key: cord-0591184-ghbqq4u9
authors: Zhou, Ruiyang; Wei, Fengying
title: An age-structured epidemic model with vaccination
date: 2022-05-08
journal: nan
DOI: nan
sha: 0f26ef04f32ed141af5f363761ac35d89217d3da
doc_id: 591184
cord_uid: ghbqq4u9

In our article, we construct an age-structured model for COVID-19 with vaccination and analyze it from multiple perspectives. We derive the unique disease-free equilibrium point and the basic reproduction number $ mathscr{R}_0 $, then we show that the disease-free equilibrium is locally asymptotically stable when $ mathscr{R}_0<1 $, while is unstable when $ mathscr{R}_0>1 $. We also investigate the properties of our model in endemic equilibrium. We work out endemic equilibrium points and reveal the stability. We turn to sensitivity analysis to explore how parameters influence $ mathscr{R}_0 $. Sensitivity analysis helps us develop more targeted strategies to control epidemics. Finally, this model is used to discuss the cases in Shijiazhuang, Hebei Province at the beginning of 2021.

COVID-19 is an infectious disease caused by the SARS-CoV-2 virus. The main clinical characteristics have been figured out [1] . Many people infected with the virus will experience mild to moderate respiratory illness and recover without requiring special treatment. However, some will become seriously ill and require medical attention. Older people and those with underlying medical conditions like cardiovascular disease, diabetes, chronic respiratory disease, or cancer are more likely to develop serious illnesses. Anyone can get sick with COVID-19 and become seriously ill or die at any age. There exist various kinds of mathematical models to study epidemics, such as SIR, SIRS, SEIR, and SEIRS on the transmission of diseases. Some of them is age-structured model, such as in COVID-19 epidemic [2, 3] , tuberculosis transmission [4] , and measles epidemics [5] . For COVID-19 epidemic, people in different ages have different clinical characteristics, so it's necessary to include age-structure into the model. COVID-19 vaccines can produce protection against the disease. Developing immunity through vaccination means there is a reduced risk of developing the illness and its consequences. This immunity helps people fight the virus if exposed. Getting vaccinated can also protect people around. This is particularly important to protect people at increased risk for severe illness from COVID-19, such as healthcare providers, older or elderly adults, and people with other medical conditions. Some articles have studied the importance and the impact of COVID-19 vaccination, such as in [6] [7] [8] .

In this article, we study the dynamics of an age-structured epidemic model with vaccination. In this model, individuals are distinguished both by age and the stage of the disease. The formulation of the model is addressed in Section 2. We discuss the Properties of the model near the disease-free equilibrium point in Section 3, including the disease-free equilibrium point, the basic reproduction number, and the stability of disease-free equilibrium. We study the endemic equilibrium in Section 4. We find the endemic equilibrium points and their stability. Sensitivity analysis of the model parameters is given in Section 5 to show the importance of vaccination, city lock-down, and other factors. In Section 6, we apply our model to analyze the cases in Shijiazhuang, Hebei Province. Finally, the conclusion is drawn in Section 7.

In this section, we consider an age-structured epidemic model in which individuals are distinguishable both by their age and the stage of the disease. In the model, the total size of the population N (t) contains 2 age groups, we use subscripts i = 1, 2 to stand for the younger age group (0 to 59 years) and older age group (60 years and older) respectively. The population in the same age group is divided into 5 clusters, including the susceptible individuals (S i ), the latent individuals (E i ), individuals with infectious (I i ), the recovered individuals (R i ) and those with vaccination (V i ). The latent class consist of individuals infected, but without an infectious status, while the infectious class consists of those with infectious status. The transmission diagram is given in Figure 1 . We divide vaccinated individuals separately for figuring out the effects of vaccination in an age-structured population.

Our model is derived as follows:

(2.1)

All of the parameters are assumed to be constants. We assume that µ i > 0 (i = 1, 2), while the others are nonnative. The description of the parameters used in the model (2.1) is shown on Table 1 .

We note the solution as X = (S 1 , S 2 , E 1 , E 2 , I 1 , I 2 , R 1 , R 2 , V 1 , V 2 ) T , and the total population size is 3 Properties in disease-free equilibrium

Then, model (2.1) can be simplified as follows:

(3.1)

The system of equations (3.1) is linear with respect to S 0 1 , S 0 2 , V 0 1 and V 0 2 , so the solution (3.2) can be expressed by

which gives a unique disease-free equilibrium point

Further, the initial total population size is N 0 = S 0 1 + S 0 2 + V 0 1 + V 0 2 to model (2.1). The basic reproduction number R 0 describes the expected number of secondary case from primary case during the infectious period of the primary case, which is an important threshold parameter in studying the epidemics according to [9] . We use the method provided in [9] to calculate the basic reproduction number R 0 .

where k 11 , k 12 , k 21 , k 22 are given in equation (3.11) .

Firstly, we pick up the compartments E and I and denote Y = (E 1 , E 2 , I 1 , I 2 , S 1 , S 2 , R 1 , R 2 , V 1 , V 2 ), and the appearance of new infective individuals in compartments I and E:

the transfer of individuals out of compartment I:

where 

with

,

(3.8)

which then yields that

where Σ 11 = α 2 β 12 g 2 + α 1 β 11 µ 2 2 + α 1 α 2 β 11 d 2 + α 1 α 2 β 12 g + α 1 α 2 β 11 γ 2 + α 2 β 12 d 1 g + α 2 β 12 gγ 1 +α 1 α 2 β 11 µ 2 + α 1 β 11 d 2 µ 2 + α 1 β 12 gµ 2 + α 2 β 12 gµ 1 + α 1 β 11 γ 2 µ 2 , Σ 21 = α 2 β 22 g 2 + α 1 β 21 µ 2 2 + α 1 α 2 β 21 d 2 + α 1 α 2 β 22 g + α 1 α 2 β 21 γ 2 + α 2 β 22 d 1 g +α 2 β 22 gγ 1 + α 1 α 2 β 21 µ 2 + α 1 β 21 d 2 µ 2 + α 1 β 22 gµ 2 + α 2 β 22 gµ 1 + α 1 β 21 γ 2 µ 2 ,

Now we consider the eigenvalues of the matrix F V −1 , for an identity matrix E 4 of order 4, the corresponding determinant is

which gives

then the eigenvalues of (3.13):

with

The basic reproduction number R 0 is the spectral radius of the next generation matrix, so we can get:

where the positive parameters k ij (i, j = 1, 2, 3, 4) are defined in (3.11) .

We next study the stability by the method provided in [10] . Firstly, let the Jacobian matrix J 10×10 be

then the Jacobian matrix turns into J 0 at disease-free equilibrium point X = X 0 , that is

To find the eigenvalues of the Jacobian matrix J 0 , the determinant λE 10 − J 0 is computed as:

By careful computation, we derive

which implies that the characteristic equation of M 1 is 

The conclusions (3.23) are valid provided that the parameters in (3.22) satisfy:

We turn to M 4 :

M 4 is a lower triangular matrix, the eigenvalues is equal to the elements along the main diagonal, so the eigenvalues of M 4 are negative.

In summary, we can state the following result which shows the stability at the disease-free equilibrium point.

The disease-free equilibrium point X 0 is locally asymptotically stable when R 0 < 1, while it is unstable when R 0 > 1, when the parameters satisfy equation (3.24).

In this section, we consider the properties of our model near the endemic equilibrium points, which can be used to reveal us how the disease spreads. In the endemic equilibrium points, all the compartments S,

Theorem 4 The endemic equilibrium points X * exist and are unique, when the parameters satisfy specific conditions discussed below.

Proof We consider a general case to study the nature of the model near endemic equilibrium points, in which the population number of five competent are all greater than zero.

Let right hand side of model (2.1) be zero, the equalities are derived as follows:

Taking sum of (4.1a) and (4.1c), (4.1b) and (4.1d) respectively, which then gives two equations as follows:

so S * 1 and S * 2 are expressed linearly by E * 1 and E * 2 . Similarly, I * 1 , I * 2 , R * 1 , R * 2 , V * 1 , V * 2 are linearly dependent on E * 1 and E * 2 by (4.1e)-(4.1j). The expressions are written as follows:

(4.

3)

The total population size N * is given

= u 10 + u 20 + y 10 + y 20 + (u 11 + u 21 + v 11 + v 21 + x 11 + x 21 + y 11 + y 21 ) E *

(4.4)

Together with (4.3) and (4.4), equations (4.1c) and (4.1d) can be rewritten as:

with

The endemic equilibrium points to equations (4.5) are about to discuss by cases as follows.

Case 1 a 11 = 0 and b 11 = 0.

Equations (4.5) determine a unique endemic equilibrium point as ∆ 1 = ∆ 2 = 0, that is:

then we figure out E * 1 for f 1 (E * 1 ) = 0 and f 2 (E * 1 ) = 0 respectively in (??) as follows:

which further implies

substituting (4.8) into (4.7) gives

(4.9)

We denote

thus E * 1 and E * 2 could be simplified as:

such that the following expressions are valid

We substitute E * 2 into the equation (4.6) to find the relationship between E * 1 andE * 2 as follows:

which further gives

(4.14)

Case 2 a 11 = 0 and b 11 = 0.

The first equation of (4.15) is linear, while the second one is quadratic. As ∆ 2 = 0, we can work out E * 1 from the second equation:

and combine with the first equation and (4.16), we can get:

If a 12 b 12 = 0, as ∆ = (a 12 b 1 + a 1 b 12 − a 2 b 11 ) 2 − 4a 12 b 12 a 1 b 1 = 0, we can get:

If a 12 b 12 = 0, then

Case 3 a 11 = 0 and b 11 = 0.

(4.20)

The second equation of (4.20) is linear, while the first one is quadratic. As ∆ 1 = 0, we can work out E * 1 from the first equation: 

Case 4 a 11 = b 11 = 0. equations (4.5) can be simplified as follows:

(4.25)

The first equation of (4.25) are linear in variable E * 1 , so we can get 

(4.28)

If 2a 12 b 22 = 0, then we can work out

Theorem 5 The endemic equilibrium point X * is locally asymptotically stable as R 0 < 1, and the endemic equilibrium point X * is unstable as R 0 > 1, under the conditions that the parameters satisfy (4.36).

Proof We construct the Jacobian matrix for adults and elders respectively as follows:

(4.31)

Because the same structure for elements in J a and J e , we denote them as M as follows:

The corresponding characteristic equation of M is described as follows: Based on Descartes' Rule of Signs [11] [12] [13] , the number of negative roots of the characteristic equation 

The conclusions in (4.35) are valid provided that the parameters satisfy:

then conditions (4.36) can be simplified as k 12 k 13 k 21 k 44 − k 2 11 k 22 k 53 > 0, k 11 k 22 − k 12 k 21 > 0, k 13 k 44 − k 11 k 53 > 0. (4.37) then condition (4.35) is satisfied as well, so λ 1 , λ 2 , λ 3 , λ 4 and λ 5 < 0.

In short, the characteristic values of the equation system in our model are all negative with constraint (4.36), so the endemic equilibrium point is stable global asymptotic.

In this article, we wonder what and how much the parameters affect the epidemic variable we are interested in. For instance, we want to figure out how much some important parameters influence R 0 , which can show the importance of vaccination, city lock-down, and some other factors.

To reach the goal, we turn to sensitivity analysis [4, 14, 15] . We define the sensitivity index Γ P in which the variable R 0 depends on the parameter P . When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives as follows:

As we have an explicit formula of R 0 in (3.3), we derive an analytical expression for the sensitivity of it, to each of the seventeen different parameters described in Table ? ?. By this way, we can compare the sensitivity indices of basic reproduction number R 0 with respect to some parameters.

In figure group 2, we find that R 0 is significantly impacted by s 1 , s 2 , v 1 , v 2 , and β ij (i, j = 1, 2), while the impact of the other parameters are much smaller. In this sense we can draw the conclusion that the value change of R 0 are mainly caused by them.

According to figure 2(a) and 2(b), the basic reproduction ratio show different sensitivity in the young and the elderly group, because of the proportion of young people is several times higher than that of the elderly.

On the other hand, due to different scales for horizontal axes between figure 2(a) and 2(b), actually if there is a small change in the proportion of the elderly group, the basic reproduction ratio will have notable change.

Considering that the age structure of a region generally does not change in the short term, strengthening the protection of the elderly population will effectively reduce the value of R 0 .

According to figure 2(c) and 2(d), we can find that to increase vaccination rate can decrease R 0 , but only when the rate is high enough, it can make significant effect on R 0 . This reveals a situation where the vaccine alone can completely stop the spread of the virus only if the percentage of the immunized population is high enough; otherwise, the authorities should take other measures to control the outbreak.

From figure 2(e), 2(f), 2(g) and 2(h), we can find that for any βij(i, j = 1, 2), they all have positive influence to R 0 , we can make such explanation to this phenomenon: the higher value of β represents a higher infectivity of the virus and, therefore, a higher value of R 0 . On the other hand, due to the proportion between young people and the elderly and population-to-population differential infectivity, their value is different.

Hence, we can say that our age-structured model provides a realistic description of COVID-19 transmission in different age groups in the population. We collect data from the website of Health Commission of Hebei Province, and then we analysis the data in many viewpoints. We plot the number of reported daily cases in figure 3 (a), we cam find that most cases was reported between January 6th and January 19th. We also statistics the locations where the cases reported in figure 3(b) , the result is shown that most cases are reported nearby the Xiaoguozhuangcun, while other cases distribute loosely in the rest area of Shijiazhuang city. Figure 5(a) shows the change of the number of people in each compartments over time.

In the figure 5(b) , the curve fits the actual cases very well. Based on the well result of simulation, we can consider the characteristics of case propagation in terms of the setting of parameters. The infectious rate is a two-stage segmentation function, the first stage is from December 20 to January 11, while the second stage is from January 11, which means that the control is also divided into two stages. Before January 11, the virus spreads unaffected in the city, while after January 11, the dissemination is severely restricted. In this sense, we find that the city lock-down measure in January 7 didn't work immediately, which didn't prevent the spread in the villages and towns around Xiaoguozhuangcun. However, the isolation of whole villages in January 11 immediately work and lower the infectious rate because cases are most reported around there. 

In our article, we construct an age-structured model for COVID-19 with vaccination and analyze it from multiple perspectives. We derive the unique disease-free equilibrium point and the basic reproduction number R 0 , then we show that the disease-free equilibrium is locally asymptotically stable when R 0 < 1, while is unstable when R 0 > 1. We also investigate the properties of our model in endemic equilibrium.

We work out endemic equilibrium points and reveal the stability. We turn to sensitivity analysis to explore how parameters influence R 0 . Sensitivity analysis can help us develop more targeted strategies to control epidemics. Finally, this model is used to discuss the cases in Shijiazhuang, Hebei Province at the beginning of 2021.

We define η 1 = I 1 β 11 + I 2 β 12 , η 2 = I 1 β 21 + I 2 β 22 .

(A.1)

Now the block matrices in matrix (3.16) can be written as follows: 

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