Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Volume 2, Number 2, June, pp.134-148 https://doi.org/10.5206/mase/13889

A COVID-19 EPIDEMIC MODEL PREDICTING

THE EFFECTIVENESS OF VACCINATION

GLENN WEBB

Abstract. A model of a COVID-19 epidemic is developed to predict the effectiveness of vaccination.

The model incorporates key features of COVID-19 epidemics: asymptomatic and symptomatic infec-

tiousness, reported and unreported cases, and social measures that decrease infection transmission.

The model incorporates key features of vaccination: vaccination efficiency, vaccination scheduling, and

relaxation of social measures that increase infection transmission as vaccination is implemented. The

model is applied to predict vaccination effectiveness in the United Kingdom.

1. Introduction

The objective of this paper is to predict the outcome of vaccine implementation for the mitigation

of COVID-19 epidemics. Currently, vaccine policies are underway throughout the world, and their

outcomes offer great hope for curtailment and elimination of the COVID-19 pandemic. There is to some

extent, controversy about the implementation strategies underway, in terms of the vaccine effectiveness

and the consequence of resumption of normal social behaviour as the number of vaccinated people

increases. This paper addresses these issues with a mathematical model incorporating key features of

COVID-19 epidemics and key features of COVID-19 vaccination implementation. Related works can

be found in [1]-[43].

The organization of this paper is as follows: In Section 2, an ordinary differential equations model

of a COVID-19 epidemic is developed to analyse the transmission dynamics of the epidemics, based on

current reported cases data and current vaccination data. In Section 3, the model is used to predict

the outcome of vaccine implementation in the United Kingdom, from April, 2021 to January, 2022.

The transmission dynamics of the model are advanced forward from the current data in the UK. The

projection examines the roles of vaccine efficiency, numbers of people vaccinated, and the restoration of

normal social practices in predicting the future of the epidemic in the UK.

2. A General COVID-19 Model Based on Current Daily Data

The compartments of the model are S(t) = susceptible individuals at time t, I(t) = asymptomatic

infectious individuals at time t, R(t) = symptomatic infectious individuals at time t who will be reported,

U(t) = symptomatic infectious individuals at time t who will not be reported, and V (t) = vaccinated

Received by the editors 23 April 2021; revised 11 May 2021; accepted 11 May 2021, published online 14 May 2021.

2010 Mathematics Subject Classification. Primary 92D30; Secondary 92C60.

Key words and phrases. epidemic, cases, transmission, asymptomatic, symptomatic, vaccination.

134



PREDICTING THE EFFECTIVENESS OF VACCINATION 135

individuals at time t. The equations of the model are

S′(t) = −τ(t,S(t),I(t),R(t)) −v(t)S(t), t ≥ t0, (2.1)

I′(t) = τ(t,S(t),I(t),R(t)) − (ν1 + ν2)I(t), t ≥ t0, (2.2)

R′(t) = ν1I(t) −ηR(t), t ≥ t0, (2.3)

U′(t) = ν2I(t) −ηU(t), t ≥ t0, (2.4)

V ′(t) = v(t)S(t), t ≥ t0. (2.5)

Figure 1. Flow diagram of the model

The parameters of the model are as follows:

• The transmission rate is time-dependent. Before the most recent day of daily reported cases,
it is defined in terms of the rolling weekly average of the daily reported cases data. After

this time, the transmission rate has the form τ(t) S(t)(I(t) + (1 + f)R(t) ), where f is a fixed

ratio of unreported symptomatic cases to reported symptomatic cases. The total number of

symptomatic cases at time t is (1 + f) R(t). It is assumed that asymptomatic cases, unreported

symptomatic cases, and reported symptomatic cases have equal likelihood of transmission to

susceptibles. The function τ(t) incorporates time dependent relaxation of social distancing

behavior as vaccination is implemented.

• Asymptomatic infectious individuals I(t) are infectious for an average period of 1/ν days.
• The fraction f of asymptomatic infectious become reported symptomatic infectious at rate
ν1 = f ν, and the fraction 1−f become unreported symptomatic infectious at rate ν2 = (1−f) ν,
where ν = ν1 + ν2.

• A reported symptomatic individual is reported, on average, after 1/η days, and no longer creates
transmissions due to isolation.

• Unreported symptomatic individuals U(t) are infectious for an average period of 1/η days.
• Susceptible individuals are removed from the possibility of infection as a result of vaccination,

at a rate of v(t) per day, where this time dependent rate incorporates the first and second

vaccine doses, and their effectiveness in preventing infection.



136 GLENN WEBB

The time-dependent transmission rate in the model before the last date of daily reported cases,

is obtained from the daily reported cases data. Since the daily reported cases data is typically very

erratic, a rolling weekly average of the daily reported cases data is used to smooth this data. Let

dr(t1),dr(t2), . . . be the rolling weekly average number of daily reported cases each day, from the first

week up to the last day of daily reported cases, where time t1, t2, . . . is discrete, day by day. In the

model, the continuum version DR(t) of dr(t1),dr(t2), . . . , can be assumed to satisfy

DR′(t) = ν1 I(t) −DR(t) ⇒ I(t) =
(
DR′(t) + DR(t)

ν1

)
. (2.6)

Then, Equation (2.2) for I(t) in the model

I′(t) = τ(t,S(t),I(t),R(t)) −νI(t),
implies the transmission rate τ(t,S(t),I(t),R(t)) satisfies, until the last day of reported cases data,

τ(t,S(t),I(t),R(t)) = I′(t) + ν I(t)

=
DR′′(t) + DR′(t)

ν1
+ ν

(
DR′(t) + DR(t)

ν1

)
.

Similar methods have been used in [9], [16], [17], [28], [29] to relate reported cases data to model

dynamics.

The discrete smoothing of the daily reported cases data to rolling weekly average values, can be

interpolated by a continuum cubic spline curve CS(t). This curve is constructed by defining cubic poly-

nomials on successive pairs of intervals [t1, t2], [t2, t3], [t3, t4], [t4, t5], . . . , where the interpolation agrees

with the rolling weekly average daily cases data at the integer values, and is three times differentiable

from the first to last day of rolling weekly average daily cases. Then, DR(t) in (2.6) for the model can

be equated to CS(t), and the derivatives DR′(t) = CS′(t) and DR′′(t) = CS′′(t) can also be obtained.

Thus, the continuum interpolation CS(t) derived from the rolling weekly average daily data agrees

exactly with this data at discrete day by day values, and has continuous first and second derivatives on

its domain. The continuum time-dependent transmission rate in the model before the last date of daily

reported cases, is thus given by

τ(t,S(t),I(t),R(t)) =
CS′′(t) + CS′(t)

ν1
+ ν

(
CS′(t) + CS(t)

ν1

)
. (2.7)

The model with this form for the transmission dynamics provides information about S(t), I(t), R(t),

and U(t) up to the last date of daily reported cases. After this date, the transmission dynamics can be

extrapolated, based on their most recent history before this last date, and the dynamics of the epidemic

can be projected forward in time.

The asymptotic behaviour of the solutions of Equations (2.1), (2.2) and (2.3) is obtained as follows:

add Equations (2.1), (2.2) and (2.3) and integrate from t0 to t to obtain

S(t) + I(t) + R(t) +

∫ t
t0

v(s) S(s)ds + (ν −ν1)
∫ t
t0

I(s)ds + η

∫ t
t0

R(s)ds = S(t0) + I(t0) + R(t0).

Then,
∫ t
t0
I(s)ds < ∞, and since I′(t) is continuous and bounded, limt→∞I(t) = 0. Similarly,

limt→∞R(t) = 0. Integration of Equation (2.6) yields

DR(t) = e−t
(
DR(t0) + ν1

∫ t
t0

esI(s)ds

)
,



PREDICTING THE EFFECTIVENESS OF VACCINATION 137

which implies limt→∞DR(t) = 0.

3. Application of the COVID-19 model to the United Kingdom

A chronology of the COVID-19 epidemic in the United Kingdom, starting in February, 2020, is given

below [45]:

• February: First cases reported.
• March: The government imposed stay-at-home order banning all non-essential travel and closing

most gathering places.

• April: First wave of daily reported cases.
• Late April, May, June: Number of cases slowed, and the government eased the lock-down

restrictions.

• July and August: Cases remained at relatively low levels.
• September and October: Second wave of daily reported cases and the government re-imposed

lock-down measures.

• November 25: 696 deaths reported, the highest since May.
• Late November: Number of cases and deaths slowed.
• December: Third wave of daily reported cases.
• December 8: Vaccination began with a 2-dose Pfizer vaccine regimen.
• December 30: The NHS delayed the second dose for the more than 500,000 people receiving the

first dose up to that date, in order to provide a first dose to as many people as possible. Also

on December 30, AstraZeneca vaccine was approved, and began implementation in January,

with the same policies as the Pfizer vaccine. The second dose for both was supposed to be

approximately 12 weeks after the first dose.

• December 30: New government restrictions imposed across the country.
• January, February, March: Daily cases subsided.
• By January 2021 testing was running at approximately 4,000,000 tests per week, and by mid-

February 2021, approximately 75,000,000 tests had been conducted.

• February 21: Prime Minister Boris Johnson announced vaccination goal to give first dose to all
over the age of 50 by mid-April and all adults by end of July.

• February 22: Government projected that all lockdown measures would be ended by June 21
at the earliest: Schools re-open in mid-March, travel permitted outside of local areas in late

March, opening of non-essential retail and personal services in mid-April, outdoor social contact

restrictions eased by mid-May, all lockdown limits removed by June 21 or later. The Prime

Minister announced that international travel would resume on May 17, at the earliest, to allow

free international travel of vaccinated travellers.

The daily reported cases data in the UK is graphed in Figure 2, from March 1, 2020 to April 1, 2021.

A reported case means at least one positive laboratory test result or a lateral-flow device test result

([44]). The rolling weekly average daily cases data in the UK is also graphed in Figure 2, as is the cubic

spline interpolation CS(t) of this discrete rolling weekly averaged daily cases data.

In Figure 3 the daily number of first dose vaccinations and the daily number of second dose vacci-

nations in the UK, from January 11, 2021 to April 1, 2021, are graphed ([44]). From January 11, 2021

to April 1, 2021, 31, 318, 262 people had received a first dose vaccination, and 4, 958, 874 people had

received a second dose vaccination.



138 GLENN WEBB

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Figure 2. Vertical bars are daily reported cases from March 7, 2020 to April 1, 2021,

red dots are discrete rolling weekly averaged daily reported cases, and the green graph

is the continuum cubic spline interpolation CS(t) of the red dots.

Figure 3. Vaccinations in the UK, beginning January 11, 2021, through April 1,

2021. Daily first dose step function v1(t) (top). Daily second dose step function v2(t)

(bottom)

3.1. Model parameters for the COVID-19 epidemic in the United Kingdom. Set the initial

susceptible population S(0) = 68, 000, 000, the current population of the United Kingdom. Set ν = 1/7,



PREDICTING THE EFFECTIVENESS OF VACCINATION 139

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Figure 4. Graph of the model transmission rate from March 1, 2020 to April 1, 2021,

as in (2.7). ν2/ν2 = 1.5.

which means that asymptomatic infectious individuals remain infectious for 7 days. Set η = 1/7, which

means that reported symptomatic individuals R(t) are infectious for an average period of 1/η = 7 days,

as are unreported symptomatic individuals U(t). The values for ν and η are uncertain, but reasonable

for this application. The value of η = 1/7 is consistent with the assumption that the daily reported

cases data can be replaced by rolling weekly averaged values.

The ratio of unreported cases to reported cases will be assumed as 3 to 2, 2 to 1, and 3 to 1 in

simulations of the model. Before April 1, 2021, the model transmission rate, based on the rolling

weekly average of daily data as in (2.7), is graphed in Figure 4 and Figure 6 for the case that this ratio

is 3 to 2 (ν1 = .4/7, ν2 = .6/7, ν2/ν1 = 1.5). The case that this ratio is 2 to 1 is graphed in Figure 7

and the case that this ratio is 3 to 1 is graphed in Figure 8.

3.2. Simulation of the Model COVID-19 Epidemic in the United Kingdom. In Figure 5, the

graph of symptomatic reported cases R(t) and the graph of cumulative reported cases CR(t), from the

model simulation with the above parameters, are shown for March 7, 2020 through April 1, 2021. The

graphs are the same for ν1 = .4/7, ν1 = .3333/7, and ν1 = .25/7.

3.3. Projecting the Model Forward from the Last Date of Reported Cases Data. The last

date of daily reported cases data and vaccination data is tD = April 1, 2021. From January 11, 2021 to

April 1, 2021, the daily vaccination rate is v1(t) per day, as in Figure 3 (top) for the first dose, and v2(t)

per day, as in Figure 3 (bottom) for the second dose. For the model projections forward from April 1,

2021, the vaccination rate is assumed as v1(t) = 200, 000 vaccinations per day for the first dose, and

v2(t) = 200, 000 vaccinations per day for the second dose. In Equation (2.1), set

v(t) =
0.9 v1(t) + 0.05 v2(t)

S0
, t > tD.

Vaccination removes individuals from the susceptible population at time t at a rate proportional to the

remaining susceptible individuals at time t. In Equation (2.1), the loss of susceptibles per day due to



140 GLENN WEBB

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1×106

2×106

3×106

4×106

Data

CR(t)
R(t)

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Figure 5. Graphs of the daily reported cases R(t) and the rolling weekly averaged cu-

mulative daily reported cases CR(t) from the model simulation. CR(t) = approximately

4,370,000 on April 1, 2021. The graphs are the same for ν2/ν1 = 1.5, 2.0, 3.0.

vaccination

v(t) S(t) =

(
0.9 v1(t) + 0.05 v2(t)

)
S(t)

S(0)
, t > tD

involves the fraction S(t)/S(0) of unvaccinated population and prior infected population, still susceptible

at time t. The form of v(t) incorporates the effectiveness of vaccination, through both doses.

After time tD = April 1, 2021, a time t1 = April 15, 2021 is set such that there is an increasing

return to normalcy of social distancing behaviour. After April 15, the increase in the transmission rate

per day involves a linear scaling factor ω, and lasts until time t2 = July 1, 2021. Thus, the transmission

rate has the following form:

from day t0 = March 1, 2020 until time tD = April 1, 2021, the transmission rate is as in (2.7):

τ(t,S(t),I(t),R(t)) =
DR′′(t) + DR′(t)

ν1
+ ν

(
DR′(t) + DR(t)

ν1

)
, t0 ≤ t ≤ tD;

for tD = April 1, 2021 < t ≤ t1 = April 15, 2021:

τ(t,S(t),I(t),R(t)) = τ(tD,S(tD),I(tD),R(tD))

(
(I(t) + (1 + ν2

ν1
) R(t)) S(t)

(I(tD) + (1 +
ν2
ν1

) R(tD)) S(tD)

)
;

for t1 = April 15, 2021 < t ≤ t2 = July 1, 2021:

τ(t,S(t),I(t),R(t)) =

(
1.0 + ω (t−t1)

)
τ(tD,S(tD),I(tD),R(tD))

(
(I(t) + (1 + ν2

ν1
)R(t)) S(t)

(I(tD) + (1 +
ν2
ν1

) R(tD)) S(tD)

)
;

for t2 = July 1, 2021 < t, the scaling factor ω term, corresponding to social distancing relaxation, is

maximized at t2 − t1:

τ(t,S(t),I(t),R(t)) =

(
1.0+ω (t2−t1)

)
τ(tD,S(tD),I(tD),R(tD))

(
(I(t) + (1 + ν2

ν1
)R(t)) S(t)

(I(tD) + (1 +
ν2
ν1

) R(tD)) S(tD)

)
.



PREDICTING THE EFFECTIVENESS OF VACCINATION 141

The transmission rate is continuous, and in particular, continuous at day tD = April 1, 2021, day

t1 = April 15, 2021, and day t2 = July 1, 2021. The magnitude of the parameter ω, corresponding to

resumption of normal social distancing behaviour, is critical for resurgence of the epidemic.

In Figure 6, the graphs of the model transmission rate τ(t,S(t),I(t),R(t)) are shown from March 7,

2020 to January 1, 2022, with the ratio of unreported cases to reported cases 3 to 2 (ν1 = .4/7, ν2 =

.6/7), and ω = .03, .035 and .04.

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Figure 6. Graphs of the transmission rate τ(t,S(t),I(t),R(t)), with relaxation of

social distancing measures beginning on April 15, 2021. ν2/ν1 = 1.5. The relaxation

scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple).

ν2/ν1 = 1.5 ω = .03 ω = .035 ω = .04

τ(t,S(t),I(t),R(t)) 5,713 5,713 5,713

on tD = April 1, 2021

Max of τ(t,S(t),I(t),R(t)) after tD 2,738 10,812 40,709

occurring on Aug 19, 2021 Sept 18, 2021 Sept 26, 2021

τ(t,S(t),I(t),R(t)) on January 1, 2022 274 2,313 6,639

Table 1. Transmission rate τ(t,S(t),I(t),R(t)) for the ratio of unreported cases to

reported cases ν2/ν1 = 1.5, and the social behaviour relaxation parameter ω.

In Figure 7, the graphs of the model transmission rate τ(t,S(t),I(t),R(t)) are shown from March 7,

2020 to January 1, 2022, with the ratio of unreported cases to reported cases 2 to 1 (ν1 = .3333/7, ν2 =

.6667/7), and ω = .03, .035 and .04.



142 GLENN WEBB

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Figure 7. Graphs of the transmission rate τ(t,S(t),I(t),R(t)), with relaxation of

social distancing measures beginning on April 15, 2021. ν2/ν1 = 2.0. The relaxation

scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple).

ν2/ν1 = 2.0 ω = .03 ω = .035 ω = .04

τ(t,S(t),I(t),R(t)) 6,856 6,856 6,856

on day tD = April 1, 2021

Max of τ(t,S(t),I(t),R(t)) after tD 3,230 12,400 44,028

occurring on Aug 18, 2021 Sept 16, 2021 Sept 21, 2021

τ(t,S(t),I(t),R(t)) on January 1, 2022 310 2,399 5,744

Table 2. Transmission rate τ(t,S(t),I(t),R(t)) for the ratio of unreported cases to

reported cases ν2/ν1 = 2.0, and the social behaviour relaxation parameter ω.

In Figure 8, the graphs of the model transmission rate τ(t,S(t),I(t),R(t)) are shown from March 7,

2020 to January 1, 2022, with the ratio of unreported cases to reported cases 3 to 1 (ν1 = .25/7, ν2 =

.75/7), and ω = .03, .035 and .04.

ν2/ν1 = 3.0 ω = .03 ω = .035 ω = .04

τ(t,S(t),I(t),R(t)) 9,141 9,141 9,141

on day tD = April 1, 2021

Max of τ(t,S(t),I(t),R(t)) after tD 4,149 15,054 48,395

occurring on Aug 16, 2021 Sept 12, 2021 Sept 13, 2021

τ(t,S(t),I(t),R(t)) on January 1, 2022 363 2,367 4,248

Table 3. Transmission rate τ(t,S(t),I(t),R(t)) for the ratio of unreported cases to

reported cases ν2/ν1 = 3.0, and the social behaviour relaxation parameter ω.



PREDICTING THE EFFECTIVENESS OF VACCINATION 143

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Figure 8. Graphs of the transmission rate τ(t,S(t),I(t),R(t)), with relaxation of

social distancing measures beginning on April 15, 2021. ν2/ν1 = 3.0. The relaxation

scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple).

In Figure 9, the graphs of the model simulation are shown from March 7, 2020 to January 1, 2022 for

daily reported cases, with the ratio of unreported cases to reported cases 3 to 2 (ν1 = .4/7, ν2 = .6/7),

and ω = .03, .035, and .04.

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Figure 9. Graphs of daily reported cases DR(t) for the model simulations with relax-

ation of social distancing measures beginning on April 15. ν1 = .4/7, ν2 = .6/7. The

relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple).



144 GLENN WEBB

ν2/ν1 = 1.5 ω = .03 ω = .035 ω = .04

Number of susceptibles on January 1, 2022 17,612,000 16,781,000 14,242,000

Cumulative reported cases April 1, 2021 - January 1, 2022 234,600 600,200 1,953,000

New asymptomatic cases on January 1, 2022 142 1,145 3,438

Table 4. Model simulation output for the ratio of unreported cases to reported cases

ν2/ν1 = 3 to 2, and the social behaviour relaxation parameter ω.

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Figure 10. Graphs of daily reported cases DR(t) for the model simulations with relax-

ation of social distancing measures beginning on April 15. ν1 = .3333/7, ν2 = .6667/7.

The relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple).

In Figure 10, the graphs of the model simulation are shown from March 7, 2020 to January 1, 2022

for daily reported cases, with the ratio of unreported cases to reported cases 2 to 1 (ν1 = .3333/7, ν2 =

.6667/7), and ω = .03, .035, and .04.

In Figure 11, the graphs of the model simulation are shown from March 7, 2020 to January 1, 2022 for

daily reported cases, with the ratio of unreported cases to reported cases 3 to 1 (ν1 = .25/7, ν2 = .75/7),

and ω = .03, .035, and .04.

4. Summary

A general model of COVID-19 epidemics has been developed to predict the effectiveness of vaccina-

tion. The model incorporates basic elements of COVID-19 dynamics: transmission due to asymptomatic

and symptomatic infected individuals, transmission due to reported and unreported cases, and trans-

mission mitigation due to social distancing measures. Because the daily reported cases data is typically

very erratic, a rolling weekly averaging process is used to provide better consistency with the model

dynamics. The model formulation is constructed so that the daily reported cases DR(t) in the model

agrees exactly with the rolling weekly averaged daily reported cases data.



PREDICTING THE EFFECTIVENESS OF VACCINATION 145

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Figure 11. Graphs of daily reported cases DR(t) for the model simulations with re-

laxation of social distancing measures beginning on April 15. ν1 = .25/7, ν2 = .75/7.

The relaxation scaling values are ω = .03 (green), ω = .035 (red), ω = .04 (purple).

The objective of the modelling process is to project forward in time from the last day tD of daily

reported cases data, the effectiveness of vaccination in controlling the epidemic. The transmission rate

in the model, forward from day tD, is based on the model transmission rate on this day. As time

proceeds forward from tD, the transmission rate is moderated, in correspondence with a restoration of

normal social distancing, as the number of susceptible individuals is reduced due to vaccination. Two

dates t1 and t2 are set, such that tD < t1 < t2, and the transmission rate increases from t1 to t2 with a

scaling factor ω, that corresponds to the reduction of social distancing measures.

The model is applied to the COVID-19 epidemic in the United Kingdom. The last day of daily

reported cases tD = April 1, 2021. The restoration of normal social distancing is from t1 = April 15,

2021 to t2 = July 1, 2021. The model outputs are analysed with ratios of unreported case to reported

cases (ν2/ν1) as 3 to 2, 2 to 1, and 3 to 1, and with of restoration of normal social distancing scaling

parameters ω = .03, .035, and .04. The model output shows the following dependence of the cumulative

reported cases CR(t) between April 1, 2021 and January 1, 2022:

(1) the cumulative reported cases CR(t) between April 1, 2021 and January 1, 2022 are decreasing as

the ratio of unreported to reported cases ν2/ν1 increases;

(2) the cumulative reported cases CR(t) between April 1, 2021 and January 1, 2022 are increasing as

the scaling factor ω corresponding to relaxation of social behaviour restrictions increases.

These results are consistent with the results in [24] and [34], which were based on extensive data input

for the COVID-19 epidemic in the United Kingdom.

The general model is applicable to COVID-19 epidemics in all locations. The parameterisation of

the model is based on daily reported cases data and daily vaccination data, which is readily available

in all locations. The model parameters ν,ν1,ν2,η,tD, t1, t2 can be adjusted to specific locations, and

predictions based on vaccination implementation and restoration of normal social distancing can be

provided. In future work the model will be applied to COVID-19 epidemics in other locations.



146 GLENN WEBB

ν2/ν1 = 2.0 ω = .03 ω = .035 ω = .04

Number of susceptibles on January 1, 2022 16,837,000 15,906,000 13,312,000

Cumulative reported cases April 1, 2021 - January 1, 2022 231,500 631,200 1,751,000

New asymptomatic cases on January 1, 2022 134 999 2,537

Table 5. Model simulation output for the ratio of unreported cases to reported cases

ν2/ν1 = 2.0, and the social behaviour relaxation parameter ω.

ν2/ν1 = 3.0 ω = .03 ω = .035 ω = .04

Number of susceptibles on January 1, 2022 15,300,000 14,221,000 11,623,000

Cumulative reported cases April 1, 2021 - January 1, 2022 224,800 575,600 1,425,000

New asymptomatic cases on January 1, 2022 118 753 1,444

Table 6. Model simulation output for the ratio of unreported cases to reported cases

ν2/ν1 = 3.0, and the social behaviour relaxation parameter ω.



PREDICTING THE EFFECTIVENESS OF VACCINATION 147

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Mathematics Department, Vanderbilt University, Nashville, TN, USA