# Interplay between COVID-19 vaccines and social measures for ending the SARS-
CoV-2 pandemic

Glenn Young  Department of Mathematics, Kennesaw State University, Marietta,
GA, USA  Pengcheng Xiao  Department of Mathematics, Kennesaw State University,
Marietta, GA, USA  Kenneth Newcomb  Center for Global Health Infectious
Disease Research, University of South Florida, Tampa, FL, USA  Edwin Michael
Center for Global Health Infectious Disease Research, University of South
Florida, Tampa, FL, USA

###### Abstract

The development and authorization of COVID-19 vaccines has provided the
clearest path forward to eliminate community spread hence end the ongoing
SARS-CoV-2 pandemic. However, the limited pace at which the vaccine can be
administered motivates the question, to what extent must we continue to adhere
to social intervention measures such as mask wearing and social distancing? To
address this question, we develop a mathematical model of COVID-19 spread
incorporating both vaccine dynamics and socio- epidemiological parameters. We
use this model to study two important measures of disease control and
eradication, the effective reproductive number $R_{t}$ and the peak intensive
care unit (ICU) caseload, over three key parameters: social measure adherence,
vaccination rate, and vaccination coverage. Our results suggest that, due to
the slow pace of vaccine administration, social measures must be maintained by
a large proportion of the population until a sufficient proportion of the
population becomes vaccinated for the pandemic to be eradicated. By contrast,
with reduced adherence to social measures, hospital ICU cases will greatly
exceed capacity, resulting in increased avoidable loss of life. These findings
highlight the complex interplays involved between vaccination and social
protective measures, and indicate the practical importance of continuing with
extent social measures while vaccines are scaled up to allow the development
of the herd immunity needed to end or control SARS-CoV-2 sustainably.

##  1 Introduction

The advent of COVID-19 vaccines and mass vaccinations of populations have led
to widespread public expectation that we may be able to end the ongoing SARS-
CoV-2 pandemic in some economically advanced countries by as early as the of
beginning 2022 [1]. While the pace at which these vaccines have been developed
and authorized by governments for population-wide usage has been unprecedented
[2] reflecting the desire to fast track the ending or control of the pandemic
given the socio-economic costs of protracted non-pharmaceutical interventions
(NPIs), such as cyclical lockdowns and social distancing measures [3], it is
also clear that several features of the current vaccines and vaccination
strategies for achieving this goal remain unresolved [4, 5].

First, it is important to consider that vaccines serve two major purposes: to
protect the individual from contracting the disease and to stop the
transmission of community infection. While the initial vaccine trial data for
the 3 major vaccines approved for use thus far in developed world settings,
viz. Pfizer-Biontech, Moderna, and AstraZeneca-Oxford, indicate that these
could induce very high levels of protection (70-90%) against symptomatic
disease [6], more recent data with regard to the AstraZeneca vaccine suggests
that vaccination may also reduce community transmission of the virus
significantly [7]. These data suggest that both disease outcomes and
transmission could be significantly reduced in communities as a result of mass
deployments of these vaccines. A key factor, however, is that in both cases
current vaccines are not 100% protective. Second, it is apparent that the
number of vaccines initially available and the logistical challenges connected
with their delivery will hamper the rapid vaccination of a population, which
will prolong the time to eradicating the disease through vaccination in
populations [8]. An added challenge is the reduced effectiveness of these
vaccines as currently formulated against newly emerging virus variants [9].

A third important factor is to consider the epidemiological and social
contexts in which vaccinations will take place. This is important because many
populations undergoing vaccinations will have already experienced one or more
waves of COVID-19, as a result of which some level of natural immunity to
SARS-CoV-2 will likely to be in operation in these communities. Such pre-
existing immunity could indicate that the vaccine coverages to end the
pandemic (reduce the prevailing effective reproduction number, $R_{t}$ to
below 1 sustainably) need not be too high even if vaccine effectiveness is not
perfect [10], increasing the prognosis for using the current vaccines for
ending the pandemic. On the other hand, these populations are also currently
experiencing various levels of NPIs [3]. Such social mitigation or containment
measures, while protecting the susceptible fraction from infection will act
also to depress the development of natural immunity in the population. These
outcomes suggest that there may be complex interactions between the two
interventions, a better understanding of which will be crucial for determining
how best to optimally deploy these tools for controlling or ending the
pandemic. Further, investigating such interactions will also be important to
fully understanding the implications of relaxing these NPIs as vaccinations
roll out [10, 11].

Here, we extend our existing data-driven socio-epidemiological SEIR-based
COVID-19 mathematical model [12] by incorporating imperfect vaccination
dynamics in order to undertake a theoretical investigation of this topic. To
enhance realism we use the basic model calibrated to infection data on the
course of the pandemic in the Tampa Bay region, and use the resulting model to
investigate the interplay between vaccination and social protective measures
for effective control or elimination of the pandemic. In particular, we
explore the dynamical implications of imperfect vaccine effectiveness, vaccine
rollout rates, and coverages of vaccination and social measures on the
eradication of the disease in the community, and the effects these will have
for virus transmission and critical care requirements. We also inspect the
effect that the current vaccination roll out will have on the extent to which
a population may relax social measures to return to normalcy.

##  2 Methods

###  2.1 Model formulation

Here we develop an extended SEIR model to assess population-level disease
dynamics of COVID transmission. We consider a population of fixed size that we
divide into eleven interacting subpopulations: susceptible without access to a
vaccine ($S$), vaccinated susceptible ($V$), unvaccinated susceptible (with
access to a vaccine) ($X$), exposed but not intectious ($E$), asymptomatic
infected ($I_{a}$), pre-symptomatic infected ($I_{p}$), mildly symptomatic
infected ($I_{m}$), hospitalized infected ($I_{h}$), critical care infected
($I_{c}$), recovered ($R$), and died ($D$). Since the population size is
fixed, we can impose the condition that
$S+X+V+E+I_{a}+I_{p}+I_{m}+I_{h}+I_{c}+R+D=1,$ and each population can
therefore be interpreted as a proportion of the total population.

Our model is visualized as a diagram in Figure 1. Importantly, we assume that
asymptomatic, pre-symptomatic, and mildly symptomatic individuals transmit the
disease at the same rate $\beta$, and that these are the only sources of
transmission. The transmission rate is reduced due to social measures (face
masks, social distancing, self quarantine, etc) by a factor of $1-ac$ among
all infectious individuals, where $c$ is the efficacy of social measures and
$a$ is the proportion of the population adhering to social measures.

Figure 1: Model diagram

The vaccine becomes available to the population at rate $\mu$, and individuals
can choose whether or not to get the vaccine. Specifically, proportion $\phi$
of the susceptible population $S$ enters the vaccinated population $V$ at rate
$\mu$, while proportion $1-\phi$ of the susceptible population enters the
unvaccinated class $X$ at the same rate. This parameter $\phi$ allows us to
analyze the effects of vaccine coverage, specifically as a consequence of
individuals who are unwilling to receive the vaccine [13]. Vaccinated
individuals are infected at a rate reduced by $1-\xi$, where $\xi$ is the
efficacy of the vaccine, while the unvaccinated class $X$ has the same
dynamics as the susceptible without access to a vaccination class $S$.

With these assumptions, our model is given by the following system of eleven
ordinary differential equations:

$\displaystyle S^{\prime}$ $\displaystyle=-(1-d)\beta S(I_{a}+I_{p}+I_{m})-\mu
S$ (1) $\displaystyle X^{\prime}$ $\displaystyle=\mu(1-\phi)S-(1-d)\beta
X(I_{a}+I_{p}+I_{m})$ $\displaystyle V^{\prime}$
$\displaystyle=\mu\phi-(1-\xi)(1-d)\beta V(I_{a}+I_{p}+I_{m})$ $\displaystyle
E^{\prime}$
$\displaystyle=(1-d)\beta(I_{a}+I_{p}+I_{m})\left[S+X+(1-\xi)V\right]-\sigma
E$ $\displaystyle I_{a}^{\prime}$ $\displaystyle=\sigma\rho E-\gamma_{a}I_{a}$
$\displaystyle I_{p}^{\prime}$ $\displaystyle=\sigma(1-\rho)E-\delta_{1}I_{p}$
$\displaystyle I_{m}^{\prime}$
$\displaystyle=\delta_{1}I_{p}-x_{1}\delta_{2}I_{m}-(1-x_{1})\gamma_{m}I_{m}$
$\displaystyle I_{h}^{\prime}$
$\displaystyle=x_{1}\delta_{2}I_{m}-x_{2}\delta_{3}I_{h}-(1-x_{2})\gamma_{h}I_{h}$
$\displaystyle I_{c}^{\prime}$
$\displaystyle=x_{2}\delta_{3}I_{h}-(1-x_{3})\gamma_{c}I_{c}-x_{3}mI_{c}$
$\displaystyle R^{\prime}$
$\displaystyle=\gamma_{a}I_{a}+(1-x_{1})\gamma_{m}I_{m}+(1-x_{2})\gamma_{h}I_{h}+(1-x_{3})\gamma_{c}I_{c}$
$\displaystyle D^{\prime}$ $\displaystyle=x_{3}mI_{c}$

The state variables and parameters are defined in Tables 1 and 2,
respectively. The parameters are estimated by the methods described in
Appendix 2.2.

State Variable
---
$S$ | Susceptible population without access to a vaccine
$X$ | Susceptible, unvaccinated population with access to a vaccine
$V$ | Susceptible, vaccinated population
$E$ | Exposed population (pre-infectious)
$I_{a}$ | Asymptomatic, infectious population
$I_{p}$ | Pre-symptomatic, infectious population
$I_{m}$ | Symptomatic, infectious population without need of hospitalization
$I_{h}$ | Hospitalized infected population
$I_{c}$ | Infected population in intensive care unit (ICU)
$R$ | Recovered population
$D$ | Deceased population
Table 1:  State variables. Each variable should be interpreted as a proportion
of the total population.

###  2.2 Parameter estimation

We used a Monte Carlo-based Bayesian Melding approach to parameterize the base
model using case notification, death, and mobility data reported for the Tampa
Bay region for the period between March 10th and August 24th 2020. (details of
methods provided in Newcomb et al. 2020). Briefly, all social and
epidemiological model parameters that could not be fixed at initiation were
sequentially updated using 10-day blocks or segments of data between the above
estimated period from their initial prior values using this procedure, and the
final model thus estimated was used for the simulations carried out in this
paper. Confirmed case data for the four counties comprising Tampa Bay, viz.
Hillsborough, Pasco, Pinelles and Polk, were obtained from the Johns Hopkins
University Coronavirus Resource Center (Dong et al. 2020). Mobility data serve
as an estimate for population mixing and the fraction of the population under
restricted movement in the counties concerned were obtained from the location
data firm, Unacast (https://www.unacast.com/covid19/social-distancing-
scoreboard).

Parameter | Value
---|---
$\beta$ | Transmission rate | 0.76
$a$ | Proportion of population adhering to social measures | varies
$c$ | Reduction in transmission due to social measures | 0.85
$\phi$ | Vaccine coverage | varies
$\xi$ | Vaccine efficacy | 0.9
$\mu$ | Rate of vaccination | 0.02
$\sigma$ | Rate of symptom onset among exposed | 0.32
$\rho$ | Proportion of expose who never develop symptoms | 0.37
$\gamma_{a}$ | Rate of recovery among $I_{a}$ | 0.23
$\gamma_{m}$ | Rate of recovery among $I_{m}$ | 0.23
$\gamma_{h}$ | Rate of recovery among $I_{h}$ | 0.22
$\gamma_{c}$ | Rate of recovery among $I_{c}$ | 0.23
$\delta_{1}$ | Rate of progressing from $I_{p}$ to $I_{m}$ | 0.69
$\delta_{2}$ | Rate of progressing from $I_{m}$ to $I_{h}$ | 0.15
$\delta_{3}$ | Rate of progressing from $I_{h}$ to $I_{c}$ | 0.55
$x_{1}$ | Proportion of mild cases that progress to hospital | 0.18
$x_{2}$ | Proportion of hospital cases that process to ICU | 0.25
$x_{3}$ | Proportion of ICU cases that die | 0.51
$m$ | Mortality rate among ICU patients | 0.37
Table 2:  Parameters. The value for each parameter is used throughout this
work unless stated otherwise. All rate parameters are measured in units of
(day)-1.

###  2.3 Effective reproduction number

The effective reproductive number $R_{t}$ quantifies the average number of
secondary infections caused by each new infection. We can calculate $R_{t}$
using the next generation matrix method developed in [14]. The next-generation
matrices for system (1) are

$F=\begin{bmatrix}0&\lambda&\lambda&\lambda\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}$

and

$V=\begin{bmatrix}-\sigma&0&0&0\\\ -\rho\sigma&\gamma_{A}&0&0\\\
-(1-p)\sigma&0&\delta_{1}&0\\\
0&0&-\delta_{1}&x_{1}\delta_{1}+(1-x_{1})\gamma_{M}\end{bmatrix},$

where

$\lambda=(1-ac)\beta\left(S+X+(1-\xi)V\right)$

is the force of infection.

Then the effective reproduction number $R_{t}$ is exactly the spectral radius
of $FV^{-1}$:

$R_{t}=\lambda\left(\frac{1-\rho}{\delta_{1}}+\frac{1-\rho}{(1-x_{1})\gamma_{M}+x_{1}\delta_{2}}+\frac{\rho}{\gamma_{A}}\right).$
(2)

We use this formula to assess the viability of vaccination programs on disease
eradication in the following sections.

##  3 Results

###  3.1 Disease control dynamics

We begin by determining the extent to which a population can relax social
measures once a vaccine becomes available and still control the virus. Here,
we use simulations of the number of individuals requiring intensive care as a
measure of control, and thus consequently, we consider the relationship
between vaccine coverage $\phi$ and social measure compliance $a$ on ICU
hospitalization to address this question. Figure 2 shows ICU hospitalizations
over time under various vaccine coverages and social measures. The red curve
shows ICU cases without any vaccine or social measures, serving as a baseline
for comparison. The black curve shows these cases with some vaccination and no
social measures; the green curve shows ICU cases with strong social measures
and no vaccine; the blue curve shows ICU cases with both vaccine coverage and
social measures. Of course, greater vaccine coverage and stronger social
measures result in fewer cases; however, social measures reduce the incidence
of cases requiring intensive care much more than vaccine coverage. This is due
to the slow vaccination rate $\mu$, as we show in Section 3.3.

Figure 2: ICU cases under various vaccination coverages and social measures.
The vaccination rate and efficacy are $\mu=0.02$ and $\xi=0.94$, respectively.
The right panel shows the same blue and green curves from the left panel, but
zoomed in. All simulations performed using XPPAUT [15].

###  3.2 Conditions for eradication

We use the effective reproduction number $R_{t}$ to study strategies by which
the virus can be eradicated. Importantly, $R_{t}=R_{t}(t)$ is a function of
time due to its dependence on $S$, $X$, and $V$: as the susceptible
populations decrease, so does $R_{t}$. We consider the temporal dynamics of
$R_{t}$ in the following section. Here, we consider the idealized case in
which proportion $\phi$ of the population is instantaneously vaccinated,
$V=\phi S_{r}$ and $X=(1-\phi)S_{r}$, where $S_{r}$ is a new parameter
representing the remaining susceptible proportion of the population at the
time the vaccine is administered (that is, $1-S_{r}$ is the proportion of the
population who are currently infected, recovered, or deceased). In this case,
$\lambda$ can be written

$\lambda=(1-ac)\beta S_{r}\left(1-\xi\phi\right),$

and so

$R_{t}=(1-ac)\beta
S_{r}\left(1-\xi\phi\right)\left(\frac{1-\rho}{\delta_{1}}+\frac{1-\rho}{(1-x_{1})\gamma_{M}+x_{1}\delta_{2}}+\frac{\rho}{\gamma_{A}}\right).$
(3)

The ideal goal in eradicating the disease is to permanently reduce the
effective reproductive number below the threshold $R_{t}=1$. Figure 3 shows
curves in parameter space satisfying $R_{t}=1$ with $\xi=0.94$. On the left,
curves are plotted over vaccine coverage $\phi$ and social measure compliance
$a$ and remaining susceptible population $S_{r}=1$ (red curve) and $S_{r}=0.8$
(blue curve). In order to achieve $R_{t}<1$, $(\phi,a)$ pairs must lie above
the curve. Note that disease eradication without social measures ($a=0$) would
require more than $80\%$ of the population to receive a vaccine that is $94\%$
effective in a naive population ($S_{r}=1$); a population in which 20% have
already been exposed would require approximately 75% to receive a vaccine in
order to eradicate the disease without social measures. On the right, we show
curves defined by $R_{t}=1$ over vaccine coverage $\phi$ and remaining
susceptible population $S_{r}$ for varied social measures. For each fixed
value of $a$, $(\phi,S_{r})$ pairs must be below the curve to drive $R_{t}<1$.
Without social measures in place (red curve), approximately 75% of the
population would have to be exposed to the virus before the disease is
eradicated without a vaccine. In general, the larger the remaining susceptible
population, the higher the vaccination coverage required to eradicate the
disease. Importantly, the vaccine coverage threshold necessary to drive
$R_{t}<1$ decreases with increased social measures.

Figure 3: Vaccination and social measure thresholds to achieve $R_{t}<1$. Each
curve defines $R_{t}=1$ over varied parameters with $\xi=0.94$. (Left)
$R_{t}=1$ plotted over vaccine coverage $\phi$ and social measure compliance
$a$. The region above each curve represents parameter pairs for which
$R_{t}<1$. (Right) $R_{t}=1$ plotted over vaccine coverage $\phi$ and
remaining susceptible population $S_{r}$ with no, medium, and high proportions
of social measure compliance. The region below each curve represents parameter
pairs for which $R_{t}<1$.

###  3.3 Time to eradication: interactions between vaccination coverage and
rate and social measures

Vaccine rollout will not be instantaneous; it will likely take months to
vaccinate a majority of the population. We therefore must consider the
simultaneous effects of infection dynamics and slow vaccination rates on
disease eradication, which requires considering the effective reproduction
number as a function of time:

$R_{t}(t)=(1-ac)\beta\left(S(t)+X(t)+(1-\xi)V(t)\right)\left(\frac{1-\rho}{\delta_{1}}+\frac{1-\rho}{(1-x_{1})\gamma_{M}+x_{1}\delta_{2}}+\frac{\rho}{\gamma_{A}}\right).$
(4)

As the three susceptible populations $S$, $X$, and $V$ change over time via
infection and vaccination dynamics, $R_{t}(t)$ strictly decreases. We denote
the time at which $R_{t}(t)$ decreases below 1 by $t_{h}$; that is,
$R_{t}(t_{h})=1$. Time $t_{h}$ marks the beginning of the end of community
spread, and we therefore refer to $t_{h}$ as the time to eradication. The left
panel of Figure 4 shows $t_{h}$ as a function of vaccination coverage $\phi$
for varied social measure compliance $a$ with vaccination rate $\mu=0.02$. The
time to eradication remains nearly constant over all $\phi$ for low social
measure compliance ($a=0,0.5$; red and blue curves): this is due to the slow
vaccination rate $\mu$, and the comparably fast infection rate due to low
social measure compliance. That is, for sufficiently small $a$, the virus
spreads quickly through the population, infecting the susceptible population
$S$ much more quickly than the susceptibles become vaccinated. Thus,
eradication is achieved primarily through infection, rather than through
vaccination. For high social measure compliance ($a=0.8$; black curve),
$t_{h}$ is large for small $\phi$. This is because, for large $a$, the
infection dynamics are slowed down, but because $\phi$ is small, the
vaccination rate is also slow. The two mechanisms by which eradication is
achieved (infection and vaccination) are therefore both slow, and so $t_{h}$
is large. As $\phi$ increases, however, $t_{h}$ decreases dramatically. The
effective reproduction number $R_{t}$ decreases as $a$ increases, and thus
achieving $R_{t}=1$ requires less vaccination and infection for large $a$.
Thus, despite the slow vaccination rate $\mu$, there is a critical $\phi$
value past which the susceptible population becomes vaccination quickly enough
so that $R_{t}$ decreases below 1 due primarily to vaccine administration. In
other words, the vaccination timescale overtakes the infection rate timescale
for sufficiently large $\phi$.

The right panel of Figure 4 shows the maximum ICU case load (that is, the
peaks of the curves in Figure 2) as a function of vaccination coverage for
varied $a$. Unsurprisingly, as $\phi$ increases, the maximum ICU load
decreases for each $a$. However, the peak ICU load remains comparatively high
for $a=0$ and $a=0.5$ (red and blue curves, respectively) compared with
$a=0.8$ (black curve). For this latter case, the peak ICU load decreases
dramatically from $\phi=0$ until around $\phi=0.4$, then remains low for all
larger $\phi$. ICU capacity in most states is between $10^{-4}$ and $3\times
10^{-4}$: peak ICU cases only remain below this threshold for high social
measure compliance and vaccination coverage.

Figure 4: Controlling the pandemic with a vaccine depends on broad coverage
and sustained social measure compliance. (Left) Time until $R_{t}<1$ over
$\phi$. (Right) Peak ICU cases over $\phi$ (log scale). In both panels,
$\mu=0.02$, and the red curve corresponds to no social measures ($a=0$), the
blue curve corresponds to moderate social measure compliance ($a=0.5$), and
and the black curve corresponds to high compliance ($a=0.8$). Details on how
these figures were made are in Appendix A.

Figure 4 suggests that sustained social measures help to eradicate the disease
more efficiently than vaccination programs. This at least in part due to a
relatively slow vaccination rate: $\mu=0.02$. We now investigate the influence
of vaccination rate $\mu$ on the time to eradication and on peak ICU cases.
The left panel of Figure 5 shows the time until the disease is eradicated,
$t_{h}$, as a function of vaccination rate, $\mu$, with $\phi=0.6$. Without or
with sufficiently low social measure compliance ($a=0$ and $a=0.5$; red and
blue curves), the time until eradication increases with $\mu$. For both cases,
vaccination coverage $\phi$ is too low to to eradicate the disease in a
completely susceptible population (Figure 3), and consequently a non-
negligible percentage of the population must become infected before $R_{t}<1$.
As $\mu$ increases, the infection dynamics slow down, causing the time it
takes for $R_{t}$ to drop below 1 to increase. For high social measure
compliance ($a=0.8$; black curve), the time to eradication decreases with
$\mu$. When $a$ is sufficiently large, $\phi=0.6$ is large enough to eradicate
the disease through vaccination alone (Figure 3), and increasing the
vaccination rate therefore reduces the time to eradication. Thus, rapid
eradication of the virus is only achievable with sustained, widely obeyed
social measures.

The right panel of Figure 5 shows the peak ICU load over $\mu$. Naturally, the
faster the vaccine is administered to the population, the lower the peak ICU
case load will be. However, peak ICU load only remains below typical ICU
capacity ($1-3\times 10^{-4}$) for small $\mu$ when social measure compliance
is high (black curve). This again suggests that social measures must remain in
place throughout the vaccination program in order to avoid hospital strain and
associated loss of life.

Figure 5: Controlling the pandemic with a vaccine depends on fast dispersal
and sustained social measure compliance. (Left) Time until $R_{t}<1$ over
$\mu$. (Right) Peak ICU cases over $\mu$ (log scale). In both panels,
$\phi=0.6$, and the red curve corresponds to no social measures ($a=0$), the
blue curve corresponds to moderate social measure compliance ($a=0.5$), and
and the black curve corresponds to high compliance ($a=0.8$). Details on how
these figures were made are in Appendix A.

##  4 Discussion and Conclusion

We introduce an extended SEIR socio-epidemiological model incorporating
vaccination dynamics to evaluate the interactions between vaccination and
social measures for controlling or ending the spread of COVID-19. Following
standard analytical techniques [14], we derived an explicit form for the
effective reproduction number $R_{t}$. This value is of central concern to
controlling the pandemic: through a combination of natural infection, social
measures, and vaccination administration, we must drive $R_{t}<1$ in order to
eradicate the disease. Our analysis therefore focused on the influence of
social measures and vaccination rates on the time until the disease is the
eradicated, but also considered hospital demand as a function of these
interventions. Importantly, we show that while eliminating social measures
entirely might help eradicate the disease faster, the hospital demand, and
therefore death toll, are reduced dramatically with even partial adherence to
social intervention strategies.

Our analysis focused on three parameters: the proportion of the population
willing to receive a vaccine $\phi$, the proportion of the population willing
to adhere to social measures $a$, and the rate at which vaccines are
administered to the population $\mu$. Figures 3-5 summarize the major results
arising from these interactions, and suggest that, with low vaccination rate
$\mu$, sustained social measures become increasingly important to keep the
hospitalization rates low, even if a large proportion of the population are
willing to receive the vaccine (that is; even if $\phi$ is large). This
finding is consistent with previous studies under varied assumptions [10, 16,
17].

The interplay between social measures and vaccine administration is perhaps
most complicated when considering the time until eradication. When the
proportion of the population who adhere to social measures is small, the time
to eradication is relatively fast (Figures 4 and 5, left panels). This is
because, without social measures, the virus spreads quickly, thereby
increasing the number of individuals with infection-conferred immunity (or who
die due to the disease). On the other hand, when social measure adherence is
high, the time to eradication is large for low vaccination coverage and rate,
but decreases with both parameters. For low vaccination coverage or rate,
population-level immunity is primarily being conferred via infection, and
infection rates are low due to high levels of social measures. As vaccination
coverage or rate increases, however, the rate at which individuals become
vaccinated begins to outpace the rate at which individuals become infected,
and the time until eradication becomes small. Importantly, only in the case of
high social measure compliance and high vaccination rate or coverage do the
number of ICU cases remain manageable (Figures 4 and 5, right panels).

We developed our model under a set of assumptions that captured important
features of COVID-19 transmission. However, we omitted one or more realities
of COVID-19 dynamics that could quantitatively influence our results. First,
we did not include any age structure to our model. It is well known that
mortality rates due to COVID-19 are disproportionately high among elderly
populations. By including age structure, on can study targeted vaccination
programs in which the elderly are given earlier access to the vaccine. Second,
we include only a single vaccine in our model, while many with varying
efficacies are likely to enter the market before the pandemic is over [18, 19,
20]. While incorporating these features into our model would surely result in
quantitative differences, the qualitative predictions of our current model
would likely remain unchanged; that is, social measures must remain in place
throughout the vaccination campaign in order to mitigate hospital and
mortality rates. An important consideration for future work is the degree to
which variants of the SARS-CoV-2 will increase community spread [21], which
can be incorporated into our model as a second set of infected classes.

Notwithstanding these limitations, this work highlights several major
implications for the use of vaccination for either controlling or eradicating
the current pandemic. The first is that slow vaccination roll out rates mean
that continuing with currently applied social measures is imperative to
containing the clinical outcomes (demand for ICU care and deaths) of the
pandemic for a population. Only a ramped up vaccination rate will allow easing
of these social measures. The second important finding arising from the
present results is that while social measures under the current slow rate of
vaccinations will be crucial to prevent hospitalizations and the death toll
from the virus, this intervention will also delay the development of herd
immunity in the population. However, two critical results here are that at
high levels of social measures, the numbers of individual that are required to
be vaccinated to achieve $R_{t}<1$ can be significantly small, and that there
might be a vaccination coverage level past which this can be achieved. We term
this as “herd immunity due to social measures”, which will be much lower than
the corresponding herd immunity in the absence of social measures. Note,
however, that the imposition of social measures will keep a large fraction of
the population continuing to be susceptible, and while achieving the lower
level of herd immunity through vaccination under these measures will allow
interruption of transmission, any relaxation of the latter in the presence of
infected individuals, or if infected individuals were to arrive into an area
lifting such restrictions, would seed resurgences of infection. This
conclusion suggests that only by ramping up vaccinations to achieve natural
herd immunity (i.e., the higher level of herd immunity that will be required
to prevent transmission in the absence of any social containment measures)
will the pandemic be fully suppressed over the longer-term.

Data accessibility. All parameter data and code can be found at
https://github.com/EdwinMichaelLab/COVID-Vaccination-Paper.

## Appendix

##  Appendix A Time $R_{t}=1$ and Peak ICU cases

To determine the time until $R_{t}<1$, and, separately, to determine the peak
ICU cases as a function of $\phi$ (Figure 4) and $\mu$ (Figure 5), we consider
system (1) extended to include an auxiliary variable $\tau$:

$\displaystyle S^{\prime}$ $\displaystyle=-(1-d)\beta S(I_{a}+I_{p}+I_{m})-\mu
S$ (5) $\displaystyle X^{\prime}$ $\displaystyle=\mu(1-\phi)S-(1-d)\beta
X(I_{a}+I_{p}+I_{m})$ $\displaystyle V^{\prime}$
$\displaystyle=\mu\phi-(1-\xi)(1-d)\beta V(I_{a}+I_{p}+I_{m})$ $\displaystyle
E^{\prime}$
$\displaystyle=(1-d)\beta(I_{a}+I_{p}+I_{m})\left[S+X+(1-\xi)V\right]-\sigma
E$ $\displaystyle I_{a}^{\prime}$ $\displaystyle=\sigma\rho E-\gamma_{a}I_{a}$
$\displaystyle I_{p}^{\prime}$ $\displaystyle=\sigma(1-\rho)E-\delta_{1}I_{p}$
$\displaystyle I_{m}^{\prime}$
$\displaystyle=\delta_{1}I_{p}-x_{1}\delta_{2}I_{m}-(1-x_{1})\gamma_{m}I_{m}$
$\displaystyle I_{h}^{\prime}$
$\displaystyle=x_{1}\delta_{2}I_{m}-x_{2}\delta_{3}I_{h}-(1-x_{2})\gamma_{h}I_{h}$
$\displaystyle I_{c}^{\prime}$
$\displaystyle=x_{2}\delta_{3}I_{h}-(1-x_{3})\gamma_{c}I_{c}-x_{3}mI_{c}$
$\displaystyle R^{\prime}$
$\displaystyle=\gamma_{a}I_{a}+(1-x_{1})\gamma_{m}I_{m}+(1-x_{2})\gamma_{h}I_{h}+(1-x_{3})\gamma_{c}I_{c}$
$\displaystyle D^{\prime}$ $\displaystyle=x_{3}mI_{c}$
$\displaystyle\tau^{\prime}$ $\displaystyle=0.$

This additional variable provides an additional degree of freedom allowing us
to solve the system as a boundary value problem (BVP) with a terminal,
parameter-dependent boundary condition (BC). In particular, the variable
$\tau$ in solutions of system (5) with BCs

$\displaystyle S(0)$ $\displaystyle=S_{0}$ (6) $\displaystyle E(0)$
$\displaystyle=1-S_{0}$ $\displaystyle R_{t}(\tau)$ $\displaystyle=1$
$\displaystyle X(0)$
$\displaystyle=V(0)=I_{a}(0)=I_{p}(0)=I_{m}(0)=I_{h}(0)=I_{c}(0)=R(0)=D(0)=0,$

where $R_{t}(t)$ is defined in equation (4), represents the time at which
$R_{t}(t)=1$. Since $R_{t}(t)$ is strictly decreasing over time, $R_{t}(t)<1$
for all $t>\tau$, and $\tau$ is therefore the time at which the disease is
eradicated.

Next, consider system (5) with BCs

$\displaystyle S(0)$ $\displaystyle=S_{0}$ (7) $\displaystyle E(0)$
$\displaystyle=1-S_{0}$ $\displaystyle 0$
$\displaystyle=x_{2}\delta_{3}I_{h}(\tau)-(1-x_{3})\gamma_{c}I_{c}(\tau)-x_{3}mI_{c}(\tau)$
$\displaystyle X(0)$
$\displaystyle=V(0)=I_{a}(0)=I_{p}(0)=I_{m}(0)=I_{h}(0)=I_{c}(0)=R(0)=D(0)=0.$

The third boundary condition is equivalent to $I_{c}^{\prime}(\tau)=0$ and
therefore requires $I_{c}$ to be at a local maximum at time $\tau$. The
variable $\tau$ is therefore the time at which $I_{c}$ reaches its maximum
[22]. Evaluating $I_{c}$ at $t=\tau$ for any solution of BVP (5)-(7) provides
the peak ICU cases for the fixed parameter set.

Using AUTO [15], we can numerically continue solutions to system (5) with BCs
(6) or (7) over any system parameter; doing so over $\phi$ and $\mu$ produced
the Figures 4 and 5, respectively. In both figures, $S_{0}=0.99$.

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