cord-005350-19za0msu	2013	Using the stochastic differential equation, we can obtain analytical expressions for statistical signatures of leading indicators and early warning signals, including the power spectrum and autocorrelation function (see Appendix A for details). To investigate the results of this theory for a particular parameter set (Table 7) , we calculated leading indicators of elimination and emergence, assuming alternatively that (a) the mean proportion of infectious individuals is given by the deterministic endemic equilibrium ( → 0 theory) or (b) assuming it is given by the current state of the fast-slow system approaching a transition. We also compared the elimination indicators with those calculated assuming that the mean proportion of infectious individuals was given by the deterministic endemic equilibrium from the limiting case models with no immigration. The goal of our study was to develop the theory of such early warning signals and leading indicators for infectious disease transmission systems that meet the assumptions of the familiar SIS and SIR models and which are forced through a critical transition by changes in transmission.
cord-007399-qbgz7eqt	2016	6, 7 In typical seasonally forced models of infectious diseases, the transmission rate (i.e., the per capita rate at which a susceptible individual interacts with an infectious individual and acquires a new infection) is modulated in a periodic fashion using a sine or cosine function. One of the characteristic features of temporally forced systems is that they possess multiple coexisting attractors: the dynamics sensitively depends on the initial values of state variables. [15] [16] [17] Coexistence of multiple attractors or states has also been observed in the dynamics of epidemic models when the transmission rates are periodically forced. As hinted in the beginning of this paragraph, the models considered here vary in internal feedbacks on disease dynamics in terms of the effect of immune response in regulating the build-up of susceptibles during inter-epidemic periods necessary for fuelling next outbreaks. Previous studies, investigating the effects of periodic modulation in the transmission rate, showed the coexistence of multiple attractors in the dynamics of the SIR family of epidemic models.
cord-007404-s2qnhswe	2015	The existing studies have provided different theoretical predictions for epidemic threshold of the susceptible-infected-recovered (SIR) model on complex networks, while the numerical verification of these theoretical predictions is still lacking. To understand the effectiveness of the variability measure, the distribution of outbreaks sizes is investigated near the epidemic threshold on random regular networks. Considering that the existing theories more or less have some limitations (e.g., the HMF theory neglects the quenched structure of the network; QMF theory ignores dynamical correlations 14 ) , some numerical methods such as the finite-size scaling analysis, 15 susceptibility, 16 and lifetime measure 17 have been proposed to check the accuracies of different theoretical predictions for the SIS model. In this work, we perform extensive numerical simulations of the SIR model on networks with finite size, and present a numerical identification method by analyzing the peak of the epidemic variability 24,25 (i.e., the maximal value of the epidemic variability) to identify the epidemic threshold.
cord-010715-91fob3ax	2016	We derive the percolation transition points for the SIR model with multiple seeds to show that as the infection rate increases epidemic clusters generated from each seed percolate before a single seed can induce a global outbreak. To evaluate the time evolution of the SIR dynamics and the total densities of the susceptible and removed nodes in the final states, we consider the approximate master equations (AMEs) [12, 14] . In particular, the gap between λ c1 and λ SIR c indicates that as the infection rate increases, the epidemic clusters generated from each seed percolate before a single seed can induce a global outbreak. We have numerically and analytically shown that the present model with multiple seeds on the RRG percolates at a lower infection rate than the epidemic threshold. The SIR model with numerous seeds shows the percolation transition of the removed and susceptible nodes at λ c1 and λ c2 , respectively.
cord-010719-90379pjd	2017	We also consider the SIR model on structured networks and study the effect of topology on threshold points in a non-Markovian dynamics. In all previous works, the authors rarely discuss the effect of fractional order differential equations and memory on the epidemic thresholds and the macroscopic behavior of epidemic outbreaks. This means that the initial time for taking into account the disease control memory is shifted toward more recent times: thereafter, the dynamics is evolving with a new fraction of susceptible and infected individuals, different from that predicted by the solution of the differential equations. In Sec. II, following Caputo''s approach, we convert the differential equations of the standard SIR model to the fractional derivatives, thereby allowing us to consider memory effects. In order to observe the influence of memory effects, first we rewrite the differential equations (1) in terms of time-dependent integrals as follows:
cord-016965-z7a6eoyo	2017	In addition for infected sites to transmit the disease to neighboring susceptible lattice sites, every now and then (with a probability of 1%) they can also Fig. 19 .1) geographic distance to the initial outbreak location is no longer a good predictor of arrival time, unlike in systems with local or spatially limited host mobility infect randomly chosen lattice sites anywhere in the system. A visual inspection of the air-transportation system depicted in Fig. 19 .1 is sufficiently convincing that the significant fraction of long-range connections in global mobility will not only increase the speed at which infectious diseases spread but, more importantly, also cause the patterns of spread to exhibit high spatial incoherence and complexity caused by the intricate connectivity of the air-transportation network. Figure 19 .7 shows that also the model epidemic depicts only a weak correlation between geographic distance to the outbreak location and arrival time.
cord-029725-px209lf0	2020	Using the available data of the number of COVID-19 positive cases reported in the state of Kerala, and in India till 26th April, 2020 and 12th May 2020, respectively, the parameter estimation problem is converted into an optimization problem with the help of a least squared cost function. Using the estimated set of parameters, the model predicts that in the state of Kerala, by using certain interventions the pandemic can be successfully controlled latest by the first week of July, whereas the [Formula: see text] value for India is still greater than 1, and hence lifting of lockdown from all regions of the country is not advisable. In such a scenario, assuming the present levels of testing, the model predicts that the number of cases in Kerala can be controlled by the first week of June (as shown in Fig. 8 ).
cord-034824-eelqmzdx	2020	In this paper, we propose the EnRenew algorithm aimed to identify a set of influential nodes via information entropy. Compared with the best state-of-the-art benchmark methods, the performance of proposed algorithm improved by 21.1%, 7.0%, 30.0%, 5.0%, 2.5%, and 9.0% in final affected scale on CEnew, Email, Hamster, Router, Condmat, and Amazon network, respectively, under the Susceptible-Infected-Recovered (SIR) simulation model. The impressive results on the SIR simulation model shed light on new method of node mining in complex networks for information spreading and epidemic prevention. defined the problem of identifying a set of influential spreaders in complex networks as influence maximization problem [57] , and they used hill-climbing based greedy algorithm that is within 63% of optimal in several models. Besides, to make the algorithm practically more useful, we provide EnRenew''s source code and all the experiments details on https://github.com/YangLiangwei/Influential-nodes-identification-in-complex-networksvia-information-entropy, and researchers can download it freely for their convenience.
cord-102966-7vdz661d	2020	In their landmark 1927 publication Contribution to the Mathematical Theory of Epidemics, 1, 2 Kermack and McKendrick developed a general, if elaborate model structure to capture the dynamics of a fixed-size population comprising compartments of individuals susceptible (S) to a spreading infection, infectious (I), and removed (R) from the preceding two compartments by recovery or death. Starting with the assumption that individuals leave the infectious group at time after infection, we develop in this paper a corresponding mathematical model structure, named delay SIR (dSIR), in the form of a single delay differential equation (DDE) for , and two associated delay algebraic equations, for and in terms of . It turns out (Appendix A) that the following simple remedy can be used to retain the ODE structure of the standard SIR model, while better approximating the DDE dynamics of the more realistic dSIR model structure: The SIR equations for { ′ , ′ }, eqns.
cord-103598-8umv06ox	2020	This article describes a simple Susceptible Infected Recovered (SIR) model fitting with COVID-19 data for the month of march 2020 in New York (NY) state. The model is a classical SIR, but is non-autonomous; the rate of susceptible people becoming infected is adjusted over time in order to fit the available data. In this short article, we first introduce a simple SIR model, in which we adjust a key parameter k standing for a control on the Susceptible-Infected rate, and secondarily the death rate, in order to fit the data of the pandemic in NY state in March 2020, and provide predictions for a near future. Accordingly, the main key points of this article are that, 1) it highlights the dynamics and epidemiological characteristics which have been discussed in press and health policies; It highlights qualitatively how lockdown policies have decreased the spread of the virus and provides prediction and explanation of an upcoming apex, 2) it fits real data provided for the New York state and 3) it fits the data of NJ state by considering coupled equations taking into account the daily fluxes between NY and NJ.
cord-104158-l7s2utqb	2020	• We demonstrate that the SIR dynamics map to the well-known Lotka-Volterra (LV) system [8] on interpreting infectious patients as predators and susceptible contacts (i.e., the product of transmission rate and susceptible population) as the prey under specific conditions on the transmission rate. • We derive optimal control policy for transmission rate (CoSIR) using control-Lyapunov functions [45] based on the energy of the system, that is guaranteed to converge to the desired equilibrium, i.e., target infectious levels from any valid initial state. We also discuss extensions to compartmental model variants that involve an incubation period (e.g., delayed SIR, SEIR) as well as control of the infectious period that is influenced by testing and quarantine policy. We now consider the problem of controlling the transmission rate β for the LVSIR model (Fig 2(c) ) to nudge the infectious levels to a desired equilibrium.
cord-121428-79wyxedn	2020	The kinetic description leads to study the evolution over time of Boltzmann type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical compartmental model in epidemiology. Inspired by the model considered in [14] for describing a social attitude and making use of the SIR dynamics, we present here a model composed by a system of three kinetic equations, each one describing the time evolution of the distribution of the number of contacts for the subpopulation belonging to a given epidemiological class. Once the system of Fokker-Planck type equations has been derived, in Section 4 we close the SIR-type system of kinetic equations around the Gamma-type equilibria to obtain a SIR model in which the presence and consequently the evolution of the social contacts leads to a non-linear incidence rate of the infectious disease satisfying the compatibility conditions introduced in [33] .
cord-131678-rvg1ayp2	2020	
cord-140977-mg04drna	2020	Based on a proposed parametrization model appropriate for implementation to an epidemic in a large population, we focused on the disease spread and we studied the obtained curves, as well as, we investigated probable correlations between the country''s characteristics and the parameters of the parametrization. where the function c(t) applied in an epidemic spread represents the rate of the infected individuals as the new daily reported cases (DRC) and coincides with the function I(t) in the SIR model, as we can see in the following. The more analytical approach, in the general case from the mathematical point of view, comes from the fundamental study of the epidemic growth and includes a number of terms in a form of double summation related to the inverse Laplace Transform of a rational function given in [8] , referring to the "Earlier stages of an epidemic in a large population".
cord-146213-924ded7t	2020	We propose a rigorous hybrid model-and-data-driven approach to risk scoring based on a time-varying SIR epidemic model that ultimately yields a simplified color-coded risk level for each community. We show how this risk score can be estimated using another useful metric of infection spread, $R_t$, the time-varying average reproduction number which indicates the average number of individuals an infected person would infect in turn. First, we obtain the daily effective reproduction number R t of a time-varying SIR model as well as the corresponding confidence Interval. Our code for infection risk calculation uses this data in conjunction with a time-varying SIR-based Bayesian mathematical model to obtain risk estimates and prediction for different communities. A well-known parameter in the classical SIR model is called R0, the effective reproductive number, which measures the average number of infections caused by infectious individuals at the beginning of the epidemic.
cord-152881-k1hx1m61	2020	This paper aims to help decision making by building a mathematical epidemic model, estimating it using the up-to-date data of COVID-19 cases around the world, making out-of-sample predictions, and discussing optimal policy and economic impact. Due to the high transmission rate and lack of herd immunity, in the absence of mitigation measures such as social distancing, the virus spreads quickly and may infect around 30 percent of the population at the peak of the epidemic. 4 Although the fraction of cases c(t) is likely significantly underestimated because infected individuals do not appear in the data unless they are tested, it does not cause problems for estimating the parameter of interest (the transmission rate β) because under-reporting is absorbed by the constant y 0 in (2.3b), which only affects the onset of the epidemic by a few weeks without changing the overall dynamics (see Figure 5 ).
cord-153905-qszvwqtj	2020	En la sección II se considera el mismo modelo SIR pero en el cual se introduce una ya reconocida propiedad de la presente epidemia: la razón k entre el número de infectados observados (por los sistemas de salud) y el total de infectados, es un número que se estima en el intervalo de 0.1 a 0.2 [5, 12] . Sin embargo, los datos de que se disponen para resolver las ecuaciones a partir de sus condiciones iniciales, en muchos casos son solo la población total del país y los números de infectados y recuperados que detecta el sistema de Salud. Consideremos la solución del sistema de ecuaciones (1,2,3) que describa aproximadamente la lista de valores para el número de los infectados activos y sus incrementos diarios observados por el Sistema de Salud de Cuba entre los días 11.03.20 y 3.04.20.
cord-155015-w3k7r5z9	2020	To understand social distancing dynamics it is important to combine basic epidemiology models for viral unfold (like SIR) with game theory tools, such as a utility function that quantifies individual or government forecast for epidemic damage and economy cost as the functions of social distancing. The work proceeds with the presentation of the SIR model with induced transitions (SIRIT), almost analytical treatment of this model, the calibration of an epidemic and economy parameters of the model using time series of active cases and causalities during the 1st wave of COVID-19 in Austria, Israel, and Germany, followed by a discussion of obtained results and their implications. The work introduces SIRIT, a standard Susceptible-Infected-Recovered (SIR) model extended with a utility function that predicts induced transitions (IT) of social distancing. The fit of the first social distancing transitions (deviations from SIR model) after COVID-19 climax in Austria, Germany, and Israel, but, demonstrates that β remains constant during the transitions.
cord-159425-fgbruo9l	2020	This method consists of unsupervised and supervised parts and is capable of solving inverse problems formulated by DEs. We also propose an extension of the SIR model to include a passive compartment P , which is assumed to be uninvolved in the spread of the pandemic (SIRP), presenting a novel machine learning technique for solving inverse problems and improving disease modeling. Then, we introduce the SIRP model and study the pandemic''s evolution by applying the semi-supervised approach to real data, capturing the populations infected and removed by COVID-19 in Switzerland, Spain, and Italy. We examined the effectiveness of the SIRP model and the semi-supervised method by fitting data obtained during the COVID-19 pandemic for three countries: Switzerland, Spain, and Italy [3] . We applied the proposed semi-supervised method on real data to study the COVID-19 spread in Switzerland, Spain, and Italy.
cord-167454-ivhqeu01	2020	Specifically, we employ daily data on the number of individuals positive to COVID-19 at the municipality level, focusing on a period in which the entire country was subject to a lockdown. Note: fit between data and the corresponding SIR model for Lombardy region (left) and the most affected municipalities at the beginning of our period of interest in absolute and per capita terms, respectively (center, right). In order to shed light on this indeterminacy, we proceed to simulating the SIR model for each municipality until the predicted size of the infected population decreases below either (i) 0.1 cases for one thousands inhabitants or (ii) 0.1 cases 6 and we consider the number of periods elapsed as the outbreak duration. We show that in Lombardy, during a lockdown, the basic reproduction number for COVID-19 reacts negatively to the initial size of an outbreak at the municipality level, an effect which cannot be explained by the population having reached herd immunity.
cord-174036-b3frnfr7	2020	Based on simulations of unrestricted COVID-19 diffusion in 19 U.S cities, we conclude that heterogeneity in population distribution can have large impacts on local pandemic timing and severity, even when aggregate behavior at larger scales mirrors a classic SIR-like pattern. These results demonstrate the potential for spatial network structure to generate highly non-uniform diffusion behavior even at the scale of cities, and suggest the importance of incorporating such structure when designing models to inform healthcare planning, predict community outcomes, or identify potential disparities. In this paper, we examine the potential impact of local spatial heterogeneity on COVID-19, modeling the diffusion of SARS-CoV-2 in populations whose contacts are based on spatially plausible network structures. The disease diffuses through the contact network, with currently infectious individuals infecting susceptible neighbors as a continous time Poisson process with a rate estimated from mortality data (see supplement); recovered or deceased individuals are not considered infectious for modeling purposes.
cord-175366-jomeywqr	2020	We analyse the structural identifiability and observability of all of the models, considering all plausible choices of outputs and time-varying parameters, which leads us to analyse 255 different model versions. It should be taken into account that in the present work we are interested in assessing structural identifiability and observability both with constant and continuous time-varying model parameters (or equivalently, with unknown inputs), as explained in Remark 1. The recovered state (R) is almost never observable unless it is directly measured (D.M.) as output; the only exceptions are two SEIR models, 31 and 38, for which R is observable under the assumption of time-varying parameters. Changing β from a constant to a time-varying parameter (or equivalently an unknown input) does not change its observability nor that of the other variables in SIR models. Considering the recovery rate γ (Fig. 7) or the latent period κ (Fig. 6) individually as time-varying parameters generally leads to greater observability, except for model 31 (1) .
cord-184685-ho72q46e	2020	We present the development of a forecasting model using local fine-grained hospital-level data to track the changes in hospitalization and mortality rates owing to reopening orders in the greater Houston area encompassing nine counties in the state of Texas, USA. We demonstrated our new approach using a policy-aware risk-Stratified Susceptible-Infectious-Recovered Hospitalization-Critical-Dead (SSIR-HCD) model, which compared favorably to existing methods (including our neural network latent space modeling, a nonlinear extension of SIR-HCD). • Epidemiology based dynamic models based on grouping populations into a discrete set of compartments (i.e., states), and defining ordinary differential equations (ODE) rate equations describing the movement of people between compartments: SEIR (Susceptible, Exposed, Infected, Recovered) models and their myriad variants are examples in this category. Our SSIR-HCD model forecasts fine-grained COVID-19 hospitalization and mortality by accounting for the impact of local policies.
cord-186927-b8i85vo7	2020	
cord-187462-fxuzd9qf	2020	We started from a simple Susceptible, Infected, Recovered (SIR) model and we added the condition that, after a certain time, the basic reproduction number $R_0$ exponentially decays in time, as empirically suggested by world data. Hence, at a given time t from the beginning of the spreading of the epidemic, I(t) and S(t) are the number of infected people present in the population and the number of vulnerable people that have not contracted the virus yet, respectively, while R(t) is the sum of the ones that have developed immunity (recovered) or deceased and are therefore removed from the susceptible count. Although a simulation with the standard SIR appears to be adequate to describe an epidemic spreading in a sample where all the initial conditions remain constant throughout the period of time, it is not sufficient when it comes to a more complex and realistic situation such as the population of a given country, where the parameters of the model are influenced by other external factors.
cord-187700-716af719	2020	In this paper, we present a susceptible-infected-recovered (SIR) model with individuals wearing facial masks and individuals who do not. The disease transmission rates, the recovering rates and the fraction of individuals who wear masks are all time dependent in the model. We determine the fraction of individual who wear masks by a maximum likelihood estimation, which maximizes the transition probability of a stochastic susceptible-infected-recovered model. We develop a bond percolation analysis to predict the eventual fraction of population who are infected, assuming that parameters of the SIR model do not change anymore. Specifically, we propose a time dependent susceptible-infected-recovered (SIR) model with two types of individuals. From the data published by John Hopkins University [5] we progressively estimate the time dependent disease transmission rates and the recovery rates of the SIR model. In this report, we presented a time dependent SIR model, in which some individuals wear facial masks and some do not.
cord-188958-id9m3mfk	2020	Recently [12] , we have proposed an extension of the SIR model based on dynamical density functional theory (DDFT) [13] [14] [15] [16] that incorporates social distancing in the form of a repulsive interaction potential. In this work, we use the SIR-DDFT model and an extended susceptible-infected-recovereddead (SIRD) model with hysteresis to investigate the effects of various containment strategies with model parameters adapted to the current COVID-19 outbreak in Germany. We compare the effects of face masks and social distancing/isolation and of various threshold values (of the number of infected persons) for imposing and lifting restrictions. This is an important advantage, since it allows to distinguish the effects of two of the main NPIs that were implemented against the COVID-19 outbreak: Face masks and other hygiene measures such as frequent hand washing reduce c, i.e., they decrease the probability of an infection in case of contact.
cord-189434-nrkvbdu4	2020	In both cases, Hamilton''s equations in terms of a suited Hamiltonian as well as Hamilton''s principle in terms of a suited Lagrangian are considered in minimal and extended phase and state space coordinates, respectively. Taken together, the time re-parameterized SIR model obeys Hamiltonian structure and identifies the relation between the gradient G(Z) ∈ R 2 of the Hamiltonian H(Z) in minimal phase space coordinates and the forcing term F (Z) ∈ R 2 as Clearly, the Euler-Lagrange equation in minimal state space coordinates coincides with the single, non-linear ODE formulation of the time re-parameterized SIR model in Eq. 7. Using again Hamilton''s equation Q • = 2Λ · J and exploiting the skew-symmetry of the symplectic matrix, i.e. Λ · J = −J · Λ, recovers once more the already previously established relation between the gradient G(Q) ∈ R 2 of the Hamiltonian H(Q) (in extended state space coordinates) and the forcing term
cord-190296-erpoh5he	2020	This contribution starts in section 2 with a rather elementary reconciliation of the standard SIR model for epidemics, featuring the central notions like Basic Reproduction Number, Herd Immunity Threshold, and Doubling Time, together with some critical remarks on their abuse in the media. To run this hidden model with constant N = S + M + H + C, one needs initial values and good estimates for β and γ, which are not the ones of the Johns Hopkins Data Model of section 3.3. These yield estimates for the parameters of the full SIR model that replace the earlier time series from the Johns Hopkins Data Model in section 3.3. Note that the only ingredients beside the Johns Hopkins data are the number k for the k-day rule, the Infection Fatality Rate γ IF from the literature, and the backlog m for estimation of constants from time series.
cord-190495-xpfbw7lo	2020	We introduce a methodology to guarantee safety against the spread of infectious diseases by viewing epidemiological models as control systems and by considering human interventions (such as quarantining or social distancing) as control input. We consider a generalized compartmental model that represents the form of the most popular epidemiological models and we design safety-critical controllers that formally guarantee safe evolution with respect to keeping certain populations of interest under prescribed safe limits. The parameters β 0 = 0.33 day −1 , γ = 0.2 day −1 and N = 33 × 10 6 of the SIR model were fitted following the algorithm in [10] to the recorded number of confirmed cases I + R [31] between March 25 and August 9, 2020, while the control input u(t), that represents the level of quarantining and social distancing, was identified from mobility data [32] based on the medium time people spent home.
cord-191574-1g38scnj	2020	The series representations of the time evolution of the SIR model with vital dynamics are compared with the exact numerical solutions of the model, and we find that, at least for a specific range of parameters, there is a good agreement between the Adomian and Laplace-Adomian semianalytical solutions, containing only a small number of terms, and the numerical results. In the present work we consider the possibility of obtaining some accurate semianalytical solutions of the equations of the SIR model with vital dynamics by using the Adomian and the Laplace-Adomian Decomposition Methods, respectively. In order to obtain some approximate solutions of the basic evolution equation we will apply to it both the Adomian and the Laplace-Adomian Decomposition Methods, We obtain in each case the recurrence relations giving the successive terms in the Adomian series representation as a function of the Adomian polynomials.
cord-194157-ak2gc3nz	2020	We introduce a simple algorithm that uses the early infection times from a sample path of the SIR model to estimate the parameters this model, and we provide a performance guarantee in the setting of locally tree-like graphs. Section 3 gives the proof of our main result: that our approach provides decent estimates of λ and µ in the setting of locally tree-like graphs. For example, it is known that for every fixed choice of d ∈ N with d > 1 and c ∈ (0, 1 4 ), there exists γ > 0 such that a random d-regular graph on n vertices is (c log d−1 n, n −γ )-locally tree-like with probability approaching 1 as n → ∞; see Proposition 4.1 in [2] . For each n, consider the SIR model on G n with parameters λ and µ, and let E ∞ denote the event that U (∞) contains a vertex of distance greater than r from U (0).
cord-212912-t5v11gs0	2020	Especially a good choice of $beta$ as the number of others that one infected person encounters per unit time (per day) influences the adequateness of the results of the model. We use the European Centre for Disease Prevention and Control [2] as a data for the COVID-19 infected people for the period from December 31st 2019 to April 8th 2020. For the iterative Gauss-Newton method we guessed the respective periods for every country by a visual inspection of the graphs of the infected people over days. The numerical tests showed that a very early start of the lockdown resulting in a reduction of the infection rate β results in the typical Gaussian curve to be delayed by I; however, the amplitude (maximum value of I) doesn''t really change. The interesting points in time are those where the acceleration of the numbers of infected people increases or decreases, respectively.
cord-220116-6i7kg4mj	2020	To determine the robustness of our results and compare the influence of different contact characteristics, we ran our stochastic model on four distinct spatially structured architectures, namely i) regular two-dimensional square lattices, wherein individuals move slowly and with limited range, i.e., spread diffusively; ii) two-dimensional small-world networks that in addition incorporate substantial long-distance interactions and contaminations; and finally on iii) random as well as iv) scale-free social contact networks. For both the two-dimensional regular lattice and small-world structure, a similar sudden drop in the total number of infected individuals ( Figure 6B ) requires a considerably longer mitigation duration: In these dynamical networks, the repopulation of nodes with infective individuals facilitates disease spreading, thereby diminishing control efficacy. In this study, we implemented social distancing control measures for simple stochastic SIR epidemic models on regular square lattices with diffusive spreading, two-dimensional Newman-Watts small-world networks that include highly infective long-distance connections, and static contact networks, either with random connectivity or scale-free topology.
cord-222193-0b4o0ccp	2020	Based on the classical SIR model, we derive a simple modification for the dynamics of epidemics with a known incubation period of infection. We use the proposed model to analyze COVID-19 epidemic data in Armenia. Moreover, it is crucial to consider the final incubation period of the disease to construct a correct model for the COVID-19 case. In this study, we derive a system of integro-differential equations based on the rigorous master equation that adequately describes infection dynamics with an incubation period, e.g., COVID-19. In fact, the real data allows us to measure three main parameters: the exponential growth coefficient at the beginning of the epidemic; the minimum period of time, in which an infected person can transmit the infection; and the maximum period, when an infected person ceases to transmit the infection. In this paper, we introduced a version of SIR model for infection spreading with known incubation period. This model was applied to analyze the COVID-19 epidemic data in Armenia.
cord-229937-fy90oebs	2020	The Death or ''D'' model is a simplified version of the SIR (susceptible-infected-recovered) model, which assumes no recovery over time, and allows for the transmission-dynamics equations to be solved analytically. The evolution of the COVID-19 pandemic in several countries (China, Spain, Italy, France, UK, Iran, USA and Germany) shows a similar behavior in concord with the D-model trend, characterized by a rapid increase of death cases followed by a slow decline, which are affected by the earliness and efficiency of the lockdown effect. These results are in agreement with more accurate calculations using the extended SIR model with a parametrized solution and more sophisticated Monte Carlo grid simulations, which predict similar trends and indicate a common evolution of the pandemic with universal parameters. Additionally, D-model calculations are benchmarked with more sophisticated and reliable calculations using the extended SIR (ESIR) and Monte Carlo Planck (MCP) models -also developed in this work -which provide similar results, but allow for a more coherent spatial-time disentanglement of the various effects present during a pandemic.
cord-241596-vh90s8vi	2020	
cord-243070-0b06zk1q	2020	This results in a system of forward backward stochastic differential equations, which is amenable to numerical solution via Monte Carlo simulations. In this note we study the problem of optimal control of an epidemic modeled by means of a stochastic extension of the SIR model (see Section 2 for definition). The optimal control problem is recast as the stochastic minimum principle problem and formulated in terms of a system of forward backward stochastic differential equations (FBSDE). If a vaccine against the disease is unavailable, we set u 1 = 0 in the equation above, which yields the following controlled process: Using Ito''s lemma, we verify that these two conditions lead to the following nonlinear partial differential equation for the value function, namely the stochastic Hamilton-Jacobi-Bellman equation: Under this running cost function, the optimal policy is to implement a draconian isolation regime, which leads to a rapid drop in infections, while keeping the susceptible fraction of the population at a very high level.
cord-247144-crmfwjvf	2020	We observe that the time series of active cases in individual countries (the difference of the total number of confirmed infections and the sum of the total number of reported deaths and recovered cases) display a strong agreement with polynomial growth and at a later epidemic stage also with a combined polynomial growth with exponential decay. Note that the inflection points of the function I = I(t) are located at T ± I = (α ± √ α) T G , particularly the time t = T − I plays an important role in the observed epidemic data as it corresponds to a moment at which the growth of the number of active cases reaches its maximum and starts to decrease. The simple PGED model, i.e., the universal scaling 7 and nonlinear fitting of the parameters from the data, can be used for as a predictive tool for the number of the reported active cases, particularly in countries in the growth phase.
cord-248050-apjwnwky	2020	title: Effects of social distancing and isolation on epidemic spreading: a dynamical density functional theory model We present an extended model for disease spread based on combining an SIR model with a dynamical density functional theory where social distancing and isolation of infected persons are explicitly taken into account. In this article, we present a dynamical density functional theory (DDFT) [18] [19] [20] [21] for epidemic spreading that allows to model the effect of social distancing and isolation on infection numbers. While DDFT is not an exact theory (it is based on the assumption that the density is the only slow variable in the system [50, 51] ), it is nevertheless a significant improvement compared to the standard diffusion equation as it allows to incor-porate the effects of particle interactions and generally shows excellent agreement with microscopic simulations.
cord-253461-o63ru7nr	2020	Intent of this research is to explore how a specific class of mathematical models namely Susceptible-Infected-Removed model can be utilized to forecast peak outbreak timelines of COVID-19 epidemic amongst a population of interest starting from the date of first reported case. With this in mind, SIR model is explored in current research to forecast peak COVID-19 outbreak over a large population in India. DISCUSSION This research was conducted to evaluate the feasibility of application of SIR model to predict peak COVID-19 outbreak timeline from the date of first reported case for the 10 largest states in India which together constitute more than 74% or almost 3/4 th of total population in India. For 9 out of 10 largest states in India included in the research, chosen SIR model could predict peak outbreak timeline from the date of the first reported case with error of +/-6 days or less and Standard Deviation (SD) in error = 5.83 day.
cord-258018-29vtxz89	2020	The data in [29] for China, South Korea, India, Australia, USA, Italy and the state of Texas (communities) are organised in the form of time-series where the rows are recordings in time (from January to June, 2020), and the three columns are, the total cases I d tot (first column), number of infected individuals I d (second column) and deaths D d (third column). Assuming the published data are reliable, the SIR model (1) can be applied to assess the spread of the COVID-19 disease and predict the number of infected, removed and recovered populations and deaths in the communities, accommodating at the same time possible surges in the number of susceptible individuals. In this work, we have augmented the classic SIR model with the ability to accommodate surges in the number of susceptible individuals, supplemented by recorded data from China, South Korea, India, Australia, USA and the state of Texas to provide insights into the spread of COVID-19 in communities.
cord-264248-wqkphg2e	2020	For the study in hand, this model is used to forecast the infection rate if the lockdown measures are reduced by 25% on the 1st of June 2020 or the 1st of July 2020; hence, the impact of delaying this step is also investigated. 23.20111526 doi: medRxiv preprint in lockdown measures and predicts the evolution of the number of infected cases until the end of 2020 following the assumed conditions. . https://doi.org/10.1101/2020.05.23.20111526 doi: medRxiv preprint more affected by reopening as they have not fully conquered COVID-19 yet; hence, the hasty reduction of quarantine measures might lead to even higher infection rates that has happened before during the Spanish flu [21; 22] . In conclusion, this study offers a quantifiable prediction of how reducing the lockdown measures shall lead to the second wave of COVID-19 in the United States, Germany, the United Kingdom, Italy, Spain, and Canada.
cord-270519-orh8fd1c	2020	
cord-273429-dl6z8x9h	2020	Figure 2 shows the comparison of the model-estimated infected and recovered case counts with actual Covid-19 data for the highest affected European countries as of 1 June 2020, namely: Russia, UK, Spain and Italy, in that order. Figure 6 shows reasonably good match between the model-estimated infected and recovered case counts with actual Covid-19 data for the highest affected North American states (including states from Mexico, the United States, and Canada) as of 1 June 2020, namely: New York, New Jersey, Illinois and California. Figure 10 shows reasonably good match between the model-estimated infected and recovered case count with actual Covid-19 data for the highest affected Asian countries as of 1 June 2020, namely: India, China and South Korea. Figure 13 shows reasonably good match between the model-estimated infected and recovered case count with actual Covid-19 data for the highest affected South American countries as of 1 June 2020, namely: Brazil, Chile and Peru.
cord-277094-2ycmxcuz	2020	Since the epidemic of COVID-19 was declared in Wuhan, Hubei Province of China, and other parts of the world, several studies have been carried out over several regions to observe the development of the epidemic, to predict its duration, and to estimate its final size, using complex models such as the SEIR model or the simpler ones such as the SIR model. Also, as the number of infected cases is increasing, it is necessary for modellers to estimate the severity of the epidemic in terms of the total number of people infected, the total number of confirmed cases, the total number of deaths, and basic reproduction and to predict the duration of the epidemic, the arrival of its peak, and its final size. Our simulation study on the optimization of the final size of COVID-19 epidemic evolution in the Kingdom of Morocco, with the SIR model, has allowed us to accurately predict the peak of the infected and death cases (Table 2) , although the number of people tested is very low, about 3,079, until 31 March 2020.
cord-279112-ajdkasah	2020	This comment shows that data regarding cumulative confirmed cases from the coronavirus COVID-19 disease outbreak, in the period December 31, 2019–June 29, 2020 of some countries reported by the European Centre for Disease Prevention and Control, can be adjusted by the exact solution of the Kermack – McKendrick approximation of the SIR epidemiological model. In a recent article published in this journal [1] , after some (unnecessary) considerations, the author presents the logistic function (equation (8) in [1] ) as an alternative solution of the differential equation known as the Kermack and McKendrick 1927 approximation [2] of the SIR epidemiological model [3, 4] in order to fit data regarding the cumulative confirmed of COVID-19 infected cases from some countries.
cord-280683-5572l6bo	2020	
cord-288884-itviia7v	2020	We fit this model to the latest epidemic data with an approximate Bayesian computation (ABC) technique. Within this SIR-ABC framework, we extrapolate long-term infection curves for several regions and evaluate their steepness. Armed with the ability to generate stochastic infection and recovery curves from starting parameters, we turn to fitting the starting parameters from real-world epidemic data. We therefore employ an approximate Bayesian computation (ABC) technique to compare our simulations to observations and recover the posterior distributions of β and γ (Figure 1 ). The general goal of ABC is to sample the posterior distributions of simulation parameters such that the simulations match the observed data. Given a simulated epidemic and the observed data, we quantify the difference between both the infectious and recovered population curves to obtain a distance In this proof-of-concept study, we apply approximate Bayesian computation to fit stochastic epidemic models to real world data.
cord-289325-jhokn5bu	2016	
cord-293148-t2dk2syq	2020	In the more realistic scenario of a core-periphery structure with multiple locations, we unexpectedly find that the time spent by agents in their base location does not influence the endemic prevalence in the SIS model and the epidemic size in the SIR model, which are measures of the overall fraction of population that is affected by the disease. Here, we propose a one-dimensional model that provides some analytical intuitions on the influence that the randomness α, the probability of jumping outside the base location p, and the presence of a core-periphery structure have in the evolution of SIS and SIR epidemic processes. We consider the two-dimensional agent-based model and numerically study the influence of the randomness α, the probability of jumping outside the base location p, and the presence of a core-periphery structure on the evolution of SIS and SIR epidemic processes.
cord-297161-ziwfr9dv	2020	The model thereby enables a dynamic inspection of the pandemic and allows estimating key figures, like the number of overall detected and undetected COVID-19 cases and the infection fatality rate. Such models allow describing the dynamics of mutually exclusive states such as Susceptible (S) which for COVID-19 is assumed to be the entire population of a country, a region or city, the number of Infected (I) and Removed (R) that often combines (deaths and recovered), as well as the number of Exposed (E) for SEIR models. As the number of performed tests strongly influences the dynamic analysis of the COVID-19 pandemic in a country or region, we developed a novel SIR based epidemiological model (SIVRT, Figure 1 ) which allows the integration of this key information. In summary, the novel testing informed SIVRT model structure allows to describe and analyze the COVID-19 pandemic data of Luxembourg in dependency of the number of performed tests.
cord-303030-8unrcb1f	2020	In this paper we discuss the different effects of these ingredients on the epidemic dynamics; the discussion is conducted with the help of two simple models, i.e. the classical SIR model and the recently introduced variant A-SIR (arXiv:2003.08720) which takes into account the presence of a large set of asymptomatic infectives. In the SIR model [1] [2] [3] [4] [5] , a population of constant size (this means the analysis is valid over a relatively short time-span, or we should consider new births and also deaths not due to the epidemic) is subdivided in three classes: Susceptibles, Infected (and by this also Infectives), and Removed. Acting on α or on β to get the same γ will produce different timescales for the dynamics; see Fig. 1 , in which we have used values of the parameters resulting from our fit of early data for the Northern Italy COVID-19 epidemic [7] .
cord-310863-jxbw8wl2	2020	We use the procedure to fit a set of SIR and SIRD models, with time dependent contact rate, to Covid-19 data for a set of 45 most affected countries. We find that SIR and SIRD models with constant transmission coefficients cannot fit Covid-19 data for most countries (mainly because social distancing, lockdown etc., make those time dependent). Some of the most important problems related to Covid-19 research are (1) estimating the controlling parameters of the pandemic, (2) making short term predictions using mathematical-statistical modeling which can help in mitigating policies (3) simulating the growth of the epidemic by taking into account as many contributing effects as possible and (4) quantifying the impact of mitigation measures, such as lockdown etc [ea20j] . One of the main reasons to consider these models has been that the Covid-19 data is available only for the Susceptible, Infected, Recovered and Dead compartments (for the notations used here and other places in the present work see table (1)).
cord-311183-5blzw9oy	2020	
cord-314725-og0ybfzf	2020	Abstract This work deals with the inverse problem in epidemiology based on a SIR model with time-dependent infectivity and recovery rates, allowing for a better prediction of the long term evolution of a pandemic. The method is used for investigating the COVID-19 spread by first solving an inverse problem for estimating the infectivity and recovery rates from real data. This work aims to create a method that can accurately identify the time dependent parameters of the SIR system using real data and then use the computed parameter values to predict the spread of the epidemics. The inverse problem for estimating the time-dependent transmission and removal rates in the SIR epidemic model is derived and solved. The inverse problem for estimating the time-dependent transmission and removal rates in the SIR epidemic model is derived and solved.
cord-316393-ozl28ztz	2020	
cord-318525-nc5rtwtd	2020	Studies on the outbreak of COVID-19 in the Hubei province and the rest of mainland China show that the temporal evolution of confirmed cases can be classified in three distinct regimes: 1) an initial exponential growth phase, 2) an extended phase of power law growth kinetics indicative of a small world network structure, with a universal growth exponent of µ ≈ 2.1, and 3) a slow inflection to a plateau phase, following a parabolic profile in double logarithmic scale [1] . This model was recently extended to include symptomatic quarantined individuals (X), resulting in the ''SIR-X'' model, which was successfully applied to predict the spreading kinetics and assess containment policies for COVID-19 in China [4] , and is currently being used to monitor the number of confirmed COVID-19 cases in various countries [5] .
cord-318688-ditadt8l	2020	
cord-319435-le2eifv8	2020	To estimate the impact of social distancing we assumed eight different scenarios, the predicted results confirmed the positive impact of this type of control strategies suggesting that by strict social distancing and lockdown, COVID-19 infection can be under control and then the infection cases will steadily decrease down to zero. In this study, we attempt to estimate the final epidemic size of COVID-19 using the classic compartmental susceptible-infected-recovered (SIR) model [9] . The SIR model presents the increase of decrease information of an outbreak based on some initial data i.e. total given population (N), the infection rate of the infectious disease (β), the recovery rate of the disease (Ɣ), initial susceptible population (S0), initial infected population (I0) and the initial recovered population (R0). The SIR model base prediction of infection curve was compared with the confirmed cases ( Figure 02 ).
cord-320912-jfeu4tho	2020	
cord-321984-qjfkvu6n	2020	Despite relying on a valid infectious diseases mechanism, deterministic approaches have several drawbacks: (i) the actual population in each compartment at a given time is never accurately measured because we only obtain an observation around the mean; (ii) the nature of disease transmission and recovery is stochastic on the individual level and thus never certain; and (iii) without random component in the model, it is neither possible to learn model parameters (e.g. R 0 ) from available data nor to assess prediction uncertainty. In an early stage of the current COVID-19 pandemic, the daily infection and death counts reported by health agencies are highly influenced by the availability of testing kits, reporting delays, reporting and attribution schemes, and under-ascertainment of mild cases in public health surveillance databases (see discussions in Angelopoulos et al., 2020; Banerjee et al., 2020) ; both disease transmission rate and time to recovery or death are also highly uncertain and vary by population density, demographic composition, regional contact network structure and non-uniform mitigation schemes (Ray et al., 2020) .
cord-324993-hs66uf1u	2020	Using the Susceptible Infectious Recovered (SIR) model, the spread of the COVID 19 under 3 intervention scenarios (low, moderate, high) is simulated and predicted in indigenous tribe populations. While, in the scenario of high intervention, the COVID 19 peaks can be reduced to values ranging from 53% to 15% .To conclude, the simulated interventions tested by SIR model have reduced the pandemic peak and flattened the COVID 19 curve in indigenous populations. The COVID 19 SIR model of indigenous tribe populations living in remote Yasuni rainforest enclaves with simulated 25% (low), 50% (moderate), and 75% (high) interventions (x axis: days, y axis: proportion of total population). 1101 The indigenous tribe populations and COVID 19 cases in Amazon rainforest enclaves including Lagartococha, Callarú, and Yasuni are presented in the Figure 2 . The proposed SIR model in this study simulates the widespread of COVID throughout indigenous tribe populations living in remote Lagartococha and Yasuni rainforests.
cord-326631-7gd3hjc3	2006	More recent analyses have established that the standard final size formula is valid regardless of the distribution of infectious periods, but that it fails to be correct in the presence of certain kinds of heterogeneous mixing (e.g., if there is a core group, as for sexually transmitted diseases). We then proceed to generalize these results in three new directions, showing that the standard formula remains valid (i) regardless of the number of distinct infectious stages, (ii) if the mean contact rate is itself arbitrarily distributed and (iii) for a large class of spatially heterogeneous contact structures. Since this substage trick can be applied equally well to any infectious stage, Anderson and Watson''s (1980) conclusion that the final size in an SIR model with Gamma distributed infectious periods is given by the usual formula (5) now generalizes to an arbitrary number of stages, each with Gamma distributed durations.
cord-332922-2qjae0x7	2020	
cord-335141-ag3j8obh	2020	The British Association of Plastic, Reconstructive and Aesthetic Surgeons, the British Society for Surgery of the Hand and the Royal College of Surgeons of England, have all issued guidance: both encouraging patients to avoid risky pursuits, which could result in accidental injuries and to members how to prioritise and optimise services for trauma and urgent cancer work. We have adapted our Hand Trauma Service to a ''One Stop Hand Trauma and Therapy'' clinic, where patients are assessed, definitive surgery performed and offered immediate post-operative hand therapy where therapists make splint and give specialist advice on wound care and rehabilitation including an illustrated hand therapy guide. Local assessment of our practice is ongoing but we have found that this model has enabled a cohort of vulnerable plastic surgery trainees to successfully continue to work whilst reducing the risk of exposure to COVID-19 and providing gold standard care for patients.
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