A Physics-Informed Neural Network approach for compartmental epidemiological models.

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.

Beyond SIRD models: a novel dynamic model for epidemics, relating infected with entries to health care units and application for identification and restraining policy.

Epidemic models of susceptibles, exposed, infected, recovered and deceased (SΕIRD) presume homogeneity, constant rates and fixed, bilinear structure. They produce short-range, single-peak responses, hardly attained under restrictive measures. Tuned via uncertain I,R,D data, they cannot faithfully represent long-range evolution. A robust epidemic model is presented that relates infected with the entry rate to health care units (HCUs) via population averages. Model uncertainty is circumvented by not presuming any specific model structure, or constant rates. The model is tuned via data of low uncertainty, by direct monitoring: (a) of entries to HCUs (accurately known, in contrast to delayed and non-reliable I,R,D data) and (b) of scaled model parameters, representing population averages. The model encompasses random propagation of infections, delayed, randomly distributed entries to HCUs and varying exodus of non-hospitalized, as disease severity subdues. It closely follows multi-pattern growth of epidemics with possible recurrency, viral strains and mutations, varying environmental conditions, immunity levels, control measures and efficacy thereof, including vaccination. The results enable real-time identification of infected and infection rate. They allow design of resilient, cost-effective policy in real time, targeting directly the key variable to be controlled (entries to HCUs) below current HCU capacity. As demonstrated in ex post case studies, the policy can lead to lower overall cost of epidemics, by balancing the trade-off between the social cost of infected and the economic contraction associated with social distancing and mobility restriction measures.

Computation of random time-shift distributions for stochastic population models.

Even in large systems, the effect of noise arising from when populations are initially small can persist to be measurable on the macroscale. A deterministic approximation to a stochastic model will fail to capture this effect, but it can be accurately approximated by including an additional random time-shift to the initial conditions. We present a efficient numerical method to compute this time-shift distribution for a large class of stochastic models. The method relies on differentiation of certain functional equations, which we show can be effectively automated by deriving rules for different types of model rates that arise commonly when mass-action mixing is assumed. Explicit computation of the time-shift distribution can be used to build a practical tool for the efficient generation of macroscopic trajectories of stochastic population models, without the need for costly stochastic simulations. Full code is provided to implement the calculations and we demonstrate the method on an epidemic model and a model of within-host viral dynamics.

Inference of epidemic dynamics in the COVID-19 era and beyond.

The COVID-19 pandemic demonstrated the key role that epidemiology and modelling play in analysing infectious threats and supporting decision making in real-time. Motivated by the unprecedented volume and breadth of data generated during the pandemic, we review modern opportunities for analysis to address questions that emerge during a major modern epidemic. Following the broad chronology of insights required - from understanding initial dynamics to retrospective evaluation of interventions, we describe the theoretical foundations of each approach and the underlying intuition. Through a series of case studies, we illustrate real life applications, and discuss implications for future work.

Source-Sink Dynamics in a Two-Patch SI Epidemic Model with Life Stages and No Recovery from Infection.

This study presents a comprehensive analysis of a two-patch, two-life stage SI model without recovery from infection, focusing on the dynamics of disease spread and host population viability in natural populations. The model, inspired by real-world ecological crises like the decline of amphibian populations due to chytridiomycosis and sea star populations due to Sea Star Wasting Disease, aims to understand the conditions under which a sink host population can present ecological rescue from a healthier, source population. Mathematical and numerical analyses reveal the critical roles of the basic reproductive numbers of the source and sink populations, the maturation rate, and the dispersal rate of juveniles in determining population outcomes. The study identifies basic reproduction numbers R 0 for each of the patches, and conditions for the basic reproduction numbers to produce a receiving patch under which its population. These findings provide insights into managing natural populations affected by disease, with implications for conservation strategies, such as the importance of maintaining reproductively viable refuge populations and considering the effects of dispersal and maturation rates on population recovery. The research underscores the complexity of host-pathogen dynamics in spatially structured environments and highlights the need for multi-faceted approaches to biodiversity conservation in the face of emerging diseases.

Prevention and control of Ebola virus transmission: mathematical modelling and data fitting.

The Ebola virus disease (EVD) has been endemic since 1976, and the case fatality rate is extremely high. EVD is spread by infected animals, symptomatic individuals, dead bodies, and contaminated environment. In this paper, we formulate an EVD model with four transmission modes and a time delay describing the incubation period. Through dynamical analysis, we verify the importance of blocking the infection source of infected animals. We get the basic reproduction number without considering the infection source of infected animals. And, it is proven that the model has a globally attractive disease-free equilibrium when the basic reproduction number is less than unity; the disease eventually becomes endemic when the basic reproduction number is greater than unity. Taking the EVD epidemic in Sierra Leone in 2014-2016 as an example, we complete the data fitting by combining the effect of the media to obtain the unknown parameters, the basic reproduction number and its time-varying reproduction number. It is shown by parameter sensitivity analysis that the contact rate and the removal rate of infected group have the greatest influence on the prevalence of the disease. And, the disease-controlling thresholds of these two parameters are obtained. In addition, according to the existing vaccination strategy, only the inoculation ratio in high-risk areas is greater than 0.4, the effective reproduction number can be less than unity. And, the earlier the vaccination time, the greater the inoculation ratio, and the faster the disease can be controlled.

Stratified epidemic model using a latent marked Hawkes process.

We extend the unstructured homogeneously mixing epidemic model introduced by Lamprinakou et al. (2023) to a finite population stratified by age bands. We model the actual unobserved infections using a latent marked Hawkes process and the reported aggregated infections as random quantities driven by the underlying Hawkes process. We apply a Kernel Density Particle Filter (KDPF) to infer the marked counting process, the instantaneous reproduction number for each age group and forecast the epidemic's trajectory in the near future. Taking into account the individual inhomogeneity in age does not increase significantly the computational cost of the proposed inference algorithm compared to the cost of the proposed algorithm for the homogeneously unstructured epidemic model. We demonstrate that considering the individual heterogeneity in age, we can derive the instantaneous reproduction numbers per age group that provide a real-time measurement of interventions and behavioural changes of the associated groups. We illustrate the performance of the proposed inference algorithm on synthetic data sets and COVID-19-reported cases in various local authorities in the UK, and benchmark our model to the unstructured homogeneously mixing epidemic model. Our paper is a "demonstration" of a methodology that might be applied to factors other than age for stratification.

Stochastic two-strain epidemic model with saturated incidence rates driven by Lévy noise.

In this paper, we introduce a stochastic two-strain epidemic model driven by Lévy noise describing the interaction between four compartments; susceptible, infected individuals by the first strain, infected ones by the second strain and the recovered individuals. The forces of infection, for both strains, are represented by saturated incidence rates. Our study begins with the investigation of unique global solution of the suggested mathematical model. Then, it moves to the determination of sufficient conditions of extinction and persistence in mean of the two-strain disease. In order to illustrate the theoretical findings, we give some numerical simulations.

Multiple endemic equilibria in an environmentally-transmitted disease with three disease stages.

We construct, analyze and interpret a mathematical model for an environmental transmitted disease characterized for the existence of three disease stages: acute, severe and asymptomatic. Besides, we consider that severe and asymptomatic cases may present relapse between them. Transmission dynamics driven by the contact rates only occurs when a parameter R∗>1, as normally occur in directly-transmitted or vector-transmitted diseases, but it will not adequately correspond to a basic reproductive number as it depends on environmental parameters. In this case, the forward transcritical bifurcation that exists for R∗<1, becomes a backward bifurcation, producing multiple steady-states, a hysteresis effect and dependence on initial conditions. A threshold parameter for an epidemic outbreak, independent of R∗ is only the ratio of the external contamination inflow shedding rate to the environmental clearance rate. R∗ describes the strength of the transmission to infectious classes other than the I-(acute) type infections. The epidemic outbreak conditions and the structure of R∗ appearing in this model are both responsible for the existence of endemic states.

Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer.

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

Mathematical analysis of simple behavioral epidemic models.

COVID-19 highlighted the importance of considering human behavior change when modeling disease dynamics. This led to developing various models that incorporate human behavior. Our objective is to contribute to an in-depth, mathematical examination of such models. Here, we consider a simple deterministic compartmental model with endogenous incorporation of human behavior (i.e., behavioral feedback) through transmission in a classic Susceptible-Exposed-Infectious-Recovered (SEIR) structure. Despite its simplicity, the SEIR structure with behavior (SEIRb) was shown to perform well in forecasting, especially compared to more complicated models. We contrast this model with an SEIR model that excludes endogenous incorporation of behavior. Both models assume permanent immunity to COVID-19, so we also consider a modification of the models which include waning immunity (SEIRS and SEIRSb). We perform equilibria, sensitivity, and identifiability analyses on all models and examine the fidelity of the models to replicate COVID-19 data across the United States. Endogenous incorporation of behavior significantly improves a model's ability to produce realistic outbreaks. While the two endogenous models are similar with respect to identifiability and sensitivity, the SEIRSb model, with the more accurate assumption of the waning immunity, strengthens the initial SEIRb model by allowing for the existence of an endemic equilibrium, a realistic feature of COVID-19 dynamics. When fitting the model to data, we further consider the addition of simple seasonality affecting disease transmission to highlight the explanatory power of the models.

Leveraging advances in data-driven deep learning methods for hybrid epidemic modeling.

Mathematical modeling of epidemic dynamics is crucial to understand its underlying mechanisms, quantify important parameters, and make predictions that facilitate more informed decision-making. There are three major types of models: mechanistic models including the SEIR-type paradigm, alternative data-driven (DD) approaches, and hybrid models that combine mechanistic models with DD approaches. In this paper, we summarize our work in the COVID-19 Scenario Modeling Hub (SMH) for more than 12 rounds since early 2021 for informed decision support. We emphasize the importance of deep learning techniques for epidemic modeling via a flexible DD framework that substantially complements the mechanistic paradigm to evaluate various future epidemic scenarios. We start with a traditional curve-fitting approach to model cumulative COVID-19 based on the underlying SEIR-type mechanisms. Hospitalizations and deaths are modeled as binomial processes of cases and hospitalization, respectively. We further formulate two types of deep learning models based on multivariate long short term memory (LSTM) to address the challenges of more traditional DD models. The first LSTM is structurally similar to the curve fitting approach and assumes that hospitalizations and deaths are binomial processes of cases. Instead of using a predefined exponential curve, LSTM relies on the underlying data to identify the most appropriate functions, and is capable of capturing both long-term and short-term epidemic behaviors. We then relax the assumption of dependent inputs among cases, hospitalizations, and death. Another type of LSTM that handles all input time series as parallel signals, the independent multivariate LSTM, is developed. Independent multivariate LSTM can incorporate a wide range of data sources beyond traditional case-based epidemiological surveillance. The DD framework unleashes its potential in big data era with previously neglected heterogeneous surveillance data sources, such as syndromic, environment, genomic, serologic, infoveillance, and mobility data. DD approaches, especially LSTM, complement and integrate with the mechanistic modeling paradigm, provide a feasible alternative approach to model today's complex socio-epidemiological systems, and further leverage our ability to explore different scenarios for more informed decision-making during health emergencies.

Dynamical analysis of a stochastic maize streak virus epidemic model with logarithmic Ornstein-Uhlenbeck process.

To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein-Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus.

Global stability and optimal control of an age-structured SVEIR epidemic model with waning immunity and relapses.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number R 0 and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of R 0 is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

Approximating reproduction numbers: a general numerical method for age-structured models.

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

An SIS epidemic model with individual variation.

We study an extension of the stochastic SIS (Susceptible-Infectious-Susceptible) model in continuous time that accounts for variation amongst individuals. By examining its limiting behaviour as the population size grows we are able to exhibit conditions for the infection to become endemic.

SpatialWavePredict: a tutorial-based primer and toolbox for forecasting growth trajectories using the ensemble spatial wave sub-epidemic modeling framework.

BACKGROUND: Dynamical mathematical models defined by a system of differential equations are typically not easily accessible to non-experts. However, forecasts based on these types of models can help gain insights into the mechanisms driving the process and may outcompete simpler phenomenological growth models. Here we introduce a friendly toolbox, SpatialWavePredict, to characterize and forecast the spatial wave sub-epidemic model, which captures diverse wave dynamics by aggregating multiple asynchronous growth processes and has outperformed simpler phenomenological growth models in short-term forecasts of various infectious diseases outbreaks including SARS, Ebola, and the early waves of the COVID-19 pandemic in the US. RESULTS: This tutorial-based primer introduces and illustrates a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using an ensemble spatial wave sub-epidemic model based on ordinary differential equations. Scientists, policymakers, and students can use the toolbox to conduct real-time short-term forecasts. The five-parameter epidemic wave model in the toolbox aggregates linked overlapping sub-epidemics and captures a rich spectrum of epidemic wave dynamics, including oscillatory wave behavior and plateaus. An ensemble strategy aims to improve forecasting performance by combining the resulting top-ranked models. The toolbox provides a tutorial for forecasting time-series trajectories, including the full uncertainty distribution derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score. CONCLUSIONS: We have developed the first comprehensive toolbox to characterize and forecast time-series data using an ensemble spatial wave sub-epidemic wave model. As an epidemic situation or contagion occurs, the tools presented in this tutorial can facilitate policymakers to guide the implementation of containment strategies and assess the impact of control interventions. We demonstrate the functionality of the toolbox with examples, including a tutorial video, and is illustrated using daily data on the COVID-19 pandemic in the USA.

The influence of cross-border mobility on the COVID-19 epidemic in Nordic countries.

Restrictions of cross-border mobility are typically used to prevent an emerging disease from entering a country in order to slow down its spread. However, such interventions can come with a significant societal cost and should thus be based on careful analysis and quantitative understanding on their effects. To this end, we model the influence of cross-border mobility on the spread of COVID-19 during 2020 in the neighbouring Nordic countries of Denmark, Finland, Norway and Sweden. We investigate the immediate impact of cross-border travel on disease spread and employ counterfactual scenarios to explore the cumulative effects of introducing additional infected individuals into a population during the ongoing epidemic. Our results indicate that the effect of inter-country mobility on epidemic growth is non-negligible essentially when there is sizeable mobility from a high prevalence country or countries to a low prevalence one. Our findings underscore the critical importance of accurate data and models on both epidemic progression and travel patterns in informing decisions related to inter-country mobility restrictions.

Dynamics of simplicial SEIRS epidemic model: global asymptotic stability and neural Lyapunov functions.

The transmission of infectious diseases on a particular network is ubiquitous in the physical world. Here, we investigate the transmission mechanism of infectious diseases with an incubation period using a networked compartment model that contains simplicial interactions, a typical high-order structure. We establish a simplicial SEIRS model and find that the proportion of infected individuals in equilibrium increases due to the many-body connections, regardless of the type of connections used. We analyze the dynamics of the established model, including existence and local asymptotic stability, and highlight differences from existing models. Significantly, we demonstrate global asymptotic stability using the neural Lyapunov function, a machine learning technique, with both numerical simulations and rigorous analytical arguments. We believe that our model owns the potential to provide valuable insights into transmission mechanisms of infectious diseases on high-order network structures, and that our approach and theory of using neural Lyapunov functions to validate model asymptotic stability can significantly advance investigations on complex dynamics of infectious disease.

An SEIR network epidemic model with manual and digital contact tracing allowing delays.

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered such that each infected network contact is reported, tested and isolated with some probability and after a random delay. Additionally, digital tracing (based on a tracing app) is triggered if the diagnosed individual is an app-user, and then all of its app-using infectees are immediately notified and isolated. The early phase of the epidemic with manual and/or digital tracing is approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of both contact tracing mechanisms is numerically quantified through the reduction of the reproduction number. This shows that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The relative effectiveness of manual tracing compared to digital tracing increases if: more of the transmission occurs on the network, when the tracing delay is shortened, and when the network degree distribution is heavy-tailed. For realistic values, the combined tracing case can reduce R0 by 20%-30%, so other preventive measures are needed to reduce the reproduction number down to 1.2-1.4 for contact tracing to make it successful in avoiding big outbreaks.