# Cost Effective Reproduction Number Based Strategies for Reducing Deaths from
COVID-19

Christopher THRON  Vianney MBAZUMUTIMA  LVTLuis V. TAMAYO  Léonard TODJIHOUNDE
Department of Sciences and Mathematics, Texas A $\&$ M University-Central
Texas, Killeen TX 76549 USA, Austin, USA  Institute of Mathematics and
Physical Sciences, IMSP-Bénin, Abomey Calavi University, Porto-Novo, Bénin
Department of Sciences and Mathematics, Texas A&M University-Central Texas,
Killeen TX 76549 USA, Austin, USA  Institute of Mathematics and Physical
Sciences, IMSP-Bénin, Abomey Calavi University, Porto-Novo, Bénin

###### Abstract

In epidemiology, the effective reproduction number $R_{e}$ is used to
characterize the growth rate of an epidemic outbreak. If $R_{e}>1$, the
epidemic worsens, and if $R_{e}<1$, then it subsides and eventually dies out.
In this paper, we investigate properties of $R_{e}$ for a modified SEIR model
of COVID-19 in the city of Houston, TX USA, in which the population is divided
into low-risk and high-risk subpopulations. The response of $R_{e}$ to two
types of control measures (testing and distancing) applied to the two
different subpopulations is characterized. A nonlinear cost model is used for
control measures, to include the effects of diminishing returns. Lowest-cost
control combinations for reducing instantaneous $R_{e}$ to a given value are
computed. We propose three types of heuristic strategies for mitigating
COVID-19 that are targeted at reducing $R_{e}$, and we exhibit the tradeoffs
between strategy implementation costs and number of deaths. We also consider
two variants of each type of strategy: basic strategies, which consider only
the effects of controls on $R_{e}$, without regard to subpopulation; and high-
risk prioritizing strategies, which maximize control of the high-risk
subpopulation. Results showed that of the three heuristic strategy types, the
most cost-effective involved setting a target value for $R_{e}$ and applying
sufficient controls to attain that target value. This heuristic led to
strategies that begin with strict distancing of the entire population, later
followed by increased testing. Strategies that maximize control on high-risk
individuals were less cost-effective than basic strategies that emphasize
reduction of the rate of spreading of the disease. The model shows that
delaying the start of control measures past a certain point greatly worsens
strategy outcomes. We conclude that the effective reproduction can be a
valuable real-time indicator in determining cost-effective control strategies.

Coronavirus 2019,

Control strategies,

Testing,

Distancing,

Effective reproduction number,

Reproduction number,

Spectral radius,

at-risk subpopulation,

###### keywords:

Research

[id=n1]Equal contributor

##  1 Introduction

One of the major concerns of the World Health Organization(WHO) is the
prevention of large epidemics or pandemics. Various techniques and different
human, economic, and material resources are deployed in order to eradicate
epidemics as soon as possible. For example, smallpox destabilized the world
for centuries [1], but was completely eliminated worldwide by 1977 thanks to
efforts by WHO and other organizations [2, 3]. However, in spite of man’s best
efforts, some diseases have evaded control, and continue to threaten entire
populations both locally and internationally. A recent example of this is
Coronavirus-19, which was discovered in Wuhan City, China, in December 2019
[4, 5], spread throughout the world within a few weeks, and was declared a
pandemic by WHO[6] on 30th January 2020.

COVID-19 poses special difficulties in that a significant proportion of
infectious cases are asymptomatic. Infectious asymptomatic cases may spread
the infection without being detected. Also, asymptomatic cases may persist
even after all known cases of the disease have been eradicated. Asymptomatic
cases of COVID-19 may constitute a large proportion of the infected
individuals. There is a wide range of estimates of the proportion of cases
that are asymptomatic. Early reports from China from testing of residents and
overseas arrivals suggested that $40\%-80\%$ of infections showed no symptoms
[7, 8]. Comprehensive testing was performed in the city of Vo’ before and
after lockdown showed that about $43\%$ of infections detected were
asymptomatic [9]. The authors of [10] reviewed of $41$ studies with a total of
$50,155$ confirmed COVID-19 cases, and found the pooled percentage of
asymptomatic infection was $15.6\%(95\%CI:10.1\%‐23.0\%)$. In [11], infection
rates due to contact with asymptomic carriers was estimated at $4.11\%$ (6
infections from 146 contacts) compared to $6.30\%$ for symptomatic cases (126
infections from 2001 contacts). Besides asymptomatic cases, a presymptomatic
infectious phase of $1-4$ days is estimated in [12], while the estimated
asymptomatic infectious period was $4-9.5$ days. Since the discovery of
COVID-19, numerous measures and resources were deployed for its eradication.
Vaccines and curative medicines were lacking, but some measures such as social
distancing and testing were effective in slowing its spread. Health
specialists recommend using both strategies. The effectiveness and costs
associated with both strategies are discussed below.

There exist many various types of social distancing, but the most basic
involves maintaining distance in public spaces, mask-wearing, and quarantine
for symptomatic individuals and their contacts. More severe measures include
banning public gatherings, restricting population movement, closing
businesses, and stay-at-home orders. The effectiveness of distancing measures
has been investigated by researchers using simulations based on mathematical
models. In [13] an SIR model that includes lockdown policies was studied
analytically. The authors concluded that the optimal policy depends only on
the shadow price difference between infected and susceptible individuals. They
furthermore concluded that more extreme measures applied over a short time
horizon are more effective than less extreme measures over a longer horizon.
In [14], an age-structured mathematical model was developed for investigating
the effectiveness of social distancing interventions to stop the spread of
COVID-19 using 4 scenarios. It was found that the number of the new
infections, hospitalisations and deaths were all decreased by distancing
measures. [15] uses the data from Wuhan city and found that when the contact
patterns are changed as a result of distancing, the epidemic peak is delayed
and new cases of coronavirus disease 2019 are decreased. Other research works
that have explored the importance of using the social distance strategy
against COVID-19 are [16, 17, 18, 19, 20]. Although social distancing measures
have saved many lives, they also have incurred significant costs for society.
Economic activity has decreased, producing widespread hardship and
unemployment. Several researchers have investigated these costs in order to
better understand economic consequences of the COVID-19 epidemic, especially
the costs associated with distancing and testing measures. [21] uses a SIR
model to simulate transmission rates for various countries under various
social distancing strategies, and estimates the associated prices during an
epidemic period. [22] estimates the economic costs caused by social distancing
strategy in fighting against COVID-19, where five main social-distancing
policies have been considered. Results show a decline in average income in the
range of $4.6-18.6\%$, depending on level of distancing. As an alternative and
supplement to distancing, testing and tracing is another viable and
complementary approach. The effectiveness of tests to detect the presence of
SARS-CoV-2 virus and antibodies to SARS-CoV-2 has also been studied by a
number of researchers[23, 24, 25]. Reference[26] established a mathematical
model of SARS-CoV-2 that includes PCR(Polymerase Chain Reaction) testing, and
estimates the reduction in the effective reproduction number achieved by
testing and isolating symptomatic individuals, regular screening of high-risk
groups irrespective of symptoms, and quarantine of contacts of laboratory-
confirmed cases identified through test-and-trace protocols.

Although testing avoids the economic slowdown and social costs associated with
distancing, it is still not without costs. In [27], it is estimated the cost
of $51 or $100 per diagnostic test depending on the type of test, while [28]
gives an approximate cost of $100 paid by medicare for each laboratory tests
for detecting SARS-CoV-2, and [29] quotes a price of $5 per test, but without
mentioning the false positive and false negative rates.

###  1.1 Basic reproduction number and effective reproduction number

In theoretical epidemiology, the basic parameter used to characterize the rate
of spread of a disease is called the _reproduction number_. The basic
reproduction number $R_{0}$ is defined as the average number of secondary
infections which one typical infected individual would generate if the
population were completely susceptible[30]. In multicompartment models of
disease dynamics, $R_{0}$ is computed as the dominant eigenvalue of a positive
linear operator. The concept of basic reproduction number ($R_{0}$) was first
introduced in 1886 [31] and has been used in multitudes of studies of
infectious diseases. Some recent examples include [32, 33, 34, 35, 36, 37, 38,
39]. For the recent pandemic COVID-19, many researchers have estimated $R_{0}$
using different approaches for various countries and regions [40, 41, 42, 43,
44, 45, 46].

In addition to $R_{0}$, the _effective reproduction number_ (denoted by
$R_{e}$) is also of interest. $R_{e}$ is defined as the number of secondary
infections produced by a single infectious when the population has both
susceptible and non-susceptible individuals (non-susceptible may include
infectious, immune, vaccinated, etc.) and/or control methods have been
implemented. Several previous studies have estimated $R_{e}$ for various
scenarios. The effective reproduction number for COVID-19 of India and its
states has been determinated in [47] using Real-Time Bayesian Method. [48]
determines the effective reproduction number for COVID-19 at the first 10 days
of Latin American countries where the highest was in Ecuador($R_{e}=3.95$) and
the smallest in Peru ($R_{e}=2.36$) and make a comparison with one of
Spain($R_{e}=2.9$) and Italy($R_{e}=2.83$). The effective reproduction number
is evaluated in [49] by using a probabilistic methodology which consider only
the daily death statistics of a given country.

In our work, we will focus on the effects of different levels of testing and
distancing measures on the effective reproduction number of COVID-19, as well
as these measures’ economic costs. The organization of the paper is as
follows. Section 2 describes the COVID-19 model with and without controls,
computes basic and effective reproduction numbers, and estimates model
parameters from available data. Section 3 gives simulations and
interpretations of daily and long-term scenarios, including sensitivity
analysis and comparison of different control strategies. Finally, Section 4
summarizes our findings and gives concluding remarks.

##  2 COVID-19 epidemic model formulation and mathematical properties

In this section, we present the multicompartment models of COVID-19 and,
identify the parameters and estimated control costs used in simulation. In
addition, using the next generation matrix, we calculate the basic and
effective reproduction number.

###  2.1 Multicompartment model

In this section, we describe the deterministic compartmental model that was
used to model the transmission dynamics of COVID-19. The model is based on
[50], and partitions the entire population into subpopulations according to
age and risk group and, where each subpopulation is further subdivided into
the following compartments: susceptible$(S)$, exposed $(E)$, pre-symptomatic
infectious $(P^{Y})$, pre-asymptomatic infectious $(P^{A})$, symptomatic
infectious $(I^{Y})$, asymptomatic infectious $(I^{A})$, symptomatic
infectious that are hospitalized $(I^{H})$, recovered $(R)$, and deceased
$(D)$. In our model, two subpopulations are identified, namely low risk and
high risk. It is assumed that the survivors have permanent immunity, and that
dead individuals are not infectious. The subpopulation model is diagrammed in
Figure 1,

Figure 1: COVID-19 transmission schema[50]

and the explicit equations are as follows:

$\displaystyle\frac{dS_{j}}{dt}$
$\displaystyle=-\sum_{i=0}^{1}\frac{1}{N_{i}}\left(I^{Y}_{i}\omega^{Y}+I^{A}_{i}\omega^{A}+P^{Y}_{i}\omega^{PY}+P^{A}_{i}\omega^{PA}\right)\phi_{ji}\beta
S_{j},$ (1) $\displaystyle\frac{dE_{j}}{dt}$
$\displaystyle=\sum_{i=0}^{1}\frac{1}{N_{i}}\left(I^{Y}_{i}\omega^{Y}+I^{A}_{i}\omega^{A}+P^{Y}_{i}\omega^{PY}+P^{A}_{i}\omega^{PA}\right)\phi_{ji}\beta
S_{j}-\sigma E_{j},$ $\displaystyle\frac{dP^{A}_{j}}{dt}$
$\displaystyle=(1-\tau)\sigma E_{j}-\rho^{A}P^{A}_{j},$
$\displaystyle\frac{dP^{Y}_{j}}{dt}$ $\displaystyle=\tau\sigma
E_{j}-\rho^{Y}P^{Y}_{j},$ $\displaystyle\frac{dI^{A}_{j}}{dt}$
$\displaystyle=\rho^{A}P^{A}_{j}-\gamma^{A}I^{A}_{j},$
$\displaystyle\frac{dI^{Y}_{j}}{dt}$
$\displaystyle=\rho^{Y}P^{Y}_{j}-(1-\Pi_{j})\gamma^{Y}I^{Y}_{j}-\Pi_{j}\eta
I^{Y}_{j},$ $\displaystyle\frac{dI^{H}_{j}}{dt}$ $\displaystyle=\Pi_{j}\eta
I^{Y}_{j}-(1-\nu_{j})\gamma^{H}I^{H}_{j}-\mu\nu_{j}I^{H}_{j},$
$\displaystyle\frac{dR_{j}}{dt}$
$\displaystyle=\gamma^{A}I^{A}_{j}+(1-\Pi_{j})\gamma^{Y}I^{Y}_{j}+(1-\nu_{j})\gamma^{H}I^{H}_{j}+$
$\displaystyle~{}~{}-\mu\nu_{j}I_{j}^{H}r\left(1-\frac{\theta/r}{\max(\nu_{0}I_{0}^{H}+\nu_{1}I_{1}^{H},\theta/r)}\right),$
$\displaystyle\frac{dD_{j}}{dt}$
$\displaystyle=\mu\nu_{j}I_{j}^{H}\left(1+r\left(1-\frac{\theta/r}{\max(\nu_{0}I_{0}^{H}+\nu_{1}I_{1}^{H},\theta/r)}\right)\right),$

where $j$ is the subpopulation index (0=low risk, 1=high risk), and

$N_{i}=S_{i}+E_{i}+P^{A}_{i}+P^{Y}_{i}+I^{A}_{i}+I^{Y}_{i}+I^{H}_{i}+R_{i}.$
(2)

The initial conditions require all compartment populations to be nonnegative.
The interpretation and numerical values of the parameters in (1) are listed in
Table 1. The model (1) has non-negative solutions contained in the feasible
region
$\Gamma=\\{S_{j},E_{j},P^{A}_{j},P^{Y}_{j},I^{A}_{j},I^{Y}_{j},I^{H}_{j},R_{j},D_{j}\\}\in\mathbb{R}^{18}_{+}$.

The contact matrix $\Phi=[\phi_{ji}]$, $(i,j)\in\\{0,1\\}$ is defined by

$\displaystyle\Phi$ $\displaystyle=\left(\begin{matrix}\phi_{00}&\phi_{01}\\\
\phi_{10}&\phi_{11}\\\ \end{matrix}\right)=\left(\begin{matrix}10.52&2.77\\\
9.4&2.63\\\ \end{matrix}\right),$ (3)

where $\phi_{ji}$ represents the mean number of contacts per day experienced
by individuals in group $j$ from individuals of group $i$. The matrix values
in (3) were obtained by averaging the contacts between low and high risk
individuals over all age groups, using Tables A.4.1-4 and Figure A3 in [50]
for contact rates and age-specific high risk proportions, respectively.

Table 1: Baseline parameters used in the model (based on [50]).

Parameters | Interpretation | Values
---|---|---
$\beta$ | baseline transmission rate | 0.0640
$\gamma^{A}$ | recovery rate on asymptomatic compartment | Equal to $\gamma^{Y}$
$\gamma^{Y}$ | recovery rate on symptomatic non-treated compartment | $\frac{1}{\gamma^{Y}}=4.0$
$\tau$ | symptomatic proportion | $0.55$
$\sigma$ | exposed compartment exit rate | $\frac{1}{\sigma}\sim 2.9$
$\rho^{A}$ | pre-asymptomatic compartment exit rate | Equal to $\rho^{Y}$
$\rho^{Y}$ | pre-symptomatic compartment exit rate | $\frac{1}{\rho^{Y}}=2.3$
$P$ | proportion of pre-symptomatic transmission | $0.44$
$\omega^{Y}$ | relative infectiousness of symptomatic individuals | $1.0$
$\omega^{A}$ | relative infectiousness of infectious individuals in compartment $I^{A}$ | $0.66$
$\omega^{P}$ | relative infectiousness of pre-symptomatic individuals | $\omega^{P}=\frac{P}{1-P}\frac{\tau\omega^{Y}[YHR/\eta+(1-YHR)/\gamma^{Y}]+(1-\tau)w^{A}/\gamma^{A}}{\tau\omega^{Y}/\rho^{Y}+(1-\tau)\omega^{A}/\rho^{A}}$
$IFR$ | infected fatality ratio, age specific (%) | $[0.6440,6.440]$
$YFR$ | symptomatic fatality ratio, age specific (%) | $[1.130,11.30]$
$\gamma^{H}$ | recovery rate in hospitalized compartment | $\frac{1}{\gamma^{H}}\sim 10.7$
$YHR$ | Symptomatic case hospitalization rate % | $[4.879,48.79]$
$\Pi$ | rate of symptomatic individuals go to hospital, age-specific | $\Pi=\frac{\gamma^{Y}*YHR}{\eta+(\gamma^{Y}-\eta)YHR}$
$\eta$ | rate from symptom onset to hospitalized | $0.1695$
$\mu$ | rate at which terminal patients die | $\frac{1}{\mu}=8.1$
$HFR$ | hospitalized fatality ratio, age specific (%) | $[4,23.158]$
$\nu$ | death rate on hospitalized individuals, age specific | $\nu=\frac{\gamma^{H}HFR}{\mu+(\gamma^{H}-\mu)HFR}$
$\theta$ | total ventilator capacity in all hospitals | 3000 [51]
$1/r$ | number of deaths from people who are put on respirators | 1/3

There are a few differences between our model and the model in [50]. In our
model the definition of $N_{i}$ in (2) does not include $D_{i}$, since the
individuals who have died are no longer in the active population. In addition,
the last two equations in our model include additional terms that reflect the
additional mortality that occurs when the ventilator capacity (represented by
the parameter $\theta$) is exceeded. Note that this change does not affect the
reproduction number of the system. The modifications are derived based on the
following assumptions:

  * •

All patients that are at risk of dying are put on respirators, if respirators
are available;

  * •

A fixed fraction of patients that need respiration and are put on respirators
nonetheless die. According to the literature, this fraction is about 1/3; In
the model we introduce the parameter $r$, which is the inverse of this
fraction, thus $r\approx 3$;

  * •

All patients that need respiration but are not put on respirators will die;

  * •

Respirators are allocated proportionately to the low and high risk patients
who need them.

Let $I_{0}^{H},~{}I_{1}^{H}$ be the number of each group that is hospitalized.
We already have that $\nu_{0},\nu_{1}$ are the death rates for hospitalized
low and high risk, respectively. It follows that there are $r\nu_{0}I_{0}^{H}$
and $r\nu_{1}I_{1}^{H}$ low and high risk patients respectively that need
respirators. Letting $n_{0},~{}n_{1}$ be the number of patients in each group
who are on respirators, it follows that the number of terminal patients that
die without respiration is
$(r\nu_{0}I_{0}^{H}+r\nu_{1}I_{1}^{H}-n_{0}-n_{1})$.

It remains to solve for $n_{0}$ and $n_{1}$. The constraint on total
respirators gives $n_{0}+n_{1}\leq\theta.$ According to our assumption of
proportionate respirator allocation,
$n_{0}/~{}n_{1}=\nu_{0}I_{0}^{H}/\nu_{1}I_{1}^{H}$. We may distinguish two
cases. First, if $r(\nu_{0}I_{0}^{H}+\nu_{1}I_{1}^{H})\leq\theta$, then we
have $n_{j}=r\nu_{j}I_{j}^{H}$ for $j=0,1$. Otherwise, $n_{0}+n_{1}=\theta$
which implies

$n_{j}=\frac{\theta\nu_{j}I_{j}^{H}}{(\nu_{0}I_{0}^{H}+\nu_{1}I_{1}^{H})},~{}~{}j=0,1.$
(4)

We may combine these two cases into the single equation:

$n_{j}=\frac{r\theta\nu_{j}I_{j}^{H}}{\max(r\nu_{0}I_{0}^{H}+r\nu_{1}I_{1}^{H},\theta)},~{}~{}j=0,1,$
(5)

so that the number of terminal patients in group $j$ that are denied
respirators is $\omega_{j}$, where

$\omega_{j}=r\nu_{j}I_{j}^{H}-\frac{r\theta\nu_{j}I_{j}^{H}}{\max(r\nu_{0}I_{0}^{H}+r\nu_{1}I_{1}^{H},\theta)}.$
(6)

Low- and high-risk patients that not denied respirators are terminal at rates
$\nu_{0}$ and $\nu_{1}$, respectively. Therefore we have:

$\displaystyle\frac{dD_{j}}{dt}$
$\displaystyle=\mu\nu_{j}I_{j}^{H}+\mu\omega_{j}(1-\nu_{j})=\mu\nu_{j}I_{j}^{H}\left(1+r\left(1-\frac{\theta/r}{\max(\nu_{0}I_{0}^{H}+\nu_{1}I_{1}^{H},\theta/r)}\right)\right),$
(7)

which is identical to the equation for $\frac{dD_{j}}{dt}$ in (1). The
equation for $\frac{dR_{j}}{dt}$ must be similarly adjusted by an amount
$-\mu\omega_{j}(1-\nu_{j})$ to offset the increased number of deaths due to
insufficient respirators.

###  2.2 COVID-19 epidemic model formulation under controls

The use of the control measures has an important effect at a certain level on
the spread of the COVID-19 epidemic. In order to study disease mitigation, we
introduce the effects of two controls: social distancing and COVID testing.
Social distancing (denoted by $v_{j}$) will reduce the overall infectivity,
while COVID testing (denoted by $u_{j}$) will reduce the infectivity of the
asymptomatic and presymptomatic infectious compartments. The model with
controls is identical to (1), except the first two equations are modified as
follows:

$\displaystyle\frac{dS_{j}}{dt}$
$\displaystyle=-\sum\limits_{i=0}^{1}\frac{1}{N_{i}}\left(I^{Y}_{i}\omega^{Y}+(1-u_{i})[I^{A}_{i}\omega^{A}+\omega^{P}(P^{Y}_{i}\omega^{Y}+P^{A}_{i}\omega^{A})]\right)(1-v_{j})\beta\phi_{ji}S_{j},$
(8) $\displaystyle\frac{dE_{j}}{dt}$
$\displaystyle=\sum\limits_{i=0}^{1}\frac{1}{N_{i}}\left(I^{Y}_{i}\omega^{Y}+(1-u_{i})[I^{A}_{i}\omega^{A}+\omega^{P}(P^{Y}_{i}\omega^{Y}+P^{A}_{i}\omega^{A})]\right)(1-v_{j})\beta\phi_{ji}S_{j}-\sigma
E_{j}.$

We shall use $X$ to denote the vector $X=[X_{0},X_{1}]$ of all infected
classes, where we define
$X_{j}=[E_{j},P^{A}_{j},P^{Y}_{j},I^{A}_{j},I^{Y}_{j},I^{H}_{j}]$ and
$X^{\prime}=[X^{\prime}_{0},X^{\prime}_{1}]$ the vector of all uninfected
classes with $X^{\prime}_{j}=[S_{j},R_{j}]$ where $j=0,1$ corresponds to low
and high risk subpopulations respectively, with susceptible$(S_{j})$, exposed
$(E_{j})$, pre-symptomatic infectious $(P_{j}^{Y})$, pre-asymptomatic
infectious $(P_{j}^{A})$, symptomatic infectious $(I_{j}^{Y})$, asymptomatic
infectious $(I_{j}^{A})$, symptomatic infectious that are hospitalized
$(I_{j}^{H})$, recovered $(R_{j})$, and deceased $(D_{j})$.

In order to take into account the cost associated to the model (8), we define

$\displaystyle\alpha_{j}(u_{j},N^{A}_{j})$
$\displaystyle=\begin{cases}0&~{}~{}\text{if}~{}~{}u_{j}=0,\\\
a_{j0}+a_{j1}N^{A}_{j}u_{j}+a_{j2}N^{A}_{j}(u_{j})^{2}&~{}~{}\text{if}~{}~{}0<u_{j}\leq
u_{j}^{(max)},\end{cases}$ (9) $\displaystyle\beta_{j}(v_{j})$
$\displaystyle=b_{j1}N_{j}v_{j}+b_{j2}N_{j}(v_{j})^{n}\qquad\qquad\qquad\text{if}~{}~{}0<v_{j}\leq
v_{j}^{(max)},$

where $N^{A}_{j}=S_{j}+E_{j}+P^{A}_{j}+P^{Y}_{j}+I^{A}_{j}$ is the number of
asymptomatic individuals in subpopulation. The functions $\alpha_{j}$ and
$\beta_{j}$ are intended to model the costs associated with COVID testing and
social distancing respectively for subpopulations $j=0,1$. The coefficient
$a_{j0}$ represents the fixed cost when the testing program is implemented;
$a_{j1}$ is the testing cost per person, $u_{j}$ is the fraction of
asymptomatic individuals in population j that are subject to testing, and
$a_{j2}$ represents the increasing marginal cost per person incurred as the
testing program becomes more intensive (reflecting the law of diminishing
returns). The cost function $\beta_{j}$ in (9) reflects the economic cost of
social distancing measures, and $b_{j1}$, $b_{j2}$ reflect per-capita costs.
$\beta_{j}$ is modeled as a nonlinear function, since marginal costs will
increase as the severity of distancing measures increases (for example, low-
level distancing measures such as wearing masks incurs much less expense than
serious measures such as closing stores and stay-at-home orders). The exponent
$n>1$ is chosen to reflect these nonlinear effects. The parameter $v_{j}$
expresses the proportionate reduction in contacts that result from the
implemented measures.

###  2.3 Estimation of basic and effective reproduction numbers

In this section, we will present the calculation of basic and effective
reproduction numbers ($R_{0}$ and $R_{e}$, respectively) using the next
generation matrix technique of Van Den Driessche.

####  2.3.1 Computation of basic reproduction number

The basic reproduction number $(R_{0})$ is the average number of secondary
infections produced by a typical case of an infection in a population where
everyone is susceptible. It is affected by the following factors: the rate of
contacts in the host population, the probability of infection being
transmitted during contact, the duration of infectiousness. Using the next
generation matrix defined in [37] (see also [52]), we establish $R_{0}$ for
the system in (1). Recall that $X$ and $X^{\prime}$ are vectors representing
infected and uninfected compartments respectively. We have

$\displaystyle\frac{dX}{dt}$ $\displaystyle=$
$\displaystyle\mathcal{F}(\mathbb{X})-\mathcal{V}(\mathbb{X}),$ (10)
$\displaystyle\frac{dX^{\prime}}{dt}$ $\displaystyle=$
$\displaystyle\mathcal{W}(\mathbb{X}),$ (11)

where $\mathbb{X}=(X,X^{\prime})$, $\mathcal{F}(\mathbb{X})$ represents the
vector of in-flows into infected compartments (including new infections) and
$\mathcal{V}(\mathbb{X})$ is the vector of out-flows. The functions
$\mathcal{F}$ and $\mathcal{V}$ are chosen so that
$\mathcal{F}(\mathbb{X})\geq 0$ and $\mathcal{V}(\mathbb{X})\geq 0$. We denote
the disease free equilibrium by $(0,\bar{X^{\prime}})$. Replacing in Equation
(10), we have $\mathcal{F}(0,\bar{X^{\prime}})=0$ and
$\mathcal{V}(0,\bar{X^{\prime}})=0$. The next generation matrix is given by
$FV^{-1}$, where

$F=\left.\left(\frac{\partial\mathcal{F}}{\partial
X}\right)\right|_{(0,\bar{X^{\prime}})}~{}\text{and}~{}V=\left.\left(\frac{\partial\mathcal{V}}{\partial
X}\right)\right|_{(0,\bar{X^{\prime}})}.$

The basic reproduction number $R_{0}$ is computed as the spectral radius of
the matrix $FV^{-1}$. From System (1), $F$ and $V$ may be evaluated as (note
in the disease-free case, $S_{j}=N_{j}$):

$\displaystyle F$ $\displaystyle=\left(\begin{matrix}F_{00}&F_{01}\\\
F_{10}&F_{11}\\\ \end{matrix}\right),$ (12)

with

$\displaystyle F_{jj}$
$\displaystyle=\left(\begin{matrix}0&\beta\omega^{P}\omega^{A}\Phi_{jj}&\beta\omega^{P}\omega^{Y}\phi_{jj}&\beta\omega^{A}\phi_{jj}&\beta\omega^{Y}\phi_{jj}\\\
(1-\tau)\sigma&0&0&0&0\\\ \tau\sigma&0&0&0&0\\\ 0&\rho^{A}&0&0&0\\\
0&0&\rho^{Y}&0&0\\\ \end{matrix}\right),$ (13)

$\displaystyle F_{1-j,j}$
$\displaystyle=\left(\begin{matrix}0&\frac{\beta\omega^{P}\omega^{A}N_{1-j}}{N_{j}}\phi_{1-j,j}&\frac{\beta\omega^{P}\omega^{Y}N_{1-j}}{N_{j}}\phi_{1-j,j}&\frac{\beta\omega^{A}N_{1-j}}{N_{j}}\phi_{1-j,j}&\frac{\beta\omega^{Y}N_{1-j}}{N_{j}}\Phi_{1-j,j}\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{matrix}\right),$ (14)

and

$\displaystyle V$ $\displaystyle=\left(\begin{matrix}V_{00}&V_{01}\\\
V_{10}&V_{11}\\\ \end{matrix}\right),$ (15)

with

$\displaystyle V_{jj}$ $\displaystyle=\left(\begin{matrix}\sigma&0&0&0&0\\\
0&\rho^{A}&0&0&0\\\ 0&0&\rho^{Y}&0&0\\\ 0&0&0&\gamma^{A}&0\\\
0&0&0&0&(1-\Pi_{j})\gamma^{Y}+\Pi_{j}\eta\\\ \end{matrix}\right);\qquad
V_{1-j,j}=\textbf{0}_{5\times 5}.$ (16)

The following theorem guarantees that the spectral radius is equal to the
dominating eigenvalue $FV^{-1}$, which is real, positive, and has an
associated eigenvector that is also real and positive:

######  Theorem 1.

(Perron-Frobenius theorem)[53]: Suppose $A\in\mathbb{R}^{n\times n}$ is
nonnegative and regular, $i.e.A^{k}>0$ for some $k$, then

  1. $\bullet$

there is an eigenvalue $\lambda_{pf}$ of $A$ with geometric multiplicity 1
that is real and positive, with positive left and right eigenvectors

  2. $\bullet$

for any other eigenvalue $\lambda$, we have $|\lambda|<\lambda_{pf}$.

####  2.3.2 Effective reproduction number

As an epidemic progresses, there will be an increasing proportion of the
population which has recovered from the disease and hence has some degree of
immunity. When this happens, the basic reproduction does not accurately
reflect the number of secondary cases produced by an infection. Our
calculation of $R_{0}$ also does not include the introduction of controls. In
order to obtain an estimate of the effective reproduction number $R_{e}$, we
modify (13)-(14) based on (8) as follows:

$\displaystyle F_{jj}$
$\displaystyle=\scriptsize\left(\begin{matrix}0&\frac{(1-u_{j})(1-v_{j})\beta\omega^{P}\omega^{A}S_{j}}{N_{j}}\Phi_{jj}&\frac{(1-u_{j})(1-v_{j})\beta\omega^{P}\omega^{Y}S_{j}}{N_{j}}\Phi_{jj}&\frac{(1-u_{j})(1-v_{j})\beta\omega^{A}S_{j}}{N_{j}}\Phi_{jj}&\frac{(1-v_{j})\beta\omega^{Y}S_{j}}{N_{j}}\Phi_{jj}\\\
(1-\tau)\sigma&0&0&0&0\\\ \tau\sigma&0&0&0&0\\\ 0&\rho^{A}&0&0&0\\\
0&0&\rho^{Y}&0&0\\\ \end{matrix}\right),$ (17)

$\displaystyle F_{1-j,j}$ $\displaystyle=\frac{(1-v_{1-j})\beta
S_{1-j}}{N_{j}}\Phi_{1-j,j}\left(\begin{matrix}0&(1-u_{j})\omega^{P}\omega^{A}&(1-u_{j})\omega^{P}\omega^{Y}&(1-u_{j})\omega^{A}&\omega^{Y}\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ \end{matrix}\right).$ (18)

The effective reproduction number $R_{e}$ is computed as the spectral radius
of the matrix $FV^{-1}$, where $F$ and $V$ are specified using (12) and (15),
where $F_{jj}$ and $F_{1-j,j}$ are given by Eqs. (17) and (18), respectively.

###  2.4 Control cost estimates

In order to optimize control costs according to the model described above, it
is necessary to find realistic estimates of the coefficients $a_{jk}$ and
$b_{jk}$ in (9). In this section, we present results from the literature on
which our coefficient estimates are based.

In [54], the reduction in U.S. Gross Domestic Product (GDP) due to distancing
was estimated as $7.2 trillion over a 30-year period, assuming a 3% discount
rate. This corresponds to a $3 per day cost per person and the distancing was
assumed to reduce contact rate by 40%. The article also presents an estimate
by Goldman-Sachs of 6% reduction in U.S. GDP due to mortality, morbidity, and
productivity losses due to distancing measures. Since the mean yearly income
in Houston is $31,576 per person [55], this would translate to a cost of
roughly $5/day/person which includes mortality and morbidity impacts in
addition to distancing. Reference [22] estimates between 4.6–18.6% decline in
income in Texas for distancing measures, depending on severity of measures. If
we suppose that these declines correspond to contact rate reductions of 40%
and 80% respectively, this would indicate that doubling the severity of
distancing measures roughly quadruples the cost. This would correspond to a
purely quadratic cost function $\beta_{j}$ in (9), which implies $b_{j1}=0$
and $n=2$. This quadratic cost function reflects the economic principle of
diminishing returns: increasingly strict methods incur disproportionately
higher costs.

Economic cost is only part of the total cost incurred by distancing measures.
There are social and political costs as well. People feel oppressed by
distancing measures, and many people view such measures as infringements on
their freedom[56]. Supportive connections with family and friends are
disrupted [57]. In the U.S., many anti-distancing demonstrations have taken
place [58] as people are not convinced that such severe measures are
necessary. The costs of distancing are also not distributed evenly among the
population, and low-income individuals are often the hardest hit when retail
sales diminish and restaurants and shops are closed down as part of distancing
measures.

Testing does not have the same social costs as distancing, but has its own
problems. The gold standard test for COVID is the molecular Polymerase Chain
Reaction (PCR) test, which requires lab analysis. Results of the test are not
immediately available, and often take up to five days to obtain. The lag means
that any infection acquired between the test administration and the results
will go undetected. Therefore, testing also involves some quarantining, which
has its own costs. Additionally, a testing program is most effective if it is
coupled with contact tracing, so that others exposed to possible infection are
identified and tested as well. Some contacts may missed, which diminishes the
effectiveness of the control strategy. In addition, PCR tests can yield both
false negative and false positive results: estimates of error rates range from
2-33% and 0.8-4% for false negative and false positive rates per test,
respectively [59]. PCR tests are also relatively expensive, at about $50 per
test. A cheaper test with 15-minute turnaround is available (the BinaxNOW
Covid-19 Ag Card, produced by Abbott Laboratories), but has relatively high
false negative and false positive rates of 2.9% and 1.5% respectively [60],
and is thus most useful in locations where prevalence is high [61].

From a study by Campbell et al [62] on testing costs in Canada, we may infer a
per-test total cost estimate (including test + personnel) of $57 for testing
only, and $69 for testing plus tracing [62]. The article also mentions the
possibility of reducing costs by 40% through introduction of pooled sampling
and other efficiency measures. One must also take into consideration that the
economic cost of testing is of a somewhat different nature from the economic
costs of distancing. Distancing costs correspond to lost productivity and
reduced consumption, while testing costs are paid to companies which will
produce higher wages and profits, thus returning benefits to the economy.

Two questions remain vis-a-vis testing, namely the frequency of testing and
the effectiveness. A study by Jiang et al. [63] recommends performing three
PCR tests before discharging patients into the general population (their
conclusion is based on presumed false negative rate of 29%, which is high
compared to other studies). Many studies appear to recommend weekly or monthly
testing of key subpopulations ([62], [64]). In [64], the most intensive
testing regimen was estimated to produce a $63\%$ reduction in infections, but
at a cost that is 4 times as high as a less strict regimen. The cost of the
most intensive regimen was between 2–2.5 billion dollars per million people
per 180 days, which comes out to between $11–$14 per day per person. Reference
[22] estimates a decline in income of 4.6–18.6% due to distancing measures,
depending on severity. Using a figure of $31,576 for mean income per person in
Houston registers, this gives a daily cost $4–$16. Based on those simple cost
examples, we arrived at baseline cost coefficients for testing and distancing
controls as specified in Table 2 that will be used on the rest of the work.

Table 2: Testing and social distancing control cost and level parameters

Parameters | Interpretation | Values
---|---|---
$a_{00},a_{10}$ | minimum testing cost per person | $0
$a_{01},a_{11}$ | linear testing cost coefficient | $2.3/person/day
$a_{02},a_{12}$ | quadratic rate of increase of per capita testing cost | $27/person/day2
$b_{01},b_{11}$ | constant per capita social distancing costs | $0
$b_{02},b_{12}$ | quadratic rate of increase of per capita social distancing cost | $40/person/day2
$u_{0}^{max},u_{1}^{max}$ | maximum testing control level | $0.66$
$v_{0}^{max},v_{1}^{max}$ | maximum social distancing control level | $0.8$

##  3 Simulations results and discussion

A number of simulations were performed to analyze the relations $R_{e}$,
control levels, and control costs, and to explore the use of $R_{e}$ in
determining cost-effective strategies. All simulations used the model
described in Section 2, with the parameters given in Tables 1 and 2. The
simulations performed can be classified into three groups. In the first group
of simulations, we first characterize the response of $R_{e}$ and control cost
to individual control levels. In the second group we investigate the
sensitivity of these relationships to important parameters. In the third
groups, we simulate long-term strategies for epidemic mitigation that are
based on the findings from previous simulations, and determine their cost-
effectiveness.

###  3.1 Dependence of $R_{e}$ on control level and control cost for
individual control strategies

In this subsection we investigate the behavior of $R_{e}$ depending on the
level of testing and distancing controls applied to low and high risk
population groups. Our computations were based on (12)–(18), with high-risk
and low-risk populations of 1.34 million and 423,000 corresponding to
demographics of the city of Houston, TX.

Figure 2 shows $R_{e}$ as a function of control level for six different
control strategies, at two different population immunity levels. The first
four strategies employ a single nonzero control (either testing or distancing)
on a single group (either low- or high-risk). In the last two strategies, both
population groups are subject to the same control (testing and distancing,
respectively). Solid curves correspond to these six strategies when the entire
population has $0\%$ immunity (i.e. $100\%$ of the population is susceptible),
and the dotted curves are for a population with $67\%$ immunity respectively
($33\%$ susceptible). The curves show that when no herd immunity is present,
none of the six strategies is sufficient to bring $R_{e}$ below 1. When herd
immunity reaches $67\%$ of the population, only high levels of distancing for
either the low-risk population or the entire population can bring $R_{e}$
below $1$. The figure shows that controls that are applied only to the high-
risk group do not significantly reduce $R_{e}$ . Note however that these
graphs do not take into account the reduced deaths in the high-risk
population, because they only show the effect on overall $R_{e}$ and not the
number of deaths incurred by infection. It is also clear that applying the
same strategy to the entire population rather than just the low-risk group
greatly increases the effectiveness of the strategy.

Figure 2: Dependence of $R_{e}$ on control level for six control strategies at
$0\%$ and $66.6\%$ immunity

Figures 3(a) and (b) show the daily cost associated with each level of the six
control strategies defined above, applied at $0\%$ and 66.6% immunity
respectively. We notice that costs for distancing are consistently higher than
corresponding costs for testing: for example, distancing costs for the entire
population (which is the most expensive strategy, for a given control level)
is higher than the cost for testing the entire population at the same level of
control. Comparing Figures 3(a) and (b), we see that distancing costs do not
depend on immunity level, but testing costs are reduced by more than 50% when
immunity is increased from $0\%$ to $66.6\%$.

(a) Control costs at 0% immunity

(b) Control costs at 67% immunity

Figure 3:  Control cost as a function of control level for six strategies, for
two levels of immunity.

In Figure 4, $R_{e}$ is plotted as function of daily implementation cost for
the different control strategies, where the control costs depend on the
control levels through the relations (1)-(8). At $0\%$ immunity, the most
cost-effective strategy is distancing applied to the entire population: this
is true regardless of expenditure level. However, the situation changes at
$67\%$ immunity: in this case, testing of the entire population is most cost-
effective. However, the effects of universal testing are limited, and cannot
reduce $R_{e}$ below 1 even at the highest possible testing level. As with
Figure 2, these results do not account for the greater percentage of deaths
among the high risk population, because they only include implementation costs
and not the costs associated with hospitalization or death.

Figure 4: Dependence of $R_{e}$ on daily implementation cost for six
strategies at $0\%$ and $67\%$ immunity

Figures 2 \- 4 above show that strategies applied to the entire population
have a much greater effect and are more cost-effective than strategies applied
to a single population group. Therefore, in the following analysis, we
consider only testing and distancing strategies that are applied to the entire
population, and not to individual subpopulations.

Figures 5(a),(b) show the dependence of $R_{e}$ and daily implementation cost
on social distance and testing control levels, for two different levels of
population immunity. The white contour lines represent different values of
$R_{e}$ which are achieved by the different distancing and testing control
levels indicated on the $x$ and $y$ axes respectively. The color scale
indicates the cost for that combination. As control levels increase, $R_{e}$
decreases but daily costs increase. The shapes of the constant-$R_{e}$
contours in Figure 5(a) (0% immunity) resemble those in Figure 5(b) (67%
immunity), but the $R_{e}$ values for corresponding contours differ by a ratio
of roughly $2/3$. Similarly, cost contours in the two figures resemble each
other, but the cost values differ by a similar factor. According to Figure
5(a), a daily control cost of over 60 million dollars is required to reduce
$R_{e}$ below $1$, while Figure 5 (b) shows that only about $10$ million
dollars per day is required to achieve the same $R_{e}$ level at $67\%$
immunity. The black stars on each $R_{e}$ contour line mark the (distancing,
testing) control combinations that achieve minimum cost for the corresponding
$R_{e}$ value. Note that these optimum points may be visually identified as
the points on the contour lines where the contour lines are parallel to the
nearest constant-cost contour, which is represented as a boundary between two
colors.

(a) Control costs and $R_{e}$ values at 0% immunity

(b) Control costs and $R_{e}$ values at 67% immunity

Figure 5: $R_{e}$ and cost dependence on testing and social distancing control
levels at two different population immunity levels: Stars locate points on
$R_{e}$ contours corresponding to minimum cost for the given value of $R_{e}$.

###  3.2 Sensitivity of effective reproduction number and control costs to
model parameters

In this section we analyse the sensitivity of the effective reproduction
number $R_{e}$ and control costs to important parameters and cost
coefficients. Three key parameters that we will analyze are the infectivity
$\beta$; symptomatic proportion $\tau$; and relative infectiousness of
asymptomatic individuals $\omega^{A}$. These parameters are difficult to
estimate exactly, so it is important to determine their effect on model
outcomes. In the following analysis, perturbations of $\pm 25\%$ are applied
to each parameter, under two different levels of herd immunity.

Figure 6 shows the sensitivity of $R_{e}$ at $0\%$ immunity under different
values of $\beta,\tau$ and $\omega^{A}$. The six colored curves represent the
control levels that produce an effective $R_{e}=1.5$ when the three parameters
are individually varied by $\pm 25\%$. The two curves for $\beta=1\pm.25$
times the baseline value are widely separated, showing that the predicted
effect of controls on costs depends strongly on the value of $\beta$ used in
the model. Indeed, $25\%$ shifts in the value of $\beta$ produce changes in
control levels that exceed $25\%$. In contrast, the model parameter
$\omega^{A}$ has little effect on control level estimates, while $\tau$ only
has a large effect when the testing control level is high. These same
observations apply to Figure 7, which shows the effect of $\beta,\tau$ and
$\omega^{A}$ on the $R_{e}=1$ curve in a population with $67\%$ immunity.

From the positions of the blue arrows in Figures 6 and 7, we may conclude that
the values of $\beta$ and $\tau$ used in the model have a much greater effect
on the optimum distancing controls than on testing. For example, at 0%
immunity a variation of $\pm 25\%$ in $\beta$ gives an optimal distancing
control range of $0.34\pm 40\%$, and a variation in optimal testing control of
$0.25\pm 30\%$. These results indicate that it may be difficult in practice to
accurately determine optimal control levels that can produce a given $R_{e}$
value for the system.

Figure 6: Sensitivity of $R_{e}$ at $0\%$ immunity under different values of
$\beta,\tau$ and $\omega^{A}$. Blue arrows indicate minimum-cost control
solutions for different contours. Figure 7: Sensitivity of $R_{e}$ at $66.6\%$
immunity under different values of $\beta,\tau$ and $\omega^{A}$. Blue arrows
indicate minimum-cost control solutions for different contours.

Figures 8–9 represent the sensitivity of costs to changes in the quadratic
cost coefficients for testing and distancing ($a_{j2}$ and $b_{j2}$
respectively). The horizontal and vertical axis scales represent control costs
for social distancing and testing respectively (as in the previous figures,
both distancing controls have the same values, as do both testing controls).
Each figure shows two white contours showing constant values of $R_{e}$ with
the baseline cost parameter values. Both $a_{j2}$ and $b_{j2}$ are varied by
$\pm 25\%$, corresponding to the red and blue contours respectively. The
shades of color indicate total cost for each mix of testing and distancing
strategies: in this case, the lines of constant cost are straight lines.
Optimum (cost-minimizing) operating points for the different values of $R_{e}$
are indicated by arrows, as in previous figures. Regardless of immunity level,
the shifts in costs and optimum strategy point are much greater when $b_{j2}$
is varied than when $a_{j2}$, indicating a greater sensitivity of the system
to distancing quadratic costs than testing quadratic costs. For example, for
$R_{e}=1.2$ in Figure 8, with baseline parameters the control costs along the
contour vary from $35-40$ million dollars per day. When the quadratic testing
cost $a_{j2}$ is increased by $25\%$, then control costs still lie in the same
range. However, when the quadratic distancing cost $b_{j2}$ is increased by
the same percentage, the cost range shifts upwards to $40-45$ million dollars
per day. The two figures closely resemble each other: however, it should be
noted that Figure 9 which portrays $67\%$ immunity is showing $R_{e}$ values
that are only about $67\%$ as big as the $R_{e}$ values shown in Figure 8
which shows $0\%$ immunity. Also, the testing cost scale for Figure 9 has been
reduced by roughly $67\%$, although the distancing cost scale remains the
same.

Figure 8: Cost sensitivity from quadratic testing cost $a_{j2}$ and quadratic
distancing cost $b_{j2}$ at $0\%$ herd immunity, for two different levels of
$R_{e}$. Figure 9: Cost sensitivity from quadratic testing cost $a_{j2}$ and
quadratic distancing cost $b_{j2}$ at $0\%$ herd immunity, for two different
levels of $R_{e}$.

###  3.3 Instantaneous control strategies for optimizing effective
reproduction number

Equations (12)-(18) can be used to find optimum control levels associated with
a given value of $R_{e}$. This optimization was implemented in Python using
the minimize function from the scipy.optimize package in Python for
constrained minimization. Figures 10(a) and 11(a) show the optimal levels of
four controls (low and high risk testing, low and high risk distancing)
associated with different instantaneous $R_{e}$ values for 0% and 67%
population immunity, respectively. In the figures, solid lines indicate
optimal control levels when all four controls are allowed to vary
independently; while dashed and dotted lines show control levels when the
controls on low and high risk groups are constrained to be the same. The costs
associated with these optimal control strategies (both strategies where all
four controls vary independently and those for which high and low risk
controls are the same) as a function of $R_{e}$ are shown in Figures 10(b) and
11(b) for 0% and 67% immunity, respectively. As expected, lower values of
$R_{e}$ require higher levels of control, and incur greater costs. At all
levels, distancing controls are applied at a higher level than testing,
especially for values of $R_{e}$ near 0.7 where the optimal testing levels
show a dip. When four controls are allowed to vary independently, social
distancing for the low-risk subpopulation is applied at a higher level than
for the high-risk subpopulation. This is due to the fact that according to (3)
low-risk individuals have greater contact rates, and thus are more influential
in spreading the disease. However, when the same level of control is applied
to both low and high risk subpopulations, the costs are nearly the same as
shown in Figures 10(b) and 11(b). This indicates that costs are not highly
sensitive to the specific strategies used.

(a) Optimal control levels vs. $R_{e}$

(b) Control costs vs. $R_{e}$ for instantaneous optimal strategies

Figure 10: Optimal control levels (a) an associated control costs (b) as a
function of population’s current $R_{e}$ value for populations with $0\%$
immunity. Solid lines correspond to strategies in which both controls can be
applied at different levels on the low-risk and high-risk population
subgroups, while dashed lines are for strategies in which the same control
levels are applied to both subgroups.

(a) Optimal control levels vs. $R_{e}$

(b) Control costs vs. $R_{e}$ for instantaneous optimal strategies

Figure 11: Optimal control levels (a) an associated control costs (b) as a
function of population’s current $R_{e}$ value for populations with $0\%$
immunity. Solid lines correspond to strategies in which both controls can be
applied at different levels on the low-risk and high-risk population
subgroups, while dashed lines are for strategies in which the same control
levels are applied to both subgroups.

###  3.4 Optimizing long-term strategies that target effective reproduction
number reduction

We may define three different classes of long-term strategies that target
$R_{e}$ reduction. For all strategies, control measures are begun at a certain
time, and continue until the total infective population is reduced below a
given level, to prevent resurgence of the disease. All strategies set control
measure levels on a daily basis, so that the intensity of measures varies from
day to day.

In the first class of control strategies, a maximum daily budget is fixed to
spend on control measures. Daily expenditure is constant, except in cases
where the maximum possible controls cost is less than the allocated budget.
The strategy for each day is determined as the set of control measures that
does not exceed the budget, but which reduces $R_{e}$ as much as possible. The
user-defined parameters for these strategies are the daily maximum budget and
the date at which control starts.

In the second class of control strategies, during the active control period a
combination of distancing and testing measures are used to reduce the $R_{e}$
level to a constant fraction of the level that would be achieved without
control. For example if at day 40 the computed $R_{e}$ value without control
is 1.4 and the target fraction is $0.8$, then sufficient testing and
distancing controls are applied to reduce $R_{e}$ to a value of $1.4\cdot
0.8=1.12$. The combination of testing and distancing controls used to achieve
this value is computed using the same algorithm as was used to compute Figures
5. The user-defined parameters for these strategies are the $R_{e}$ ratio and
the date at which control starts.

The third class of strategy resembles the second, except that instead of
targeting a given fraction of $R_{e}$, the daily control measures are chosen
so as to achieve a fixed $R_{e}$ value between 0 and 1. If it is not possible
to achieve the target $R_{e}$ even with the maximum control limits, then
maximum controls are applied. The user-defined parameters for these strategies
are the $R_{e}$ target value and the date at which control starts.

We used simulations to evaluate and compare the effectiveness of these three
classes of strategies. Simulations used the parameters in Table 1. In
addition, the simulation assumed an exposed population of 150 low-risk and 50
high-risk individuals at time $t=0$, out of a total population of 1.34 million
low-risk and 423,000 high-risk individuals. Treatment is continued until the
number of exposed and infectious individuals reaches 10, at which point it is
assumed that the disease can be contained by targeted measures without the
need for population-wide control. The simulation was continued for 180 days.
Control strategies were updated on a daily basis.

We also simulated three parallel strategies in which high-risk individuals
were given maximum protection. In these strategies, applying controls to the
high-risk population was prioritized. Specifically, controls were only applied
to the low-risk population if the target budget or $R_{e}$ value could not be
met through control measures applied to the high-risk population. For example,
suppose that we use the third strategy and the target $R_{e}$ value on day 20
is 0.9. In a case where distancing and testing applied only to the high-risk
population is sufficient to achieve the target, then on day 20 no controls are
applied to the low-risk population. On the other hand, in a case where maximum
distancing and testing on the high-risk population still fails to reach
$R_{e}=0.9$, then on day 20 maximum control would be applied to the high-risk
population, and additional controls would also be placed on the low-risk
population so that $R_{e}=0.9$ can be achieved.

Figures 12–14 show the cost and timing characteristics of the two variants of
the three types of strategies considered. Each plot shows the costs (color
level) and deaths (white contour lines) for each combination of policy start
day ($x$ axis) and policy severity level ($y$ axis). For each figure,
Subfigure (a) shows the regular case where controls are chosen to reduce
$R_{e}$ at the lowest cost; while Subfigure (b) shows the results of policies
that first prioritize controls on high-risk individuals.

Figure 12 shows results for daily budget-based strategies. The black stars on
each white contour indicate the control policy start date and daily budget
that minimize the total control cost for the number of deaths specified by the
contour. From Figure 12(a) we see that regular strategies give a death range
of 1000–80,000 and a cost range of about 7.5–0.6 billion USD. Optimum start
dates range from day 0 for 1000 deaths to day 19 for 60,000 deaths. For the
high-risk prioritizing strategies showed in Figure 12(b), costs are about 0.5
billion USD higher, while the optimum policy start dates are slightly later,
ranging up to day 23 for 80,000 deaths. However, regardless of start date and
death level the regular strategy will cost less than a high-risk prioritizing
strategy with the same start date and deaths. Both figures show the critical
role of start date. For example, if a regular strategy with a daily budget $30
million is started at day 15, then deaths are limited to 40,000 and the total
cost is $5.5 billion. However, if control is delayed for one week, then to
acheive the same number of deaths the daily budget must be raised to $35
million, and the total cost is about $6.5 billion. If the start date is after
day 30, it is not possible to achieve less than 80,000 deaths if the daily
budget is limited to $50 million. On the other hand, if the control policy is
started too early then the total cost is also increased: a control policy
starting at day 0 that obtains 40,000 deaths has a daily budget limit of $30
million, and total cost of about $5.5 billion which is $0.5 more than the
policy starting on day 20. In general, a lower death target will require an
earlier start date for the cost-minimizing strategy.

(a) Strategies that minimize $R_{e}$

(b) Strategies that prioritize high-risk

Figure 12: Daily budget and starting day for adjusting the number of deaths.

Figure 13 shows results for strategies that produce a constant proportional
reduction in $R_{e}$ on a daily basis. Most of the observations for budget-
based strategies also apply to these strategies as well. From 13(a), we see
that an early intervention is critical in reducing the number of deaths for
aggressive strategies that produce large decreases in $R_{e}$; for example, if
control starts after day 18 it is not possible to obtain less than 5,000
deaths. Start date is much less important for milder strategies: for basic
strategies that attain 60,000 or more deaths, day 18 is optimal. High-risk
prioritizing strategies are expensive: reducing to 5,000 deaths requires at
least $7.5 billion USD, regardless of start date. Compared to basic
strategies, optimal start dates are delayed: for 60,000 deaths, the optimal
start date for the basic strategy is day 18, with total cost about $4 billion,
while the optimal high-risk prioritizing strategy for 60,000 deaths starts on
day 23, with total cost about $4.5 billion. For high-risk prioritizing
strategies, between start dates 20 and 33 the number of deaths increases
rapidly, while the cost decreases rapidly. In this start date range, both
deaths and costs are nearly independent of the target $R_{e}$ fraction.

(a) Strategies that minimize $R_{e}$

(b) Strategies that prioritize high-risk

Figure 13: Deaths number and costs depending on target $R_{e}$ fraction and
control starting day strategies.

Figures 14 (a) and (b) show costs and deaths for control strategies that
target fixed $R_{e}$ values, for basic control strategies and strategies that
prioritize the high risk group respectively. These figures resemble each
other, showing that strategies targeting high-risk give nearly the same
results as basic strategies. The cost and start date for optimal strategies at
each death level is not strongly dependent on the target value of $R_{e}$,
although for most death levels setting the target $R_{e}=1$ gives the lowest
cost control. Optimal strategies producing lower deaths must be initiated
earlier (e.g. to attain 1000 deaths, it is best to start on day 10 using
target $R_{e}=1$, while the optimal strategy corresponding to 80,000 deaths
begins around day 30, with $R_{e}\approx 0.9$.

(a) Strategies that minimize $R_{e}$

(b) Strategies that prioritize high-risk

Figure 14: Deaths number and costs depending on target $R_{e}$ value and
control starting day strategies.

Figures 15(a) and (b) show the impact of starting control day on the deaths
and costs that can be obtained from the different types of optimal strategies.
In each figure, the three strategy types (constant budget, constant $R_{e}$
fraction, and constant $R_{e}$ target) are compared at 3 different levels of
total control cost: $2, $4 and $6 billion USD. The vertical axis shows deaths
resulting from each of the strategies at the given control start date with the
given total control cost. As before, Subfigures (a) and (b) correspond to
basic and high-risk prioritizing strategies, respectively.

Figure 15(a) shows that for all three cost levels, the basic $R_{e}$ target
strategy achieves the lowest number of deaths at the latest optimal start
date, indicating the superior performance of this type of strategy. However,
if the $R_{e}$ target strategy is delayed past the optimal start date, the
effectiveness is drastically reduced. For example, a target $R_{e}$ strategy
with total cost $4 billion starting at day 10 can reach 33,000 deaths, but if
the start date is delayed by 4 additional days the numbers of deaths rises to
almost 60,000. Apart from $R_{e}$ target strategies, the other two strategies
produce similar deaths for the same cost and start date. For all strategies,
earlier optimal start times are associated with higher-cost strategies that
attain fewer deaths.

Figure 15(b) compares strategies that apply maximum control to high-risk
individuals. Similar relations between strategies hold as were noted for
Figure 15(a). A comparison between Figures (a) and (b) shows that high-risk
prioritizing strategies yield higher deaths for similar costs, and are thus
less effective: for high-risk prioritizing strategies at the $2, $4, and $6
billion USD level respectively, optimal deaths for $R_{e}$ target strategies
are 57,000, 33,000, and 17,000 compared to 53,000, 30,000, and 15,000 which
may be obtained by basic $R_{e}$ target strategies at the same cost levels.
Once again, the constant budget and $R_{e}$ fraction strategies yield similar
outcomes.

(a) Strategies that minimize $R_{e}$

(b) Strategies that prioritize high-risk

Figure 15: Number of the deaths obtained from different control strategy types
depending on the control start day, for different budget levels.

Figures 16 shows Pareto fronts for the 3 strategy types, where deaths and
control costs are the two competing factors. For each strategy type, two
Pareto fronts are shown: one front is based on overall optimal strategies,
while the other front restricts strategies to those which begin after 21 days.
The $x$ axis representes the control cost while the vertical $y$ axis shows
the number of deaths depending on the current cost. As above, Subfigures (a)
and (b) give results for basic and high risk-prioritizing strategies,
respectively.

Figure 16(a) In most cases, target $R_{e}$ is best except for very high
control costs and low deaths. The advantage of target $R_{e}$ is especially
large for mid-range strategies that produce about 30,000 deaths at a cost of
$4 billion USD. In contrast, to obtain the same number of deaths with the
other two strategy types costs $5.5 billion. Alternatively, using the $4
billion for constant-budget or target $R_{e}$ fraction will produce an
additional 20,000 deaths compared to a strategy with target $R_{e}$ value.
Delaying the control start date has large costs: compared to the above-
mentioned target $R_{e}$ strategy which reaches 30,000 deaths and costs $4
billion, a 21-day delayed target $R_{e}$ strategy will either cost an
additional $1.5 billion at the same death level, or will result in an
additional 9,000 deaths for the same cost. The observations for 16(a) are
still valid for 16(b) except that the cost-to-death tradeoffs are slightly
more unfavorable.

(a) Basic strategies that minimize $R_{e}$

(b) Strategies that prioritize high-risk

Figure 16: Number of the deaths associated to control cost depending on the
starting control date for the three different strategies.

Figure 17 indicates the application of the four different controls (low and
high risk testing and distancing) for the basic versions of the three
different strategy types, for optimal controls associated with different death
outcomes. On each figure, the vertical axis indicates the number of deaths
acheived by the strategy; the horizontal axis gives the date and the color
indicates the level of each control, according to the colorbar accompanying
each figure. For example, for the optimal $R_{e}$ fraction strategy that
achieves 80,000 deaths, the time progression of low risk social distancing may
be obtained from the third plot in the second row by looking across the plot
at the 80,000 level on the vertical axis. The $R_{e}$ target strategies (last
row of plots) begin with high levels of low-risk (third plot in the row) and
high-risk distancing (fourth plot), and then transition towards testing in the
later stages (first two plots). In contrast, the other two strategy types
prioritize distancing (especially distancing of the low-risk group) over
testing throughout the period of control. In budget-based strategies,
typically each control is applied at a nearly constant level throughout the
period of control as evidenced by the horizontal color patterns in the first
row of plots. For between 20,000-120,000 deaths, the optimal start dates for
all three strategy types are very close to day 20. The figures show that for
deaths below about 50,000, the control continues up the end of the period,
indicating that the disease is still extant and herd immunity has not yet been
reached.

Figure 18 is analogous to Figure 17, and represents the control history for
the four controls for optimal high-risk prioritizing controls at different
death levels. As reflected in the figures, testing and distancing for the high
risk group is at maximum level throughout the control period, while testing
and distancing for the low risk group resembles Figure 17 but at somewhat
lower levels.

Figure 19 shows the progression over time of current infected, current
hospitalized, cumulative recovered, and cumulative deaths corresponding to the
three basic strategies shown in Figure 17. Both deaths and recovered increase
with decreasing intensity of control. All three strategies show a peak of
infected around 30 days, and a hospitalized peak around 40 days: both peaks
are flatter with the $R_{e}$ target strategy compared to the constant budget
and $R_{e}$ fraction strategies. For strategies with high levels of control,
infections and hospitalizations persist up to the end of the 180-period. For
example, for constant $R_{e}$ strategies that reduce deaths below 60,000, even
at 180 days, current infection and hospitalizations up to about 50,000 and
5,000 respectively may be experienced. Figure 20 is analogous to Figure 19,
except that current infected, current hospitalized, cumulative recovered, and
cumulative deaths are shown for the three strategy types where high risk
individuals are subjected to maximum control. The above observations made for
Figure 19 also apply to Figure 20.

Figure 17: Control levels (low risk testing, high risk testing, low risk
distancing, high risk distancing) for constant budget (first row), constant
$R_{e}$ fraction (second row), and constant $R_{e}$ target (third row) basic
control strategies. The color bar indicates control levels (0 to 0.8 for
distancing, 0 to 0.66 for testing) over time (horizontal axis) for strategies
that achieve total deaths from 0 to 120,000 (vertical axis).

Figure 18: Control levels (low risk testing, high risk testing, low risk
distancing, high risk distancing) for constant budget (first row), constant
$R_{e}$ fraction (second row), and constant $R_{e}$ target (third row) high
risk prioritizing control strategies. Plot arrangement and color scales are
the same as in Figure 17

Figure 19: Infections, hospitalizations, recovered, and deaths as a function
of time (horizontal axis) and total number of deaths (vertical axis), for
constant budget (first row), constant $R_{e}$ fraction (second row), and
constant $R_{e}$ target (third row) basic control strategies. Color scales for
plots correspond to infections (first column), hospitalizations (second
column), recovereds (third column) and deaths (fourth column).

Figure 20: Infections, hospitalizations, recovered, and deaths as a function
of time and total number of deaths, for constant budget (first row), constant
$R_{e}$ fraction (second row), and constant $R_{e}$ target (third row) control
strategies that maximize controls on the high risk subpopulation. The plots
are arranged as in Figure 19.

##  4 Conclusion

In this paper, an SEIR epidemic model of COVID-19 in the city of Houston, TX
USA is presented under testing and social distancing controls with low and
high risk population groups. The basic and effective reproduction numbers for
the model have been calculated, and the effective reproduction number has been
explored as a key parameter in understanding the dynamics of disease and its
relationship with different control measures and strategies. Comprehensive
graphical representations of the dependence of effective reproduction number
$R_{e}$ and control cost on control levels have been presented under different
levels of population immunity (Figures 2-5). Restricted strategies that used
only one control (either distancing or testing) and/or targeted only part of
the population (either high or low risk) were incapable of reducing $R_{e}$
below 1, implying that such strategies are not sufficient to prevent disease
spreading except in cases of high levels of population immunity.

A sensitivity analysis was performed, which showed that both costs and $R_{e}$
as well as optimal distancing levels are highly sensitive to the baseline
transmission rate, and less sensitive to symptomatic proportion and the
relative infectiousness of asymptomatic individuals (Figures 6-7). Hence given
the difficulty in obtaining exact values for baseline transmission rate, it
may be difficult to determine precisely the best distancing policy for given
conditions. Overall costs were also found to be highly sensitive to model cost
parameters, particularly the increasing costs associated with diminishing
returns from social distancing (Figures 8-9).

Optimal instantaneous strategies which combined distancing and testing were
computed (Figures 10-11). The results showed that optimal strategies utilized
distancing primarily (especially at high control levels), and were applied
nearly equally to both population groups. Three different types of long-term
control strategies based on $R_{e}$ were simulated. The simulations confirm
that the starting date of the control has an enormous effect on the
effectiveness of the strategies in preventing deaths, and the minimum number
of deaths increases rapidly if controls are delayed past a certain point
(Figures 12-14). However, the results showed that it is not most cost-
effective to begin serious controls too soon: for example, it was found that
the best $R_{e}$ targeting strategy that can reduce deaths to 30,000 was begun
on day 9 (Figure 15). Although the more intensive (and more costly) strategies
reduced the number of deaths, they also do not entirely eliminate the
infection, and it was found that all strategies which reduced deaths below
60,000 required continuing control past the 180 day period of the simulation
(Figures 17-20). Strategies that set a target value for $R_{e}$ were found to
be most cost-effective, even when started later than other strategies (Figure
15). These strategies are characterized by an initial very high level of
distancing, which is later reduced and replaced by higher levels of testing
(17-18). Strategies that focused primarily on applying controls to the high-
risk population were found to be less cost-effective than strategies that were
applied evenly across the entire population (Figure 15).

The situation with the COVID epidemic, as with previous epidemics, is
continuously changing. The development of vaccines introduces new
possibilities for control. The baseline model we have developed in this
research can readily be modified to accommodate such changes.

## Abbreviations

  1. 1.

COVID-19 : Coronavirus disease 2019,

  2. 2.

SEIR: Susceptible, Exposed, Infectious, Recovered,

  3. 3.

TX, USA: Texas, United States of America,

  4. 4.

WHO: World Health Organization,

  5. 5.

SIR: Susceptible, Infectious, Recovered,

  6. 6.

SARS-CoV-2: Severe Acute Respiratory Syndrome Coronavirus 2,

  7. 7.

PCR: Polymerase Chain Reaction,

  8. 8.

COVID: Coronavirus disease .

## Acknowledgements

The support of this research through the German Academic Exchange Service
(DAAD) and the anonymous reviewers are hereby acknowledged.

## Conflict of interest

The authors declare that they have no conflict of interest.

## Author’s contributions

All authors evenly contributed to the whole work. All authors read and
approved the final manuscript.

## Availability of data and materials

The software used for the simulations in this paper is contained in the
project “COVID19 Cost Effectiveness Of Control Measures”,
DOI:10.5281/zenodo.4474866
(https://github.com/LuisEVT/COVID19_Cost_Effectiveness_Of_Control_Measures).

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