###### Abstract

It is evident that increasing the intensive-care-unit (ICU) capacity and
giving priority to admitting and treating younger patients will reduce the
number of COVID-19 deaths, but quantitative assessment of these measures has
remained inadequate. We develop a comprehensive, non-Markovian state
transition model, which is validated through accurate prediction of the daily
death toll for two epicenters: Wuhan, China and Lombardy, Italy. The model
enables prediction of COVID-19 deaths in various scenarios. For example, if
treatment priorities had been given to younger patients, the death toll in
Wuhan and Lombardy would have been reduced by 10.4% and 6.7%, respectively.
The strategy depends on the epidemic scale and is more effective in countries
with a younger population structure. Analyses of data from China, South Korea,
Italy, and Spain suggest that countries with less per capita ICU medical
resources should implement this strategy in the early stage of the pandemic to
reduce mortalities.

## Introduction

The continuous and recently accelerated spreading of COVID-19 in more than 145
different countries and regions of the world has placed an unprecedented
burden on the corresponding healthcare systems. As of December 1, 2020, there
have been more than 63.3 million confirmed cases with over 1.4 million deaths
reported, and the daily number of deaths [1] has exceeded 8000. To mobilize
the available medical resources to the maximum degree to reduce the COVID-19
fatalities has become the top priority in many hospitals and healthcare
facilities. In a hospital, there are two types of care resources for
inpatients: General Ward (GW) and Intensive Care Unit (ICU) beds, with the
latter currently serving mostly COVID-19 patients. Increasing the ICU beds
would undoubtedly reduce the COVID-19 fatalities. When a hospital is
overwhelmed with COVID-19 patients so that the ICU beds are in a serious
shortage, a selective ICU admission policy must be invoked to allow specific
groups of patients to receive the ICU care. In this regard, from a sheer
mathematical point of view, giving priority to the young age group will reduce
the fatality rate, as patients in this group have a better chance of being
cured. Stretching the ICU resources to their limit and implementing selective
ICU admission policy are becoming inevitable and even absolutely necessary in
many parts of the world as the COVID-19 cases have continued to skyrocket in
recent months.

It is intuitively evident that enhancing ICU capabilities and selectively
admitting patients into ICU can reduce the COVID-19 mortalities, but we lack a
modeling framework to quantitatively assess and characterize their effects.
The main goal of this paper is to address this issue that is critically
important to maximizing the usage and effectiveness of the limited medical
resources to minimize COVID-19 deaths.

There have been intense modeling efforts on early warning, prevention and
control of the COVID-19 pandemic [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. For
example, Kraemer et al. [2] found that travel restriction in the early stage
of the COVID-19 outbreak can effectively prevent the infection imported from
known sources. Once cases begin to spread in the community, the contribution
of newly imported cases tends to diminish, requiring a set of control measures
including travel restrictions, detection, tracking, and isolation to mitigate
the pandemic. Kissler et al. [3] established a SARS-CoV-2 transmission model
with seasonal variations, immune duration, and cross immunization, where the
peak of SARS-CoV-2 infection is assumed to be low in spring and summer and a
larger epidemic breakout can occur in autumn and winter. A finding was that,
with respect to imposing one time social distancing, intermittent social
alienation measures can prevent the overload of public health resources. Hao
et al. [4] found that the new coronavirus has two characteristics: high
infectivity and concealment. If 87% of the infected individuals are not
detected, without any prevention and control measures, after 14 consecutive
days of zero confirmed cases, the probability of a second wave of epidemics
will be 32%. If only 53% of the infected are undetected, the probability of an
epidemic rebound will drop to 6%. Premature removal of prevention and control
measures will greatly increase the possibility of a second outbreak of the
epidemic. Long et al. [5, 6, 7] developed a time-delay transmission model that
can accurately simulate and predict the epidemic development in countries or
regions, and the model enables evaluation of the impact of virus detection and
social distancing intensity on the epidemic, as well as an accurate estimate
of the time zero point of the epidemic in various countries and regions. In
particular, it was found [7] that community-based transmission occurred in
Europe and in the United States in early January 2020. The study of Flaxman et
al. [8] revealed that the outbreak interventions implemented in 11 European
countries have reduced about 1.3 million deaths and these measures are enough
to bring down the basic reproduction number $R_{0}$ to less than one. Dehning
et al. [9] studied the spread of COVID-19 in Germany by combining
epidemiological modeling with Bayesian inference, and found that the change
point of the effective growth rate of new infection is closely related to the
time point when the intervention measures were imposed.

There have also been recent studies on the interplay between the medical
resources and COVID-19 mortality. Particularly, in comparison with the common
influenza virus, a higher proportion of the patients infected with SARS-CoV-2
need to be hospitalized. The aggravated growth in the number of infected
individuals in recent months has stretched the capacities of the medical
systems in many countries and regions to their limit. A shortage of medical
resources, such as the respiratory support devices, will result in a large
number of severely ill patients not getting timely and effective treatment,
exposing them to a greater risk of mortality. Ferguson et al. [12, 13]
evaluated the demand for medical resources of the non-pharmaceutical
interventions type, predicting that, even when the most effective mitigation
measures are implemented, the numbers of hospital GW and ICU beds in the UK
still need to be expanded by more than eight times to meet the needs of
patient care. Miller et al. [14] used the age-specific mortality and
demographic data to predict the cumulative COVID-19 cases and medical resource
burden in different regions of the United States, pointing out that medical
resources are relatively scarce in remote areas. When the medical system is
overloaded, the nursing standard for the patients will be compromised, which
would affect the treatment outcome. Without proper and rigorous care, the
condition of critically ill patients with COVID-19 will further deteriorate,
and the shortage of ICU facilities will aggravate the high mortality rate of
such population [15, 16]. These studies [12, 13, 14, 15, 16] made evident the
potential significant impact of the medical resource availability on the
COVID-19 mortality.

In this paper, we develop a comprehensive state transition model to predict
the COVID-19 death and its evolution over time for a regional healthcare
system with limited medical resources and selective ICU admission policy. The
typical systems are a city or a region, e.g., Wuhan city in China or the
Lombard region of Italy. Realistic time delays associated with various state
transitions are fully incorporated into the model, rendering it non-Markovian
to better describe the real world situations. The model is based on and
validated with empirical data such as the number of confirmed cases, clinical
data, demographics, and the amount of medical resources, and it enables such
questions to be addressed as: if the ICU resources were deployed certain days
earlier or if the ICU capacity was doubled or tripled, how much reduction in
the death toll would be achieved? The findings of this paper can be best
described with concrete numbers. For example, for Wuhan city, if the ICU
resources had been deployed a week in advance or if the number of ICU beds had
been doubled, the death toll would have been reduced by 5% or 13%,
respectively. For Lombardy, the corresponding numbers are similar: 3% or 14%.
For both Wuhan and Lombardy, tripling the ICU capacity would have resulted in
a 21% reduction in the mortalities. With respect to selective ICU admission
policy, prioritizing younger patients would have reduced the death toll in
Wuhan and Lombardy by 10.4% and 6.7%, respectively. As illustrated by the
exemplary numbers, our model provides a framework to predict COVID-19 deaths
in arbitrary scenarios. Because the model has been fully validated with real
data, we expect the model predictions to be reliable and accurate. A further
analysis of the data from China, South Korea, Italy and Spain indicates that
the age-selective admission strategy is more effective in countries with a
younger age structure, while the countries with less per capita ICU medical
resources should implement the selective admission strategy in the early stage
of the epidemic to suppress mortalities.

## Model

Supplementary Figure S1:  Schematic diagram of the proposed COVID-19 patient
admission model under limited medical resources. Each node represents a
possible state of the patients. The specific definition of each state and the
transition paths among them (light blue and light red arrows with time
representing the average delay of the transition) are described in detail in
the text. The various quantities $P_{\cdot\rightarrow\cdot}$ denote the
fractions of the state transition paths, and the time delays follow a normal
distribution. The light red dotted arrows indicate the two admission paths of
ICU, where ICU admission I and II have the first and second priority,
respectively. The light blue dotted arrows indicate the two treatment paths of
GW, which have a lower priority than ICU admission: GW admission I and II have
the third and fourth priority, respectively.

Our COVID-19 patient admission model under limited medical resources has ten
dynamical states, as shown in Fig. S1. It describes the basic transition
process of a diagnosed individual’s state from onset to recovery or death.
Time delays associated with the various state transitions are taken into
account, rendering non-Markovian the dynamical evolution. In particular, state
M denotes that a patient is infected with COVID-19, state F means that the
patient needs to be hospitalized but is currently under home isolation or
centralized isolation, state C describes that the patient needs ICU support
treatment but is currently isolated, states G and U indicate an inpatient
occupying a GW and an ICU bed, respectively, a patient in state W currently
occupies a GW bed but needs ICU treatment, a patient in the $G_{U}$ state is
transferred out of ICU who does not need to occupy a GW bed, a patient in the
$M_{x}$ state is discharged from the hospital with lessened symptoms, and the
R and D states represent two clinical outcomes: recovery and death.

The medical resources can be divided into two categories: GW and ICU resources
denoted as $Q_{G}$ and $Q_{U}$, respectively, which can be measured, e.g., by
the corresponding numbers of beds. That the resources are limited is modeled,
as follows. when the GW beds are all used up, new patients will no longer be
admitted. When all ICU beds have been occupied, patients who require ICU care
either continue to wait at GW (W-state) or continue to wait outside the
hospital (C-state). A common hospital admission policy is “first-come first-
serve” (FCFS), which gives priority to patients who have had a long waiting
time. The admission policies of GW and ICU thus consist of four admission
paths with different priorities: GW admission I/II and ICU admission I/II,
where ICU admission I for W-state patients has the highest priority, followed
by ICU admission II for C-state patients, GW admission I for the remaining
C-state patients in GW, and GW admission II for the F-state patients. The
priorities define the execution order of the corresponding paths. For M-state
patients, two different transitions can occur: some patients deteriorating
into the F-state so as to require hospitalization and others slowly recovering
into the R-state. For patients in the F-state, there are three types of
transitions: (i) when the GW resources are available, the patients are
admitted to the hospital through GW admission II, (ii) a proportion of the
remaining F-state patients deteriorate to C-state, requiring ICU treatment,
and (iii) the rest are cured so they switch to the $M_{X}$-state. Patients in
state C enter ICU through ICU admission II to transition into U-state, or
through GW admission I to enter GW to become W-state, while the rest are in
D-state. For G-state patients, some switch to $M_{X}$-state and no longer
occupy GW resources, while others deteriorate to the W state, requiring ICU
care. W-state patients will enter ICU through ICU admission I, and the
remaining will be in the D state. Some of the U-state patients occupying ICU
resources will transition to the $G_{U}$ state, and the others will switch to
the D state. For patients in the $G_{U}$ state, the symptoms are relieved
after a period time and they enter the $M_{X}$ state. Patients in the $M_{X}$
state recover to the R state after a period of time.

Let $\bigtriangleup Q_{G}$ and $\bigtriangleup Q_{U}$ denote the changes in
the GW and ICU resources, respectively. A decrease in the GW resources is the
sum of the number of patients admitted through the GW admission I/II pathway,
and an increase is the sum of the number of ICU admission I, the number of
patients in G (W)-state cured (or died), and the amount of newly deployed GW
resources. Likewise, a decrease in ICU resources is the sum of the number of
patients admitted through ICU admission I/II pathway, and an increase is the
sum of the number of patients cured (or died) in state U and any newly
deployed ICU resources. The state dynamical evolution described by a set of
difference equations with time delays characterizing the state ages
[Supplementary Note 1 (SN 1)].

## Results

We validate our ten-state model by simulating the trend of the death tolls in
Wuhan and Lombardy, using the daily number of confirmed cases, clinical and
demographic data. The model then enables us to assess the overall effects of
varying the timing of resource deployment and its amount on the COVID-19
mortalities. At a more detailed level, we divide the patients into several age
groups and calculate the death-toll trend in each age group. This allows us to
assess the impact of resource allocation scheme of different age groups on the
number of deaths, and to obtain the optimal ICU admission strategy in terms of
the age structure and outbreak scale with limited medical resources.

### Impact of limited medical resource deployment

#### Model validation

To simulate the trend of the death tolls in Wuhan and Lombardy using the the
second-order difference equations in our modeling framework, three types of
data are required: time delays associated with state transitions, patient
morbidity data, and local medical resource deployment data. The details of
these data are described in SN2 and Supplementary tables S1-S5. which include
information such as the average number of days in the transition delay from
M-state to F-state and the average mortality rate of patients in ICU. Most of
the data were obtained through references and official reports [17, 18, 19].
As detailed in SN2, the values of some model parameters need to be estimated
in an optimal way, e.g., at the 95% confidence level, through model simulation
and empirical data such as the average fraction of the patients switching from
G-state to W-state, denoted as $|P|_{G\rightarrow W}$, and that from F-state
to C-state, denoted as $|P|_{F\rightarrow C}$. We use the cumulative days of
insufficient GW and ICU resources to quantify the level of stress on the
healthcare system, defined as the GW overload days $O_{G}$ and ICU overload
days $O_{U}$ (see SN3 - medical system stress level indicators).

Supplementary Figure S2:  Simulation results of the daily evolution of death
tolls for the two COVID-19 epicenters. (a,b) Actual data of cumulative death
toll with time in Wuhan and Lombardy, respectively. The insets show the daily
numbers of new deaths. The open black squares and red crosses represent the
actual data and simulation results, respectively. The pink shaded area marks
the 95% confidence interval. There is a good agreement between the simulated
daily number of deaths and the actual data, validating our ten-state model.

With the optimal parameter values and empirical data as inputs to the model,
we simulate the cumulative death toll and new deaths over time in Wuhan (from
December 8, 2019 to April 15, 2020) and Lombardy (from January 28 to June 14,
2020), as shown in Fig. S2. It can be seen that the model predicted trends of
the cumulative and daily new death tolls agree with the actual data very well,
validating the model. The model also gives that the values of $O_{G}$ are 14
and 17 days for Wuhan and Lombardy, respectively, and the corresponding values
of $O_{U}$ are 49 and 74 days.

#### Deployment of medical resources

Since the onset of COVID-19 pandemic, many countries (most notably the USA)
and regions have missed the best time window to control the spreading. At the
present, there is a skyrocketing increase in the demand for ICU beds and
medical resources such as ventilators and other special medical devices. Many
hospitals and healthcare facilities have reached or will soon reach the limit
of their operating capacities.

Optimizing the usage of medical resources by deploying them as early as
possible and augmenting them as much as possible are key to reducing the
mortality rate. Let DT and RI denote the two key parameters: the deployment
time and the resource input, where $\mbox{DT}=0$ and $\mbox{RI}=1$ represent
the actual deployment time and the available normalized amount of resources,
respectively. If the resources are deployed seven days ahead of the actual
time, we have $\mbox{DT}=-7$. Likewise, if the resources are doubled (e.g.,
twice as many ICU beds as in the actual case), we have $\mbox{RI}=2$ (see SN3
for COVID-19 special medical resource deployment plans). A virtue of our
modeling framework lies in its ability to provide a quantitative picture of
the dependence of the mortalities on the two key parameters.

To uncover the impact of deploying medical resources on the patient mortality
rate in a concrete way, we consider the following three scenarios: (i) varying
the GW resources only, (ii) varying ICU resources only, and (iii) varying both
resources simultaneously. Some representative results are presented below
(more results in SN4).

Supplementary Figure S3:  Impact of timing of medical resource deployment and
resource input on COVID-19 mortalities. Each panel shows the number of deaths
(color coded) in the parameter plane of deployment time and resource input:
panels (a,b) are for Wuhan, China, and (c,d) are for Lombardy region, Italy.
In panels (a,c), the ICU resource amount is fixed and the resource that varies
is GW. In panels (b,d), the GW resource amount is fixed and the resources that
vary are ICU. The red hexagon indicates the true death toll under the actual
resource deployment, and the white circles mark the contour of the actual
death toll in the parameter plane.

Figure S3(a) shows, for Wuhan, the impact of varying GW resource deployment on
the number of deaths for fixed ICU resource deployment, where the red hexagon
indicates the true death toll under the actual resource deployment. If the GW
resources had been deployed one week in advance or if its amount had been
doubled, the death toll would have been reduced by 10%. While remarkable, this
10% figure is about the maximum reduction that can be achieved by varying the
GW resource deployment. For example, deploying the GW resources earlier than
one week or an increase in its investment over three times would not reduce
the death toll further. The maximum reduction in the deaths that can be
achieved is determined approximately by the contour of the actual death toll
\- the curve connecting the white circles Fig. S3(a). These results indicate
that deploying the GW resources earlier and/or augmenting them have only
limited effects on reducing the deaths.

In contrast, deploying the ICU resources earlier and/or augmenting them can be
more effective at reducing the deaths in Wuhan, as shown in Fig. S3(b), where
the GW resource deployment is fixed. If the ICU resources had been deployed
one week ahead of the actual time or if the ICU resource amount had been
doubled, the death toll would have been decreased by 5% and 13% respectively.
While these figures are similar to those that would have been achieved with
the same adjustment in the GW resources [Fig. S3(a)], deploying the ICU beds
earlier or increasing their number can have a much more significant effect on
the number of deaths. For example, if the number of ICU beds had been tripled,
the death toll would have been reduced by 21%! The maximum reduction in the
deaths is achieved for $\mbox{DT}=-14$ and $\mbox{RI}=2.5$, which is about 30%
(about three times more effective than that which can be achieved by
stretching the GW resources and their deployment). The corresponding ICU
overload days would have been decreased to less than seven days (comparing
with the actual 49 days for Wuhan). Similar results are obtained for Lombardy.
We find that adjusting the GW resource deployment would have only limited
effects in reducing the death toll. For example, as shown in Fig. S3(c), if
the GW resources had been deployed one week or two weeks earlier or if the
number of GW beds had been doubled or tripled, the reduction in the deaths
would have been only about 1%. This indicates that the deployment of the GW
resources in Lombardy was timely and its amount was appropriate, a fact that
can also be seen from the value of the GW overload days $O_{G}$: to make it
less than a week, the resources would need to be deployed only five days
earlier or the number of GW beds would need to be only 1.4 times higher. In
contrast, deploying the ICU resources earlier or augmenting them would have
been much more effective at r educing the deaths. For example, if the number
of ICU beds had been tripled, the death toll would have been reduced by 21%,
as shown in Fig. S3(d). The maximum reduction that can be achieved is about
22% and $O_{U}$ can be decreased to less than a week.

SN3 presents results on the impact of varying the GW and ICU resource
deployment simultaneously on COVID-19 death toll for both Wuhan and Lombardy.

### Admission strategy based on age groups

To improve the efficiency of treatment and to reduce the mortality rate, it is
essential and imperative to allocate the available resources as reasonably as
possible because as they are not unlimited. For COVID-19, the hospitalization
and mortality rates of elderly patients are higher than those of younger
patients [20]. A common strategy is to divide the patients into distinct age
groups. Our modeling framework provides a rigorous way to calculate the
mortalities for different age groups.

#### Age groups

Supplementary Figure S4:  Evolution of death toll with time for different age
groups. (a) Simulated evolution trend of the death toll of the three age
groups in Wuhan; (b) the actual mortality rate of each age group in Wuhan in
comparison with model prediction; (c,d) the corresponding results for
Lombardy. The insets in (a) and (c) display the actual daily death toll and
model predictions for Wuhan and Lombardy, respectively. In panels (b) and (d),
the light blue and pink columns represent the actual and simulated mortality
rate of each group, respectively.

Individuals in any age group have the possibility of getting infected by
COVID-19 with certain death risk [21]. Currently available data indicate that,
in most countries, the risk of hospitalization and death increases with age
[22, 23]. A direct manifestation is that the fraction of hospitalized patients
and ICU mortality vary among the age groups.

For Wuhan, the available data divide patients into three groups: [0-69] years
old, [70-79] years old, and [80+] years old. A similar division scheme applies
to Lombardy. We use a linear regression to obtain the estimates of the
fractions of various state transitions for each age group: $P_{M\rightarrow
F}$, $P_{G\rightarrow W}$, $P_{F\rightarrow C}$ and $P_{U\rightarrow D}$. (SN4
provides more details.)

We input the parameters associated with each age group into the model and
calculate the death tolls as a function of time for Wuhan and Lombardy. Figure
S4 shows that the mortalities obtained from the simulation agree well with
those of the actual data. Figures S4(a) and S4(c) show that the death toll of
patients over 80 years old in Wuhan is the lowest among the age groups, while
it is the highest in Lombardy: about 59%. There are two possible reasons for
this difference. First, the aging populations in the two regions are
different: the fraction of people over 65 years old is 14.06% in Wuhan and 20%
in Lombardy [24, 25]. The elderly have weak autoimmunity and most of them have
preexisting, underlying diseases, resulting in an excessively high death rate
in Lombardy. The second reason is that more extensive tests were conducted in
China (near 100%), enabling more young people with mild symptoms to be
detected and included in the statistics. In contrast, as of April 29, 2020,
there were about 1910761 people in Italy who had been tested, of whom 203591
were diagnosed with COVID-19 - about 11% only [26]. Nonetheless, Figs. S4(b)
and S4(d) show that the mortality rate of patients over 70 years old is higher
than that of patients under 70 for both Wuhan and Lombardy. Especially, in
Lombardy, the mortality rate of patients over 80 years old reached 41%.

When the population is divided into nine age groups, our model generates
essentially the same phenomenon that the risk of death increases with age
(SN5).

#### Group weighting strategy

People of all ages have certain risk of being infected with COVID-19. Setting
priority of admission for certain age groups can reduce the number of deaths.
In Wuhan, the ICU resources were more scarce than GW resources, so making ICU
admission policy dependent on age is especially important for suppressing the
mortality rate. That is, the conventional FCFS admission policy is not
suitable for COVID-19 under limited ICU resources. To set the priority of
admission for different age groups, we shift the state age of each group by a
priority weight. We then change the admission order of different age groups
with the goal to find an optimal set of weights that minimizes the number of
deaths. In particular, we set the priority weight of the $i_{th}$ age group as
$w_{i}$ and record $\tau_{i}$ as the state age of the this age group. After
weighting, the new state age $\tau_{i}^{{}^{\prime}}$ becomes,

$\tau_{i}^{{}^{\prime}}=\tau_{i}+w_{i}.$ (1)

The state age of patients in the $i_{th}$ age group is increased by $w_{i}$
days. The patients, after incorporating the weighted state age, are admitted
according to FCFS.

To carry out the optimization procedure, we fix the priority weight of the
second age group, i.e., the [70, 80) age group, to be $w_{2}=30$. As the state
age of each group does not exceed 15, we implement different admission
strategies for the age groups by changing the value of $w_{1}$ and $w_{3}$
(see SN5 for a specific method).

Supplementary Figure S5:  Dependence of death toll on weights. (a,b) Model
generated death toll and its dependence on $w_{1}$ and $w_{3}$ for fixed
$w_{2}=30$ for Wuhan and Lombardy, respectively, where the blue and purple
colors correspond to relatively fewer deaths. In the regions marked by Roman
numerals, the death toll changes slowly with weights. The insets show the
distribution of the number of deaths for different weight combinations, where
the numerical values represent the differences in the number of deaths than
that from the FCFS treatment strategy. (c,d) Daily number of new deaths under
different admission strategies for Wuhan and Lombardy, respectively, where red
circles, gray boxes, and blue-purple forks denote the results from the three
strategies: preferential treatment for elderly patients, FCFS, and preference
for young patients, respectively. Giving priority to young patients results in
the lowest daily death curve.

Figure S5 shows the effect of the state age of groups 1 and 3 on the number of
deaths when the weight of group 2 is fixed at constant (30). The larger the
state age, the higher priority of admission is. When adjusting the weights of
the first and third groups, their correspond state ages will vary and so the
admission order of each age group will change accordingly. Different orders of
treatment will lead to different death tolls. Figure S5 reveals, in the
parameter plane of $(w_{1},w_{3})$, five distinct death toll regions for
Wuhan. For Lombardy, there are six such regions, as shown in Fig. S5(b). The
weight range in region V in Fig. S5(a) corresponds to the ranges in regions V
and VI in Fig. S5(b), due to the relatively small fraction of patients in the
third group in Wuhan. When patients in the first group are given the priority,
the treatment order of the patients in the third group has little impact on
the total number of deaths. While the patients in the third group in Lombardy
have a large proportion, giving the priority to the first group will have a
significant impact. In this case, the order of treatment for patients in the
third group will affect the total number of deaths.

In Fig. S5, the number of deaths in Wuhan and Lombardy in parameter region I
increases by 4.8% and decreases by 3.2%, respectively, in comparison with the
death tolls from the FCFS treatment strategy, for $w_{2}>w_{1}>w_{3}$. Area II
in Fig. S5 shows the number of deaths for $w_{2}>w_{3}>w_{1}$. Comparing with
the FCFS strategy, the number of deaths increases by 10.5% and 4.1% for Wuhan
and Lombardy, respectively. In area III in Fig. S5, the weighting order is
$w_{3}>w_{2}>w_{1}$ and the largest death tolls are observed in Wuhan and
Lombardy, as characterized by an increase of 11% and 5.7%, respectively, in
comparison with the FCFS case. In area IV in Fig. S5 where the order of
priority is $w_{3}>w_{1}>w_{2}$, the death tolls in Wuhan and Lombardy
increase by 2.5% and 4.5%, respectively. In area V in Fig. S5(a), we have
$w_{1}>w_{3}>w_{2}$, and the number of deaths in Wuhan is the lowest as
represented by a decrease of 10.4% relative to that associated with the FCFS
strategy. In the same area V for Lombardy, the corresponding decrease in the
death toll is by 5.3%, as shown in Fig. S5(b). In area VI in Fig. S5(b), the
priority ordering is $w_{1}>w_{2}>w_{3}$ and there is a decrease of 6.7% in
the death toll. The implication of these results is that giving the treatment
priority to young patients will reduce the number of deaths but giving
priority to elderly patients will increase the death toll. Due to the resource
limit, this strategy is intuitively reasonable and has in fact been commonly
practiced in many hospitals and healthcare facilities. Our results provide a
validation at a quantitative level.

Figures S5(c) and S5(d) demonstrate the evolution of new deaths over time in
Wuhan and Lombardy, respectively, under different admission strategies. In
both cases, there is no significant difference in the number of deaths in the
early stages of the epidemic under three treatment strategies. When the
epidemic has lasted for a period of time and the death toll increases sharply,
giving priority to young patients can significantly reduce the number of
deaths. At the end of the epidemic, again the differences among the three
treatment strategies diminish. Taken together, these results verify that,
under limited medical resources, during the rapidly increasing phase of
COVID-19 infection, admitting and treating young patients are necessary to
reduce the final death toll.

#### Optimization efficiency

When the ICU resources are in a serious shortage, medical staff have to face
the hard choice of adopting the strategy of giving priority to treating young
patients with a higher survival probability. The strategy’s effectiveness
notwithstanding, it has serious implications for medical ethics. It is thus
worth evaluating this admission/treatment strategy further. To this end, we
exploit our model to study more extensively the impacts of different
strategies for four countries: China, South Korea, Italy, and Spain. For each
country, we collect information about the numbers of GW and ICU beds per
100000 population and about the age distribution of patients diagnosed with
COVID-19. For example, in China and Italy, the numbers of ICU beds per 100000
individuals are 3.6 and 12.5, respectively. (A detailed display of the
information can be found in SN6). We define the time-dependent optimization
efficiency as

$E(t)=\frac{V_{Q}(t)}{V_{Y}(t)},$ (2)

where $V_{Q}(t)$ and $V_{Y}(t)$, respectively, are the numbers of new deaths
per day with the FCFS and the age-selective strategy. In the asymptotic time
limit $t\rightarrow\infty$ when the system has reached a steady state,
$E(\infty)$ characterizes the final optimization efficiency. Since information
about the average state transition fractions for patients in all four
countries is not available, we use the model parameter values for Wuhan and
Lombard to calculate the value of $E(\infty)$ for four countries versus the
average daily patient size.

Supplementary Figure S6:  Asymptotic optimization efficiency in China, Korea,
Italy, and Spain. Shown is the steady-state efficiency $E(\infty)$ versus the
cases per ten million people. (a) The result for China based on the parameter
values from Wuhan. Inset shows the corresponding results for Italy, South
Korea, and Spain. (b) The result for China based on the parameter values from
Lombardy and the results for the other three countries are displayed in inset.

Based on the per capita medical resource data of the four countries as well as
the age distribution data of confirmed patients in each country, making use of
the relevant model parameter values for Wuhan and Lombard, we calculate
$E(\infty)$ of the four countries versus the daily average patient size. We
assume that up to 50% of the medical resources can be deployed to treating
patients with COVID-19. As shown in Fig. S6, when the number of patients is
below 100, the value of $E(\infty)$ for the four countries is equal to one. In
this case, there is no need to implement the age-selective admission strategy.
When the number of patients exceeds a critical value, $E(\infty)$ for all four
countries increases rapidly as the number of patients increases through the
critical value and then reaches a peak value. In fact, for a wide range of
variation of the patient size, the values of $E(\infty)$ are high. Note that
the peak values of $E(\infty)$ for China and South Korea are slightly below
1.3, but those for Italy and Spain are lower than 1.2 with a relatively slow
increasing rate to reach the peak value. A plausible reason lies in the
difference in the age distribution of patients diagnosed with COVID-19 in the
eastern and western countries. In particular, in China and South Korea, less
than 12% of the patients are over 70 years old, while those in Italy and Spain
account for 37% and 34%, respectively (SN6). The value of $E(\infty)$ is
highly correlated with the availability of the ICU beds. For $E(\infty)=1$,
ICU beds are fully sufficient, but $E(\infty)>1$ signifies a shortage of ICU
beds. Figure S6(a) reveals that, with the per capita medical resources of
South Korea, Italy, and Spain, the need for ICU of 1,000 cases per tens of
millions of people per day can be met for a long time, while the capacity of
China may be less than half of the capacity of those countries. Indeed, the
number of ICU beds per capita in China is 3.6 per 100000 persons, while the
corresponding numbers in Italy, South Korea, and Spain are 12.5, 10.6, and
9.7, respectively (SN 6). We conclude that, while the age-selective admission
strategy is more effective in countries with a younger population structure,
countries with less per capita ICU medical resources need to implement this
strategy, particularly in the early stage of the pandemic when the number of
patients is relatively small.

## Discussion

With the resurgence of COVID-19 cases in many countries and regions, a severe
shortage of medical resources for treating the disease is inevitable,
potentially leading to a significant death toll. To study the impact of
limited medical resources on the patient mortality at a quantitative level can
provide insights into developing optimal resource allocation schemes to reduce
the number of deaths. Based on the COVID-19 data, medical resources and other
relevant information, we have developed an admission treatment model subject
to limited medical resources, which enables a quantitative and systematic
assessment of the mortality rate associated with various resource allocation
scenarios. Using empirical data from Wuhan and Lombardy, we validate the model
by demonstrating that it can reproduce accurately the evolution of daily death
toll in both places. The validated model is then used to assess the impact of
different scenarios of medical deployment (including deployment time and
resource investment) on the number of deaths. In general, the ICU resources
have a significant impact on the mortality rate, and it is intuitively evident
that deploying the ICU resources earlier and/or augmenting them will reduce
the death toll. A virtue of our model is that it enables an accurate
quantification of such intuitive expectations. For example, we find that, if
the deployment of ICU resources had been made one week earlier or if they had
been doubled, the death toll in Wuhan would have been reduced by 5% or 13%,
and that in Lombardy by 3% and 14%, respectively. If the number of ICU beds
had been tripled, the death toll in both Wuhan and Lombardy would have been
reduced by 21%.

Our model enables an accurate prediction of the COVID-19 death toll for any
combination of the two parameters: the amount of advance or delay in the
timing of ICU deployment and the enhancement factor of the ICU resources. Our
model asserts that simultaneously advancing the timing of ICU resource
deployment and enhancing the ICU resources will reduce the death toll
significantly. For example, if the ICU resources had been augmented by a
factor of 2.5 and deployed two weeks earlier than the actual date, the final
death toll would have been reduced by 29% in Wuhan. In Lombardy, if the number
of ICU beds had been three times the actual number and they had been placed
into service one week earlier, the final death toll would have been reduced by
22% and the ICU ward overload days would have been suppressed to less than one
week.

Likewise, our model is fully capable of assessing the impacts of GW resources
on COVID-19 mortalities. For example, if the GW resources in Wuhan had been
deployed one week in advance or if the GW resources had been doubled, the
death toll would have been reduced by 10% comparing with the actual toll.
However, the model predicts that deploying GW resources earlier than one week
or augmenting them further would have little effect on suppressing the death
toll further. For Lombardy, the GW resources play an even less role in
reducing the death toll: augmenting them or deploying them earlier would
reduce the death toll by about 1% at maximum. This indicates that the GW
resources in Lombardy were sufficient. This should be contrasted with the
scenario of significantly enhancing the ICU resources, e.g., by a factor of
four, where the death toll would be reduced by 26% and 22% for Wuhan and
Lombardy, respectively. The implication is that, in comparison with the GW
resources, the actual ICU resources in Wuhan and Lombardy were far from
sufficient.

The morbidity and mortality of patients infected with COVID-19 increase with
age. We have studied the role of age selective admission and treatment
strategies in the death toll. In general, giving admission priorities to
patients in different age groups would result in different mortality rate. In
particular, if priorities had been given to the younger age groups, the number
of deaths in Wuhan and Lombardy would have been reduced by 10.4% and 6.7%,
respectively, in comparison with that from the normal FCFS strategy. In
contrast, if priority had been given to elderly patients, the number of deaths
would have increased by 11.5% and 5.7% for the two places, respectively.

The optimal admission/treatment strategies also depend on the scale of
COVID-19 outbreak, the age structure of patients in the general population,
and the per capita medical resources. We have quantified these effects by
defining the steady-state optimization efficiency $E(\infty)$ and calculated
this quantity for four countries: China, South Korea, Italy, and Spain. One
finding is that, when the number of patients exceeds a threshold value, the
values of $E(\infty)$ for the four countries first rise rapidly with the
increase of the epidemic scale and then reaches a peak value. Remarkably,
$E(\infty)$ can be maintained at a high value in a wide range of the epidemic
scale. The peak values of Italy and Spain are both lower than 1.2, and the
rising rate is slower. However, the peak values of China and South Korea are
both close to 1.3. This discrepancy between the two eastern Asian and two
western countries can be explained by the age distribution of the patients
diagnosed with COVID-19: in China and South Korea, less than 12% of the
diagnosed patients are over 70 years old, while those in Italy and Spain
account for about 35%. The per capita medical resources of South Korea, Italy
and Spain can meet the long term demand of 1000 cases per tens of million
people per day, while the caring capacity of China may be less than half of
that of the these countries. This is due to the small number of ICU beds per
capita in China, which is 3.6 per 100000 persons, compared with 12.5, 10.6,
and 9.7 in Italy, South Korea, and Spain, respectively. We conclude that the
age-selective admission strategy is more effective in countries with a younger
age structure, while the countries with less per capita ICU medical resources
may need to implement this admission strategy in the early stage of the
epidemic when the number of patients is relatively small.

Our model provides a general evaluation framework for assessing, at a
quantitative level, the necessary medical resource deployment and admission
strategy. It can be used to predict and articulate, under limited medical
resources, optimal scenarios with respect to resource deployment and hospital
admission/treatment strategies to minimize the death toll for future outbreaks
of infectious diseases.

## Data Availability

All relevant data are available from the authors upon request.

## Code Availability

All relevant computer codes are available from the authors upon request.

## Acknowledgments

We thank Zhai Zhengmeng and Long Yongshang for correcting the procedure, Kang
Jie for collecting data, and Lin Zhaohua and Han Lilei for discussing the
experimental design. This work was supported by the National Natural Science
Foundation of China (Grant Nos. 11975099, 11675056, 61802321 and 11835003),
the Natural Science Foundation of Shanghai (Grant No. 18ZR1412200), and the
Science and Technology Commission of Shanghai Municipality (Grant No.
14DZ2260800). YCL would like to acknowledge support from the Vannevar Bush
Faculty Fellowship program sponsored by the Basic Research Office of the
Assistant Secretary of Defense for Research and Engineering and funded by the
Office of Naval Research through Grant No. N00014-16-1-2828.

## Author Contributions

M.T. designed research; Y.-Q.Z., L.Z. performed research; Y.-Q.Z., L.Z., M.T.,
Y.L., Z.L., and Y.-C.L. analyzed data; Y.-Q.Z., L.Z., M.T., Z.L., and Y.-C.L.
wrote the paper.

## Competing Interests

The authors declare no competing interests.

## Correspondence

To whom correspondence should be addressed. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Supplementary Information for

Quantitative assessment of the effects of resource optimization and ICU
admission policy on COVID-19 mortalities

Ying-Qi Zeng, Lang Zeng, Ming Tang∗, Ying Liu, Zong-Hua Liu, and Ying-Cheng
Lai∗

## Supplementary Note 1: Equations governing non-Markovian dynamical evolution
of states

Because of the non-Markovian nature of the dynamical evolution of the states,
it is convenient to describe the underlying process in terms of difference
equations over infinitesimal intervals in both time delay and time. In the
following, we derive, one by one, the difference equations for all the ten
states in our model.

### M state

The evolution equation of the number of M-state patients with state age $\tau$
at time $t$ is

$\displaystyle M(\tau+d\tau;t+dt)$ $\displaystyle=[1-{\omega_{M\rightarrow
F}}(\tau)d\tau]\frac{{{P_{M\rightarrow
F}}\int_{\tau}^{+\infty}{{f_{M\rightarrow
F}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}{{{P_{M\rightarrow
F}}\int_{\tau}^{+\infty}{{f_{M\rightarrow
F}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}+(1-{P_{M\rightarrow
F}})\int_{\tau}^{+\infty}{{f_{M\rightarrow
R}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}M(\tau;t)$ (3)
$\displaystyle+[1-{\omega_{M\rightarrow
R}}(\tau)d\tau]\frac{{(1-{P_{M\rightarrow
F}})\int_{\tau}^{+\infty}{{f_{M\rightarrow
R}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}{{{P_{M\rightarrow
F}}\int_{\tau}^{+\infty}{{f_{M\rightarrow
F}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}+(1-{P_{M\rightarrow
F}})\int_{\tau}^{+\infty}{{f_{M\rightarrow
R}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}M(\tau;t),$

where $M(\tau;t)$ is the density function of M-state patients with state age
$\tau$ at time $t$, so the number of nodes in the M state with the state age
in the interval $(\tau,\tau+d\tau)$ is $M(\tau;t)d\tau$. A node in the M state
enters the F state (R state) at the conditional rate $\omega_{M\to F}(\tau)$
[$\omega_{M\to R}(\tau)$], which are given by

$\displaystyle\omega_{M\to F}(\tau)$ $\displaystyle=$
$\displaystyle\frac{f_{M\to F}(\tau)}{\int_{\tau}^{+\infty}f_{M\to
F}(\tau^{\prime})d\tau^{\prime}},$ $\displaystyle\omega_{M\to R}(\tau)$
$\displaystyle=$ $\displaystyle\frac{f_{M\to
R}(\tau)}{\int_{\tau}^{+\infty}f_{M\to R}(\tau^{\prime})d\tau^{\prime}},$

where $f_{M\rightarrow F}(\tau)$ and $f_{M\rightarrow R}(\tau)$ are the
probability density functions of the time delay from the M state to the F and
R states, respectively, each representing the probability of a state
transition within the time interval $(t,t+dt)$. Since the M state can
transition to two different states: $M\rightarrow F$ and $M\rightarrow R$, the
two processes will compete with each other, so the remaining probabilities of
the state transitions at time $t$ are

$\displaystyle\frac{{{P_{M\rightarrow
F}}\int_{\tau}^{+\infty}{{f_{M\rightarrow
F}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}{{{P_{M\rightarrow
F}}\int_{\tau}^{+\infty}{{f_{M\rightarrow
F}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}+(1-{P_{M\rightarrow
F}})\int_{\tau}^{+\infty}{{f_{M\rightarrow
R}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}},$ (4)

and

$\displaystyle\frac{{(1-{P_{M\rightarrow
F}})\int_{\tau}^{+\infty}{{f_{M\rightarrow
R}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}{{{P_{M\rightarrow
F}}\int_{\tau}^{+\infty}{{f_{M\rightarrow
F}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}+(1-{P_{M\rightarrow
F}})\int_{\tau}^{+\infty}{{f_{M\rightarrow
R}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}},$ (5)

respectively.

The newly increased patients enter the M state. Suppose that $Z_{M}(t)$ new
patients are added within $(t,t+dt)$, we have

$\displaystyle M(0;t+dt)=Z_{M}(t).$ (6)

The state age of the newly added M-state patients is zero.

### F state

The evolution equation of the number of F-state patients with state age $\tau$
at time $t$ is

$\displaystyle F(\tau+d\tau;t+dt)$ $\displaystyle=[1-{\omega_{F\rightarrow
C}}(\tau)d\tau]\frac{{{P_{F\rightarrow
C}}\int_{\tau}^{+\infty}{{f_{F\rightarrow
C}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}{{{P_{F\rightarrow
C}}\int_{\tau}^{+\infty}{{f_{F\rightarrow
C}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}+(1-{P_{F\rightarrow
C}})\int_{\tau}^{+\infty}{{f_{F\rightarrow
M_{X}}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}F(\tau;t)$ (7)
$\displaystyle+[1-{\omega_{F\rightarrow
M_{X}}}(\tau)d\tau]\frac{{{(1-P_{F\rightarrow
C})}\int_{\tau}^{+\infty}{{f_{F\rightarrow
M_{X}}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}{{{P_{F\rightarrow
C}}\int_{\tau}^{+\infty}{{f_{F\rightarrow
C}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}+(1-{P_{F\rightarrow
C}})\int_{\tau}^{+\infty}{{f_{F\rightarrow
M_{X}}}(\tau^{{}^{\prime}})}d\tau^{{}^{\prime}}}}F(\tau;t),$

with notations similar in their meanings to those of the M-state equation.

After admission of patients in the C state, the remaining available GW
resources are $\Delta Q_{G}^{{}^{\prime}}(t)$. There are two cases: (i) the
new resources available are sufficient to accommodate all F-state patients;
(ii) new available resources can only accept some or none of the F-state
patients. In these two cases, patients in the F state are admitted according
to the strategy of FCFS (first-come first-serve). The number of F-state
patients changing to the G state in $(t,t+dt)$ is denoted as
$\bigtriangleup_{F\rightarrow G}(t)$.
In the first case where the GW resources are sufficient to accommodate all
current F-states patients, if

$\Delta
Q_{G}^{{}^{\prime}}(t)(t)\geq\int_{0}^{+\infty}F(\tau+d\tau;t+dt)d\tau,$

then all F-state individuals will change to the G state. We have
$F(\tau+d\tau;t+dt)=0$ for all $\tau$ $\in$ [0,$+\infty$), and

$\bigtriangleup_{F\rightarrow
G}(t)=\int_{0}^{+\infty}F(\tau+d\tau;t+dt)d\tau.$

In the second case where the GW resources can hold only some or none of the
current F-state patients, if

$\Delta
Q_{G}^{{}^{\prime}}(t)(t)=\int_{\tau_{c}}^{+\infty}F(\tau+d\tau,t+dt)d\tau,$

then some F-state individuals will change to the G state. We have
$F(\tau+d\tau;t+dt)=0$ for $\tau$ $\in$ [$\tau_{c}$,$+\infty$), and

$\bigtriangleup_{F\rightarrow G}(t)=\Delta Q_{G}^{{}^{\prime}}(t)(t).$

When the patients in the M state enter into the F state, the initial state age
is set to be zero. The number of newly added F-state patients is given by

$\displaystyle F(0;t+dt)=\int_{0}^{+\infty}\omega_{M\to F}(\tau)\frac{P_{M\to
F}\int_{\tau}^{+\infty}f_{M\to F}(\tau^{\prime})d\tau^{\prime}}{P_{M\to
F}\int_{\tau}^{+\infty}f_{M\to F}(\tau^{\prime})d\tau^{\prime}+(1-P_{M\to
F})\int_{\tau}^{+\infty}f_{M\to
R}(\tau^{\prime})d\tau^{\prime}}M(\tau;t)d\tau.$ (8)

### C state

The evolution equation of the number of C-state patients with state age $\tau$
at time $t$ is

$\displaystyle C(\tau+d\tau;t+dt)=[1-\omega_{C\to D}(\tau)d\tau]C(\tau;t).$
(9)

To get the number of C-state patients admitted to the hospital, we note that
there are two admission paths: (a) through ICU admission II to the U state;
(b) through GW admission I to the W state.

ICU admission II path. After admission of W-state patients, the remaining
available ICU resources are $\Delta Q_{U}^{\prime}(t)$. There are two cases:
(i) the newly available resources are sufficient to accommodate all the
C-states patients and (ii) the newly available resources are able to
accommodate some or none of the C-state patients. The number of C-state
patients who switch to the U state in $(t,t+dt)$ time interval is denoted as
$\bigtriangleup_{C\rightarrow U}(t)$.

In case (i), if

$\Delta Q_{U}^{\prime}(t)\geq\int_{0}^{+\infty}C(\tau+d\tau;t+dt)d\tau,$

all C-state patients will change to the U state. We have
$C(\tau+d\tau;t+dt)=0$ for $\tau\in$ [0,$+\infty$), and

$\bigtriangleup_{C\rightarrow
U}(t)=\int_{0}^{+\infty}C(\tau+d\tau;t+dt)d\tau.$

In case (ii), if

$\Delta Q_{U}^{\prime}(t)=\int_{\tau_{c}}^{+\infty}C(\tau+d\tau;t+dt)d\tau,$

and $0\leq\tau_{c}\leq+\infty$, then the C-state individuals whose state ages
are greater than $\tau_{c}$ will enter the U state. We have
$C(\tau+d\tau,t+dt)=0$ for $\tau\in$ [$\tau_{c}$,$+\infty$) and

$\bigtriangleup_{C\rightarrow U}(t)=\Delta Q_{U}^{\prime}(t).$

The remaining C-state patients can switch to the W state through the GW
admission I pathway.

GW admission I path. Considering that the remaining C-state patients are in
critical conditions, when the ICU resources are unavailable to them, they will
be admitted to GW and change into the W state with the highest priority. The
available resources within the time interval $(t,t+dt)$ are denoted as $\Delta
Q_{G}(t)$. There are two cases: (i) the available GW resources are enough to
accommodate all the remaining C-state patients, and (ii) the available GW
resources can accommodate some or none of the remaining C-state patients. The
number of patients in the C state who change to the W state in $(t,t+dt)$ is
denoted as $\bigtriangleup_{C\rightarrow W}(t)$.

In case (i), we have

$\Delta Q_{G}(t)\geq\int_{0}^{+\infty}C(\tau+d\tau;t+dt)d\tau.$

The amount of the remaining available resources through F-state patients for
GW admission II is

$\Delta Q^{\prime}_{G}(t)=\Delta
Q_{G}(t)-\int_{0}^{+\infty}C(\tau+d\tau;t+dt)d\tau.$

Individuals in the C state will change to the W state, so we have
$C(\tau+d\tau;t+dt)=0$ for $\tau\in$ [0,$+\infty$), and

$\bigtriangleup_{C\rightarrow
W}(t)=\int_{0}^{+\infty}C(\tau+d\tau;t+dt)d\tau.$

In case (ii), we have

$\Delta Q_{G}(t)=\int_{\tau_{c}}^{+\infty}C(\tau+d\tau;t+dt)d\tau.$

The amount of the remaining available resources for GW admission II is $\Delta
Q^{\prime}_{G}(t)=0$. The C-state individuals with state ages greater than
$\tau_{c}$ will enter the W state. We have $C(\tau+d\tau,t+dt)=0$ for
$\tau\in$ [$\tau_{c}$,$+\infty$) and

$\bigtriangleup_{C\rightarrow W}(t)=\Delta Q_{G}(t).$

When the F-state patients enter the C state, the initial state age is set to
be zero. The number of newly added C-state patients is

$\displaystyle C(0;t+dt)=\int_{0}^{+\infty}\omega_{F\to C}(\tau)\frac{P_{F\to
C}\int_{\tau}^{+\infty}f_{F\to C}(\tau)^{\prime}d\tau^{\prime}}{P_{F\to
C}\int_{\tau}^{+\infty}f_{F\to C}(\tau)^{\prime}d\tau^{\prime}+[1-P_{F\to
C}]\int_{\tau}^{+\infty}f_{F\to
R}(\tau)^{\prime}d\tau^{\prime}}F(\tau;t)d\tau.$ (10)

###  0.1 G state

The evolution equation of the number of G-state patients with state age $\tau$
at time $t$ is

$\displaystyle G(\tau+d\tau;t+dt)=[1-\omega_{G\to W}(\tau)d\tau]\frac{P_{G\to
W}\int_{\tau}^{+\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}}{P_{G\to
W}\int_{\tau}^{+\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}+(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to M_{X}}(\tau^{\prime})d\tau^{\prime}}G(\tau;t)$
(11) $\displaystyle+[1-\omega_{G\to M_{X}}(\tau)d\tau]\frac{(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to M_{X}}(\tau^{\prime})d\tau^{\prime}}{P_{G\to
W}\int_{\tau}^{+\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}+(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to
M_{X}}(\tau^{\prime})d\tau^{\prime}}G(\tau;t).$

The number of newly added G-state patients is

$\displaystyle G(0;t+dt)=\bigtriangleup_{F\rightarrow G}(t).$ (12)

###  0.2 W state

The evolution equation of the number of W-state patients with state age $\tau$
at time $t$ is

$\displaystyle W(\tau+d\tau;t+dt)=(1-\omega_{W\to D}(\tau)d\tau)W(\tau;t).$
(13)

To have the number of W-state patients admitted to the hospital, we denote the
available ICU resources at time $t$ as $\Delta Q_{U}(t)$. There are two cases:
(i) there are sufficient resources to accommodate all current W-state
patients, and (ii) the available resources can accommodate some or none of the
W-state patients. The number of W-state patients who change to the U state in
$(t,t+dt)$ is $\bigtriangleup_{W\rightarrow U}(t)$.

In case (i), if

$\Delta Q_{U}(t)\geq\int_{0}^{+\infty}W(\tau+d\tau;t+dt)d\tau,$

the amount of the remaining available resources is

$\Delta Q^{\prime}_{U}(t)=\Delta
Q_{U}(t)-\int_{0}^{+\infty}W(\tau+d\tau;t+dt)d\tau.$

We have

$\displaystyle W(\tau+d\tau;t+dt)=0\ for\
\tau\in[0,+\infty),\bigtriangleup_{W\rightarrow
U}(t)=\int_{0}^{+\infty}W(\tau+d\tau;t+dt)d\tau.$ (14)

In case (ii), if

$\Delta Q_{U}(t)=\int_{\tau_{c}}^{+\infty}W(\tau+d\tau,t+dt)d\tau$

and $0\leq\tau_{c}<+\infty$, the amount of the remaining available resources
is $\Delta Q^{\prime}_{U}(t)=0$. We get

$\displaystyle W(\tau+d\tau;t+dt)=0\ for\
\tau\in[\tau_{c},+\infty),\bigtriangleup_{W\rightarrow U}(t)=\Delta Q_{U}(t).$
(15)

The sources of the W-state patients are the G-state patients whose conditions
have deteriorated and the C-state patients admitted through GW admission I.
The number of newly added W-state patients is

$\displaystyle W(0;t+dt)=\bigtriangleup_{C\rightarrow
W}(t)+\int_{0}^{+\infty}\omega_{G\to W}(\tau)\frac{P_{G\to
W}\int_{\tau}^{\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}}{P_{G\to
W}\int_{\tau}^{\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}+(1-P_{G\to
W})\int_{\tau}^{\infty}f_{G\to
M_{X}}(\tau^{\prime})d\tau^{\prime}}G(\tau;t)d\tau.$ (16)

### U state

The evolution equation of the number of U-state patients with state age $\tau$
at time $t$ is

$\displaystyle U(\tau+d\tau;t+dt)=[1-\omega_{U\to D}(\tau)d\tau]\frac{P_{U\to
D}\int_{\tau}^{+\infty}f_{U\to D}(\tau^{\prime})d\tau^{\prime}}{P_{U\to
D}\int_{\tau}^{+\infty}f_{U\to D}(\tau^{\prime})d\tau^{\prime}+(1-P_{U\to
D})\int_{\tau}^{+\infty}f_{U\to G_{U}}(\tau^{\prime})d\tau^{\prime}}U(\tau;t)$
(17) $\displaystyle+[1-\omega_{U\to G_{U}}(\tau)d\tau]\frac{(1-P_{U\to
D})\int_{\tau}^{+\infty}f_{U\to G_{U}}(\tau^{\prime})d\tau^{\prime}}{P_{U\to
D}\int_{\tau}^{+\infty}f_{U\to D}(\tau^{\prime})d\tau^{\prime}+(1-P_{U\to
D})\int_{\tau}^{+\infty}f_{U\to
G_{U}}(\tau^{\prime})d\tau^{\prime}}U(\tau;t).$

Patients in the U state are from the W state and the admitted C-state
patients. The number of newly added U-state patients is

$\displaystyle U(0;t+dt)=\bigtriangleup_{W\rightarrow
U}(t)+\bigtriangleup_{C\rightarrow U}(t).$ (18)

###  $G_{U}$ state

The evolution equation of the number of $G_{U}$-state patients with state age
$\tau$ at time $t$ is

$\displaystyle G_{U}(\tau+d\tau;t+dt)=[1-\omega_{G_{U}\to
M_{X}}(\tau)d\tau]G_{U}(\tau;t).$ (19)

The number of newly added $G_{U}$-state patients is

$\displaystyle G_{U}(0;t+dt)=\int_{0}^{+\infty}\frac{(1-P_{U\to
D})\omega_{U\rightarrow G_{U}}(\tau)\int_{\tau}^{+\infty}f_{U\to
G_{U}}(\tau^{\prime})d\tau^{\prime}}{P_{U\to D}\int_{\tau}^{+\infty}f_{U\to
D}(\tau^{\prime})d\tau^{\prime}+(1-P_{U\to D})\int_{\tau}^{+\infty}f_{U\to
G_{U}}(\tau^{\prime})d\tau^{\prime}}U(\tau,t)d\tau.$ (20)

###  $M_{X}$ state

The evolution equation of the number of $M_{X}$-state patients with state age
$\tau$ at time $t$ is

$\displaystyle M_{X}(\tau+d\tau;t+dt)=[1-\omega_{M_{X}\to
R}(\tau)d\tau]M_{X}(\tau;t).$ (21)

The sources of $M_{X}$ state patients are three: patients from the F, G, and
$G_{U}$ states. The number of newly added $M_{X}$-state patients is given by

$\displaystyle M_{X}(0,t+dt)$
$\displaystyle=\int_{0}^{+\infty}\omega_{G_{U}\to
M_{X}}(\tau)G_{U}(\tau;t)d\tau$ (22)
$\displaystyle+\int_{0}^{+\infty}\frac{(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to M_{X}}(\tau^{\prime})d\tau^{\prime}}{P_{G\to
W}\int_{\tau}^{+\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}+(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to
M_{X}}(\tau^{\prime})d\tau^{\prime}}\omega_{G\to M_{X}}G(\tau;t)d\tau$
$\displaystyle+\int_{0}^{+\infty}\frac{(1-P_{F\to
C})\int_{\tau}^{+\infty}f_{F\to M_{X}}(\tau^{\prime})d\tau^{\prime}}{P_{F\to
C}\int_{\tau}^{+\infty}f_{F\to C}(\tau^{\prime})d\tau^{\prime}+(1-P_{F\to
C})\int_{\tau}^{+\infty}f_{F\to
M_{X}}(\tau^{\prime})d\tau^{\prime}}\omega_{F\to M_{X}}F(\tau;t)d\tau.$

### R state

The R-state patients come from the M and $M_{X}$ states. We have

$\displaystyle R(t+dt)$ $\displaystyle=R(t)+\int_{0}^{+\infty}\omega_{M_{X}\to
R}M_{X}(\tau;t)d\tau$ (23) $\displaystyle+\int_{0}^{+\infty}\frac{(1-P_{M\to
F})\int_{\tau}^{+\infty}f_{M\to R}(\tau^{\prime})d\tau^{\prime}}{P_{M\to
F}\int_{\tau}^{+\infty}f_{M\to F}(\tau^{\prime})d\tau^{\prime}+(1-P_{M\to
F})\int_{\tau}^{+\infty}f_{M\to R}(\tau^{\prime})d\tau^{\prime}}\omega_{M\to
R}M(\tau;t)d\tau.$

### D state

The D-state patients come from C, W, and U states. We have

$\displaystyle D(t+dt)$ $\displaystyle=D(t)+\int_{0}^{+\infty}\omega_{C\to
D}C(\tau;t)d\tau+\int_{0}^{+\infty}\omega_{W\to D}W(\tau;t)d\tau$ (24)
$\displaystyle+\int_{0}^{+\infty}\frac{P_{U\to D}\int_{\tau}^{+\infty}f_{U\to
D}(\tau^{\prime})d\tau^{\prime}}{P_{U\to D}\int_{\tau}^{+\infty}f_{U\to
D}(\tau^{\prime})d\tau^{\prime}+(1-P_{U\to D})\int_{\tau}^{+\infty}f_{U\to
G_{U}}(\tau^{\prime})d\tau^{\prime}}\omega_{U\to D}(\tau)U(\tau;t)d\tau.$

### Calculation of available ICU resources $\bigtriangleup Q_{U}(t)$

The available ICU resources are those that have been newly added, those that
are released when patients go from the U state to the D state or the
$G_{U}$-state patients, and the resources consumed by the W-state and C-state
patients. We have

$\displaystyle\Delta Q_{U}(t+dt)$ $\displaystyle=\Delta
Q_{U}(t)+Z_{U}(t)-\Delta_{W\rightarrow U}(t)-\Delta_{C\rightarrow U}(t)$ (25)
$\displaystyle+\int_{0}^{+\infty}\frac{P_{U\to D}\int_{\tau}^{+\infty}f_{U\to
D}(\tau^{\prime})d\tau^{\prime}}{P_{U\to D}\int_{\tau}^{+\infty}f_{U\to
D}(\tau^{\prime})d\tau^{\prime}+(1-P_{U\to D})\int_{\tau}^{+\infty}f_{U\to
G_{U}}(\tau^{\prime})d\tau^{\prime}}\omega_{U\to D}(\tau)U(\tau;t)d\tau$
$\displaystyle+\int_{0}^{+\infty}\frac{(1-P_{U\to
D})\int_{\tau}^{+\infty}f_{U\to G_{U}}(\tau^{\prime})d\tau^{\prime}}{P_{U\to
D}\int_{\tau}^{+\infty}f_{U\to D}(\tau^{\prime})d\tau^{\prime}+(1-P_{U\to
D})\int_{\tau}^{+\infty}f_{U\to
G_{U}}(\tau^{\prime})d\tau^{\prime}}\omega_{U\to G_{U}}(\tau)U(\tau;t)d\tau.$

### Calculation of available GW resources $\Delta Q_{G}(t)$

The available GW resources are those that have been newly added, those
released when patients switch from the G state to the $M_{x}$ state and from
the W state to the D state or the U state, and the resources consumed by the
C-state and F-state patients. We have

$\displaystyle\Delta Q_{G}(t+dt)$ $\displaystyle=\Delta
Q_{G}(t)+Z_{G}(t)-\Delta_{C\rightarrow W}(t)-\Delta_{F\rightarrow
G}(t)+\int_{0}^{+\infty}W(\tau;t)\omega_{W\to D}(\tau)d\tau$ (26)
$\displaystyle+\int_{0}^{+\infty}\frac{(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to M_{X}}(\tau^{\prime})d\tau^{\prime}}{P_{G\to
W}\int_{\tau}^{+\infty}f_{G\to W}(\tau^{\prime})d\tau^{\prime}+(1-P_{G\to
W})\int_{\tau}^{+\infty}f_{G\to
M_{X}}(\tau^{\prime})d\tau^{\prime}}\omega_{G\to M_{X}}(\tau)G(\tau;t)d\tau.$

## Supplementary Note 2: Parameter estimation

### Average delay and fraction of state transition

As shown in Tab. S1, we assume that the time delay of patients switching from
state M to F follows the normal distribution with the average of five days
[17]. The distributions of the time delays associated with other state
transitions are also assumed to be normal. The average time delays from the U
state to the D and $G_{u}$ states are set as seven days [18, 19] and eight
days [18], respectively, and that from the G state to the W state is three
days [17]. In our model setting, the clinical symptoms of the F-state and
G-state patients are identical, and the difference lies in whether the
patients are admitted (i.e., occupying GW beds), and the same rule applies to
the C-state and W-state patients.

The average time delay from the F to the C state is two days [17, 27], and
that from the G and F states to the $M_{X}$ state is eight days [12]. The
average time delays from the W and C states to the D state are assumed to be
three days and one day, respectively.

As shown in Tab. S2, we set the average mortality rate of ICU patients as
61.5% [18, 28]. During the epidemic, due to the different testing methods and
conditions, the fraction of M-state patients in the F state is different in
different time periods. According to the analysis of 32583 cases of
laboratory-confirmed patients in Wuhan and reconstruction of the epidemic
trend [29, 4], we divide the epidemic process in terms of the fraction of the
transition from the M to the F state into five stages. As shown in Table S3,
the time periods and the transition fractions of the five stages are [29]:
from 8 December 2019 to 10 January 2020 with the fraction $|P|_{M\rightarrow
F}(1)=53.10\%$; from January 11 to January 22 with $|P|_{M\rightarrow
F}(2)=35.10\%$; from January 23 to February 1 with $|P|_{M\rightarrow
F}(3)=23.50\%$; from February 2 to February 16 with $|P|_{M\rightarrow
F}(4)=15.90\%$; and after February 17 with $|P|_{M\rightarrow F}(5)=10.30\%$
[29].

For Lombardy, the early epidemic can be divided into three stages [30]. We add
a new time point: March 21, 2020, leading to a division into four stages.
After (including) this date, the intervention measures in Lombardy and Italy
as a whole reached maximum [31]. As shown in Table S4, the time periods and
the fractions associated with the four stages are: from 28 January 2020 to 19
February 2020 with $|P|_{M\rightarrow F}(1)=63\%$; from February 20 to
February 25 with $|P|_{M\rightarrow F}(2)=61\%$; from February 26 to March 20
with $|P|_{M\rightarrow F}(3)=56\%$ [30]; after March 21 with
$|P|_{M\rightarrow F}(4)=25\%$.

By using the weighted least-squares method, we obtain the optimal estimates of
the parameters. In particular, for the Wuhan scenario, we obtain the optimal
set of parameters $\Xi=(|P|_{F\rightarrow C},|P|_{G\rightarrow W})$ by
minimizing the sum of the weighted difference squares between the reported
death curve [32, 23] $D(t_{i})$ and the model fitting value $F(t_{i},\Xi)$
$(i=0,1,\ldots,n-1)$, where

$\displaystyle\widehat{\Xi}=\mbox{argmin}\sum_{i}w_{t_{i}}[F(t_{i},\Xi)-D(t-i)]^{2}.$
(27)

To quantify the uncertainties of parameter estimation, we resort to the
general bootstrap method [33, 34] and then obtain the 95% confidence interval
of the estimated parameter values. As shown in Tab. S2, the optimal values and
95% confidence interval of the average transition fractions from G-state to
W-state and from F-state to C-state in Wuhan are $|P|_{F\rightarrow
C}^{*}=48.3\%(41.06\%,54.97\%)$ and $|P|_{G\rightarrow
W}^{*}=27.4\%(26.24\%,28.51\%)$, respectively.

For Lombardy, the required optimal parameters are $\Xi=(|P|_{M\rightarrow
F}(4),|P|_{F\rightarrow C},|P|_{G\rightarrow W})$. As shown in Tabs. S2 and
S4, the optimal values and the 95% confidence intervals of the average
transition fractions in Lombardy are $|P|_{M\rightarrow F}^{*}=25\%\
(24.29\%,25.65\%)$, $|P|_{F\rightarrow C}^{*}=88.86\%\ (76.8\%,100\%)$, and
$|P|_{G\rightarrow W}^{*}=63.77\%\ (62.9\%,64.71\%)$, respectively.

### Incidence dates estimated from the confirmed data

According to the average time delay from the onset date to the diagnosis date
of laboratory-confirmed cases in the literature [29, 30], we determine the
distributions of state transition delay from onset to diagnosis for Wuhan and
Lombardy using a backtracking method based on non-Markov processes that we
have developed, where the incidence dates are deduced from the confirmed
cases. We designate a new state, the J state, to denote that a patient has
been confirmed. The dynamic process of J state to M state is described by the
following difference equations:

$\displaystyle J(\tau+d\tau;t-dt)$ $\displaystyle=[1-\omega_{J\rightarrow
M}]J(\tau;t),$ (28) $\displaystyle Z_{M}(t-dt)$
$\displaystyle=\int_{0}^{+\infty}J(\tau;t)\omega_{J\rightarrow M}(\tau)d\tau
and$ $\displaystyle J(0;t-dt)$ $\displaystyle=Z_{J}(t),$

where $\omega_{J\rightarrow M}(\tau)$ is the conditional rate of transition
from J to M states with state age $\tau$, with the specific form

$\omega_{J\to M}(\tau)=\frac{f_{J\to M}(\tau)}{\int_{\tau}^{+\infty}f_{J\to
M}(\tau^{\prime})d\tau^{\prime}},$

with $f_{J\rightarrow M}(\tau)$ being the probability density function of the
transition time delay from J state to M state. For Wuhan and Lombardy, the
state transition delays are assumed to follow the Gamma distribution with the
mean value of 9.5 and 7.3 days, respectively. The quantities $Z_{M}(t)$ and
$Z_{J}(t)$ are the numbers of new patients in the M and J states in the time
interval $(t,t+dt)$, respectively, where the former is our estimated value and
the latter is the newly reported, daily confirmed data. We validate the
accuracy of this method using data from early laboratory-confirmed cases in
Lombardy (January 28-February 27, 2020) [35], as shown in Fig. S7.

There are 50,008 laboratory/clinically confirmed cases in Wuhan, and 93,901
such cases in Lombardy. We use the same backtracking method to trace the onset
time of the non-laboratory confirmed cases. The incidence dates of all
confirmed cases is April 15 for Wuhan and June 30 for Lombardy.

### Actual medical resource deployment

In response to an emerging public health event, the government usually adopts
the policy of gradual deployment of medical resources to designated hospitals.
The dedicated medical resources in an area typically slowly increase with
time.

As shown in Tabs. S5 and S6, we collect medical resource deployment data
dedicated to COVID-19 patients in Wuhan and Lombardy, which include GW and ICU
beds [32, 36]. For Wuhan, we count the number of open beds in critical
hospitals. As the official reports do not distinguish GW beds from ICU beds,
we assume that ICU beds account for 4% of the total beds [37].

## Supplementary Note 3: Parameters of medical resource

### Health care system stress metrics

In addition to the final number of deaths, the load of local special medical
resources for COVID-19 patients is also an important indicator to measure the
stress of the medical system. We simulate and obtain the full-load working
days of GW and ICU in Wuhan and Lombardy during the recovery stage, denoted as
$O_{G}$ and $O_{U}$, respectively. When the GW or ICU is under full load, no
new patients can be admitted.

In Fig. S8, the remaining ICU resources after ICU admission I/II at time $t$
is denoted as $\bigtriangleup Q_{U}(t)$, and the corresponding quantity for GW
is $\bigtriangleup Q_{G}(t)$, which are given by

$\displaystyle O_{U}$ $\displaystyle=\int_{T_{s}}^{T_{e}}\delta[\triangle
Q_{U}(t)]dt\ \ \mbox{and}$ (29) $\displaystyle O_{G}$
$\displaystyle=\int_{T_{s}}^{T_{e}}\delta[\triangle Q_{G}(t)]dt,$

where $\delta(*)$ is the Dirac-$\delta$ function, the integration interval is
the whole recovery stage, $T_{s}$ and $T_{e}$ represent the starting and
ending time of the recovery stage, respectively. For example, $T_{s}$ in Wuhan
is December 8, 2019 and $T_{e}$ is April 16, 2020. The time interval is set to
be $dt=0.01$ day.

### Deployment plan of dedicated medical resources for COVID-19

The deployment plan of local officially dedicated medical resources is
determined by two key factors: the deployment time DT and resource input RI,
where DT is the time for local authorities to start the deployment of the
dedicated medical resources and RI represents the financial and personnel
investment of the dedicated medical resources as characterized by the open GW
and ICU beds.

Figure S9 shows the effects of varying ICU deployment plan in Lombardy, where
the black curve represents Lombardy’s current dedicated ICU deployment plan.
The actual deployment time serves as the benchmark $\mbox{DT}=0$ and the
actual resource input is $\mbox{RI}=1$. The left blue arrow indicates the
scenario where the Lombard authorities had delayed the deployment time by two
weeks ($\mbox{DT}=14$ and $\mbox{RI}=1$), and the red up arrow corresponds to
the scenario where the official resource input had been increased by 50%
($\mbox{DT}=0$ and $\mbox{RI}=1.5$).

### Effects of varying medical resource deployment on death toll

We study the impacts of varying both GW and ICU resources on the number of
deaths. As shown in Tab S7 and Fig. S10, if resources had been deployed one
week in advance, the death toll in Wuhan and Lombardy would have been reduced
by 14% and 3.4%, respectively. If the input of resources had been doubled, the
death toll would have been suppressed by 22% and 15%, respectively. If the
deployment had been one week ahead of the actual time and the input of
resources had been doubled, the death toll would have been reduced by 30% and
17% in Wuhan and Lombardy, respectively, indicating that varying the two
factors simultaneously can be more effective at reducing the death toll.
Figure S11 demonstrates the change in the ward overload days when GW
deployment is modified while ICU deployment is unchanged. Figure S12 displays
the change in the ward overload days when GW deployment is unchanged but the
ICU deployment plan is modified.

## Supplementary Note 4: Estimation of state transition fractions for
different age groups

In the main text, we divide the patients in Wuhan and Lombardy into three age
groups: [0-69], [70-79] and [80+], at intervals of 10%. The transition
fractions from M state to F state, from G state to W state, from F state to C
state and the ICU mortality rates are different for different age groups. We
articulate a linear regression method to estimate the state transition
fractions for different age groups. Take as an example the transition from M
state to F state. We first introduce the matrix of average transition
fraction, as (detailed in SN2)

$\displaystyle|P|_{M\rightarrow
F}=\left(\begin{array}[]{ccc}|p|_{1}^{M\rightarrow F}\\\ \vdots\\\
|p|_{1}^{M\rightarrow F}\\\ \end{array}\right),$ (30)

where $|p|_{i}^{M\rightarrow F}$ represents the average transition fraction
from M state to F state at stage $i$. The age distribution of patients in the
M state at different stages is

$\displaystyle
D_{M}=\left(\begin{array}[]{ccc}d_{1,1}^{M}&\cdots&d_{1,y}^{M}\\\
\vdots&\ddots&\vdots\\\ d_{s,1}^{M}&\cdots&d_{s,y}^{M}\\\ \end{array}\right),$
(31)

where $d_{i,j}^{M}$ is the fraction of patients in age group $j$ among all
patients at stage $i$. We then define the matrix of average transition
fraction from M state to F state in different age groups as

$\displaystyle R_{M\rightarrow F}=[r_{1}^{M\rightarrow F}\cdots
r_{y}^{M\rightarrow F}],$ (32)

where $r_{j}^{M\rightarrow F}$ is the average transition fraction from M state
to F state in the $j$th age group during the whole recovery period.

The parameters can be obtained empirically. We make a linear transformation of
matrix $R_{M\rightarrow F}$ to obtain the estimation of the fraction of the
M-state patients to F-state whose weighted average transition fraction is
$|P|_{M\rightarrow F}$. The linear transformation LT of different stages is
given by

$\displaystyle\mbox{LT}=\frac{|P|_{M\rightarrow F}}{D_{M}\cdot R_{M\rightarrow
F}^{T}},$ (33)

Finally, the transition fractions from M state to F state for different age
groups are

$\displaystyle P_{M\rightarrow F}=LT\cdot\mbox{ext}(1,y)\cdot R_{M\rightarrow
F}=\left(\begin{array}[]{ccc}p_{1,1}^{M\rightarrow
F}&\cdots&p_{1,y}^{M\rightarrow F}\\\ \vdots&\ddots&\vdots\\\
p_{s,1}^{M\rightarrow F}&\cdots&p_{s,y}^{M\rightarrow F}\\\
\end{array}\right),$ (34)

where the rows represent stages, the columns represent groups, and
$p_{i,j}^{M\rightarrow F}$ denotes the transition fraction from M-state to
F-state in the $j$th age group at stage $i$. $\mbox{ext}(n,m)$ is an auxiliary
matrix of $n$ rows and $m$ columns whose elements are all one. The patient age
distribution matrix $D_{F}$ is

$\displaystyle D_{F}=\left(\begin{array}[]{ccc}p_{1,1}^{M\rightarrow F}\cdot
d_{1,1}^{M}&\cdots&p_{1,y}^{M\rightarrow F}\cdot d_{1,y}^{M}\\\
\vdots&\ddots&\vdots\\\ p_{s,1}^{M\rightarrow F}\cdot
d_{s,1}^{M}&\cdots&p_{s,y}^{M\rightarrow F}\cdot d_{s,y}^{M}\\\
\end{array}\right).$ (35)

Where $R$ and $D_{M}$ and $\mid P\mid$ are known (see Supplementary Table S3
and Supplementary Table S8.). Similarly, we can obtain the estimation of the
transition fractions among other states for different age groups at different
stages. To simplify the model, we assume that the age distribution of M-state
patients at each stage is
identical($d_{1,j}^{M}=d_{2,j}^{M}=\ldots=d_{s,j}^{M},j=1,2,\ldots,y$), as
shown in Table S8. The reference transition fraction matrix among the states
is also given in Table S8. Finally, as shown in Fig. S13, we divide the
patient population into nine groups.

## Supplementary Note 5: A scheme to realize different admission strategies –
translation weighting

We have developed a translation weighting scheme to achieve different
admission strategies. We set the priority weight of the $i$th age group as
$w_{i}$ and denote $\tau_{i}$ as the state age of the $i$th age group. After
weighting, the new state age $\tau_{i}^{\prime}$ becomes,

$\displaystyle\tau_{i}^{\prime}=\tau_{i}+w_{i}.$ (36)

The state age of patients in the age group $i$ is increased by $w_{i}$ units
(days). The patients are then admitted according to the weighted state age
under FCFS.

We divide the patients into three age groups, which are patients younger than
70 years old, patients between 70 and 80 years old, and patients older than 80
years. Figure S14 presents examples of several weight combinations. For
example, in Fig. S14(b), the weights of the three age groups are $w_{1}=0$,
$w_{2}=30$ and $w_{3}=0$, respectively. The state ages of the three age groups
after weighting are also illustrated.

## Supplementary Note 6: Medical resources per capita and the age distribution
of confirmed cases in four countries

We collect data on ward resources per 100,000 people per capita in China,
South Korea, Italy and Spain, as well as the fractions of COVID-19 confirmed
cases in each age group, as shown in Tables S9 and S10.

## Supplementary Tables

Supplementary Table S1: State transition time delay parameter setting

Parameter(delay) | Estimate/assumption | Definition | Justification
---|---|---|---
$\mu_{M\rightarrow R}$ | Wuhan/Lombard | 14d | Average delay from M state to report recovery | [1]
$\mu_{M\rightarrow F}$ | Wuhan/Lombard | 5d(std=6.67) | Average delay from M state to F state | [17]
$\mu_{F\rightarrow M_{X}}$ | Wuhan/Lombard | 8d(std=4.4477) | Average delay from F state to $M_{X}$ state | [12]
$\mu_{F\rightarrow C}$ | Wuhan/Lombard | 2d(std=3.7064) | Average delay from F state to C state | [17, 27]
$\mu_{G\rightarrow M_{X}}$ | Wuhan/Lombard | 8d(std=4.4477) | Average delay from G state to $M_{X}$ state | [12]
$\mu_{G\rightarrow W}$ | Wuhan/Lombard | 3d(std=3.7064) | Average delay from G state to W state | [17]
$\mu_{U\rightarrow G_{U}}$ | Wuhan/Lombard | 8d(std=4.4477) | Average delay from U state to $G_{U}$ state | [18]
$\mu_{U\rightarrow D}$ | Wuhan/Lombard | 7d(std=5.93) | Average delay from U state to D state | [18, 19]
$\mu_{C\rightarrow D}$ | Wuhan/Lombard | 1d(std=2.2239) | Average delay from C state to D state | Assume
$\mu_{W\rightarrow D}$ | Wuhan/Lombard | 3d(std=3.7064) | Average delay from W state to D state | Assume
$\mu_{G_{U}\rightarrow M_{X}}$ | Wuhan/Lombard | 8d(std=4.4477) | Average delay from $G_{U}$ state to $M_{X}$ state | [12]
$\mu_{M_{X}\rightarrow R}$ | Wuhan/Lombard | 14d | Average delay from $M_{X}$ state to report recovery | [1]

Supplementary Table S2: Parameter setting of state transition fraction

Parameter(delay) | Estimate/assumption | Definition | Justification
---|---|---|---
$P_{M\rightarrow F}$ | Wuhan/Lombard | see Tab S3.1 and S3.2 | Transition fraction from M state to F state |
$P_{F\rightarrow C}$ | Wuhan | 48.3% (41.06%, 54.97%) | Transition fraction from F state to C state | Fitting according to the reported death data
Lombard | 88.86% (76.8%, 100%)
$P_{G\rightarrow W}$ | Wuhan | 27.4% (26.24%, 28.51%) | Transition fraction from G state to W state
Lombard | 63.77% (62.9%, 64.71%)
$P_{U\rightarrow D}$ | Wuhan/Lombard | 61.5% | Transition fraction from U state to D state | [18, 28]

Supplementary Table S3: Setting of transition fraction from M state to F state
in five stages in Wuhan

Date | Dec 8, 2019-Jan 9, 2020 | Jan 10-Jan 22 | Jan 23-Feb 1 | Feb 2-Feb 16 | Feb 17–
---|---|---|---|---|---
$P_{M\rightarrow F}(Wuhan)$ | 53.10% | 35.10% | 23.50% | 15.90% | 10.30%
Justification | [29]

Supplementary Table S4: Setting of transition fraction from M state to F state
in four stages in Lombardy

Date | Jan 28, 2020-Feb 19, 2020 | Feb 20-Feb 25 | Feb 26-Mar 20 1 | Mar 21–
---|---|---|---|---
$P_{M\rightarrow F}(Lombardy)$ | 63.00% | 61.00% | 56.00% | 25%(24.29%, 25.65%)
Justification | [30] | Fitting according to the reported death data

Supplementary Table S5: Actual medical resource deployment plan for Wuhan,
China [32]

Date | Beds(ICU+General ward)
---|---
2020/2/1 | 842
2020/2/2 | 842
2020/2/3 | 1962
2020/2/4 | 2234
2020/2/5 | 2264
2020/2/6 | 2515
2020/2/7 | 2657
2020/2/8 | 2911
2020/2/9 | 3263
2020/2/10 | 4307
2020/2/11 | 4461
2020/2/12 | 5873
2020/2/13 | 6078
2020/2/14 | 6636
2020/2/15 | 6926
2020/2/16 | 7035
2020/2/17 | 7067
2020/2/18 | 7225
2020/2/19 | 7296
2020/2/20 | 7560
2020/2/21 | 7560
2020/2/22 | 7844
2020/2/23 | 7844
2020/2/24 | 7936
2020/2/25 | 8194

Supplementary Table S6: Actual medical resource deployment in Lombardy, Italy
[36]

Date | Beds(ICU) | Beds(General ward)
---|---|---
2020/2/24 | 19 | 76
2020/2/25 | 25 | 79
2020/2/26 | 25 | 79
2020/2/27 | 41 | 172
2020/2/28 | 47 | 235
2020/2/29 | 80 | 256
2020/3/1 | 106 | 406
2020/3/2 | 127 | 478
2020/3/3 | 167 | 698
2020/3/4 | 209 | 877
2020/3/5 | 244 | 1169
2020/3/6 | 309 | 1622
2020/3/7 | 359 | 1661
2020/3/8 | 399 | 2217
2020/3/9 | 440 | 2802
2020/3/10 | 466 | 3319
2020/3/11 | 560 | 3852
2020/3/12 | 605 | 4247
2020/3/13 | 650 | 4435
2020/3/14 | 732 | 4898
2020/3/15 | 767 | 5500
2020/3/16 | 823 | 6171
2020/3/17 | 879 | 6953
2020/3/18 | 924 | 7285
2020/3/19 | 1006 | 7387
2020/3/20 | 1050 | 7735
2020/3/21 | 1093 | 8258
2020/3/22 | 1142 | 9439
2020/3/23 | 1183 | 9266
2020/3/24 | 1194 | 9711
2020/3/25 | 1236 | 10026
2020/3/26 | 1263 | 10681
2020/3/27 | 1292 | 11137
2020/3/28 | 1319 | 11152
2020/3/29 | 1328 | 11613
2020/3/30 | 1330 | 11815
2020/3/31 | 1334 | 11883
2020/4/1 | 1342 | 12009
2020/4/2 | 1351 | 12009
2020/4/3 | 1381 | 12009

Supplementary Table S7: Comparison of different medical resource deployment
plans (simultaneously adjusting GW and ICU resources)

| Simulated death toll
---|---
Wuhan | Lombard
The actual deployment | 2553 | 16363
Sufficient resources | Decrease by 33% | Decrease by 22%
Deploy 7 days in advance | Decrease by 14% | Decrease by 3.4%
Deploy 14 days in advance | Decrease by 18% | Decrease by 4%
7 days delay in deployment | Increase by 36% | Increase by 14%
14 days delay in deployment | Increase by 57% | Increase by 28%
Resource input *0.5 | Increase by 43% | Increase by 26%
Resource input *2 | Decrease by 22% | Decrease by 15%
Resource input *5 | Decrease by 32% | Decrease by 22%
Deploy 7 days in advance and invest resources *2 | Decrease by 30% | Decrease by 17%

Supplementary Table S8: Reference transition fraction matrix among states and
age distribution of patients in M state

Parameter(Reference matrix) | Estimate/assumption | Definition | Justification
---|---|---|---
$R_{M\rightarrow F}$ | Wuhan | [0.0953 0.2430 0.2730] | Reference transformation matrix from M state to F state | [12]
Lombard | [0.0827 0.2430 0.2730]
$R_{F\rightarrow C/G\rightarrow W}$ | Wuhan | [0.0982 0.1419 0.2393] | Reference transformation matrix from F/G state to C/W state | Fitting according to the reported death data fitting
Lombard | [0.1249 0.2240 0.2393]
$R_{U\rightarrow D}$ | Wuhan/Lombard | [0.2297 0.7727 0.8571] | Reference transformation matrix from U state to D state | [19]
$D_{M}$ | Wuhan | [0.8240 0.1260 0.0500] | Age group distribution matrix of M-state patients | [21]
Lombard | [0.6050 0.1420 0.2530]

Supplementary Table S9:  Average wards per 100,000 people

Country/Resource | General ward beds per 100000 people | ICU beds per 100000 people
---|---|---
China | 400 | 3.6
Korea | 530 | 10.6
Italy | 333 | 12.5
Spain | 269 | 9.7
Justification | [38]

Supplementary Table S10:  Proportion of confirmed cases in each age group in
each country

Age distribution of confirmed cases
---
Groups/Country | China | Italy | Korea | Spain
0-9 | 0.009312 | 0.006245 | 0.012344 | 0.002639
10-19 | 0.01229 | 0.008122 | 0.052919 | 0.005433
20-29 | 0.081013 | 0.04061 | 0.272823 | 0.05018
30-39 | 0.170129 | 0.069164 | 0.106507 | 0.096366
40-49 | 0.191865 | 0.128128 | 0.133589 | 0.150725
50-59 | 0.224033 | 0.198045 | 0.183732 | 0.186374
60-69 | 0.192134 | 0.173838 | 0.126316 | 0.165787
70-79 | 0.087706 | 0.185173 | 0.066411 | 0.158476
80+ | 0.031519 | 0.188301 | 0.045359 | 0.18402
Justification | [39]

## Supplementary Figures

Supplementary Figure S7:  Validation of our backtracking method. The
validation method is based on the data of early laboratory-confirmed cases in
Lombardy area (January 28 - February 27, 2020). The incidence data of patients
are estimated from the laboratory reported confirmed cases and the
distribution of the time delay between the onset of the disease and the
reported confirmation. The blue dots and black crosses represent,
respectively, the incidence estimation curve and the retrospective survey data
of laboratory-confirmed cases in the literature [35]. The light blue shaded
region represents the 95% confidence interval of the estimation. Supplementary
Figure S8:  Remaining resources of GW and ICU in Wuhan after two admission
processes. The ordinate is on a logarithmic scale, and the red and blue dots
indicate that the amounts of remaining resources of Wuhan ICU and GW at time
$t$ are $\Delta Q_{U}(t)$ and $\Delta Q_{G}(t)$, respectively. The missing
areas of the red and blue dots represent the period of $\Delta Q_{U}=0$ and
$\Delta Q_{G}=0$, respectively. Supplementary Figure S9:  Effects of varying
ICU deployment plan in Lombardy, Italy. The black curve shows Lombardy’s
actual ICU medical resource deployment plan, as displayed in Tab. S4. The blue
and red curves show the two cases where Lombardy’s government would delay the
deployment of ICU resources by two weeks ($\mbox{DT}=14$ and $\mbox{RI}=1$)
and the ICU resource is expanded by a factor of 1.5 ($\mbox{DT}=0$ and
$\mbox{RI}=1.5$), respectively. Supplementary Figure S10:  Changes in the
number of deaths and overload days for simultaneous GW and ICU resource
deployment for Wuhan and Lombardy. (a,b) Numbers of deaths in Wuhan and
Lombardy, respectively. (c,d) GW ward overload days in Wuhan and Lombardy,
respectively. (e,f) ICU ward overload days in Wuhan and Lombardy,
respectively. Supplementary Figure S11:  Change in overload days for fixed ICU
resource deployment and varying GW resource deployment. (a,b) Overload days of
GW in Wuhan and Lombardy, respectively. (c,d) Overload days of ICU in Wuhan
and Lombardy, respectively. Supplementary Figure S12:  Change in overload days
for fixed GW resource deployment and varying ICU resource deployment. (a,b)
Overload days of GW in Wuhan and Lombardy, respectively. (c,d) Overload days
of ICU in Wuhan and Lombardy, respectively. Supplementary Figure S13:
Mortality rate of different age groups and evolution of the number of deaths.
(a) Actual data of mortality rate of different age groups in Wuhan versus
model prediction. (b) Change in the number of deaths by age groups in Wuhan.
(c) Actual mortality data for all age groups in Lombardy versus model
prediction. (d) Number of deaths in Lombardy in each age group. In (a) and
(c), the legends from top to bottom are age [0-9] to [80]. In (b) and (d), the
light blue and pink cylinders, respectively, represent the actual and
simulated mortalities in each group, where the $x$-axis marks the groups with
growing age. Supplementary Figure S14:  Weighted state age of each age group.
(a) Initial state age of the three groups of patients. (b) The state age of
the second group is translated by 30 units and fixed as a benchmark. (c) The
state age of the first group is translated by 44 units, and that of the third
group remains unchanged. (d) The state age of the third group is translated by
44 units, and that of the first group remains unchanged.

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