key: cord-0794428-xh3vx2bp
authors: Zhao, Shi
title: A simple approach to estimate the instantaneous case fatality ratio: Using the publicly available COVID-19 surveillance data in Canada as an example
date: 2020-08-15
journal: Infect Dis Model
DOI: 10.1016/j.idm.2020.08.002
sha: 5b01774c3d7d73d96e6f078e8859c4bba09b02a7
doc_id: 794428
cord_uid: xh3vx2bp

The case fatality ratio (CFR) is one of the key measurements to evaluate the clinical severity of infectious diseases. The CFR may vary due to change in factors that affect the mortality risk. In this study, we developed a simple likelihood-based framework to estimate the instantaneous CFR of infectious diseases. We used the publicly available COVID-19 surveillance data in Canada as a demonstrative example. The mean fatality ratio of reported COVID-19 cases (rCFR) in Canada was estimated at 6.9% (95%CI: 4.5–10.6). We emphasize the extensive implementation of the constructed instantaneous CFR that is to identify the key determinants affecting the mortality risk.

The mortality risk is one of the key measurements to evaluate the clinical severity of diseases. 26 The case fatality ratio (CFR) quantifies the mortality risk of infectious disease when being infected, 27 which is commonly calculated as a constant. However, changes in some external factors may vary 28 the scale of CFR, e.g., pathogenic evolution, health status, changes in treatment strategies or 29 medication (1), the supply of healthcare and critical care resources (2, 3), and exposure to the 30 environmental factors (4). In such situations, variation in CFR likely occurs. To explore changing 31 dynamics on CFR, the instantaneous, or time-varying, CFR is of importance in understanding the 32 patterns of the mortality risk (5-7). 33

Recently, the coronavirus disease 2019 (COVID-19), caused by the severe acute respiratory 34 syndrome coronavirus 2 (SARS-CoV-2), emerged in Wuhan, China at the end of 2019 (7-12). The 35

COVID-19 spread to over 200 foreign countries as of July 2020 (12, 13). The World Health 36

Organization (WHO) declared the outbreak to be a public health emergency of international concern 37 on January 30, 2020 (14 In this study, we developed a simple likelihood-based framework to estimate the 46 instantaneous case fatality ratio of infectious diseases. We used the publicly available COVID-19 47 surveillance data in Canada as an example for demonstration. 48

For the time interval (denoted by g) between being reported as case and the death (if the 50 death occurs), we use f(•) to denote the probability distribution function (PDF) of this time interval. 51

For convenience, we denote the time interval between onset and death by s following the PDF of h(s), 52 and the time interval between onset and death by q following the PDF of δ(q). Following the 53 previous study (6), the g is modelled as the difference of s minus d, i.e., g = s -q, and we remark that 54 g is not necessarily positive. Then, the PDF of g, i.e., f(g), is formulated as in Eqn (1). 55

Thus, if one case is reported at time τ who dies eventually, the value of f(g) is considered as the 56 relative likelihood of death at time (τ + g). 57

Since each disease-related death is diagnosed as a case at the first place, each individual case 58 is considered as a 'source' (or 'pool') of the death, i.e., subjects at mortality risk. We consider all 59

reported cases as the 'pool' to generate deaths, and we model this candidate pool as a time-varying 60 function denoted by Φ(t) at time t. Then, at time t, the i-th case, who is reported at time τ i , contributes 61

For the contribution from all reported cases, the Φ(t) is summated as in Eqn (2). 62 J o u r n a l P r e -p r o o f

Hence, the reported case fatality ratio (rCFR), i.e., the fatality ratio of reported cases, at time t can be 63 calculated by rCFR t = d t / Φ t . Here, the d t is the observed number of deaths at time t, and Φ t is the 64 discretised Φ(t) at time t. 65

To construct the likelihood profile, we model d t as a binomial process with sizes at Φ t 66 (rounding to the closest integer) and successful probabilities at rCFR t to be estimated. As such, by 67

fitting to the daily number of deaths time series, the rCFR t can be estimated by using the maximum 68 likelihood estimation approach. The 95% confidence intervals (95%CI) of rCFR t are calculated by 69 using the profile likelihood estimation framework with a cutoff threshold determined by a Chi-square 70

quantile (21), as well as previously adopted in (3, 10, 22-25). 71

For demonstration, we used the publicly available COVID-19 surveillance data in Canada as 73 an example to construct the instantaneous rCFR t series. The daily reported number of COVID-19 74 cases and deaths time series were collected from the COVID-19 public surveillance platform 75 released by the WHO, accessed via https://covid19.who.int/region/amro/country/ca. Fig 1A and B  76 show the epidemic curve of COVID-19 cases and deaths in Canada from February to July 2020. 77

To set up the initial conditions of the model framework in Eqn (1), we set h(s) as a Gamma 78 distribution with mean (±SD) at 20 days (±10) referring to (17, 18), and δ(q) as another Gamma 79 distribution with mean (±SD) at 7 days (±4) referring to (6). We remark that slight changes and 80 similar alternative settings in the initial conditions will not affect our main results. 81

Since the first COVID-19 death reported on March 11, 2020, we estimated the instantaneous 83 rCFR t ranging from 0% to 23.8%, see Fig 1C. The mean rCFR of COVID-19 was estimated at 6.9% 84 (95%CI: 4.5 -10.6). Our rCFR estimate is largely consistent with previous estimates, e.g., 5.3% in 85

Wuhan, China (20) Linked to the change in the temporal trends of rCFR, we suspect that the decreasing trends in 102 rCFR after May 3, 2020, might be due to the increase in the ascertainment rate in Canada during the 103 same period. Speculatively, the increasing trends in rCFR before May 3 might be partially associated 104

with insufficient intensive care preparedness during the early phase of the outbreak. However, 105

further studies are warranted to explore evidence for these two speculative hypotheses. This study 106

proposed an analytical approach to construct the instantaneous rCFR that can be adopted to further 107 examine the associations with its potential determinants, e.g., pathogenic evolution, change in the 108 cases ascertainment rate (27-29), the supply of critical care resources (2), and exposure to the 109 environmental factors (4). 110

The analytical approach proposed in this study has limitations. First, we presume the real 111 number of disease-induced deaths and the time of each death are correctly reported. This setting is 112 practically reasonable since mortality is considered as a serious clinical outcome, which is under 113 more rigorous surveillance, and thus is unlikely mis-ascertained. Second, as a data-driven analysis, 114

our estimates are relying on both statistical framework and consistency in the reported COVID-19 115 cases data. Note that the current framework requires fixed distributions of both the reporting delay, 116 δ(q), and lag between onset and death, h(s), and temporal variation in either of them may undermine 117 the statistical unbiasedness of the rCFR estimates. Nevertheless, our framework can be extended in a 118 more complex context to address this issue. Last, most essentially, merely construct the 119 instantaneous CFR is less important from the public health point of view, but we emphasize its 120 extensive implementation, which is to identify the key determinants affecting the disease mortality 121 risk, e.g., PM2.5 was found positively associated with the crude CFR of COVID-19 (4). 

The COVID-19 surveillance data were collected via the public domains, and thus neither ethical 127 approval nor individual consent was applicable. 128

All data used in this work were publicly available via https://covid19.who.int/region/amro/country/ca. 130

Not applicable. 132

Funding 133

This work is not funded. 134

None. 136

The funding agencies had no role in the design and conduct of the study; collection, management, 138 analysis, and interpretation of the data; preparation, review, or approval of the manuscript; or 139 decision to submit the manuscript for publication. 140

The author declares no conflict of interest. 

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