key: cord-0835508-xkqltk57 authors: Yamamoto, Nao; Jiang, Bohan; Wang, Haiyan title: Quantifying compliance with COVID-19 mitigation policies in the US: A mathematical modeling study date: 2021-03-04 journal: Infect Dis Model DOI: 10.1016/j.idm.2021.02.004 sha: d0f42cb75b46a867025774e5c1ad9c158735fa75 doc_id: 835508 cord_uid: xkqltk57 The outbreak of COVID-19 disrupts the life of many people in the world. In response to this global pandemic, various institutions across the globe had soon issued their prevention guidelines. Governments in the US had also implemented social distancing policies. However, those policies, which were designed to slow the spread of COVID-19, and its compliance, have varied across the states, which led to spatial and temporal heterogeneity in COVID-19 spread. This paper aims to propose a spatio-temporal model for quantifying compliance with the US COVID-19 mitigation policies at a regional level. To achieve this goal, a specific partial differential equation (PDE) is developed and validated with short-term predictions. The proposed model describes the combined effects of transboundary spread among state clusters in the US and human mobilities on the transmission of COVID-19. The model can help inform policymakers as they decide how to react to future outbreaks. The Coronavirus Disease 2019 pandemic is an unprecedented global crisis, and the US has become the center of the crisis. By December 20, 2020 , the total number of reported COVID-19 cases exceeded 17,000,000, with over 300,000 deaths in the US alone [1] . As the number of confirmed cases in the US continued to rise in early 2020, many states declared states of emergency and issued shutdown orders or stay-at-home orders to slow the spread of COVID-19 [2] . Many schools, workplaces, and public gathering spaces across the US were closed for an extended time. Although such measures might have saved lives, they have come at a high cost socially and economically. To balance 10 various health, economic, and social concerns, governors across the US made decisions to gradually reopen the economy in summer 2020, which resulted in the increase of the number of COVID-19 cases in many states. While the first wave of COVID-19 in the early spring was mainly in coastal cities, the second wave was observed among the states in the Sun Belt. Although the geographic 15 location was one of the main factors to identify the epidemic trend, almost all the states are still setting weekly records for new cases. As the third wave of COVID-19 threatens communities' health across the nation, governors are considering another round of lockdowns. However, because of trade-offs between health and economy, when and how a state should 20 impose and/or ease restrictions are not trivial. Different states have taken very different approaches to tackle the pandemic. Because businesses are only advised to follow federal guidance on social distancing, many businesses do not fully implement social distancing measurements. The absence of a national level mask mandate is thought to be escalating the spread of the virus. As delays in 25 policy implementation could produce significant harm to public health, rigorous quantification of the non-pharmaceutical interventions to slow the spread of the disease is urgently necessary. The other important factor determining the success of those policies is the levels of compliance. [3] shows that compliance with policies depends on the level In this paper, we quantify compliance with the COVID-19 mitigation policies at a regional level during the first two waves of the pandemic, which is peaked on April 10 and July 24 ( Figure 1 ). We use a spatio-temporal model, 40 specifically, a partial differential equation (PDE) model. Our analysis is based on ten regions defined by the US Department of Health and Human Services (HHS) because the clusters represent different geographical and social characteristics regarding the spread of COVID-19. The proposed model describes the combined effects of transboundary spread among regional clusters and human 45 activities on the transmission of COVID-19, enabling us to model the regional risk disparities and validate the COVID-19 spread. The localized results of the spatio-temporal model could provide valuable information to the local governments and public health officials to closely monitor new COVID-19 outbreaks and quickly reinstating mobility restrictions. In response to the current COVID-19 pandemic, many mathematical models have been proposed. Many of them use ordinary differential equations (ODE) ( [4, 5, 6, 7] ). The classical susceptible-infectious-recovered model (SIR) ( [8] ) and susceptible-exposed-infectious-recovered model (SEIR) ([9, 10, 11] ) are the most widely adopted ones for characterizing the outbreak of COVID-19. The 55 extension of the classical SEIR model with the age-stratified model ( [12] ) and the meta-population model ( [13] ) were also introduced. Our previous work ([14, 15, 16] ) applies PDE models to make a regional level of influenza with geo- infectious diseases ( [16, 17, 18, 19, 20] ), to the best of our knowledge, this work is the first attempt to apply PDE models on COVID-19 short-term prediction incorporating COVID-19 mitigation policies and its compliance together with human mobility data. We divide the country into ten regions defined by the HHS, with a regional office located within each of the regions (Figure 2 ). There are several reasons for us to use the ten regions: (i) this enables us to capture the geographical and social characteristics regarding the spread of the virus while avoiding the high computational cost, (ii) the Centers for Disease Control and Prevention (CDC) 80 uses the ten regions to report weekly influenza activities. We compute three time-series data of each region by accumulating the data of all states belonging to a region. adding the changes of all states belonging to a region. We generated two timeseries of data sets from the GCMR; one is for the activities outside of the home, and the other is for stay-at-home activities. Former is the sum of five categories (i.e., retail & recreation, groceries & pharmacies, parks, transit stations, workplaces), and the latter is the data of residential activities. This section introduces a specific PDE model to characterize the spatiotemporal dynamics of the US COVID-19 cases at a regional level. To apply a PDE model to the interaction of the dynamics of COVID-19 cases, one needs to embed the ten regions into Euclidean space in such a way that the ten regions 120 stay as close as possible to ensure that the continuous model can capture the spread of COVID-19 cases between them. Here we embed the ten regions onto 6 J o u r n a l P r e -p r o o f the x-axis of the Cartesian coordinates at x = 1, 2, · · · , 10 in the east-west direction of US as shown in Figure 4 . One might use some algorithms discussed in [14] for a slight improvement. Because the accuracy of the prediction is divided into two processes: an internal (local) process within each region and an external (global) process between different regions. Similar derivation for the PDE model has been used in our previous work for PDE models for COVID-19 infection in Arizona and information diffusion in online social networks ( [14, 17] ). (1) Among those functions, m(x, t − 10) , p(x, t − 10) and ψ(x) takes data and d(x), r(t), l(x), and E(x) are to be estimated. Following is a detailed explanation of each term. • Neumann boundary condition ∂C ∂x (1, t) = ∂C ∂x (10, t) = 0, t > 1 is applied in ( [26] ). For simplicity, we count the cases imported from neighbor states as local US cases and assume that no COVID-19 spreads across the 180 boundaries at x = 1, 10. • Initial function C(x, 1) = ψ(x) describes the initial states of COVID-19 in every US region, which can be constructed from the historical data of COVID-19 cases by cubic spline interpolation. The basic mathematical properties of the proposed PDE model in Equation (1), such as existence and uniqueness, can be established from the standard theorems for parabolic PDEs in [27] . Below, we evaluate the robustness of our cases with the observed COVID-19 cases for the ten regions, which are the ground truth. The mean absolute percentage error is applied to measure the prediction accuracy, where x real is the observed COVID-19 cases at every data collection time point and x predict is the predicted cases. The average relative accuracy of the ten regions with one 225 day prediction are well acceptable with 93% and above as in Table 2 . To further justify the model, we also perform 7 and 14 days ahead predictions 13 J o u r n a l P r e -p r o o f as in Table 2. As demonstrated in the table, the average accuracy for 7 and 14 days ahead are about 86% and 69% respectively. • Pseudo-code: -Begin with first time frame for prediction * Input data to update m(x, t−10), p(x, t−10) and ψ(x). initialize We also find that region 1 (Connecticut, Maine, Massachusetts, New Hampshire, Rhode Island, and Vermont) and region 4 share similar policy scores, but the infection level of region 4 significantly exceeds that of region 1. The main reason is that the compliance index of region 1 is greater than region 4, which means the mitigation strategies in region 1 get more social support. In The average relative accuracy of the ten regions with three days prediction were well acceptable with 95% and above. Our results highlight that social distancing policy and its compliance can reduce the number of cases and help end the epidemic more quickly. In the US there was a large degree of heterogeneity in the social distancing policy and 305 its compliance throughout the states, cluster level analysis captured the trend while saving the computational cost. The effectiveness of mitigation measures also vary in different areas due to many social factors. Every precautionary measure needs time for citizens to accept. For example, public mask-wearing is now widely viewed as a low-cost and effective measure for reducing COVID-19 trans- To the best of our knowledge, this paper is the first study to validate COVID-19 cases in the US with local policies and its compliance using a specific PDE. This work demonstrates the influences and effectiveness of various social precautions such as stay-at-home order, face masks mandate, and practicing social distance. The proposed framework provides the measurement of localized poli-325 cies. Thus, medical workers and governors will have better preparation for the coming COVID-19 waives. In conclusion, we have developed a specific PDE model taking into account social distancing policy, its compliance, and human mobility -all issues which are crucial to disentangle the COVID-19 epidemic. The model fits the current 330 data remarkably well with one, seven and 14 day ahead predictions. We believe that our model can help inform policymakers as they decide how to react to future pandemic waves. 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