Why case fatality ratios can be misleading: individual- and population-based mortality estimates and factors influencing them
Lucas Böttcher Mingtao Xia Tom Chou
q-bio.PE 92C60, 92D30, 91G70





















^1Dept. of Computational Medicine, UCLA, Los Angeles, CA 90095-1766
^2Dept. of Mathematics, UCLA, Los Angeles, CA 90095-1555
^3Beijing Computational Science Research Center, Beijing, China





Different ways of calculating mortality ratios during epidemics
have yielded very different results, particularly during the current COVID-19
pandemic.  









We formulate both a survival probability model and an associated
infection duration-dependent SIR model to define individual- and
population-based estimates of dynamic mortality ratios. The key
parameters that affect the dynamics of the different mortality
estimates are the incubation period and the time individuals
were infected before confirmation of infection. We stress that none
of these ratios are accurately represented by the often misinterpreted
case fatality ratio (CFR), the number of deaths to date divided by the
total number of confirmed infected cases to date. Using data on the
recent SARS-CoV-2 outbreaks, we estimate and
compare the different dynamic mortality ratios and highlight their
differences. Informed by our modeling, we propose more systematic
methods to determine mortality ratios during epidemic outbreaks and
discuss sensitivity to confounding effects and uncertainties in the data.





Why case fatality ratios can be misleading: individual- and
  population-based mortality estimates and factors influencing them
    Tom Chou^1,2,3
    December 30, 2023
===============================================================================================================================




§ INTRODUCTION






The mortality ratio is a key metric describing the severity of a viral
disease <cit.>.  These metrics typically change in
time before converging to a constant value and can be defined in a
number of ways.  One commonly used metric is the (confirmed) case fatality ratio
(CFR or cCFR), the total number of deaths to date, D(t), divided by
the total number of all confirmed cases to date
N(t) <cit.>.
Infection fatality ratios (IFR), the number of deaths to date divided
by the number of all infecteds, have also been used
<cit.> although the IFR requires an
estimate of the number of unconfirmed infecteds.


Both the CFR and IFR have been widely estimated from aggregated
population data from past outbreaks <cit.> as well as from
those of the recent SARS-CoV-2 outbreaks
<cit.>. We
show examples of CFR curves in Fig. <ref> and in the
Supplemental Information (SI). As of March 31, 2020, the global
CFR(t)=D(t)/N(t) = 42,158/858,892 ≈
4.9% <cit.>, while CFRs in individual regions vary
significantly. Clearly, this estimate would correspond to the actual
mortality ratio if (i) all infecteds were tested and (ii) all
remaining unresolved individuals recover.  However, some infected
patients will die, increasing the estimated mortality ratio over
time. Despite the underestimation of this type of population-based
measurement, it is still commonly being used by various health
officials and is often inconsistently defined as deaths/(deaths +
recovereds) even though this difference has been clearly
distinguished <cit.>.



During the severe 2003 acute respiratory syndrome (SARS) outbreak in
Hong Kong, the World Health Organization (WHO) also used the
aforementioned estimate to obtain a CFR of 4.5% while the final
values, after resolution of infecteds, approached
17.0% <cit.>. For the ongoing
SARS-CoV-2 outbreaks, analyses by WHO and many others still use the
CFR=D(t)/N(t) metric
<cit.>, (see
Table <ref>), or time-shifted variants
<cit.>.  Since true mortality rates are
critical for assessing the risks associated with epidemic outbreaks,
typical underestimations by CFRs may lead to insufficient
countermeasures and a more severe
epidemic <cit.>.

An unambiguous definition of mortality ratio is the probability that a
single, newly infected individual will eventually die of the
disease. If there are sufficient individual-level or cohort data,
these probabilities can be further stratified according to patient
age, gender, health condition,
etc. <cit.>. This intrinsic mortality ratio,
or probability of death, should be an intrinsic property of
the virus and the infected individual, depending on age, health,
access to health care, etc., and ought not directly depend on
the population-level dynamics of infected and recovered individuals.
Thus, it can be framed as a survival probability of a single infected
individual.  Whether this individual infects others does not directly
affect his probability of eventually dying [Of course, at the
  population level, if there are many deaths, medical facilities are
  stressed which indirectly leads to an increase in death rates.].

In the Results, we derive a model describing the probability
M_1(t) that an infected individual dies or recovers before time
t. Importantly, these models incorporate the duration of infection
(including an incubation period) before a patient tests positive at
time t=0.  However, the CFR and other mortality measures are
typically reported based on population data. Do these population-based
measures, including CFR, provide reasonable measures of the
probability of death of an individual?  Also in the Results, we
describe how mortality ratios are defined within population-level
models, specifically, a disease duration-structured SIR model. We will
show that population-based estimates are typically not a meaningful
measure of mortality, but that under simplifying assumptions, the
mortality ratio M_ p(t) = D(t)/(D(t)+R(t)), where R(t) is the
number of recovereds up to time t, is more closely related to the
true probability of death M_1(t) <cit.>.  The
simplest population-level mortality ratio is currently (as of March
31, 2020) 42,158/(42,158+178,100)≈ 19.1% <cit.>,
significantly higher than the March 31, 2020 CFR≈ 4.9%
estimate.

We use the same estimates for the rate parameters in our individual
and population models to compute the different mortality ratios. Note
that in general, both the individual mortality probability M_1(t)
and the population-based estimates M_ p(t) depend on the time
of measurement t. By critically analyzing these estimates, the CFR,
and a “delayed” case fatality ratio CFR_ d, we
illustrate and interpret the differences among these measures and
discuss how changes or uncertainty in the data affect them.  In
the Discussion, we summarize our results and identify a
correction factor to transform population-level mortality estimates
into individual mortality probabilities.






§ RESULTS






 §.§ Intrinsic individual mortality rate





Consider an individual that, at the time of positive testing (t=0),
had been infected for a duration τ_1. A “survival”
probability density can be defined such that P(τ, t|τ_1)τ is the probability that the patient is still alive
and infected (not recovered) at time t>0 and has been infected for a
duration between τ and τ +τ.  Since τ_1 is
unknown, it must be estimated or averaged over some distribution.
















The individual survival probability evolves according to 


    ∂ P(τ, t|τ_1)∂  t
    + ∂ P(τ, t|τ_1)∂τ
     = -(μ(τ, t|τ_1)+c(τ, t|τ_1))P(τ, t|τ_1),


where the death and recovery rates, μ(τ, t|τ_1) and
c(τ, t|τ_1), depend explicitly on the duration of
infection at time t and implicitly on patient health and age
a [The timescale of infections are much smaller than the
  aging process. In other words, the aging process across the disease
  time scale is negligible and the age-dependent transport terms
  ∂ P/∂ a can be neglected.]. They may also depend
explicitly on time t to reflect changes in clinical policy or
available health care.  For example, enhanced medical care may
decrease the death rate μ, giving the individual's intrinsic
physiological processes a chance to cure the patient. 




If we assume an initial condition of one individual having been
infected for time τ_1 at the time of confirmation,
Eq. <ref> can be solved using the method of
characteristics (see the SI).








From the solution P(τ=t+τ_1, t|τ_1) (see the
Methods) one can derive the probabilities of death and
recovery by time t as


    P_ d(t|τ_1) = ∫_0^t s μ(τ_1+s,s)
    P(τ_1+s, t|τ_1),   
    P_ r(t|τ_1) = ∫_0^t s  c(τ_1+s, s)
    P(τ_1+s, t|τ_1).












The probability that an individual died before time t, 
conditioned on resolution (either death or recovery), is
then defined as


    M_1(t|τ_1) = P_ d(t|τ_1)  P_ d(t|τ_1)
    +P_ r(t|τ_1).






Equations (<ref>) and (<ref>) also
depend on all other relevant patient attributes such as age,
accessibility to health care, etc. In the long-time limit, when
resolution has occurred (P_ d(∞) +P_ r(∞) = 1),
the individual mortality ratio is simply M_1(∞) = P_
  d(∞). 





In order to capture the dependence of death and recovery rates on the
time an individual has been infected, we propose a constant recovery
rate c and a piece-wise constant death rate μ(τ|τ_1) that is not explicitly a function of time t:


    c(τ,t|τ_1) = c,   μ(τ|τ_1) =
    {[          0   τ≤τ_ inc;        μ_1 τ > τ_ inc ].,


where τ_ inc is the incubation time during which the patient
is asymptomatic, has zero death rate, but can recover by clearing the
virus. In other words, some patients fully recover without ever
developing serious symptoms.




For coronavirus infections, the incubation period appears to be highly
variable with a mean of τ_ inc≈
6.4 days <cit.>.  We can estimate μ_1 and
c using individual patient data where 19 patients (outside Hubei)
had been tracked from the date on which their first symptoms occurred
until the disease resolved <cit.>.

Two out of 19 patients died, on average, 20.5 days after first
symptoms occurred and the mean recovery time of the remaining 17
patients is 16.8 days. We show the recovery-time distribution in
Fig. <ref>(a). Since we know that the
mortality ratio in this dataset is 2/19, we can determine the
dependence between μ_1 and c according to μ_1/(μ_1+c)
≈ 2/19 (or c/μ_1 ≈ 8.5).  The constant recovery and
after-incubation period death rates <cit.> are
estimated to be




    c = 1/20.5/day≈ 0.049/day  and  μ_1 = c / 8.5 ≈ 0.006/day.



Using these numbers, the recovery and death rate functions
c(τ,t|τ_1) and μ(τ|τ_1) are plotted as
functions of τ in Fig. <ref>(b). We show
the evolution of M_1(t|τ_1) at different values of
τ_1 in Fig. <ref>(c). The corresponding
long-time limit M_1(∞) is readily apparent in
Fig. <ref>(d): for τ_1 ≥τ_inc, M_1(∞)=μ_1/(μ_1+c)≈ 0.105,
while M_1(∞) < μ_1/(μ_1+c) when τ_1 <
τ_inc. The smaller expected mortality associated with
early identification of infection arises from the remaining incubation
time during which the patient has a chance to recover without
possibility of death. When conditioned on testing positive at or after
the incubation period, the patient immediately suffers a positive
death rate, increasing his M_1(∞).

Finally, in order to infer M_1 (and also indirectly μ and c)
during an outbreak, a number of statistical issues must be considered.
First, if the outbreak is ongoing, there may not be sufficient
long-time cohort data. Second, τ_1 is unknown. Since testing
typically occurs at the onset of symptoms, most positive patients will
have been infected a few days earlier. The uncertainty in τ_1
can be represented by a probability density ρ(τ_1) for the
individual. The expected mortality can then be constructed as an
average over ρ(τ_1):


    M̅_1(t) = P̅_ d(t) P̅_ d(t)+P̅_ r(t),


where P̅_ d(t) and P̅_ r(t) are the
τ_1-averaged probabilities death and cure probabilities.

















Some properties of the distribution ρ(τ_1) can be inferred
from the behavior of patients. Before symptoms arise, only very few
patients will know they have been infected, seek medical care, and get
their case confirmed (i.e., ρ(τ_1) ≈ 0 for
τ_1≈ 0). The majority of patients will contact
hospitals/doctors when they have been infected for a duration of
τ_ inc. The distribution ρ(τ_1) thus reaches its
maximum near or shortly after τ_ inc. Since patients are
most likely to test positive after experiencing symptoms, we choose a
gamma distribution



    ρ(τ_1; n, γ)= γ^n/Γ(n)τ_1^n-1 e^-γτ_1


with shape parameter n=8 and rate parameter γ=1.25/day
so that the mean n/γ is equal to τ_ inc=6.4. 






Upon using the rates in Eqs. (<ref>) and
averaging over ρ(τ_1), we derived expressions for
P̅(t), P̅_ d(t), and P̅_ r(t) which are
explicitly given in the SI. Using the values in Eq. (<ref>)
we find an expected individual mortality ratio M̅_1(t) (which
are subsequently plotted in Fig. <ref>) and its asymptotic
value M̅_1(∞) = P̅_ d(∞)=0.101. Of
course, it is also possible to account for more complex time-dependent
forms of c and μ_1 <cit.>, but we will
primarily use Eqs. (<ref>) in our
subsequent analyses.

In the next section, we define population-based estimates for
mortality ratios, M_ p(t), and explore how they can be computed
using SIR-type models. By comparing M_1(t) to M_ p(t), we
gain insight into whether population-based metrics are good proxies
for individual mortality ratios. We will outline the mathematical
differences and additional errors that confound population-level
estimates.
























 §.§ Infection duration-dependent SIR model







While individual mortalities can be estimated by tracking many
individuals from infection to recovery or death, oftentimes, the
available data are not resolved at the individual level and only total
populations are given. Typically, one has the total number of cases
accumulated up to time t, N(t), the number of deaths to date
D(t), and the number of cured/recovered patients to date R(t) (see
Fig. <ref>). The CFR is simply D(t)/N(t).  Note
that N(t) includes unresolved cases and that N(t) ≥ R(t) +
D(t). Resolution (death or recovery) of all patients, N(∞) =
R(∞) +D(∞), occurs only well after the epidemic passes.

A variant of the CFR commonly used in the
literature <cit.> is the delayed
CFR


    CFR_ d(t,τ_ res) = D(t)/N(t-τ_ res),


where τ_ res is a corresponding time lag that accounts for
the duration from the day when first symptoms occurred to the day of
cure/death. Many estimates of the COVID-19 mortality ratio assume that
τ_ res=0 <cit.> and
thus underestimate the number of death cases D(t) that result from a
certain number of infected individuals. Similar underestimations using
CFR_ d have been reported in previous epidemic outbreaks
of SARS <cit.> and
Ebola <cit.>.

Alternatively, a simple and interpretable population-level mortality
ratio is M_ p(t) = D(t)/(R(t) + D(t)), the death ratio of all
resolved cases. To provide a concrete model for D(t) and
R(t), and hence M_ p(t), we will use a variant of the
standard infection duration-dependent susceptible-infected-recovered
(SIR)-type model described by <cit.>


    S(t) t    =  
    -S(t)∫_0^∞τ' β(τ',t)I(τ',t),
    ∂ I(τ, t)∂ t+∂ I(τ,t)∂τ    =
    -(μ(τ,t)+c(τ,t))I(τ,t),


and R(t)/ t = ∫_0^∞τ c(τ,t)I(τ,t),
where S(t) is the number of susceptibles, I(τ,t) is density of
individuals at time t who have been infected for time τ, and
R(t) is the number of recovered individuals.  The rate at which an
individual infected for time τ at time t transmits the
infection to a susceptible is denoted by β(τ, t)S(t).  For
simplicity, we assume only community spread and neglect immigration of
infected, which can be straightforwardly incorporated
<cit.>.

Note that the equation for I(τ,t) is identical to the equation for
the survival probability described by Eq. (<ref>).  It
is also equivalent to McKendrick age-structured models
<cit.>
[In both the individual model (Eq. (<ref>))
  and population models (Eq. (<ref>)), the death and cure
  rates are insensitive to changes in age a over the time scale of
  epidemic if it occurs over, say, only 1-2 years. In this limit, we
  consider only infection-duration dependence in the population
  dynamics.]. Infection of susceptibles is described by the boundary
condition


    I(τ = 0, t) = S(t)∫_0^∞τ' β(τ', t)I(τ', t),


which is similar to that used in age-structured models to represent
birth <cit.>. Finally, we use an initial condition
consistent with the infection duration density given by
Eq. (<ref>): I(τ,0) = ρ(τ;n=8,γ=1.25). Note
that Eq. (<ref>) assumes that all newly infected individuals
are immediately identified; i.e., these newly infected
individuals start with τ_1=0. After solving for the infected
population density, the total number of deaths and recoveries to date
can be found via


    D_0(t) = ∫_0^t t' ∫_0^∞τ μ(τ, t')I(τ,t'),    R_0(t) =
    ∫_0^t t'∫_0^∞τ 
    c(τ, t')I(τ,t').


The corresponding total number of cases N(t) in Eq. (<ref>) is 


    N_0(t)=R_0(t)+D_0(t) + ∫_0^∞τ 
    I(τ, t).


In the definitions of D_0(t), R_0(t), and N_0(t), we account for
all possible death and recovery cases to date (see SI) and that newly
infected individuals are immediately identified. We use these case
numbers as approximations of the reported case numbers to study the
evolution of mortality-ratio estimates. Mortality ratios based on
these numbers underestimate the actual individual mortality M_1 (see
the previous “Intrinsic individual mortality rate” subsection) since
they involve individuals that have been infected for different
durations τ, particularly recently infected individuals who have
not yet died.

An alternative way to compute populations is to exclude the newly
infecteds and consider only the initial cohort. The corresponding
populations in this case are defined as


    D_1(t) = ∫_0^t t'∫_t'^∞τ μ(τ, t')I(τ,t'),    R_1(t) =
    ∫_0^t t' ∫_t'^∞τ 
    c(τ, t')I(τ,t').









Since D_1(t) and R_1(t) do not include infecteds with τ < t,
they exclude the effect of newly infected individuals, but may yield
more accurate mortality-ratios as they are based on an initial cohort
of individuals in the distant past. The infections that occur after
t=0 contribute only to I(τ < t, t); thus, D_1(t) and R_1(t)
do not depend on the transmission rate β, possible immigration
of infecteds, or the number of susceptibles S(t).  Note that all the
populations derived above implicitly average over ρ(τ_1; n,
γ) for the first cohort of identified infecteds (but not
subsequent infecteds). Moreover, the population density I(τ≥ t,
t) follows the same equation as P̅(t|τ_1) provided
the same ρ(τ_1; n, γ) is used in their respective
calculations.

The two different ways of partitioning populations
(Eqs. (<ref>) and (<ref>)) lead to two
different population-level mortality ratios


    M_p^0(t) = D_0(t)  D_0(t)+ R_0(t)  and   M_p^1(t) = D_1(t) D_1(t)+R_1(t).


Since the populations D_0(t) and R_0(t), and hence
M_p^0(t), depend on disease transmission through
β(τ,t) and S(t), we expect M_ p^0(t) to carry a
different interpretation from M_1(t) and M_ p^1(t).

In the special case in which μ and c are constants, the
time-integrated populations ∫_0^t
t' ∫_0^∞τ  I(τ,t') and ∫_0^t
t' ∫_t'^∞τ I(τ,t') factor out of M_
  p^0(t) and M_ p^1(t), rendering them time-independent and


    M_ p^0,1 = μ_1 μ_1 + c = M_1.


Thus, only in the special time-homogeneous case do both
population-based mortality ratios become independent of the
population (and transmission β) and coincide with the individual
death probability.

To illustrate the differences between M_1(t), M_ p^0,1(t),
and CFR_ d(t,τ_ res) in more general cases, we
use the simple death and cure rate functions given by
Eqs. (<ref>) in solving
Eqs. (<ref>) and (<ref>). For
β(τ,t) in Eq. (<ref>), we account for incubation
effects by neglecting transmission during the asymptomatic incubation
period (τ≤τ_ inc) and assume


    β(τ, t) = {[           0    τ≤τ_ inc;         β_1 τ > τ_ inc. ].


We use the estimated basic reproductive number R_0 =
β_1S(0)/(μ_1+c) ≈ 2.91 <cit.> to fix
β_1S(0) = (μ_1+c) R_0≈ 0.158/ day. We
also first assume that the susceptible population does not change
appreciably before quarantine and set S(t) = S(0). Thus, we only
need to solve for I(τ,t) in Eqs. (<ref>) and
(<ref>). We solve Eqs. (<ref>) and
(<ref>) numerically (see the Methods section for
further details) and use these numerical solutions to compute
D_0,1(t), R_0,1(t), and N_0,1(t) (see
Fig. <ref>(a) and (b)), which are then used in
Eqs. (<ref>) and CFR_ d(t-τ_ res). To
determine a realistic value of the time lag τ_ res, we use
data on death/recovery periods of 36 tracked
patients <cit.> and find that patients
recover/die, on average, τ_ res=16.5 days after first
symptoms occurred.




We show in Figs. <ref>(c) and (d) that M_p^1(t)
approaches the individual mortality ratio M̅_1(∞)≈
0.1 given in the “Intrinsic individual mortality rate” subsection
above. This occurs because the model for P(τ, t) and I(τ,t)
are equivalent and we assumed the same initial distribution
ρ(τ;8,1.25) for both quantities.  However, the
population-level mortality ratios CFR_d(t,τ_
  res) and M_p^0(t) also take into account recently
infected individuals who may recover before symptoms. This difference
yields different mortality ratios because newly infecteds are
implicitly assumed to be detected immediately and all have
τ_1=0.  Thus, the underlying infection-time distribution is not
the same as that used to compute M̅_p^1(t) (see the
SI for further details).  The mortality ratios CFR_d(t,τ_ res) and M_p^0(t) should
not be used to quantify the individual mortality probability of
individuals who tested positive.  Moreover, due to evolution of the
disease, D(t), R(t), and N(t) do not change with the same rates
during an outbreak, the population-level mortality measures CFR_d(t,τ_ res) and M_p^0(t) reach
their final steady state values only after sufficiently long times
(see Fig. <ref>(c) and (d)).






The evolution of the mortality ratios in Fig. <ref>
qualitatively resembles the behavior of the mortality-ratio estimates
in Fig. <ref>. As shown in
Fig. <ref>, the population-based estimates for
coronavirus varies, decreasing in time for China but fluctuating for
Italy. These changes could result from changing practices in data
collecting, or from explicitly time-inhomogeneous parameters
μ(τ, t), c(τ, t), and/or β(τ,t).

Although population-level quarantining does not directly affect the
individual mortality M_1(t|τ_1) or M̅_1(t), it
can be easily incorporated into the SIR-type population dynamics
equations through changes in β(τ, t)S(t).  For example, we
have set S(t>t_q) = 0 to represent implementation of a
quarantine after t_q=50 days of the outbreak. After
t_q=50 days, no new infections occur and the estimates
CFR_d(t,τ_ res) and M_p^0(t)
start converging immediately towards their steady-state values (see
Fig. <ref>(d)). Since the number of deaths decreases after the
implementation of quarantine measures, the delayed CFR_d(t,τ_ res=17) is first decreasing until
t=t_q+τ_ res=67. For t > 67, the CFR_d(t,τ_ res=17) measures no new cases and is
thus equal to the CFR.



§ DISCUSSION AND SUMMARY

























During an epidemic, it is important to assess the severity of the
disease by estimating its mortality and other disease
characteristics. Assuming accurate data, the often-used CFR typically
underestimates the true, final death ratio. For example, during the
SARS outbreaks in Hong Kong, the WHO first estimated the fatality rate
to 2.5% (March 30, 2003) whereas the final estimates reached values
of about 17.0% (June 30, 2003) (see
Fig. <ref>(a)) <cit.>. Standard metrics like
the CFR are seen to be easily confounded by and sensitive to
uncertainty in intrinsic disease parameters such as the incubation
period and the time τ_1 a patient had been infected before
clinical confirmation of infection. For the recent COVID-19 outbreaks,
CFR-based measures may still provide reasonable estimates of the
actual mortality across different age classes due to a counter-acting
error in the numbers of unreported mild-symptom cases.

Here, we stress that more mechanistically meaningful and interpretable
metrics can be defined and, as easily, estimated from population data as
CFRs. Our proposed mortality ratios for viral epidemics are defined in
terms of (i) individual survival probabilities and (ii) population
ratios using numbers of deaths and recovered individuals. Both of
these measures are based on the within-host evolution of the disease,
and in the case of M_ p^0,1(t), the population-level
transmission dynamics.  







Among the metrics we describe, M_ p^1(t) is structurally
closest to the individual mortality M̅_1(t) in that both are
independent of disease transmission since new infections are not
considered. Both of these mortality ratios converge after an
incubation time τ_ inc to a value smaller than or equal to
μ_1/(μ_1+c). 






The most accurate estimates of M_1 can be obtained if we keep
track of the fate of cohorts that were infected within a small time
window in the past. By following only these individuals, one can track
how many of them died as a function of time. As more cases arise, one
should stratify them according to their estimated times since
infection to gather improving statistics for M_1(∞).  With
the further spread of SARS-CoV-2 in different countries, data on more
individual cases of death and recovery can be more easily stratified
according to other central factors in COVID-19 mortality: age, sex,
health condition.

Besides accurate cohort data, for which at present there are few for
coronavirus, cumulative population data has been used to estimate the
mortality ratio. The population-level metrics M_p^0(t)
and CFR(t) implicitly depend on new infections and the
transmission rate β. Despite this confounding factor,
M_p^0(t) and CFR_ d(t,τ_ res)
approach e^-c τ_ incμ_1/(μ_1+c) as t→∞, where e^-c τ_ inc is the probability that no
recovery occurred during the incubation time τ_ inc. Based
on these results, we can establish the following connection between
the different mortality ratios for initial infection times with
distribution ρ(τ_1;n,γ) and mean τ̅=n/γ:


    CFR_
        d(∞)=M_ p^0(∞) ≈ e^- c τ̅ M_
        p^1(∞)=e^- c τ̅M̅_1(∞) .












According to Eq. (<ref>), population-level mortality
estimates (e.g., CFR and M_ p^0) can be transformed,
at least approximately, into individual mortality probabilities using
the correction factor e^- c τ̅ with τ̅≈τ_ inc.

Besides the mathematical differences between M_1(t) and M_
  p^0(t), CFR, estimating M_ p^0(t) and CFR(t)
from aggregate populations implicitly incorporate a number of
confounding factors that contribute to their variability. In
Fig. <ref>(b), we plot the population-level mortality-ratio
estimates M_p^0 against the CFR for different regions and
observe large variations and very little correlation between
countries <cit.>. As of March 31, 2020, the value of
M_ p^0 in Italy is almost 45% and can increase further if the
current conditions (e.g., treatment methods, age group proportion of
infecteds, etc.) do not change. Differences between the mortality
ratios in China and Italy (see Figs. <ref>(b) and
(c)) might be a result of varying medical treatment strategies,
different practices in data collecting (e.g., post-mortem testing),
differences in the age demographics between the countries, and/or 
inaccuracies in reporting. 

Even if the cohort initially tested was only a fraction of the total
infected population, tracking M̅_1(t) or M_ p^1(t) of
this cohort still provides an accurate estimation of the mortality
rate. However, underreporting newly infecteds can confound CFR and
M_ p^0(t). Current estimates show that only a minority of
SARS-CoV-2 infections are reported (e.g., f≈ 14% in China
before January 23, 2020) <cit.>. At early times
(Fig. <ref>(a)) most patients, tested or untested, have
not resolved. A reported/tested fraction f<1 would not
directly affect the CFRs or mortality ratios if the
unreported/untested individuals die and recover in the same proportion
as the tested infecteds (Fig. <ref>(b)).  Undertesting
would overestimate the true M_ p^0(t) and infection fatality
ratio (IFR) if untested (presumably mildly or asymptomatic infected)
individuals are less likely to die than the tested infecteds
(Fig. <ref>(c)).  If untested infecteds do not die at
all, the true long-time mortality M_ p^0,1(∞)
≈ f M_ p^0,1(∞) (see the SI). In the less likely
scenario in which untested individuals do not receive medical care and
hence die at a faster rate than tested patients (see
Fig. <ref>(d)), M_ p^0,1(∞) and CFR based
on the tested fraction would underestimate the true long-time
mortality M_ p^0,1(∞) and IFR, respectively.



















Besides underreporting, the delay in transmission after becoming
infected will also affect M_ p^0(t). Although we have assumed
that transmission occurs only after the incubation period when
symptoms arise, there is evidence of asymptomatic transmission of
coronavirus <cit.>. Asymptomatic transmission
can be modeled by setting β(τ) > 0 even for τ < τ_
  inc. An undelayed transmission in a no-quarantine scenario causes
relatively more new infecteds who have not had the chance to die yet,
leading to a smaller mortality ratio M_ p^0(t).
Within our SIR model, delaying transmission reduces the number of
infected individuals and deaths at any given time but
increases the measured mortality ratio M_ p^0(t).
Without quarantine, the asymptotic values M_p^0(∞)
and CFR(∞) will also change as a result of changing the
transmission latency period, as shown in the SI.  With perfect
quarantining, the asymptote M_ p^0(∞) is eventually
determined by a cohort that does not include new infections and is
thus independent of the transmission delay.

In this work, we have explicitly defined a number of interpretable
mathematical metrics that represent the probability of dying from a disease 




By rigorously defining these metrics, we are able to reveal the
inherent assumptions and factors that affect their estimation. Within
survival probability and SIR-type models, we explicitly illustrate how
physiologically important parameters such as incubation time, death
rate, cure rate, and transmissibility influence the temporal evolution
and asymptotic values of mortality ratios. We also discussed how
statistical factors such as time of testing after infection
(τ_1) and testing ratio (f) affect our estimates.






In practice, the mortality ratios M_1(t) and M_ p^1(t)
may provide good estimates of mortality of patients who have tested
positive. In addition to our metrics and mathematical models, we
emphasize the importance of curating individual cohort data. These
data are more directly related to the probability of death M_1(t)
and are subject to the fewest confounding factors and statistical
uncertainty.





§ DATA AVAILABILITY




The datasets that we used in this study are stored in the publicly
accessible repositories of
Refs. <cit.>.



§ ACKNOWLEDGEMENTS





LB acknowledges financial support from the SNF Early Postdoc.Mobility
fellowship on “Multispecies interacting stochastic systems in
biology”. The authors also acknowledge financial support from the
Army Research Office (W911NF-18-1-0345), the NIH (R01HL146552), and
the National Science Foundation (DMS-1814364).




§ COMPETING INTERESTS




The authors declare no competing interests.



§ AUTHOR CONTRIBUTIONS




LB and TC developed the analyses and wrote the manuscript. LB and MX analyzed data,
performed numerical computations, and edited the manuscript.















§ METHODS







 §.§ Numerical scheme




To numerically solve Eqs. (<ref>) and (<ref>), we
use a uniform discretization τ_k = kΔτ, k=0,1,…,
K. A backward difference operator [I(τ_k, t) - I(τ_k-1,
  t)]/(Δτ) is used to approximate
∂_τI(τ, t) and a predictor-corrector Euler scheme
is used to advance time <cit.>. Setting the cut-offs
I(-Δτ, t) ≡ 0 and I(KΔτ, t) ≡ 0, the
resulting discretized equations for the full SIR model are


    S(t+Δt) =     S(t) - ΔtS(t)∑_k=0^Kβ(τ_k, t) I(τ_k, t)
    Δτ, 
    Ĩ(τ_k, t)  =     I(τ_k, t) - ΔtI(τ_k, t) 
    - I(τ_k-1, t)Δτ - Δ t (c(τ_k, t) + μ(τ_k, t))I(τ_k, t),
    
    I(τ_k, t+Δt) =     I(τ_k, t) -Δ t2[I(τ_k, t) 
    - I(τ_k-1, t)Δτ + (c(τ_k, t) + μ(τ_k, t)) 
    I(τ_k, t). 
            + . Ĩ(τ_k, t) - Ĩ(τ_k-1, t)Δτ 
    + (c(τ_k, t+Δ t) + μ(τ_k, t+Δ t))Ĩ(τ_k, t)] 
           + δ_k,0Δ t/Δτ S(t)
    ∑_j=0^Kβ(τ_j,t)I(τ_j,t) Δτ,


where Ĩ is the initial predicted guess, and the last term
proportional to δ_k,0 encodes the boundary condition
Eq. (<ref>). Note that we use ∑_k=0^Kβ(τ_k,
t)I(τ_k, t) Δτ to indicate the numerical evaluation of
∫_0^∞dτ' β(τ',t) I(τ',t). Quadrature
methods such as Simpson's rule and the trapezoidal rule can be used to
approximate the integral more efficiently.

The total deaths, recovereds, and infecteds at time t are found by


    D_0(mΔ t)     = 1 2∑_j=0^m∑_k=0^K
    c(kΔτ, jΔ t)[I(kΔτ, jΔ t)+
    Ĩ(kΔτ, jΔ t)]ΔτΔt, 
    
    R_0(t)     = 1 2∑_j=0^m∑_k=0^Kμ(kΔτ, jΔ t)[I(jΔτ, jΔ t)+
    Ĩ(kΔτ, jΔ t)]ΔτΔt, 
    
    I(mΔ t)     = ∑_k=0^K I(kΔτ, mΔ t)Δτ,


with analogous expressions for D_1(mΔ t) and R_1(mΔ
t). To obtain a stable integration scheme, the time steps Δt
and Δτ have to satisfy Δ t/(2 Δτ)<1. In all
of our numerical computations, we thus set Δt=0.002,
Δτ = 0.02, and K=10^4. In the SI, we show additional
plots of the magnitude of I(τ, t) in the t-τ plane.






 §.§ Solutions for τ_1-averaged probabilities






Using the method of characteristics, we find the formal solution to
Eq. (<ref>):


    P(τ, t|τ_1) = δ(τ-t-τ_1)e^-∫_0^t
    (μ(τ-t+s,s|τ_1)+ c(τ-t+s,s|τ_1))ds,

which can be used to construct the death and cure probabilities 


    P_ d(t|τ_1)     = ∫_0^t t' μ(τ_1+t', t')
    e^-∫_0^t'(μ(τ_1+s,s)+ c(τ_1+s,s))
    ds
    
    P_ r(t|τ_1)     = ∫_0^t t'  c(τ_1+t', t')
    e^-∫_0^t'(μ(τ_1+s,s)+ c(τ_1+s,s))ds.




If we now invoke  the functional forms of μ and c given in
Eq. (<ref>), we find explicitly


    P_ d(τ, t|τ_1)= {[                                μ_1μ_1+c(1-e^-(μ_1+c)t)                                       τ > t+τ_
        inc;                                                      0                                         τ_ inc≥τ > τ_1; μ_1e^-c(τ_ inc-τ_1)μ_1+c( 1 -
        e^-(μ_1+c)(τ-τ_inc))                                       τ >
      τ_ inc≥τ_1 ].


and

    P_ r(τ,t|τ_1) = {[                                                  c μ_1+c(1-e^-(μ_1+c)t)                                                           τ 
    > t+τ_ inc;                                                                 1-e^-ct                                                          τ_ inc≥τ > τ_1; 1-e^-c(τ_ inc-τ_1) + 
    c e^-c(τ_ inc-τ_1)μ_1+c(
    1 - e^-(μ_1+c)(τ-τ_inc))                                                         τ > τ_ inc≥τ_1. ].





Finally, we can also find the τ_1-averaged 
probabilities for τ≥ t by weighting over ρ(τ_1; n,γ).
For example, 


    P̅(τ, t) = {[           ρ(τ-t;n,γ)e^-(μ_1+c)t                     τ≥ t+τ_ inc;                 ρ(τ-t;n,γ)e^-ct                    τ_ inc≥τ > t; ρ(τ-t;n,γ)e^-cte^-μ_1(τ-τ_ inc)              t+τ_ inc≥τ >τ_ inc ]. .


These solutions hold for the different regions shown in the phase plot
of Fig. <ref> and are equivalent to those for
I(τ>t,t). Corresponding expressions for P̅_ d(t) and
P̅_ r(t) can be found and used to construct M_
  p^1(t).










Fig. <ref>(a) shows the magnitude of I(τ, t) in the
t-τ plane when we set S(t)=S constant (so that the first
equation in Eq. (<ref>) does not apply) such that
β_1S ≈ 0.158/day. In this case, the epidemic continues
to grow in time, but the mortality rates M_ p^0,1(t)
nonetheless converge as t→∞. In Fig. <ref>(b), we
set β_1 S = 0 for t > t_q to model strict
quarantining after t_q=50 days. We observe no new
infections after the onset of strict quarantine measures. In both
cases (quarantine and no quarantine), we use
ρ(τ;n=8,γ=1.25) (see Eq. (<ref>) in the main
text) to describe the initial distribution of infection times τ.
As time progresses, more of the distribution of τ moves towards
smaller values until quarantine measures take effect (see
Fig. <ref>(c) and (d)).





 
   
  




§ SUPPLEMENTARY INFORMATION







 §.§ Additional examples of mortality-ratio evolutions










In Fig. <ref>, we show additional examples of
mortality-ratio estimates for Iran, South Korea, Spain, Germany,
Switzerland, and the United Kingdom. As in
Fig. <ref> in the main text, we observe that, by
definition, the population-based mortality ratio M_p^0(t)
is significantly larger than the corresponding CFR in all cases.





 §.§ Effects of undertesting




Note that I(τ,t) in the SIR equations determines the dynamics of
the actual infected population. However, (i) typically only a fraction
f of the total number of infecteds might be tested and confirmed
positive and (ii) the testing of newly infecteds may also be delayed
by a distribution ρ(τ; n,γ).

If positive tests represent only a fraction f of the total infected
population, and the confirmation of newly infecteds occurs
immediately, the known infected density is given by I^*(τ,t) =
fI(τ,t) where I(τ,t) is the true total infected
population. If testing of newly infecteds occurs after a distribution
ρ(τ;n,γ) of infection times, I^*(τ,t) = f
∫_0^τI(t-τ+τ_1,
t)ρ(τ_1;n,γ)τ_1.


In our development of M_ p^0,1(t)
and CFR_ d(t,τ_ res) in the manuscript, we
assumed the entire infected population was tested and confirmed. Thus,
M_ p^0,1(t) and CFR_ d(t,τ_ res) were
computed using f=1 and more accurately represent the mortality
ratios of the population conditioned on being tested
positive.



To estimate the mortality ratio of the population conditioned simply
on being infected, we have to estimate the larger number of recovereds
that went untested.  For the most likely scenario in which untested
infecteds have negligible death rate, as shown in
Fig. <ref>(c), we can employ the SIR model without death
for the untested pool of infecteds,


    S(t) t    =  
    -S(t)∫_0^∞τ' β(τ',t)(I^*(τ',t)+ 
    I^ u(τ',t)),
    ∂ I^*(τ, t)∂ t+∂ I^*(τ,t)∂τ    =
    -(μ(τ,t)+c(τ,t))I^*(τ,t), 
    ∂ I^ u(τ, t)∂ t+∂ I^ u(τ,t)∂τ    =
    - c(τ,t)I^ u(τ,t), 
     R(t) t    = ∫_0^∞τ c(τ,t)
    (I^*(τ,t)+I^ u(τ,t)),


where I^*(τ,t)+I^ u(τ,t)=I_ T(τ, t), the total
density of infecteds, and the production of tested and untested
infecteds follow the boundary conditions


    I^*(0,t)     = f S(t) ∫_0^∞τ β(τ,t)I_ T(τ,t) 
    
    I^ u(0,t)     = (1-f) S(t) ∫_0^∞τ β(τ,t)I_ T(τ,t).

Here, we have assumed that testing occurs only for the infected who
can die. In this scenario, the IFR, or the true CFR of all infecteds,
is then IFR(t) = D_0^*(t)/N_ T(t), where in analogy
to Eqs. (<ref>) and (<ref>),


    D_0^*(t) = ∫_0^t t' ∫_0^∞τ μ(τ, t')I^*(τ,t'),    R_0(t) =
    ∫_0^t t'∫_0^∞τ 
    c(τ, t')I_ T(τ,t'),


and 


    N_ T(t)=D_0^*(t) + R_0(t) + ∫_0^∞τ 
    I_ T(τ, t).


The true mortality ratio is also straightforwardly defined by, for
example,


    M_ p^0(t) = D^*_0 D_0^*(t) + R_0^*(t) + R_0^ u(t),


where 


    R_0^*(t) =  ∫_0^∞τ∫_0^t t'  c(τ, t')I^*(τ, t')
          
    R_0^ u(t) = ∫_0^∞τ∫_0^t t'  c(τ, t')
    I^ u(τ, t'),


with analogous expressions for D_1^*(t), R^*_1(t), and 
R^ u_1(t). At long times, after resolution of all infecteds, 
the untested recovered population is


    R^ u_0,1(∞) = (1 f -1)(D_0,1^*(∞) + R_0,1^*(∞)),


which yields the asymptotic true ratio M_ p^0,1(∞)
= fM_ p^0,1(∞) as described in the Discussion and
Summary. In this simple rescaling to account for untested populations,
we have assumed that all deaths come from the tested pool and that the
recovery rate c is the same in the tested and untested pools.













 §.§ Influence of different transmission rates




In Fig. <ref> of the main text, we observe that the
population-level mortality ratio M_p^0(t) approaches a
plateau during the initial exponential growth phase of an epidemic
(i.e., for S(t)≈ S_0). If the number of new
infections decreases (e.g., due to quarantine measures),
M_p^0(t) starts growing until it reaches its asymptotic
value M_p^0(∞). Interestingly, the pre-asymptotic
values of M_p^0(t) are smaller for larger infection rates
β_1 (see Fig. <ref>(a)). This counter-intuitive effect
arises because larger values of β_1 generate relatively larger
numbers of new infected which have a lower chance of dying before
τ_ inc (see Eq. (<ref>) in the
main text).



A similar effect occurs for non-delayed transmission (i.e.,
τ_β≈ 0). As the transmission delay decreases, more
secondary cases will result from one infection, leading to 
smaller values of M_p^0(t) in the initial exponential
growth phase of an epidemic (see Fig. <ref>(b)).









