The European Physical Journal Plus

One of the central tools to control the COVID-19 pandemics is the knowledge of its spreading dynamics. Here we develop a fractal model capable of describe this dynamics, in term of daily new cases, and provide quantitative criteria for some predictions. We propose a fractal dynamical model using conformed derivative and fractal time scale. A Burr-XII shaped solution of the fractal-like equation is obtained. The model is tested using data from several countries, showing that a single function is able to describe very different shapes of the outbreak. The diverse behavior of the outbreak on those countries is presented and discussed. Moreover, a criterion to determine the existence of the pandemic peak and a expression to find the time to reach herd immunity are also obtained.

A relevant problem in the study of the Covid-19 pandemic is the study of its temporal evolution. Such evolution depends on a number of factors, among which the average rate of contacts between susceptible and infected individuals, the duration of infectiousness and the transmissibility, that is the probability of infection after a contact between susceptible and infected individuals. In a previous study, we analyzed the potentiality of a number of distributions to describe the evolution of the pandemic and the potentiality of each distribution to mathematically predict the evolution of the pandemic in Italy. Since the number of daily tests was changing and increasing with time, we used the ratio of the new daily cases per swab. We considered distributions of the type of Gauss (normal), Gamma, Beta, Weibull, Lognormal and in addition of the type of the Planck blackbody radiation law. The Planck law, describing the amount of energy of the electromagnetic radiation emitted by a black body at each wavelength or at each frequency, marked in 1900 the beginning of Quantum Mechanics. The result of our analysis was that, among the considered distributions, the Planck law has the best potentiality to mathematically predict the evolution of the pandemic and the best fitting capability. In this paper, we analyze the time evolution of the second wave of the Covid-19 pandemic in Italy and in particular we predict the ratio of the new daily cases per swab at Christmas 2020 using the data in the interval from 17 Oct to 21 Nov. According to Figure 4 and Figure 8, the prediction for such a ratio around Christmas is approximately within 6% and 7%. In this study there is also an attempt to account for the effects of the governmental containment measures.

Diffusion comes anytime and everywhere. If there is a gradient or a potential difference of a quantity a diffusion process happens and this ends if an equilibrium is reached only. The concentration of a species maybe such quantity, or the voltage. An electric currant will be driven by a voltage difference for example. In this COVID-19 pandemic one observes both regions with low incidence and other ones with high incidence. The local different people density could be a reason for that. In populous areas like big cities or congested urban areas higher COVID-19 incidences could be observed than in rural regions. The aim of this paper consists in the application of a diffusion concept to describe one possible issue of the the COVID-19 propagation. This will be discussed for the German situation based on the quite different incidence data for the different federal states of Germany. With this ansatz some phenomenoms of the actual development of the pandemic could be confirmed. The model gives a possibility to investigate certain scenarios like border-crossings or local spreading events and their influence on the COVID-19 propagation as well.

This study shows that the disease free equilibrium (E0) for COVID-19 coronavirus does not satisfy the criteria for a locally or globally asymptotic stability. This implies that as a pandemic as declared by WHO (2020) the COVID-19 coronavirus does not have a curative vaccine yet and precautionary measures are advised through quarantine and observatory procedures. Also, the Basic Reproductive number (R0 < 1) by Equation (33) shows that there is a chance of decline of secondary infections when the ratio between the incidence rate in the population and the total number of infected population quarantined with observatory procedure. The effort to evaluate the disease equilibrium shows that unless there is a dedicated effort from government, decision makers and stakeholders, the world would hardly be reed of the COVID-19 coronavirus and further spread is eminent and the rate of infection will continue to increase despite the increased rate of recovery because of the absence of vaccine at the moment.

We show that the standard SIR model is not effective to predict the 2019-20 coronavirus pandemic propagation. We propose a new model where the logarithm of the detected population number follows a linear dynamical system. We estimate the parameters of this system and compare models obtained with data observed from different countries. Based on the given estimator and results obtained with the Pr. Raoults treatment, we affirm with a reasonable degree of confidence that his "test-treat-noconfine" policy was less expensive in human lives than the"confine and wait for a proved treatment" policy adopted by the French government.

In this short paper we describe a simple technique aimed to forecast the course of the spreading of Covid-19 virus infection in Italy. Data released every day by the Italian "Protezione Civile" organization have been processed with reference of the the SIR classical mathematical model by Kermack and McKendrick. The model provides a rough estimate of the time needed to completely block virus spread. The above assumptions will be valid if Covid-19 will not be recognized as capable of establishing a chronic productive infection in significant fraction of the population.

This article presents a mathematical infection model that is designed to estimate the course of coronavirus infection in Germany for several days in advance: How many people become ill or die, what is the temporal development? If the contact restriction is perfect, then the model predicts the development of the virus infection after the initial subsidence of the infection. However, since this restriction cannot always be strictly adhered to, the model is dynamically adapted to the development. This makes it possible to estimate the number of infected people, the number of new infections and deaths in Germany about a week in advance.

Concerning the two approaches to the Covid-19 case mortality rate published in the literature, namely computing the ratio of (a) the daily number of deaths to a time delayed daily number of confirmed infections; and (b) the cumulative number of deaths to confirmed infections up to a certain time, both numbers having been acquired in the middle of an outbreak, it is shown that each suffers from systematic error of a different source. We further show that in the absence of detailed knowledge of the time delay distribution of (a), the true case mortality rate is obtained by pursuing method (b) at the end of the outbreak when the fate of every case has decisively been rendered. The approach is then employed to calculate the mean case mortality rate of 13 regions of China where every case has already been resolved. This leads to a mean rate of 0.527 {+/-} 0.001 %.

An asymmetric version of the classical Kermack-McKendrick description of an epidemic evolution is presented in terms of four independent parameters. This is enough to obtain an accurate description of the different stages of the COVID-19 pandemia in any country for the reported daily and total number of casualties due to the infection. The asymmetry accounts for lockdown effects introduced to reduce the impact of the epidemic outburst. A set of new variables allows for an analytic study of the evolution of the system before and after the lockdown measures are put in place. A continuous matching is possible for all variables in the system apart from the time dependence of the infection rate. An analytic expression is obtained for this discontinuity which is proposed as a good quantity to gauge the efficiency of the lockdown measures. A study of this variable for different countries is performed.

Predictions about the further development of the Corona pandemic are of great public interest but many approaches demand a large number of country specific parameters and are not easily transferable. A special interest of simulations on the pandemic is to trace the effect of politics for reducing the virus spread, since these measures have had an enormous impact on economy and daily life. Here a simple yet powerful algorithm is introduced for fitting the infection numbers by simple analytic functions. This way, the increase of the case numbers in periods with different regulations can be distinguished, and by extrapolating the fit functions, a forecast for the maximum numbers and time scales are possible. The effect of the restraints such as lock down are demonstrated by comparing the resulting infection history with the likely unconstrained virus spread, and it is shown that a delay of 1-4 weeks before imposing measures aiming at social distancing could have led to a complete infection of the respective populations. The approach is simply transferable to many different states. Here data from six E.U. countries, the UK, Russia, two Asian countries, the USA and ten states inside the USA with significant case numbers are analyzed, and striking qualitative similarities are found. Keywords: Covid-19, forecast, analytic fit, France, Germany, Italy, Spain South Korea, New York, Washington, Florida, Michigan, Poland, Sweden, USA, Pennsylvania, China, Russia, UK, California, Illinois, Indiana, Maryland, North Carolina.

We consider the impact of superspreading on the course of the COVID-19 epidemic. A two-component model of the epidemic has been developed, in which all infected are divided in two groups. The groups are asymptomatic superspreaders spreading the infection and sensitive persons which can only get infection. Once infected the sensitive exhibit clear symptoms and become isolated. It is shown that the ratio of increment of the number of daily cases in the beginning of the epidemic and decrement at the end of the epidemic is equal to the ratio of the spreading rates of the infection transmission from the superspreaders to potential superspreaders and to the sensitive persons, respectively. On the basis of data from 12 countries and territories it is found that the superspreaders transmit the infection to potential superspreaders approximately 4 times more often then to the sensitive persons. Specific measures to limit the epidemical incidence are proposed. The possibility of an allergic component in the disease is discussed.

A study is presented on the use of real-time information about symptomatic infectious individuals to adjust restrictions of human contacts at two basic levels, the stricter being on the symptomatic infectious group. Explicit analytical formulas as well as numerical results are presented to rapidly elucidate what-if questions on averting resurgence of the coronavirus epidemic after the first wave wanes. Implementation of related ideas would rely on a mix of several factors, including personal initiative and sophisticated technology for monitoring and testing. For robust decision making on the subject, detailed multidisciplinary studies remain indispensable.

The SIR-model is adapted to the covid-19 pandemic through a modification that consists in making the basic reproduction number variable. Independent of it, another reproduction number is introduced, which is defined similarly to the usual net reproduction number. Due to its simple analytic form, it enables a clear interpretation for all values. A further parameter, provisionally called acceleration parameter, is introduced and applied, which enables a more differentiated characterization of the infection number dynamics. By a variable transformation the 3 equations of the modified SIR-model can be reduced to 2. The latter are solved up to ordinary integrations. The solutions are evaluated for current situations, yielding a pretty good match with the data reported. Encouraged by this, a variety of possible future developments is examined, including linear and exponential growth of the infection numbers as well as sub- and super-exponential growth. In particular, the behavior of the two reproduction numbers and the acceleration parameter is studied, which in some cases leads to surprising results. With regard to the number of unreported infections it is shown, that from the solution for a special one solutions for others can be derived by similarity transformations.

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.

Using a stochastic epidemic model explicitly considering the entire population of Guadeloupe (1), we explore the domain of solutions presenting an efficient slowing down of the COVID-19 epidemic spread during the post-containment period. The considered model parameters are the basic reproduction number R0 to simulate the effects of social distancing, the time delay{delta} TO_SCPLOWQC_SCPLOW elapsed between the detection of a symptomatic person and her/his placement in quarantine to suppress her/his contagiousness, and the number Na of asymptomatic people tested positively and isolated. We show that acceptable solutions are obtained for a wide range of parameter values. Thanks to a good control of the initial epidemic spread resulting from an early containment and efficient communication by the sanitary and administrative authorities, the present situation corresponds to a pre-epidemic state. The most safe solutions are a combinations of social distancing, numerous testing to perform a systematic isolation of symptomatic patients and guided detection of asymptomatic people in the entourage of localised symptomatic patients.

The existence of a universal law which maps the bell curve of daily cases to a sigmoid curve for cumulative ones is used for making robust estimations about the final outcome of a disease. Computations of real time effective reproduction rate are presented and its limited usefulness is derived. After using methods ESE & EDE we are able to find the inflection point of the cumulative curve under consideration and study its time evolution. Since mortality processes tend to follow a Gompertz distribution, we apply the properties of it and introduce novel estimations for both the time remaining after inflection time and the capacity of the curve. Special properties of sigmoid curves are used for assessing the quality of estimation and as indices for the cycle completion. Application is presented for COVID-19 evolution for most affected countries and the World.

Bimolecular cell membranes play a crucial role in many biological processes and possess a unique set of physical properties. Bimolecular membranes and monomolecular films can be considered as a "two-dimensional fluid" because the diffusion of molecules along the membrane or film is a hydrodynamic process. On the other hand, the bending of the cell membrane is controlled by its stiffness and elastic tension. The aim of this work is to adapt the Navier-Stokes hydrodynamic equations, obtained using the classical Chapman-Enskog method, to the case of two-dimensional membranes. The hydrodynamic equation system is complemented by an elasticity equation for the bending oscillations of the membrane. The obtained system of equations for the dynamics of the cell membrane is linearized for the case of disturbances with small amplitude. Dispersion equations for stable and unstable linear oscillations of cell membranes are investigated, and conditions for the onset of instabilities are derived.

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.

A discrete-time deterministic epidemic model is proposed with the aim of reproducing the behaviour observed in the incidence of real infectious diseases. For this purpose, we analyse a SIRS model under the framework of a small world network formulation. Using this model, we make predictions about the peak of the Covid-19 epidemic in Italy. A Gaussian fit is also performed, to make a similar prediction.

In order to quantitatively characterize the epidemic of COVID-19, useful relations among parameters describing an epidemic in general are derived based on the Kermack-McKendrick model. The first relation is 1/{tau}grow =1/{tau}trans-1/{tau}inf, where{tau} grow is the time constant of the exponential growth of an epidemic,{tau} trans is the time for a pathogen to be transmitted from one patient to uninfected person, and the infectious time{tau} inf is the time during which the pathogen keeps its power of transmission. The second relation p({infty}) {approx} 1-exp(-(R0-1)/0.60) is the relation between p({infty}), the final size of the disaster defined by the ratio of the total infected people to the population of the society, and the basic reproduction number, R0, which is the number of persons infected by the transmission of the pathogen from one infected person during the infectious time. The third relation 1/{tau}end = 1/{tau}inf-(1-p({infty}))/{tau}trans gives the decay time constant{tau} end at the ending stage of the epidemic. Derived relations are applied to influenza in Japan in 2019 for characterizing the epidemic.