PolSIRD: Modeling Epidemic Spread Under Intervention Policies

Abstract

Epidemic spread in a population is traditionally modeled via compartmentalized models which represent the free evolution of disease in the absence of any intervention policies. In addition, these models assume full observability of disease cases and do not account for under-reporting. We present a mathematical model, namely PolSIRD, which accounts for the under-reporting by introducing an observation mechanism. It also captures the effects of intervention policies on the disease spread parameters by leveraging intervention policy data along with the reported disease cases. Furthermore, we allow our recurrent model to learn the initial hidden state of all compartments end-to-end along with other parameters via gradient-based training. We apply our model to the spread of the recent global outbreak of COVID-19 in the USA, where our model outperforms the methods employed by the CDC in predicting the spread. We also provide counterfactual simulations from our model to analyze the effect of lifting the intervention policies prematurely and our model correctly predicts the second wave of the epidemic.

Appendix: GraphPolSIRD: Spatial Transport with PolSIRD

In this section, we describe the more detailed GraphPolSIRD model which collectively models both the temporal and the spatial evolution of a pandemic. Consider a set of M heterogeneous populations (e.g., the M = 50 for the US states) as nodes in a graph $\mathbb {G}$. We assume that the population on a node can be assumed to mix homogeneously within itself while the populations between two nodes require traveling to mix. There exists a directed edge from node i to node j in the graph if people can travel from i to j. The edge weight denoted by f_ij is proportional to the frequency at which residents of node i travel to node j and come back. Note that f_ij may not be equal f_ji since some states like New York may be popular tourist/business destinations while others like Alaska might experience a limited number of visitors (hence f_AK,NY ≫ f_NY,AK). An estimate of these travel frequencies is required from travel data for the GraphPolSIRD model. For our experiment results in Section 4.2, we used the PlaceIQ movement data derived from anonymized, aggregated smartphone movements provide by PlaceIQ [27]. Note that while we assume these travel frequencies f_ij to be averaged constants independent of time, in practice this may not be the case. If non-aggregated daily travel frequencies are available, our GraphPolSIRD model also admits using them instead. However, for our application to COVID-19 modeling, only aggregated travel frequencies were available and hence we describe a simpler version of GraphPolSIRD with constant graph edge weights f_ij independent of time.

At any time t, the state G(t) of graph $\mathbb {G}$ is described by the compartment populations at all the nodes, i.e., $G(t) = \{S_{i}(t), I_{i}(t), \tilde {I}_{i}(t), R_{i}(t), \tilde {D}_{i}(t), \tilde {C}_{i}(t)\}_{i \in [M]}$. The population conservation equation still applies independently at all nodes:

$$ \begin{array}{@{}rcl@{}} S_{i}(t) + I_{i}(t) + \tilde{I}_{i}(t) + R_{i}(t) + \tilde{D}_{i}(t) = N_{i}, \quad \forall i \in [M], \forall t. \end{array} $$

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The evolution from time t to t + 1 is now governed by a cascade of two operators, denoted as $\mathbb {T}$ (the temporal evolution operator) and $\mathbb {S}$ (the spatial evolution operator), i.e.,

$$ \begin{array}{@{}rcl@{}} G(t+1) = \mathbb{T}(\mathbb{S}(G(t))), \forall t. \end{array} $$

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The temporal operator $\mathbb {T}$ applies to each node individually and is governed by the same set of (6)–(13) described in Section 3. Here, we describe the spatial operator $\mathbb {S}$ which applies to the graph $\mathbb {G}$ as a whole and models the effect of epidemic spread due to travel between the heterogeneously mixing populations at different nodes.

The spatial operator $\mathbb {S}$ affects only the susceptible and the unreported infectious compartments at a node due to the presence of infected population traveling to it from adjacent nodes. We assume that reported infected individuals refrain from traveling (or the number of such individuals traveling is small enough to be neglected); hence, the inter-node spread happens primarily due to unreported infected individuals traveling between nodes. Hence, the spatial operator reduces the susceptible population and increases the infected population at each time step at every node i ∈ [M] as follows:

$$ \begin{array}{@{}rcl@{}} S_{i,\mathbb{S}}(t) &=& S_{i}(t) - \sum\limits_{j \in [M]\setminus\{i\}} \sigma f_{ji} \frac{I_{j}(t)}{N_{j} - \tilde{I}_{j}(t) - \tilde{D}_{j}(t)} \end{array} $$

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$$ \begin{array}{@{}rcl@{}} I_{i,\mathbb{S}}(t) &=& I_{i}(t) + \sum\limits_{j \in [M]\setminus\{i\}} \sigma f_{ji} \frac{I_{j}(t)}{N_{j} - \tilde{I}_{j}(t) - \tilde{D}_{j}(t)}, \end{array} $$

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where the term being subtracted from S_i(t) and being added to I_i(t) are the number of individuals getting affected at node i at time step t due to unreported infected individuals traveling from neighboring nodes j ∈ [M] ∖{i}. Note that the term $N_{j} - \tilde {I}_{j}(t) - \tilde {D}_{j}(t)$ is the total traveling population (since reported infected individuals and dead individuals do not travel) and the term $\frac {I_{j}(t)}{N_{j} - \tilde {I}_{j}(t) - \tilde {D}_{j}(t)}$ gives the number of unreported infected population as a fraction of the total population traveling. When multiplied by the average number of people travelling from node j to node i, i.e., f_ji (a.k.a. travel frequency), the term $f_{ji} \frac {I_{j}(t)}{N_{j} - \tilde {I}_{j}(t) - \tilde {D}_{j}(t)}$ gives the number of unreported infected people traveling from node j to node i. The constant $\sigma \in (0, \infty )$ is an additional learnable parameter of the GraphPolSIRD model which stands for the number of people infected at the destination node per traveling person from a source node. This parameter is learnt end-to-end with gradient descent along with the other learnable parameters described in Section 3.4. Note that since we have the learnable parameter σ multiplying with the travel frequencies f_ji, it can account for any magnitude changes in the f_ij coefficients. This implies that we do not necessarily need exact travel frequencies. Instead, one can use any available quantities proportional to them. This is a very useful property since the PlaceIQ movement data [27] does not provide us exact travel frequencies but rather the fraction of smartphones in state A on the current day that pinged in any of the other states in the last 15 days. This quantity is roughly proportional to the travel frequencies but does not reflect their exact magnitude; however, our learnable parameter σ can to adapt to any scaling in the magnitudes.

Finally, the spatial operator does not change any other compartments or observables other than S and I. Hence, after the spatial operator $\mathbb {S}$, we can apply the temporal operator $\mathbb {T}$ to the graph state $G(t) = \{S_{i,\mathbb {S}}(t), I_{i,\mathbb {S}}(t), \tilde {I}_{i}(t),R_{i}(t), \tilde {D}_{i}(t), \tilde {C}_{i}$ (t)}_i∈[M] as described in the main text to model the transition to time step t + 1.