# Studying the mixed transmission in a community with age heterogeneity:
COVID-19 as a case study

Xiaoying Wang 1∗, Qing Han2 & Jude Dzevela Kong2
1 Department of Mathematics, Trent University
Peterborough, ON K9L 0G2, Canada
2 Africa-Canada Artificial Intelligence and Data Innovation Consortium
(ACADIC)
Laboratory for Industrial and Applied Mathematics (LIAM)
Department of Mathematics and Statistics, York University
Toronto, ON M3J 1P3, Canada
∗ Corresponding author<EMAIL_ADDRESS>
###### Abstract

COVID-19 has been prevalent worldwide for about 2 years now and has brought
unprecedented challenges to our society. Before vaccines were available, the
main disease intervention strategies were non-pharmaceutical. Starting
December 2020, in Ontario, Canada, vaccines were approved for administering to
vulnerable individuals and gradually expanded to all individuals above the age
of 12. As the vaccine coverage reached a satisfactory level among the eligible
population, normal social activities resumed and schools reopened starting
September 2021. However, when schools reopen for in-person learning, children
under the age of 12 are unvaccinated and are at higher risks of contracting
the virus. We propose an age-stratified model based on the age and vaccine
eligibility of the individuals. We fit our model to the data in Ontario,
Canada and obtain a good fitting result. The results show that a relaxed
between-group contact rate may trigger future epidemic waves more easily than
an increased within-group contact rate. An increasing mixed contact rate of
the older group quickly amplifies the daily incidence numbers for both groups
whereas an increasing mixed contact rate of the younger group mainly leads to
future waves in the younger group alone. The results indicate the importance
of accelerating vaccine rollout for younger individuals in mitigating disease
spread.

Keywords: COVID-19, Age-stratified Model, Mixed Social Contact Pattern,
Disease Mitigation Strategy

##  1 Introduction

COVID-19 has been prevalent worldwide for around 2 years now since the initial
identification in Wuhan, China in December 2019. The global pandemic has
caused more than 328,000,000 total infections and over 5,500,000 deaths
worldwide [31]. In Ontario, Canada alone, the total confirmed case number of
COVID-19 surpassed 970,000 and more than 10,000 population deceased due to
complications of COVID-19 infection [4].

The novel pneumonia disease can be transmitted via close contact between
susceptible and infected populations, similar to other pneumonia diseases such
as influenza. However, recent evidences show that the majority of COVID-19
transmission may be attributed to the aerosol droplets [1, 30]. The
transmission via aerosol implies that the spread of the disease may occur in a
long-range, which signifies the difficulties of mitigating the disease spread.

In Ontario, before vaccines were approved, the main disease mitigation
strategies were non-pharmaceutical, such as closing non-essential businesses,
practising social distancing, setting limitations of indoor gathering size,
requiring mandatory face-covering etc. The non-pharmaceutical strategies are
effective in mitigating the disease spread but disrupt normal social
activities and cause huge economic loss. Therefore, aggressive strategies can
only be imposed in a short period and are not practical in long term.
Progressing into the second year of the pandemic, vaccines were approved by
Health Canada in December 2020 and quickly rolled out across the province.

In the initial stage of the vaccine rollout program, vaccines were
administered to the senior population or other vulnerable individuals with
underlying health conditions. Starting in late May 2021, Health Canada
expanded the criteria so that all individuals above the age of 12 were
eligible for receiving vaccines. The vaccine for children between the age of 5
to 11 was approved recently in late November 2021 but the rollout takes time
and only $3.2\%$ of the children in the age group are fully vaccinated up to
now [2].

Clinical trail evidences show that vaccines offer a high protection efficacy
for vaccinated individuals that reduces the probability of infection to a
large extent [31]. As the vaccine coverage steadily increased and reached a
satisfactory level among eligible individuals, the province lifted majority of
the restrictive measures in July 2021. During this time, unvaccinated children
were on the summer break and weren’t on high risks of transmitting the disease
because of limited contacts within the household.

Starting in September 2021, schools reopen and in-person learning resumes. In-
person learning environment inevitably creates a larger social gathering size,
larger contact numbers between the children, and a higher infection
probability in the indoor setting. Evidences show that children are at a lower
risk of developing serious symptoms after contracting the virus and therefore
the within-group transmission may not raise serious concerns. However,
children may transmit the disease to elder family members within the household
after school. Infected elder individuals are more likely to develop serious
symptoms and need medical attention, which causes stress on the provincial
health system.

Mathematical models have been used extensively in studying COVID-19 since the
initial emergence of the disease, see [13, 15, 18, 19, 21, 25, 27, 29, 32] for
example. The aforementioned studies focus on modeling the initial wave of the
pandemic and provide valuable insights in short-term predictions and disease
intervention strategies. Age-stratified models were proposed to study the
heterogeneity in social contact patterns among a susceptible population in
early stage of the disease [5, 7, 8, 20]. Statistical analyses were applied in
studying various aspects of the disease spread, such as identifying the impact
of undiagnosed cases in COVID-19 [10], the importance of household
transmission [24], and suggestions for testing policy [16]. As the disease
evolves, particularly with the rollout of vaccines and resumption of normal
social activities, models need to be revisited to gain better understanding of
the disease spread.

In this paper, we propose an age-stratified model where individuals in the
older group are eligible for receiving vaccines while the individuals in the
younger group are not. We fit the model to the COVID-19 data in Ontario,
Canada for illustration. In Ontario, the schools reopen starting in September
2021 and the vaccines for children haven’t been approved by then. The
individuals above the age of 12 are eligible for receiving vaccines but
breakthrough infections may occur due to relaxed social distancing. The main
objective of this paper is to identify the relative importance of within-group
transmission and between-group transmission and shed light on future disease
mitigation strategies.

##  2 Methods

###  2.1 Data

We accessed the COVID-19 data in Ontario from the online data catalogue in
Ontario government [26]. The data contains demographic information of
individuals who are confirmed of positive COVID-19 infection among all 34
health units across Ontario, such as age, sex, and location of the reporting
health unit. Figure 1 shows the daily incidence number between Aug. 1, 2021
and Oct. 25, 2021, based on the estimated symptom onset date recorded in [26].
In the aforementioned time window, individuals who are above the age of 12 are
eligible for receiving vaccines but individuals who are below the age of 12
haven’t been approved for receiving vaccines yet. Therefore, in this study, we
divide the entire population in Ontario into two groups: the older group and
the younger group, where individuals are above/below the age of 12
respectively. Figure 1 demonstrates the daily incidence case number for each
group separately.

Figure 1: The daily incidence number for older/younger group in Ontario
between August 1, 2021 and October 25, 2021.

###  2.2 The model

We propose a compartmental model that stratifies the entire population into
the following compartments: susceptible class ($S$), vaccinated class ($V$),
exposed class ($E$), asymptomatic class ($A$), unreported symptomatic class
($U$), confirmed infected class ($I$), recovered class ($R$), and the deceased
class ($D$). Moreover, within each compartment, we further stratify
individuals into the older group and the younger group, as denoted by $S_{i}$
for $i=1,2$ for example. Figure 2 demonstrates the flowchart of the model.
Note that within the time window between August and October, the individuals
below the age of 12 are not eligible for receiving vaccines and therefore, the
compartment $V$ comprises of vaccinated individuals of the older group only.

$S_{i}$ $E_{i}$ $A_{i}$ $I_{i}$ $R_{i}$ $U_{i}$ $D_{i}$ $V$ Figure 2: The flow
chart for model.

In the model, individuals in the exposed class ($E$) are individuals who have
been infected with the COVID-19 virus but are still in the incubation period
and are not contagious yet. Vaccinated individuals are at a lower risk of
contracting the virus but may still become infected because the vaccine
efficacy is not $100\%.$ Asymptomatic individuals in class ($A$) are those who
are contagious but show no symptoms of infection. On the other hand,
individuals in class ($U$) show mild symptoms and are also contagious but do
not seek medical resources for diagnosis. Individuals in class ($I$) are those
who are confirmed with the COVID-19 infection and are in quarantine at home if
they experience mild to moderate symptoms or are in the hospital if they show
severe symptoms. Because individuals in class ($I$) follow the public health
protocols, we assume that they maintain a very low possibility of transmitting
the disease. Individuals in all the asymptomatic classes, unreported
symptomatic class, and reported infected class may recover and are no longer
transmissible and therefore move to the recovered class ($R$).

Based on the aforementioned discussions, the model for the older group, where
individuals are above the age of 12 and are eligible to be vaccinated is

$\displaystyle\frac{dS_{1}}{dt}=-\Lambda_{11}-\Lambda_{12}-\omega S_{1},$
$\displaystyle\frac{dV}{dt}=\omega S_{1}-\Lambda_{11}^{v}-\Lambda_{12}^{v},$
$\displaystyle\frac{dE_{1}}{dt}=\Lambda_{11}+\Lambda_{12}+\Lambda_{11}^{v}+\Lambda_{12}^{v}-\mu_{1}E_{1},$
$\displaystyle\frac{dA_{1}}{dt}=\mu_{1}\rho_{1}E_{1}-\eta_{11}A_{1},$ (2.1)
$\displaystyle\frac{dU_{1}}{dt}=\mu_{1}\rho_{2}E_{1}-\eta_{12}U_{1}-\tau_{1}U_{1},$
$\displaystyle\frac{dI_{1}}{dt}=\mu_{1}\left(1-\rho_{1}-\rho_{2}\right)E_{1}+\tau_{1}U_{1}-\eta_{13}I_{1}-\delta_{1}I_{1},$
$\displaystyle\frac{dR_{1}}{dt}=\eta_{11}A_{1}+\eta_{12}U_{1}+\eta_{13}I_{1},$
$\displaystyle\frac{dD_{1}}{dt}=\delta_{1}I_{1}.$

In (2.2), $\Lambda_{11}=\beta_{11}A_{1}S_{1}+\beta_{12}U_{1}S_{1}$ represents
the within-group transmission of the susceptible population, where
$\beta_{11},\beta_{12}$ are the infectious contact rates between susceptible
individuals and asymptomatic/unreported symptomatic individuals in the older
group. Moreover, $\Lambda_{12}=\beta_{13}A_{2}S_{1}+\beta_{14}U_{2}S_{1}$
represents the between-group transmission, where $\beta_{13},\beta_{14}$ are
the contact rates between susceptible individuals in the older group and the
asymptomatic/unreported infected individuals in the younger group.

Similarly,
$\Lambda_{11}^{v}=\beta_{11}^{v}(1-\epsilon)A_{1}V+\beta_{12}^{v}(1-\epsilon)U_{1}V$
is the within-group transmission of the vaccinated individuals, where
$\beta_{11}^{v},\beta_{12}^{v}$ are the contact rates between vaccinated
individuals and asymptomatic/unreported symptomatic individuals in the older
group. Furthermore,
$\Lambda_{12}^{v}=\beta_{13}^{v}(1-\epsilon)A_{2}V+\beta_{14}^{v}(1-\epsilon)U_{2}V$
is the between-group transmission of the vaccinated individuals, where
$\beta_{13}^{v},\beta_{14}^{v}$ are the contact rates between the vaccinated
individuals and the asymptomatic/unreported symptomatic individuals in the
younger group. Because vaccinated individuals are at lower risk of
transmitting the disease, we introduce the factor $(1-\epsilon)$ in
$\Lambda_{11}^{v},\Lambda_{12}^{v}$ to indicate the reduced probability of
infection where $\epsilon$ is the vaccine efficacy and $0<\epsilon<1.$

The model for the younger group is similar but without the vaccinated
compartment because individuals below the age of 12 are not eligible for
receiving vaccines. The model is

$\displaystyle\frac{dS_{2}}{dt}=-\Lambda_{21}-\Lambda_{22},$ (2.2)
$\displaystyle\frac{dE_{2}}{dt}=\Lambda_{21}+\Lambda_{22}-\mu_{2}E_{2},$
$\displaystyle\frac{dA_{2}}{dt}=\mu_{2}\rho_{1}E_{2}-\eta_{21}A_{2},$
$\displaystyle\frac{dU_{2}}{dt}=\mu_{2}\rho_{2}E_{2}-\eta_{22}U_{2}-\tau_{2}U_{2},$
$\displaystyle\frac{dI_{2}}{dt}=\mu_{2}\left(1-\rho_{1}-\rho_{2}\right)E_{2}+\tau_{2}U_{2}-\eta_{23}I_{2}-\delta_{2}I_{2},$
$\displaystyle\frac{dR_{2}}{dt}=\eta_{21}A_{2}+\eta_{22}U_{2}+\eta_{23}I_{2},$
$\displaystyle\frac{dD_{2}}{dt}=\delta_{2}I_{2},$

where $\Lambda_{21}=\beta_{21}A_{2}S_{2}+\beta_{22}U_{2}S_{2}$ is the within-
group transmission of the susceptible individuals, and
$\Lambda_{22}=\beta_{23}A_{1}S_{2}+\beta_{24}U_{1}S_{2}$ is the between-group
transmission of the susceptible individuals.

In (2.2)-(2.2), $\rho_{1}$ is the proportion of the exposed individuals who
move to the asymptomatic class and $\rho_{2}$ is the proportion of the exposed
individuals who move to the unreported symptomatic class. This leaves
$1-\rho_{1}-\rho_{2}$ as the proportion of the exposed individuals who
eventually move to the reported symptomatic class. We assume that
$\rho_{1},\rho_{2}$ are the same for both the older and the younger groups
because there is no evidence so far that suggests the proportions of
asymptomatic individuals and symptomatic individuals differ significantly
among different age groups.

Susceptible individuals in the older group transfer to the vaccinated class at
a rate of $\omega.$ The parameter $\omega$ will be estimated below from the
data in Figure 1. Exposed individuals in the older and the younger group move
to either the asymptomatic class/unreported symptomatic class/reported
infected class at a rate of $\mu_{i}$ for $i=1,2$ respectively. Based on [14,
17], we fix $\mu_{i}=1/3$ for $i=1,2$ because evidence shows that individuals
who have contracted the virus generally have an incubation period of 2-3 days
before they become contagious.

Asymptomatic individuals in the older group and the younger group move to the
recovered class at a rate of $\eta_{11}$ and $\eta_{21}$ respectively.
Similarly, the unreported infected individuals and the confirmed infected
individuals in the older/younger group recover from the infection at a rate of
$\eta_{i2}$ and $\eta_{i3}$ for $i=1,2$. We assume that $\eta_{i,j}=1/7$ where
$i=1,2,j=1,2,3$ based on [18], which indicates that in average, infected
individuals may transmit the disease within a period of 7 days until they
recover and are no longer contagious.

Unreported infected individuals may seek diagnosis from medical facilities if
the symptoms persist or their heath deteriorates over time and therefore move
to the reported infected class at a rate of $\tau.$ In [17], the authors
analyzed the patients’ data and recorded that infected individuals waited for
an average of 4.6 days after observing symptoms before they sought diagnosis.
Hence, in our study, we assume that $\tau_{i}=1/4.6$ for $i=1,2.$

###  2.3 The reproduction number

We first calculate the basic reproduction number $\mathcal{R}_{0}$ for the
age-structured model (2.2)-(2.2). Direct calculations show that

$\frac{d(S_{1}+V+E_{1}+A_{1}+U_{1}+I_{1}+R_{1}+D_{1})}{dt}=0,$

which leads to the total population of the older group as a constant $N_{1}.$
Similarly, the total population in the younger group also remains as a
constant $N_{2}.$ Hence, the disease-free equilibrium of (2.2)-(2.2) is

$\displaystyle\left(S_{1},V,E_{1},A_{1},U_{1},I_{1},R_{1},D_{1},S_{2},E_{2},A_{2},U_{2},I_{2},R_{2},D_{2}\right)$
$\displaystyle=\left(0,N_{1},0,0,0,0,0,0,N_{2},0,0,0,0,0,0\right).$

Following the next generation matrix method for compartmental models in [28],
we obtain

$\displaystyle
F_{1}=\Lambda_{11}+\Lambda_{12}+\Lambda_{11}^{v}+\Lambda_{12}^{v},\quad
F_{2}=0,\quad F_{3}=0,\quad F_{4}=0,\quad F_{5}=\Lambda_{21}+\Lambda_{22},$
$\displaystyle F_{6}=0,\quad F_{7}=0,\quad F_{8}=0,$ $\displaystyle
V_{1}=\mu_{1}E_{1},\quad V_{2}=\eta_{11}A_{1}-\mu_{1}\rho_{1}E_{1},\quad
V_{3}=\eta_{12}U_{1}+\tau_{1}U_{1}-\mu_{1}\rho_{2}E_{1},$ $\displaystyle
V_{4}=\eta_{13}I_{1}+\delta_{1}I_{1}-\mu_{1}(1-\rho_{1}-\rho_{2})E_{1}-\tau_{1}U_{1},\quad
V_{5}=\mu_{2}E_{2},\quad V_{6}=\eta_{21}A_{2}-\mu_{2}\rho_{1}E_{2},$
$\displaystyle V_{7}=\eta_{22}U_{2}+\tau_{2}U_{2}-\mu_{2}\rho_{2}E_{2},\quad
V_{8}=\eta_{23}I_{2}+\delta_{2}I_{2}-\mu_{2}(1-\rho_{1}-\rho_{2})E_{2}-\tau_{2}U_{2}.$

It follows that

$\mathcal{R}_{0}=\frac{J_{11}+J_{55}+\sqrt{(J_{11}+J_{55})^{2}-4(J_{11}J_{55}-J_{15}J_{51})}}{2},$
(2.3)

where

$\displaystyle
J_{11}=\frac{\beta_{11}^{v}(1-\epsilon)\rho_{1}N_{1}}{\eta_{11}}+\frac{\beta_{12}^{v}(1-\epsilon)\rho_{2}N_{1}}{\eta_{12}+\tau_{1}},\quad
J_{15}=\frac{\beta_{13}^{v}(1-\epsilon)\rho_{1}N_{1}}{\eta_{21}}+\frac{\beta_{14}^{v}(1-\epsilon)\rho_{2}N_{1}}{\eta_{22}+\tau_{2}},$
$\displaystyle
J_{51}=\frac{\beta_{23}\rho_{1}N_{2}}{\eta_{11}}+\frac{\beta_{24}\rho_{2}N_{2}}{\eta_{12}+\tau_{1}},\quad
J_{55}=\frac{\beta_{21}\rho_{1}N_{2}}{\eta_{21}}+\frac{\beta_{22}\rho_{2}N_{2}}{\eta_{22}+\tau_{2}}.$

To predict the severity of the disease spread at an arbitrary time of the
epidemics, we calculate the effective reproduction number
$\mathcal{R}_{t}=(S_{t}/S_{0})\mathcal{R}_{0}$, where $S_{t}$ is the total
susceptible population at time t and $S_{0}$ is the total susceptible
population at the initial time when the epidemic starts [6, 23]. Biologically,
the effective reproduction number implies that the disease persists if
$R_{t}>1$ and diminishes if $R_{t}<1.$

###  2.4 Parameter estimation

Based on the census data in Ontario, the total population of the older group
at the beginning of the pandemic is 12,932,471 and the total population of the
younger group is 1,801,543. The number of cumulative infections until July 31,
2021 for the older group is 555,583 whereas the total infection case for the
younger group is 50,156 [2]. The vaccine coverage data shows that a total
number of 10,582,731 individuals received at least one dose and a total of
9,147,534 individuals are fully vaccinated with two doses. We average the data
to obtain the estimated initial population of the vaccinated compartment
$V_{0}=9,865,132.$ It follows that the initial susceptible population for the
older group is $S_{1}(0)=2,511,756.$ Similarly, the initial susceptible
population for the younger group is $S_{2}(0)=1,751,387.$ On August 1, the
reported infection number for the older group is 200 and the reported
infection number for the younger group is 18, which leads to $I_{1}(0)=200$
and $I_{2}(0)=18$ respectively.

We fit the model (2.2), (2.2) to the daily incidence data in Figure 1 by the
Markov Chain Monte Carlo (MCMC) method and adopt the adaptive Metropolis-
Hastings algorithm to carry out this approach [9]. We run the algorithm for
10,000 iterations with a burn-in of the first 7,000 iterations. Geweke
convergence test is employed to diagnose the convergence of the Markov chains.
The estimated parameters and the initial data are listed in Table 1.

Table 1: Parameter estimates for the COVID-19 epidemics in Ontario, Canada

Parameter | Definition |  Estimated Mean Value |  Standard Deviation |  Data Source
---|---|---|---|---
$\beta_{11}$ |  Contact rate between $S_{1}$ and $A_{1}$ | $2.1772\times 10^{-8}$ | $1.708\times 10^{-9}$ | Fitted
$\beta_{13}$ |  Contact rate between $S_{1}$ and $A_{2}$ | $4.8079\times 10^{-7}$ | $3.5895\times 10^{-8}$ | Fitted
$\omega$ |  Transition rate from $S_{1}$ to $V$ | $3.7286\times 10^{-2}$ | $4.3604\times 10^{-3}$ | Fitted
$\beta_{11}^{v}$ |  Contact rate between $V$ and $A_{1}$ | $5.5607\times 10^{-7}$ | $2.4585\times 10^{-8}$ | Fitted
$\beta_{13}^{v}$ |  Contact rate between $V$ and $A_{2}$ | $1.4986\times 10^{-8}$ | $1.1856\times 10^{-9}$ | Fitted
$\epsilon$ |  Contact rate reduction between $V$ and $A_{i}/U_{i}$ | $0.96783$ | $4.423\times 10^{-3}$ | Fitted
$\mu_{1}$ |  Transition rate from $E_{1}$ to $A_{1}/U_{1}/I_{1}$ | $1/3$ | $-$ | [17]
$\mu_{2}$ |  Transition rate from $E_{2}$ to $A_{2}/U_{2}/I_{2}$ | $1/3$ | $-$ | [17]
$\rho_{1}$ |  Proportionality of transferred $E_{i}$ to $A_{i}$ | $0.3$ | $-$ | [22]
$\rho_{2}$ |  Proportionality of transferred $E_{i}$ to $U_{i}$ | $0.3$ | $-$ | [22]
$\eta_{i1}$ | Recovery rate of $A_{i}$ | $1/7$ | $-$ | [18]
$\eta_{i2}$ | Recovery rate of $U_{i}$ | $1/7$ | $-$ | [18]
$\eta_{i3}$ | Recovery rate of $I_{i}$ | $1/7$ | $-$ | [18]
$\tau_{i}$ |  Transition rate from $U_{i}$ to $I_{i}$ | $0.21739$ | $-$ | [18]
$\delta_{1}$ | Disease death rate of $I_{1}$ | $1.0928\times 10^{-5}$ | $6.1912\times 10^{-7}$ | Fitted
$\beta_{21}$ |  Contact rate between $S_{2}$ and $A_{2}$ | $4.079\times 10^{-9}$ | $3.765\times 10^{-10}$ | Fitted
$\beta_{23}$ |  Contact rate between $S_{2}$ and $A_{1}$ | $7.0164\times 10^{-8}$ | $2.5621\times 10^{-9}$ | Fitted
$\delta_{2}$ | Disease death rate of $I_{2}$ | $2.9753\times 10^{-6}$ | $4.1384\times 10^{-7}$ | Fitted
Initial Value | Definition | Estimated Mean Value | Standard Deviation | Data Source
$S_{1}(0)$ | Initial susceptible population of group-1 | $2.5117\times 10^{6}$ | $-$ | [2]
$V(0)$ | Initial vaccinated population | $9.8651\times 10^{6}$ | $-$ | [2]
$E_{1}(0)$ | Initial exposed population of group-1 | 912.8571 | $-$ | Fitted
$A_{1}(0)$ | Initial asymptomatic population of group-1 | $678.83$ | $63.497$ | Fitted
$U_{1}(0)$ | Initial unreported population of group-1 | $42.791$ | $4.3679$ | Fitted
$I_{1}(0)$ | Initial reported case number of group-1 | $200$ | $-$ | [2]
$S_{2}(0)$ | Initial susceptible population of group-2 | $1.7513\times 10^{6}$ | $-$ | [2]
$E_{2}(0)$ | Initial exposed population of group-2 | $287.1429$ | $-$ | Fitted
$A_{2}(0)$ | Initial asymptomatic population of group-2 | $165.88$ | $15.296$ | Fitted
$U_{2}(0)$ | Initial unreported population of group-2 | $184.81$ | $32.996$ | Fitted
$I_{2}(0)$ | Initial reported case number of group-2 | $18$ | $-$ | [2]

##  3 Results

(i@) Fitting results

Figure 3 shows the fitting result of the daily incidence data in Figure 1 to
the older group (2.2) and the younger group (2.2) respectively. As shown in
Figure 3, the daily incidence data increases initially but then declines. The
data shows the fourth wave of COVID-19 in Ontario, which is before the new
variant emerges in Canada. The initial increase of the daily incidence case
number may be attributed to the reopening in August and September, when the
vaccine program steadily rolls out. It is then followed by a gradual decline,
which is due to a combination of a few factors, such as a high vaccine
coverage among eligible individuals, implementation of health protocols etc.
Figure 3 indicates that the daily incidence data changes in the same pattern
for both the older and the younger groups but the infected case number peaks
slightly later in the younger group than the older group.

Under the current set of parameters in Table 1, the prediction of (2.2), (2.2)
indicates that the disease dies out eventually. The prediction is confirmed by
the effective reproduction number $\mathcal{R}_{t}=0.31727<1$ by substituting
the parameter values in Table 1. However, starting in November and afterwards,
vaccinated individuals may experience higher probabilities of infection
because of the gradual decline of vaccine protection efficacy. Meanwhile,
before December, individuals in the younger group are still not eligible for
receiving vaccines but are at higher risks of infection. This is mainly
because children stay indoors more often due to the weather condition and will
contract the virus more easily if they follow the same health protocols as
before. We will explore different scenarios below by increasing the within-
group contact rate and the between-group contact rate.

(a)

(b)

Figure 3: The fitting result of model (2.2)-(2.2) to the daily incidence data
between August 1 and October 25 in Ontario, Canada. Figure 3a shows the
fitting of the daily incidence data to the older group. Figure 3b shows the
fitting of the daily incidence data to the younger group.

(ii@) Comparing the relative importance of the within-group and between-group
transmission

Next, we analyze the impact of $\beta_{11}$ and $\beta_{13}$ on future
epidemic waves and compare the relative importance of the within-group contact
rate and the between-group contact rate on the final epidemic size. By
increasing the contact rate $\beta_{11}$ or $\beta_{13}$, we obtain a similar
pattern of the future waves for both the older group and the younger group and
hence show only the result of the older group.

Figure 4a shows that the daily incidence number of the older group increases
and initiates a further wave if the within-group contact rate $\beta_{11}$
increases. The further epidemic wave achieves a higher peak if the contact
rate $\beta_{11}$ is larger. However, the peak size of epidemics is relatively
small compared to the peak size of the fourth wave even if the contact rate is
larger and is on an order of different magnitude.

However, Figure 4b shows that an increasing between-group contact rate
$\beta_{13}$ triggers a future epidemic wave of a large size. The peak size of
future epidemic waves possibly exceeds the peak size of the fourth wave if the
between-group contact rate of the older group is sufficiently large. Figure 4
shows that the between-group contact rate of the older group imposes a much
larger impact on future epidemic waves compared to the within-group contact
rate.

Next, we increase the within-group contact rate of the younger group
$\beta_{21}.$ Figure 5a demonstrates that the daily incidence number of the
older group declines monotonically even if $\beta_{21}$ increases
significantly. However, the daily incidence number of the younger group shifts
to a gradual increase from the original decline if $\beta_{21}$ increases, as
shown in Figure Figure 5b. More importantly, the daily incidence number of the
younger group increases rapidly and forms a new epidemic wave of the size much
larger than the fourth wave if $\beta_{21}$ is relatively large.

We also examine how the between-group contact rate of the younger group
$\beta_{23}$ may trigger the future epidemic waves. Figure 6a shows that the
daily incidence number of the older group decreases monotonically if
$\beta_{23}$ increases. However, by comparing Figure 5a and Figure Figure 6a,
we observe that the daily incidence number of the older group is more
sensitive to $\beta_{23}$ than $\beta_{21}$ even though a steady decline holds
for both varying $\beta_{21}$ and $\beta_{23}$ in reasonable ranges. Figure 6b
indicates that an increasing between-group contact rate of the younger group
leads to a rapid increase in the daily incidence number of the younger group.
The new wave reaches the peak of a larger size than the fourth wave in a short
time and then declines.

Overall, Figure 4, Figure 5, Figure 6 demonstrate that an increasing between-
group contact rate in either the older group or the younger group may trigger
future epidemic waves. The difference is that an increasing between-group
contact rate of the older group leads to rapid growths of the daily incidence
numbers of both groups whereas an increasing between-group contact rate of the
younger group impacts more heavily on the younger group alone. The results
indicate that a mixed transmission in different age groups plays an important
role in triggering future waves and hence confirms the importance of
vaccinating the younger group. The results also shed light on the demographic
structure among the infected individuals in future waves: reported cases in
the younger group will constitute a heavier proportion in the total infected
individuals.

(a)

(b)

Figure 4: Daily incidence number of the older group for different contact rate
$\beta_{11}$ and $\beta_{13}$ respectively. All the parameters remain
unchanged as shown in Table 1 except $\beta_{11}$ and $\beta_{13}$
respectively.

(a)

(b)

Figure 5: Daily incidence number of the older group and the younger group
respectively for different contact rate $\beta_{21}$. All the parameters
remain unchanged as shown in Table 1 except $\beta_{21}.$

(a)

(b)

Figure 6: Daily incidence number of the older group and the younger group
respectively for different contact rate $\beta_{23}$. All the parameters
remain unchanged as shown in Table 1 except $\beta_{23}.$

(iii@) Sensitivity analysis

In this section, we conduct sensitivity analysis to investigate the
sensitivity of the total cumulative infection and infection number in each
age-group on the estimated parameters. For each parameter, Latin Hypercube
Sampling [3, 12] is adopted to generate parameter values with assumed ranges
and distributions as specified in Table 2. We generate 3000 sets of parameter
values for the analysis. By using these sets of values, partial rank
correlation coefficients (PRCC) are calculated to determine the impact of
varying parameters on the number of cumulative infections of COVID-19 between
August and October, 2021 [11]. The PRCC indices range between -1 and 1, with
positive (negative) values indicating a positive (negative) relationship and
magnitudes indicating the relative level of impact on the quantity of
interest, with a magnitude of 0 having almost no impact and 1 having the most
influential impact.

Table 2: Parameter ranges and distributions for sensitivity analysis.

Parameter | Lower Bound | Upper Bound | Distribution
---|---|---|---
$\beta_{11}=\beta_{12}$ | 0 | $10^{-5}$ | T
$\beta_{13}=\beta_{14}$ | 0 | $10^{-5}$ | T
$\beta^{v}_{11}=\beta^{v}_{12}$ | 0 | $10^{-5}$ | T
$\beta^{v}_{13}=\beta^{v}_{14}$ | 0 | $10^{-5}$ | T
$\epsilon$ | 0 | 1 | T
$\beta_{21}=\beta_{22}$ | 0 | $10^{-5}$ | T
$\beta_{23}=\beta_{24}$ | 0 | $10^{-5}$ | T
$\omega$ | 0 | $0.3$ | T
$\tau_{1}$ | 0 | 0.3 | T
$\tau_{2}$ | 0 | 0.3 | T

  * •

$T$ indicates triangular distribution with its peak value from Table 1.

Figure 7 shows the PRCC for cumulative incidence number of COVID-19 infection
for the older group, the younger group, and the entire population in Ontario
between August 1 and October 25, 2021. Figure 7a, Figure 7b, Figure 7c
demonstrate that the transition rate $\tau_{1},\tau_{2}$ from unreported
infected class to reported infected class in respective group and vaccination
rate $\omega$ in the older group have negative impact on the number of
cumulative infections. The result is not surprising because individuals in the
reported infected class are excluded from social activities due to self-
isolation or hospitalization and therefore unable to transmit the pathogen
whereas vaccinated individuals gain immunity and become less susceptible to
infection. Comparing $\tau_{1},\tau_{2},\omega$, vaccination rate $\omega$ has
a stronger influence than the reporting rates $\tau_{1}$ and $\tau_{2}$, which
implies that vaccination is a more effective means to control the spread of
disease than reporting infected cases. Moreover, reporting rate of the older
group $\tau_{1}$ has a relatively larger impact than reporting rate of the
younger group $\tau_{2}$ because the majority of infected population are
adults.

(a)

(b)

(c)

Figure 7: Partial rank correlation coefficients (PRCC) calculated using
parameter values from Latin Hypercube Sampling with respect to cumulative
infections for population 12+ years of age, 0-12 years of age and all from
Aug. 1, 2021 to Oct. 25, 2021 in Ontario, Canada.

##  4 Conclusion and Discussion

Since the initial identification of COVID-19 in Wuhan, China, the
transmissible disease quickly escalates and becomes a global pandemic for
about 2 years up to now. In Ontario, Canada, during the first year of the
disease prevalence, the main intervention strategies have been
nonpharmaceutical, such as provincial-wise lock down, keeping social
distancing etc. due to the lack of effective vaccines. Such interventions are
effective in mitigating the disease spread but cause significant economic loss
at the same time and therefore are not long-lasting.

Progressing to the second year of the pandemic, in Ontario, starting from
December 2020, the vaccine program gradually rolls out. The initial phase of
the vaccine rollout targets the seniors population or vulnerable individuals
and then gradually expands to individuals who are above the age of 12. The
vaccine for children between the age of 5 to 11 was approved by Health Canada
in late November but the rollout takes time and only $3.2\%$ of the children
in the age of 5-11 are fully vaccinated by now [2].

In this paper, we propose an age-stratified model that divides the entire
population in Ontario into two groups: the older group where individuals are
above the age of 12 and are eligible to receive vaccines and the younger group
where individuals are below the age of 12 and are not eligible to receive
vaccines. We fit the model to the daily incidence data of each group in
Ontario between August 1, 2021 and October 25, 2021 and obtain a good fitting
result.

The results demonstrate that between-group contact rate plays a more important
role in triggering future waves than the within-group contact rate. The
increasing between-group contact rate of the older group shifts the daily
incidence number of either group from the decline to a rapid growth. However,
an increasing between-group contact rate of the younger group mainly leads to
a new epidemic wave in the younger group only. The results confirm the
importance of achieving a high vaccine coverage in the younger group in order
to mitigate the disease spread.

A new variant B.1.1.529 emerges in late November, 2021 in multiple African
countries and quickly spreads to Canada in December. Early data indicate that
the new variant is highly transmissible compared to other variants. Moreover,
data show that currently, vaccines offer a much lower protection efficacy for
even fully vaccinated individuals. Our model can be extended to study the
invasion of the new variant, which leaves as future work.

## Acknowledgement

XW is grateful for funding from the NSERC of Canada (RGPIN-2020-06825 and
DGECR-2020-00369).

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