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September  2020, 25(9): 3491-3521. doi: 10.3934/dcdsb.2020070

Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China

Hongyong Zhao 1,, , Peng Wu 1, and  Shigui Ruan 2,

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146-4250, USA

* Corresponding author: Hongyong Zhao, Email: Hyzho1967@126.com

Received  July 2019 Revised  October 2019 Published  September 2020 Early access  April 2020

Fund Project: The first author is supported by NSFC grants [11571170, 11971013]

Based on the fact that HIV/AIDS manifests different transmission characteristics and pathogenesis in different age groups, and the proportions of youth and elderly HIV infected cases in total are increasing in China, we classify the whole population into three age groups, youth (15-24), adult (25-49), and elderly ($ \geqslant $50), and establish a three-age-class HIV/AIDS epidemic model to investigate the transmission dynamics of HIV/AIDS in China. We derive the explicit expression for the basic reproduction number via the next generation matrix approach. Qualitative analysis of the model including the local, global behavior and permanence is carried out. In particular, numerical simulations are presented to reinforce these analytical results and demonstrate HIV epidemiological discrepancy among different age groups. We also formulate an optimal control problem and solve it using Pontryagin's Maximum Principle and an efficient iterative numerical methods. Our numerical results of optimal controls for the elderly group indicate that increasing the condom use and decreasing the rate of the formerly HIV infected persons converted to AIDS patients are important measures to control HIV/AIDS epidemic among elderly population.

Keywords: Three-age-classes, HIV/AIDS, permanence, optimal control, sensitivity analysis.
Mathematics Subject Classification: Primary: 92D30, 34D20; Secondary: 49J15.
Citation: Hongyong Zhao, Peng Wu, Shigui Ruan. Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3491-3521. doi: 10.3934/dcdsb.2020070
References:
[1]

A. Babiker, S. Darby, et al., Time from HIV-1 seroconversion to AIDS and death before widespread use of highly-active antiretroviral therapy: A collaborative re-analysis, Lancet, 355 (2000), 1131–1137. doi: 10.1016/S0140-6736(00)02061-4.

[2]

N. Bacaër, X. Abdurahman and J. Ye, Modelling the HIV/AIDS epidemic among injecting drug users and sex workers in Kunming, China, Bull. Math. Biol., 68 (2006), 525-550.  doi: 10.1007/s11538-005-9051-y.

[3]

CCDC, HIV Among people aged 50 and over last update: February 12, 2018., Available from: https://www.cdc.gov/hiv/group/age/olderamericans/index.html.

[4]

C. C. Chavez, Z. Feng and W. Huang, On the computation of ${R}_0$ and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Disease: An Introduction, IMA Vol. Math. Appl., 125, Springer, New York, 2002, 229–250. doi: 10.1007/978-1-4757-3667-0_13.

[5]

S. Chen, X. Yang, H. Li and Y. Qiu, AIDSand syphilis infection in the elderly and status of knowledge and behavior, Occup. Health, 31 (2015), 272-275. 

[6]

China Association for the Prevention and Control of STDs and AIDS, Increasing by 14% of New HIV/AIDS Infection Cases Were Reported in 2018, China, 2018. Available from: http://2018aids.medmeeting.org/cn.

[7]

CHPC, China Statistical Yearbook 1998-2012, 2003. Available from: http://www.stats.gov.cn.

[8]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[9]

O. Diekman, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[10]

P. Driessche and R. Watmongh, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[11]

ECDC, HIV/AIDS Surveillance in Europe - 2016 Data, WHO Gevena, 2017.

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C. Emlet and K. Farkas, A descriptive analysis of older adults with HIV/AIDS in California, Health Social Work, 26 (2001), 226–234. doi: 10.1093/hsw/26.4.226.

[13]

W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, 1, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/978-1-4612-6380-7.

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L. Gao, J. Fu and S. Li, A systematic analysis of HIV epidemic characteristics and related risk factors in people over 50 years old, J. Dermatol and Venereol, 38 (2016), 36-42. 

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Q. He, Y. Wang and P. Lin, et al., Potential bridges for HIV infection to men who have sex with men in Guangzhou, China, AIDS and Behavior, 10 (2006), 17–23. doi: 10.1007/s10461-006-9125-3.

[16]

F. Hei, L. Wang, Q. Qin, Z. Ding and L. Wang, Epidemiological analysis on the characteristics and related factors of HIV/AIDS in 50-year and older Chinese population, Chin. J. Epidemiol., 32 (2011), 526-527. 

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L. Jie, X. Chen and B. Qin, Investigation of HIV-related risk factors among elderly HIV-positive patients, Pract. Prev. Med., 17 (2010), 227–229.

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R. Kumar, et al., Trends in HIV-1 in young adults in South India from 2000 to 2004: A prevalence study, Lancet HIV, 367 (2006), 1164–1172. doi: 10.1016/S0140-6736(06)68435-3.

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A. Li and Z. Li, The epidemiological characteristics, diagnosis and therapy of HIV/AIDS among the elderly, Med. Recap., 22 (2016), 4479-4482. 

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Y. Li, J. Xu, H. Qian and B. You, High prevalence of HIV infection and unprotected and intercourse among older men who have sex with men in China: A systematic review and meta-analysis, BMC Infect. Dis., 14 (2014), 531–540. doi: 10.1186/1471-2334-14-531.

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[23]

Y. Lu, J. Wu and Y. Hu, HIVand syphilis prevalence among high school and college students in China: A mate-analysis, Chin. J. AIDS STD, 23 (2017), 524–528.

[24]

H. Ma, Adolescent and AIDS, J. Peking University (Health Sciences), 48 (2016), 385-388. 

[25]

A. Mokdad and M. Forouzanfar, et al., Global burden of diseases, injuries, and risk factors for young people's health during 1990-2013: A systematic analysis for the Global Burden of Disease Study 2013, Lancet HIV, 387 (2016), 2383–2401. doi: 10.1016/S0140-6736(16)00648-6.

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NCAIDS and CCDC, Update on the AIDS/STD epidemic in China in December, 2016, Chin. J. AIDS STD., 23 (2017).

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NCAIDS and CCDC, Update on the AIDS/STD epidemic in China in December, 2017, Chin. J. AIDS STD, 24 (2018).

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NCAIDS and CCDC, Update on the AIDS/STD epidemics in China and main response in control and prevention in December, 2012, Chin. J. AIDS STD, 19 (2013). doi: 10.1002/mma.2599.

[29]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China and main response in control and prevention in December, 2013, Chin. J. AIDS STD., 20 (2014).

[30]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China and main response in control and prevention in December, 2014, Chin. J. AIDS STD., 21 (2015).

[31]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China and main response in control and prevention in December, 2015, Chin. J. AIDS STD., 22 (2016).

[32]

L. Pang, S. Ruan, S. Liu, Z. Zhao and X. Zhang, Transmission dynamics and optimal control of measles epidemics, Appl. Math. Comput., 256 (2015), 131-147.  doi: 10.1016/j.amc.2014.12.096.

[33]

J. Z. Perez and R. D. Moore, Greater effect of highly active antiretroviral therapy on survival in people aged more than 50 years compared with younger people in an urban observational cohort, Clin. Infec. Dis., 36 (2003), 212-218.  doi: 10.1086/345669.

[34]

H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Vaccination models and optimal control strategies to dengue, Math. Biosci., 247 (2014), 1–12. doi: 10.1016/j.mbs.2013.10.006.

[35]

S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics, Kluwer Academic Publishers, Boston, MA, 2000.

[36]

Shangli County CCDC, HIV/AIDS epidemic and control strategy, Shangli County CCDC, 2016.

[37]

S. Siringi, HIV/AIDS on the rise in young people in Kenya, Lancet HIV, 10 (2010). doi: 10.1016/S1473-3099(10)70037-2.

[38]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[39]

UNAIDS, Report on the Global Epidemic, WHO Geneva, 2006.

[40]

H. Wang, R. Xu, Z. Wang and H. Chen, Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal. Model. Control, 20 (2015), 21-37.  doi: 10.15388/NA.2015.1.2.

[41]

L. Wang, The Elder Infected HIV/AIDS, A High-Risk Population That May Be Neglected, 2017. Available from: http://www.cas.cn/kx/kpwz/201704/t20170421_4597865.shtml.

[42]

L. Wang, Z. Ding, R. Xuan and D. Li, HIVprevalence among population at risk, using sentinel surveillance date from 1995 to 2009 in China, Chin. J. Epidemiol., 32 (2011), 20–24.

[43]

L. Wang, Q. Qin, W. Zheng, L. Li and X. Hei, Current case reporting of HIV/AIDS epidemic in 2010, China, Chin. J. AIDS STD., 17 (2014), 275-278. 

[44]

Y. Wang, Q. Qin and L. Ge, Characteristics of HIV infection among over 50-year olds population in China, Chin. J. Epidemiol., 24 (2018).

[45]

Y. Wang, Q. Qian, W. Zheng, C. Cai and Y. Cui, Analysis of epidemiological characteristics of AIDS cases in children under 15 years of age in China, Dis Surveil, 32 (2017), 227–231.

[46]

World Health Organization, 10 Facts About HIV/AIDS, HIV Fact Sheet No. 204, 2018. Available from: https://www.who.int/news-room/facts-in-pictures/detail/hiv-aids.

[47]

Y. Xiao, S. Tang, Y. Zhou, R. Smith, J. Wu and N. Wang, Predicting the HIV/AIDS epidemic and measuring the effect of mobility in mainland China, J. Theoret. Biol., 317 (2013), 271-285.  doi: 10.1016/j.jtbi.2012.09.037.

[48]

J. Xing, Y. Li, W. Tang, W. Guo and et al., HIV/AIDS epidemic among older adults in China during 2005-2012: Results from trend and spatial analysis, Clin. Infect. Dis., 59 (2014), 53-60.  doi: 10.1093/cid/ciu214.

[49]

J. Xu, The Number of HIV Infected Cases Among Adolescents Increased Rapidly in China, Which 80 Percents Infected Cased by Homosexual Transmission, 2016. Available from: http://edu.qq.com/a/20160808/003918.htm.

[50]

Q. Xu, F. Lv, H. Zhu and Y. Yuan, Analysis of HIV/AIDS prevalence of the older persons in China, Population & Economic, 6 (2005), 1-5. 

[51]

Z. Yang, Z. Huang, Z. Dong, S. Zhang, J. Han and M. Jin, Prevalence of high-risky behaviors in transmission of HIV among high school and college student MSM in China: A meta-analysis, BMC Public Health, 15 (2015), 1272–1274. doi: 10.1186/s12889-015-2614-4.

[52]

X. Yuan and W. Lu, The prevalence characteristics and risk factors of AIDS among people fifty years or older, at home and abroad, Chin. J. Epidemiol., 32 (2011), 166–1169.

[53]

T. Zhang, M. Jia, H. Luo, Y. Zhou and N. Wang, Study on a HIV/AIDS model with application to Yunnan province, China, Appl. Math. Model., 35 (2011), 4379-4392.  doi: 10.1016/j.apm.2011.03.004.

[54]

T. Zhang and Y. Zhou, Mathematical model of transmission dynamics of human immune-deficiency virus: A case study for Yunnan, China, Appl. Math. Model., 40 (2016), 4859-4875.  doi: 10.1016/j.apm.2015.12.022.

[55]

X. Zhang, et al., The HIV/AIDS epidemic among young people in China between 2005 and 2012: Results of a spatial temporal analysis, HIV Med, 18 (2016), 141–151. doi: 10.1111/hiv.12408.

[56]

X. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:
[1]

A. Babiker, S. Darby, et al., Time from HIV-1 seroconversion to AIDS and death before widespread use of highly-active antiretroviral therapy: A collaborative re-analysis, Lancet, 355 (2000), 1131–1137. doi: 10.1016/S0140-6736(00)02061-4.

[2]

N. Bacaër, X. Abdurahman and J. Ye, Modelling the HIV/AIDS epidemic among injecting drug users and sex workers in Kunming, China, Bull. Math. Biol., 68 (2006), 525-550.  doi: 10.1007/s11538-005-9051-y.

[3]

CCDC, HIV Among people aged 50 and over last update: February 12, 2018., Available from: https://www.cdc.gov/hiv/group/age/olderamericans/index.html.

[4]

C. C. Chavez, Z. Feng and W. Huang, On the computation of ${R}_0$ and its role on global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Disease: An Introduction, IMA Vol. Math. Appl., 125, Springer, New York, 2002, 229–250. doi: 10.1007/978-1-4757-3667-0_13.

[5]

S. Chen, X. Yang, H. Li and Y. Qiu, AIDSand syphilis infection in the elderly and status of knowledge and behavior, Occup. Health, 31 (2015), 272-275. 

[6]

China Association for the Prevention and Control of STDs and AIDS, Increasing by 14% of New HIV/AIDS Infection Cases Were Reported in 2018, China, 2018. Available from: http://2018aids.medmeeting.org/cn.

[7]

CHPC, China Statistical Yearbook 1998-2012, 2003. Available from: http://www.stats.gov.cn.

[8]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[9]

O. Diekman, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[10]

P. Driessche and R. Watmongh, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[11]

ECDC, HIV/AIDS Surveillance in Europe - 2016 Data, WHO Gevena, 2017.

[12]

C. Emlet and K. Farkas, A descriptive analysis of older adults with HIV/AIDS in California, Health Social Work, 26 (2001), 226–234. doi: 10.1093/hsw/26.4.226.

[13]

W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, 1, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/978-1-4612-6380-7.

[14]

L. Gao, J. Fu and S. Li, A systematic analysis of HIV epidemic characteristics and related risk factors in people over 50 years old, J. Dermatol and Venereol, 38 (2016), 36-42. 

[15]

Q. He, Y. Wang and P. Lin, et al., Potential bridges for HIV infection to men who have sex with men in Guangzhou, China, AIDS and Behavior, 10 (2006), 17–23. doi: 10.1007/s10461-006-9125-3.

[16]

F. Hei, L. Wang, Q. Qin, Z. Ding and L. Wang, Epidemiological analysis on the characteristics and related factors of HIV/AIDS in 50-year and older Chinese population, Chin. J. Epidemiol., 32 (2011), 526-527. 

[17]

L. Jie, X. Chen and B. Qin, Investigation of HIV-related risk factors among elderly HIV-positive patients, Pract. Prev. Med., 17 (2010), 227–229.

[18]

R. Kumar, et al., Trends in HIV-1 in young adults in South India from 2000 to 2004: A prevalence study, Lancet HIV, 367 (2006), 1164–1172. doi: 10.1016/S0140-6736(06)68435-3.

[19]

P. D. Leenheer and H. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905.

[20]

A. Li and Z. Li, The epidemiological characteristics, diagnosis and therapy of HIV/AIDS among the elderly, Med. Recap., 22 (2016), 4479-4482. 

[21]

Y. Li, J. Xu, H. Qian and B. You, High prevalence of HIV infection and unprotected and intercourse among older men who have sex with men in China: A systematic review and meta-analysis, BMC Infect. Dis., 14 (2014), 531–540. doi: 10.1186/1471-2334-14-531.

[22]

S. Lindau, L. Schumm and E. Laumann, et al., A study of sexuality and health among older adults in the United States, New England J. Med., 357 (2007), 762–774. doi: 10.1056/NEJMoa067423.

[23]

Y. Lu, J. Wu and Y. Hu, HIVand syphilis prevalence among high school and college students in China: A mate-analysis, Chin. J. AIDS STD, 23 (2017), 524–528.

[24]

H. Ma, Adolescent and AIDS, J. Peking University (Health Sciences), 48 (2016), 385-388. 

[25]

A. Mokdad and M. Forouzanfar, et al., Global burden of diseases, injuries, and risk factors for young people's health during 1990-2013: A systematic analysis for the Global Burden of Disease Study 2013, Lancet HIV, 387 (2016), 2383–2401. doi: 10.1016/S0140-6736(16)00648-6.

[26]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China in December, 2016, Chin. J. AIDS STD., 23 (2017).

[27]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China in December, 2017, Chin. J. AIDS STD, 24 (2018).

[28]

NCAIDS and CCDC, Update on the AIDS/STD epidemics in China and main response in control and prevention in December, 2012, Chin. J. AIDS STD, 19 (2013). doi: 10.1002/mma.2599.

[29]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China and main response in control and prevention in December, 2013, Chin. J. AIDS STD., 20 (2014).

[30]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China and main response in control and prevention in December, 2014, Chin. J. AIDS STD., 21 (2015).

[31]

NCAIDS and CCDC, Update on the AIDS/STD epidemic in China and main response in control and prevention in December, 2015, Chin. J. AIDS STD., 22 (2016).

[32]

L. Pang, S. Ruan, S. Liu, Z. Zhao and X. Zhang, Transmission dynamics and optimal control of measles epidemics, Appl. Math. Comput., 256 (2015), 131-147.  doi: 10.1016/j.amc.2014.12.096.

[33]

J. Z. Perez and R. D. Moore, Greater effect of highly active antiretroviral therapy on survival in people aged more than 50 years compared with younger people in an urban observational cohort, Clin. Infec. Dis., 36 (2003), 212-218.  doi: 10.1086/345669.

[34]

H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Vaccination models and optimal control strategies to dengue, Math. Biosci., 247 (2014), 1–12. doi: 10.1016/j.mbs.2013.10.006.

[35]

S. P. Sethi and G. L. Thompson, Optimal Control Theory. Applications to Management Science and Economics, Kluwer Academic Publishers, Boston, MA, 2000.

[36]

Shangli County CCDC, HIV/AIDS epidemic and control strategy, Shangli County CCDC, 2016.

[37]

S. Siringi, HIV/AIDS on the rise in young people in Kenya, Lancet HIV, 10 (2010). doi: 10.1016/S1473-3099(10)70037-2.

[38]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[39]

UNAIDS, Report on the Global Epidemic, WHO Geneva, 2006.

[40]

H. Wang, R. Xu, Z. Wang and H. Chen, Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal. Model. Control, 20 (2015), 21-37.  doi: 10.15388/NA.2015.1.2.

[41]

L. Wang, The Elder Infected HIV/AIDS, A High-Risk Population That May Be Neglected, 2017. Available from: http://www.cas.cn/kx/kpwz/201704/t20170421_4597865.shtml.

[42]

L. Wang, Z. Ding, R. Xuan and D. Li, HIVprevalence among population at risk, using sentinel surveillance date from 1995 to 2009 in China, Chin. J. Epidemiol., 32 (2011), 20–24.

[43]

L. Wang, Q. Qin, W. Zheng, L. Li and X. Hei, Current case reporting of HIV/AIDS epidemic in 2010, China, Chin. J. AIDS STD., 17 (2014), 275-278. 

[44]

Y. Wang, Q. Qin and L. Ge, Characteristics of HIV infection among over 50-year olds population in China, Chin. J. Epidemiol., 24 (2018).

[45]

Y. Wang, Q. Qian, W. Zheng, C. Cai and Y. Cui, Analysis of epidemiological characteristics of AIDS cases in children under 15 years of age in China, Dis Surveil, 32 (2017), 227–231.

[46]

World Health Organization, 10 Facts About HIV/AIDS, HIV Fact Sheet No. 204, 2018. Available from: https://www.who.int/news-room/facts-in-pictures/detail/hiv-aids.

[47]

Y. Xiao, S. Tang, Y. Zhou, R. Smith, J. Wu and N. Wang, Predicting the HIV/AIDS epidemic and measuring the effect of mobility in mainland China, J. Theoret. Biol., 317 (2013), 271-285.  doi: 10.1016/j.jtbi.2012.09.037.

[48]

J. Xing, Y. Li, W. Tang, W. Guo and et al., HIV/AIDS epidemic among older adults in China during 2005-2012: Results from trend and spatial analysis, Clin. Infect. Dis., 59 (2014), 53-60.  doi: 10.1093/cid/ciu214.

[49]

J. Xu, The Number of HIV Infected Cases Among Adolescents Increased Rapidly in China, Which 80 Percents Infected Cased by Homosexual Transmission, 2016. Available from: http://edu.qq.com/a/20160808/003918.htm.

[50]

Q. Xu, F. Lv, H. Zhu and Y. Yuan, Analysis of HIV/AIDS prevalence of the older persons in China, Population & Economic, 6 (2005), 1-5. 

[51]

Z. Yang, Z. Huang, Z. Dong, S. Zhang, J. Han and M. Jin, Prevalence of high-risky behaviors in transmission of HIV among high school and college student MSM in China: A meta-analysis, BMC Public Health, 15 (2015), 1272–1274. doi: 10.1186/s12889-015-2614-4.

[52]

X. Yuan and W. Lu, The prevalence characteristics and risk factors of AIDS among people fifty years or older, at home and abroad, Chin. J. Epidemiol., 32 (2011), 166–1169.

[53]

T. Zhang, M. Jia, H. Luo, Y. Zhou and N. Wang, Study on a HIV/AIDS model with application to Yunnan province, China, Appl. Math. Model., 35 (2011), 4379-4392.  doi: 10.1016/j.apm.2011.03.004.

[54]

T. Zhang and Y. Zhou, Mathematical model of transmission dynamics of human immune-deficiency virus: A case study for Yunnan, China, Appl. Math. Model., 40 (2016), 4859-4875.  doi: 10.1016/j.apm.2015.12.022.

[55]

X. Zhang, et al., The HIV/AIDS epidemic among young people in China between 2005 and 2012: Results of a spatial temporal analysis, HIV Med, 18 (2016), 141–151. doi: 10.1111/hiv.12408.

[56]

X. Zhao, Dynamical Systems in Population Biology, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

Figure 1.  (a): The new HIV/AIDS infection cases among youth (15-24) from 2005 to 2017 (see Table 2), (b): The new HIV/AIDS infection cases among elderly ($ \geq50 $) from 2005 to 2017 (see Table 2)
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Figure 2.  The diagram of transmission among three epidemiological classes
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Figure 3.  Dynamics of the solutions of system $ (1) $ when $ R_0 = 1.73415 $, where the variables and parameters are given in Table 1
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Figure 4.  The fitting curves of total new HIV/AIDS infection cases
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Figure 5.  The fitting curves of new HIV/AIDS infection cases among youth, elderly, adult, respectively
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Figure 6.  $ PRCC-R^{(2)}_0 $
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Figure 7.  $ R^{(2)}_0 $ trends with respect to different parameters, the other parameters are listed in Table 1
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Figure 8.  $ R^{(2)}_0 $ trends with respect to different parameters, the other parameters are listed in Table 1
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Figure 9.  (a): Optimal solutions for the system $ (1) $, $ u^{*}_2 $ in green full line and $ u^{*}_1 $ in purple full line. (b): $ A_e(t) $ trends in regard to the optimal controls, where $ A_e $ without control strategy in red solid line, $ A_e $ under the optimal control in blue dashed line. Other parameters are the same in Table 1
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Figure 10.  (a): Optimal control with respect to $ I_a(0) $, (b): Optimal control with respect to $ I_y(0) $, (c): Optimal control with respect to $ I_e(0) $. Other parameters are the same in Table 1
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Figure 11.  (a): The influence of $ R^{(1)}_0 $ on $ u^*_1 $. (b): The influence of $ R^{(1)}_0 $ on $ u^*_2 $, other parameters are the same in Table 1
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Figure 12.  (a): Optimal control strategy $ u_2(t)\neq 0, \ u_1(t) = 0 $. (b): The impact on $ A_e(t) $ with the optimal control strategy $ u_2(t)\neq 0, \ u_1(t) = 0 $, other parameters are the same in Table 1
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Figure 13.  (a): Optimal control strategy with $ u_1(t)\neq 0, \ u_2(t) = 0 $. (b): The impact on $ A_e(t) $ with the optimal control strategy $ u_1(t)\neq 0, \ u_2(t) = 0 $, other parameters are the same in Table 1
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Table 1.  The parameters and numerical values
Parameters Description Range $ (\%) $ Value $ (year^{-1}) $ Source
$ \Lambda $ Recruitment of the youth class - 3075405 Assume
$ \mu_y $ Natural death rate of youth [0.066-0.087] 0.765‰ [7]
$ \mu_a $ Natural death rate of adult [0.1-0.327] 1.852‰ [7]
$ \mu_e $ Natural death rate of elderly [0.19-0.45] 2.07‰ [7]
$ r_{y} $ Average remove rate from $ I_y $ to $ A_y $ - 1/12.5 [46]
$ r_{a} $ Average remove rate from $ I_a $ to $ A_a $ - 1/10 [46]
$ r_{e} $ Average remove rate from $ I_e $ to $ A_e $ - 1/7.9 [46]
$ d_y $ AIDS-related death rate among youth [1.0-4.3] 2.3% [12]
$ d_a $ AIDS-related death rate among adult [5.03-12.1] 9.7% [12]
$ d_e $ AIDS-related death rate among elderly [12.1-17.5] 16% [52]
$ c_{11}, c_{12} $ Condom use rate of youth [34-68] 57.5% [51]
$ c_{21}, c_{22}, c_{23} $ Condom use rate of adult [30-40] 34.5% [51]
$ c_{33} $ Condom use rate among elderly [17.5-37.5] 20.0% [17]
$ \alpha_y $ Transfer rate of youth group - 0.043 Fit
$ \alpha_a $ Transfer rate of adult group - 0.031 Fit
$ \delta_{11}, \delta_{12} $ Infected rate among youth [0.012-0.065] 0.014% [23]
$ \delta_{21} $ Infected rate from adult to young $ [0.13-9.9] $ 2.8% [42]
$ \delta_{22} $ Infected rate among adult $ [0.13-9.9] $ 4.2% [42]
$ \delta_{23} $ Infected rate from adult to elderly $ [0.13-9.9] $ 6.6% [42]
$ \delta_{33} $ Infected rate among elderly [2.7-30.6] 25% [21]
$ \beta_{y} $ Transfer rate from $ I_y $ to $ A_a $ - 0.036 Fit
$ \beta_{a} $ Transfer rate from $ I_a $ to $ A_e $ - 0.041 Fit
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Parameters Description Range $ (\%) $ Value $ (year^{-1}) $ Source
$ \Lambda $ Recruitment of the youth class - 3075405 Assume
$ \mu_y $ Natural death rate of youth [0.066-0.087] 0.765‰ [7]
$ \mu_a $ Natural death rate of adult [0.1-0.327] 1.852‰ [7]
$ \mu_e $ Natural death rate of elderly [0.19-0.45] 2.07‰ [7]
$ r_{y} $ Average remove rate from $ I_y $ to $ A_y $ - 1/12.5 [46]
$ r_{a} $ Average remove rate from $ I_a $ to $ A_a $ - 1/10 [46]
$ r_{e} $ Average remove rate from $ I_e $ to $ A_e $ - 1/7.9 [46]
$ d_y $ AIDS-related death rate among youth [1.0-4.3] 2.3% [12]
$ d_a $ AIDS-related death rate among adult [5.03-12.1] 9.7% [12]
$ d_e $ AIDS-related death rate among elderly [12.1-17.5] 16% [52]
$ c_{11}, c_{12} $ Condom use rate of youth [34-68] 57.5% [51]
$ c_{21}, c_{22}, c_{23} $ Condom use rate of adult [30-40] 34.5% [51]
$ c_{33} $ Condom use rate among elderly [17.5-37.5] 20.0% [17]
$ \alpha_y $ Transfer rate of youth group - 0.043 Fit
$ \alpha_a $ Transfer rate of adult group - 0.031 Fit
$ \delta_{11}, \delta_{12} $ Infected rate among youth [0.012-0.065] 0.014% [23]
$ \delta_{21} $ Infected rate from adult to young $ [0.13-9.9] $ 2.8% [42]
$ \delta_{22} $ Infected rate among adult $ [0.13-9.9] $ 4.2% [42]
$ \delta_{23} $ Infected rate from adult to elderly $ [0.13-9.9] $ 6.6% [42]
$ \delta_{33} $ Infected rate among elderly [2.7-30.6] 25% [21]
$ \beta_{y} $ Transfer rate from $ I_y $ to $ A_a $ - 0.036 Fit
$ \beta_{a} $ Transfer rate from $ I_a $ to $ A_e $ - 0.041 Fit
Table 2.  Numbers of youth, adult, the elderly and total new reporting HIV/AIDS cases in China $ (2005-2017) $. Adult cases are calculated by the total cases minus other groups cases, where the data on children under 15 years old from 2005 to 2017 are counted in [45]
Year Total cases Source The youth cases (proportion) Source The adult cases (proportion) The elderly cases (proportion) Source
2005 40711 [43] 4186 (10.28%) [55] 33430 (82.12%) 2563 (6.30%) [48]
2006 44070 [43] 4872 (11.06%) [55] 34227 (77.67%) 3437 (7.80%) [16]
2007 45151 [43] 5524 (12.23%) [55] 34449 (76.30%) 4515 (10.06%) [16]
2008 50081 [43] 6628 (13.23%) [55] 36064 (72.01%) 6599 (13.18%) [44]
2009 53249 [43] 7416 (13.93%) [55] 35916 (67.45%) 9016 (16.93%) [44]
2010 64108 [43] 7875 (13.28%) [55] 44696 (69.72%) 11537 (18.00%) [44]
2011 74517 [36] 8925 (11.98%) [55] 48983 (65.73%) 16609 (22.30%) [44]
2012 82434 [28] 10195 (12.37%) [55] 52718 (63.95%) 19521 (23.68%) [44]
2013 90119 [29] 10800 (13.49%) [36] 56253 (62.42%) 23066 (25.61%) [44]
2014 103501 [30] 15000 (14.61%) [36] 60139 (58.10%) 27520 (26.60%) [44]
2015 114656 [31] 16986 (14.81%) [49] 63308 (55.22%) 33522 (29.24%) [41]
2016 124555 [26] 18437 (15.00%) [49] 70356 (56.49%) 35762 (28.71%) [41]
2017 134551 [27] 21250 (15.79%) [49] 72468 (53.86%) 40833 (30.35%) [41]
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Year Total cases Source The youth cases (proportion) Source The adult cases (proportion) The elderly cases (proportion) Source
2005 40711 [43] 4186 (10.28%) [55] 33430 (82.12%) 2563 (6.30%) [48]
2006 44070 [43] 4872 (11.06%) [55] 34227 (77.67%) 3437 (7.80%) [16]
2007 45151 [43] 5524 (12.23%) [55] 34449 (76.30%) 4515 (10.06%) [16]
2008 50081 [43] 6628 (13.23%) [55] 36064 (72.01%) 6599 (13.18%) [44]
2009 53249 [43] 7416 (13.93%) [55] 35916 (67.45%) 9016 (16.93%) [44]
2010 64108 [43] 7875 (13.28%) [55] 44696 (69.72%) 11537 (18.00%) [44]
2011 74517 [36] 8925 (11.98%) [55] 48983 (65.73%) 16609 (22.30%) [44]
2012 82434 [28] 10195 (12.37%) [55] 52718 (63.95%) 19521 (23.68%) [44]
2013 90119 [29] 10800 (13.49%) [36] 56253 (62.42%) 23066 (25.61%) [44]
2014 103501 [30] 15000 (14.61%) [36] 60139 (58.10%) 27520 (26.60%) [44]
2015 114656 [31] 16986 (14.81%) [49] 63308 (55.22%) 33522 (29.24%) [41]
2016 124555 [26] 18437 (15.00%) [49] 70356 (56.49%) 35762 (28.71%) [41]
2017 134551 [27] 21250 (15.79%) [49] 72468 (53.86%) 40833 (30.35%) [41]
Table 3.  The PRCC of the parameters in model $ (1) $
Parameters p value PRCC Parameters p value PRCC
$ c_{11} $ 0.3721 -0.1622 $ \delta_{22} $ 0.6886 0.7230
$ c_{12} $ 0.6879 -0.2595 $ \alpha_{y} $ 0.6152 0.3462
$ c_{21} $ 0.2277 -0.4457 $ \alpha_{a} $ 0.7502 -0.2868
$ c_{22} $ 0.8075 -0.6452 $ \beta_{y} $ 0.5915 -0.0771
$ \delta_{11} $ 0.4428 0.0172 $ \beta_{a} $ 0 -0.8105
$ \delta_{21} $ 0.3971 0.5004 $ r_{y} $ 0.2674 -0.3981
$ \delta_{12} $ 0.8916 -0.0008 $ r_{a} $ 0.2412 -0.3063
$ \delta_{23} $ 0.8646 -0.0036 $ c_{23} $ 0.1928 -0.0292
$ \delta_{33} $ 0.3956 0.0190 $ c_{33} $ 0.5424 -0.0135
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Parameters p value PRCC Parameters p value PRCC
$ c_{11} $ 0.3721 -0.1622 $ \delta_{22} $ 0.6886 0.7230
$ c_{12} $ 0.6879 -0.2595 $ \alpha_{y} $ 0.6152 0.3462
$ c_{21} $ 0.2277 -0.4457 $ \alpha_{a} $ 0.7502 -0.2868
$ c_{22} $ 0.8075 -0.6452 $ \beta_{y} $ 0.5915 -0.0771
$ \delta_{11} $ 0.4428 0.0172 $ \beta_{a} $ 0 -0.8105
$ \delta_{21} $ 0.3971 0.5004 $ r_{y} $ 0.2674 -0.3981
$ \delta_{12} $ 0.8916 -0.0008 $ r_{a} $ 0.2412 -0.3063
$ \delta_{23} $ 0.8646 -0.0036 $ c_{23} $ 0.1928 -0.0292
$ \delta_{33} $ 0.3956 0.0190 $ c_{33} $ 0.5424 -0.0135
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