{'fraction_non_alphanumeric': 0.04311205805979093, 'fraction_numerical': 0.027731540049270923, 'mean_word_length': 4.442833846711361, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 4, 'https://': 1, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
DYNAMICAL ANALYSIS IN DISEASE TRANSMISSION AND FINAL EPIDEMIC SIZE

. We propose two compartment models to study the disease transmission dynamics, then apply the models to the current COVID-19 pandemic and to explore the potential impact of the interventions, and try to provide insights into the future health care demand. Starting with an SEAIQR model by combining the eﬀect from exposure, asymptomatic and quarantine, then extending the model to the one with ages below and above 65 years old, and classify the infectious individuals according to their severity. We focus our analysis on each model with and without vital dynamics. In the models with vital dynamics, we study the dynamical properties including the global stability of the disease free equilibrium and the existence of endemic equilibrium, with respect to the basic reproduction number. Whereas in the models without vital dynamics, we address the ﬁnal epidemic size rigorously, which is one of the common but diﬃcult questions regarding an epidemic. Finally, we apply our models to estimate the basic reproduction number and the ﬁnal epidemic size of disease by using the data of COVID-19 conﬁrmed cases in Canada and Newfoundland & Labrador province.

pandemic. The data surrounding the biology, epidemiology, and clinical characteristics of the COVID-19 virus have been growing daily. Up to June 20, 2020, there have been more than 8.8 million confirmed cases, including 465K deaths globally. At present, many researchers have made rapid responses to the current epidemic [25,26,30] and established mathematical models about the dynamics and transmission of COVID-19. It is well known that the basic reproduction number, which is defined by the expected number of new infective individuals infected by an typical infective individual during its entire period of infectiousness in a fully susceptible population, is an important index to measure the transmission potential of a disease. The use of mathematical models to estimate the basic reproduction number of COVID-19, ranging from 1.4 − 6.49, gives a sharp threshold property for their global dynamics [10,24], helps to determine the potential and severity of outbreaks [28,22].
There has been growing recognition that, different from SARS, MERS coronavirus [31,9], there is a long incubation period of the new coronavirus (14 days on average), which reflects the importance of quarantine. Quarantine/isolation (we mix them for simplicity in this paper) is one of the commonest methods of controlling the spread of diseases. In the light of the experience of China, quarantine is considered as one of the most effective ways to prevent people from getting infected by the virus. In previous pandemics such as SARS, the number of asymptomatic or mildly symptomatic people after infection was quite low, making it easier to contact trace the infected cases, and isolate them. We do not yet have enough information as to whether this is the case with COVID-19, although a large number of facts indicate that one of the characteristics of COVID-19 is the presence of an asymptomatic infected population.
One of the common questions regarding an epidemic is its final size [1,3,5,6]. The final size of an epidemic can be defined informally as the total number of people experiencing infection [11] throughout the outbreak. The probability distribution of final sizes is of particular interest to statistical epidemiologists. For a deterministic epidemic in a closed, homogeneous population, the final size equation gives the number (or frequency) of susceptible hosts at the end of the epidemic. The result in [18] shows that in most of such models, the final size relation involves only a single parameter -the basic reproduction number. With interventions such as quarantine and reduction of contact are included in the compartmental model, does this claim still hold?
There is limited information regarding the new coronavirus risk factors for severe disease. From the official site of the Centers for Disease Control and Prevention (CDC), based on currently available information and clinical expertise, we know that older adults (with age above 65) and people of any age who have serious underlying medical conditions might be at higher risk for severe illness. The majority of patients (accounting for 80% of the USA and Canada deaths from the disease) dying with COVID-19 are elderly and the large majority of the deceased may have severe underlying diseases. As the epidemic has the potential to collapse the health care system, it is essential for the effective allocation of emergency resources in areas with high levels of infection.
Given the above considerations, to improve understanding of the complex dynamics in the emerging outbreak of COVID-19, we propose two different compartmental models to study the transmission dynamics of the virus and to explore the potential impact of the quarantine interventions in the population of Canada and Newfoundland & Labrador province within Canada, and to disentangle how transmission is affected in different age group. First, we establish a one-group SEAIQR model to investigate the dynamics of the disease transmission by combining the effect from exposure, asymptomatic, and quarantine classes. Since COVID-19 poses a significant threat to anyone (especially for older people) unfortunate enough to be hospitalized with it, and critical cases can be admitted to the intensive care unit and possibly hooked up to ventilators, we further extend and modify the one-group model to two-groups with ages below and above 65 years old, and classify the infectious individuals to mild symptoms who needs self-isolation, patient in hospital and patient in intensive care, try to provide some insights into infectious disease trends in different age group. We focus our analysis on each model with and without vital dynamics. As we know that, in a population with vital dynamics, new births can provide more susceptible individuals to the population, sustaining an epidemic or allowing new introductions to spread throughout the population. Without the implementation of strict containment measures and movement restrictions, the disease dynamics will reach a steady state. In this category, we work on the dynamical properties including the global stability of the disease-free equilibrium and the existence of endemic equilibrium, with respect to the basic reproduction number. On the other side, if the virus transmission over time is much faster than the natural birth and death rates in the population, vital dynamics (birth and death) can be ignored. Because of the rapid spreading of the COVID-19 virus, in the models without vital dynamics, we explore the final epidemic size which is one of the common but difficult questions regarding an epidemic. Finally, using the data of COVID-19 confirmed cases in Canada and Newfoundland & Labrador province, we parameterize the model to estimate the basic reproduction number and the final epidemic size of disease. This paper is structured as follows. After the introduction, in Section 2, we propose a one-group SEAIQR model, study the dynamical properties involved in the model with vital dynamics, give the relation of the basic reproduction number to the system parameters, explore the final epidemic size when ignoring the natural birth and death rates and provide theoretical computation for the final size. With the concern of high risk in elder people, in Section 3, we modify the previous onegroup model to two-groups by separating the population with age below or above 65 years, together with the consideration of the severity of infectious individuals. Parallel dynamical analysis and final epidemic size computation are carried. In Section 4, we implement the numerical simulations to supplement the global dynamics beyond the theoretical analysis and compute the final epidemic size for one-and two-group models by using the real data in Canada and Newfoundland & Labrador province. Conclusion and discussion are drawn in Section 5, where the recovery rate incorporating social healthy system and medical resources is addressed.

Model description.
We propose a deterministic "Susceptible-Exposed-Infectious without or with symptoms-Quarantined-Recovered" (SEAIQR) compartmental model based on the clinical progress and interventions of the disease, the flow diagram of the disease transmission is shown in Figure 1.
Here, we divide the total population (N ) into six classes: susceptible (S), exposed (E), infected but not yet symptomatic (A), infectious with symptoms (I), quarantined (Q), and recovered (R), that is, N = S + E + A + I + R + Q. We assume that, after exposure to the virus, the individuals will be quarantined in each class. With vital dynamics, the disease transmission is governed by the following nonlinear system of differential equations: where λ(t) is quarantine-adjusted infection rate or the force of infection: Here we take the term β 3 Q(t) into account in the infection rate, since there exists the household infection in reality. For instance, it is fairly serious at the early stage of the outbreak in Wuhan China. In addition, due to the rapid spread of COVID-19, medical resources have been extremely saturated, many patients with mild or no symptoms are put at home to isolate which inevitably may cause family infection. This situation was particularly serious in The United States at the beginning of the outbreak. Many mild patients were not hospitalized, but isolated at home, which happens to infect the family members, since the infection chain is not interrupted. The description of the parameters and their value ranges are provided in the following table (See Table 1   Disease-induced death rate for I 0.04227 [22] (ii) From (2.1), we have We know that the disease-free equilibrium point (DFE) M 0 = ( Π µ , 0, 0, 0, 0, 0) always exists in (2.1).
In (2.1), we denote Applying the formula in [28], we can calculate the basic reproduction number by [q3(σα1 + m2α2) + m3(q2α1 + m2q1)] m1m2m3m4 =:RA + RI + RQ. (2. 3) The basic reproduction number obtained in (2.3) breaks down into three components: secondary infections generated from the infectious people without symptom, with symptom, and from the quarantined individuals respectively. Taking the basic reproduction number R 0 as a threshold index, we can obtain the global stability of the DFE M 0 .
What dynamical behaviour involves in the system when R 0 > 1? For the existence of endemic equilibrium point, we have the following result.
Proof. If the endemic equilibrium point M * exists, the right-hand side functions in (2.1) must be zero at M * . By denoting we have Finally λ * S * = m 1 E * results which determines E * , by (2.4), as Thus, E * exists whenever R 0 >1. Followed from (2.4), we know that the endemic equilibrium point M * exists and is unique when R 0 >1.
About the stability of the endemic equilibrium point M * , we will provide some numerical simulation results later, instead of giving tedious and complicated relation with respect to the model parameters.
2.3. The final epidemic size without vital dynamics. Due to the rapid spreading of the COVID-19 virus, we can ignore the host mortality and recruitment in the model (2.1), that is Π = 0, µ = 0. Then (2.1) becomes, (2.5) How many people will experience COVID-19 infection throughout the outbreak? From the viewpoint of mathematical modelling, it is related to the final epidemic size. While during the pandemic outbreak, the social distancing and behavioural responses can significantly reduce the spreading of the virus. Such interventions are key to flattening the epidemic curve. For the model (2.5), it is easy to obtain the same basic reproduction number R 0 as that in (2.3). According to the relationship between R 0 and β i , i = 1, 2, 3, it is obvious that when the contact rate β i decreases, the basic reproduction number R 0 decreases as well.
To explore the impact of intervention strategies in model (2.1), we take the effective contact rates β i , (i = 1, 2, 3) in the classes A, I and Q as the control parameters and assume that they are time-dependent and the intervention occurs Further, we assume that the total population N is constant for t T (by counting the disease-related death together), denoted by N T := N (T ), and denote as an effective reproduction number.
To compute the final epidemic size rigorously, from the model (2.5), we have Since all the solutions in the model (2.5) are non-negative and bounded, from (2.7) we can deduce that, as t → ∞, When t ≥ T , define the function We can show that W 1 (t) is invariant for t T by calculating the derivatives along with the solution of (2.5), which is W 1 (t) = 0. Therefore we have W 1 (T ) = W 1 (∞). An epidemic ends when no infections remain. So with the further assumption and from (2.8) we have Since the total number of individuals is assumed to be constant when t ≥ T , so Using (2.6), we find QT . (2.12) (2.12) provides a final size relation after the intervention is started at time T , implying that taking the intervention into account can, indeed, lead to estimate of the final epidemic size much smaller and more realistic than estimate that by ignoring the intervention. More specifically, when T > 0 (for example, T = 10, 20, 30, · · · ), the numbers of susceptible (S T ), infectious without or with symptoms (A T , I T ), quarantined (Q T ) and recovered groups (R T ) at some time T can be confirmed through the statistics of the health department, then we can use the formula (2.12) to calculate the value of S ∞ , so the final size of infected people (N − S ∞ ) can be obtained.
which presents a proportion of susceptible class from the starting of disease transmission to the end. Obviously, the estimate from (2.13) is larger than that by (2.12).
It is interesting to observe that, the formula obtained in (2.13) is different from most of the results in the literature (see [1,3,5,6,11,18]). What we can see is that quarantine is involved in our model which is part of the control measure in preventing the spreading of disease, while no such compartment(class) appeared in the mentioned references. While in the existed models with quarantine, there is no final size computation to our best knowledge! 3. Dynamics in the model with two age groups.
3.1. Model description. Although diseases can make anyone sick, there is an increased risk of more severe outcomes for vulnerable populations including the people aged 65 and over. The chance that a COVID-19 patient would develop symptoms severe enough to require hospitalization, especially for respiratory support, also rises sharply with age. In [8], Del Valle et al. combined age-related residual immunity with mixed effects to develop an age-structured model of disease transmission. They confirmed that reducing the number of contacts in the population can slow down the spread of the disease, and found that if vaccination is necessary, it should first be given to the most vulnerable age groups. Miller et al. found a consistent trend in the distribution of COVID-19 cases in the United States and elsewhere, with a significantly higher incidence in the older group in [19]. After further adjusting for demographic and uncertainty factors, it was found in [17,29] that the case fatality rate was higher in the elderly group than in the other groups. Therefore the strategies focusing specifically on protecting high-risk elderly individuals should be considered in managing the pandemic. In this section, we separate the population into two age groups, one with the age below 65 years old (denote as group-I), and another one with age 65 and over (denote as group-II). To capture the dynamics of the infectious virus more accurately, with the fact that hospitalization and intensive care unit admission rates are also higher among the elders, we further classify the infectious individuals into different stages including mild symptoms who need selfisolation, patient in hospital and patient in intensive care, according to the severity of infection. With the interaction between these two groups, the architecture of the disease transmission flow is given in Figure 2, there the natural mortality rate is not shown in the figure for simplicity. R I II Figure 2. Architecture of disease transmission flow with two age groups. Mathematically, the model can be represented by the following system of ordinary equations:Ṡ where i = 1, 2 is the index of the group, the description of all the variables is given in Table 2.
Infectious population without symptoms I 1 i (t) Infectious population with mild symptoms Recovered population The quarantine-adjusted infection rate function λ(t) has the following form: = 1, 2). The description of the parameters and their value ranges are shown in Table 3. Being short of real data, most of the parameter values are chosen by assumption there.  . With tedious computation, we can obtain the explicit relation of the basic reproduction number R 0 with respect to the parameters, that is, for i = 1, 2. Note that, here R 0 in (3.2) is the basic reproduction number, but R i0 , (i = 1, 2) in (3.3) are only notations used in the analysis and computation, not the basic reproduction number to each group. Similar to the model (2.1), within the closed set we have the following global stability result at M 0 : Theorem 3.1. The disease-free equilibrium (DFE) M 0 in the model (3.1) is globally asymptotically stable (GAS) in D 1 whenever R 10 + R 20 < 1.
Remark 1. (i) With Π 1 = Π 2 and µ 1 = µ 2 , this global stability condition be- . (ii) We should mention that the condition R 10 + R 20 < 1 is strong, at least for the case in (i), comparing with the normal condition R 0 < 1, although it is acceptable due to the complexity of the model. How to construct an appopriate Lyapunov function or use some other methods to obtain weaker condition (e.g., R 0 < 1) to ensure the global stability of DFE is not trivial and open to discussion.
Different from one-group model (2.1), in addition to the DFE M 0 , there exists one-group disease free (or partial disease free) equilibrium in this two-group model (3.1). Regarding the high risk for the group with age 65 and over, we discuss the condition for the existence of group-I disease free equilibrium M * 1 = (P * 1 , P * 2 ), where P * 1 = ( Π1 µ1 , 0, 0, 0, 0, 0, 0, 0) and . That is the case where disease only exists in the group with age 65 or over.
Although we can discuss the existence of positive endemic equilibrium theoretically, we neglect it here due to the complexity of the model and the lengthy analysis. We should mention that, in general, older people have lower immunity and are more likely to be infected. If they come into contact with younger people, the two groups can infect each other. Different from the one-group model, in the two-group model, we divide the patients into infectious population with mild symptoms, patient in hospital, and patient in intensive care. A detailed division of the patients by their degree of illness can help to formulate strategies to combat the spread of infectious diseases. To compensate for the limitations of theoretical analysis, we will give a numerical simulation of diseased equilibrium. We find that the diseased equilibrium is asymptotically stable under certain conditions. 3.3. The final epidemic size without vital dynamics. First we notice that with Π i = µ i = 0, i = 1, 2, the disease-free equilibrium is M 0 = (P 1 ,P 2 ) withP i = (S 0 i , 0, 0, 0, 0, 0, 0, 0), where S 0 i = N 0 i is the initial number of susceptible individuals in each group.
Consequently, we can obtain the corresponding basic reproduction number as which is different from that given in (3.2) although they have the same R i0 in (3.3). By using similar intervention strategies as that in the previous section for onegroup model, and denote β ij to ignore the tilde as the control parameters. Assuming that N 1 , N 2 are constants for t T , denoted by N T 1 := N 1 (T ), N T 2 := N 2 (T ), and From the model without vital dynamics, we can deduce that, as To find the final epidemic size relation, we define the following function: where the quantities of B i1 , B i2 , B i3 , B i4 , B i5 , B i6 , and B i7 are given in Appendix.
To show that W 2 (t) is invariants for t T , we calculate, , and the effective reproduction number R c = R ef f (T ) as We then can explore the relation of the final epidemic size from the invariance of W 2 (t). Since W 2 (T ) = W 2 (∞), that is, by (3.8), Since the total number of individuals at time T is Using (3.9), we find (3.12) which determine the final size relation to the epidemic situation at the intervention time T , effective basic reproduction number and the system parameters. Although it is tedious and intricate to estimate the final size of infected people in such complicated model, the formula (3.12), at least, can provide a way to approximate the final size from N − S 1 (∞) − S 2 (∞). In practice, several different mechanisms, including differences in susceptibility or transmission potential between different age groups, may contribute to the variety. When T = 0, assuming that S 0

Numerical simulation.
Previously we have done some theoretical analysis for the models (2.1) and (3.1), due to the complexity of the models, in this section, we implement some numerical simulation to explore more dynamic properties and estimate the final epidemic sizes in Canada and Newfoundland & Labrador. I. The stability of epidemic equilibrium. For the one-group model (2.1), based on the data in Table 1, we can calculate the basic reproduction number R 0 = 5.2515, and know that the endemic equilibrium M * exists and is unique. When choosing different initial values, each system ((2.1) and (3.1)) approaches the same positive equilibrium point M * (see Figure 3 for the one-group model and Figure 4 for the two-group model). We can conjecture that M * may be globally asymptotically stable, while the theoretical proof is non trivial. In the two-group model (3.1), we assume that the initial population ratio is 5:1 for the first group (under 65) and the second group (above 65). From the numerical simulation we can observe that, in Figure 4, the system approaches to a steady state with the total patients number 89 and 43 respectively, implying the ratio is about 2:1. The simulation result indicates that the older adults are more susceptible to infection.
II. The final epidemic size estimation. Through theoretical analysis, we have obtained the final epidemic size relationship of the prognosis. These analytical formulas (2.12) and (3.12) can be used to predict the total number of infected cases The initial values are (500, 0, 0, 0, 0, 1, 0, 0, 100, 0, 0, 0, 0, 1, 0, 0). over the entire outbreak period and help us understand what might happen and possibly drive public health action to achieve the best possible outcome.
a. Data based on Canada: Since the first four confirmed COVID-19 cases were reported in Canada on January 31, 2020, the number was increased to 15 on February 29. With the spreading of the virus, the confirmed number is blowing. Up to June 20, 2020, the number of confirmed cases had raised to 101K including 8410 deaths, from www.canada.ca/en/public-health/services/diseases/2019-novelcoronavirus-infection. In a released report by Government of Canada on April 9, 2020, the projected infection number will reach from 940000 to 1879000, under different control intensities.
The total population in Canada is N 1 = 37590000 from the website of Statistics Canada. To apply our model (2.1), we take January 31, 2020 as the initial time, and the initial condition is I(0) = 4, E(0) = A(0) = Q(0) = R(0) = 0, and S(0) = 37589996. We assume the intervention is carried after 30 days (at Feb. 29, 2020), implying T = 30. Using the same parameter values given in Table 1 except Π = µ = 0, γ 3 = 0.51624, we can obtain the basic reproduction number R 0 = 1.6768, and numerical simulation from (2.1) gives S T = 37589888, E T = 25, A T = 1, I T = 14, Q T = 7, R T = 56, here I T = 14 is almost consistent with the reported confirmed cases 15 from Government of Canada from Jan. 31 to Feb. 29, 2020.
The reflection of the public intervention (school closing, work from home, wear the mask, avoid group gathering, etc.) in our model (2.1) is assumed to be the reduction of contact rate β i in each class. If we assume the overall contact rates are dropped to 20∼80%, correspondingly we can calculate a final size of S ∞ which can determine a total number of disease cases throughout the epidemic (see Table  4), and draw the relationship between the reduction of contact rate and infection cases in Canada, see Figure 5.
Obviously, the reduction of the contact rate is one of the powerful measures to combat the spread of COVID-19 across Canada and flattening the pandemic curve. In addition, we find 50-55% of the contact rate is a considerably better estimate, comparing with the real data. In our prediction, when the contact rate is reduced by about 51.3% or 52%, the associated number of infection cases is 934929 or 1919191 respectively, which is approximately consistent with the prediction in the national report: 940000, 1879000. In particular, when the contact rate is reduced by about 50.71%, the disease cases is 109652, which is roughly consistent with the reported  cases in Canada so far. This model simulation indicates that reducing the contact rate by half is probably the current reality, comparing with the normal situation before COVID-19 time. From Figure 5, we can see there is a big jump in the number of cases around a critical percentage of contact rates, which is close to 50% as well. From the Statistics Canada website, we know that the proportion of the population with age 65 and above is about 17.2% in Canada. When applying our two group model (3.1), we have N 0 1 ≈ 31124520, N 0 2 ≈ 6465480. According to news reports (see https://globalnews.ca/news), we know the first four confirmed cases were under the age of 65, so the initial condition in (3.1) is, (31124516, 0, 0, 4, 0, 0, 0, 0) and (6465480, 0, 0, 0, 0, 0, 0, 0) for each group. Taking the intervention time T = 30 again, we can obtain S T 1 = 31124456, Table 3 Applying the final size relation (3.12), we calculate a final size of S 1 (∞) and S 2 (∞), giving a total number of disease cases N 0 i − S i (∞) over the course of the epidemic, with different contact rate (see Table 5). From Table 5, we can see that 40-60% of contact rate deduction gives a better match for the real data. In our prediction, when the reduction of the contact rates is around 50.5% and 55% (see Figure 6), the corresponding number of infected cases is 943765 and 1844531, respectively, which has good agreement with the result obtained from the model (2.1). The proportion of infected people for each group is 1 − Si(∞) N 0 i . The prediction is given in Table (5) confirms the factor that the proportion of infectious people in group-I was lower than that in group-II with aged over 65. Table 5. Cases over time in Canada for two groups % S1(∞) S2(∞) Disease cases (I) % Disease cases (II) % Total cases βij N 0 1 − S1(∞) 1 − S 1 (∞) b. Data based on Newfoundland & Labrador. In Canada, Newfoundland & Labrador is among the oldest provinces with respect to median age and has a high incidence of chronic disease. Due to its particular location and cluster effect at the beginning of spreading of COVID-19 virus, since March 14, 2020, the province reported the first case, the number was jumped to 152 on March 31. By April 30, the total number of confirmed cases was 258. During those days the province experienced what was at the time the biggest single outbreak in Canada, related to "the Caul's cluster" which infected 167 people -64 percent of the total cases in the province. The number 261 has been lasted for more than 20 days. Up to June 18, 2020, all infected people are recovered, except three death.

by using the parameters in
The total population in Newfoundland & Labrador (NL) is N 2 = 521542 from Statistics Canada. Since the few cases in the province, we only do the simulation to the model (2.1). Taking March 14, 2020 as the initial time with initial condition I(0) = 1, E(0) = A(0) = Q(0) = R(0) = 0 and S(0) = 521541. With the same intervention time T = 30, we have S T = 521515, A T = 1, I T = 4, Q T = 2, R T = 14 from (2.1). Following the final size relation (2.12), we can estimate the number of the infected case, which is shown in Table 6 and Figure 7, with different contact rates reduction. We can see that 40-50% of contact rate reduction is more reliable. Specifically, when the percentage is 48%, the disease cases is 268, which is roughly consistent with the reported cases.  Our model projections suggest that as long as cases are reported in any country, intervention strategies cannot be ignored. If control measures are strengthened, the number of infectious cases will be reduced. Therefore suppression interventions are key to flattening the epidemic curve.

5.
Conclusion and discussion. According to the transmission mechanism of COVID-19, we have proposed two kinds of compartment models to study the transmission dynamics of the COVID-19 virus and to explore the potential impact of interventions, to disentangle how transmission is affected in different age groups. We can obtain the basic reproduction numbers and final epidemic sizes in each model, which could be used to estimate the number of infected people during the outbreak. Our model predictions are very consistent with ongoing disease data and the final epidemic size, implying that intervention strategies cannot be ignored. If control measures or population behaviors are relaxed, the second wave of infection may appear in the future [7]. We must point out that, mathematical models cannot predict what will happen exactly, but rather can help us understand what might happen.