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PLoS Comput Biol
PLoS Comput Biol
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10.1371/journal.pcbi.1012387
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A Physics-Informed Neural Network approach for compartmental epidemiological models
A PINN approach for epidemiological models
https://orcid.org/0000-0002-9743-4547
Millevoi Caterina Conceptualization Methodology Visualization Writing – original draft 1 *
https://orcid.org/0000-0001-6892-9826
Pasetto Damiano Conceptualization Validation Writing – review & editing 2
https://orcid.org/0000-0002-5077-1394
Ferronato Massimiliano Conceptualization Supervision Writing – review & editing 1
1 Department of Civil, Environmental and Architectural Engineering, University of Padova, via Marzolo 9, Padova, Italy
2 Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University of Venice, Via Torino 155, Venezia Mestre, Italy
Scarpino Samuel V. Editor
Northeastern University, UNITED STATES OF AMERICA
The authors have declared that no competing interests exist.

* E-mail: caterina.millevoi@unipd.it
9 2024
5 9 2024
20 9 e101238715 12 2023
5 8 2024
© 2024 Millevoi et al
2024
Millevoi et al
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Compartmental models provide simple and efficient tools to analyze the relevant transmission processes during an outbreak, to produce short-term forecasts or transmission scenarios, and to assess the impact of vaccination campaigns. However, their calibration is not straightforward, since many factors contribute to the rapid change of the transmission dynamics. For example, there might be changes in the individual awareness, the imposition of non-pharmacological interventions and the emergence of new variants. As a consequence, model parameters such as the transmission rate are doomed to vary in time, making their assessment more challenging. Here, we propose to use Physics-Informed Neural Networks (PINNs) to track the temporal changes in the model parameters and the state variables. PINNs recently gained attention in many engineering applications thanks to their ability to consider both the information from data (typically uncertain) and the governing equations of the system. The ability of PINNs to identify unknown model parameters makes them particularly suitable to solve ill-posed inverse problems, such as those arising in the application of epidemiological models. Here, we develop a reduced-split approach for the implementation of PINNs to estimate the temporal changes in the state variables and transmission rate of an epidemic based on the SIR model equation and infectious data. The main idea is to split the training first on the epidemiological data, and then on the residual of the system equations. The proposed method is applied to five synthetic test cases and two real scenarios reproducing the first months of the Italian COVID-19 pandemic. Our results show that the split implementation of PINNs outperforms the joint approach in terms of accuracy (up to one order of magnitude) and computational times (speed up of 20%). Finally, we illustrate that the proposed PINN-method can also be adopted to produced short-term forecasts of the dynamics of an epidemic.

Author summary

During the recent COVID-19 pandemic, we all became familiar with the reproduction number, a crucial quantity to determine if the number of infections is going to increase or decrease. Understanding the past changes of this quantity is fundamental to produce realistic forecasts of the epidemic and to plan possible containment strategies. There are several methods to infer the values of the reproduction number and, thus, the number of new infections. Statistical methods are based on the analysis of the collected epidemiological data. Instead, modeling approaches (such as the popular SIR model) attempt constructing a set of mathematical equations whose solution aims at approximating the dynamics underlying the data.

In this paper, we explore the use of a recently developed technique called Physics-Informed Neural Network, which tries to combine the two approaches and to simultaneously fit the data, infer the dynamics of the unknown parameters, and solve the model equations.

The proposed PINN implementations are tested in different scenarios using both synthetic and real-world data referred to the COVID-19 pandemic outbreak in Italy. The promising results can pave the way for a wider use of PINNs in epidemiological applications.

The author(s) received no specific funding for this work. PLOS Publication Stagevor-update-to-uncorrected-proof
Publication Update2024-09-17
Data AvailabilityThe epidemiological data for the Italian COVID-19 epidemic are available at the following link: https://www.epicentro.iss.it/en/coronavirus/sars-cov-2-integrated-surveillance-data. The source code in Python is available at the following repository: https://github.com/cmillevoi/EpiPINN.
Data Availability

The epidemiological data for the Italian COVID-19 epidemic are available at the following link: https://www.epicentro.iss.it/en/coronavirus/sars-cov-2-integrated-surveillance-data. The source code in Python is available at the following repository: https://github.com/cmillevoi/EpiPINN.
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pmc1 Introduction

Epidemiological models are nowadays fundamental to assist and guide policy makers in the fight against the spreading of diseases. This has been evident during the recent COVID-19 pandemic, when epidemiologists and scientists all over the world devoted their research to develop ad-hoc transmission models. Focusing, for example, on Italy, where the European outbreak started in February 2020, epidemiological models have been adopted to analyze different aspects of the epidemic: to determine the urgency to impose regional restrictions [1]; to analyze the impact of the national lockdown [2, 3]; to explore the results of transmission scenarios after the release of the restrictions [4]; to study the impact of the different variants and the vaccination campaign [5–7]; and to compute optimal strategies for the vaccine deployment in order to minimize the number of cases or deaths [8, 9]. Most of these studies describe the SARS-CoV-2 transmission using different variations of compartmental models. The basic SIR model is at the core of those more-complex epidemiological models. It subdivides the population of interest into compartments indicating the infectious status of each individual (i.e. susceptible, infected, or recovered individuals). The dynamic describes the mean contacts between susceptible and infected individuals, and thus, the average rate at which susceptible individuals transit to the infected compartment. The main model parameter is the rate of transmission of the infection, β. This is strictly related to the well known basic reproduction number, R0, representing the average number of secondary infections generated by one infected individual in a totally susceptible population. The value of this quantity changes during an outbreak due to the temporal variations in human behavior (caused, for example, by changes in individual awareness or social distancing policies) and in the infectiousness of the virus. The effective reproduction number, Rt, aims at describing the ongoing transmission in a changing system.

Data-driven methods provide effective estimates of Rt based on the renewal equation [10–12], i.e., a convolution on the reported cases having as kernel the serial interval (the time interval between the symptom onset of an individual and its secondary infections). These data-driven estimates do not explicitly provide a relationship between the changes in Rt and its possible causes, such as the implemented non-pharmaceutical interventions or the vaccination campaigns. Compartmental models give a deeper understanding of the ongoing spreading of the disease and, at the same time, allow the computation of Rt using the spectral radius of the next generation matrix [13–15]. However, they require the assessment and calibration of time-dependent parameters.

Tracking the temporal variations in the model parameters is an essential but complex problem to follow and predict the spreading of a disease. Many studies tackle this problem using Bayesian inference, i.e., searching for the posterior distribution of the unknown parameters based on the available reported cases and the prior distribution. Among these approaches, we recall the iterative particle filter [16], sequential data-assimilation schemes [17], or the use of subsequent Markov chain Monte Carlo (MCMC) [4, 7]. Being based on random sampling, these approaches might result in low quality results and large computational times, due to the slow Monte Carlo convergence.

Here, we propose to adopt a deterministic approach based on Physics-Informed Neural Networks (PINNs). The idea behind PINNs is to exploit the universal approximation property of Neural Networks (NNs) [18, 19] to estimate the solution of differential equation [20]. In practice, this is done by describing the state variables and, in case, the time-dependent parameters using NNs. The parameters of the NNs are trained by seeking the minimum of a loss function based on both the misfit on the available data, and the residual of the differential equations governing the problem at hand, i.e., the SIR model equations in our case. Thus, the PINN functions fit the data and, at the same time, provide good approximations of the solutions of the differential equations. The use of the epidemiological model equations is fundamental in PINNs and constitutes the main innovation with respect to simpler NNs or Deep Neural Networks (DNNs), which are completely data-driven.

The application of PINNs to epidemiological models became particularly relevant during the COVID-19 pandemic. Many studies used PINNs as an inverse-problem solver, to calibrate the parameters of epidemiological compartmental models. However, the model parameters has frequently been considered constant in time, e.g, [21, 22], or with particular periodic dependencies on time [23]. Schiassi et al. [24] showed the computational efficacy of using PINNs to estimate constant parameters of different basic compartmental models under increasing levels of noise in the data. Long et al. [25] considered a more realistic scenario, and used PINNs to accurately identify the time-varying transmission parameter in a SIRD model of COVID-19 when assimilating the reported infected cases in three USA states. Feng et al. [26] proposed a similar approach to predict the number of active cases and removed cases in the US. Olumoyin et al. [27] used PINNs to track the changes in transmission rate and the number of asymptomatic individuals for COVID-19. Ning et al. [28] and He et al. [29] presented applications of PINNs to COVID-19 outbreaks in Italy and China, respectively. Bertaglia et al. [30] constrained PINNs to satisfy an asymptotic-preservation property to avoid poor results caused by the multiscale nature of the residual terms in the loss function.

Building on top of these examples, our work aims to deeper explore the properties of PINNs as an inverse solver for the estimation of time-dependent transmission rates or reproduction numbers in SIR models. Our analysis aims on further showing some benefits of using PINNs that are not directly available with more traditional approaches such as: the simultaneous estimation of multiple parameters that change in time, the inference using jointly different types of data, the possibility of providing a future projection for the evaluated parameters, the possibility of training the model even if there are gaps or large errors or uncertainties on the quality of the data.

In particular, we propose two modifications of the PINNs algorithm that grant faster convergence and more stable results, thus providing a step forward in the use of PINNs in real epidemiological models.

The first modification splits the PINN implementation in two steps. The motivation for this approach is that in the common PINN implementation for SIR-like models, the NNs representing the model state variables and, if present, the time-dependent parameters, are calibrated together through the minimization of the loss function on the data and the model residual. This inverse problem is particularly complex and many epochs might be required to achieve convergence. Starting from the idea that the available epidemiological data, which are typically the daily or weekly reported infections, is directly associated to a model state variable, the split PINN approach is based on the following two steps: as first, construct the NN of the state variable associated to the data, e.g., the infected compartment, by minimizing the loss function based on the data; as second, calibrate the other NNs for the remaining state variables and parameters based on the NN computed in the first step and the minimization of the residuals of the governing equations. We will refer to the traditional PINN approach as joint approach, in contrast to the described split approach. A graphical sketch of the two approaches is shown in Fig 1.

10.1371/journal.pcbi.1012387.g001 Fig 1 Diagram of the workflow of the joint (a) and split (b) PINN approaches.

The second proposed modification reduces the number of NNs considered in the PINN approximation and, consequently, simplifies the structure of the loss function. This simplification is possible because, in simple SIR-based models, the transmission parameter and the infected compartment control the system dynamic. In fact, these functions allow to directly evaluate the other state variables, which are then redundant in the formulation of the loss function.

Our analysis compares the joint, split, and reduced approaches in a sequence of synthetic test cases where we progressively challenge the structure of the transmission rate from constant, to a sinusoidal-like dependence on time, to a real scenario, and increase the noise on the synthetic reported data. The proposed test cases assume model parameters that are inspired by the first months of the COVID-19 outbreak in Italy. As an example of application, the PINN strategies are adapted in order to fit the real epidemiological data reported in Italy. Due to the large uncertainties that characterize the real data on the reported infections, in this last setting we propose to include in the loss function also the data on the daily hospitalizations, which are a more reliable representation of the number of individuals with severe symptoms. Finally, we consider this scenario to explore the accuracy of the short-term forecasts produced by PINNs.

The paper is organized as follows. Section 2 presents the mathematical formulation of the proposed methods. It starts with the equations of the SIR model (Section 2.1), then it describes the joint and split implementations of PINNs (Section 2.2), and finishes with the modified schemes for the reduced approach (Section 2.3) and the extension to the hospitalized data (Section 2.4). The numerical results of the application of the proposed PINNs to seven test cases are illustrated in Section 3. The method is tested and validated on synthetic cases (Section 3.1) and then applied in a real-world scenario (Section 3.2) for both parameter estimation and forecast. Finally, Section 4 presents the discussion of the results and sums up the main conclusions.

2 Methods

2.1 The basic SIR model

The well-known SIR model is largely adopted for the theoretical analysis of epidemics, and lies at the core of several more complex epidemiological models for real applications. At a given time t [T], the individuals in a population of dimension N [–] are subdivided into compartments on the basis of their epidemiological status, in this case the susceptible (S), the infected (I), and the recovered (R) individuals. The number of individuals in the three compartments changes in time under the assumption that, in a well mixed population, any susceptible individual can enter in contact with any infected individual, thus possibly becoming infected itself.

From a mathematical point of view, the strong form of the ordinary differential problem governing these dynamics can be stated as follows. Let T=[t0,tf]⊆R+∪{0} be the time domain of interest, with t0 and tf [T] the initial and final times of the simulation, respectively. Given the continuous functions β(t):T→R+ and δ(t):T→R+, find S(t):T→[0,N], I(t):T→[0,N], and R(t):T→[0,N] such that: {S˙(t)=-βNI(t)S(t)I˙(t)=βNI(t)S(t)-δI(t)R˙(t)=δI(t),∀t∈T, (1)

and satisfying the initial conditions: {S(t0)=N-I0I(t0)=I0R(t0)=0 (2)

In Eqs (1) and (2) β [T−1] is the transmission rate controlling the average rate of the infection, δ [T−1] is the mean rate of removal of the infected individuals that become recovered. Another relevant quantity used to set up the model is D = δ−1, i.e., the mean reproduction period [T] representing the average time spent by an individual in compartment I. Initial conditions for the spreading of a new disease assume that the population at the initial time is completely susceptible besides a small number I0 of infected individuals (typically 1, but not necessarily).

The basic reproduction number R0 [-] associated to this model reads R0=β(t0)/δ(t0) and provides an estimate of the number of secondary infections generated by one infectious individual in a susceptible population, i.e. at the beginning of the epidemic. The threshold R0>1 indicates the occurrence of an outbreak, while R0<1 indicates that the number of infected individuals is rapidly decreasing. Note that, in a real population, the number of individuals in each compartment is a discrete variable, whose dynamic can be described by stochastic approaches, e.g., the Gillespie method or discrete Markov chains. Hence, the continuous deterministic model in Eqs (1) and (2) is a valuable representation of the mean process in large populations.

Standard numerical ODE solvers, such as Runge-Kutta-based methods, can provide an accurate solution to the differential problem (1) and (2). For R0>1 and constant parameters, the solution depicts an initial exponential-like increase in the number of infections up to a peak, and then a fast decrease due to the depletion of susceptible individuals. However, it is clear that this dynamic does not correspond to what happens during an outbreak. The main challenge when using a model based on (1) to describe a real epidemic is that the transmission rate β and the mean reproduction period δ−1 can change in time because of many factors: social behaviors (individual awareness, increase or decrease of gatherings, mobility, social distancing), non-pharmaceutical interventions (use of devices that reduce transmission—such as masks, introduction of lock-downs), changes in the pathogen infectiousness due to new variants, reduction of the susceptibility of the population due to vaccination campaigns. In this evolving scenario, the effective reproduction number Rt [-] is the critical quantity that controls the spreading of the disease. Rt is the equivalent of R0 in time, i.e., Rt=β(t)/δ(t)·S(t)/N, taking into account that the number of susceptible individuals decreases and the main parameters controlling the spreading of the disease generally change. An essential element for a reliable simulation is therefore the assessment of Rt, hence β(t) and δ(t) along with the compartment S(t), from the available epidemiological data. In the following we will assume that δ is constant in time, assumption done in many epidemiological applications (see e.g., [2, 4, 5]).

2.2 PINN solution to the SIR model

Here we develop and analyze a PINN-based approach to simultaneously solve the problem (1) and (2) and estimate the temporal values of the reproduction number by using a time series of infectious individuals as basic epidemiological information.

A standard NN aims to reconstruct an unknown function u from the knowledge of some training data points. The NN approximating a generic u, denoted throughout this work by u^, is the recursive composition of the function: Σ(l)(x(l))=ϕ(l).(W(l)x(l)+b(l)), (3)

where W(l)∈Rnl×nl-1, b(l)∈Rnl, and ϕ(l) are weights, biases, and activation functions of the l-th layer, respectively. The last layer is the output layer, the others are the hidden layers. We denote with nl the number of neurons in layer l. Activation functions are user-specified functions with limited range, which are generally non linear in order to provide a source of non linearity to the NN and maintain low weight values. The Matlab-inspired notation ϕ.(x) means that the function ϕ is applied to each component of the vector x. Let L be the number of hidden layers. If u:T→R is the solution of an ordinary differential equation in the domain T, the input of the first layer reads x(1)=t∈T, so n0 = 1, and the output of the last layer u^ is a scalar, so nL+1 = 1. Then, the NN for u formally reads: u^(t)=Σ(L+1)∘Σ(L)∘⋯∘Σ(1)(t). (4)

The NN depends on the set of weights and biases, which are trained through an optimization algorithm so as to minimize an appropriate loss function defined as the mean squared error of u^ over the set of training points. In the case of PINNs, the information from the governing equations of the physical system is introduced in a weak way in the loss function by adding the residual of the differential equations evaluated at some collocation points [31, 32].

For the SIR model (1) and (2), we assume that the training data points for the fitting are the reported infections. Let I˜j be the number of reported infected individuals at times t˜j, j = 1, …, ND. This might be subject to reporting errors, thus, in general I˜j≠I(t˜j). The residual of the governing equations is computed over NC collocation points.

We aim at finding a NN representation for the susceptible, infected, and recovered individuals (S^(t), I^(t), and R^(t), respectively) along with the transmission rate (β^(t)). Since the state variables S, I, R span an extremely wide range of values (from zero to the population size N > 106), the functional search is optimized by a proper scaling: S(t)=CSs(ts),I(t)=CIs(ts),R(t)=CRs(ts), (5)

where C [-] is an appropriate constant and ts is the dimensionless scaled temporal variable, ts = (t − t0)/(tf − t0). The system of ODEs (1) for the scaled variables becomes: {S˙s(ts)=-C1βs(ts)Is(ts)Ss(ts)I˙s(ts)=C1βs(ts)Is(ts)Ss(ts)-C2Is(ts)R˙s(ts)=C2Is(ts),ts∈[0,1], (6)

where βs(ts):[0,1]→R+, C1 = (tf − t0)C/N and C2 = (tf − t0)δC. The initial conditions (2) are correspondingly scaled as well as the infectious data I˜j=CI˜s,j at times t˜s,j=(t˜j-t0)/(tf-t0).

The SIR model (6) does not consider death and birth processes and assumes a negligible mortality rate of the disease. Thus, the total population N is constant in time and equal to N = S + I + R. Under these hypotheses, the PINN model needs only two NNs representing the behavior of the population: one for the state variable of the susceptible individuals S^s, and one for the infected individuals I^s. The number of recovered individuals is computed as R^s=NC-I^s-S^s. A third NN is included for the estimation of the transmission rate β^s. In this way, the number of parameters to be tuned during the training is consistently reduced.

It is important to underline that the state variables represent the number of individuals in a compartment, thus they all have positive outputs. Training the model without imposing this condition could lead to nonphysical negative NN outputs. The non-negative constraint can be imposed in the NN in two alternative ways: inserting a penalty term for the negative values of the NNs (weak constraint) or building the NN architecture so as to allow for positive values only (hard constraint). The latter prescription can be met by setting the output activation function, i.e., the one related the last layer, ϕ(L+1), equal for example to the square function. An experimental comparison between the two approaches shows that the latter is generally more effective and provides more robust results. The numerical outcomes that follow are therefore obtained by using the hard constraint prescription for the non-negativity of the solution. The same constraint is adopted to entail a positive value for βs.

The selection of the loss function is one of the most sensitive steps in the PINN approach, given the multi-objective nature of the method. Using the Mean Squared Error (MSE) as loss measure, the objective is to minimize the mismatch on the ND data: LD(I^s)=ωD1ND∑j=1ND[I^s(t˜s,j)-I˜s,j]2, (7)

the squared norm of the residual of Eq (6) evaluated on NC collocation points {t¯s,i}i=1NC: LODE(S^s,I^s,β^s)=1NC∑i=1NCωS[dS^sdts+C1β^sI^sS^s]2|t¯s,i+1NC∑i=1NCωI[dI^sdts-C1β^sI^sS^s+C2I^s]2|t¯s,i+1NC∑i=1NCωR[dR^sdts-C2I^s]2|t¯s,i, (8)

and the misfit on the initial conditions: LIC(S^s,I^s)=ωS0[S^s(0)-N-I0C]2+ωI0[I^s(0)-I0C]2+ωR0R^s2(0), (9)

where ω* are proper weights needed to balance the relative importance of the entries arising from each contribution to the global MSE value. Fig 2 shows a diagram of the PINN implementation for the solution of the scaled SIR model (6).

10.1371/journal.pcbi.1012387.g002 Fig 2 Diagram of the PINN model for the SIR equations with unknown β(t).

The parameters of the NNs for β, S, I are obtained by minimizing the loss functions on the infectious data, and on the residual and initial conditions of the model equations.

We explore two possible approaches for the construction of the PINN model, indicated as joint or split. The joint approach aims to simultaneously calibrate S^s, I^s, and β^s by minimizing the joint loss function corresponding to the sum of LD, LODE, and LIC: Ljoint(S^s,I^s,β^s)=LD(I^s)+LODE(S^s,I^s,β^s)+LIC(S^s,I^s). (10)

By distinction, the split approach subdivides the overall problem. First, I^s is independently calibrated on the data error LD (7) only. In this case, a standard NN is used with weight ωD = 1, thus obtaining a differentiable regression function for the data. The only-data regression is followed by a fully-physics-informed regression, where the parameters defining S^s and β^s are trained by minimizing: Lsplit(S^s,β^s)=LODE(S^s,β^s)+LIC(S^s). (11)

It is important to underline that in standard data-driven NNs a regularization term is frequently added to the loss function to avoid overfitting on the data. The term related to the residual in the loss function (Eq 9 in our case) acts as a regularization in PINNs, therefore no additional regularization has been added (see [20] for more details).

2.3 Reduced SIR model

The system of ODEs in (1) can be further reduced by directly considering the definition of the effective reproduction number Rt. By easy developments, the model (1) becomes: {I˙(t)=δ(Rt-1)I(t)S˙(t)=-δRtI(t),t∈[t0,tf]. (12)

where the unknown functions are I(t) and S(t), and the state variable R(t) is simply obtained from the consistency relationship R(t) = N − S(t) − I(t). The initial conditions (2) still hold. The new system (12) can be solved sequentially by integrating the upper equation first and then computing S(t) from the second equation.

This approach reduces the number of functions that are approximated by NNs to two, i.e., I and Rt, and eliminates any redundant term in the loss function minimized in the PINN approach. The same scaling as in Eq (5) is used for the state variable I, so that the upper equation in (12) reads: I˙s(ts)=δ(tf-t0)(Rt-1)Is(ts),ts∈[0,1]. (13)

The NNs approximating the variables of interest, i.e., I^s and R^t, can be obtained by minimizing the mismatch on data LD (Eq (7)) and the squared norm of the residual of Eq (13) on NC collocation points: Lr,ODE(I^s,R^t)=1NC∑i=1NC[dI^sdts-δ(tf-t0)(R^t-1)I^s]2|t¯s,i (14)

Notice that in this case the contributions in LD and Lr,ODE have a consistent size, hence there is no need for introducing the weight parameters ω* to balance the loss function terms. For this reason, we simply set ωD = 1 in the expression (7).

The joint and split approaches can be formulated for this PINN-based model as well. The joint approach consists in training simultaneously the NNs I^s and R^t by minimizing the total loss function: Lr,joint(I^s,R^t)=LD(I^s)+Lr,ODE(I^s,R^t). (15)

By distinction, the split approach implies training I^s on the data only by the minimization of LD in Eq (7). Then, the time-dependent parameter R^t is obtained by minimizing: Lr,split(R^t)=Lr,ODE(R^t). (16)

Notice that in the reduced PINN model no initial condition is set, but we let the model deduce it from the data. From a theoretical viewpoint, initial conditions are not necessary because I^s is obtained from the data, while the governing differential Eq (13) is used to calibrate R^t. This outcome is relevant because it replicates what typically happens in a real-case scenario, where there is no actual knowledge about the instant of beginning of the outbreak. In fact, the case 0 in most outbreaks is unknown and the conventional start of the epidemic has a number of infected individuals that is usually largely underestimated. The use of the reduced modeling approach makes it possible to remove the term related to the initial condition from the loss function.

2.4 SIR model with the hospitalization compartment

The reported infections can be often affected by large uncertainties. Especially at the beginning of an epidemic outbreak, the disease cannot be easily recognized, either because of the difficulty of correctly identifying the symptoms, or the absence of well-established detection and surveillance procedures, or the impossibility of reaching and testing all the people infected by the disease. Moreover, these data can be strongly affected by territorial peculiarities and the logistic of testing facilities. Hence, founding an epidemiological model on these pieces of information can undermine its reliability. A much less uncertain epidemiological datum is the daily number of individuals that require to be hospitalized. This fraction of the overall number of infected individuals is representative of the entire I compartment by assuming that hospitalization is needed over a certain common threshold level of symptoms in the population.

We introduce a new variable, H, defined as: H(t)=δσI(t), (17)

where σ represents the fraction of infected individuals moving to the hospitalized compartment. Note that also parameter σ might change in time, for example because of the insurgence of more aggressive variants or the improvement of home treatment. A more convenient formulation uses the cumulative number ΣH of hospitalized individuals: ΣH(t)=∫t0tH(z)dz. (18)

The new formulation of the updated SIR model can be therefore stated as follows. Given Rt:T→R+, σ(t):T→[0,1], and δ(t):T→R+, find ΣH(t):T→[0,N], I(t):T→[0,N], and S(t):T→[0,N] such that: {Σ˙H(t)=δσI(t)I˙(t)=δ(Rt-1)I(t)S˙(t)=-RtδI(t),∀t∈T, (19)

with R(t) = N − I(t) − S(t), the initial conditions (2) and ΣH(t0) = 0. The available information from the actual epidemiological data is the daily variation ΔH of the cumulative number of hospitalized individuals: ΔH(t)=ΣH(t)-ΣH(t-1)≃Σ˙H(t), (20)

whose values represents the training dataset for the PINN approximation of system (19). As previously done, the functional search of the approximating NNs is carried out on the properly scaled quantities I(t) = CIs(ts) (see Eq (5)) and: ΔH(t)=CHΔH,s(ts), (21)

with CH the scaling factor. The upper equation in system (19) with the scaled quantities reads: CHΔH,s(ts)=δCσs(ts)Is(ts), (22)

with σs:T→[0,1], while the second scaled equation is the same as in (13). Hence, the NNs needed to solve the SIR model with hospitalization data are Δ^H,s, I^s, σ^s, and R^t. The training data points for the fitting are both the scaled reported infections I˜s,j and the hospitalizations Δ˜H,s,j at the scaled times t˜s,j, j = 1, …, ND. The NNs can be obtained by minimizing the mismatch (7) on the infection data and on the hospitalization data: LH=1ND∑j=1ND[Δ^H,s(t˜s,j)-Δ˜H,s,j]2, (23)

and the squared norm of the residuals of Eqs (13) and (22) on NC collocation points: LH,ODE(Δ^H,s,I^s,σ^s,R^t)=1NC∑i=1NC[dI^sdts-δ(tf-t0)(R^t-1)I^s]2|t¯s,i+1NC∑i=1NC[Δ^H,s-δCσ^sCHI^s]2|t¯s,i. (24)

The joint approach consists in the simultaneous estimate of Δ^H,s, I^s, σ^s, and R^t by finding the minimum to the functional: LH,joint(Δ^H,s,I^s,σ^s,R^t)=LD(I^s)+LH(Δ^H,s)+LH,ODE(Δ^H,s,I^s,σ^s,R^t). (25)

As for the PINN solution to the reduced SIR model, it is not necessary to include the mismatch on the initial conditions into the global loss function (25) because they are met through the available training data. Moreover, also the use of non-unitary weights ω* for the different contributions to LH,joint is not required since all terms are likely to have a similar magnitude.

In the split approach, Δ^H,s is directly trained with the hospitalization data only by minimizing LH in Eq (23). Then, I^s is computed from (22) as: I^s=CHδCΔ^H,sσ^s, (26)

and σ^s and R^t are trained by minimizing: LH,split(σ^s,R^t)=1ND∑j=1ND[CHδCΔ^H,s(t˜s,j)σ^s(t˜s,j)-I˜s,j]2+1NC∑i=1NC{CHC[ddts(Δ^H,sδσ^s)-(tf-t0)(R^t-1)Δ^H,sσ^s]}2|t¯s,i. (27)

In real-world scenarios, the new daily infections is a more common piece of information than the total number of infected individuals. In order to include these data in the PINN model, we introduce the cumulative number ΣI of infected individuals: ΣI(t)=∫t0tI(z)dz. (28)

The variation of ΣI in time coincides with negative variation of the class of susceptible individuals S(t), so we can simply update the SIR model with hospitalization data (19) by replacing the last equation with: Σ˙I(t)=δRtI(t). (29)

Since the available information is the daily variation ΔI of the cumulative number of infected individuals: ΔI(t)=ΣI(t)-ΣI(t-1)≃Σ˙I(t), (30)

we use these values as training data set. As usual, scaled values are considered such as ΔI = CΔI,s and we assume that the set of scaled values Δ˜I,s,j is available at the training scaled times t˜s,j, j = 1, …, ND, instead of I˜s,j. The mismatch of Δ^I,s, i.e., the NN approximating ΔI,s, with the data is measured by: LI=1ND∑j=1ND[Δ^I,s(t˜s,j)-Δ˜I,s,j]2, (31)

while the squared norm of the residual reads: LHI,ODE(Δ^H,s,Δ^I,s,I^s,σ^s,R^t)=1NC∑i=1NC[dI^sdts-δ(tf-t0)(R^t-1)I^s]2|t¯s,i+1NC∑i=1NC[Δ^H,s-δCσ^sCHI^s]2|t¯s,i+1NC∑i=1NC[Δ^I,s-δR^tI^s]2|t¯s,i. (32)

Hence, with the joint approach we aim at minimizing the functional: LHI,joint(Δ^H,s,Δ^I,s,I^s,σ^s,R^t)=LI(Δ^I,s)+LH(Δ^H,s)+LHI,ODE(Δ^H,s,Δ^I,s,I^s,σ^s,R^t). (33)

By distinction, with the split approach we first train Δ^H,s by the available data (see Eq (23)). Then, we use Eq (26) for I^s and: Δ^I,s=CHCR^tΔ^H,sσ^s (34)

for Δ^I,s, and minimize the functional: LHI,split(σ^s,R^t)=1ND∑j=1ND[CHCR^tΔ^H,s(t˜s,j)σ^s(t˜s,j)-Δ˜I,s,j]2+1NC∑i=1NC{CHC[ddts(Δ^H,sδσ^s)-(tf-t0)(R^t-1)Δ^H,sσ^s]}2|t¯s,i. (35)

This choice for the split approach is based on the fact that hospitalization data are usually more reliable than infected individuals, hence they are more appropriate for an only-data regression training.

2.5 Simulation setup

The PINN-based approaches are here implemented by making use of the SciANN software library [33], a Keras and TensorFlow wrapper specifically developed for physics-informed deep learning. We analyze the performance of the PINN-based approaches to estimate the state variables and identify the governing parameters of an epidemiological model mimicking the setup of the first 90 days of a COVID-like disease outbreak in Italy. The total population is set to N = 56 × 106 and the mean infectious period to D = 5 days, which is an estimate used for COVID-19 [4]. The initial value of infectious individuals I0 is set to 1. The accuracy of the trained NNs is evaluated by the 2-norm of the error with respect to the 2-norm of the reference solution: er=∥y^-yref∥2∥yref∥2, (36)

where y can be either one of the state variables, or a time-dependent parameter. The relative error (36) is numerically computed by using 90 points equally spaced in the domain. We consider a number of scenarios, summarized in Table 1, differing for the reference SIR model and state variables of interest, the selection of the estimated governing parameters, and the available training data. The first five scenarios are used to validate the numerical model, while the last two consist of a real application to the Italian COVID-19 epidemic.

10.1371/journal.pcbi.1012387.t001 Table 1 Scenarios adopted to analyze the proposed PINN-based approaches.

	State variables	Estimated parameters	Reference values	Training data	
Case 1	S,I,R	β (constant)	β0 = 0.6 d−1	I˜j (Poisson error)	
Case 2	S,I,R	β (time-dependent)	Synthetic β	I˜j (Poisson error)	
Case 3	S,I,R	β (time-dependent)	COVID-19 Rt	I˜j (Poisson error)	
Case 4	I	Rt	Synthetic β	I˜j (40% Gaussian error)	
Case 5	I,ΔH	Rt ,σ (time-dependent)	Synthetic β,σ	I˜j (40% Gaussian error), Δ˜H,j (Poisson error)	
Case 6	ΔI,ΔH	Rt ,σ (constant)	COVID-19 Rt	Δ˜I,j , Δ˜H,j (COVID-19 dataset)	
Case 7	ΔI,ΔH	Rt ,σ (time-dependent)	COVID-19 Rt	Δ˜I,j , Δ˜H,j (COVID-19 dataset)	

For the estimation of the transmission rate β(t) in the basic SIR model (1) and (2), we consider three different scenarios:

Case 1: constant β. We use this scenario to compare the efficiency of the joint and split approaches (10) and (11), respectively.

Case 2: synthetic time-dependent β(t), where the reference values are provided as an analytical function.

Case 3: the reference β(t) is obtained from the estimates of Rt in the first months of the COVID-19 epidemic outbreak in Italy.

In each case, the training data are the number of infectious individuals per day. These are synthetically generated by numerically integrating the system (1) using the selected reference function for β(t). In particular, we used ND = 90 training data points (one value per day). To take into account possible reporting errors, the data I˜j for each time t˜j are obtained by sampling from a Poisson distribution having as mean I(t˜j). This kind of Poisson error is frequently assumed on data arising from a counting process. The infectious data and the epidemiological model are scaled by a factor C = 105 (see Eq (5)).

The reduced SIR model (12) is then used to explore a more realistic scenario with strongly perturbed data on the infected individuals and accurate data on the number of hospitalizations. The joint and split approaches (Eqs (15) and (16)) are used to estimate the governing parameter Rt in the following inverse problems:

Case 4: synthetic time-dependent β(t) (as in Case 2), subject to a larger error noise on the infectious data.

Case 5: an adaptation of Case 4, considering also the hospitalization data and a time-dependent hospitalization fraction σ (to be estimated).

In Case 5, the synthetic data of the daily hospitalizations, {Δ˜H,j}j=1ND, are obtained by sampling from a Poisson distribution having as mean value the reference solution. The scaled values are obtained by setting CH = 103.

Finally, we applied the PINN approaches to the infected and hospitalized data reported in Italy during the first months of the COVID-19 pandemic:

Case 6: infers a time-dependent Rt while considering σ as a constant.

Case 7: simultaneously infers Rt and σ as functions of time.

In these scenarios we consider the epidemiological data provided by the Italian surveillance system [34] from February 21st, 2020 to May 20th, 2020. The period coincides with the advent of the disease and its initial spread. The vaccination campaign was not started yet and possible reinfections are negligible. The Italian dataset contains the number of new daily hospitalizations and reported infections, {Δ˜H,j}j=1ND and {Δ˜I,j}j=1ND, respectively, and supplies an estimate of the COVID-19 reproduction number Rt based on [10].

The scaled values are obtained by setting C and CH equal to the maximum experimented values for ΔI and ΔH, respectively, in the 90 days taken into consideration. New infections are multiplied by a reporting ratio αr = 6, following the estimate from Italian Institute of Statistic based on the sierological data [35].

2.6 Implementation details

The best architecture of the NNs is typically dependent on the desired application. On the one hand, the number of neurons and hidden layers should be large enough to make the PINN-based model able to reconstruct the epidemic dynamics. On the other hand, parsimonious NNs are required to contain the number of parameters, limit the computational times of the training process, and avoid overfitting.

To select an adequate architecture for the considered PINN model, we compared the error, the training times and the number of parameters obtained in Cases 1 and 2 using different number of layers (4, 10) and neurons (5, 25, 50, and 100). The details about this sensitivity analysis are reported in Appendix A in S1 Text. Neural Network architectures. Hence, the NNs for S^s and I^s are built with 4 hidden layers, 50 neurons for each and tanh as activation function. In Case 1 the constant transmission rate is treated as a single parameter in the training. In Cases 2 and 3, β^s has 4 hidden layers with 100 neurons. In Cases 4 and 5 the NN for R^t has the same architecture described for β^s. In Cases 6 and 7 the NN for R^t has 4 hidden layers with 100 neurons each. In Cases 5 and 7 the NN for σ^s has 10 hidden layers with 5 neurons each. In all scenarios we consider NC = 6000 collocation points randomly sampled in [t0, tf] from a uniform distribution. The NNs are initialized using Glorot initialization [36] and trained using Adam optimization algorithm [37] with a reduced-on-plateau learning rate schedule, which is initialized to 0.001 and halved if learning stagnates.

In the joint approach we trained the NNs for 5000 epochs with batches containing 100 training points. In the split approach we set the number of epochs to 3000 for the data fitting training and to 1000 for the fully-physics-informed regression, with a batch size equal to 10 and 100, respectively. When the total number of training points (ND + NC) is large, it is a good practice to use a mini-batch gradient descent as optimization algorithm, which splits the training dataset into small batches that are used to calculate the loss function and update the model coefficients at each epoch. The weights ω* are calibrated in the fitting process using the eigenvalues of the Neural Tangent Kernel (NTK) [38], with the specific SciANN built-in function. The approach leverages the NTK to dynamically and adaptively tune the loss-term weights, enhancing the performance and robustness of PINNs in solving differential equations and other physics-informed tasks. Indeed, not using NTK adaptive weights results in larger errors as it is shown in detail in Appendix A in S1 Text. All simulations were performed on a machine with two Intel(R) Xeon(R) CPU E5–2680 v2 @ 2.80GHz and 256GB of RAM. We report the version of the main libraries that were used: Tensorflow = 2.5.3, Keras = 2.5.0, SciANN = 0.6.6.1.

3 Numerical results

3.1 Model validation

3.1.1 Case 1: Constant transmission rate

The first scenario assumes a constant transmission rate during the simulation. The reference value for the parameter is fixed to β = β0 = 0.6 d−1. The corresponding basic reproduction number is R0=3, which is an estimate of the basic reproduction number in the COVID-19 epidemic in Italy. The resulting system dynamic is shown in Fig 3a. The evaluation of a constant parameter does not require an additional NN for β, and it is implemented through an object of the SciANN parameter class with the additional non-negative constraint. Both the joint and split approaches obtain solutions that match well the reference dynamic (see Fig 3a for the results of the split approach). Fig 3b shows that the convergence of the joint approach to the reference value is slower than the split approach. In fact, there are strong oscillations during training. The split approach mitigates these oscillations and reaches a faster convergence to the reference value.

10.1371/journal.pcbi.1012387.g003 Fig 3 Case 1: Constant transmission rate.

Comparison between (a) reference solutions of the SIR model (1) and PINN approximations in the split approach; (b) β0 identification during training of the joint and the split methods.

Table 2 provides a quantitative comparison of the resulting errors for each state variable along with the training times. Note that for the split approach the sum of the only-data and physics-informed training is reported. In both approaches the PINN solution reaches an acceptable accuracy, with a relative error on the order of 10−3 for all state variables, resulting in a good estimate of the unknown parameter β as well. The split approach reduces the total training time about by a factor 3, and improves the accuracy with respect to the joint approach by one order of magnitude on average, hence it appears to be in this case largely preferable.

10.1371/journal.pcbi.1012387.t002 Table 2 Case 1: Constant transmission rate.

Training time, approximation errors, and estimations of the joint and split methods for β = 0.6 d−1. The daily and weekly infection data correspond to ND = 90 and ND = 13, respectively.

	Daily data	Weekly data	
Joint	Split	Joint	Split	
Training time [s]	2223	719	1977	493	
Error S	2.763 × 10−3	6.221 × 10−4	2.833 × 10−2	2.247 × 10−2	
Error I	3.846 × 10−3	8.444 × 10−4	4.544 × 10−2	4.245 × 10−2	
Error R	3.258 × 10−3	7.434 × 10−4	3.276 × 10−2	2.491 × 10−2	
Error β	5.086 × 10−3	3.587 × 10−4	4.693 × 10−2	5.187 × 10−2	
PINN β^	0.59738	0.60007	0.57323	0.56888	

A second test is carried out by changing the amount of available data for the training. For instance, we consider only weekly values for the infection data, which are more likely to compensate the errors and oscillations of daily data. This reduces the number of training points to ND = 13. The model is built and trained as stated in Section 3, with the difference that the training of I^s in the split method is performed at each epoch on the whole data set and the mini-batches are not needed. Moreover, given the overall small size of the training dataset, we can decrease the maximum number of epochs in the data regression from 3000 to 1000. Training times and approximation errors reported in Table 2 show an equivalent outcome in terms of accuracy, but with a strong gain in training time for the split approach. By reducing the number of samples the amount of information is smaller, hence the accuracy decreases with respect to a daily updated information. The convergence of the parameter β has a similar behavior to the one shown previously (Fig 4), with the split method reducing both the oscillations and the convergence time.

10.1371/journal.pcbi.1012387.g004 Fig 4 Case 1: Constant transmission rate.

Comparison between β0 identification during the training of the joint and split methods for the weekly infection data (ND = 13).

3.1.2 Cases 2 and 3: Time-dependent transmission rates

Case 2 consists of a simulation of disease spread according to the time-dependent transmission rate plotted in Fig 5. This β(t) behavior implies two waves of infection, with a maximum number of infectious individuals two orders of magnitude lower than in Case 1.

10.1371/journal.pcbi.1012387.g005 Fig 5 Case 2: Synthetic transmission rate.

Comparison between the reference solution of the SIR model and the PINN approximations with the joint (a) and split (b) approach. Grey bands provide the confidence interval of one standard deviation.

Fig 5 shows the reference values of the unknowns and the corresponding NN approximations. The dashed lines correspond to the mean outcome from 10 different runs, while the grey bands provide the confidence interval of plus/minus a standard deviation.

At most times the split method provides more stable and accurate results, with smaller variations from one run to another. The higher stability and speed of convergence of the split approach can be also appreciated from the error behavior on β^s during the training (Fig 6). The overall performance of the PINN approaches is summarized in Table 3.

10.1371/journal.pcbi.1012387.g006 Fig 6 Case 2: Synthetic transmission rate.

Comparison between the errors on β^s(t) during the training with the joint and split approach.

10.1371/journal.pcbi.1012387.t003 Table 3 Cases 2–3: Time-dependent transmission rates.

Training time and approximation errors for the state variables and the estimated parameters with the joint and split approach.

	Joint	Split	
Case 2			
Training time [s]	1936	717	
Error S	1.814 × 10−3	3.381 × 10−4	
Error I	2.501 × 10−2	7.754 × 10−3	
Error R	2.288 × 10−1	4.251 × 10−2	
Error β	2.678 × 10−1	4.127 × 10−1	
Error β (last 70d)	5.979 × 10−2	1.563 × 10−2	
Case 3			
Training time [s]	1906	726	
Error S	1.935 × 10−3	5.031 × 10−4	
Error I	4.729 × 10−3	6.006 × 10−3	
Error R	2.805 × 10−2	7.101 × 10−3	
Error Rt	4.692 × 10−1	6.988 × 10−1	
Error Rt (last 70d)	5.341 × 10−2	6.701 × 10−2	

Both approaches, however, fail to estimate the initial values of β (Fig 5). The low number of infected individuals and the presence of perturbed data produce an initial dynamic that the NNs erroneously learn by considering a larger number of initial infected individuals and a lower initial transmission rate. Errors of this kind represent a common hurdle in epidemiology, as the evolution of a disease is extremely difficult to be identified at the beginning of the epidemic outbreak under the assumption of a temporal-depending transmission rate. For this reason, Table 3 provides also the error for β during the last 70 days of simulation. These small errors confirm the accuracy of the PINN estimates when the data provide a clear signal.

Case 3 considers the Rt estimates supplied by the Italian health institute “Istituto Superiore della Sanità” (ISS) [34] as reference values for the effective reproduction number. These values, depicted in Fig 7, are used to evaluate the associated transmission rate β in the model and to synthetically generate the epidemiological data of infected individuals from February 21st to May 20th, 2020.

10.1371/journal.pcbi.1012387.g007 Fig 7 Case 3: Transmission rate of the Italian COVID-19.

Comparison between the reference solution of the SIR model and the PINN approximations with the joint (a) and split (b) approach. Grey bands provide the confidence interval for one standard deviation.

Note that for the first 20 days the values have been kept equal to 3.012, in order to simulate the free transmission of the pathogen in a completely susceptible population without restrictions in contacts. Fig 7 and Table 3 contain the outcome of the joint and split methods. Similarly to Case 2, the estimation of the temporal evolution of the transmission rate is hard for the initial times. Both methods can achieve a good accuracy after the first 20 days, as it can be deduced from Fig 7 and from the errors associated to the last 70 days of the simulation. The matching is quite accurate, with relative errors on the order of 10−3, and the variance from 10 different runs is also limited. While the gain in accuracy of the split method is not so clear in this case, the benefit in the training duration is still relevant. Similar results in terms of accuracy have been obtained on these test cases when using the reduced implementation (12) of the split and joint approaches.

3.1.3 Cases 4 and 5: Reduced model and hospitalization data

Case 4 is used to investigate the performance of the joint and split approaches for the reduced model (12) in a synthetic scenario where the data are subject to large errors. The unknown time-dependent transmission rate is the same as the one of Case 2, whose corresponding reproduction number is shown in Fig 8.

10.1371/journal.pcbi.1012387.g008 Fig 8 Case 4: Synthetic transmission rate with large data errors.

Reference solution and PINN approximation with the joint (a) and split (b) approach in the reduced SIR model (12) and noisy data.

The time domain is extended to tf = 120 days, which implies two complete waves of infection. To simulate the typically large uncertainties on the reported daily infections, we generate synthetic noisy data by perturbing the numerical solution for I(t˜j) with a Gaussian error having zero mean and coefficient of variation of 40%. The data are then rounded to the closest integer with the negative values set to 0.

The outcome of the PINN approximations are provided in Fig 8 and Table 4. The larger errors on the data reduce the accuracy in the estimate of the reproduction number. The two approaches are almost equivalent in terms of accuracy, but the split one has faster training times. Clearly, a large uncertainty in the training data reduces the reliability of the split approach, which mostly relies on this piece of information, while the joint approach compensate possible errors on data with the residuals of the governing equations. Nevertheless, the accuracy in the reproduction of the Rt appears to be in any case fairly satisfactory, given the large reported errors. It is worth noting that the solution with PINNs of the proposed reduced model clearly outperforms the one of the full SIR model when the data are subject to large errors, as reported in Appendix B in S1 Text.

10.1371/journal.pcbi.1012387.g009 Fig 9 Case 5: Synthetic data of infections and hospitalizations.

Reference solution and PINN approximation in the joint (a) and split (b) approaches.

10.1371/journal.pcbi.1012387.t004 Table 4 Cases 4–5: Reduced model and hospitalization data.

Training time and approximation errors for the state variables and the estimated parameters with the joint and split approaches when considering the hospitalization data (Case 5) or not (Case 4).

	Joint	Split	
Case 4			
Training time [s]	1658	1053	
Error I	1.411 × 10−1	1.331 × 10−1	
Error Rt	4.961 × 10−1	4.744 × 10−1	
Error Rt (last 100d)	3.141 × 10−1	3.336 × 10−1	
Case 5			
Training time [s]	1829	1445	
Error ΔH	8.783 × 10−3	4.722 × 10−3	
Error I	1.209 × 10−1	7.434 × 10−2	
Error Rt	4.407 × 10−1	4.992 × 10−1	
Error Rt (last 100d)	2.410 × 10−1	1.240 × 10−1	
Error σ	3.858 × 10−1	1.417 × 10−1	
Error σ (last 100d)	2.626 × 10−1	1.145 × 10−1	

Case 5 aims at improving the estimate of Rt by introducing the information on the daily hospitalizations. This entails that also the unknown parameter σ, which is fraction of infected individuals that become hospitalized, becomes a NN to be trained. The reference σ for this case is shown in Fig 9. The results of the joint and split approaches are summarized in Fig 9 and Table 4. The inclusion of more reliable data implies a better estimate of Rt with respect to Case 4. This is particularly evident for the split approach, which is able to provide good estimates of both temporal depending parameters (Rt and σ), and thus, lower errors (Table 4).

3.2 Application to real data

3.2.1 Cases 6 and 7: Italian COVID-19 surveillance data

Cases 6 and 7 apply the procedure adopted in Case 5 to the real setting of the Italian COVID-19 epidemic outbreak. The fraction of the number of infected individuals that becomes hospitalized, σ, is assumed to be constant in time in Case 6 and time-dependent in Case 7. The goal is to estimate Rt and σ by means of the presented PINN approaches.

The results of the joint and the split approaches are shown in Figs 10 and 11 for Cases 6 and 7, respectively. We use the reproduction number evaluated by ISS (Fig 10) as a reference value for comparison. However, it is important to keep in mind that its values have been obtained with a data driven approach (renewal equation, [10]) on the symptomatic infected individuals. In Case 6 the joint approach provides an acceptable accuracy on the new infection data, while the peak of the daily hospitalizations is underestimated (Fig 10a). The split approach, instead, is firstly trained on the daily hospitalization well retrieving these high-fidelity data. The second part of the training, considering both the daily infections and the reduced model equations, does not achieve the same level of accuracy for the infection data with respect to the joint approach (Fig 10b). For what concerns the estimation of Rt, the joint approach strongly underestimates its value at the beginning of the epidemic. By distinction, the split approach provides a trend of Rt that is fully consistent with the ISS estimates, i.e., a decrease during the first weeks of the epidemic, followed by an almost stationary value around 1 during the recession phase. The overall performance of the PINN approaches are summarized in Table 5.

10.1371/journal.pcbi.1012387.g010 Fig 10 Case 6: Constant hospitalization rate.

PINN approximations with the joint (a) and split (b) methods for the Italian COVID-19 outbreak with constant σ. The blue trajectory of Rt is the official estimate by ISS.

10.1371/journal.pcbi.1012387.g011 Fig 11 Case 7: Time-dependent hospitalization rate.

PINN approximations with the joint (a) and split (b) methods for the Italian COVID-19 outbreak with time-dependent σ. The blue trajectory of Rt is the official estimate by ISS.

10.1371/journal.pcbi.1012387.t005 Table 5 Cases 6–7: Application to the Italian COVID-19 data.

Training time and approximation errors for the state variables and the estimated parameters with the joint and split approach.

	Joint	Split	
Case 6			
Training time [s]	1655	1042	
Error Rt	3.881 × 10−1	2.495 × 10−1	
Estimated σ^	0.0686	0.0849	
Case 7			
Training time [s]	1778	1101	
Error Rt	3.646 × 10−1	2.236 × 10−1	

Considering a time-dependent σ (Case 7) helped both approaches to improve the estimate of both hospitalized and infectious data. Both approaches depict a similar trend for σ, where the fraction of infected individuals that became hospitalized decreases during March 2020 from a peak of about 23% to about 3%, which is a realistic outcome in the framework of Italian COVID-19 outbreak. The main differences among the two approaches are still at the beginning of the outbreak, where the joint approach suggests a lower value of σ, while underestimating the values of Rt. The training time required by the split approach was about 40% of the time for the joint approach (Table 5).

3.2.2 Forecasting

Cases 1–7 use the complete set of data to infer the past dynamics of the unknown state variables and parameters. This section analyzes the performance of the PINN methods in a realistic forecasting scenarios where PINNs are trained using only a portion of the data, and then employed to produce a short-term forecast of the epidemic. TNote that the training still consists of an inverse problem, since the time-dependent parameters of the model are still unknown.

In the Italian COVID-19 setting of Case 7, we trained the PINN model on four temporal windows of increasing length. As first, the PINN model is trained on the data from February 21st, 2020 (day 0) to March 6th, 2020 (day 15). The NNs are further trained on the data of the other windows (days [0, 30], [0, 45], [0, 60]), i.e. the training of the same architectures is carried on by sequentially adding new data corresponding to longer temporal widows. This approach has the advantage of improving the previously trained NNs, instead of starting new NNs from scratch. Short term forecasts of 15 days are produced at the end of each training window using the neural networks in extrapolation. Fig 12 compares the trained solutions and the forecasts at each window with the reported data. The PINN solution obtained by training the NNs on the whole set of data (result of Case 7) is shown as reference solution.

10.1371/journal.pcbi.1012387.g012 Fig 12 Case 7: Forecasting.

PINN predictions of the Italian COVID-19 evolution using the split method on subsequent training time intervals: (a) 0–15 days, (b) 0–30 days, (c) 0–45 days, (d) 0–60 days. The blue dashed lines, here used as reference solution, are the outputs of the split approach computed in Case 7 (Fig 11b).

As one could expect, the projections are not accurate when few data are available or when the model is close to the peak of the epidemic (Fig 12a and 12b). Good outcomes are achieved during the recession (Fig 12c and 12f).

Similar forecast results are obtained using the joint PINN approach (see Appendix C in S1 Text).

4 Discussion and conclusions

This work proposes two innovative ideas to improve the application of PINNs for the solution of SIR-based epidemiological models, and to estimate the time-dependent transmission rate, or the effective reproduction number, of an epidemic. The first idea consists in splitting the training of the NNs in two steps: the first step provides the fit on the epidemiological data, while the second step minimizes the residual on the model equations. The performance of the split approach has been compared to a standard PINN application, which trains simultaneously the NNs on the joint loss function on data and residual. The second idea consists in implementing a modification to the basic model equations, possibly removing the state variables that are not directly related in the disease transmission and the associated redundant terms in the loss function. This reduced PINN model has been extended to include both infection cases and hospitalization data, which are usually more reliable pieces of information.

Synthetic test Cases 1–3 showed that, when infectious data are subject to small errors, both the split and joint PINN approaches are able to retrieve with high accuracy the system dynamics. The initial training on the data of the split approach provides a clear advantage when minimizing the residual on the model equations and estimating the reproduction number. In fact, it achieves lower errors (up to an order of magnitude) with faster computational times (speed up larger than 60% in Cases 2 and 3). This is probably due to more stable results during the training epochs, as depicted in Fig 3b for a constant transmission (Case 1), and Fig 6 for a time-dependent transmission (Case 2). However, the simultaneous estimate of the initial conditions and the initial transmission proved to be particularly challenging for both PINN approaches (Figs 5 and 7). In fact, even small errors on the data become particularly relevant when there are low number of infections, such as at the beginning of the epidemic. Model results could improve by assuming a constant initial transmission rate, which is generally in agreement with the free circulation of the pathogen in absence of interventions.

Besides this inaccuracies in the early times of the outbreak, the Rt estimated by PINN in Case 3 has a similarly accuracy to the one obtained with the renewal equation [10], approach that is commonly adopted during outbreaks (see Appendix D in S1 Text).

The large errors that typically characterize the data of the reported daily infections might deteriorate the retrieval of the temporal changes of the transmission rate and the associated effective reproduction number (Case 4, Fig 8 and G in S1 Text). For this reason, numerous epidemiological analysis are based on data that are less biased by the surveillance system, such as daily hospitalization data, e.g., [4]. Case 5 shows that the use of daily hospitalizations and infections into the reduced PINN model allows to improve the accuracy in the estimation of the uncertain time-varying parameters, in particular both the effective reproduction number Rt and the fraction of infected individuals requiring hospitalization σ. The split approach still outperforms the joint counterpart with 20% savings in the training cost, however both approaches still present limitations at the beginning of the outbreak.

The application to the Italian COVID-19 epidemic (Cases 6 and 7) emphasizes the importance of considering the fraction of hospitalized individuals, σ, as a temporal-dependent parameter. In fact, the PINN approximations were not able to accurately follow both time series of daily hospitalized and infectious data when considering a constant σ (Fig 10). Results notably improve in the case of a temporal-dependent σ (Fig 11). Many modeling studies assume a constant σ, with possible temporal variations assigned only on the arrival of new disease variants or after the deployment of vaccines. Other processes that might directly impact this parameter are typically neglected. For example, in the early stages of the outbreak, the fear of the new disease might prompt many symptomatic infected individuals to seek health care at the hospital (thus generating a large value of σ). The subsequent overcrowding of the hospitals and improvement of treatment at home might reduce the value of σ in time. The time-dependent σ values estimated by the PINN approaches (Fig 11) show exactly such a dynamic, with small differences at the beginning. We argue that also in this application the split approach outperforms the joint one. Besides the advantages in the computational times (40% faster), the effective reproduction number obtained with the split approach depicts a closer trajectory to the reference Rt estimated by the Italian Institute of Health (Fig 11).

The shorter training times and higher accuracy that we consistently obtained for the split approach in each test case can be attributed to the structure of the loss functions. Splitting the training implies the minimization of two loss functions which are simpler, thus sparing the complex solution of a multi-objective optimization problem.

The results presented in Cases 1–7 demonstrate the ability of the PINNs model to retrieve the past dynamics of an epidemic, to infer the temporal changes in the parameters, and to possibly fill the gaps among the data (Fig 4). A clear benefit of PINNs with respect to other traditional epidemiological approaches is that PINNs can directly produce short-term forecasts and projections of the epidemics. In fact, the calibrated PINNs are functions of time and can be extrapolated outside the training window. This is explored by analyzing the forecasts produced by the split PINN approach in the framework of the Italian COVID-19 pandemic (Fig 12).

In general, forecasts based on extrapolation might be extremely far from the real data due to possible strong fluctuations of the functions outside the training window. Our results show that the PINNs show smooth behaviors also in extrapolation, which in general are consistent with the dynamics of the disease spread. We attribute this consistency to the fact that the NNs are trained using the residual of the ODEs, which informs the model of the ongoing trend. PINNs, thus, can be particularly interesting for short-term forecasting as well. As one could expect, the quality of the forecast changes for different training window. If the training interval stops in conjunction with a peak of the cases, the model hardly predicts the fall in infections in the following days, as in Fig 12b. Instead, predictions during recession are more accurate (Fig 12c-d). This kind of results are common to many models that attempt predicting the dynamics of an outbreak. Improvements could be achieved, for example, by using universal models trained on scaled data [39].

A major limitation of the PINN approach with respect to traditional statistical methods for Rt inference (such as [10]), is that the deterministic nature of PINN does not provide a quantification of the uncertainty. For example, when dealing with strongly perturbed data as in Case 4, it would be reasonable to expect a large confidence interval around the estimated Rt values (see Fig G in S1 Text). Uncertainty quantification (UQ) is a fundamental ingredient in epidemiological analysis. Unfortunately, it is frequently missing in the PINNs results. Future developments of the proposed split PINN approach should consider UQ in more complex compartmental models, for example following the framework proposed by Linka et al. [40] which combines NNs and Bayesian inference.

This study focuses on the standard and simple SIR model. While the split approach can be easily adapted to more complex compartmental models (which, for example, consider an exposed compartment, deaths, re-infections, and vaccinations), the reduced equations described in (12) will require ad-hoc formulations depending on the model. It is important to underline that in this simple setting it would be possible to directly use purely data-driven DNNs to attempt predicting the future data as done in Case 7 (Fig 12). However a comparison between the results of PINN and DNN has not been presented in this manuscript because PINN has a wider goal with respect to DNN: PINNs in fact allow the user to also estimate and predict the dynamics of model parameters and/or state variables that are not possible to link to the data if not using the model equations. This information is not available in more standard DNNs or statistical approaches such as the renewal equation, and it constitutes the main advantage of PINNs methods.

In conclusion, the proposed split PINN-based approach is a robust and easy-to-implement tool to monitor the initial spreading of a disease. It provides estimates of the temporal changes in the model parameters, which is essential to produce more accurate short-term forecasts.

Supporting information

S1 Text Appendix A. Neural Network architectures. Appendix B. Full SIR model with large data errors. Appendix C. Forecasting using the joint PINN approach. Appendix D. Comparison with the renewal equation.

(PDF)

The authors are members of the Gruppo Nazionale Calcolo Scientifico—Istituto Nazionale di Alta Matematica (GNCS-INdAM) and C.M. is part of the project CUP_E53C23001670001. C.M. and M.F. acknowledge the support provided by BIRD2023 and ICEA Department of the University of Padova through the project “SurMoDeL: Deep Learning Surrogate Models for reservoir characterization”. D.P. acknowledges the support provided by the PRIN project “Hydro-ROM, Reduced order models of hydraulic protection systems for extreme water hazards” H53D23001350006.

10.1371/journal.pcbi.1012387.r001
Decision Letter 0
Pitzer Virginia E. Section Editor
Scarpino Samuel V. Academic Editor
© 2024 Pitzer, Scarpino
2024
Pitzer, Scarpino
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version0
19 Feb 2024

Dear Dr Millevoi,

Thank you very much for submitting your manuscript "A Physics-Informed Neural Network approach for compartmental epidemiological models" for consideration at PLOS Computational Biology.

As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. In light of the reviews (below this email), we would like to invite the resubmission of a significantly-revised version that takes into account the reviewers' comments.

I agree with both reviewers that this work potentially represents an important contribution to the field of computational epidemiology and infectious disease modeling; however, substantial work is needed to clarify why specific modeling techniques were selected and how the results may compare to more established methods. The authors should clarify whether true out-of-sample performance was measured (not necessarily prospectively, but at least on fully held-out data). If such out-of-sample testing was not performed doing so will be essential for fully addressing the reviewer comments. I also would like the authors to revise their work to make it more accessible to a broad audience. Please pay close attention to the detailed comments from the two reviewers.

We cannot make any decision about publication until we have seen the revised manuscript and your response to the reviewers' comments. Your revised manuscript is also likely to be sent to reviewers for further evaluation.

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Sincerely,

Samuel V. Scarpino

Academic Editor

PLOS Computational Biology

Virginia Pitzer

Section Editor

PLOS Computational Biology

***********************

I agree with both reviewers that this work potentially represents an important contribution to the field of computational epidemiology and infectious disease modeling; however, substantial work is needed to clarify why specific modeling techniques were selected and how the results may compare to more established methods. The authors should clarify whether true out-of-sample performance was measured (not necessarily prospectively, but at least on fully held-out data). If such out-of-sample testing was not performed doing so will be essential for fully addressing the reviewer comments. I also would like the authors to revise their work to make it more accessible to a broad audience. Please pay close attention to the detailed comments from the two reviewers.

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: The paper introduces an innovative approach to address inverse problems related

to epidemiological data and utilizing PINNs. Specifically, two simple innovative

modifications, named the reduced and split approaches, are introduced

and compared in a comprehensive campaign of numerical results involving both

synthetic and realistic scenarios.

Strengths

• The authors have shown a clear understanding of the existing state-of-theart

about Machine Learning techniques for solving Inverse Problems and

in particular the application to the epidemiological context;

• The proposed methods introduce little modifications adapted to the epidemiological

framework to the well-established framework of PINNs.

Major Comments

• I believe there is a lack of explicit justification for employing Neural Networks

with large dimensions, as indicated in lines 369-373, particularly

regarding their depth and width. Could this choice be somehow linked to

the recent progress in infinitely-wide neural networks?

• It would be beneficial to include a thorough sensitivity analysis for the parameters

that define the topology of the neural network (like some comparison

between different architectures, even with smaller neurons or layer).

This is because the number of parameters used to calibrate PINNs’ weights

and biases significantly impacts the overall complexity of the training process.

• In lines 619–620, the authors assess that the proposed split PINN-based

approach is a robust and straightforward tool for monitoring the initial

spread of a disease. However, I am unable to evaluate the goodness of this

tool in the prediction sense, since, as I understand it, the numerical results

are retrospective. They approximate the dynamics within the same period

as the data used for training and have not been utilized to extrapolate

future behavior. Could you provide some clarification on this aspect?

Recommendation

I recommend that the paper addresses the previous points, and then it can be

accepted.

Reviewer #2: In this study, the authors apply a neural network approach informed by differential equations (Physics-Informed Neural Networks) to infer the parameters’ values that regulate the dynamics of an SIR model.

The results show that the approach adopted by the authors, based on a reduced-split of the training set, outperforms the methods that have been previously proposed in the literature both in terms of accuracy and computational times.

The study presents a few strengths and weaknesses at the same time, which I summarize in the following. Overall, the paper quality does not currently meet the standards to be published in PLOS Computational Biology and I think the paper requires a major revision before it can be considered for publication PCB.

I am suggesting a number of changes that I hope could improve the manuscript.

Strengths:

- The use of PINN in infectious disease modeling is promising – yet understudied – and the method proposed by the author improves on previous results, thus showing a more effective way of using PINN in this field.

- The study considers a range of application scenarios that include real-world data, and more realistic cases than previous studies.

Weaknesses:

- The main result of the study is the introduction of a novel training approach for PINNs applied to the SIR model. While I consider it interesting and useful, in terms of impact I am not sure it will be of high interest to the broad audience of PCB.

- For the readers who are not familiar with PINNs, it will be difficult to assess the quality of the results with respect to more traditional approaches. The paper is often technical, and it is hard to compare the results against more traditional infectious disease modeling techniques.

- The presentation of the results and the structure of the manuscript follows a bullet list style (Case 1, Case 2, etc.) that is not very engaging. Figures and tables repeat the same structure over and over, and they are sometimes hard to interpret.

- The paper suffers from an overly technical notation that can appear obscure to biologists, epidemiologists and infectious disease modellers.

Comments and suggestions

1. It would be important for infectious disease modelers to understand if PINNs provide benefits with respect to classic methods that are used to estimate Rt in real world cases. For instance, the widely adopted Rt estimation method proposed by Cori et al. AJE 2013. Can PINNs replace these approaches to some extent? In general, it is not obvious for those who work in IDM what tasks can be accomplished by PINN with better or similar performance than commonly used methods.

2. On a similar note, I did not fully grasp what is the portion of the epidemic curve that is used to train the PINN in each scenario. Does the model need the full curve as training input? In this case, its use would be strongly limited in the early phase of an outbreak.

3. Choosing a fixed value for the average infectious period (5 days) seems a strong assumption. Is this kept constant in all experiments? Often, there is a large uncertainty affecting estimates of the infectious period. How sensitive are the results to changes in the infectious period value?

4. The section Methods currently describes in detail the mathematical structure of the PINN. I would suggest extending the section with a description of the numerical methods that right now appear at the beginning of the results. All the information regarding the numerical experiments (size of training, testing, error metrics, etc.) should be part of the methods.

5. A graphic description of the numerical methods could also improve the readability of the paper. A summary figure describing how the epidemiological data are used in the training phase, what is the output of the model, etc.

6. I would recommend changing the structure of the manuscript with sections title that do not just report the names “Case 1, Case 2, Case 3…” For instance, Results could be grouped into two main parts. One considering synthetic data and the other considering COVID-19 data.

7. One major limitation of the PINN approach is to fit a deterministic SIR model to the data. The deterministic model leads to parameter estimates with almost exact values that are difficult to interpret when compared to traditional statistical methods used in the literature that provide a confidence interval of credible values. Is there a way to overcome this limitation by extending the range of possible outcomes of the model? I understand this is an open question that may lack a satisfying answer, but at least I would discuss this limitation more extensively at the end of the paper.

**********

Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data and code should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

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Attachment Submitted filename: review_PLOS.pdf

10.1371/journal.pcbi.1012387.r002
Author response to Decision Letter 0
Submission Version1
19 Apr 2024

Attachment Submitted filename: Response letter -PLOS CB.pdf

10.1371/journal.pcbi.1012387.r003
Decision Letter 1
Pitzer Virginia E. Section Editor
Scarpino Samuel V. Academic Editor
© 2024 Pitzer, Scarpino
2024
Pitzer, Scarpino
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version1
30 May 2024

Dear Dr Millevoi,

Thank you very much for submitting your manuscript "A Physics-Informed Neural Network approach for compartmental epidemiological models" for consideration at PLOS Computational Biology.

As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. In light of the reviews (below this email), we would like to invite the resubmission of a significantly-revised version that takes into account the reviewers' comments.

I appreciate the authors' effort on the revisions and agree with the reviewers that the manuscript has been strengthened as a result. I agree with Reviewer 3's comments that the work needs some improvements. In particular, the authors should pay careful attention to addressing concerns about whether regularization is sufficient, whether it is appropriate to use a Python package that is core to the method but is not longer being actively maintained, whether off-the-shelf deep neural networks could perform as well or better than what is proposed here, and whether the forecasting objectives are sufficiently complicated to meaningfully test the approach. I also agree that there are many places where the methodology can be more clearly specified. Lastly, the Reviewers raised concerns about whether all necessary code/data is available to reproduce the entire study. If the authors cannot provide all necessary code/data they must formally request a waiver of the Journal's policies, which I cannot promise will be granted. I believe this work is interesting and could be impactful, so I hope that authors pay careful attention to the detailed comments from Reviewer 3 and address the code/data availability concerns.

We cannot make any decision about publication until we have seen the revised manuscript and your response to the reviewers' comments. Your revised manuscript is also likely to be sent to reviewers for further evaluation.

When you are ready to resubmit, please upload the following:

[1] A letter containing a detailed list of your responses to the review comments and a description of the changes you have made in the manuscript. Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

[2] Two versions of the revised manuscript: one with either highlights or tracked changes denoting where the text has been changed; the other a clean version (uploaded as the manuscript file).

Important additional instructions are given below your reviewer comments.

Please prepare and submit your revised manuscript within 60 days. If you anticipate any delay, please let us know the expected resubmission date by replying to this email. Please note that revised manuscripts received after the 60-day due date may require evaluation and peer review similar to newly submitted manuscripts.

Thank you again for your submission. We hope that our editorial process has been constructive so far, and we welcome your feedback at any time. Please don't hesitate to contact us if you have any questions or comments.

Sincerely,

Samuel V. Scarpino

Academic Editor

PLOS Computational Biology

Virginia Pitzer

Section Editor

PLOS Computational Biology

***********************

I appreciate the authors' effort on the revisions and agree with the reviewers that the manuscript has been strengthened as a result. I agree with Reviewer 3's comments that the work needs some improvements. In particular, the authors should pay careful attention to addressing concerns about whether regularization is sufficient, whether it is appropriate to use a Python package that is core to the method but is not longer being actively maintained, whether off-the-shelf deep neural networks could perform as well or better than what is proposed here, and whether the forecasting objectives are sufficiently complicated to meaningfully test the approach. I also agree that there are many places where the methodology can be more clearly specified. Lastly, the Reviewers raised concerns about whether all necessary code/data is available to reproduce the entire study. If the authors cannot provide all necessary code/data they must formally request a waive of the Journal's policies, which I cannot promise will be granted. I believe this work is interesting and could be impactful, so I hope that authors pay careful attention to the detailed comments from Reviewer 3 and address the code/data availability concerns.

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: This work has strongly improved after the revision and it represents a significant contribution about the interplay of Machine Learning Schemes and epidemiology.

By my side, all the raised questions have been answered.

Reviewer #2: I thank the authors for revising the manuscript in light of my remarks. All my comments have been addressed.

Reviewer #3: 

The authors propose the use of PINNs to regress time-dependent epidemiological parameters from data. They propose two training strategies called joint and split and also propose a method independent of the initial condition where the initial conditions do not enter the loss function. The article has been thoroughly reviewed before and the epidemiological models used are time-tested and well established in the literature. The innovation proposed by the authors is in the design and training of the PINNs. To do this the authors use a publicly available Python module called SciANN based on Tensorflow. After the first round of reviews, the authors added an example of forecasting on held-out data. And improved several parts of the work and the draft. However, I still find the draft lacking in some respects, and the methods used by the authors are not sufficient.

While the core idea proposed by the authors is interesting, the implementation needs more rigor and some of the conclusions drawn by the authors might be affected by this lack of rigor. The authors introduce and use concepts that are not made clear to the audience this is aimed at and the training of the NN models has not been done following best-practices from proper ablation studies. The authors have added some studies of the NN architecture in the supplementary document but thia is not sufficient. The authors have not explored learning rate schedules, regularization, or dependence on activation functions.

Comments:

- In lines 423-425, the authors mention calibrating the weights in the loss function using eigenvalues of the NTK and cite a reference. The authors do not put any details on what they do to calibrate the weights using an NTK. From the code, it seems like they use a built-in function in the SciANN module to do this. However, they do not show whether their errors improve by using the NTK calibration or not. I am guessing it might help with the solution of the inverse problem but not with the out-of-sample prediction.

- From the supplementary material on the NN ablation study, it is not clear why the authors get larger errors for models of higher complexity. The authors seem to train the models on a fixed number of epochs. If they do so, more complex models are bound to overfit as the authors do not mention using any explicit regularization of the weights of the NN. It is straightforward to understand that NNs of greater complexity will be prone to overfitting if no implicit or explicit regularization is used. I would suggest the authors use early stopping as an implicit regularizer and/or add regularization to the loss function and check how that affects the NN training. This can potentially reduce training times too.

- The actual time taken by the NNs to train is not meaningful as the authors do not specify the computer architecture they use. The code provided by the authors is not well documented and is a single Python notebook that does not allow one to reproduce all the results. It looks more like a sample. Moreover, the authors use the package SciANN which is limited to Python <= v3.10 and works only with Tensorflow < v2.11. The training times and accuracy of the NN can be highly dependent on the Tensorflow version. The authors do not specify which version of SciANN/Tensorflow they use. If the authors want their work to be useful in the future they should write a code that can be used without much trouble in the future. The authors could try implementing a PINN in Tensorflow or Pytorch. I know the objective of the paper is about training strategies, but many of the conclusions the authors come to can be dependent on the fact that they use a deprecated framework for their NNs (SciANN is no longer maintained as of 02/2023).

- In SI S3, the authors show that using the renewal equation approach of Cori et al. leads to comparable or better results than using PINNs. Given the complexity of the approach proposed by the authors, what is the real advantage of using their method other than just applying machine learning to this problem?

- In their response, the authors emphasize: “A final important consideration is that PINNs benefit from the so-called ‘transfer learning’ typical of Neural Networks, that is the possibility to adapt an already trained NN to new data. This has been explored, for example, in the application to forecasting (Section 2.2.2).” The authors are not clear about what they call “transfer learning”. It's a very loaded term and can mean several different things. The authors do not specify their strategy for transfer learning. Do they freeze the weights and add layers to the NN or do they continue the training with the same architecture as before just adding more data? If it is the latter, then it is not really transfer learning as data from the same domain and modality is being used for further training the same architecture without any fixed weights. I suspect that without using their version of “transfer learning” the NNs would do equally well leading to a lack of reason for using this transfer learning method. Please refer to A Comprehensive Survey on Transfer Learning by Zhuang et al. for more details. I would strongly recommend the authors to not incorrectly refer to their PINNS using transfer learning.

- The authors claim: “As one could expect, the projections are not accurate when few data are available or when the model is close to the peak of the epidemic (Figs. 12a and 12b). Good outcomes are achieved during the recession (Figs. 12c and 12f).” How is the approach proposed in this paper better than that proposed in Emergence of universality in the transmission dynamics of COVID-19 by Paul et al. or any other models that perform significantly better during recession than during the growth of the epidemic? If not, then the PINN approach proposed in this work is, at best, a descriptive approach and not a predictive approach which makes the use of PINNs in this case a bit questionable.

- I do not understand why the authors create their own accuracy metric (eq. 36) rather than using standardized ones like R^2 which has a very clear interpretation.

- The simulated scenarios are very simple and I suspect even a DNN can be used to find the solution. The authors do not demonstrate that their PINN outperforms a simple DNN implementation.

- I do not see a comparison between split and reduced-split nor where the reduced-split method significantly outperforms the split method. The code does not seem to have any examples of the two separately.

- For the case of split training, it seems like the SciANN module is using a reduce-on-plateau learning rate schedule where the learning rate is reduced by a factor of 2 on plateau of the validation loss. Hence, the authors’ statement in section 1.6 that they use a learning rate of 0.001 in all cases is incorrect. Secondly, it is well known that learning rate schedules significantly improve training. I’d encourage the authors to explore the use of learning rate schedules in training the PINNs.

- Using Tensorflow 2.11 and Keras 2.11 I get an order of magnitude smaller errors that the authors get with the code they have shared in their repository. Since the authors did not specify the versions of the modules they used, it is impossible for me to say whether the improvement is due to the module version or the initialization of random numbers (unlikely) that have not been seeded in the provided notebook.

- The authors have provided no insight as to why the split method works better than the join. In a real-world scenario, it might be imaginable that the infection rate is affected by real-world uncertainties like testing strategies etc., in which case a data-driven approach to this estimating the infection rate might work better. However, for simulated data, this is not the case. So, some insights into why split training works better would be useful.

The fork of the repository run with TensorFlow 2.11.0 can be found here where the discrepancy in error estimates can be verified (the requirements file specifies the module versions): https://github.com/anonymousacademicc/EpiPINN

**********

Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data and code should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: None

Reviewer #2: Yes

Reviewer #3: No: The authors have only provided sample code and not all the code necessary to reproduce their work. They have specified a link to the data but have not provided code for the ETL of the data. The link they provide simply leads to a landing page: https://www.epicentro.iss.it/en/coronavirus/. With the information and the code that they provide it is impossible to reproduce their results more so because they have not specified the version of the python modules used. Moreover, they use a python module that is no longer supported by the developers and is deprecated. The module depends on interpreters and modules that will be deprecated in the near future limiting the utility of the computational method that they develop and inhibiting the reproducibility of their work.

**********

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Reviewer #1: No

Reviewer #2: No

Reviewer #3: No

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10.1371/journal.pcbi.1012387.r004
Author response to Decision Letter 1
Submission Version2
5 Jul 2024

Attachment Submitted filename: Response Letter EPIPINN.pdf

10.1371/journal.pcbi.1012387.r005
Decision Letter 2
Pitzer Virginia E. Section Editor
Scarpino Samuel V. Academic Editor
© 2024 Pitzer, Scarpino
2024
Pitzer, Scarpino
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Submission Version2
5 Aug 2024

Dear Dr Millevoi,

We are pleased to inform you that your manuscript 'A Physics-Informed Neural Network approach for compartmental epidemiological models' has been provisionally accepted for publication in PLOS Computational Biology.

The reviewer's comments are included below for your consideration, although we feel they are beyond the scope of the current analysis and do not need to be addressed at this time.

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Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology. 

Best regards,

Samuel V. Scarpino

Academic Editor

PLOS Computational Biology

Virginia Pitzer

Section Editor

PLOS Computational Biology

***********************************************************

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #3: I thank the authors for the detailed response to the review and the corrections. I have a few major concerns

1. The authors say that there are three sources of errors in an NN. I agree. The last two that they cite in that list can be mitigated by regularization. Yet, the authors claim that regularization is unnecessary for PINNs. I am not sure the two statements are not contradictory. Yes, it is true that certain implementations of PINN show a reduced necessity for regularizers. However this is not true for any and every PINN regardless of their complexity. While the additional terms in the loss function can help regularize the neural network, even in the paper the authors cite, it does not say that regularization is not necessary. Unfortunately, in the world of neural networks such things can only be confirmed by empirical evidence. The authors should show that it is not necessary if they wish to claim so.

2. I disagree with the authors that a computational paper proposing a new neural network based method can be published without supporting code to reproduce their work. I understand that dependencies tend to get deprecated over time. However, it is not acceptable that reproducing the work is tedious even during the review process a few months after the completion of the work. I maintain that the burden of reproducibility of the work and the creating of a code base that others can further develop lies on the authors. The early retirement of a lead developer of a code not maintained by the authors cannot be an excuse for the reproduction of the work being unnecessarily painful.

3. The authors claim that forecasting is not their goal. I can accept that. Then I do not understand why it is presented in the paper even though I do understand that the previous reviewers asked for it. It can be simply stated as being something outside the scope of the work. If the goal is just parameter estimation, I do not see how this approach is doing much better than other approaches. The authors themselves show that the renewal equation works as well. The authors make four cases for the advantage of PINNs over the renewal model but none of them can lead to significantly better or quicker estimation of parameters. Also, the authors already admitted that forecasting is not the objective. I do not see what advantage their model is providing. Even if the authors could show that they are doing better than a simple DNN, it might have held some credibility. I understand that a simple DNN would miss out on the underlying epidemiological dynamics, but then the entire point of using a simple DNN is to see how well one can do without using physics information, i.e., how much better a DNN can do when knowledge of the dynamics is added.

Given these reasons, I do not find a compelling case for recommending this paper for publication.

**********

Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

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Reviewer #3: Yes

**********

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Reviewer #3: Yes: Ayan Paul

10.1371/journal.pcbi.1012387.r006
Acceptance letter
Pitzer Virginia E. Section Editor
Scarpino Samuel V. Academic Editor
© 2024 Pitzer, Scarpino
2024
Pitzer, Scarpino
https://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
28 Aug 2024

PCOMPBIOL-D-23-02037R2

A Physics-Informed Neural Network approach for compartmental epidemiological models

Dear Dr Millevoi,

I am pleased to inform you that your manuscript has been formally accepted for publication in PLOS Computational Biology. Your manuscript is now with our production department and you will be notified of the publication date in due course.

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Anita Estes

PLOS Computational Biology | Carlyle House, Carlyle Road, Cambridge CB4 3DN | United Kingdom ploscompbiol@plos.org | Phone +44 (0) 1223-442824 | ploscompbiol.org | @PLOSCompBiol
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References

1 Guzzetta G , Poletti P , Ajelli M , Trentini F , Marziano V , Cereda D , et al . Potential short-term outcome of an uncontrolled COVID-19 epidemic in Lombardy, Italy, February to March 2020. Eurosurveillance. 2020;25 (12 ):2000293. doi: 10.2807/1560-7917.ES.2020.25.12.2000293 32234117
2 Marziano V , Guzzetta G , Rondinone BM , Boccuni F , Riccardo F , Bella A , et al . Retrospective analysis of the Italian exit strategy from COVID-19 lockdown. Proceedings of the National Academy of Sciences. 2021;118 (4 ):e2019617118. doi: 10.1073/pnas.2019617118 33414277
3 Gatto M , Bertuzzo E , Mari L , Miccoli S , Carraro L , Casagrandi R , et al . Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. Proceedings of the National Academy of Sciences. 2020;117 (19 ):10484–10491. doi: 10.1073/pnas.2004978117 32327608
4 Bertuzzo E , Mari L , Pasetto D , Miccoli S , Casagrandi R , Gatto M , et al . The geography of COVID-19 spread in Italy and implications for the relaxation of confinement measures. Nature Communications. 2020;11 (1 ). doi: 10.1038/s41467-020-18050-2 32848152
5 Marziano V , Guzzetta G , Mammone A , Riccardo F , Poletti P , Trentini F , et al . The effect of COVID-19 vaccination in Italy and perspectives for living with the virus. Nature Communications. 2021;12 (1 ):7272. doi: 10.1038/s41467-021-27532-w 34907206
6 Gozzi N , Chinazzi M , Davis JT , Mu K , Piontti APy , Ajelli M , et al . Anatomy of the first six months of COVID-19 vaccination campaign in Italy. PLOS Computational Biology. 2022;18 (5 ):e1010146. doi: 10.1371/journal.pcbi.1010146 35613248
7 Parolini N , Dede’ L , Ardenghi G , Quarteroni A . Modelling the COVID-19 epidemic and the vaccination campaign in Italy by the SUIHTER model. Infectious Disease Modelling. 2022;7 (2 ):45–63. doi: 10.1016/j.idm.2022.03.002 35284699
8 Lemaitre JC , Pasetto D , Zanon M , Bertuzzo E , Mari L , Miccoli S , et al . Optimal control of the spatial allocation of COVID-19 vaccines: Italy as a case study. PLoS Computational Biology. 2022;18 (7 ). doi: 10.1371/journal.pcbi.1010237 35802755
9 Ziarelli G , Dede’ L , Parolini N , Verani M , Quarteroni A . Optimized numerical solutions of SIRDVW multiage model controlling SARS-CoV-2 vaccine roll out: An application to the Italian scenario. Infectious Disease Modelling. 2023;8 (3 ):672–703. doi: 10.1016/j.idm.2023.05.012 37346476
10 Cori A , Ferguson NM , Fraser C , Cauchemez S . A new framework and software to estimate time-varying reproduction numbers during epidemics. Am J Epidemiol. 2013;178 (9 ):1505–1512. doi: 10.1093/aje/kwt133 24043437
11 Pasetto D , Lemaitre JC , Bertuzzo E , Gatto M , Rinaldo A . Range of reproduction number estimates for COVID-19 spread. Biochemical and Biophysical Research Communications. 2021;538 :253–258. doi: 10.1016/j.bbrc.2020.12.003 33342517
12 Trevisin C , Bertuzzo E , Pasetto D , Mari L , Miccoli S , Casagrandi R , et al . Spatially explicit effective reproduction numbers from incidence and mobility data. Proceedings of the National Academy of Sciences of the United States of America. 2023;120 (20 ):e2219816120. doi: 10.1073/pnas.2219816120 37159476
13 Mari L , Casagrandi R , Bertuzzo E , Pasetto D , Miccoli S , Rinaldo A , et al . The epidemicity index of recurrent SARS-CoV-2 infections. Nature Communications. 2021;12 (1 ). doi: 10.1038/s41467-021-22878-7 33980858
14 van den Driessche P , Watmough J . Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 2002;180 (1 ):29–48. doi: 10.1016/S0025-5564(02)00108-6 12387915
15 Diekmann O , Heesterbeek JAP , Metz JAJ . On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology. 1990;28 (4 ):365–382. doi: 10.1007/BF00178324 2117040
16 Ionides EL , Nguyen D , Atchadé Y , Stoev S , King AA . Inference for dynamic and latent variable models via iterated, perturbed Bayes maps. Proceedings of the National Academy of Sciences. 2015;112 (3 ):719–724. doi: 10.1073/pnas.1410597112 25568084
17 Pasetto D , Finger F , Rinaldo A , Bertuzzo E . Real-time projections of cholera outbreaks through data assimilation and rainfall forecasting. Advances in Water Resources. 2017;108 :345–356. doi: 10.1016/j.advwatres.2016.10.004
18 Cybenko G . Approximation by superpositions of a sigmoidal function. Math Control Signal Systems. 1989;2 :303–314. doi: 10.1007/BF02551274
19 Hornik K , Stinchcombe M , White H . Multilayer feedforward networks are universal approximators. Neural Networks. 1989;2 (5 ):359–366. doi: 10.1016/0893-6080(89)90020-8
20 Raissi M , Perdikaris P , Karniadakis GE . Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics. 2019;378 :686–707. doi: 10.1016/j.jcp.2018.10.045
21 Torku TK , Khaliq AQM , Furati KM . Deep-Data-Driven Neural Networks for COVID-19 Vaccine Efficacy. Epidemiologia. 2021;2 (4 ):564–586. doi: 10.3390/epidemiologia2040039 36417217
22 Malinzi J , Gwebu S , Motsa S . Determining COVID-19 Dynamics Using Physics Informed Neural Networks. Axioms. 2022;11 (3 ):121. doi: 10.3390/axioms11030121
23 Berkhahn S , Ehrhardt M . A physics-informed neural network to model COVID-19 infection and hospitalization scenarios. Advances in Continuous and Discrete Models. 2022;2022 (1 ):61. doi: 10.1186/s13662-022-03733-5 36320680
24 Schiassi E , De Florio M , D’Ambrosio A , Mortari D , Furfaro R . Physics-Informed Neural Networks and Functional Interpolation for Data-Driven Parameters Discovery of Epidemiological Compartmental Models. Mathematics. 2021;9 (17 ):2069. doi: 10.3390/math9172069
25 Long J , Khaliq AQM , Furati KM . Identification and prediction of time-varying parameters of COVID-19 model: a data-driven deep learning approach. International Journal of Computer Mathematics. 2021;98 (8 ):1617–1632. doi: 10.1080/00207160.2021.1929942
26 Feng L , Chen Z , L HA Jr , Furati K , Khaliq A , Feng L , et al . Data driven time-varying SEIR-LSTM/GRU algorithms to track the spread of COVID-19. Mathematical Biosciences and Engineering. 2022;19 (9 ):8935–8962. doi: 10.3934/mbe.2022415 35942743
27 Olumoyin KD , Khaliq AQM , Furati KM . Data-Driven Deep-Learning Algorithm for Asymptomatic COVID-19 Model with Varying Mitigation Measures and Transmission Rate. Epidemiologia. 2021;2 (4 ):471–489. doi: 10.3390/epidemiologia2040033 36417211
28 Ning X , Guan J , Li XA , Wei Y , Chen F . Physics-Informed Neural Networks Integrating Compartmental Model for Analyzing COVID-19 Transmission Dynamics. Viruses. 2023;15 (8 ):1749. doi: 10.3390/v15081749 37632091
29 He M , Tang S , Xiao Y . Combining the dynamic model and deep neural networks to identify the intensity of interventions during COVID-19 pandemic. PLOS Computational Biology. 2023;19 (10 ): e1011535. doi: 10.1371/journal.pcbi.1011535 37851640
30 Bertaglia G , Lu C , Pareschi L , Zhu X . Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread. Mathematical Models and Methods in Applied Sciences. 2022;32 (10 ):1949–1985. doi: 10.1142/S0218202522500452
31 Raissi M , Perdikaris P , Karniadakis GE . Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. 2017.
32 Raissi M , Perdikaris P , Karniadakis GE . Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations. 2017.
33 Haghighat E , Juanes R . SciANN: A Keras/TensorFlow wrapper for scientific computations and physics-informed deep learning using artificial neural networks. Computer Methods in Applied Mechanics and Engineering. 2021;373 :113552. doi: 10.1016/j.cma.2020.113552
34 EpiCentro. COVID-19 integrated surveillance: key national data; 2020. Available from: https://www.epicentro.iss.it/en/coronavirus/sars-cov-2-integrated-surveillance-data.
35 ISTAT. Primi risultati dell’indagine di sieroprevalenza sul SARS-CoV-2. Istituto Nazionale di Statistica, Ministero della Salute (Italia); 2020. Available from: https://www.istat.it/it/files//2020/08/ReportPrimiRisultatiIndagineSiero.pdf.
36 Glorot X, Bengio Y. Understanding the difficulty of training deep feedforward neural networks. In: Teh YW, Titterington M, editors. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. vol. 9 of Proceedings of Machine Learning Research. Chia Laguna Resort, Sardinia, Italy: PMLR; 2010. p. 249–256. Available from: https://proceedings.mlr.press/v9/glorot10a.html.
37 Kingma DP , Ba J . Adam: A Method for Stochastic Optimization. 2017.
38 Wang S , Yu X , Perdikaris P . When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics. 2021; p. 110768.
39 Ayan P , Jayanta Kumar B , Akshay P , Sagar C . Emergence of universality in the transmission dynamics of COVID-19 Scientific Reports. 2021;11 (1 ):18891. doi: 10.1038/s41598-021-98302-3 34556753
40 Linka K , Schäfer A , Meng X , Zou Z , Karniadakis GE , Kuhl E . Bayesian Physics Informed Neural Networks for real-world nonlinear dynamical systems. Computer Methods in Applied Mechanics and Engineering. 2022;402 :115346. doi: 10.1016/j.cma.2022.115346
