key: cord-0801406-9n23fjnz authors: Comunian, Alessandro; Gaburro, Romina; Giudici, Mauro title: Inversion of a SIR-based model: A critical analysis about the application to COVID-19 epidemic date: 2020-08-12 journal: Physica D DOI: 10.1016/j.physd.2020.132674 sha: 79c8e154907a972d60d8c04d2398bdcbeb644144 doc_id: 801406 cord_uid: 9n23fjnz Calibration of a SIR (Susceptibles-Infected-Recovered) model with official international data for the COVID-19 pandemics provides a good example of the difficulties inherent the solution of inverse problems. Inverse modeling is set up in a framework of discrete inverse problems, which explicitly considers the role and the relevance of data. Together with a physical vision of the model, the present work addresses numerically the issue of parameters calibration in SIR models, it discusses the uncertainties in the data provided by international authorities, how they influence the reliability of calibrated model parameters and, ultimately, of model predictions. Epidemic modeling is usually performed with compartmental models, often called SIR (Susceptibles-Infected-Recovered) models, which are claimed to go back to the work by Ronald Ross and Hilda P. Hudson more than one century ago [1, 2] and, ten years later, to the work of Anderson Gray McKendrick and 5 William Ogilvy Kermack [3, 4] . This class of models shares several characteristics with models of population dynamics and with conceptual lumped models this work does not aim to provide forecasts of the pandemic evolution at this 70 stage. It is in the authors' opinion that the quality of the data that are currently available does not allow to perform reliable forecast and model outcomes should be used with high prudence. It will be material of future work to further develop and refine the SIR model presented in this work and to address the issue of providing forecasts of the epidemics, when the data will be better 75 understood. For instance, the correct number of infected people "remains unknown because asymptomatic cases or patients with very mild symptoms might not be tested and will not be identified", as recognized, e.g., by [28] . In an interview published on March 23rd, 2020, by the Italian newspaper "La Repubblica", 80 Angelo Borrelli, head of Dipartimento della Protezione civile (national civil protection department) stated that a ratio of one certified case out of every 10 total cases is credible. Furthermore, different criteria have been adopted by different countries and institutions to define the various categories of infected, recovered and deceased people by or with COVID-19. This fact has been widely 85 recognized as a cause of uncertainty in the collected data. Finally, censorship on COVID-19 pandemics is reported by journalists and organizations in some of the countries affected by the pandemic. The paper is organized as follows. Section 2 contains the description of the SIR model in both the continuous and the discrete case (subsection 2.1) together 90 with a precise formulation of the inverse problem addressed in this paper in the discrete setting (subsection 2.2). In particular, inverse modeling, i.e., model calibration, is set up and discussed computationally within the framework proposed by [22] . The results obtained by applying our SIR model to the pandemic are shown in section 3. Section 4 is devoted to a discussion about the 2. Methods and materials 2.1. The continuous and the discrete models 100 We start by defining the objects involved in the continuous SIR model considered in this paper. Definition 2.1. We denote by S(t), I(t), R(t) and D(t) the number of susceptible, infected, recovered and deceased individuals of the population under study at time t, respectively, for t varying in some interval I ⊂ R. Here D includes 105 only those individuals who died while being infected, whereas the total population, at time t, is given by P (t) = S(t) + I(t) + R(t). Definition 2.2. We denote by β and δ the birth and death rate, respectively, under normal conditions, i.e., without considering deaths caused by the epidemic. We also denote by γ, ρ and φ the infection, recovery and fatality rate, 110 respectively. The dimension of these coefficients is [time −1 ] . Notice that φ accounts for the deaths related to the pandemic, i.e., it represents the increase in the death rate due to the pandemic. The normal death rate is considered through δ. Note that β and δ in definition 2.2 are rarely considered in epidemic mod-115 eling, as the time variation of P due to the normal evolution of the population is either negligible or smoother than its variation due to the presence of an epidemic. This is due to the fact that typical values of β and δ are smaller than the ones of γ, ρ and φ by one or more orders of magnitude, as shown in subsection 3.2. We keep birth and death rates in the model, in order to facilitate 120 a thorough discussion of the assumptions behind this model, which is given in section 4. We make the following assumptions. Assumption 2.1. The coefficients β, δ, γ, ρ and φ are assumed to be constant. is given by S/P , whereas (I + R)/P is the fraction of those persons who cannot be infected, as it is also assumed that recovered people are immunized. The following equations, based on the seminal papers [1, 2, 3, 4] , are used to describe the time evolution of S, I, D and R: and D(t ini ) = 0, where t ini ∈ I ⊂ R is the time at which the first individual is infected and P ini is the population at t ini . Notice that from equations (1) to (4) one can easily deduce dP and if we couple (5) with (2) where 6 J o u r n a l P r e -p r o o f We can approximate (6) to the simple system of autonomous linear ordinary differential equations for some h, 0 < h 1. This rough approximation is justified by thinking that, for h small enough, I(t) P ini S(t) and therefore IS/P I in (6). The system describes the population evolution taking into account demographic aspects only, i.e., in absence of the perturbation caused by epidemics and by assuming that the birth and death rate are constant, whereas    dI dt = (γ − α)I, in (t ini , t ini + h), describes the time evolution of the number of infected cases during a short time 145 after the beginning of the infection at time t = t ini . The solution to (10) , and, for h small enough, its linear approximation , gives a first rough explanation about why, during the first phases of the epidemics, i.e., for t t ini , the number of infected individuals, I(t), seems to grow linearly. This fact motivates the difficulties in 150 the design of an efficient early warning system. In fact, once I(t) increases to a significant level to be detected, the exponential growth had already kicked in and the containment measures can be effective only if quite drastic. The discrete model is a simple forward-time finite-differences discretization of equations (1) to (4) . For n ∈ Z, we denote the discrete time steps, at a Definition 2.3. We denote by S n , I n , R n and D n the number of susceptible, infected, recovered and deceased individuals of the population under study at time t n , respectively, for n = n ini , . . . , n ini + N (mod) − 1, where n ini is such 160 that t ini = n ini ∆t and N (mod) is the number of modeled time steps. The total population at time t n is given by P n = S n + I n + R n . Then the resulting algebraic iterative equations are of the form for n = n ini , . . . , n ini + N (mod) − 1, with initial conditions and the discrete counterpart of (5) is Here the time spacing ∆t = 1 day, in agreement with the sampling of the available data set on COVID-19 pandemic (see section 2.3). Equations (11) are implemented in a specifically designed code, developed using the Python programming language. The choice n ∈ Z allows to simplify the notation adopted in the formulation 170 of the inverse problem in section 2.2. It is important to notice that n = 0, i.e., t 0 = 0, represents the first day for which epidemic data are available and in general it does not coincide with n = n ini , which corresponds to t ini , the day when the first person was infected in a given nation, according to our model. We will call t 0 = 0 (n = 0) and t ini (n = n ini ) the monitoring initial time and the 2.2. The inverse problem: model calibration As stated in the introduction, the inverse problem addressed here is defined in the discrete setting by making use of the conceptual framework and the 180 notation of [22] . The numerical task in treating the inverse problem consists in solving iteratively (11) and matching such solutions with the data collected within a certain time-frame [t min , t max ). Such (discrete) time-varying vectorsolutions s n are collected in an array s, called the state of the system n , s n , s where N (mod) and n = n ini have been introduced in definition 2.3. s is the model 185 outcome used to forecast the number of infected, recovered and dead individuals. To this end, we also introduce the model forecast, an array y defined by for some n min , n max , with n ini ≤ n min < n max ≤ n ini + N (mod) . The available data are collected in an array d. In the specific case considered here, the subset of data denoted by d ⊂ d includes the cumulative number of the confirmed 190 infected cases, together with the number of the recovered and dead persons, released by health official organizations n , d d can include also other data, e.g., demographic data used to infer the values of some model parameters (β and δ). n = 0 represents the so-called monitoring initial time introduced in 2.1, which corresponds to the first day for which epidemic data d are available; recall that, in general, it does not coincide with 200 the day n = n ini when the first person was infected in a given country. Model calibration requires that the model forecast be close to a calibration target, an array t that collects the values which should be attained by the model forecast, if the model were physically "correct" and the model parameters were "optimal". In this specific case t is defined by where I (ref) n is given by (17) and n min , n max are such that 0 ≤ n min < n max ≤ . The model parameters are placed in an array p: where R + = (0, +∞) and we recall that ∆t = 1 day and P ini is the model initial population introduced in section 2.1. If we summarize the algebraic equations in the discrete model (11) together 210 with the initial conditions (12) with the forward problem can be stated as: given p, find the unique state s = g(p) that solves (20) . In other words, given the parameters p, the solution to the forward problem will give the state of the system, s. In order to introduce the corresponding inverse problem, it is convenient to write p as where p (fix) = (β, δ, ∆t) , p (cal) = (ρ, φ, γ, n ini , P ini ) . p (fix) and p (cal) include the model parameters whose values are fixed before the simulation and the model parameters whose values are obtained from the solution of the underlying inverse problem, which is yet to be stated, respectively. Remark 2.1. Some remarks on p, y and t are in order. 1. The array of fixed parameters is a function of d: 2. The model forecast y is a function of s, p and d: y = y (d, s, p); 3. t may depend on d and p (fix) , but must be independent of p (cal) : t = t d, p (fix) . The misfit between model predictions and the target values is computed by means of the following objective function: where O (i) for i = 1, 2, 3, where ξ ≥ 1 is a threshold-and-weight parameter and n min , n max are such that In other words, O y,t is the sum of three functions, each of which considers one of the three reference quantities, separately. The model calibration is then performed by solving the following inverse problem: Given p (fix) and d, given the solution s = g (p) to (20) , determine y (d, g (p) , p), t and find p (cal) , such that 235 p (cal) = arg min J o u r n a l P r e -p r o o f In other words, the objective of model calibration is to find the parameter values which best fit the reference data in a given time interval, n min ≤ n < n max . The parameter ξ plays a double role. First of all, it is a threshold which keeps positive the denominator of the quantity appearing in (24) , even in the particular case when t y,t is nothing but the root-mean-squared relative difference between target, t y,t reduces to the standard root-mean-squared error. Notice that the latter condition implies that the value of ξ could be very great. A sensible upper limit 250 for ξ, ξ max , could be for instance given by the upper bound of P ini , which is identified with the total population of the country for which the simulation is performed, as shown in subsection 3.2. It is worth stressing that the explicit use of an interval n min ≤ n < n max for the definition of t, y and the objective function, although somehow cumbersome, 255 is useful to assess changes in the physical parameters with time. Some examples of it will be shown in section 3.2. The application of the model introduced in section 2.1 and of the model calibration introduced in section 2.2 can be attempted thanks to publicly available 260 data on COVID-19 pandemic. The application will be performed at national level, i.e., the considered population will be the whole population of some countries. For each country, the array d is populated with data coming from two basic sources. Data on COVID-19 pandemic are available from the GitHub repository man-265 aged by the Johns Hopkins University [29] . This is a collection of publicly available data from multiple sources, which are processed and delivered by the Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE). Notice that the data are provided to the public strictly for educational and academic research purposes. The data are updated daily and the files used in this Figure 1 . The optimization algorithms that have been tested are based on constrained minimization, so that some bounds on p (cal) should be pre-280 scribed. Best results have been obtained by global optimization with the function differential evolution [30] . Since this function implements a stochastic algorithm, the pseudo-code of Figure 1 shows that several runs of the algorithms are executed in an easily parallelized loop. Aside from China, for which the starting phase is not reported, since the virus diffusion started earlier than the first date for which data are available in the data set, the number of confirmed cases (plots A in Figure 2 ) shows a first slow increase, followed by an exponential increase and possibly a slowdown after few weeks. It is highly questionable whether this behavior is related to the First of all, the behavior of the model is shown with test case 1, which 325 includes four model runs for which all the model parameters, but ρ, are kept fixed. Exploratory tests were run with diverse parameters sets. However, the selection of the parameter sets used to obtain the results presented here was inspired by a preliminary calibration test performed on the data available for Italy. In this sense, these parameters, listed in Table 1 , could be considered which differ only for the value of ρ, the total population has only a limited variation, so that approximating P to a constant value could appear reasonable. Nevertheless, the term used to compute the infection rate is directly proportional to both I and S and inversely proportional to P so that it introduces a non- In particular, this paper is focused on the results obtained with data from Italy, but the same qualitative remarks hold also for the application to data The basic properties of the performed tests are listed in Table 2 . The comparison between reference and fitted time series for test A, which is to be considered as the ideal one, because all the data are used and the standard settings are applied, is shown in Figure 5 . The discrepancy between reference and modelled values in log scale is greater for the initial phase of the epidemic; the model Notice that for tests B, C and D three subsets of data are used, corresponding to three non overlapping time intervals, each of which is 33-days-long. In particular, the first day for which data are available is January 22, 2020 and 395 the data series used in this paper ends on May 2, 2020. Therefore, the data set for test B ends on February 24, 2020, the data set for test C covers the interval 20 J o u r n a l P r e -p r o o f D e c 1 6 D e c 2 3 D e c 3 0 J a n 0 6 J a n 1 3 J a n 2 0 J a n 2 7 Italy with the parameters obtained by solution of the inverse problem for test A (see Table 2 ). The vertical dotted black lines delimit the time-frame of the data set used for model calibration, i.e., they correspond to t min and tmax. (24) is nothing but the root-mean-squared relative difference between reference and modeled values of I, R and D for i = 1, 2, 3, respectively. Therefore, test E has been designed in order to assess the dissimilarities in the inversion results due to the application of different objective 405 functions, by comparing tests A and E, which in ultimate essence are founded on absolute versus relative errors between model predictions and calibration targets, respectively. Test F is based on a subset of the data, in particular, for this test the number of dead patients only is fitted; the rationale behind this test is that D (ref) should be less uncertain than the other data of d . Finally, 410 test G is an attempt to consider the hints raised by several authorities and researchers, suggesting that official numbers could be heavily underestimated. In this test, it is assumed that the number of infected and recovered persons be 10 21 J o u r n a l P r e -p r o o f times greater than those reported in official documents; analogously the number of deaths is assumed to be twice the official value. Notice that this does not 415 mean that these estimates are more accurate than the official ones; test G is designed to be a first attempt of sensitivity analysis, by considering a data set very different from the reference one, which is used for test A. to find a minimum, also by taking into account possible bounds on p (cal) . The reader is referred to the on-line SciPy's documentation for details; here, it suffices to recall that tested methods include Nelder-Mead simplex algorithm [32] and conjugate-gradient based methods [33] . The bounds have been assigned on the basis of preliminary gross estimates from available data and they are listed 425 in Table 3 . In particular, notice that the upper bound for P ini is 10 8 , hence here ξ max = 10 8 , i.e., ξ ∈ [1, 10 8 ]. Several runs have been conducted with a routine for local minimization and the best results were obtained with the L-BFGS-B method, which is a variation of the BroydenFletcherGoldfarbShannon (BFGS) algorithm [33] to reduce As a consequence, these preliminary tests called for the application of a global 440 minimization algorithm. Global minimization by application of differential evolution [30] , even with the default settings, yielded good results, which are listed in Tables 4 and 5 . The mean value of each parameter and the corresponding standard deviation have been estimated after 10 runs of this stochastic algorithm, for which the 445 random initializing seed introduces variations among the returned results. When looking at Table 5 , it is important to recall again that t ini and P ini are integer numbers, but in the table the averages and the relative standard errors are computed after 10 runs and this explains the float numbers notation. Besides the optimal values of p (cal) listed in Tables 4 and 5 , it is important 450 and useful to consider also some properties of the inversion procedure for each test; they are listed in Table 6 . Table 4 shows that, apart from few tests, the optimal values of γ, ρ and φ are relatively similar, sharing the same order of magnitude and the relationship 23 J o u r n a l P r e -p r o o f Test Table 4 would also suggest that at the pandemic peak a large fraction of the population would have already been infected, and possibly recovered. In fact, in the continuous model, when I reaches its maximum value we have and, after simple algebraic manipulations, where α is defined in (7). The calibration results listed in Table 4 show that α is about one order of magnitude smaller than γ. In particular for the calibration tests performed in this study (Table 4 ) (γ − α) · γ −1 assumes a relatively high value, close to 0.8. Two facts should be mentioned about the results of test A shown in Table 5 : first, t ini < 0, i.e., it seems that the infection started before the official appearance of the first confirmed case; second, P ini is close to the lower bound chosen in Table 3 , so that the model predicts that the population which has been involved in the infection could be relatively small. These qualitative remarks Table 6 shows that tests A, D and G are those for which the results of different runs are more consistent with each other. This is important, because it 25 J o u r n a l P r e -p r o o f shows that the identification of p (cal) with the proposed approach appears to be robust for these tests. On the other hand, for the remaining tests, it is important to carefully check the outcomes of each single run. In fact, the initial seed could 490 introduce some bias which cannot be overcome by the differential evolution routine with its default settings and the final result could yield a local minimum, instead of the global one. This is illustrated by the comparison in Figure 6 , which shows the results of test F for the optimal parameters and those averaged among the 10 runs and listed in Tables 4 and 5 . This test was designed to fit the data 495 on the deceased people, as shown in Figure 6 (a); the fit seems extremely good, in fact, the two green curves overlap almost perfectly for a large time interval. On the other hand, from Figure 6 (b) it is evident that some of the inversion runs yielded parameters which do not permit to properly and satisfactorily reproduce the data. From Table 6 , it is also apparent that the objective function is computed a Italy with the parameters obtained by solution of the inverse problem for test F (see Table 2 ): (a) optimal parameters corresponding to the global minimum; (b) parameters averaged among the 10 inversion runs (Tables 4 and 5 ). The vertical dotted black lines delimit the time-frame of the data set used for model calibration, i.e., they correspond to t min and tmax. where δ is the death rate introduced in definition 2.2. An attempt to account for time-varying parameters, depending on varying 535 conditions, can be found in [36] . The assumption of homogeneity could be relaxed by considering distributed models, similar to those applied to the study of transport phenomena, e.g., for diffusion of contaminants in the environment. Those models can account for "diffusive" spread and for "advective" transport. However, the required 540 parametrization is often much finer than the one for lumped models, so that the number of parameters to be calibrated strongly increases, and therefore in absence of good quality data it could be difficult to perform a reliable calibration and validation of the model for a practical application. It is also assumed that the population under study is a closed system, thus Epidemic models rarely consider birth and death rates, because the cor-555 responding terms in the underlying equations are usually negligible. In this work, however, these terms have been kept, as they play a significant role in our discussion. In particular, following the assumption of population homogeneity mentioned above, it is assumed that infected pregnant women give birth to infected babies and that this occur at the same rate as for susceptible women. With regard to infection rate, which is described by the term γIS/P in (1) and (2), some remarks are in order. This term is computed by assuming that each infected individual has a given, constant number of contacts with other persons per unit time. Our model assumes that the number of persons who cannot be infected is I + R, so that the fraction of contacted persons who 565 cannot be infected is given by (I + R)/P ; on the other hand, the fraction of contacted individuals who can be infected is given by S/P . This is equivalent to assuming that recovered people become immune to the virus. Notice that timing, magnitude and longevity of immunity against SARS-CoV-2 is still an open question for the scientific community (see, e.g., [37, 38, 39] ). Moreover, 570 recovered people are assumed to be not infectious, which is the case if the response of their immune system is fast enough so that, once they come in contact with the virus again, the virus is destroyed by the immune system before it can be spread to susceptible persons. Other promising classes of models are stochastic models [40] , either under a be adapted in a relatively easy way to account for several phenomena and also to consider the role of aspects like gender, age, health and wellness on the probability of infection, recovery and decease. On the other hand, EnKF could 580 provide a firm theoretical framework to improve model predictions by means of uncertain data. Other models in the Bayesian framework [42] could be very helpful to handle discrepancies between model predictions and reference values. Unfortunately, in this case the systematic and random errors could be so high as to make it very difficult to handle them even in a stochastic framework. Several tests were performed in order to study the impact of the data, the time frames over which the data were collected and the performance of different objective functions, depending on the choice of ξ, on the calibration of the parameters (see Table 2 ). Test A can be considered as the reference one, as it difficulties we applied the "differential evolution" algorithm [30] and the results obtained were very good. Other relevant algorithms for global optimization that could be tested as part of future work are genetic algorithms [43, 35] , particle swarm optimization [44] , simulated annealing [45] . Another limitation is given by the uncertainty in the available data. Test G was performed with data ten times greater than the official ones with the 660 intent of starting a sensitivity analysis. The results obtained in test G show that the calibration of the parameters via these data, in particular the values for γ and ρ are not very consistent with tests A, D and F mentioned above. These results emphasize the ill-posed nature of the underlying inverse problem, by providing evidence about the great care that has to be given to the quality of An application of the theory of probabilities to the study of a priori pathometry.-Part I An application of the theory of probabilities to the study of a priori pathometry A contribution to the mathematical theory of epidemics Contributions to the mathematical theory of epidemics. 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Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy Early dynamics of COVID-19 in Algeria: a model-based study Modelling provincial COVID-19 epidemic data in Italy using an adjusted time-dependent SIRD model COVID-19 infection: the perspec-820 tives on immune responses The dynamics of humoral immune responses following SARS-CoV-2 infection and the potential for reinfection SARS-CoV-2: Virology, epidemiology, immunology and vaccine development Stochastic models for epidemics with special reference to AIDS The Ensemble Kalman Filter: theoretical formulation and practical implementation Bayesian dynamical esti-835 mation of the parameters of an SE(A)IR COVID-19 spread model Handbook of genetic algorithms Particle swarm optimization Optimization by simulated annealing An application of the theory of probabilities to the study of a priori pathometry.-Part I An application of the theory of probabilities to 850 the study of a priori pathometry A contribution to the mathematical theory of epidemics Contributions to the mathematical theory of epidemics. II. the problem of endemicity Game theory of social distancing in response to an epidemic Economic considerations for social distancing and behavioral 865 based policies during an epidemic The macroeconomics of epidemics, Working Paper 26882 A generalization of the Kermack-McKendrick deterministic epidemic model Mathematical structures of epidemic systems Global stability of an SIR epidemic model with time delays The mathematics of infectious diseases The effect of cross-immunity and seasonal forcing in a multi-strain epidemic model External forcing of ecological and epidemiological systems: a resonance approach Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission Complex dynamic behavior in a viral 890 model with delayed immune response Asymptotic profile of the positive steady state for an SIS epidemic reactiondiffusion model: Effects of epidemic risk and population movement Complete global stability for an SIR epidemic model with delay distributed or discrete Comparing vectorhost and SIR models 900 for dengue transmission Real time bayesian estimation of the epidemic potential of emerging infectious diseases Modeling the effect of information campaigns on the HIV epidemic in Uganda A double epidemic model for the 910 SARS propagation A conceptual framework for discrete inverse problems in geophysics Development, calibration and validation of physical models Solutions of ill-posed problems, V.H. Win-920 ston & sons Stable determination of conductivity by boundary measurements Open issues of stability for the inverse conductivity prob-925 lem Lipschitz stability at the boundary for time-harmonic diffuse optical tomography Real estimates of mortality following COVID-19 infection An interactive web-based dashboard to track COVID-19 in real time Differential evolution a simple and efficient heuristic for global optimization over continuous spaces Annuaire démographique, 69th Edition, United Nations Implementing the Nelder-Mead simplex algorithm with adaptive parameters Practical methods of optimization Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy Early dynamics of COVID-19 in Algeria: a model-based study Modelling provincial COVID-19 epidemic data in Italy using an adjusted time-dependent SIRD model COVID-19 infection: the perspectives on immune responses The dynamics of humoral immune responses fol-960 lowing SARS-CoV-2 infection and the potential for reinfection SARS-CoV-2: Virology, epidemiology, immunology and vaccine development Stochastic models for epidemics with special reference to AIDS The Ensemble Kalman Filter: theoretical formulation and practical implementation The data on COVID-19 epidemic have been downloaded from the following URL: from https://github.com/CSSEGISandData/COVID-19.Finally, the authors wish to thank the anonymous referees and the editor,