Studying the impacts of variant evolution for a generalized age-group transmission model

The differences of SARS-CoV-2 variants brought the changes of transmission characteristics and clinical manifestations during the prevalence of COVID-19. In order to explore the evolution mechanisms of SARS-CoV-2 variants and the impacts of variant evolution, the classic SIR (Susceptible-Infected-Recovered) compartment model was modified to a generalized SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) compartment model with age-group and varying variants in this study. By using of the SVEIR model and least squares method, the optimal fittings against the surveillance data from Fujian Provincial Center for Disease Control and Prevention were performed for the five epidemics of Fujian Province. The main epidemiological characteristics such as basic reproduction number, effective reproduction number, sensitivity analysis, and cross-variant scenario investigations were extensively investigated during dynamic zero-COVID policy. The study results showed that the infectivities of the variants became fast from wild strain to the Delta variant, further to the Omicron variant. Meanwhile, the cross-variant investigations showed that the average incubation periods were shortened, and that the infection scales quickly enhanced. Further, the risk estimations with the new variants were performed without implements of the non-pharmaceutical interventions, based on the dominant variants XBB.1.9.1 and EG.5. The results of the risk estimations suggested that non-pharmaceutical interventions were necessary on the Chinese mainland for controlling severe infections and deaths, and also that the regular variant monitors were still workable against the aggressive variant evolution and the emergency of new transmission risks in the future.


Introduction
COVID-19 is an infectious disease caused by the SARS-CoV-2 virus.The main clinical characteristics have been figured out by [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.
According to WHO, on November 27, 2021, there are 259 502 031 confirmed cases and 5 183 003 confirmed deaths over the world.At the same time, vaccine doses administered 7 702 859 718 on November 25, 2021.According to National Health Commission of the People's Republic of China, China has reported 98 631 confirmed cases and 4636 confirmed deaths on November 27, 2021.
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].
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 calculate R 0 in Subsection 3.1.We discuss the Properties of the model near the disease-free equilibrium point in Subsection 3.2, including the disease-free equilibrium point, the basic reproduction number, and the stability of diseasefree equilibrium.We study the endemic equilibrium in Subsection 3.3.We find the endemic equilibrium points and their stability.Sensitivity analysis of the model parameters is given in Section 4 to show the importance of vaccination, city lock-down, and other factors.In Section 5, we apply our model to analyze the cases in Shijiazhuang, Hebei Province.Finally, we make some discussion in Section 6 and the conclusion is drawn in Section 7.

Model
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 age group (< 60 yr) and age group ( 60 yr) 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.We divide vaccinated individuals separately for figuring out the effects of vaccination in an age-structured population.So the solution as T , and the total population size is The transmission diagram is given in Figure 1.
Our model is derived as follows: 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 ( 1) is shown on Table 1.

Long-term Dynamics of the Model
In this section, we give the expression of the basic reproduction number R 0 , and discuss the properties of model ( 1) around the disease-free equilibrium point and the endemic equilibrium point and their stabilities.

Basic Reproduction Number R 0
The basic reproduction number R 0 describes the expected number of secondary case from primary case during the infectious period of the primary Fig. 1: Diagram for conversions between the two age group in compartments.
case, which is an important threshold parameter in studying the epidemics according to [20].We use the method provided in [20] to calculate the basic reproduction number R 0 . Let Then, model (1) can be simplified as follows: The equations ( 2) are linear with respect to S 0 1 , S 0 2 , V 0 1 and V 0 2 , so we derive which gives a unique disease-free equilibrium point X 0 = S 0 1 , S 0 2 , 0, 0, 0, 0, 0, 0, V 0 1 , V 0 2 T to model (1).Further, the initial total population size is 1 G1 means the individuals who are under 60 years old, short for G1 (< 60 yr); 2 G2 means the individuals who are over 60 years old, short for G2 ( 60 yr).
Firstly, the appearance of new infective individuals in compartments I and E is written as: T − (P 1 , P 2 , P 3 , P 4 , P 5 ) T , where V − stands for the transfer of the individuals out of compartment I and V + stands for the transfer of the individuals into compartment I by all other means (except for the appearance of new infective individuals).The components of are respectively We denote , for i, j = 1, 2, 3, 4 and Y 0 = (0, 0, 0, 0, S 0 1 , S 0 2 , 0, 0, V 0 1 , V 0 2 ), by Lemma 1 in [20], we obtain which then yields that The characteristic equation of matrix F V −1 is followed Then, the eigenvalues of (10) are respectively (12) By the next generation matrix method, the basic reproduction number R 0 is the spectral radius of F V −1 , that is: where the positive parameters k ij (i, j = 1, 2, 3, 4) are defined in (8).

Disease-free Equilibrium and Stability
The Jacobian matrix J on Appendix A at the disease-free equilibrium point X = X 0 is with Further, by the careful computation, we derive and The characteristic equation of Jacobian matrix J 0 is equal to the product of the characteristic equations of M 1 and M 4 .Because the eigenvalues of M 4 are all negative, and other four eigenvalues depend on the characteristic equation of where According to Descartes' Rule of Signs in [21][22][23], the number of negative roots of the characteristic equation ( 18) is equal to the number of variations in the change in the coefficient sign, so (18) has 4 negative values if The conclusions (20) are valid provided that the parameters in (19) satisfy: So, we derive the following result: Theorem 1 (Disease-free equilibrium point stability) The disease-free equilibrium point X 0 is locally asymptotically stable when R 0 < 1, while X 0 is unstable when R 0 > 1, when the parameters satisfy equation (21).
We construct the Jacobian matrix for age group (< 60 yr) and age group ( 60 yr) respectively as follows: (50) 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: with (53) Based on Descartes' Rule of Signs [21][22][23], the number of negative roots of the characteristic equation ( 52) is equal to the number of variations in the change in the coefficient sign, so (52) has five negative values if The conclusions in (54) are valid provided that the parameters satisfy: Due to then conditions (55) can be simplified as (56) then condition (54) 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 (55), so the endemic equilibrium point is stable global asymptotic.
Theorem 3 (Endemic equilibrium points stability) 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 (55).

Sensitivity Analysis of Parameters
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 [4,24,25].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 ( 13), we derive an analytical expression for the sensitivity of it, to each of the seventeen different parameters described in Table 1.By this way, we can compare the sensitivity indices of basic reproduction number R 0 with respect to some parameters.
In figure group B1 shown on Appendix B, 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 B1a and B1b, the basic reproduction ratio show different sensitivity in the age group (< 60 yr) and age group (< 60 yr), because of the proportion of age group (< 60 yr) is several times higher than that of age group ( 60 yr).On the other hand, due to different scales for horizontal axes between Figure B1a and B1b, 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 B1c and B1d, 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 B1e, B1f, B1g and B1h, we can find that for any β ij (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 of the difference of the two age group 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.

Numerical Simulation of the Cases in Shijiazhuang
We investigate the reported cases in Shijiazhuang of Hebei Province from January 2 to February 28 of the year 2021, and then apply our model to analyze the cases in this area.

Cases in Shijiazhuang
At the middle of December 2020, a sudden flare-up emerged in Shijiazhuang, a rural city and the capital of Hebei province.More than 300 individualas were diagnosed with COVID-19 from January 2 to January 12 of the year 2021.The authorities had taken measures to control epidemics to have promptly reduced the spread of COVID-19.Some articles have make some statistical analysis for Shijiazhuang.[12,26,27] The control measures of Shijiazhuang government can be divided into 3 stages: 1. before January 2 of 2021, the local CDC and the local government are out of awareness for a new outbreak; 2. January 7 of 2021, Shijiazhuang announced to lock down the city; [28,29] 3. January 11 of 2021, the isolation of Xiaoguozhuang village and other villages nearby are carried out by the local government.[30] The COVID-19 in Shijiazhuang was controlled effectively as of January 27.[31] We collect data from the Health Commission of Hebei Province, and then analyze the data in many viewpoints.The number of the daily reported cases is plotted in Figure 2a from January 2 to February 3 of the year 2021, and the location where the reported cases come from is plotted in Figure 2b.It is obvious that the majority of the reported cases distributes centered around Xiaoguozhuang village, the minor of the reported cases distributes nearby within Shijiazhuang city.We reorganize the reported cases in age in Figure 3a.The reported cases over 60 take the percentage 25.81% in Figure 3b.We discuss the dynamical properties of COVID-19 based on the reported cases in Shijiazhuang city.We keep the values of the parameters in 1, others parameters and their corresponding values are presented in Table 2.We apply the parameters of Table 1 to simulate the cumulative cases in Shijiazhuang city, where the basic production number is R 0 = 8.25. Figure 4 simulates the number of the cumulative cases to fit the real reported confirmed cases for both G1 (< 60yr) and G2 ( 60yr) by days.
Based on the distributions in time and space and the media reports [30], the individuals in Xiaoguozhuang village and villages and towns nearby moved to the remote hotels for the centralized quarantine on January 11 of 2021.We assume that the infection rates are separated into two-stage step functions: the first stage is before January 11, and we take β 11 = 0.2880, β 12 = 0.3000, β 21 = 0.4400, β 22 = 0.3900 before the remote centralized quarantine; moreover, the second stage is starting from January 12 to February 6, the chances of contacting with others between the individuals are less after January 12, and we take β 11 = 0.0014, β 12 = 0.0015, β 12 = 0.0022, β 22 = 0.0020 for the second stage.Further, we provide the daily new cases and the current cases for the exposed and the infected, and the cumulative cases as well in Figure 4. On February 6, the new outbreak of COVID-19 in Shijiazhuang vanishes [31], the simulations in Figure 4 present the coincidence with the control measures of Shijiazhuang government.

More simulations for the dynamical properties of the model
Based on the cases in Shijiazhuang city, we adopt more simulations to study the dynamical properties of COVID-19 in the rest of this section.The groups  2, while others keep the same as Table 1.

Variation of infection rate
Scenario A (Risk-based control) We assume that there exist three-level control strategy after the remote centralized quarantine on January 11, and adopt the decreasing contact rates strategy.For instance, we set up a 14-day prevention and control cycle.After 32 days of free transmission from December 12 of 2020 to January 11 of 2021 (i.e., 1-32d), then two 14-day control cycles are followed, i.e., from January 12 of 2021 to January 25 of 2021 (i.e., 33-46d) and from January 26 of 2021 to February 8 of 2021 (i.e., 47-60d).Finally the transmission of the virus is reduced to the minimum level.

Scenario B (Remote quarantine late strategy)
We consider the infection rate in our model.Figure 6b and Figure 5b show that the infection rate affect the number of the cumulative cases.
Scenario C (Remote quarantine late and double contact rates strategy)

Variation of vaccination number
We consider the vaccination initial values in our model.Figure 8a shows that vaccination can suppress the the number of the cumulative cases.If the vaccination initial value is low, the number of the cumulative cases raises at the peak in Figure 8b.
Variation of population aging rate   0-39d 0.2880 0.3000 0.4400 0.3900 0-46d 0.2880 0.3000 0.4400 0.3900 40-60d 0.0014 0.0015 0.0022 0.0020 47-60d 0.0014 0.0015 0.0022 0.0020 We consider the aging rates of the whole population in Shijiazhuang to compare the different impacts to the final size of the cumulative cases.Figure 9a and Figure 9b show that the aging rate makes the slight impact on the number of G1 (< 60 yr), and makes the main impact on the number of G2 ( 60 yr).The research results reveal that the government should pay more attention on the protection of the elder individuals, which can significantly reduce the number of confirmed cases in the elderly population.0-39d 0.5760 0.6000 0.8800 0.7800 0-46d 0.5760 0.6000 0.8800 0.7800 40-60d 0.0014 0.0015 0.0022 0.0020 47-60d 0.0014 0.0015 0.0022 0.0020

Discussion
A finer age structure can reveal the more precise dynamics of COVID-19, but at the same time, more levels of age structure also implies a linear increase in the dimensionality of the equations, which make the model much more complex.As a result, the model is not suitable for the analysis of COVID-19 in the real world.A good idea is to choose the model just right, not only enough finer but also can be performed theoretically.We had no detailed information about individuals, such as exposure interval, date of symptom onset and so on.Without these information, we can only get the incubation period and serial interval by estimating rather than by statistical analysis.For example, incubation period might be overestimated.[13,32] But this inaccuracy is undercontrol by reference to others studies.
The people in the rural area frequently attended at large gatherings, and when they find theirselves had symptoms, they choosed to not to see a doctor or buy medicines in village hospital or individual clinic.This caused the outbreak to spread quickly and hard to consult early cases.The COVID-19 outbreak in Shijiazhuang is a outbreak about the rural area.[12] We also find this character in the simulation by our model, so the key to control the outbreak is to control the rural area.The governance capacity of the government is different between town and rural area.If the goverment didnot inversitigate the character mentioned above of this outbreak, they will lose the focus.

Conclusion
In summary, 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 work out endemic equilibrium points and reveal their stability.We use sensitivity analysis to find out the influence of parameters to R 0 which 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.our study suggests the  Now the block matrices of Jacobian matrix can be written as follows: ) If a 12 b 12 = 0 and ∆ = (a 12 b 1 + a 1 b 12 − a 2 b 11 ) 2 −4a 12 b 12 a 1 b 1 = 0, we obtain

Fig. 2 :
Fig. 2: Distributions of the cases reported in time and space Distribution of ages in group

Fig. 3 :
Fig. 3: Distribution of ages of the cases reported

Fig. 5 :Fig. 6 :
Fig. 5: The impacts of the infection rates with risk-based strategy

Fig. 7 :
Fig. 7: The impacts of the infection rates with remote quarantine and double contact rates strategy

Fig. 8 :
Fig. 8: The impacts of the vaccination number

Fig. 9 :
Fig. 9: The impacts of the aging rate

Table 2 :
Parameters and their value used in simulation

Table 2 :
Parameters and their value used in simulation (Continued )

Table 3 :
Infection rates for medium risk and low risk strategies

Table 4 :
Infection rates for remote quarantine 7 days late and 14 days late

Table 5 :
Infection rates for remote quarantine and double contact rates )