Abstract

This paper develops an impulsive SUI model with infection age for HIV. Firstly the basic reproduction number, which depends on the impulsive HIV-finding period and the HIV-finding proportion, is obtained by mathematical analysis. Secondly there exists a globally asymptotically stable infection- free equilibrium when the basic reproduction number is less than one. Therefore the HIV epidemic is theoretically cleared if we have the suitable HIV-finding proportion and the impulsive HIV-finding period such that the basic reproduction number is less than one.

References

[1] R.M. May, R.M. Anderson, Transmission dynamics of HIV infection, Nature. 326(6109) (1987) 137–142.

[2] S. Busenberg, C. Castillo-Chavez, A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured models for the spread of AIDS, IMA J. Math. Appl. Med. Biol. 8(1991) 1–29.
[3] J.M. Hyman, J. Li, E.A. Stanley, Threshold conditions for the spread of the HIV infection in age-structured populations of homosexual men, J. Theor. Biol. 166(1994) 9–31.
[4] H. Inaba, Endemic threshold results in an age-duration-structured population model for HIV infection, Math. Biosci. 201(2006) 15–47.
[5] J.M. Heffernan, R.J. Smith, L.M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface. 2 (4)( 2005) 281–283.
[6] R.M. May, R.M. Anderson, A.R. McLean, Possible demographic consequences of HIV/AIDS, Math. Biosci. 90(1988) 475–506.
[7] L.M. Xu, L.Q. Sun, B.H. Wu, Study on the current situation and effect of HIV / AIDS prevention after exposure in Shenzhen(in Chinese), Chinese Journal of AIDS & STD. 197(02)( 2020) 62-65.
[8] Z.Y. Wang, L.M. Yan, Q.H. Xia, J. Fu, HIV / AIDS knowledge and its influencing factors among STD outpatients in Shanghai(in Chinese), Chinese Journal of Viral Diseases. 10(04)( 2020) 62-65.
[9] S.X. Wang, M.Y. Li, X.B. Liu, Feasibility study of ARIMA model in predicting the number of newly discovered AIDS cases in China(in Chinese), Chinese Journal of AIDS & STD. 26(7) (2020) 705-708.
[10] R.J. Smith, L.M. Wahl, Distinct effects of protease and reverse transcriptase inhibition in an immunological model of HIV-1 infection with impulsive drug effects, Bull. Math. Biol. 66(5)(2004) 1259–1283.
[11] R.J. Smith, L.M. Wahl, Drug resistance in an immunological model of HIV-1 infection with impulsive drug effects, Bull. Math. Biol. 67(4)(2005) 783–813.
[12] O. Krakovska, L.M. Wahl, Optimal drug treatment regimens for HIV depend on adherence, J. Theor. Biol. 246(3)(2007) 499–509.
[13] H.L. Liu, J.Y. Yu, G.T. Zhu,. Global behaviour of an age-infection-structured HIV model with impulsive drug-treatment strategy, J. Theor. Biol. 253(4)(2008) 749–754.
[14] P. Yan, Impulsive SUI epidemic model for HIV/AIDS with chronological age and infection age, Journal of Theoretical Biology. 265(2) (2010) 177-184.
[15] L.B. Li, P. Zhang, H.L. Liu, Global stability of SIA model with pulse drug therapy strategy(in Chinese), Mathematics in Practice and Theory. 44(12) (2014) 233-240.
[16] H.L. Liu, N.Tong, Y.P. Jin, Continuous and impulsive HIV / AIDS models with age structure and treatment strategy(in Chinese), Journal of Xinyang Teachers College ( Natural Science Edition). 30(004)(2017) 517-520.
[17] H. Liu, L. Li, A Class Age-Structured HIV/AIDS Model with Impulsive Drug-Treatment Strategy, Discrete Dynamics in Nature and Society. 210(2)( 2010) 1038-1045.
[18] J. Jiao, L. Li, Y. Zhang, Dynamics of a stage-structured single population system with winter hibernation and impulsive effect in polluted environment, International Journal of Biomathematics. 09(5)( 2016) 277-294.
[19] G. Liu, X. Wang, X. Meng, et al. Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps. Complexity. 1950970( 2017) 1-15.
[20] P. Yan, Well posedness of the model for the dynamics of infectious diseases without immunity and with latent period (in Chinese). J. Biomath. 23 (2)(2008) 245–256.
[21] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations. World Scientific, Singapore. 1989.
[22] A. D’onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179(1)(2002) 57–72.