Questions on whether governments have acted promptly enough, and whether lockdown measures can be lifted soon, have since been central in public discourse. Data-driven models that predict COVID-19 fatalities under different lockdown policy scenarios are essential for addressing these questions and informing governments on future policy directions.

Explainability is valuable to ensure that the model's forecasts are credible

The advancement of deep neural network models in the last fifteen years has profoundly altered the scientific landscape of estimation, prediction and decision making, from the early success of image recognition (Krizhevsky et al., 2012; He et al., 2016), to the success of self-learning of board games (Silver et al., 2017), to machine translation (Wu et al., 2016), to generative AI (Ho et al., 2020), and to the success of protein structure prediction (Jumper et al., 2021), among many other developments. In many of these successes, there are no well-established mechanistic models to describe the underlying problem (for example, we do not fully understand how human brains translate from one language to another). As such, it is conceivable that such successes are attributable to deep neural networks' remarkable capabilities for universal function approximation. In contrast, the hand-crafted models that existed before deep neural networks (such as n-gram models (Katz, 1987; Brown et al., 1992; Bengio et al., 2000)) were too restricted to offer satisfactory approximation. How well do deep neural network models work when there are well-established mechanistic models (as in physical sciences, where decades of theoretical and experimental endeavor have yielded highly accurate mechanistic models in many cases) -- in particular, how do the inference and prediction results of deep neural network models compare to more statistical approaches in the presence of reliable mechanistic models -- is an interesting question.

We focus on the problem of uncertainty informed allocation of medical resources (vaccines) to heterogeneous populations for managing epidemic spread. We tackle two related questions: (1) For a compartmental ordinary differential equation (ODE) model of epidemic spread, how can we estimate and integrate parameter uncertainty into resource allocation decisions? (2) How can we computationally handle both nonlinear ODE constraints and parameter uncertainties for a generic stochastic optimization problem for resource allocation? To the best of our knowledge current literature does not fully resolve these questions. Here, we develop a data-driven approach to represent parameter uncertainty accurately and tractably in a novel stochastic optimization problem formulation. We first generate a tractable scenario set by estimating the distribution on ODE model parameters using Bayesian inference with Gaussian processes. Next, we develop a parallelized solution algorithm that accounts for scenario-dependent nonlinear ODE constraints. Our scenario-set generation procedure and solution approach are flexible in that they can handle any compartmental epidemiological ODE model. Our computational experiments on two different non-linear ODE models (SEIR and SEPIHR) indicate that accounting for uncertainty in key epidemiological parameters can improve the efficacy of time-critical allocation decisions by 4-8%. This improvement can be attributed to data-driven and optimal (strategic) nature of vaccine allocations, especially in the early stages of the epidemic when the allocation strategy can crucially impact the long-term trajectory of the disease.

Objective: COVID-19 has spread worldwide and made a huge influence across the world. Modeling the infectious spread situation of COVID-19 is essential to understand the current condition and to formulate intervention measurements. Epidemiological equations based on the SEIR model simulate disease development. The traditional parameter estimation method to solve SEIR equations could not precisely fit real-world data due to different situations, such as social distancing policies and intervention strategies. Additionally, learning-based models achieve outstanding fitting performance, but cannot visualize mechanisms. Methods: Thus, we propose a deep dynamic epidemiological (DDE) method that combines epidemiological equations and deep-learning advantages to obtain high accuracy and visualization. The DDE contains deep networks to fit the effect function to simulate the ever-changing situations based on the neural ODE method in solving variants' equations, ensuring the fitting performance of multi-level areas. Results: We introduce four SEIR variants to fit different situations in different countries and regions. We compare our DDE method with traditional parameter estimation methods (Nelder-Mead, BFGS, Powell, Truncated Newton Conjugate-Gradient, Neural ODE) in fitting the real-world data in the cases of countries (the USA, Columbia, South Africa) and regions (Wuhan in China, Piedmont in Italy). Our DDE method achieves the best Mean Square Error and Pearson coefficient in all five areas. Further, compared with the state-of-art learning-based approaches, the DDE outperforms all techniques, including LSTM, RNN, GRU, Random Forest, Extremely Random Trees, and Decision Tree. Conclusion: DDE presents outstanding predictive ability and visualized display of the changes in infection rates in different regions and countries.

Computing observables from conditioned dynamics is typically computationally hard, because, although obtaining independent samples efficiently from the unconditioned dynamics is usually feasible, generally most of the samples must be discarded (in a form of importance sampling) because they do not satisfy the imposed conditions. Sampling directly from the conditioned distribution is non-trivial, as conditioning breaks the causal properties of the dynamics which ultimately renders the sampling procedure efficient. One standard way of achieving it is through a Metropolis Monte-Carlo procedure, but this procedure is normally slow and a very large number of Monte-Carlo steps is needed to obtain a small number of statistically independent samples. In this work, we propose an alternative method to produce independent samples from a conditioned distribution. The method learns the parameters of a generalized dynamical model that optimally describe the conditioned distribution in a variational sense. The outcome is an effective, unconditioned, dynamical model, from which one can trivially obtain independent samples, effectively restoring causality of the conditioned distribution. The consequences are twofold: on the one hand, it allows us to efficiently compute observables from the conditioned dynamics by simply averaging over independent samples. On the other hand, the method gives an effective unconditioned distribution which is easier to interpret. The method is flexible and can be applied virtually to any dynamics. We discuss an important application of the method, namely the problem of epidemic risk assessment from (imperfect) clinical tests, for a large family of time-continuous epidemic models endowed with a Gillespie-like sampler. We show that the method compares favorably against the state of the art, including the soft-margin approach and mean-field methods.

Stock and flow diagrams are already an important tool in epidemiology, but category theory lets us go further and treat these diagrams as mathematical entities in their own right. In this chapter we use communicable disease models created with our software, StockFlow.jl, to explain the benefits of the categorical approach. We first explain the category of stock-flow diagrams and note the clear separation between the syntax of these diagrams and their semantics, demonstrating three examples of semantics already implemented in the software: ODEs, causal loop diagrams, and system structure diagrams. We then turn to two methods for building large stock-flow diagrams from smaller ones in a modular fashion: composition and stratification. Finally, we introduce the open-source ModelCollab software for diagram-based collaborative modeling. The graphical user interface of this web-based software lets modelers take advantage of the ideas discussed here without any knowledge of their categorical foundations.

Previous analysis has been performed on the daily number of cases, deaths, infected people, and people who were exposed to the virus, all of them in a timeline of 550 days. Moreover, it has made the fitting of infection spread detailing the most efficient and optimal methods with lower propagation error and the presence of statistical biases. Finally, four different prevention scenarios were proposed to evaluate the ratio of each one of the parameters related to the disease.