the course of an epidemic and public health management of disease control. In a subcritical parameter regime undetected infection, here called "imported case" or import, i.e. a susceptible individual becoming infected from outside the study area e.g., can either spark recurrent isolated outbreaks or keep the ongoing levels of infection, but cannot cause an exponential growths of infection. However, when the community transmission becomes supercritical, any index case or few "imported cases" will lead the epidemic to an exponential growths of infections, hence being distinguished from the subcritical dynamics by a critical epidemic threshold in which large fluctuations occur in stochastic versions of the considered processes. As a continuation of the COVID-19 Basque Modeling Task Force, we now investigate the role of critical fluctuations and import in basic Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered (SIR) epidemiological models on disease spreading dynamics. Without loss of generality, these simple models can be treated analytically and, when considering the mean field approximation of more complex underlying stochastic and eventually spatially extended or generalized network processes, results can be applied to more complex models used to describe the COVID-19 epidemics. In this paper, we explore possible features of the course of an epidemic, showing that the subcritical regime can explain the dynamic behaviour of COVID-19 spreading in the Basque Country, with this theory supported by empirical data data.

(Dated: September 6, 2020) A well-known characteristic of pandemics such as COVID-19 is the high level of transmission heterogeneity in the infection spread: not all infected individuals spread the disease at the same rate and some individuals (superspreaders) are responsible for most of the infections. To quantify this phenomenon requires the analysis of the effect of the variance and higher moments of the infection distribution. Working in the framework of stochastic branching processes, we derive an approximate analytical formula for the probability of an outbreak in the high variance regime of the infection distribution, verify it numerically and analyze its regime of validity in various examples. We show that it is possible for an outbreak not to occur in the high variance regime even when the basic reproduction number R0 is larger than one and discuss the implications of our results for COVID-19 and other pandemics. PACS numbers: 87.10.+e

A recently proposed temporal correlation-based- network framework applied on financial markets called Struc- tural Entropy has prompted us to utilize it as a means of analysis for COVID-19 fatalities across countries. Our observation on the resemblance of volatility of fluctuations of daily novel coronavirus related number of deaths to the daily stock exchange returns suggests the applicability of this approach.

The concern about (socio-)economic consequences of collective lockdowns in the Covid-19 pandemic calls for alternative strategies. We consider a divide and conquer strategy in which a high risk group (HRG) is put on strict isolation, whereas the remainder of the population is exposed to the virus, building up immunity against Covid-19. The question is whether this strategy may suppress the effective reproduction number below the critical value of [Formula] without further lockdown once the HRG is released from isolation. While this proposal appears already rather academic, we show that [Formula] can only be obtained provided that the HRG is less than ~ 20 - 30% of the total population. Hence, this strategy is likely to fail in countries with a HRG larger than the given upper bound. In addition, we argue that the maximum infection rate occurring in this strategy is likely to exceed realistic capacities of most health care systems. While the conclusion is rather negative in this regard, we emphasise that the strategy of stopping the curve at an early stage of the Covid-19 pandemic has a chance to work out. The required duration of the lockdown is estimated to be {tau} ~ 14 days/[Formula] (up to some order one factor) for [Formula], provided a systematic tracing strategy of new infections exists for the subsequent relaxation phase. In this context we also argue why [Formula] remains the crucial parameter which needs to be accurately monitored and controlled.

The birth and death of a pandemic can be region specific. Pandemic seems to make repeated appearance in some places which is often attributed to human neglect and seasonal change. However, difference could arise from different distributions of inherent susceptibility ({sigma}inh) and external infectivity ({iota}ext) from one population to another. These are often ignored in the theoretical treatments of an infectious disease progression. While the former is determined by the immunity of an individual towards a disease, the latter depends on the duration of exposure to the infection. Here we model the spatio-temporal propagation of a pandemic using a generalized SIR (Susceptible-Infected-Removed) model by introducing the susceptibility and infectivity distributions to comprehend their combined effects. These aspects have remained inadequately addressed till date. We consider the coupling between{sigma} inh and{iota} ext through a new critical infection parameter ({gamma}c). We find that the neglect of these distributions, as in the naive SIR model, results in an overestimation in the estimate of the herd immunity threshold. That is, the presence of the distributions could dramatically reduce the rate of spread. Additionally, we include the effects of long-range migration by seeding new infections in a region. We solve the resulting master equations by performing Kinetic Monte Carlo Cellular Automata (KMC-CA) simulations. Importantly, our simulations can reproduce the multiple infection peak scenario of a pandemic. The latent interactions between disease migration and the distributions of susceptibility and infectivity can render the progression a character vastly different from the naive SIR model. In particular, inclusion of these additional features renders the problem a character of a living percolating system where the disease cluster can survive by spatial migration.

In this work we propose the retarded logistic equation as a dynamic model for the spread of COVID-19 all over the world. This equation accounts for asymptomatic transmission, pre-symptomatic or latent transmission as well as contact tracing and isolation, and leads to a transparent definition of the instantaneous reproduction number R. For different parameter values, the model equation admits different classes of solutions. These solution classes correspond to, inter alia, containment of the outbreak via public health measures, exponential growth despite public health measures, containment despite reopening and second wave following reopening. We believe that the spread of COVID in every localized area such as a city, district or county can be accounted for by one of our solution classes. In regions where R > 1 initially despite aggressive epidemic management efforts, we find that if the mitigation measures are sustained, then it is still possible for R to dip below unity when far less than the regions entire population is affected, and from that point onwards the outbreak can be driven to extinction in time. We call this phenomenon partial herd immunity. Our analysis indicates that COVID-19 is an extremely vicious and unpredictable disease which poses unique challenges for public health authorities, on account of which "case races" among various countries and states do not serve any purpose and present delusive appearances while ignoring significant determinants.

Hundreds of predictions about the duration of the pandemic and the number of infected and dead have been carried out using traditional epidemiological tools (i.e. SIR, SIRD models, etc.) or new procedures of big-data analysis. However, the extraordinary complexity of the disease and the lack of knowledge about the pandemic (i.e. R value, mortality rate, etc.) create uncertainty about the accuracy of these estimates. However, several elegant mathematical approaches, based on physics and probability principles, like the Delta-t argument, Lindys Law or the Doomsday principle-Carters catastrophe, which have been successfully applied by scientists to unravel complex phenomena characterized by their great uncertainty (i.e. Human races longevity; How many more humans will be born before extinction) allow predicting parameters of the Covid-19 pandemic. These models predict that the COVID-19 pandemic will hit us until at least September-October 2021, but will likely last until January-September 2022, causing a minimum of 36,000,000 infected and most likely 60,000,000, as well as 1,400,000 dead at best and most likely 2,333,000. Competing Interest Statement The authors have declared no competing interest.

In the course of a large-scale infectious disease a time-dependent Reproduction rate is an important parameter for political, economic and social decisions. In this paper we focus on that parameter and introduce a mathematical implementation in addition to the mostly used definition of Robert-Koch-Institute (RKI) in Germany. Such value is of particular interest in order to serve as a criterion for possible Lock-Downs and "LockUps" in society and can provide deep insights into a pandemic event. Both the definition of the new Reproduction index and the RKIs Reproduction number are compared analytically, applied to simple model calculations and finally on real Covid19 data. Clear advantages of the new Reproduction index become apparent and some weaknesses of the RKIs Reproduction number become clearly visible. In addition we propose two additional ways of displaying pandemic data to have the pandemic behaviour at a glance. We find that some signatures of the pandemic appear now very well expressed - especially in conjunction with the new Reproduction index Ri. This all could be very helpful for future political, social and economic decisions.

Equations for infection spread in a closed population are found in discrete approximation, corresponding to the published statistical data, and in continuous time in the form of delay differential equations. We consider the epidemic as dependent upon four key parameters: the size of population involved, the mean number of dangerous contacts of one infected person per day, the probability to transmit infection due to such contact and the mean duration of disease. In the simplest case of free-running epidemic in an infinite population, the number of infected rises exponentially day by day. Here we show the model for epidemic process in a closed population, constrained by isolation, treatment and so on. The four parameters introduced here have the clear sense and are in association with the well-known concept of reproduction number in the continuous susceptible--infectious--removed, susceptible--exposed--infectious--removed (SIR, SEIR) models. We derive the initial rate of infection spread from the published statistical data for the initial stage of epidemic, when the quarantine measures were absent. On this basis, we can found the corresponding basic reproduction number mentioned above. Our approach allows evaluating the influence of quarantine measures on free pandemic process that leads to the time-dependent rate of infection and suppression of infection. We found a good correspondence of the theory and reliable statistical data. The initially formulated discrete model, describing epidemic course day by day is transferred to differential form. The conditions for saturation of epidemic are found by solving the delay differential equations. They differ essentially from ones in SIR model due to finite delay, typical for COVID-19. The proposed model opens up the possibility to predict the optimal level of social quarantine measures. The model is quite flexible and it can be extended to more complex cases.

The corona virus (SARS-CoV-2) or Covid-19 pandemic is growing alarmingly throughout the whole world. Using the power law scaling we analyze the data of different countries and three states of India up to 1st April, 2020 and explain in terms of power law exponent. We find significant reduction in growth of infections in China and Denmark ({gamma} reduced from approximately 2.18 to 0.05 and 11.41 to 6.95, respectively). Very slow reduction is also seen in Brazil and Germany ({gamma} reduced from approximately 6 to 4 and 11 to 7, respectively). Infection in India is growing ({gamma}=9.23) though lesser in number than that in the USA (highest {gamma} of 16 approximately, studied so far), Italy and a few other countries. Among three Indian states the growth in West Bengal ({gamma}=0.64) is much slower than other states like Maharashtra and Kerala ({gamma}=3.23 and 3.32, respectively). Some future predictions, though not rigid, has also been incorporated in our analysis. The earlier lock-down and stricter measures from the Governments concerned are being thought to be the only possible solutions, in the present situation, to fight against this virus.

The new coronavirus known as COVID-19 is spread world-wide since December 2019. Without any vaccination or medicine, the means of controlling it are limited to quarantine and social distancing. Here we study the spatio-temporal propagation of the first wave of the COVID-19 virus in China and compare it to other global locations. We provide a comprehensive picture of the spatial propagation from Hubei to other provinces in China in terms of distance, population size, and human mobility and their scaling relations. Since strict quarantine has been usually applied between cities, more insight about the temporal evolution of the disease can be obtained by analyzing the epidemic within cities, especially the time evolution of the infection, death, and recovery rates which affected by policies. We study and compare the infection rate in different cities in China and provinces in Italy and find that the disease spread is characterized by a two-stages process. At early times, at order of few days, the infection rate is close to a constant probably due to the lack of means to detect infected individuals before infection symptoms are observed. Then at later times it decays approximately exponentially due to quarantines. The time evolution of the death and recovery rates also distinguish between these two stages and reflect the health system situation which could be overloaded.

A kinetic approach is developed, in a "tutorial style" to describe the evolution of an epidemic with spread taking place through contact. The "infection - rate" is calculated from the rate at which an infected person approaches an uninfected susceptible individual, i.e. a potential recipient of the disease, up to a distance p, where the value of p may lie between pmin[&le;] p [&le;] pmax. We consider a situation with a total population of N individuals, living in an area A, x(t) amongst them being infected while xd(t) = {beta}'x(t) is the number that have died in the course of transmission and evolution of the epidemic. The evolution is developed under the conditions (1) a faction (t) of the [N-x(t) - xd(t)] uninfected individuals and (2) a {beta}(t) fraction of the x(t) infected population are quarantined, while the "source events" that spread the infection are considered to occur with frequency{upsilon} 0. The processes of contact and transmission are considered to be Markovian. Transmission is assumed to be inhibited by several processes like the use of "masks", "hand washing or use of sanitizers" while "physical distancing" is described by p. The evolution equation for x(t) is a Riccati - type differential equation whose coefficients are time-dependent quantities, being determined by an interplay between the above parameters. A formal solution for x(t) is presented, for a "graded lockdown" with the parameters, 0[&le;] (t), {beta}(t)[&le;]1 reaching their respective saturation values in time scales,{tau} 1,{tau} 2 respectively, from their initial values (0)={beta}(0)=0. The growth is predicted for several BBMP wards in Bengaluru and in urban centers in Chikkaballapur district, as an illustrative case. Above selections serve as model cases for high, moderate and thin population densities. It is seen that the evolution of [x(t)/N] with time depends upon (a) the initial time scale of evolution, (b) the time scale of cure and (c) on the time dependence of the Lockdown function Q(t) = {[1-(t)]{middle dot}[1-{beta}(t)]}. The formulae are amenable to simple computations and show that in order to curb the spread one must ensure that Q({infty}) must be below a critical value and the vigilance has to be continued for a long time (at least 100 to 150 days) after the decay starts, to avoid all chances of the infection reappearing.

CoVID-19 is spreading throughout the world at an alarming rate. So far it has spread over 200 countries in the whole world. Mathematical modeling of an epidemic like CoVID-19 is always useful for strategic decision making, especially it is very useful to gain some understanding of the future of the epidemic in densely populous countries like India. We use a simple yet effective mathematical model SIR(D) to predict the future of the epidemic in India by using the existing data. We also estimate the effect of lock-down/social isolation via a time-dependent coefficient of the model. The model study with realistic parameters set shows that the epidemic will be at its peak around the end of June or the first week of July with almost 108 Indians most likely being infected if the lock-down relaxed after May 3, 2020. However, the total number of infected population will become one-third of what predicted here if we consider that people only in the red zones (approximately one-third of Indias population) are susceptible to the infection. Even in a very optimistic scenario we expect that at least the infected numbers of people will be [Formula].

Within the framework of a two-component model of the COVID-19 epidemic, taking into account the special role of superspreaders, we consider the impact of the recovery factor and quarantine measures on the course of the epidemic, as well as the possibility of a second wave of morbidity. It is assumed that there is no long-term immunity in asymptomatic superspreaders who have undergone the infection, and the emergence of long-term immunity in those who have undergone severe illness. It is shown that, under these assumptions, the relaxation of quarantine measures leads to the resumption of virus circulation among asymptomatic superspreaders. Depending on the characteristics of the quarantine, its removal may or may not lead to a renewed wave of daily morbidity. A criterion for the occurrence of repeated wave of morbidity is proposed based on the analysis of the final phase of the first wave. Based on this criterion, the repeated wave of the epidemic is predicted in New Zealand. A natural explanation is given for the decrease in lethality among the infected against the background of an absolute increase in their number.

The ongoing epidemic of COVID-19 originated in China has reinforced the need to develop epidemiological models capable of describing the progression of the disease to be of use in the formulation of mitigation policies. Here, this problem is addressed using a metapopulation approach to show that the delay in the transmission of the spread between different subsets of the total population, can be incorporated into a SIR framework through a time-dependent transmission rate. Thus, the reproduction number decreases with time despite the population dynamics remains uniform and the depletion of susceptible individuals is small. The obtained results are consistent with the early subexponential growth observed in the cumulated number of confirmed cases even in the absence of containment measures. We validate our model by describing the evolution of the COVID-19 using real data from different countries with an emphasis in the case of Mexico and show that it describes correctly also the long-time dynamics of the spread. The proposed model yet simple is successful at describing the onset and progression of the outbreak and considerably improves accuracy of predictions over traditional compartmental models. The insights given here may probe be useful to forecast the extent of the public health risks of epidemics and thus improving public policy-making aimed at reducing such risks.

In this paper we fit simple modifications of the SIR compartmental model to the COVID-19 outbreak data, available from official sources for Italy and other countries. Even if the complexity of the pandemic can not be easily modelled, we show that our model, at present, describes the time evolution of the data in spite of the application of the social distancing and lock-down procedure. Finally, we discuss the reliability of the model predictions, under certain conditions, for estimating the near and far future evolution of the COVID-19 outbreak. The conditions for the applicability of the proposed models are discussed.

Delay differential equations are set up for zeroth-order pandemic models in analogy with traditional SIR and SEIR models by specifying individual times of incubation and infectiousness prior to recovery. Independent linear delay relations in addition to a nonlinear delay differential equation are found for characterizing time-dependent compartmental populations. Asymptotic behavior allows a link between parameters of the delay and traditional models for their comparison. In analogy with transformation of the traditional equations into linear form giving populations and time in parametric form, expansion in the delay provides a simple recursive solution. Also, a soliton-like solution in terms of a logistic function can be applied for accurate approximation. Otherwise, straightforward numerical solution is effected in terms of linearized boundary conditions specifying the distribution of instigators as to their initial infection progress--in contrast to traditional models specifying only initial average infectious and exposed populations. Examples contrasting asymptotically-linked traditional and delay models are given.

In this paper, we analyze the real-time infection data of COVID-19 epidemic for 21 nations up to June 30, 2020. For most of these nations, the total number of infected individuals exhibits a succession of exponential growth and power-law growth before the flattening of the curve. In particular, we find a universal [Formula] growth before they reach saturation. However, at present, India, which has I(t) ~ t2, and Russia and Brazil, which have I(t) ~ t, are yet to flatten their curves. Thus, the polynomials of the I(t) curves provide valuable information on the stage of the epidemic evolution, thus on the life cycle of COVID-19 pandemic. Besides these detailed analyses, we compare the predictions of an extended SEIR model and a delay differential equation-based model with the reported infection data and observed good agreement among them, including the [Formula] behaviour. We argue that the power laws in the epidemic curves may be due to lockdowns.