Key insight

In this study we introduce the concept of a universal geospatial risk of person-to-person transmission of influenza-like illnesses in the US; universal in the sense that it is pathogen-agnostic provided the transmission mechanism is broadly similar to that of seasonal Influenza. We call this the Universal Influenza-like Transmission (UnIT) score. Transmission dynamics in the general population is known to be modulated by diverse factors, only a few of which have been investigated, and are now beginning to be characterized. In all likelihood many unmodeled factors remain, along with the impact of non-trivial interactions between such known and unknown covariates that are hard to disentangle and account for. The UnIT score allows us to account for the impact of these unmodeled effects by automatically leveraging subtle emergent geospatial patterns underlying the seasonal flu epidemics of the past. In particular, we reduce the need for human modelers to manually identify every putative covariate that impacts the process.

Importantly, the UnIT score—once computed—has applicability beyond seasonal influenza. Validating our claim that the estimated UnIT score indeed quantifies a risk phenotype of individual counties for a disease with a flu-like transmission mechanism, we significantly improve incidence forecasts for COVID-19 over currently proposed state of the art models. We show that the UnIT score emerges as the most important factor “explaining” observed county-specific incidence trends for COVID-19 in the US, with coefficients in normalized multi-variate regression dominating those for typical covariates. Thus, our key insight is that incidence patterns from a past epidemic caused by a different pathogen can substantially inform current projections under mild assumptions on the similarity of the transmission mechanisms. We operationalize this insight by crafting a general information-theoretic principle to transfer this past knowledge to the new context of COVID-19. This is accomplished via a new computable measure of intrinsic similarity between stochastic sample paths generated by the hidden processes driving incidence.

Modeling approach

Our overall scheme is summarized in Fig 1. We leverage the county-specific incidence patterns observed for the past Influenza epidemics to compute the UnIT score, which then is used as a new fixed effect to infer a general linear model (GLM) for county-specific COVID-19 weekly count totals, alongside other putative covariates. The coefficients computed for the GLM model (stag 1, updated weekly) are then used to “correct” the COVID-19 count, replacing the observed count vector with the weighted linear combination of the socio-economic, demographic and the UnIT risk covariates. Intuitively, one may visualize this step as analogous to replacing a somewhat diffused set of observed points with a fitted line in linear regression. Finally, in stage 2 this corrected incidence vector is used to train ensemble regressors (updated weekly) that predict the next week’s county-specific count totals. Importantly, the GLM model in the first stage and the regressors in the second stage are updated weekly, while the UnIT score remains invariant. The key ingredient that makes simple ensemble regressors in the final step to perform better than more involved tools reported in the literature is the information-rich UnIT score, which potentially informs about complex transmission patterns modulating Influenza-like incidence native to each US county. Importantly, for each week, we compute one GLM model, and a fixed set of ensemble regressors, which are then used to predict case counts for individual counties, i.e., we do not have separate models for each county, and the county-specific effects are captured by the spatial variation of the putative covariates.

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Fig 1. Modeling scheme.

We use a national insurance claims database with more than 150 million people tracked over a decade (Truven Claims database) to curate geospatial incidence records for past Influenza epidemics over nearly a decade, which informs our new UnIT score. This score is then used as an additional fixed effect along with other putative socio-economic and demographic covariates obtained from US Census to infer a General Linear Model (GLM) explaining the weekly county-specific case COVID-19 case count. Using this inferred GLM model we “correct” the observed weekly case count, and use it as the only feature in an ensemble regressor to forecast county-specific count totals. The GLM model and the regressor is recomputed weekly, while the UnIT score remains invariant, representing a geospatial phenotype modulating transmission.

https://doi.org/10.1371/journal.pcbi.1009363.g001

Our ability to leverage Influenza infection patterns to inform COVID-19 modeling is not surprising. COVID-19 and Influenza are both respiratory disorders, which present as a wide range of illnesses from asymptomatic or mild through to severe disease and possible death. Both viruses are transmitted by contact, droplets and fomites [23]. Current efforts to curb the spread of COVID-19 worldwide has also reduced Influenza cases [24–26]. However, to the best of our knowledge, the current paradigms have not capitalized on this similarity between the transmission mechanisms of the two viruses. This is not simply an oversight: an effective approach to leverage flu patterns in COVID-19 modeling is non-trivial. Despite similarities outlined above, there are important empirically observed differences between the two diseases precluding a “drop-in” replacement, e.g., COVID-19 has possibly a higher reproduction number [27–29], can be spread widely by asymptomatic carriers (more so than Influenza [30, 31]), is estimated to have a potentially higher mortality rate [32], is novel, i.e., is infecting a host population with almost non-existent immunity, and the COVID-19 pandemic has induced a global trend of social distancing policies alien to the seasonal flu dynamics. Despite these challenges, the UnIT score has significant predictive value, more than manual combinations of putative factors investigated so far.

Limitations & conclusion

A source of uncertainty in our approach is the use of diagnostic codes from insurance claims to infer seasonal flu incidence. Influenza is in general hard to track, since less severe cases are seldom reported. Additionally, electronic health records are also inherently noisy, and suffers from potential coding errors by physicians, and other artifacts. Similarly the number of confirmed COVID-19 cases is also a function of how many tests are actually administered, and the fraction of the infected population who are asymptomatic. Thus, we are forecasting the number of detected cases as opposed to true disease incidence.

Additionally, our database of diagnostic codes needs to have geospatial metadata, i.e, we need to know the county in which each patient was located at the time they contracted influenza. This information has been redacted recently from the Truven database due to privacy concerns, implying that we could only use data upto 2011 to construct the UnIT risk. Leaving out random years one at a time from this 9 year period did not affect the results significantly, suggesting that the key patterns we are leveraging do not change too fast. The effect of considering newer data from other sources, will be investigated in future.

An important limitation in our modeling are current assumptions on the distribution of the response variable. While we achieved good prediction results, future work in this direction might consider application of recently reported strategies that deal with non-Poisson or non-Gaussian distributions [65] to further refine our models.

Importantly our results do not imply that Influenza and COVID-19 are similar in their clinical progression. Indeed, a limitation of our approach is its reduced ability to predict COVID-19-related deaths (S4 Fig and Tables B and C in S1 Text). Our death count forecasts are worse than the top few contributors [66] to the COVID-19 forecasthub. We hypothesize that this reduced effectiveness is attributable to differences in the clinical progression of Influenza and COVID-19: COVID-19 is a more serious disease, and while historical flu patterns may be leveraged to predict the number of cases, performance suffers when we attempt to extend the same strategy to predict the mortality.

In this study we demonstrate that leveraging the knowledge of the incidence fluctuations in one epidemic informs another with a broadly similar transmission mechanism, despite differences in the epidemiological parameters and the disease processes. The COVID-19 pandemic has highlighted the need for tools to forecast case counts early in the course of future pandemics, when only sparse data is available to train upon, by leveraging incidence pertaining to different epidemics of the past.

Forecast model

The UnIT score (ν) is a spatially varying time-invariant measure. Thus, to forecast temporal changes in weekly incidence we consider the past week’s case count as a feature in training regressors as follows (where X_t is the observed case count at time t, and is the forecast made for t at time t − 1): (3a) (3b) (3c)

Here g_t is the generalized multivariate regression model (GLM) which carries out the Poisson regression, fitted with X_t as the target variable, and ν, v₁, ⋯, v_m as exogenous variables, with a logarithmic link function (See (11)). ν is the urban-UnIT risk, and the rest of the variables v₁, ⋯, v_m (as described in Table 1) are total population, fraction of population over 65 years, fraction of minorities in the population, fraction of Hispanics, fraction of the population reported as African-American or black, fraction of the population designated to be poor, and the median household income. Including the fraction of population living in urban environments as a separate variable does not change results significantly. In Eq 3b is the estimate of X_t obtained using the inferred coefficients in g_t, and may be viewed as the noise corrected version of the current case count. Finally, we train a standard regressor between the corrected case count and the count observed in the next time step, and use it for forecasting one-week futures (Eq 3c). The choice of the specific regressor (random forest, gradient boosting, feed-forward neural networks or more complex variants) does not significantly alter our performance.

This is an exceedingly simple model compared to the approaches described in the literature, and is essentially a ensemble regressor with the UnIT-corrected case count as one of the features/inputs. Nevertheless we outperform the top state of the art models put forward by the COVID-19 modeling community (https://covid19forecasthub.org/community) in mean absolute error in county-specific incidence count estimates (See Fig 3D). As examples we illustrate the county-wise predicted and confirmed case counts for New York and California at selected weeks over the pandemic, which shows that our 1-week forecasts match up well with the counts ultimately observed (See Fig 5).

Computing similarity from sample paths

Efficiently contrasting and condomness cannot be ignored. For such learning to occur, we need to define either a measure of deviation or, more genmparing stochastic processes is the key to analyzing time-dependency in epidemiological patterns, particularly where raerally, a measure of similarity to compare stochastic time series. Examples of such similarity measures from the literature include the classical l_p distances and l_p distances with dimensionality reduction [67], the short time series distance (STS) [68], which takes into account of irregularity in sampling rates, the edit based distances [69] with generalizations to continuous sequences [70], and the dynamic time warping (DTW) [71], which is used extensively in the speech recognition community.

A key challenge in the existing techniques is differentiating complex stochastic processes with subtle variations in their generative structures and parameters. When presented with finite sample paths from non-trivial stochastic processes, the state-of-the-art techniques often focus on their point-wise distance, instead of intrinsic differences in their (potentially hidden) generating processes. Our approach addresses this issue and demonstrably differentiates data streams indistinguishable by state-of-the-art algorithms.

Our intuition follows from a basic result in information theory: if we know the true distribution p of a random variable, we could construct a code [53] with average description length h(p), where h(⋅) is the entropy of a distribution. If we used this code to encode a random variable with distribution q, we would need bits on average to describe the random variable. Thus, deviation in the distributions show up as an additional contribution from the KL divergence term . Generalizing the notion of KL divergence to processes, we can therefore quantify deviations in dynamics via an increase in the entropy rate by the corresponding divergence.

Intuitive example

As a more concrete example, consider sequences of length n generated by two iid processes and , where B(p) is the Bernoulli process with parameter p [72]. Our objective is to estimate deviations in the binary sample paths generated by these processes. Here we choose iid processes for simplicity, which is not a restriction in general for our approach. Let us generate sequences of length n and use E_ij to denote the expected Hamming distance [73] between sequences generated by and . It is easy to show that E₁₁ = E₁₂ = E₂₁ = 0.5n, which implies that two sequences both generated by B(.5) are not more alike than two sequences where one is generated by B(.5) and the other by B(.8). Denoting: and letting L(x, B(p)) denote the log-likelihood of B(p) generating x, we define: (4)

Then, by law of large numbers [74], we have: which now clearly disambiguates the two processes indistinguishable by their expected Hamming distance, and the correct generator may be identified readily as the one corresponding to the index of the smaller entry in v_x. Our approach generalizes this idea to more complex processes, where we cannot make the iid assumption a priori, thus necessitating the generalization of KL divergence from distributions to stochastic processes.

Log-likelihood of generating sample paths

In the example above, the generating models are used to evaluate log-likelihoods, which are not directly accessible in our target application. The computation of the log-likelihood L(x, G) of a sequence x generated by a process G, is simple (See Algorithm A in S1 Text) if we restrict our stochastic processes to those generated by Probabilistic Finite State Automata (PFSA) [75–78]. PFSA are semantically succinct and can model discrete-valued stochastic processes of any finite Markov order, and can approximate arbitrary Hidden Markov Models [76] (HMM). Importantly, PFSA model finite valued processes taking values in a finite pre-specified alphabet. Thus, continuous or integer valued inputs must be quantized, in a manner described later.

In the context of the above discussion, we define dissimilarity Θ between observed sequences x, y as: (5) where is a set of pre-specified PFSA generators on the same alphabet. And using PFSAs for our base models implies that this measure is easily computatble via multiple applications of Algorithm A in S1 Text for pseudocode of algorithm). In our approach, we use the set of four PFSA models shown in S5 Fig as . Using a different set of models, which generate processes that are sufficiently pairwise distinct, does not significantly alter our results. These particular “base” models are chosen randomly from all possible PFSAs (See next section) with a maximum of 4 states. For a finite number of base models, Eq (5) does not technically yield a metric. However, one can approach a metric by increasing the number of models included in the base set. S5 Fig illustrates a comparison of this approach of comparing time series with the state of the art Dynamic Time Warp (DTW) algorithm. In particular, our approach is significantly faster yet produces a higher separation ratio (ratio of the mean distance between clusters computed by the two algorithms) for the University of California Riverside (UCR) time-series classification archive [79].

Probabilistic finite automata

Definition 1 (PFSA). A probabilistic finite-state automaton G is a quadruple , where Q is a finite set of states, Σ is a finite alphabet, δ: Q × Σ → Q called transition map, and specifies observation probabilities, with .

We use lower case Greeks (e.g. σ or τ) for symbols in Σ and lower case Latins (e.g. x or y) to denote sequence of symbols, with the empty sequence denoted by λ. The length of a sequence x is denoted by |x|. The set of sequences of length d is denoted by Σ^d.

The directed graph (not necessarily simple with possible loops and multi-edges) with vertices in Q and edges specified by δ is called the graph of the PFSA and, unless stated otherwise, assumed to be strongly connected [80].

Definition 2 (Observation and Transition Matrices). Given a PFSA , the observation matrix is the |Q| × |Σ| matrix with the (q, σ)-entry given by , and the transition matrix Π_G is the |Q| × |Q| matrix with the (q, q′)-entry, written as π(q, q′), given by .

Both Π_G and are stochastic, i.e. non-negative with rows of sum 1. Since the graph of a PFSA is strongly connected, there is a unique probability vector p_G that satisfies [81], and is called the stationary distribution of G.

Definition 3 (Γ-Expression). δ and may be encoded by a set of |Q| × |Q| matrices Γ = {Γ_σ|σ ∈ Σ}, where (6)

We extend the definition of the Γ to Σ^⋆ by for x = σ₁…σ_n with Γ_λ = I, the identity matrix.

Definition 4 (Sequence-Induced Distributions). For a PFSA , the distribution on Q induced by a sequence x is given by , where 〚v〛 = v/‖v‖₁.

Definition 5 (Stochastic process Generated by PFSA). Let be a PFSA, the Σ-valued stochastic process {X_t}_t∈Σ generated by G satisfies that X₁ follows the distribution and X_t+1 follows the distribution for .

We denote the probability an PFSA G producing a sequence x by p_G(x). We can verify that .

Learning PFSA from sample paths

A key step in our approach is the abductive inferrence of a PFSA [82] from quantized incidence time series. Importantly, we do not specify the number of states, or the transition structure of the model; both the transition map and the observation probabilities are inferred from the observed data streams. A single PFSA modeling the incidence dynamics in high risk counties is obtained in step 5) of the procedure outlined in the section “Calculation of UnIT Risk” below. Importantly, we have a data stream for each county inferred to have a high initiation risk (Step 3). Thus, we infer a single PFSA from multiple data streams, where each data stream is assumed to be a sample path generated by a similar underlying process. The inference algorithm cited above is designed to take advanatge of multiple data stream inputs to identify a common model.

Sequence likelihood divergence

Definition 6 (Entropy rate and KL divergence). The entropy rate of a PFSA G is the entropy rate of the stochastic process G generates [83]. Similarly, the KL divergence of a PFSA G′ from the PFSA G is the KL divergence of the process generated by the G′ from that of G. More precisely, we have the (7) and the KL divergence (8) whenever the limits exist.

We also refer to the KL divergence between stochastic processes as the Sequence Likelihood divergence (SLD).

Definition 7 (Log-likelihood). The log-likelihood [83] of a PFSA G generating x ∈ Σ^d is given by (9)

Algorithm A in S1 Text outlines the steps in computing L(x, G). The time complexity of log-likelihood evaluation is O(|x| × |Q|) with input length x and |Q| being the size of the PFSA state set.

Theorem 1 (Convergence of Log-likelihood). Let G and G′ be two irreducible PFSA, and let x ∈ Σ^d be a sequence generated by G. Then we have in probability as d → ∞.

Proof. See proof of convergence in S1 Text.

From distance matrix to similarity matrix

Let D be the pair-wise distance matrix with d_ij = Θ(s_i, s_j), where s_i is the flu time series of county c_i. Then the affinity matrix A for spectral clustering is chosen as .

Data source: COVID-19 incidence & putative factors

Data on confirmed cases of COVID-19 were compiled and released at the COVID-19 Data Repository by the Center for Systems Science and Engineering at Johns Hopkins University (https://github.com/CSSEGISandData/COVID-19). The John Hopkins COVID-19 data represent data collated by the US Centers for Disease Control & Prevention (CDC) from individual states and local health agencies. Using the John Hopkins COVID-19 data resource, we obtained county-level confirmed new weekly case counts for all weeks upto the current point in time (2021–05-30) for 3094 US counties. We calculated COVID-19 case per capita using the 2019 population estimate provided by the US Census Bureau generated from 2010 US decennial census (https://www.census.gov/data/datasets/time-series/demo/popest/2010s-counties-detail.html).

We include five demographic independent variables: 1) total population, 2) percent of the total population aged 65+, 3) percent of Hispanics in the total population, 4) percent of black/African-American in the total population, 5) percent of minority groups in the total population. For socioeconomic factors, we consider: 1) percent of the total population in poverty and 2) median household income, Which are also obtained from the US Census Bureau, based on the 2010 US decennial census.

Data source: Seasonal influenza incidence

The source of incidence counts for seasonal flu epidemic is the Truven MarketScan database [55]. This US national database collating data contributed by over 150 insurance carriers and large, self-insuring companies, contains over 4.6 billion inpatient and outpatient service claims, with over six billion diagnostic codes. We processed the Truven database to obtain the reported weekly number of influenza cases over a period of 471 weeks spanning from January 2003 to December 2011, at the spatial resolution of US counties. Standard ICD9 diagnostic codes corresponding to Influenza infection is used to determine the county-specific incidence time series, which are: 1) 487 Influenza, 2) 487.0 Influenza with pneumonia, and 3) 487.1 Influenza with other respiratory manifestations and 4) 487.8 Influenza with other manifestations.

Discretization of incidence counts

Integer-valued incidence input is quantized to produce data streams with a finite alphabet, by choosing k − 1 cut-off points p₁ < p₂ < ⋯ < p_k−1 and replacing a value <p₁ by 0, in [p_i, p_i+1) by i, and ≥ p_k−1 by k. We call the set of cut-off points a partition. In our processing of incidence count data for flu epidemics, we obtain a binary partition by first taking a 1-step difference (i.e., transforming a length-n sequence x₁, x₂, …, x_n−1, x_n to x₂ − x₁, x₃ − x₂, …, x_n − x_n−1), and then replacing each positive value in the resulting sequence by 1 and the remaining, 0. Thus weeks with a rise in count is marked by 1, and teh remianing by 0.

Calculation of UnIT risk

We estimate the UnIT risk via the following 6 steps: 1) Compute pairwise similarity between US counties using the metric Θ introduced in Eq (5). 2) Cluster counties using this similarity measure using standard spectral clustering algorithm [56]. 3) Identify the set of counties that have high initiation risk, defined as ones that report cases within the first two weeks of each flu season. 4) Identify the cluster that has a maximal overlap with the set of high-risk counties. If we infer 4 clusters, then we found that only one cluster is sufficient to represent the set of high risk counties. If we set the parameters of the clustering algorithm to find more clusters, then more than one “high-risk” cluster might emerge, which we then collapse and treat as a single set for the next steps. 5) Generate a single PFSA G^⋆ based on the quantized incidence series from counties in the high-risk cluster cluster, using a reported abductive inference algorithm [54]. 6) Finally, estimate UnIT risk as (10)

The entropy rate is estimated as the entropy of the distribution of 0s and 1s (length 2 probability vector enumerating the fraction of 0s vs 1s), which provides an upper bound to the entropy rate [53]. Thus, our estimate for the UnIT risk gives us a lower bound, and more detailed computation only improves results marginally.

Calculation of urban-UnIT risk

In our modeling and forecasting investigations pertaining to the problem at hand, we use a scaled version of the UnIT risk denoted as the urban-UnIT risk, which is the county-wise product of the UnIT risk with the fraction of the population living in urban environment, as estimated from the 2010 US census.

UnIT correction to case count forecast

We fit a generalized linear model [40, 84] (GLM) with the assumption that the response variable (county specific weekly case counts confirmed for COVID-19) follows a Poisson distribution, and that the logarithm of its expected value can be modeled by a linear combination of unknown parameters.

Specifically, if the response Y, is assumed to be a count that follows a Poisson distribution with mean μ, then: (11) where X₁, X₂, …, X_k are explanatory variables (covariates). The counts are for all one-week periods between 2020-04-04 to 2021-05-30. This is also known as Poisson regression or a log-linear model.

To investigate the predictive contribution of the UnIT risk, we explore two models: 1) Baseline model with the following demographic and socio-economic covariates: percentage of urban population, population, percentage of population above 65 years old, percentage of minority population, percentage of black population, percentage of Hispanic population, percentage of population in poverty, median household income; and 2) UnIT-Augmented model which includes the covariates in the baseline model, with the additional urban-UnIT risk factor discussed above. Note that for the GLM modeling, we use standard score for all covariates and dependent variables with zero mean and unit variance, i.e., assuming the data for a variable is x₁, …, x_n and let and be the sample mean and sample standard deviation, respectively, we transform x_i to , so that a comparison of the magnitudes of the coefficients reflect the relative importance of the significant covariates.

As described before, we use the GLM model to obtain a “corrected” version of the county-specific case count vector, which is subsequently used to train an ensemble regressor to predict case counts 1 week into future. The precise algorithmic steps are enumerated in Algorithm B in S1 Text. To reduce variance we train a set of regressors in the final step, and report the mean. Here consists of a random forest model, an extra trees model and a feed-forward neural network model with a single hidden layer implemented through Tensor Flow.

Forecasting COVID-19-related deaths

An almost identical approach is used to forecast COVID-19-related deaths, where we use the same covariates as before, but replace the county-specific case count vector with the county-specific record of COVID-19-related deaths. The modified algorithm for forecasting deaths is enumerated in Algorithm C in S1 Text, where for training regressors, we also use the case count vectors and its corrected version produced by Algorithm B in S1 Text.