Optimal Pandemic Control Strategies And Cost Effectiveness Of Covid 19 Non Pharmaceutical Interventions In The United States, Additional File 1 (Irons And Raftery)
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Additional file 1, a supplement by Nicholas J. Irons and Adrian E. Raftery to their study Optimal pandemic control strategies and cost-effectiveness of COVID-19 non-pharmaceutical interventions in the United States. It contains further results on the cost-effectiveness of interventions, including sensitivity analysis of optimal control strategies, further results for the NPI regression model, epidemiological and economic parameter Tables S1–S3, and state-specific model plots. Figure S1 compares workplace closure over time under the OC, Obs. and Obs.−school strategies. The supplement compares its estimate of $4.6 trillion in losses during 2020 with projections of $16 trillion for the full pandemic by Cutler & Summers and Bruns & Teran.
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“Optimal Pandemic Control Strategies And Cost Effectiveness Of Covid 19 Non Pharmaceutical Interventions In The United States, Additional File 1 (Irons And Raftery).”
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1 Additional file 1 for “Optimal pandemic control
2 strategies and cost-effectiveness of COVID-19
3 non-pharmaceutical interventions in the United States.”
4 Nicholas J. Irons∗, Adrian E. Raftery†.
5 This supplement contains: further results and discussion for the cost-effectiveness of
6 interventions, including sensitivity analysis of optimal control strategies and discussion of
7 the use of ICERs in infectious disease (Section S.1); further results and discussion for the
8 NPI regression model (Section S.2); epidemiological and economic parameter Tables S1–S3
9 (Section S.3); and plots of state-specific epidemiological model estimates (Section S.4).
10 S.1 Evaluating and optimizing costs
11 S.1.1 Baseline scenario
12 Dynamic workplace closure strategies. Figure S1 plots the average strength of work-
13 place closure across states over time and the resulting viral prevalence for the OC, Obs.,
14 and Obs.−school strategies. Generally, the OC strategy involves ramping up workplace
15 closure to combat new waves of infections, with implementation peaking in response to
16 the spring, summer, and fall waves of 2020. On the other hand, relative to OC, the av-
17 erage strength of workplace closure in Obs. ramped up and peaked later in spring and
∗
Department of Statistics, Leverhulme Centre for Demographic Science, and Pandemic Sciences Insti-
tute, University of Oxford, Oxford, UK. Corresponding author. Email: nicholas.irons@stats.ox.ac.uk.
†
Departments of Statistics and Sociology, University of Washington, Seattle, USA.
1
Average strength of workplace closure vs. prevalence across policies
(a) 1.0 (b) 1.0 (c) 1.0 OC policy
OC prevalence
Obs. policy
Obs. prevalence
Obs. − school policy
0.8 0.8 0.8 Obs. − school prevalence
0.6 0.6 0.6
0.4 0.4 0.4
0.2 0.2 0.2
0.0 0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020) Month (2020)
Figure S1: Average strength of workplace closure across states over time (dashed line) and
the resulting viral prevalence (posterior median with 90% credible interval) for the (a) OC,
(b) Obs., and (c) Obs.−school strategies
18 then remained constant at a (lower) moderate level throughout the year. As a result, the
19 OC policy exhibits a flatter prevalence curve, with summer and fall waves that are muted
20 relative to Obs. and Obs. - school.
21 Comparison to the literature. We can compare our estimates of the gross impacts
22 of COVID-19 and NPIs to others in the literature. Cutler & Summers [1] and Bruns
23 & Teran [2] project the total cost of the pandemic in the US over its full duration at $16
24 trillion, which is about 3.5 times our estimate of $4.6 trillion in losses observed during 2020.
25 Flaxman et al. [3] estimated that NPIs averted 3.1 (2.8–3.5) million deaths up to May 2020
26 in 11 countries totaling 375 million population. Ferguson et al. [4] predicted that NPIs
27 would prevent 1.1 million deaths in the US over the course of the pandemic. Greenstone
28 & Nigam [5] projected that moderate social distancing would save 1.7 million lives in the
29 US by October 1, with mortality benefits of $8 trillion, or $24,000 per capita, based on the
30 same VSCD used in our baseline scenario. Highlighting the inter-generational transfer of
31 wealth stemming from the implementation of social distancing measures, they note that the
32 vast majority of the monetized benefits of social distancing accrue to people age 50 or older.
33 More conservatively, Eichenbaum et al. [6] find that containment policies, if implemented
34 optimally, would save about half a million lives in the US based on low values of key
35 epidemiological parameters—specifically, they use an IFR of 0.5% and an R0 of 1.45 for their
2
36 baseline model. Thunström et al. [7] estimated that social distancing in the US would save
37 1.24 million lives at a cost of $7.2 trillion in lost GDP, which implies that social distancing
38 measures would yield net losses for any VSCD below 7.2/1.24 ≈ 5.8 million dollars. To the
39 contrary, we find that NPIs (and social distancing measures in particular) are cost-effective
40 for a VSCD of $4.5 million. Assuming a vaccine arrives (stochastically) after a year to
41 end the pandemic, Farboodi et al. [8] estimate the per capita cost of the optimal policy
42 at $8,100, comparable to our $5,781. They find that the laissez-faire equilibrium (i.e., in
43 the absence of government intervention) would only incur a cost of $12,700 per person,
44 as compared to our $20,978. Undertaking a cost-benefit analysis of confinement policies
45 targeted toward mitigation and (strict) suppression, Gollier [9] finds that both strategies
46 incur a total cost—combining economic and life costs—equating to 15% of annual GDP,
47 or about $10,000 per capita, which is comparable to our estimates of the total costs of the
48 various containment strategies in Results. Jones et al. [10] estimate that, under an optimal
49 social distancing policy, GDP declines by 12% and 0.17% of the population (about 560,000)
50 die from COVID-19 in the first 26 weeks of the pandemic. In a simulation study, Keogh-
51 Brown et al. [11] estimate that containment strategies in the UK to suppress COVID-19
52 through the end of 2020 would incur health costs of 1.7% of GDP and economic costs of
53 29.2% of GDP, with 7.3 percentage points coming from workplace absenteeism of parents
54 affected by school closure and 21.9 percentage points from business closure. Their total
55 cost is comparable to our estimate for the US in 2020 (i.e., under Obs.), which is 21%
56 of GDP. However, relative to the results of Keogh-Brown et al. [11], we find that health
57 impacts are a much larger portion of the total, costs related to business closure are much
58 smaller, and costs related to school closure are somewhat larger.
59 S.1.2 Sensitivity analysis of optimal control strategies
60 Figures S2–S7 display the results of our sensitivity analysis across regression models (i)–
61 (iii) and cost function specifications. Overall, our qualitative findings about the structure
62 of OC policies are robust across scenarios, with the main quantitative distinction being the
3
(a) VSCD high, learning low, work high (b) VSCD high, learning high, work high
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(c) VSCD high, learning low, work low (d) VSCD high, learning high, work low
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(e) VSCD low, learning low, work high (f) VSCD low, learning high, work high
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(g) VSCD low, learning low, work low (h) VSCD low, learning high, work low
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
Figure S2: Sensitivity analysis for the optimal control results under regression model (i).
Boxplots of the average value of the optimal NPI policy over the year in each state.
4
(a) VSCD high, learning low, work high (b) VSCD high, learning high, work high
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(c) VSCD high, learning low, work low (d) VSCD high, learning high, work low
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(e) VSCD low, learning low, work high (f) VSCD low, learning high, work high
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(g) VSCD low, learning low, work low (h) VSCD low, learning high, work low
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
Figure S3: Sensitivity analysis for the optimal control results under regression model (ii).
Boxplots of the average value of the optimal NPI policy over the year in each state.
5
(a) VSCD high, learning low, work high (b) VSCD high, learning high, work high
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(c) VSCD high, learning low, work low (d) VSCD high, learning high, work low
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(e) VSCD low, learning low, work high (f) VSCD low, learning high, work high
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
(g) VSCD low, learning low, work low (h) VSCD low, learning high, work low
NPI average value NPI average value
0.8 0.8
0.4 0.4
0.0 0.0
School Work Social Test Trace Mask School Work Social Test Trace Mask
Figure S4: Sensitivity analysis for the optimal control results under regression model (iii).
Boxplots of the average value of the optimal NPI policy over the year in each state.
6
(a) VSCD high, learning low, work high (b) VSCD high, learning high, work high
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.8
3.5 4.0 4.5 5.0
4.4
4.0
3.6
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(c) VSCD high, learning low, work low (d) VSCD high, learning high, work low
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.6
4.5
4.0
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(e) VSCD low, learning low, work high (f) VSCD low, learning high, work high
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.2
4.5
3.8
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(g) VSCD low, learning low, work low (h) VSCD low, learning high, work low
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.2
4.5
3.8
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
Figure S5: Sensitivity analysis for the costs of various policies under regression model (i).
Boxplots of the log-scale total cost in USD2020 per capita incurred by the OC, Full, Obs.,
Obs.−school, and Open policies across states.
7
(a) VSCD high, learning low, work high (b) VSCD high, learning high, work high
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
3.6 4.0 4.4 4.8
3.5 4.0 4.5 5.0
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(c) VSCD high, learning low, work low (d) VSCD high, learning high, work low
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.5
4.5
4.0
3.5 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(e) VSCD low, learning low, work high (f) VSCD low, learning high, work high
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.2
4.5
3.8
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(g) VSCD low, learning low, work low (h) VSCD low, learning high, work low
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.2
4.5
3.8
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
Figure S6: Sensitivity analysis for the costs of various policies under regression model (ii).
Boxplots of the log-scale total cost in USD2020 per capita incurred by the OC, Full, Obs.,
Obs.−school, and Open policies across states.
8
(a) VSCD high, learning low, work high (b) VSCD high, learning high, work high
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.8
4.5
4.2
3.6
3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(c) VSCD high, learning low, work low (d) VSCD high, learning high, work low
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.5
4.5
4.0
3.5 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(e) VSCD low, learning low, work high (f) VSCD low, learning high, work high
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.6
4.2
4.5
3.8
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
(g) VSCD low, learning low, work low (h) VSCD low, learning high, work low
log10(Cost) ($ per capita) log10(Cost) ($ per capita)
4.6
4.2
4.5
3.8
3.4 3.5
OC Full Obs. Obs. − school Open OC Full Obs. Obs. − school Open
Figure S7: Sensitivity analysis for the costs of various policies under regression model (iii).
Boxplots of the log-scale total cost in USD2020 per capita incurred by the OC, Full, Obs.,
Obs.−school, and Open policies across states.
9
63 optimal strength of workplace closure. We also find that the relative ranking by cost of the
64 policies considered can vary across cost function specifications.
65 We vary the VSCD, the cost of learning loss, and the cost of workplace closures across
66 plausible ranges, which are given in Table S2. As noted in Methods, we do not vary the
67 cost of social distancing measures as we are primarily interested in assessing the robustness
68 of the OC strategy and the relative costs of various policies rather than variation in the
69 total cost incurred by each policy. We also omit sensitivity analysis for the costs of testing,
70 tracing, and masking as they are highly cost-effective interventions under any reasonable
71 variation in their costs. Indeed, these measures are all at least somewhat effective in
72 reducing transmission and extremely cheap compared to infections, school and workplace
73 closures, and social distancing mandates.
74 Figures S2–S4 exhibit boxplots of the average value of each NPI in the OC strategy
75 across states for models (i)–(iii), respectively. We see that, barring a few outliers in some
76 scenarios, school closures are never implemented in the OC policy. This is the case even
77 when we assume both a high VSCD equal to the VSL ($10.63 million USD2020)—which
78 does not adjust the VSCD for the age profile of COVID-19 mortality—and a low cost of
79 learning loss (9% of GDP per 0.33 years)—which assumes that distance learning is 90% as
80 effective as in-person schooling—which are shown in the upper left panels. Across scenarios,
81 workplace closures are implemented fairly consistently, but their optimal duration decreases
82 as the cost of workplace closure increases or the VSCD decreases. In particular, in Figure
83 S4 (model (iii)), we see that the median optimal strength of workplace closure across states
84 is 0 when we use a low VSCD and high cost of workplace closure. These conclusions
85 are comparable to the results of Barrot et al. [12], who find that the cost-effectiveness of
86 business closures (i.e., whether they produce a net benefit or loss) is sensitive to modeling
87 assumptions and particularly the assumed value of a life-year.
88 Figures S5–S7 show boxplots of the costs of various policies across states. Notably,
89 when the cost of learning loss is high (right column, subplots (b), (d), (f), and (h)), the
90 total cost of Obs.−school is substantially lower than Obs. and Full. If we further assume
10
(a) California (b) Texas
15
Standardized ICER Standardized ICER
10
8 10
6
4 5
2
0 0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
(c) Florida (d) New York
140
10 12
OC
Obs. − school
Standardized ICER Standardized ICER
Obs.
8 100 Full
6
60
4
2
0 0 20
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S8: Posterior median and 50% credible intervals for the cumulative standardized
ICER of various policies in the four most populous states.
91 that the VSCD is low (bottom right panels, subplots (f) and (h)), Full exceeds the cost of
92 Obs., which exceeds the cost of Open. If both VSCD and the cost of learning loss are low
93 (bottom left panels, subplots (e) and (g)), Obs.−school becomes cheaper than Obs. and
94 on par with Full. These patterns are consistent across models (i)–(iii).
95 S.1.3 ICERs in infectious disease
96 The ICER is a quantity widely used in the economic evaluation of health interventions.
97 The ICER is defined as the monetary cost of an intervention divided by the benefit it
98 produces (as measured by some target outcome) relative to a baseline, with a larger ICER
99 often (mis)interpreted to mean that an intervention is less cost-effective [13]. In public
11
(a) California (b) Texas
25 40
Standardized ICER Standardized ICER
20 30
15
20
10
5 10
0 0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
(c) Florida (d) New York
100 150 200
25 OC
Obs. − school
Standardized ICER Standardized ICER
20 Obs.
Full
15
10
50
5
0
0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S9: Posterior median and 50% credible intervals for the weekly standardized ICER
of various policies in the four most populous states.
12
100 health and infectious disease, outcomes of interest include the number of quality-adjusted
101 life years (QALYs), disability-adjusted life-years (DALYs), infections, hospitalizations, or
102 deaths averted by the intervention. Despite their ubiquity in the health economics liter-
103 ature, ICERs can be difficult to interpret and are often defined, calculated, and reported
104 inconsistently, which severely limits their practical value in cost-effectiveness analysis [13,
105 14]. As a result, ICER estimates for the same strategy can vary by orders of magni-
106 tude across studies, and rankings of interventions based on their reported ICERs can yield
107 counter-intuitive results, as evidenced by a recent systematic review of economic evaluations
108 of COVID-19 interventions [15]. For example, Podolsky et al. [15] find that the median
109 ICER of school closure across studies in their review is exceeded by that of vaccination,
110 testing, facial covering, and stay-at-home policies. (This may be due to the fact that many
111 cost-effectiveness analyses considering school closure fail to account for costs associated to
112 student learning loss.) Furthermore, the median ICER of mask mandates exceeds that of
113 school closure by an order of magnitude, and the median ICER of stay-at-home orders
114 exceeds that of school closure by nearly 3 orders of magnitude. To address these issues,
115 some have argued instead for the use of a policy’s net benefit in decision-making [13, 16,
116 17]. We take a similar approach, reporting the expected total cost of various strategies,
117 which facilitates straightforward comparison of their cost-effectiveness. Nevertheless, our
118 methodology can shed light on the appropriate use of ICERs in the economic evaluation of
119 interventions targeting infectious disease transmission.
120 We argue that ICERs in the context of infectious disease interventions should be: data-
121 driven and account for uncertainty; not based solely on (poorly calibrated, overly simplistic,
122 or deterministic) simulation models; reported based on well-defined interventions; and cal-
123 culated using the intervention’s (causal) effect on the effective reproduction number Re (t),
124 the relevant quantity governing infectious disease transmission, which is interpreted as the
125 expected number of secondary cases resulting from an infection and can be estimated from
126 clinical data in real time. Related to this last point, we note that, for a given intervention,
127 the ICER is a time- and context-specific quantity. Outcomes, such as infections averted by
13
128 an intervention, depend also on the implementation of other interventions (their timing,
129 strength, duration) as well as other factors affecting disease transmission (e.g., the duration
130 of the pandemic, the baseline R0 value, other epidemiological parameters, voluntary social
131 distancing and protective measures, exogenous shocks, new variants, etc.). Therefore, in
132 calculating and reporting the ICER, our results should account for or be as invariant as
133 possible to contextual factors. We echo Prager et al. [18], who highlight the “‘...importance
134 of including a broader set of causal factors to achieve more accurate estimates of the total
135 economic impacts of not just pandemic influenza but biothreats in general.”
136 If we consider a blanket intervention that affects the population at large (e.g., social
137 distancing measures), the intervention’s reduction in Re (t) multiplied by the estimated size
138 of the infectious population gives the number of infections averted by the intervention on
139 day t, as we demonstrate below. This forms the denominator of the ICER. (On the other
140 hand, if we consider a targeted intervention, e.g., case isolation, the number of infections
141 averted can be estimated as the reduction in Re (t) multiplied by the number of treated
142 subjects.) Finally, the numerator of the ICER is the cost of implementing the intervention
143 on day t.
144 To demonstrate, Figures S8 and S9 exhibit cumulative and weekly standardized ICERs
145 of various strategies relative to the Open policy—in which no NPIs are used—over time in
146 the four most populous US states. We calculate these standardized ICERs as follows. For
147 an NPI policy u implemented on a specific day t, let
148 Re (u, t) := R0 (u, t)S(t)
149 denote the effective reproduction number on day t under policy u, where (in an abuse of
150 notation) R0 (u, t) is the basic reproduction number defined by our NPI regression model
151 (equation (3) in Methods) and S(t) is the susceptible fraction of the population. The
152 number of infections averted by the policy u on day t relative to Open is then
153 νa (t) := N I(t)(Re (0, t) − Re (u, t)),
14
154 where N is the population size, I(t) is the infectious fraction of the population, and Re (0, t)
155 denotes the effective reproduction number under no interventions. In a given period of days
156 [T1 , T2 ], the ICER of a policy u = {u(t) : t ∈ [T1 , T2 ]} is then
cNPI (u) · N
157 ICERu (T1 , T2 ) = PT2 ,
t=T1 νa (t)
158 where cNPI (u) is the per capita cost of the strategy. Finally, to define the standardized
159 ICER plotted in Figures S8 and S9, we convert infections to their monetary value using
160 the cost of an infection cν —defined in Methods—and take the reciprocal, such that the
161 standardized ICER reports the ratio of the value of infections prevented to the cost of the
162 intervention, which is a dimensionless quantity:
PT2
t=T1 cν νa (t)
163 SICERu (T1 , T2 ) := . (1)
cNPI (u) · N
164 Based on (1), in terms of net costs, a policy u relative to no intervention in time period
165 [T1 , T2 ]: breaks even if SICER = 1; produces a net benefit if SICER > 1; and produces a
166 net loss if SICER < 1.
167 We now return to Figure S8, which displays the cumulative standardized ICER over
168 time, {SICERu (1, t)}Tt=1 for various policies u. We see that, relative to Open, the different
169 containment strategies are comparable in terms of cumulative cost-effectiveness early in the
170 pandemic, with Obs. and Full becoming less cost-effective (but still producing a net benefit
171 by the end of 2020), mainly due to the cost of school closures exceeding 16 weeks. The
172 OC strategy is generally the most cost-effective by the end of the year, while Obs.−school
173 is a close second. Note that, while OC and Obs.−school are nearly equally cost-effective
174 by the end of 2020 in Florida, this does not imply that Obs.−school is also an optimal
175 policy—this can only be determined by looking at the net benefit of each policy. Figure
176 S9 displays the standardized ICER of each policy in each week (conditional on that policy
177 also being implemented in all weeks prior) relative to Open. Results are similar: OC and
178 Obs.−school are similarly cost-effective and consistently more cost-effective than Obs. and
179 Full; all containment strategies satisfy SICER > 1 for most of the year, implying that they
15
180 produce net benefits relative to no intervention.
181 The OC strategy is determined by minimizing aggregate costs accrued over the year.
182 We can solve this problem ex post—after we have observed the pandemic play out—but,
183 in principle, we cannot derive an optimal policy ex ante, since we cannot see the future.
184 Indeed, while susceptible-infectious-removed (SIR) models have demonstrated remarkable
185 utility in helping us understand infectious disease [19], modeling studies carried out early
186 in the pandemic (e.g., Ferguson et al. [4]) generally failed to predict the complex and
187 stochastic dynamics of SARS-CoV-2 (e.g., multiple waves, super-spreader events) depicted
188 in Figure 4 in Results. As we note in Results, even when we incorporate the effects of
189 interventions into the model, we fail to explain a substantial portion of the temporal vari-
190 ation in transmission rates. At its face, this may seem like a disheartening realization.
191 However, the trends in Figures S8 and S9 have important and encouraging implications for
192 decision-making during pandemics. Specifically, they show that the OC policy—a relatively
193 simple combination of testing, tracing, masking, social distancing, and reactive workplace
194 closure—was consistently highly cost-effective on a weekly basis throughout the first year
195 of the pandemic. This implies that, had we reasonable ballpark estimates of the costs and
196 effects of interventions to work with early on, we could have determined a nearly globally
197 optimal strategy by choosing, at each point in time, the policy that greedily minimized
198 the cost incurred in the next time step. In our context, we could not predict long-run
199 transmission rates and therefore we did not need to; the myopic strategy would have pro-
200 duced a nearly globally optimal solution. A similar phenomenon has been documented in
201 the control theory community [20]: the performance of control algorithms can be highly
202 sensitive to modeling errors. Hence, if a model is misspecified, or if we can only poorly
203 understand the behavior and evolution of a control system, simple algorithms tend to be
204 more robust.
205 We turn now to the cost-effectiveness of school closure in particular. In Results, we
206 estimate that, relative to Obs.−school, Obs. saved 77,168 (12,268–235,954) lives with the
207 cost of school closures amounting to $2 trillion, yielding an ICER of $25.9 (8.4–156.4)
16
208 million per death prevented. By comparison, in a systematic review Juneau et al. [21] find
209 that school closures during the 2009 H1N1 influenza pandemic cost $9.86 million per death
210 prevented, which implies that the cost of preventing a death due to H1N1 is on par with
211 the VSL—taken to be $10.63 million, in line with Robinson et al. [22], in our study. In
212 contrast, in our sensitivity analysis in S.1.2, we find that OC policies for COVID-19 involve
213 no school closure beyond the usual 16 weeks of break per year even when the VSCD is
214 assumed equal to the VSL and the cost of learning loss remains at the low value used in
215 our baseline scenario.
216 Qualitatively, our results are consistent with the findings of Juneau et al. [21] in other
217 ways. Specifically, they find that: testing, tracing, and masking are among the most cost-
218 effective measures; workplace and school closures are effective but costly, and hence the
219 least cost-effective interventions; combinations of NPIs are more cost-effective than single
220 interventions; and NPIs are more cost-effective when implemented early. While Juneau
221 et al. [21] conclude that school closure is among the least cost-effective interventions based
222 on the ICER, the value of this finding is limited for a number of reasons.
223 Firstly, as Paulden [13] notes, it is difficult to decide based on the ICER alone if an in-
224 tervention is cost-effective in a given setting, i.e., if it would be implemented in an optimal
225 policy in conjunction with other interventions and other external factors. Indeed, as noted
226 above, the cost-effectiveness of an NPI varies over time and across contexts, particularly
227 as prevalence varies. (This is one factor explaining why earlier is better with regard to
228 the timing of NPIs: as population immunity grows, NPI implementation yields diminish-
229 ing returns.) As such, calculating and ranking the ICERs of various interventions is not
230 sufficient to determine which strategies are cost-effective [13]. However, as we demonstrate
231 in Results, we can establish that school closure is not cost-effective by deriving the optimal
232 policy, which frames the analysis in terms of the expected total cost—or, equivalently, the
233 net benefit—of a policy. Our methodology goes beyond quantifying the cost-effectiveness
234 ratios of different policies by determining which NPIs should have been used and when.
235 Secondly, the results of Juneau et al. [21] are based on analysis of school closures during
17
236 the 2009 H1N1 pandemic. The biology and epidemiology of the H1N1 flu differed from
237 SARS-CoV-2 in important ways. In particular, it was evident early in the COVID-19
238 pandemic that SARS-CoV-2 was more virulent and more transmissible than H1N1, which
239 left open the possibility that school closure would be cost-effective in combating SARS-CoV-
240 2 transmission in 2020 despite its apparent lack of cost-effectiveness in 2009, as noted by
241 Pasquini-Descomps et al. [23] and Xue et al. [24]. Similarly, studying the cost-effectiveness
242 of interventions in response to outbreaks of influenza, gastroenteritis, and chickenpox in
243 France, Adda [25] finds that school closures are not cost-effective, but they “...would become
244 beneficial for epidemics characterized by a slightly more deadly strain.” Indeed, Xue et al.
245 [24], Dauelsberg et al. [26], Perlroth et al. [27], Milne et al. [28], and Kelso et al. [29]
246 find that extended school closures are cost-effective for severe pandemics. We note that
247 Dauelsberg et al. [26], Perlroth et al. [27], Milne et al. [28], and Kelso et al. [29] do not
248 account for costs associated to student learning loss and Xue et al. [24] do not consider
249 interventions other than school closure as available tools in their model. As we show in
250 Results, in the context of COVID-19, school closures are not cost-effective.
251 S.2 NPI regression model results
252 S.2.1 Fit to data
253 Regarding the fit of the NPI regression model to the estimated transmission rates, the
254 posterior median R2 is 0.59, indicating that a substantial proportion of the variance in
255 transmission rates remains unexplained by NPIs or behavioral response to the fear of infec-
256 tions. Indeed, there are numerous factors affecting SARS-CoV-2 transmission—including
257 super-spreader events and introduction of new infections from outside the state—accounted
258 for by the error terms ε(s) (w) that are difficult to predict. Figure S10 plots the maximum
259 a posteriori (MAP) trajectory of R0 output from the epidemiological model in the four
260 most populous states—California, Texas, Florida, and New York—against the posterior
261 predictive distribution from the NPI model fit to this output. The model fits the data well,
18
(a) California (b) Texas
MAP MAP
4 50% 4 50%
90% 90%
3 3
R0(t) R0(t)
2 2
1
1
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Florida New York
(c) (d)
7
4 MAP MAP
50% 6 50%
90% 90%
3 5
4
R0(t) R0(t)
2 3
2
1 1
0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S10: Time-varying transmission rates in four states. In black, the MAP trajectory
output by the epidemiological model. In dark and light blue, respectively, the 50% and
90% credible intervals of the posterior predictive distribution from the fitted NPI regression
model.
19
262 but cannot capture unpredictable shocks in transmission, which are reflected in future pre-
263 dicted values of the transmission rate through the AR(1) term. The AR(1) parameter φ is
264 0.76 (0.68–0.83), indicating a high degree of residual autocorrelation in the weekly-varying
(s)
265 reproduction number R0 (w) across states. Finally, the posterior degrees of freedom νε for
266 the Student-t distributed residuals ε(s) (w) is 2.9 (2.3–3.7), indicating that a heavy-tailed
267 error distribution is appropriate for these data.
268 S.2.2 Total effect of NPIs
Percent reduction in R0 due to NPIs
NPI.effect
60
55
50
45
Figure S11: Posterior median total percent reduction in R0 due to NPIs by state.
269 Zooming in on the state-level results, we can quantify the total effect of NPIs on trans-
270 mission in state s by
p
X
(s)
271 α := βu(s) (k),
k=1
20
(s)
272 with p(s) = 100(1 − exp(α(s) )) representing the total percent reduction in R0 under full
273 lockdown. Figure S11 displays the geographic distribution of the posterior median of p(s)
274 across states. Overall, NPIs tend to be more effective in more urbanized and populous
275 states. Some of this variation may be explained by the literature on political polarization
276 and partisan social distancing during the pandemic [30–36]. Alternatively, and related to
277 our discussion in the previous paragraph, we note that rural states tend to have lower
(s)
278 baseline R0 values, possibly due to later importation of the virus and lower levels of social
279 mixing. With a lower ceiling in these states, NPIs have less room to suppress transmission.
280 Finally, note that the p-vector
281 ρ(s) := βu(s) /α(s)
282 consists of weights representing the proportional contribution of each NPI to the total
283 reduction of transmission. Here ρ(s) can be thought of as defining a data-driven “stringency
284 index” combining NPIs according to their strengths in a single-number summary of the
285 stringency of government restrictions, as opposed to previously defined measures, such
286 as OxCGRT’s stringency index, which average NPIs uniformly without regard for their
287 varying effects on transmission [37].
288 S.2.3 Effects of individual NPIs
289 Figure S12 shows the pooled effect of each NPI, βu (k), k = 1, . . . , p, quantified as a percent
290 reduction in R0 , across the robust log-linear regression models defined in Methods. We
291 focus discussion on model (ii), which includes the effects of deaths and removals (but not
292 infections) incident in the prior week. Mask mandates are the most effective intervention,
293 reducing R0 by 19.0% (6.1%–28.5%). As shown in Figure S12, this effect attenuates from
294 19.7% (7.3%–28.5%) to 15.4% (3.3%–26.9%) as we move from model (i) to model (iii),
295 controlling for removals and infections incident in the prior week. By comparison, Sharma
296 et al. [38] estimate that mask mandates reduced transmission rates by 12% (7%–17%)
297 in the second wave in Europe. Based on data from 190 countries between January and
21
(a)
40
Percent reduction in R0
30
20
10
0
Masking Schools Stay−at−home Tracing Work Testing Events Public Info. Transit Gatherings Movement
(b)
40
Percent reduction in R0
30
20
10
0
Masking Schools Tracing Stay−at−home Work Testing Events Transit Public Info. Gatherings Movement
(c)
40
Percent reduction in R0
30
20
10
0
Masking Tracing Schools Stay−at−home Testing Work Transit Public Info. Events Gatherings Movement
Figure S12: Posterior violin plots of global NPI effects, quantified as the percent reduction
in R0 , across regression models (i)–(iii) from (a) to (c), respectively.
22
298 April 2020, Bo et al. [39] conclude that mask mandates were associated with a 15.1%
299 (7.9%–21.8%) decline in transmission. Karaivanov et al. [40] find a 22 percent weekly
300 reduction in new COVID-19 cases due to mask mandates in Canada in the summer of
301 2020. Studying the 2020 spring wave in New York City, Yang et al. [41] find that masking
302 was associated with a 7% transmission reduction overall and up to 20% reduction for
303 people over age 65. Estimating the causal effects of a number of interventions in the US,
304 Chernozhukov et al. [42] demonstrate that masking policies were highly effective, leading to
305 a reduction in the weekly growth rate of cases and deaths by more than 10 percentage points,
306 with their conclusions holding robustly across model specifications; on the other hand, the
307 effects of stay-at-home orders and business and school closures are much more uncertain.
308 Qualitatively, our results are consistent with a number of other studies demonstrating the
309 efficacy of mask mandates and the protective effects of face mask use [43–48].
310 Behind mask mandates, in our model (ii) school closure reduces the transmission rate
311 by 8.2% (1.5%–20.2%). As shown in Figure S12, this effect attenuates from 12.0% (3.0%–
312 24.8%) to 5.0% (0.8%–14.3%) as we move from model (i) to model (iii), controlling for
313 removals and infections incident in the prior week. By comparison, Brauner et al. [49]
314 and Banholzer et al. [50] find that school closures led to 38% (16%–54%) and 17% (-2%–
315 36%) transmission reductions in the first wave of 2020, respectively, and Sharma et al. [38]
316 estimate a 7% (4%–10%) transmission reduction due to school closures in the second wave.
317 Studying influenza outbreaks, Cauchemez et al. [51] found that school holidays led to a
318 20–29% transmission reduction among children with no detectable effect on transmission
319 among adults. Qualitatively, our results are consistent with a number of other studies
320 finding school closures to be one of the NPIs most effective in reducing transmission [52–
321 58].
322 As with school closures, we are unable to rule out small effects for the remaining NPIs.
323 Workplace closure reduced transmission by 5.3% (0.9%–12.5%). By comparison, Brauner
324 et al. [49], Sharma et al. [38], and Banholzer et al. [50] find that business closures led to
325 a 27% (-3%–49%), 35% (29%–41%), and 18% (-4%–40%) reduction in R0 , respectively.
23
326 The combination of social distancing measures yields a 18.9% (10.9%–28.1%) reduction
327 in R0 , which is on par with the individual effect of mask mandates. (As in Methods, we
328 define social distancing measures as the combination of stay-at-home orders, restrictions on
329 gatherings, restrictions on internal movement, public information campaigns, public transit
330 closures, and public event cancellations.) Looking at individual social distancing measures,
331 we estimate that stay-at-home orders reduced R0 by 6.1% (1.2%–14.1%), which is compa-
332 rable to other estimates in the literature: Bodenstein et al. [59] report a 6.5% transmission
333 reduction; Brauner et al. [49] report a 13% (-5%–31%) reduction; and Banholzer et al. [50]
334 report a 4% (-6%–17%) reduction. Restrictions on gatherings reduced R0 by a modest 1.9%
335 (0.3%–5.7%). However, a number of other studies estimate large effects of strict gathering
336 restrictions: Brauner et al. [49] report a 42% (17%–60%) transmission reduction; Sharma
337 et al. [38] report a 26% (18%–32%) reduction; Banholzer et al. [50] report a 37% (21%–
338 50%) reduction; and Bo et al. [39] report a 42.9% (41.6%–44.2%) reduction associated to
339 social distancing measures more broadly. (We note that these are association studies based
340 on observational data not controlling for potential confounders.) Nevertheless, even with
341 our relatively conservative estimates of the effects of social distancing measures, we find
342 that they are cost-effective interventions in combination.
343 Finally, our estimates of the effects of testing and tracing policies—which yield 3.9%
344 (0.6%–12.3%) and 6.4% (1.1%–16.0%) reductions in R0 , respectively—allow for the pos-
345 sibility of both small and large effects on transmission. Evidence on the effectiveness of
346 contact tracing, in particular, is mixed. Rainisch et al. [60] concluded that case investiga-
347 tion and contact tracing were effective in reducing transmission based on their estimates
348 of the number of COVID-19 cases and hospitalizations averted by these measures in the
349 US. Wang et al. [61] find that testing and case isolation were effective, but the effect of
350 contact tracing is marginal due to slow follow-up times in case investigation. They note
351 that contact tracing can be more effective if follow-up is accelerated. In modeling studies,
352 Hellewell et al. [62] and Davis et al. [63] find that contact tracing can be effective if carried
353 out well—with the latter reporting a potential 15% reduction in R0 —but that its effec-
24
(a) 6
5
Transmission rate (Rt)
4
3
2
1
0
(b) 1.0
0.8
NPI stringency
0.6
0.4
0.2
0.0
Apr Jul Oct Jan
Date (2020)
Figure S13: For each US state, a single posterior trajectory of the estimated transmission
rate (a) and NPI stringency (b) over time in 2020.
354 tiveness is dependent on a number of epidemiological and implementation-related factors,
355 including tracing coverage and speed. Nevertheless, even allowing for small effects, we find
356 that testing and tracing are highly cost-effective NPIs owing to their low cost relative to
357 other interventions.
358 S.2.4 Effect of lagged NPIs
359 We assume that the NPIs in place in week w directly affect transmission rates in week w (a
360 zero-week lag). This is in line with a number of other studies estimating the effects of NPIs
361 (e.g., Flaxman et al. [3], Sharma et al. [38], and Brauner et al. [49]), which do not account
362 for lagged effects of NPIs. This is not an issue for closure policies (e.g., school, business,
363 and transit closures), as they take effect immediately. For other policies, it is reasonable
364 to assume that behavioral responses occur quickly. For example, Alexander & Karger [64]
25
Covariance between transmission rate and lagged NPI stringency Covariance between transmission rate and lagged NPI stringency
0.00 0.00
−0.02
−0.02
Covariance Covariance
−0.04
−0.04
−0.06
−0.06
−5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5
Lag Lag
(a) Simple average (b) Weighted average
Figure S14: Estimated covariance between transmission rates and lagged NPI stringency
across states, where NPI stringency is defined as (a) a simple average of NPI values, as in
OxCGRT, and (b) a weighted average of NPI values based on the estimated NPI model
regression coefficients.
365 find that mobility and consumer spending declined consistently within two days of when
366 US counties enacted stay-at-home orders.
367 Figure S13 plots transmission rates and the NPI stringency index (defined as a simple
368 average of the 11 NPI values, as in OxCGRT [37]) across US states in each week w in
369 2020. To assess the effect of lagging the NPIs, we conducted a covariance analysis, which
370 is plotted in Figure S14. The x-axis shows the lag in weeks, with lag 0 corresponding to
371 no lag (as used for the analysis in the paper), positive values implying that NPIs are being
372 compared to future transmission rates, and negative values implying that NPIs are being
373 compared to past transmission rates. Figure S14a shows the mean covariance across states
374 (with 95% error bars) between transmission rates and the (lagged) NPI stringency index,
375 defined as a simple average of NPI values. Figure S14b instead uses an NPI stringency
376 index defined as a weighted average of NPI values based on our estimated NPI model
377 regression coefficients.
378 In both plots, the covariance tends to have the largest (negative) magnitude when the
379 lag is 0 weeks, although the difference with nearby means is not statistically significant.
380 This suggests that using a lag of 0 weeks (as we do in the paper) is appropriate. Also, given
26
381 the similar covariance using a lag of 1 week, our results would be unlikely to change much in
382 this case. However, using other lag values would attenuate the relationship between NPIs
383 and transmission rates, potentially affecting our conclusions about the cost-effectiveness of
384 interventions. For example, if workplace closures or social distancing orders were estimated
385 to have smaller effects on transmission based on a 2- or 3-week lag, their cost-effectiveness
386 would decline.
27
387 S.3 Epidemiological and economic parameter specification
Parameter Value Reference
δ −1 : Mean duration of latent pe- 5.5 Xin et al. [65]
riod Linton et al. [66]
Lauer et al. [67]
Gallo et al. [68]
Wu et al. [69]
γ −1 : Mean duration of infectious 5.0 Hakki et al. [70]
period
µ−1 : Mean time to death after re- 10.5 Linton et al. [66]
moval Byrne et al. [71]
τD : Mean time from exposure to 21.0 Ward & Johnsen [72]
death
ιs : State-specific IFR Varying Irons & Raftery [73]
R0max : Upper bound on R0 (t) 6.5 Liu et al. [74]
Mean time from case reporting to 8.053 Jin [75]
death
Standard deviation in time from 4.116 Jin [75]
case reporting to death
Table S1: Epidemiological parameters. All times are in days.
28
Parameter Value Reference
2019 GDP per capita Varying ($39000– U.S. Bureau of Economic Analysis [76]
by state 211000)
2019 per capita in- Varying ($39000– U.S. Bureau of Economic Analysis [77]
come by state 85000)
2019 US median per- $35980 U.S. Census Bureau [78]
sonal income
2019 population by Varying (0.575–39.5 U.S. Bureau of Economic Analysis [79]
state million)
2019 US GDP current 4.1% U.S. Bureau of Economic Analysis [80]
dollar growth rate
VSCD Low: $4.47 million Robinson et al. [22]
High: $10.63 million
Voluntary social Varying ($1491–3199) Aum et al. [81]
distancing cost per
COVID-19 infection
by state
Productivity cost of One week of state me- U.S. Bureau of Economic Analysis [77]
COVID-19 infection dian income and Skarp et al. [82]
Average medical cost $3045 Bartsch et al. [83] and DeMartino et al.
of COVID-19 infection [84]
Net present value of Low: 9% GDP per Psacharopoulos et al. [85]
GDP loss due to learn- 0.33 school-years Hanushek & Woessmann [86]
ing loss by state High: 69% GDP per
0.33 school-years
Learning loss accrued 0.35 school-years Betthäuser et al. [87] and Fahle et al.
during COVID-19 [88]
Direct GDP loss due 0.2% GDP per four Lempel et al. [89], Sadique et al. [90],
to school closure weeks and Viner et al. [91]
Employment rate de- Low: 2%; Med: 4%; Barrot et al. [12], Gupta et al. [92], and
crease due to work- High: 6% Crucini & O’Flaherty [93]
place closure
Employment rate de- 4% Bodenstein et al. [59], Gupta et al. [92],
crease due to social Crucini & O’Flaherty [93], Baek et al.
distancing mandates [94], and Coibion et al. [95]
Table S2: Economic parameters. All costs are in 2020 US dollars. Sensitivity analysis is
conducted with respect to the VSCD, cost of learning loss, and cost of workplace closure.
29
Parameter Value Reference
Daily personal mask $0.32 Skarp et al. [82] and Bartsch et al. [96]
expenditure
Cost of a PCR test $100 Skarp et al. [82], Lo et al. [97], and
Sharfstein [98]
Daily rate of testing Varying (7–40 tests The Atlantic [99]
capacity increase by per million pop.)
state
Cost of contact trac- $66.50 Skarp et al. [82], Fields et al. [100], and
ing per index case Spencer [101]
Weekly rate of con- 4.72 cases per 100k Rainisch et al. [60] and Lash et al. [102]
tact tracing capacity pop.
increase
Table S3: Masking, testing, and tracing parameters. All costs are in USD2020.
388 S.4 State-specific epidemiological model results
30
Alabama
(a) (b)
Observed Observed
6000
Predicted Predicted
150 Mean Mean
100 4000
Deaths Cases
50 2000
0 0
(c) 2.5 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
2.0 90% 0.8 90%
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S15: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
31
Alabama
(a) (b) (c)
15
5000 100
4000 80
10
3000 60
Cases 1 κC(t) Density
2000 5
40
1000 20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S16: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
32
Alaska
(a) (b)
Observed Observed
Predicted Predicted
20 Mean 1000 Mean
15
Deaths Cases 600
10
5
200
0 0
(c) 2.5 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
2.0 90% 0.8 90%
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S17: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
33
Alaska
(a) (b) (c)
50 100
800
40 80
600
30 60
Cases 1 κC(t) Density
400
20 40
200 10 20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S18: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
34
Arizona
(a) (b)
15000
Observed Observed
200 Predicted Predicted
Mean Mean
150
10000
Deaths Cases
100
5000
50
0 0
(c) (d) 1.0
4 Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
3
0.6
R0(t) 2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S19: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
35
Arizona
(a) (b) (c)
12
100
8000 10000
10
80
8
Cases 1 κC(t) Density
60
6000
6
40
4000
4
2000
2 20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S20: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
36
Arkansas
(a) (b)
200 4000
Observed Observed
Predicted Predicted
Mean Mean
150 3000
Deaths 100 Cases 2000
50 1000
0 0
(c) (d) 1.0
Median Median
2.0 50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
1.5
0.6
R0(t) 1.0
0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S21: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
37
Arkansas
(a) (b) (c)
1000 1500 2000 2500 3000
80
25
20 60
Cases 1 κC(t) Density
15
40
10
20
500
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S22: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
38
California
(a) (b)
800 Observed Observed
Predicted Predicted
60000
Mean Mean
600
Deaths 400 Cases 40000
200 20000
0 0
(c) (d) 1.0
Median Median
5 50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
4
0.6
R0(t)
3
2 0.4
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S23: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
39
California
(a) (b) (c)
50
50000
80
40
60
30
30000
Cases 1 κC(t) Density
40
20
20
10000
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S24: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
40
Colorado
(a) (b)
10000
500 Observed Observed
Predicted Predicted
Mean Mean
400
300 6000
Deaths Cases
200
100 2000
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
6 90% 0.8 90%
0.6
R0(t)
4
0.4
2
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S25: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
41
Colorado
(a) (b) (c)
8000
100
15
6000
80
1 κC(t) Density
10
Cases
60
4000
40
2000
5
20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S26: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
42
Connecticut
(a) (b)
Observed Observed
10000
250 Predicted Predicted
Mean Mean
200
150 6000
Deaths Cases
100
50 2000
0 0
(c) 7 (d) 1.0
Median Median
6 50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
5
4 0.6
R0(t)
3
0.4
2
0.2
1
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S27: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
43
Connecticut
(a) (b) (c)
8000 50
6
40
6000
30
Cases 1 κC(t) Density
4
4000
20
2000
2
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05 0.06
Month (2020) Month (2020) Posterior θC
Figure S28: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
44
Delaware
1400
(a) (b)
20 Observed Observed
Predicted Predicted
Mean Mean
15 1000
Deaths 10 Cases
600
5
200
0 0
(c) (d) 1.0
Median Median
50% 50%
3.0 dates dates
Case ascertainment rate
90% 0.8 90%
2.0 0.6
R0(t)
0.4
1.0
0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S29: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
45
Delaware
(a) (b) (c)
80
1000
30
800 25 60
20
600
Cases 1 κC(t) Density
40
15
400
10
20
200
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S30: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
46
Florida
(a) (b)
Observed Observed
20000
Predicted Predicted
300 Mean Mean
Deaths
200
Cases
100 5000 10000
0 0
(c) 5 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
4 90% 0.8 90%
3 0.6
R0(t)
2 0.4
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S31: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
47
Florida
(a) (b) (c)
80
50
15000
60
40
10000 30
Cases 1 κC(t) Density
40
20
5000 20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05
Month (2020) Month (2020) Posterior θC
Figure S32: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
48
Georgia
(a) (b)
10000 20000 30000 40000
Observed Observed
Predicted Predicted
500
Mean Mean
Deaths 300 Cases
100
0 0
(c) (d) 1.0
Median Median
50% 50%
4 dates dates
Case ascertainment rate
90% 0.8 90%
3
0.6
R0(t)
0.4
2
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S33: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
49
Georgia
(a) (b) (c)
12
80
10
25000
8
60
Cases
15000 1 κC(t) Density
6
40
4
20
5000
2
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S34: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
50
Hawaii
(a) (b) 500
Observed Observed
Predicted Predicted
15 Mean 400 Mean
300
Deaths Cases
10
200
5
100
0 0
(c) (d) 1.0
Median Median
2.0 dates
50%
dates
50%
Case ascertainment rate
90% 0.8 90%
1.5
0.6
R0(t)
1.0
0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S35: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
51
Hawaii
(a) (b) (c)
80
20
300
60
15
200
Cases 1 κC(t) Density
40
10
100 20
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S36: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
52
Idaho
(a) (b)
Observed Observed
40 Predicted Predicted
Mean 2500 Mean
30
Deaths Cases 1500
20
10
500
0 0
(c) 3.0 (d) 1.0
Median Median
50% 50%
2.5 dates dates
Case ascertainment rate
90% 0.8 90%
2.0
0.6
R0(t) 1.5
0.4
1.0
0.2
0.5
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S37: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
53
Idaho
(a) (b) (c)
100
30
2000
80
25
1500 60
20
Cases 1 κC(t) Density
1000
15
40
10
500 20
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S38: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
54
Illinois
20000
(a) 350 (b)
Observed Observed
Predicted Predicted
Mean Mean
250 15000
Deaths Cases 10000
150
5000
50
0 0
(c) 6 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
0.8
5 90% 90%
4
0.6
R0(t) 3
0.4
2
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S39: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
55
Illinois
(a) (b) (c)
15000
70
100
60
50 80
10000
40
1 κC(t) Density
60
Cases
30
40
5000 20
20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S40: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
56
Indiana
(a) (b)
150 Observed Observed
Predicted 10000 Predicted
Mean Mean
100
6000
Deaths Cases
50
2000
0 0
1.0
6
(c) Median
(d) Median
50% 50%
5 dates dates
Case ascertainment rate
90% 0.8 90%
4
0.6
R0(t)
3
0.4
2
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S41: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
57
Indiana
(a) (b) (c)
8000 50
80
40
6000
60
30
Cases 4000 1 κC(t) Density
40
20
2000 20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S42: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
58
Iowa
(a) (b)
Observed Observed
Predicted Predicted
500 Mean 6000 Mean
4000
Deaths 300 Cases
2000
100
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S43: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
59
Iowa
(a) (b) (c)
25
5000 80
20
4000
60
15
3000
Cases 1 κC(t) Density 40
2000
10
20
1000
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S44: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
60
Kansas
(a) (b)
Observed Observed
200 Predicted Predicted
2000 4000 6000 8000
Mean Mean
150
Deaths 100 Cases
50
0 0
(c) 2.5 (d) 1.0
Median Median
50% 50%
2.0 dates dates
Case ascertainment rate
90% 0.8 90%
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S45: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
61
Kansas
(a) (b) (c)
80
2.0
6000
60
1.5
4000
Cases 1 κC(t) Density
1.0 40
2000 0.5 20
0 0.0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S46: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
62
Kentucky
(a) (b)
Observed Observed
Predicted Predicted
80
Mean 6000 Mean
60
4000
Deaths Cases
40
2000
20
0 0
(c) 3.0 (d) 1.0
Median Median
50% 50%
2.5 dates dates
Case ascertainment rate
90% 0.8 90%
2.0
0.6
R0(t) 1.5
0.4
1.0
0.2
0.5
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S47: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
63
Kentucky
(a) (b) (c)
5000 25
60
4000 20
3000
1 κC(t) Density
40
Cases
15
2000 10
20
1000 5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S48: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
64
Louisiana
(a) (b)
Observed Observed
150 Predicted 8000 Predicted
Mean Mean
6000
100
Deaths Cases
4000
50
2000
0 0
(c) (d) 1.0
Median Median
6
50% 50%
dates dates
Case ascertainment rate
5 90% 0.8 90%
4 0.6
R0(t) 3
0.4
2
0.2
1
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S49: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
65
Louisiana
(a) (b) (c)
7000 70
8
60
5000 6 50
40
Cases 1 κC(t) Density
3000
4
30
20
2
1000 10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05 0.06
Month (2020) Month (2020) Posterior θC
Figure S50: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
66
Maine
(a)
Observed
(b) 1000 Observed
25
Predicted Predicted
Mean 800 Mean
20
600
Deaths
15
Cases
10 400
5 200
0 0
(c) 2.5 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
2.0 90% 0.8 90%
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S51: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
67
Maine
(a) (b) (c)
80
25
600
20 60
1 κC(t) Density
400
Cases
15
40
10
200 20
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S52: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
68
Maryland
1000 2000 3000 4000 5000
(a) (b)
Observed Observed
200 Predicted Predicted
Mean Mean
150
Deaths Cases
100
50
0 0
(c) (d) 1.0
6 Median Median
50% 50%
dates dates
Case ascertainment rate
5 90% 0.8 90%
4
0.6
R0(t) 3
0.4
2
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S53: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
69
Maryland
(a) (b) (c)
60
80
3000 50
40 60
2000
Cases 1 κC(t) 30 Density
40
20
1000
20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S54: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
70
Massachusetts
10000
(a) 400 Observed
(b)
Observed
Predicted Predicted
Mean Mean
300
6000
Deaths 200 Cases
100
2000
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
6
90% 0.8 90%
4 0.6
R0(t)
0.4
2
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S55: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
71
Massachusetts
(a) (b) (c)
30
80
6000 25
60
20
1 κC(t) Density
4000
Cases 15 40
10
2000 20
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S56: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
72
Michigan
(a) (b)
Observed Observed
20000
Predicted Predicted
300 Mean Mean
Deaths
200
Cases
100 5000 10000
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
0.8
6 90% 90%
0.6
R0(t)
4
0.4
2
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S57: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
73
Michigan
(a) (b) (c)
50
15000
50
40
40
10000 30
Cases 1 κC(t) Density
30
20
20
5000
10 10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05 0.06
Month (2020) Month (2020) Posterior θC
Figure S58: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
74
Minnesota
(a) 140 (b)
Observed Observed
10000
Predicted Predicted
Mean Mean
80 100
Deaths 60 Cases 6000
40
20 2000
0 0
(c) (d) 1.0
Median Median
50% 50%
4 dates dates
Case ascertainment rate
90% 0.8 90%
3
0.6
R0(t)
0.4
2
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S59: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
75
Minnesota
(a) (b) (c)
70
80
8000 60
50 60
6000
40
Cases 1 κC(t) Density 40
4000 30
20
2000
20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S60: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
76
Mississippi
4000
(a) (b)
Observed Observed
100 Predicted Predicted
Mean Mean
80 3000
Deaths Cases 2000
60
40
1000
20
0 0
(c) (d) 1.0
Median Median
0.0 0.5 1.0 1.5 2.0 2.5 3.0
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
0.6
R0(t)
0.4
0.2
0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S61: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
77
Mississippi
(a) (b) (c)
3000 100
2500 15
80
2000
60
Density
10
Cases 1500 1 κC(t)
40
1000
5
20
500
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S62: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
78
Missouri
100 150 200 250 300
(a) (b)
Observed Observed
Predicted Predicted
10000
Mean Mean
Deaths Cases 6000
50 2000
0 0
1.0
4
(c) Median
(d) Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S63: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
79
Missouri
(a) (b) (c)
12 100
8000
10
80
6000
8
60
Cases 1 κC(t) 6 Density
4000 40
4
2000 20
2
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S64: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
80
Montana
(a) 50 (b)
Observed Observed
Predicted 2000 Predicted
40 Mean Mean
1500
30
Deaths Cases
1000
20
10 500
0 0
(c) 2.5 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
2.0 90% 0.8 90%
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S65: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
81
Montana
(a) (b) (c)
100
20
1500
80
15
1000
60
Cases 1 κC(t) 10 Density
40
500
5
20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03
Month (2020) Month (2020) Posterior θC
Figure S66: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
82
Nebraska
(a) (b)
80 Observed Observed
6000
Predicted Predicted
Mean Mean
60
4000
Deaths 40 Cases
20 2000
0 0
4
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S67: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
83
Nebraska
(a) (b) (c)
5000 80
15
4000
60
3000
10
Cases 1 κC(t) Density 40
2000
5
20
1000
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S68: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
84
Nevada
(a) 80 (b)
4000
Observed Observed
Predicted Predicted
Mean Mean
60
3000
Deaths Cases
40
2000
1000
20
0 0
(c) (d) 1.0
Median Median
0.0 0.5 1.0 1.5 2.0 2.5 3.0
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
0.6
R0(t)
0.4
0.2
0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S69: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
85
Nevada
(a) (b) (c)
3000 100
60
2000 2500
80
1 κC(t) Density
40 60
Cases 1500
40
1000
20
20
500
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03
Month (2020) Month (2020) Posterior θC
Figure S70: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
86
New Hampshire
30
1500
(a) (b)
Observed Observed
25 Predicted Predicted
Mean Mean
1000
20
Deaths Cases
15
10 500
5
0 0
(c) (d) 1.0
3.0 Median Median
50% 50%
dates dates
Case ascertainment rate
2.5 90% 0.8 90%
2.0 0.6
R0(t) 1.5
0.4
1.0
0.2
0.5
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S71: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
87
New Hampshire
(a) (b) (c)
70
1000 40 60
800 50
30
40
Cases 1 κC(t) Density
600
20 30
400
20
10
200
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S72: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
88
New Jersey
(a) (b)
Observed Observed
Predicted 8000 Predicted
Mean Mean
500
6000
Deaths 300 Cases
4000
2000
100
0 0
(c) 8 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
6
90% 0.8 90%
0.6
R0(t)
4
0.4
2
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S73: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
89
New Jersey
(a) (b) (c)
7000
80
80
5000
60
60
Cases 1 κC(t) Density
3000 40 40
20 20
1000
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05
Month (2020) Month (2020) Posterior θC
Figure S74: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
90
New Mexico
1000 2000 3000 4000 5000
(a) (b)
Observed Observed
80 Predicted Predicted
Mean Mean
60
Deaths 40 Cases
20
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S75: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
91
New Mexico
(a) (b) (c)
80
50
3000 40 60
30
2000
Cases 1 κC(t) Density
40
20
1000 20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S76: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
92
New York
20000
(a) (b)
Observed Observed
1500 Predicted Predicted
Mean Mean
15000
1000
Deaths Cases 10000
500
5000
0 0
(c) (d) 1.0
8 Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
6
0.6
R0(t) 4
0.4
2
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S77: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
93
New York
(a) (b) (c)
15000 80
100
60 80
10000
Cases 1 κC(t) Density
60
40
40
5000
20
20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S78: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
94
North Carolina
25000
(a) (b)
Observed Observed
200 Predicted Predicted
Mean Mean
150
15000
Deaths 100 Cases
5000
50
0 0
(c) (d) 1.0
2.5 Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
2.0
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S79: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
95
North Carolina
(a) (b) (c)
20000
20
15000
60
15
Cases 10000 1 κC(t) Density
40
10
5000 5 20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05
Month (2020) Month (2020) Posterior θC
Figure S80: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
96
North Dakota
(a) 80 (b)
Observed Observed
Predicted Predicted
60 Mean 2500 Mean
Deaths Cases 1500
40
20
500
0 0
(c) (d) 1.0
Median Median
2.0
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
1.5
0.6
R0(t) 1.0
0.4
0.5
0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S81: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
97
North Dakota
(a) (b) (c)
35
1.0
2000 30
0.8
25
1500
20 0.6
Cases 1 κC(t) Density
1000 15
0.4
10
500 0.2
5
0 0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan 0.0 0.2 0.4 0.6 0.8 1.0
Month (2020) Month (2020) Posterior θC
Figure S82: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
98
Ohio
(a)
Observed
(b) 35000 Observed
250 Predicted Predicted
Mean Mean
200 25000
Deaths Cases
150
15000
100
0 5000
50
0
(c) (d) 1.0
5 Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
4
3 0.6
R0(t)
2 0.4
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S83: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
99
Ohio
(a) (b) (c)
40 80
25000
20000 30 60
15000
Cases 1 κC(t) Density
20 40
10000
10 20
5000
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05
Month (2020) Month (2020) Posterior θC
Figure S84: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
100
Oklahoma
(a) (b)
8000
70 Observed Observed
Predicted Predicted
60 Mean Mean
50 6000
Deaths
40
Cases 4000
30
20
2000
10
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0
0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
0.6
R0(t)
0.4
0.2
0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S85: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
101
Oklahoma
(a) (b) (c)
80
6000 20
5000 60
15
4000
Cases 1 κC(t) Density
40
3000 10
2000
5 20
1000
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S86: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
102
Oregon
(a) 70 (b)
Observed Observed
60 Predicted Predicted
Mean 2500 Mean
50
40
Deaths 30
Cases 1500
20
10 500
0 0
(c) 3.0 (d) 1.0
Median Median
50% 50%
2.5 dates dates
Case ascertainment rate
90% 0.8 90%
2.0
0.6
R0(t) 1.5
0.4
1.0
0.2
0.5
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S87: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
103
Oregon
(a) (b) (c)
70
70
2000
60
60
50 50
1500
40 40
Cases 1 κC(t) Density
1000
30 30
20 20
500
10 10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S88: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
104
Pennsylvania
(a) (b)
400
Observed Observed
15000
Predicted Predicted
Mean Mean
300
10000
Deaths 200 Cases
5000
100
0 0
1.0
7
(c) Median
(d) Median
6 50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
5
4 0.6
R0(t) 3
0.4
2
0.2
1
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S89: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
105
Pennsylvania
(a) (b) (c)
50
12000 80
40
60
2000 4000 6000 8000
30
Cases 1 κC(t) Density 40
20
20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S90: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
106
Rhode Island
1000 2000 3000 4000 5000
(a) 40 (b)
Observed Observed
Predicted Predicted
Mean Mean
30
Deaths Cases
20
10
0 0
(c) (d) 1.0
Median Median
50% 50%
4 dates dates
Case ascertainment rate
90% 0.8 90%
0.6
3
R0(t)
2 0.4
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S91: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
107
Rhode Island
(a) (b) (c)
30
10
3000 25
8
20
2000
Cases 1 κC(t) Density
6
15
4
10
1000
2 5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.04 0.08 0.12
Month (2020) Month (2020) Posterior θC
Figure S92: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
108
South Carolina
(a) (b)
Observed Observed
100
Predicted Predicted
2000 4000 6000 8000
Mean Mean
80
Deaths
60
Cases
40
20
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S93: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
109
South Carolina
(a) (b) (c)
35 80
6000
30
60
25
4000
20
Cases 1 κC(t) Density 40
15
2000 10
20
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S94: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
110
South Dakota
(a) 70
(b)
Observed Observed
60
Predicted 2500 Predicted
Mean Mean
50
Deaths
1500
40
Cases
30
20
10 500
0 0
(c) (d) 1.0
Median Median
0.0 0.5 1.0 1.5 2.0 2.5 3.0
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
0.6
R0(t)
0.4
0.2
0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S95: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
111
South Dakota
(a) (b) (c)
80
2000
20
60
1500
15
Cases 1 κC(t) Density
40
1000 10
20
500 5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S96: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
112
Tennessee
15000
(a) (b)
Observed Observed
Predicted Predicted
200 Mean Mean
150 10000
Deaths Cases
100
5000
50
0 0
(c) (d) 1.0
Median Median
50% 50%
2.0 dates dates
Case ascertainment rate
90% 0.8 90%
1.5 0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S97: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
113
Tennessee
(a) (b) (c)
100
8000 10000
15
80
10
Cases 1 κC(t) Density
60
6000
4000
40
5
2000
20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03
Month (2020) Month (2020) Posterior θC
Figure S98: Overdispersion and zero-inflation in state-level clinical case data. (a) Reported
cases. (b) Posterior median and 90% credible interval for the time-varying negative binomial
overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation parameter
θC .
114
Texas
(a) (b)
Observed Observed
800 Predicted Predicted
Mean Mean
60000
600
Deaths Cases
400
200 20000
0 0
(c) (d) 1.0
Median Median
5 50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
4
0.6
R0(t)
3
0.4
2
1 0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S99: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
115
Texas
(a) (b) (c)
70000
25
80
20
50000 60
15
Cases 1 κC(t) Density
30000
40
10
20
10000
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04 0.05
Month (2020) Month (2020) Posterior θC
Figure S100: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
116
Utah
8000
(a) 40 (b)
Observed Observed
Predicted Predicted
Mean Mean
6000
30
Deaths Cases 4000
20
10 2000
0 0
(c) (d) 1.0
Median Median
2.5 50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
2.0
0.6
1.5
R0(t)
0.4
1.0
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S101: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
117
Utah
(a) (b) (c)
6000
5000
20 80
4000 15 60
Cases 3000 1 κC(t) Density
10 40
2000
5 20
1000
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S102: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
118
Vermont
100 150 200 250 300
(a) 8 (b)
Observed Observed
Predicted Predicted
Mean Mean
6
Deaths Cases
4
2
50
0 0
(c) (d) 1.0
Median Median
2.5
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
2.0
0.6
1.5
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S103: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
119
Vermont
(a) (b) (c)
80
200
15
60
150
10
Cases 1 κC(t) Density 40
100
5
50 20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S104: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
120
Virginia
(a)
Observed
(b) 7000 Observed
Predicted Predicted
100
Mean Mean
5000
80
Deaths 60 Cases
3000
40
0 1000
20
0
(c) 4 (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
3
0.6
R0(t) 2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S105: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
121
Virginia
(a) (b) (c)
5000 80
40
4000
60
30
3000
Cases 1 κC(t) Density 40
20
2000
20
1000
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S106: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
122
Washington
8000
(a) (b)
Observed Observed
200 Predicted Predicted
Mean Mean
6000
150
Deaths Cases 4000
100
50 2000
0 0
(c) (d) 1.0
4 Median Median
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
3
0.6
R0(t) 2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S107: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
123
Washington
(a) (b) (c)
100
6000
25
5000 80
20
4000
60
Cases 1 κC(t) Density
15
3000
40
2000
10
20
1000
5
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.000 0.010 0.020 0.030
Month (2020) Month (2020) Posterior θC
Figure S108: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
124
Washington DC
(a) 25 (b)
Observed Observed
Predicted Predicted
20 Mean Mean
500
15
Deaths Cases 300
10
5
100
0 0
(c) (d) 1.0
Median Median
0.0 0.5 1.0 1.5 2.0 2.5 3.0
50% 50%
dates dates
Case ascertainment rate
90% 0.8 90%
0.6
R0(t)
0.4
0.2
0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S109: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
125
Washington DC
(a) (b) (c)
500 80
20
400
60
15
300
Cases 1 κC(t) Density
40
10
200
5 20
100
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S110: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
126
West Virginia
(a) (b)
2000
Observed Observed
50 Predicted Predicted
Mean Mean
1500
40
Deaths Cases
30
1000
20
500
10
0 0
(c) (d) 1.0
Median Median
50% 50%
2.0 dates dates
Case ascertainment rate
90% 0.8 90%
1.5
0.6
R0(t)
1.0 0.4
0.5 0.2
0.0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S111: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
127
West Virginia
(a) (b) (c)
1500 60
60
50
1000 40
Cases 1 κC(t) Density
40
30
500 20
20
10
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S112: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
128
Wisconsin
(a) (b)
Observed Observed
150 Predicted 20000 Predicted
Mean Mean
100
Deaths Cases
5000 10000
50
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S113: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
129
Wisconsin
(a) (b) (c)
15000
15 60
10000
1 κC(t) Density
10
Cases
40
5000 5 20
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S114: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
130
Wyoming
(a) (b)
Observed Observed
1500
40 Predicted Predicted
Mean Mean
30
1000
Deaths 20 Cases
500
10
0 0
(c) (d) 1.0
Median Median
50% 50%
dates dates
Case ascertainment rate
3 90% 0.8 90%
0.6
R0(t)
2
0.4
1
0.2
0 0.0
Apr Jul Oct Jan Apr Jul Oct Jan
Month (2020) Month (2020)
Figure S115: SEIRD model fit to COVID-19 data. Top panels: observed (a) deaths
d(t) and (b) cases c(t) are plotted in black. Median and 90% credible intervals of the
posterior predictive distributions of d(t) and c(t) are in red. Posterior median and 90%
credible intervals of the underlying mean parameters mD (t) and mC (t) are in blue. Bottom
panels: posterior median, 50%, and 90% credible intervals for (c) the basic reproduction
number R0 (t) and (d) the case ascertainment rate CAR(t).
131
Wyoming
(a) (b) (c)
80
1200
10
1000 8 60
800
6
Cases 1 κC(t) Density
40
600
4
400
20
2
200
0 0 0
Apr Jul Oct Jan Apr Jul Oct Jan 0.00 0.01 0.02 0.03 0.04
Month (2020) Month (2020) Posterior θC
Figure S116: Overdispersion and zero-inflation in state-level clinical case data. (a) Re-
ported cases. (b) Posterior median and 90% credible interval for the time-varying negative
binomial overdispersion parameter κC (t)−1 . (c) Posterior histogram for the zero-inflation
parameter θC .
132
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