Published by De Gruyter March 9, 2022

The Numerical Approximation to a Stochastic Age-Structured HIV/AIDS Model with Nonlinear Incidence Rates

Jie Ren , Huaimin Yuan and Qimin Zhang

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2021-0154

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Abstract

In this paper, a stochastic age-structured HIV/AIDS model with nonlinear incidence rates is proposed. It is of great importance to develop efficient numerical approximation methods to solve this HIV/AIDS model since most stochastic partial differential equations (SPDEs) cannot be solved analytically. From the perspective of biological significance, the exact solution of the HIV/AIDS model must be nonnegative and bounded. Then a modified explicit Euler–Maruyama (EM) scheme is constructed based on a projection operator. The EM scheme could preserves the nonnegativity of the numerical solutions and also make the numerical solutions not outside the domain of the exact solutions. The convergence results between the numerical solutions and the exact solutions are analyzed, and some numerical examples are given to verify our theoretical results.

Keywords: Stochastic Age-Structured HIV/AIDS Model; Explicit Euler–Maruyama Scheme; Projection Operator; Convergence

MSC 2010: 60H35; 65C20; 92B05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12161068

Funding source: Natural Science Foundation of Ningxia Province

Award Identifier / Grant number: 2020AAC03065

Award Identifier / Grant number: 2021AAC03022

Funding statement: The research was supported in part by the National Natural Science Foundation of China (No. 12161068) and by the Natural Science Foundation of Ningxia Province (CN) (grant numbers 2020AAC03065 and 2021AAC03022).

A Proof of Theorem 3.2

Proof

Summing the equations of system (2.1), we have

d t ⁢ ( S + I + A ) = { - ∂ ⁡ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ∂ ⁡ a - μ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] - μ 2 ⁢ ( t , a ) ⁢ A ⁢ ( t , a ) } ⁢ d ⁢ t - σ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ⁢ d ⁢ B t ≤ { - ∂ ⁡ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ∂ ⁡ a - μ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] } ⁢ d ⁢ t - σ ⁢ ( t ) ⁢ [ S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) ] ⁢ d ⁢ B t .

Let U ⁢ ( t , a ) := S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) . According to the comparison theorem, the boundedness problem of system (2.1) is transformed into the corresponding problem of the following linear age-structured model with multiplicative noise:

(A.1) { d t ⁢ U = [ - ∂ ⁡ U ⁢ ( t , a ) ∂ ⁡ a - μ ⁢ ( t ) ⁢ U ⁢ ( t , a ) ] ⁢ d ⁢ t - σ ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ B t in ⁢ Q , U ⁢ ( t , 0 ) = ∫ 0 a ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d a , t ∈ [ 0 , T ] , U ⁢ ( 0 , a ) = U 0 ⁢ ( a ) , a ∈ [ 0 , a ¯ ] ,

where d t ⁢ U := ∂ ⁡ U ⁢ ( t , a ) ∂ ⁡ t ⁢ d ⁢ t . By integration along the characteristic lines and using the variation of constants formula, the explicit solution of equation (A.1) is as follows (see [36]):

(A.2) U ⁢ ( t , a ) = { U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s ) ⁢ d ⁢ s - ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d ⁢ B s , 0 ≤ t ≤ a , U ⁢ ( t - a , 0 ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s - ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s , 0 ≤ a < t .

Since U ⁢ ( t , 0 ) = ∫ 0 a ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d a , from (A.2), we get

(A.3) U ⁢ ( t , 0 ) = ∫ 0 a ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ a = ∫ 0 t β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ a + ∫ t t ¯ β ⁢ ( t ) ⁢ U ⁢ ( t , a ) ⁢ d ⁢ a = J ⁢ ( t ) + ∫ 0 t β ⁢ ( t ) ⁢ U ⁢ ( t - a , 0 ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s ⁢ d a ,

where

J ⁢ ( t ) = ∫ t a ¯ β ⁢ ( a ) ⁢ U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s ) ⁢ d ⁢ s ⁢ d ⁢ a - ∫ 0 t β ⁢ ( t ) ⁢ ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d ⁢ B s ⁢ d ⁢ a - ∫ t a ¯ β ⁢ ( t ) ⁢ ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s ⁢ d a , 0 ≤ t ≤ a ¯ ,

and

(A.4) K ⁢ ( t , a ) = β ⁢ ( t ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s .

Since

E ⁢ ( ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s ) = E ⁢ ( ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s ) = 0 ,

from (A.2), we know that the first moment of the solution to equation (A.1) is

(A.5) E ⁢ [ U ⁢ ( t , a ) ] = { U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s + t - a ) ⁢ d ⁢ s , 0 ≤ t ≤ a , E ⁢ [ U ⁢ ( t - a , 0 ) ] ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s , 0 ≤ a < t .

By (A.3), we have

E ⁢ [ U ⁢ ( t , 0 ) ] = E ⁢ [ J ⁢ ( t ) ] + ∫ 0 t K ⁢ ( t , a ) ⁢ E ⁢ U ⁢ ( t - a , 0 ) ⁢ d a = E ⁢ [ J ⁢ ( t ) ] + ∫ 0 t K ⁢ ( t , t - s ) ⁢ E ⁢ U ⁢ ( s , 0 ) ⁢ d s ,

where

(A.6) E ⁢ [ J ⁢ ( t ) ] = ∫ t a ¯ β ⁢ ( t ) ⁢ U 0 ⁢ ( a - t ) ⁢ e ∫ 0 t - μ ⁢ ( s ) ⁢ d ⁢ s ⁢ d a .

From equations (A.4), (A.6) and assumptions 1, 2, it follows that

0 ≤ K ⁢ ( t , a ) ≤ K ¯ , ( t , a ) ∈ Q ; 0 ≤ E ⁢ [ J ⁢ ( t ) ] ≤ J ¯ , t ∈ ( 0 , T ) ,

Using a method similar to [2, Theorem 2.1.1], we may infer that E ⁢ [ U ⁢ ( t , a ) ] ≤ C 1 , where C 1 is a positive constant.

We now discuss the second moment of U ⁢ ( t , a ) . According (A.2) and (A.5), we get

(A.7) U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] = { - ∫ 0 t σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s , 0 ≤ t ≤ a , ( U ⁢ ( t - a , 0 ) - E ⁢ [ U ⁢ ( t - a , 0 ) ] ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s - ∫ t - a t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d B s , 0 ≤ a < t .

When 0 ≤ t ≤ a , the second moment of U ⁢ ( t , a ) is

M ⁢ ( t , a ) ≜ E ⁢ ( U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] ) 2 = E ⁢ ∫ 0 t ( σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ) 2 ⁢ d s ≤ σ ¯ 2 ⁢ ∫ 0 t E ⁢ | U ⁢ ( s , a ) | 2 ⁢ d s .

Hence

E ⁢ | U ⁢ ( t , a ) | 2 ≤ ( E ⁢ | U ⁢ ( t , a ) | ) 2 + σ ¯ 2 ⁢ ∫ 0 t E ⁢ | U ⁢ ( s , a ) | 2 ⁢ d s .

By Gronwall’s lemma, we have

E ⁢ | U ⁢ ( t , a ) | 2 ≤ ( E ⁢ | U ⁢ ( t , a ) | ) 2 ⁢ e σ ¯ 2 ⁢ T ≤ C 2 ⁢ e σ ¯ 2 ⁢ T .

When 0 ≤ a < t , the second moment of U ⁢ ( t , a ) is

M ⁢ ( t , a ) ≜ E ⁢ ( U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] ) 2 = M ⁢ ( t - a , 0 ) ⁢ e ∫ 0 a - 2 ⁢ μ ⁢ ( s + t - a ) ⁢ d ⁢ s + ∫ t - a t σ 2 ⁢ ( s + a - t ) ⁢ e ∫ s t - 2 ⁢ μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ | U ⁢ ( s , s + a - t ) | 2 ⁢ d ⁢ s - 2 E [ ( U ( t - a , 0 ) - E [ U ( t - a , 0 ) ] ) e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s ⋅ ∫ t - a t σ ( s + a - t ) e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ U ( s , s + a - t ) d B s ] ,

where M ⁢ ( t , 0 ) ≜ E ⁢ ( U ⁢ ( t , 0 ) - E ⁢ [ U ⁢ ( t , 0 ) ] ) 2 . It is necessary to study the boundedness of M ⁢ ( t , 0 ) . From (A.2), we have

(A.8) U ⁢ ( t , 0 ) - E ⁢ [ U ⁢ ( t , 0 ) ] = G ⁢ ( t ) + ∫ 0 t K ⁢ ( t , a ) ⁢ [ U ⁢ ( t - a , 0 ) - E ⁢ U ⁢ ( t - a , 0 ) ] ⁢ d a ,

where K ⁢ ( t , a ) = β ⁢ ( t ) ⁢ e ∫ 0 a - μ ⁢ ( s + t - a ) ⁢ d ⁢ s , and

G ⁢ ( t ) = - ∫ 0 t β ⁢ ( t ) ⁢ ∫ t - s t σ ⁢ ( s + a - t ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d a ⁢ d B s - ∫ 0 t β ⁢ ( t ) ⁢ ∫ t a ¯ σ ⁢ ( s ) ⁢ e ∫ s t - μ ⁢ ( τ ) ⁢ d ⁢ τ ⁢ U ⁢ ( s , s + a - t ) ⁢ d a ⁢ d B s .

Then

M ⁢ ( t , 0 ) ≤ 2 ⁢ E ⁢ { [ G ⁢ ( t ) ] 2 + ( ∫ 0 t K ⁢ ( t , t - s ) ⁢ [ U ⁢ ( s , 0 ) - E ⁢ U ⁢ ( s , 0 ) ] ⁢ d a ) 2 } .

Under assumptions 1, 2 and by Gronwall’s lemma, we may infer that M ⁢ ( t , 0 ) ≤ C 2 for all ( t , a ) ∈ Q , where C 2 is a positive constant. Combining (A.7) and (A.8), we may have E ⁢ ( U ⁢ ( t , a ) - E ⁢ [ U ⁢ ( t , a ) ] ) 2 ≤ C 3 for all ( t , a ) ∈ Q , where C 3 is a positive constant. Due the boundedness of E ⁢ [ U ⁢ ( t , a ) ] , we may infer that

E ⁢ | S ⁢ ( t , a ) + I ⁢ ( t , a ) + A ⁢ ( t , a ) | 2 = E ⁢ | U ⁢ ( t , a ) | 2 ≤ C 4 for all ⁢ ( t , a ) ∈ Q ,

where C 4 is a positive constant. Due to the nonnegativity of solutions for system (2.1), we have

E ⁢ ( | S ⁢ ( t , a ) | 2 + | I ⁢ ( t , a ) | 2 + | A ⁢ ( t , a ) | 2 ) ≤ C .

The proof is complete. ∎

References

[1] R. M. Anderson, G. F. Medley, R. M. May and A. M. Johnson, A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J. Math. Appl. Med. Biol. 3 (1986), no. 4, 229–263. 10.1093/imammb/3.4.229Search in Google Scholar

[2] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer, Dordrecht, 2000. 10.1007/978-94-015-9436-3Search in Google Scholar

[3] F. Brauer and C. Castillo-Chavez, Lecture 8: Models for the transmission dynamics of HIV, Mathematical Models for Communicable Diseases, ACM, New York (2012), 163–189. 10.1137/1.9781611972429.ch8Search in Google Scholar

[4] S. Busenberg and C. Castillo-Chavez, A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS, Eur. J. Oral. Sci. 117 (1991), no. 2, 200–203. 10.1093/imammb/8.1.1Search in Google Scholar

[5] L. Cai, X. Li and J. Yu, Analysis of a delayed HIV/AIDS epidemic model with saturation incidence, J. Appl. Math. Comput. 27 (2008), no. 1–2, 365–377. 10.1007/s12190-008-0070-3Search in Google Scholar

[6] V. Capasso and G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model, Math. Biosci. 42 (1978), no. 1–2, 43–61. 10.1016/0025-5564(78)90006-8Search in Google Scholar

[7] J.-F. Chassagneux, A. Jacquier and I. Mihaylov, An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients, SIAM J. Financial Math. 7 (2016), no. 1, 993–1021. 10.1137/15M1017788Search in Google Scholar

[8] N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl. 325 (2007), no. 1, 36–53. 10.1016/j.jmaa.2006.01.055Search in Google Scholar

[9] Y. Ding, M. Xu and L. Hu, Asymptotic behavior and stability of a stochastic model for AIDS transmission, Appl. Math. Comput. 204 (2008), no. 1, 99–108. 10.1016/j.amc.2008.06.028Search in Google Scholar

[10] Y. Emvudu, D. Bongor and R. Koïna, Mathematical analysis of HIV/AIDS stochastic dynamic models, Appl. Math. Model. 40 (2016), no. 21–22, 9131–9151. 10.1016/j.apm.2016.05.007Search in Google Scholar

[11] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (2011), no. 3, 876–902. 10.1137/10081856XSearch in Google Scholar

[12] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), no. 4, 599–653. 10.1137/S0036144500371907Search in Google Scholar

[13] W. Z. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math. 52 (1992), no. 3, 835–854. 10.1137/0152047Search in Google Scholar

[14] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), no. 4, 1611–1641. 10.1214/11-AAP803Search in Google Scholar

[15] J. Hyman, J. Li and E. A. Stanley, Threshold conditions for the spread of the HIV infection in age-structured populations of homosexual men, J. Theoret. Biol. 166 (1994), no. 1, 9–31. 10.1006/jtbi.1994.1002Search in Google Scholar PubMed

[16] H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci. 201 (2006), no. 1–2, 15–47. 10.1016/j.mbs.2005.12.017Search in Google Scholar PubMed

[17] C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica J. IFAC 48 (2012), no. 1, 121–131. 10.1016/j.automatica.2011.09.044Search in Google Scholar

[18] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Appl. Math. (New York) 23, Springer, Berlin, 1999. Search in Google Scholar

[19] Y. Liang, D. Greenhalgh and X. Mao, A stochastic differential equation model for the spread of HIV amongst people who inject drugs, Comput. Math. Methods Med. 11 (2016), 1–14. 10.1155/2016/6757928Search in Google Scholar PubMed PubMed Central

[20] W. Liu and X. Mao, Strong convergence of the stopped Euler–Maruyama method for nonlinear stochastic differential equations, Appl. Math. Comput. 223 (2013), 389–400. 10.1016/j.amc.2013.08.023Search in Google Scholar

[21] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987), no. 4, 359–380. 10.1007/BF00277162Search in Google Scholar PubMed

[22] W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23 (1986), no. 2, 187–204. 10.1007/BF00276956Search in Google Scholar

[23] X. Mao, Stochastic Differential Equations and Their Applications, Horwood, Chichester, 1997. Search in Google Scholar

[24] X. Mao, The truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370–384. 10.1016/j.cam.2015.06.002Search in Google Scholar

[25] X. Mao, Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016), 362–375. 10.1016/j.cam.2015.09.035Search in Google Scholar

[26] X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl. 97 (2002), no. 1, 95–110. 10.1016/S0304-4149(01)00126-0Search in Google Scholar

[27] R. May and R. Anderson, Transmission dynamics of HIV infection, Nature 326 (1987), 137–142. 10.1038/326137a0Search in Google Scholar

[28] Z. Qi-Min, L. Wen-An and N. Zan-Kan, Existence, uniqueness and exponential stability for stochastic age-dependent population, Appl. Math. Comput. 154 (2004), no. 1, 183–201. 10.1016/S0096-3003(03)00702-1Search in Google Scholar

[29] A. Rathinasamy, M. Chinnadurai and S. Athithan, Analysis of exact solution of stochastic sex-structured HIV/AIDS epidemic model with effect of screening of infectives, Math. Comput. Simulation 179 (2021), 213–237. 10.1016/j.matcom.2020.08.017Search in Google Scholar PubMed PubMed Central

[30] J. Ren, Q. Zhang, X. Li, F. Cao and M. Ye, A stochastic age-structured HIV/AIDS model based on parameters estimation and its numerical calculation, Math. Comput. Simulation 190 (2021), 159–180. 10.1016/j.matcom.2021.04.024Search in Google Scholar

[31] S. Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18 (2013), 1–10. 10.1214/ECP.v18-2824Search in Google Scholar

[32] S. Sabanis, Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients, Ann. Appl. Probab. 26 (2016), no. 4, 2083–2105. 10.1214/15-AAP1140Search in Google Scholar

[33] H. Tuckwell and E. Corfec, A stochastic model for early HIV-1 population dynamics, J. Theoret. Biol. 195 (1998), no. 4, 451–463. 10.1006/jtbi.1998.0806Search in Google Scholar PubMed

[34] J. Wang, J. Lang and X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl. 34 (2017), 75–96. 10.1016/j.nonrwa.2016.08.001Search in Google Scholar

[35] J. Wang, R. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl. 432 (2015), no. 1, 289–313. 10.1016/j.jmaa.2015.06.040Search in Google Scholar

[36] Z. Wang and X. Li, Stability and moment boundedness of the stochastic linear age-structured model, J. Dynam. Differential Equations 31 (2019), no. 4, 2109–2125. 10.1007/s10884-018-9671-1Search in Google Scholar

[37] J. Yang, X. Wang and X. Li, Global stability of an HIV/AIDS model with stochastic perturbation, Asian-Eur. J. Math. 4 (2011), no. 2, 349–358. 10.1142/S1793557111000289Search in Google Scholar

[38] Q. Zhang, Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion, J. Comput. Appl. Math. 220 (2008), no. 1–2, 22–33. 10.1016/j.cam.2007.08.026Search in Google Scholar

[39] The Chinese Center for Disease Control and Prevention http://www.chinacdc.cn/. Search in Google Scholar

Received: 2021-08-18

Revised: 2022-01-20

Accepted: 2022-01-26

Published Online: 2022-03-09

Published in Print: 2022-07-01