Identifiability and predictability of integer- and fractional-order epidemiological models using physics-informed neural networks

A combined neural ODE-Bayesian optimization approach to resolve dynamics and estimate parameters for a modified SIR model with immune memory

We propose a novel hybrid approach that integrates Neural Ordinary Differential Equations (NODEs) with Bayesian optimization to address the dynamics and parameter estimation of a modified time-delay-type Susceptible-Infected-Removed (SIR) model incorporating immune memory. This approach leverages a neural network to produce continuous multi-wave infection profiles by learning from both data and the model. The time-delay component of the SIR model, expressed through a convolutional integral, results in an integro-differential equation. To resolve these dynamics, we extend the NODE framework, employing a Runge-Kutta solver, to handle the challenging convolution integral, enabling us to fit the data and learn the parameters and dynamics of the model. Additionally, through Bayesian optimization, we enhance prediction accuracy while focusing on long-term dynamics. Our model, applied to COVID-19 data from Mexico, South Africa, and South Korea, effectively learns critical time-dependent parameters and provides accurate short- and long-term predictions. This combined methodology allows for early prediction of infection peaks, offering significant lead time for public health responses.

Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

Density physics-informed neural networks reveal sources of cell heterogeneity in signal transduction

The transduction time between signal initiation and final response provides valuable information on the underlying signaling pathway, including its speed and precision. Furthermore, multi-modality in a transduction-time distribution indicates that the response is regulated by multiple pathways with different transduction speeds. Here, we developed a method called density physics-informed neural networks (Density-PINNs) to infer the transduction-time distribution from measurable final stress response time traces. We applied Density-PINNs to single-cell gene expression data from sixteen promoters regulated by unknown pathways in response to antibiotic stresses. We found that promoters with slower signaling initiation and transduction exhibit larger cell-to-cell heterogeneity in response intensity. However, this heterogeneity was greatly reduced when the response was regulated by slow and fast pathways together. This suggests a strategy for identifying effective signaling pathways for consistent cellular responses to disease treatments. Density-PINNs can also be applied to understand other time delay systems, including infectious diseases.

A computational approach to identifiability analysis for a model of the propagation and control of COVID-19 in Chile

A computational approach is adapted to analyze the parameter identifiability of a compartmental model. The model is intended to describe the progression of the COVID-19 pandemic in Chile during the initial phase in early 2020 when government declared quarantine measures. The computational approach to analyze the structural and practical identifiability is applied in two parts, one for synthetic data and another for some Chilean regional data. The first part defines the identifiable parameter sets when these recover the true parameters used to create the synthetic data. The second part compares the results derived from synthetic data, estimating the identifiable parameter sets from regional Chilean epidemic data. Experiments provide evidence of the loss of identifiability if some initial conditions are estimated, the period of time used to fit is before the peak, and if a significant proportion of the population is involved in quarantine periods.

Data-Driven Deep Learning Neural Networks for Predicting the Number of Individuals Infected by COVID-19 Omicron Variant

Infectious disease epidemics are challenging for medical and public health practitioners. They require prompt treatment, but it is challenging to recognize and define epidemics in real time. Knowing the prediction of an infectious disease epidemic can evaluate and prevent the disease's impact. Mathematical models of epidemics that work in real time are important tools for preventing disease, and data-driven deep learning enables practical algorithms for identifying parameters in mathematical models. In this paper, the SIR model was reduced to a logistic differential equation involving a constant parameter and a time-dependent function. The time-dependent function leads to constant, rational, and birational models. These models use several constant parameters from the available data to predict the time and number of people reported to be infected with the COVID-19 Omicron variant. Two out of these three models, rational and birational, provide accurate predictions for countries that practice strict mitigation measures, but fail to provide accurate predictions for countries that practice partial mitigation measures. Therefore, we introduce a time-series model based on neural networks to predict the time and number of people reported to be infected with the COVID-19 Omicron variant in a given country that practices both partial and strict mitigation measures. A logistics-informed neural network algorithm was also introduced. This algorithm takes as input the daily and cumulative number of people who are reported to be infected with the COVID-19 Omicron variant in the given country. The algorithm helps determine the analytical solution involving several constant parameters for each model from the available data. The accuracy of these models is demonstrated using error metrics on Omicron variant data for Portugal, Italy, and China. Our findings demonstrate that the constant model could not accurately predict the daily or cumulative infections of the COVID-19 Omicron variant in the observed country because of the long series of existing data of the epidemics. However, the rational and birational models accurately predicted cumulative infections in countries adopting strict mitigation measures, but they fell short in predicting the daily infections. Furthermore, both models performed poorly in countries with partial mitigation measures. Notably, the time-series model stood out for its versatility, effectively predicting both daily and cumulative infections in countries irrespective of the stringency of their mitigation measures.

Transmission dynamics informed neural network with application to COVID-19 infections

Since the end of 2019 the COVID-19 repeatedly surges with most countries/territories experiencing multiple waves, and mechanism-based epidemic models played important roles in understanding the transmission mechanism of multiple epidemic waves. However, capturing temporal changes of the transmissibility of COVID-19 during the multiple waves keeps ill-posed problem for traditional mechanism-based epidemic compartment models, because that the transmission rate is usually assumed to be specific piecewise functions and more parameters are added to the model once multiple epidemic waves involved, which poses a huge challenge to parameter estimation. Meanwhile, data-driven deep neural networks fail to discover the driving factors of repeated outbreaks and lack interpretability. In this study, aiming at developing a data-driven method to project time-dependent parameters but also merging the advantage of mechanism-based models, we propose a transmission dynamics informed neural network (TDINN) by encoding the SEIRD compartment model into deep neural networks. We show that the proposed TDINN algorithm performs very well when fitting the COVID-19 epidemic data with multiple waves, where the epidemics in the United States, Italy, South Africa, and Kenya, and several outbreaks the Omicron variant in China are taken as examples. In addition, the numerical simulation shows that the trained TDINN can also perform as a predictive model to capture the future development of COVID-19 epidemic. We find that the transmission rate inferred by the TDINN frequently fluctuates, and a feedback loop between the epidemic shifting and the changes of transmissibility drives the occurrence of multiple waves. We observe a long response delay to the implementation of control interventions in the four countries, while the decline of the transmission rate in the outbreaks in China usually happens once the implementation of control interventions. The further simulation show that 17 days' delay of the response to the implementation of control interventions lead to a roughly four-fold increase in daily reported cases in one epidemic wave in Italy, which suggest that a rapid response to policies that strengthen control interventions can be effective in flattening the epidemic curve or avoiding subsequent epidemic waves. We observe that the transmission rate in the outbreaks in China is already decreasing before enhancing control interventions, providing the evidence that the increasing of the epidemics can drive self-conscious behavioural changes to protect against infections.

Physics-Informed Neural Networks Integrating Compartmental Model for Analyzing COVID-19 Transmission Dynamics

Modelling and predicting the behaviour of infectious diseases is essential for early warning and evaluating the most effective interventions to prevent significant harm. Compartmental models produce a system of ordinary differential equations (ODEs) that are renowned for simulating the transmission dynamics of infectious diseases. However, the parameters in compartmental models are often unknown, and they can even change over time in the real world, making them difficult to determine. This study proposes an advanced artificial intelligence approach based on physics-informed neural networks (PINNs) to estimate time-varying parameters from given data for the compartmental model. Our proposed PINNs method captures the complex dynamics of COVID-19 by integrating a modified Susceptible-Exposed-Infectious-Recovered-Death (SEIRD) compartmental model with deep neural networks. Specifically, we modelled the system of ODEs as one network and the time-varying parameters as another network to address significant unknown parameters and limited data. Such structure of the PINNs method is in line with the prior epidemiological correlations and comprises the mismatch between available data and network output and the residual of ODEs. The experimental findings on real-world reported data data have demonstrated that our method robustly and accurately learns the dynamics and forecasts future states. Moreover, as more data becomes available, our proposed PINNs method can be successfully extended to other regions and infectious diseases.

SEINN: A deep learning algorithm for the stochastic epidemic model

Stochastic modeling predicts various outcomes from stochasticity in the data, parameters and dynamical system. Stochastic models are deemed more appropriate than deterministic models accounting in terms of essential and practical information about a system. The objective of the current investigation is to address the issue above through the development of a novel deep neural network referred to as a stochastic epidemiology-informed neural network. This network learns knowledge about the parameters and dynamics of a stochastic epidemic vaccine model. Our analysis centers on examining the nonlinear incidence rate of the model from the perspective of the combined effects of vaccination and stochasticity. Based on empirical evidence, stochastic models offer a more comprehensive understanding than deterministic models, mainly when we use error metrics. The findings of our study indicate that a decrease in randomness and an increase in vaccination rates are associated with a better prediction of nonlinear incidence rates. Adopting a nonlinear incidence rate enables a more comprehensive representation of the complexities of transmitting diseases. The computational analysis of the proposed method, focusing on sensitivity analysis and overfitting analysis, shows that the proposed method is efficient. Our research aims to guide policymakers on the effects of stochasticity in epidemic models, thereby aiding the development of effective vaccination and mitigation policies. Several case studies have been conducted on nonlinear incidence rates using data from Tennessee, USA.

Dynamics-informed deconvolutional neural networks for super-resolution identification of regime changes in epidemiological time series

The ability to infer the timing and amplitude of perturbations in epidemiological systems from their stochastically spread low-resolution outcomes is crucial for multiple applications. However, the general problem of connecting epidemiological curves with the underlying incidence lacks the highly effective methodology present in other inverse problems, such as super-resolution and dehazing from computer vision. Here, we develop an unsupervised physics-informed convolutional neural network approach in reverse to connect death records with incidence that allows the identification of regime changes at single-day resolution. Applied to COVID-19 data with proper regularization and model-selection criteria, the approach can identify the implementation and removal of lockdowns and other nonpharmaceutical interventions (NPIs) with 0.93-day accuracy over the time span of a year.

The impact of multiple non-pharmaceutical interventions for China-bound travel on domestic COVID-19 outbreaks

Objectives

Non-pharmaceutical interventions (NPIs) implemented on China-bound travel have successfully mitigated cross-regional transmission of COVID-19 but made the country face ripple effects. Thus, adjusting these interventions to reduce interruptions to individuals' daily life while minimizing transmission risk was urgent.

Methods

An improved Susceptible-Infected-Recovered (SIR) model was built to evaluate the Delta variant's epidemiological characteristics and the impact of NPIs. To explore the risk associated with inbound travelers and the occurrence of domestic traceable outbreaks, we developed an association parameter that combined inbound traveler counts with a time-varying initial value. In addition, multiple time-varying functions were used to model changes in the implementation of NPIs. Related parameters of functions were run by the MCSS method with 1,000 iterations to derive the probability distribution. Initial values, estimated parameters, and corresponding 95% CI were obtained. Reported existing symptomatic, suspected, and asymptomatic case counts were used as the training datasets. Reported cumulative recovered individual data were used to verify the reliability of relevant parameters. Lastly, we used the value of the ratio (Bias2/Variance) to verify the stability of the mathematical model, and the effects of the NPIs on the infected cases to analyze the sensitivity of input parameters.

Results

The quantitative findings indicated that this improved model was highly compatible with publicly reported data collected from July 21 to August 30, 2021. The number of inbound travelers was associated with the occurrence of domestic outbreaks. A proportional relationship between the Delta variant incubation period and PCR test validity period was found. The model also predicted that restoration of pre-pandemic travel schedules while adhering to NPIs requirements would cause shortages in health resources. The maximum demand for hospital beds would reach 25,000/day, the volume of PCR tests would be 8,000/day, and the number of isolation rooms would reach 800,000/day within 30 days.

Conclusion

With the pandemic approaching the end, reexamining it carefully helps better address future outbreaks. This predictive model has provided scientific evidence for NPIs' effectiveness and quantifiable evidence of health resource allocation. It could guide the design of future epidemic prevention and control policies, and provide strategic recommendations on scarce health resource allocation.

Epi-DNNs: Epidemiological priors informed deep neural networks for modeling COVID-19 dynamics

Differential equations-based epidemic compartmental models and deep neural networks-based artificial intelligence (AI) models are powerful tools for analyzing and fighting the transmission of COVID-19. However, the capability of compartmental models is limited by the challenges of parameter estimation, while AI models fail to discover the evolutionary pattern of COVID-19 and lack explainability. This paper aims to provide a novel method (called Epi-DNNs) by integrating compartmental models and deep neural networks (DNNs) to model the complex dynamics of COVID-19. In the proposed Epi-DNNs method, the neural network is designed to express the unknown parameters in the compartmental model and the Runge-Kutta method is implemented to solve the ordinary differential equations (ODEs) so as to give the values of the ODEs at a given time. Specifically, the discrepancy between predictions and observations is incorporated into the loss function, then the defined loss is minimized and applied to identify the best-fitted parameters governing the compartmental model. Furthermore, we verify the performance of Epi-DNNs on the real-world reported COVID-19 data on the Omicron epidemic in Shanghai covering February 25 to May 27, 2022. The experimental findings on the synthesized data have revealed its effectiveness in COVID-19 transmission modeling. Moreover, the inferred parameters from the proposed Epi-DNNs method yield a predictive compartmental model, which can serve to forecast future dynamics.

Uncertainty quantification in mechanistic epidemic models via cross-entropy approximate Bayesian computation

This paper proposes a data-driven approximate Bayesian computation framework for parameter estimation and uncertainty quantification of epidemic models, which incorporates two novelties: (i) the identification of the initial conditions by using plausible dynamic states that are compatible with observational data; (ii) learning of an informative prior distribution for the model parameters via the cross-entropy method. The new methodology's effectiveness is illustrated with the aid of actual data from the COVID-19 epidemic in Rio de Janeiro city in Brazil, employing an ordinary differential equation-based model with a generalized SEIR mechanistic structure that includes time-dependent transmission rate, asymptomatics, and hospitalizations. A minimization problem with two cost terms (number of hospitalizations and deaths) is formulated, and twelve parameters are identified. The calibrated model provides a consistent description of the available data, able to extrapolate forecasts over a few weeks, making the proposed methodology very appealing for real-time epidemic modeling.

A physics-informed neural network to model COVID-19 infection and hospitalization scenarios

In this paper, we replace the standard numerical approach of estimating parameters in a mathematical model using numerical solvers for differential equations with a physics-informed neural network (PINN). This neural network requires a sequence of time instances as direct input of the network and the numbers of susceptibles, vaccinated, infected, hospitalized, and recovered individuals per time instance to learn certain parameters of the underlying model, which are used for the loss calculations. The established model is an extended susceptible-infected-recovered (SIR) model in which the transitions between disease-related population groups, called compartments, and the physical laws of epidemic transmission dynamics are expressed by a system of ordinary differential equations (ODEs). The system of ODEs and its time derivative are included in the residual loss function of the PINN in addition to the data error between the current network output and the time series data of the compartment sizes. Further, we illustrate how this PINN approach can also be used for differential equation-based models such as the proposed extended SIR model, called SVIHR model. In a validation process, we compare the performance of the PINN with results obtained with the numerical technique of non-standard finite differences (NSFD) in generating future COVID-19 scenarios based on the parameters identified by the PINN. The used training data set covers the time between the outbreak of the pandemic in Germany and the last week of the year 2021. We obtain a two-step or hybrid approach, as the PINN is then used to generate a future COVID-19 outbreak scenario describing a possibly next pandemic wave. The week at which the prediction starts is chosen in mid-April 2022.

A comparison and calibration of integer and fractional-order models of COVID-19 with stratified public response

The spread of SARS-CoV-2 in the Canadian province of Ontario has resulted in millions of infections and tens of thousands of deaths to date. Correspondingly, the implementation of modeling to inform public health policies has proven to be exceptionally important. In this work, we expand a previous model of the spread of SARS-CoV-2 in Ontario, "Modeling the impact of a public response on the COVID-19 pandemic in Ontario, " to include the discretized, Caputo fractional derivative in the susceptible compartment. We perform identifiability and sensitivity analysis on both the integer-order and fractional-order SEIRD model and contrast the quality of the fits. We note that both methods produce fits of similar qualitative strength, though the inclusion of the fractional derivative operator quantitatively improves the fits by almost 27% corroborating the appropriateness of fractional operators for the purposes of phenomenological disease forecasting. In contrasting the fit procedures, we note potential simplifications for future study. Finally, we use all four models to provide an estimate of the time-dependent basic reproduction number for the spread of SARS-CoV-2 in Ontario between January 2020 and February 2021.

Fractional SEIR model and data-driven predictions of COVID-19 dynamics of Omicron variant

We study the dynamic evolution of COVID-19 caused by the Omicron variant via a fractional susceptible-exposed-infected-removed (SEIR) model. Preliminary data suggest that the symptoms of Omicron infection are not prominent and the transmission is, therefore, more concealed, which causes a relatively slow increase in the detected cases of the newly infected at the beginning of the pandemic. To characterize the specific dynamics, the Caputo-Hadamard fractional derivative is adopted to refine the classical SEIR model. Based on the reported data, we infer the fractional order and time-dependent parameters as well as unobserved dynamics of the fractional SEIR model via fractional physics-informed neural networks. Then, we make short-time predictions using the learned fractional SEIR model.

Hamiltonian neural networks for solving equations of motion

There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven machine learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. The model learns solutions that satisfy, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. The choice of an appropriate activation function drastically improves the predictability of the network. Moreover, an error analysis is derived and states that the numerical errors depend on the overall network performance. The Hamiltonian network is then employed to solve the equations for the nonlinear oscillator and the chaotic Hénon-Heiles dynamical system. In both systems, a symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network to achieve the same order of the numerical error in the predicted phase space trajectories.

Learn bifurcations of nonlinear parametric systems via equation-driven neural networks

Nonlinear parametric systems have been widely used in modeling nonlinear dynamics in science and engineering. Bifurcation analysis of these nonlinear systems on the parameter space is usually used to study the solution structure, such as the number of solutions and the stability. In this paper, we develop a new machine learning approach to compute the bifurcations via so-called equation-driven neural networks (EDNNs). The EDNNs consist of a two-step optimization: the first step is to approximate the solution function of the parameter by training empirical solution data; the second step is to compute bifurcations using the approximated neural network obtained in the first step. Both theoretical convergence analysis and numerical implementation on several examples have been performed to demonstrate the feasibility of the proposed method.

Deep-Data-Driven Neural Networks for COVID-19 Vaccine Efficacy

Vaccination strategies to lessen the impact of the spread of a disease are fundamental to public health authorities and policy makers. The socio-economic benefit of full return to normalcy is the core of such strategies. In this paper, a COVID-19 vaccination model with efficacy rate is developed and analyzed. The epidemiological parameters of the model are learned via a feed-forward neural network. A hybrid approach that combines residual neural network with variants of recurrent neural network is implemented and analyzed for reliable and accurate prediction of daily cases. The error metrics and a k-fold cross validation with random splitting reveal that a particular type of hybrid approach called residual neural network with gated recurrent unit is the best hybrid neural network architecture. The data-driven simulations confirm the fact that the vaccination rate with higher efficacy lowers the infectiousness and basic reproduction number. As a study case, COVID-19 data for the state of Tennessee in USA is used.