key: cord-344994-68j6ekiy authors: Lyu, M.; Hall, R. title: Dynamic Modeling of Reported Covid-19 Cases and Deaths with Continuously Varying Case Fatality and Transmission Rate Functions date: 2020-09-27 journal: nan DOI: 10.1101/2020.09.25.20201905 sha: doc_id: 344994 cord_uid: 68j6ekiy In this paper, we propose an enhanced SEIRD (Susceptible-Exposed-Infectious-Recovered-Death) model with time varying case fatality and transmission rates for confirmed cases and deaths from COVID-19. We show that when case fatalities and transmission rates are represented by simple Sigmoid functions, historical cases and fatalities can be fit with a relative-root-mean-squared-error accuracy on the order of 2% for most American states over the period from initial cases to July 20 (2020). We find that the model is most accurate for states that so far had not shown signs of multiple waves of the disease (such as New York), and least accurate for states where transmission rates increased after initially declining (such as Hawaii). For such states, we propose an alternate multi-phase model. Both the base model and multi-phase model provide a way to explain historical reported cases and deaths with a small set of parameters, which in the future can enable analyses of uncertainty and variations in disease progression across regions. COVID-19 has challenged the world to react to a new contagious virus in the absence of 2 effective medical treatment and vaccines. Over the course of nine months from the 3 outbreak in December 2019, when the first cases were confirmed in Wuhan, China, until 4 April 24th 2020, 213 countries and territories reported nearly 28 million confirmed cases 5 and a death toll exceeding 900,000 persons [1] . In the meantime, waiting for effective 6 clinical care and vaccination, countries have reacted to the pandemic by controlling 7 travel, implementing large-scale quarantine and restricting gatherings and contact 8 among people, as well as requiring hygiene measures and screening for possible cases. In this paper, we seek to improve understanding of the dynamics of how COVID-19 21 is spread, utilizing a variation of the Susceptible-Exposed-Infectious-Recovered-Death 22 (SEIRD) model. Our key innovation is representation of the transmission rate and case 23 fatality rate as continuously varying functions, which are optimally fit to historical data 24 on confirmed cases and deaths. We surmise that neither parameter is static, as they are 25 influenced by the enactment and adherence to public health measures and medical care, 26 neither of which is constant over time. We have applied our model to all 50 American 27 states to derive insights into how the disease has spread in different localities, which are 28 influenced by population health, disease exposure, localized public health interventions 29 and messaging, in addition to other place specific factors. 30 Prior Research 31 Prior research on COVID-19 has estimated disease-specific parameters, such as the 32 basic reproduction number and latent period [3] [4] [5] , demonstrating why the disease is interventions, such as closures of schools and restaurants [7] . Another use of disease transmission models has been to predict and plan for future 40 demands on the health care system, such as demands for hospital beds (ICU in 41 particular) and needs for health care resources, such as ventilators. Toward that 42 goal, [8] provides a statistical model of death data to predict future fatalities, assuming 43 that social distancing measures are maintained. From the projected fatality data, they 44 estimated hospital utilization with an individual-level microsimulation model based on Due to the limits of testing methods, the long incubation period, and cases with 50 mild or no symptoms and delayed reporting, there is potentially a huge (and unknown) 51 number of unreported cases, the extent to which could affect the future evolution of the 52 epidemic. Some researchers, therefore, have used the SIR (symptomatic-infectious 53 -recovered) model and SEIRD to estimate the number of undetected cases [10] and [11] . 54 Some approaches also incorporate transportation information (such as human migration 55 data and community mobility data) to analyze the impact of travel on disease 56 transmission and thus the effect of travel restriction [12] [13] [14] . However, most studies 57 using SEIRD or SIR assume the transmission rate and death rate to be constant over 58 time. With improvement in clinical treatment and changing intervention policies, the 59 transmission rate and fatality rate will be time variable. Therefore, SEIRD models with 60 constant parameters cannot accurately depict the spread of disease. [15] and [16] In our research we explore use of a concise formulation through which continuously 69 time varying transmission and case fatality rates are modeled with a small number of 70 parameters, which are fit to historical data. Like [7] , we utilize a type of logistic 71 function (i.e., a Sigmoid function), but not simply to model reported deaths over time, 72 but to instead model transmission rate and case fatality rate within the SEIRD model. 73 Our objective is to improve the classical SEIRD model through an approach that adapts 74 to the dynamic pattern of transmission under different epidemic scenarios. Thus, we 75 provide insights into transmissibility of the disease while modeling historical data on 76 confirmed cases and confirmed deaths. The Proposed Time Varying Model 78 We draw from the SEIRD compartmental model, which divides the population into five 79 groups: susceptible(S), exposed(E), infected(I), recovered(R) and dead(D). SEIRD infectious people and turn into the state of exposed. Exposed people are in a latent 83 state and then progress to the infectious state with a rate inversely related to the 84 incubation period (thus, exposed is defined as a state in which people are not yet 85 infectious). Infected people eventually progress into either the dead state, if they 86 succumb to the disease, or the recovered state, with different rates. Those who have 87 recovered are assumed to be no longer susceptible to contracting the disease. 88 We introduce death rate α (t) as a time varying function, representing the 89 proportion of infectious individuals who eventually die from the disease, by date. Those 90 who eventually die transfer from the infected to the died state at a rate of ρ, 91 representing the inverse of the time from becoming infectious until time of death. In our 92 model, ρ is assumed to be constant over time. Those who eventually recover do so at 93 the γ, representing the inverse of the time from becoming infectious until recovery. We 94 will also later derive the effective reproduction number Rep (t), representing the average 95 number of persons who are exposed to the disease by each infectious person, as a 96 function of time. Taking these factors into account, the system of equations of the proposed SEIRD 98 model is given by:: where: preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this this version posted September 27, 2020. . https://doi.org/10.1101/2020.09.25.20201905 doi: medRxiv preprint R (t) = number of people who have recovered at time t 105 N = total number of people 106 β (t) = transmission rate at time t 107 σ = transformation rate from exposed to infectious, which is the reciprocal of the 108 incubation period While, in theory, these functions may change erratically as a consequence of discrete 115 events, such as new public health measures, we hypothesize that such discrete events do 116 not suddenly alter either function. Therefore, we seek to understand whether a simple 117 continuous model, with a minimal set of parameters, might accurately represent 118 historical data. For illustration, at the enactment of a new intervention policy, the 119 public may not react immediately, and neither do the transmission parameters. The 120 public will get used to the policy after a period of adaptation, and eventually the 121 effective reproduction number will stabilize. In addition, the public responds to both 122 government policies and communication about the disease. Communication comes from 123 many, sometimes conflicting, sources. How the public at large absorbs and responds to 124 such often confusing messages may be gradual. A natural function to describe this pattern of change is the Sigmoid function. Thus, we define the function for transmission rate and death rate as: September 25, 2020 4/15 All rights reserved. No reuse allowed without permission. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this this version posted September 27, 2020. . As mentioned in the CDC reports [17] [18] [19] , the median incubation period is 4 days, preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this this version posted September 27, 2020. . (as of March 1 st few states had reported cases). Prior research suggests that the initial 171 effective reproduction number is around 3 [5] , equivalent to a transmission rate of 0.75, 172 which we use for initialization. Because transmission rates vary significantly among The number of reported deaths is smaller than the number of reported cases in all 179 locations. Thus, treating errors in death estimation and case estimation the same will 180 lead to underfitting of the death data, in preference to minimizing the errors in case 181 data. Therefore, considering the accuracy of the reported death data and the fitting 182 accuracy, we optimized a weighted sum of squared death and case data, multiplying w 183 by deaths during the fitting process. The adjusted objective function is: The parameters are estimated by solving the nonlinear constrained least-squares Reporting has also shown a consistent day-of-week variation across many locations, 201 with weekend data differing from weekday data. This variation is more likely the 202 consequence of different patterns of healthcare staffing, and differences in how patients 203 present for testing by day of week, rather than differences in disease transmission by day 204 of week. To smooth out these effects, we model the moving 7-day average data instead 205 of the daily reported data where y N is the case/death number on the N t h day. Wyoming. Figure 2 shows that the proposed SEIRD model with time-dependent preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this this version posted September 27, 2020. . and Hawaii, the CDC data follows a pattern of two phases, which is not as well 229 captured by our model. Especially for Hawaii, the curve flattened for a period and then 230 rose. As discussed later, our model characterizes the transmission dynamic for a period 231 with one phase, i.e. the curve should become flat at most once. All rights reserved. No reuse allowed without permission. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Effective reproduction number at any time t, which we define as Rep (t), is the average 234 number of people in a population who are infected per infectious case, where everyone is 235 susceptible to the disease. Rep (t) measures the transmission potential of infectious 236 diseases [23] . When Rep (t) > 1, the rate of new cases will increase over time, until the 237 population loses susceptibility to the disease. When Rep (t) < 1, the rate of new cases 238 will decline over time. Rep (t) can be estimated with the next generation matrix method [24] . We define X 240 as the vector of infected class (i.e. E, I) and Y as the vector of uninfected class (i.e. S, 241 R, D). Then the effective reproduction number is the spectral radius of M, expressed as: At the beginning of the epidemic, Rep (t) reflects the natural transmissibility of preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this this version posted September 27, 2020. We divide the Hawaii timeline into two periods, the first from March 6 th until May 287 7 th , and the second from May 7 th to July 28 th . We fit the first stage with the 288 initialization of only one exposed people at the start. To initialize the second phase, we 289 use the predicted number of exposed people, infectious people and recovered people 290 from the first phase, combined with the reported deaths as of May 7 th . With this 291 modification, the RRMSE for cases declines below 2.5% and the RRMSE for deaths 292 declines below 2.7%. The fitting results in Figure 9 show that our two-phase model 293 captures the transmission pattern more precisely than the single-phase model. The history of effective reproduction number and death rate are shown in Figure 10 . 295 The first phase showed a decline in the reproduction number after the initial 296 announcement of the stay-at-home order. However, with the reopening, the 297 reproduction number increased, explaining increases in case rates. Death rates, by 298 contrast, exhibit a peculiar behavior, increasing over time in each phase, with a 299 discontinuity when transitioning from the first phase to the second. Beyond exhibiting 300 two phases, Hawaii has a small number of deaths, with no deaths occurring in the 301 transition period between phases. We surmise that the function, while representing the 302 data well, is peculiar because of the unusual pattern in deaths within Hawaii. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. We have developed an extension of the SEIRD model that represents changing 305 transmission and death rates over time as a continuous Sigmoid function, under the 306 hypothesis that these rates change gradually, rather than immediately upon 307 implementation of public health policies or treatments. The model fits historical data 308 for the United States well for most states. Those with poorer fits exhibit patterns of 309 multiple waves of the disease. Using Hawaii as an example, a multi-phase extension of 310 the model provides a more accurate fit, where the transition from one phase to the next 311 is defined by a change in public health policy. An advantage of the model is that it is defined by a small set of parameters. Thus, it 313 provides an efficient method for quantifying differences among regions in the spread and 314 outcomes of the disease. In the future, we intend to examine the predictive value for the 315 model, taking into consideration ranges of uncertainty in model estimates. By 316 examining historical trends, we can understand how variations in simple parameters can 317 lead to fewer or more cases and deaths. In the future, we will develop multi-region 318 extensions of the model, which permit representation of spread of disease from one 319 region to another, or perhaps within sub-regional groups. Our research is premised on the method of modeling case and death data as they are 321 reported. We recognize that the true number of cases may differ from reported values, 322 as might the number of deaths, both of which are unknown. The variations from state 323 to state reflect in part the actual spread and outcomes of disease, as well as the extent 324 to which cases are detected and reported, as well as how deaths have been classified. 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