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June  2022, 27(6): 3177-3211. doi: 10.3934/dcdsb.2021181

Dynamics of a stochastic HIV/AIDS model with treatment under regime switching

1. 

School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

2. 

College of Science, China University of Petroleum (East China), Qingdao 266580, China

3. 

Key Laboratory of Unconventional Oil and Gas Development, China University of Petroleum (East China), Ministry of Education, Qingdao 266580, China

4. 

Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

5. 

Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan

* Corresponding author: Daqing Jiang

Received  April 2020 Revised  April 2021 Published  June 2022 Early access  July 2021

Fund Project: This work is supported by the National Natural Science Foundation of China under Grant No. 11871473 and Natural Science Foundation of Shandong Province under Grant No. ZR2019MA010

This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.

Citation: Miaomiao Gao, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi, Bashir Ahmad. Dynamics of a stochastic HIV/AIDS model with treatment under regime switching. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3177-3211. doi: 10.3934/dcdsb.2021181
References:
[1]

S. Al-SheikhF. Musali and M. Alsolami, Stability analysis of an HIV/AIDS epidemic model with screening, Int. Math. Forum., 6 (2011), 3251-3273. 

[2]

P. J. Birrell, A. M. Presanis and D. D. Angelis, Multi-state Models of HIV Progression in Homosexual Men: An Application to the CASCADE Collaboration, Technical report, MRC Biostatistics Unit, 2012.

[3]

L. CaiS. Guo and S. Wang, Analysis of an extended HIV/AIDS epidemic model with treatment, Appl. Math. Comput., 236 (2014), 621-627.  doi: 10.1016/j.amc.2014.02.078.

[4]

L. CaiX. LiM. Ghosh and B. Guo, Stability analysis of an HIV/AIDS epidemic model with treatment, J. Comput. Appl. Math., 229 (2009), 313-323.  doi: 10.1016/j.cam.2008.10.067.

[5]

L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos Solitons Fractals, 41 (2009), 175-182.  doi: 10.1016/j.chaos.2007.11.023.

[6]

Y. CaiY. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221-240.  doi: 10.1016/j.amc.2017.02.003.

[7]

Collaborative Group on AIDS Incubation and HIV Survival including the CASCADE EU Concerted Action, 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.

[8]

N. H. DangN. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029.

[9]

T. Feng and Z. Qiu, Global anaiysis of a stochastic TB model with vaccination and treatment, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2923-2939.  doi: 10.3934/dcdsb.2018292.

[10]

R. M. GranichC. F. GilksC. DyeK. M. D. Cock and B. G. Williams, Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: A mathematical model, Lancet., 373 (2009), 48-57.  doi: 10.1016/S0140-6736(08)61697-9.

[11]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.

[12]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[13]

T. D. HollingsworthR. M. Anderson and C. Fraser, HIV-1 transmission, by stage of infection, J. Infect. Dis., 198 (2008), 687-693.  doi: 10.1086/590501.

[14]

S. D. Hove-Musekwa and F. Nyabadza, The dynamics of an HIV/AIDS model with screened disease carriers, Comput. Math. Methods Med., 10 (2009), 287-305.  doi: 10.1080/17486700802653917.

[15]

H.-F. HuoR. Chen and X.-Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Math. Model., 40 (2016), 6550-6559.  doi: 10.1016/j.apm.2016.01.054.

[16]

H.-F. Huo and L.-X. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Model., 37 (2013), 1480-1489.  doi: 10.1016/j.apm.2012.04.013.

[17]

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017.

[18]

C. Ji, The threshold for a stochastic HIV-1 infection model with Beddington-DeAngelis incidence rate, Appl. Math. Model., 64 (2018), 168-184.  doi: 10.1016/j.apm.2018.07.031.

[19]

J. Jia and G. Qin, Stability analysis of HIV/AIDS epidemic model with nonlinear incidence and treatment, Adv. Difference Equations, 2017 (2017), 136. doi: 10.1186/s13662-017-1175-5.

[20]

M. E. KretzschmarM. F. S. van der LoeffP. J. BirrellD. D. Angelis and R. A. Coutinho, Prospects of elimination of HIV with test-and-treat strategy, Proc. Natl. Acad. Sci., 110 (2013), 15538-15543.  doi: 10.1073/pnas.1301801110.

[21]

H. Kunita, Itô's stochastic calculus: Its surprising power for applications, Stoch. Proc. Appl., 120 (2010), 622-652.  doi: 10.1016/j.spa.2010.01.013.

[22]

J. A. Levy, Pathogenesis of human immunodeficiency virus infection, Microbiol. Rev., 57 (1993), 183-289.  doi: 10.1128/mr.57.1.183-289.1993.

[23]

D. LiS. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066.

[24]

X. LinH. W. Hethcote and P. van den Driessche, An epidemiological model for HIV/AIDS with proportional recruitment, Math. Biosci., 118 (1993), 181-195.  doi: 10.1016/0025-5564(93)90051-B.

[25]

D. Liu and B. Wang, A novel time delayed HIV/AIDS model with vaccination and antiretroviral therapy and its stability analysis, Appl. Math. Model., 37 (2013), 4608-4625.  doi: 10.1016/j.apm.2012.09.065.

[26]

H. LiuX. Li and Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002.

[27]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic HIV-1 infection model with distributed delay and logistic growth, J. Nonlinear Sci., 30 (2020), 369-395.  doi: 10.1007/s00332-019-09576-x.

[28]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Threshold behavior in a stochastic delayed SIS epidemic model with vaccination and double diseases, J. Franklin Inst., 356 (2019), 7466-7485.  doi: 10.1016/j.jfranklin.2018.11.055.

[29]

Q. LiuD. JiangT. HayatA. Alsaedi and B. Ahmad, Dynamics of a multigroup SIQS epidemic model under regime switching, Stoch. Anal. Appl., 38 (2020), 769-796.  doi: 10.1080/07362994.2020.1722167.

[30]

X. MaoG. Marion and E. Renshaw, Environmental Brownian noise suppresses explosion in population dynamics, Stoch. Process. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[31] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.
[32] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.  doi: 10.1515/9780691206912.
[33]

X. MengS. ZhaoT. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2015), 227-242.  doi: 10.1016/j.jmaa.2015.07.056.

[34]

C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.  doi: 10.1016/S0025-5564(02)00149-9.

[35]

M. A. Nowak and R. M. May, Virus Dynamics, Mathematical Principles of Immunology and Virology, Oxford University, Oxford, 2000.

[36]

M. U. Nsuami and P. J. Witbooi, A model of HIV/AIDS population dynamics including ARV treatment and pre-exposure prophylaxis, Adv. Difference Equations, 2018 (2018), 11. doi: 10.1186/s13662-017-1458-x.

[37]

M. U. Nsuami and P. J. Witbooi, Stochastic dynamics of an HIV/AIDS epidemic model with treatment, Quaest. Math., 42 (2019), 605-621.  doi: 10.2989/16073606.2018.1478908.

[38]

B. $\varnothing$ksendal, Stochastic Differential Equations: An Introduction with Applications, 6$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2005.

[39]

O. M. Otunuga, Global stability for a $2n+1$ dimensional HIV/AIDS epidemic model with treatments, Math. Biosci., 299 (2018), 138-152.  doi: 10.1016/j.mbs.2018.03.013.

[40]

S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.  doi: 10.1016/j.spa.2005.08.004.

[41]

K. Qi and D. Jiang, The impact of virus carrier screening and actively seeking treatment on dynamical behavior of a stochastic HIV/AIDS infection model, Appl. Math. Model., 85 (2020), 378-404.  doi: 10.1016/j.apm.2020.03.027.

[42]

K. Qi and D. Jiang, Threshold behavior in a stochastic HTLV-I infection model with CTL immune response and regime switching, Math. Methods Appl. Sci., 41 (2018), 6866-6882.  doi: 10.1002/mma.5198.

[43]

A. RathinasamyM. 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.  doi: 10.1016/j.matcom.2020.08.017.

[44]

A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243.  doi: 10.1016/j.amc.2014.07.012.

[45]

C. A. Stoddart and R. A. Reyes, Models of HIV-1 disease: A review of current status, Drug Discovery Today Dis. Models, 3 (2006), 113-119.  doi: 10.1016/j.ddmod.2006.03.016.

[46]

The CASCADE Collaboration, Survival after introduction of HAART in people with known duration of HIV-1 infection, Lancet., 355 (2000), 1158-1159.  doi: 10.1016/S0140-6736(00)02069-9.

[47]

T. D. TuongD. H. NguyenN. T. Dieu and K. Tran, Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate, Nonlinear Anal. Hybrid Syst., 34 (2019), 121-130.  doi: 10.1016/j.nahs.2019.05.008.

[48]

D. Wanduku, The stochastic extinction and stability conditions for nonlinear malaria epidemics, Math. Biosci. Eng., 16 (2019), 3771-3806.  doi: 10.3934/mbe.2019187.

[49]

World Health Organization Data on the Size of the HIV/AIDS Epidemic, Available from: https://www.who.int/data/gho/data/themes/hiv-aids/GHO/hiv-aids.

[50]

X. Zhang and H. Peng, Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching, Appl. Math. Lett., 102 (2020), 106095. doi: 10.1016/j.aml.2019.106095.

[51]

Y. ZhaoD. JiangX. Mao and A. Gray, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1277-1295.  doi: 10.3934/dcdsb.2015.20.1277.

[52]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.  doi: 10.1137/060649343.

show all references

References:
[1]

S. Al-SheikhF. Musali and M. Alsolami, Stability analysis of an HIV/AIDS epidemic model with screening, Int. Math. Forum., 6 (2011), 3251-3273. 

[2]

P. J. Birrell, A. M. Presanis and D. D. Angelis, Multi-state Models of HIV Progression in Homosexual Men: An Application to the CASCADE Collaboration, Technical report, MRC Biostatistics Unit, 2012.

[3]

L. CaiS. Guo and S. Wang, Analysis of an extended HIV/AIDS epidemic model with treatment, Appl. Math. Comput., 236 (2014), 621-627.  doi: 10.1016/j.amc.2014.02.078.

[4]

L. CaiX. LiM. Ghosh and B. Guo, Stability analysis of an HIV/AIDS epidemic model with treatment, J. Comput. Appl. Math., 229 (2009), 313-323.  doi: 10.1016/j.cam.2008.10.067.

[5]

L. Cai and J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence, Chaos Solitons Fractals, 41 (2009), 175-182.  doi: 10.1016/j.chaos.2007.11.023.

[6]

Y. CaiY. Kang and W. Wang, A stochastic SIRS epidemic model with nonlinear incidence rate, Appl. Math. Comput., 305 (2017), 221-240.  doi: 10.1016/j.amc.2017.02.003.

[7]

Collaborative Group on AIDS Incubation and HIV Survival including the CASCADE EU Concerted Action, 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.

[8]

N. H. DangN. H. Du and G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Equations, 257 (2014), 2078-2101.  doi: 10.1016/j.jde.2014.05.029.

[9]

T. Feng and Z. Qiu, Global anaiysis of a stochastic TB model with vaccination and treatment, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2923-2939.  doi: 10.3934/dcdsb.2018292.

[10]

R. M. GranichC. F. GilksC. DyeK. M. D. Cock and B. G. Williams, Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: A mathematical model, Lancet., 373 (2009), 48-57.  doi: 10.1016/S0140-6736(08)61697-9.

[11]

A. GrayD. GreenhalghL. HuX. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.  doi: 10.1137/10081856X.

[12]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.

[13]

T. D. HollingsworthR. M. Anderson and C. Fraser, HIV-1 transmission, by stage of infection, J. Infect. Dis., 198 (2008), 687-693.  doi: 10.1086/590501.

[14]

S. D. Hove-Musekwa and F. Nyabadza, The dynamics of an HIV/AIDS model with screened disease carriers, Comput. Math. Methods Med., 10 (2009), 287-305.  doi: 10.1080/17486700802653917.

[15]

H.-F. HuoR. Chen and X.-Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Math. Model., 40 (2016), 6550-6559.  doi: 10.1016/j.apm.2016.01.054.

[16]

H.-F. Huo and L.-X. Feng, Global stability for an HIV/AIDS epidemic model with different latent stages and treatment, Appl. Math. Model., 37 (2013), 1480-1489.  doi: 10.1016/j.apm.2012.04.013.

[17]

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017.

[18]

C. Ji, The threshold for a stochastic HIV-1 infection model with Beddington-DeAngelis incidence rate, Appl. Math. Model., 64 (2018), 168-184.  doi: 10.1016/j.apm.2018.07.031.

[19]

J. Jia and G. Qin, Stability analysis of HIV/AIDS epidemic model with nonlinear incidence and treatment, Adv. Difference Equations, 2017 (2017), 136. doi: 10.1186/s13662-017-1175-5.

[20]

M. E. KretzschmarM. F. S. van der LoeffP. J. BirrellD. D. Angelis and R. A. Coutinho, Prospects of elimination of HIV with test-and-treat strategy, Proc. Natl. Acad. Sci., 110 (2013), 15538-15543.  doi: 10.1073/pnas.1301801110.

[21]

H. Kunita, Itô's stochastic calculus: Its surprising power for applications, Stoch. Proc. Appl., 120 (2010), 622-652.  doi: 10.1016/j.spa.2010.01.013.

[22]

J. A. Levy, Pathogenesis of human immunodeficiency virus infection, Microbiol. Rev., 57 (1993), 183-289.  doi: 10.1128/mr.57.1.183-289.1993.

[23]

D. LiS. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, J. Differential Equations, 263 (2017), 8873-8915.  doi: 10.1016/j.jde.2017.08.066.

[24]

X. LinH. W. Hethcote and P. van den Driessche, An epidemiological model for HIV/AIDS with proportional recruitment, Math. Biosci., 118 (1993), 181-195.  doi: 10.1016/0025-5564(93)90051-B.

[25]

D. Liu and B. Wang, A novel time delayed HIV/AIDS model with vaccination and antiretroviral therapy and its stability analysis, Appl. Math. Model., 37 (2013), 4608-4625.  doi: 10.1016/j.apm.2012.09.065.

[26]

H. LiuX. Li and Q. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810.  doi: 10.1016/j.sysconle.2013.06.002.

[27]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic HIV-1 infection model with distributed delay and logistic growth, J. Nonlinear Sci., 30 (2020), 369-395.  doi: 10.1007/s00332-019-09576-x.

[28]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Threshold behavior in a stochastic delayed SIS epidemic model with vaccination and double diseases, J. Franklin Inst., 356 (2019), 7466-7485.  doi: 10.1016/j.jfranklin.2018.11.055.

[29]

Q. LiuD. JiangT. HayatA. Alsaedi and B. Ahmad, Dynamics of a multigroup SIQS epidemic model under regime switching, Stoch. Anal. Appl., 38 (2020), 769-796.  doi: 10.1080/07362994.2020.1722167.

[30]

X. MaoG. Marion and E. Renshaw, Environmental Brownian noise suppresses explosion in population dynamics, Stoch. Process. Appl., 97 (2002), 95-110.  doi: 10.1016/S0304-4149(01)00126-0.

[31] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006.  doi: 10.1142/p473.
[32] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001.  doi: 10.1515/9780691206912.
[33]

X. MengS. ZhaoT. Feng and T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2015), 227-242.  doi: 10.1016/j.jmaa.2015.07.056.

[34]

C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.  doi: 10.1016/S0025-5564(02)00149-9.

[35]

M. A. Nowak and R. M. May, Virus Dynamics, Mathematical Principles of Immunology and Virology, Oxford University, Oxford, 2000.

[36]

M. U. Nsuami and P. J. Witbooi, A model of HIV/AIDS population dynamics including ARV treatment and pre-exposure prophylaxis, Adv. Difference Equations, 2018 (2018), 11. doi: 10.1186/s13662-017-1458-x.

[37]

M. U. Nsuami and P. J. Witbooi, Stochastic dynamics of an HIV/AIDS epidemic model with treatment, Quaest. Math., 42 (2019), 605-621.  doi: 10.2989/16073606.2018.1478908.

[38]

B. $\varnothing$ksendal, Stochastic Differential Equations: An Introduction with Applications, 6$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2005.

[39]

O. M. Otunuga, Global stability for a $2n+1$ dimensional HIV/AIDS epidemic model with treatments, Math. Biosci., 299 (2018), 138-152.  doi: 10.1016/j.mbs.2018.03.013.

[40]

S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.  doi: 10.1016/j.spa.2005.08.004.

[41]

K. Qi and D. Jiang, The impact of virus carrier screening and actively seeking treatment on dynamical behavior of a stochastic HIV/AIDS infection model, Appl. Math. Model., 85 (2020), 378-404.  doi: 10.1016/j.apm.2020.03.027.

[42]

K. Qi and D. Jiang, Threshold behavior in a stochastic HTLV-I infection model with CTL immune response and regime switching, Math. Methods Appl. Sci., 41 (2018), 6866-6882.  doi: 10.1002/mma.5198.

[43]

A. RathinasamyM. 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.  doi: 10.1016/j.matcom.2020.08.017.

[44]

A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243.  doi: 10.1016/j.amc.2014.07.012.

[45]

C. A. Stoddart and R. A. Reyes, Models of HIV-1 disease: A review of current status, Drug Discovery Today Dis. Models, 3 (2006), 113-119.  doi: 10.1016/j.ddmod.2006.03.016.

[46]

The CASCADE Collaboration, Survival after introduction of HAART in people with known duration of HIV-1 infection, Lancet., 355 (2000), 1158-1159.  doi: 10.1016/S0140-6736(00)02069-9.

[47]

T. D. TuongD. H. NguyenN. T. Dieu and K. Tran, Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate, Nonlinear Anal. Hybrid Syst., 34 (2019), 121-130.  doi: 10.1016/j.nahs.2019.05.008.

[48]

D. Wanduku, The stochastic extinction and stability conditions for nonlinear malaria epidemics, Math. Biosci. Eng., 16 (2019), 3771-3806.  doi: 10.3934/mbe.2019187.

[49]

World Health Organization Data on the Size of the HIV/AIDS Epidemic, Available from: https://www.who.int/data/gho/data/themes/hiv-aids/GHO/hiv-aids.

[50]

X. Zhang and H. Peng, Stationary distribution of a stochastic cholera epidemic model with vaccination under regime switching, Appl. Math. Lett., 102 (2020), 106095. doi: 10.1016/j.aml.2019.106095.

[51]

Y. ZhaoD. JiangX. Mao and A. Gray, The threshold of a stochastic SIRS epidemic model in a population with varying size, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1277-1295.  doi: 10.3934/dcdsb.2015.20.1277.

[52]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46 (2007), 1155-1179.  doi: 10.1137/060649343.

Figure 1.  The solution of subsystem with state 1. (Color figure online)
Figure 2.  The solution of subsystem with state 2. (Color figure online)
Figure 3.  The pictures (a), (b) and (c) are the solution of system (3). The picture (d) is the corresponding Markov chain with $ \pi = (\frac{3}{5},\frac{2}{5}) $. (Color figure online)
Figure 4.  The solution of subsystem with state 1. (Color figure online)
Figure 5.  The solution of subsystem with state 2. (Color figure online)
Figure 6.  The pictures (a), (b) and (c) are the solution of system (3). The picture (d) is the corresponding Markov chain with $ \pi = (\frac{3}{5},\frac{2}{5}) $. (Color figure online)
Figure 7.  The diagrams track the variation trends of $ S(t) $ and $ I_{k}(t),k = 1,2 $ with different transmission rate $ \lambda(m),m = 1,2 $. (Color figure online)
Figure 8.  The diagrams track the variation trends of $ I_{k}(t) $ and $ T_{k}(t),k = 1,2 $ with different treatment rate $ \tau $. (Color figure online)
Table 1.  List of the biological parameters
Parameter Definition Value Source
$\rho_{k}$ Transition rates per year from stage $k$ to stage $k+1$ for untreated individuals $\rho_{1}=1/0.271, \rho_{2}=1/8.31$ [7,2,13]
$\gamma_{k}$ Transition rates per year from stage $k$ to stage $k+1$ for treated individuals $\gamma_{1}=1/8.21, \gamma_{2}=1/54$ [2,46]
$\tau$ Rate per year of moving from the untreated to the treated population range 0-100$\%$ [20]
$\phi$ Rate of moving from the treated back to the untreated population range 0-100$\%$ [20]
$\epsilon$ Infectivity of individuals under treatment around 0.01 [10]
$h_{k}$ Infectivity of untreated individuals in stage $k$ of infection per year around 2.76 for $h_{1}$, around 0.106 for $h_{2}$ [13]
Parameter Definition Value Source
$\rho_{k}$ Transition rates per year from stage $k$ to stage $k+1$ for untreated individuals $\rho_{1}=1/0.271, \rho_{2}=1/8.31$ [7,2,13]
$\gamma_{k}$ Transition rates per year from stage $k$ to stage $k+1$ for treated individuals $\gamma_{1}=1/8.21, \gamma_{2}=1/54$ [2,46]
$\tau$ Rate per year of moving from the untreated to the treated population range 0-100$\%$ [20]
$\phi$ Rate of moving from the treated back to the untreated population range 0-100$\%$ [20]
$\epsilon$ Infectivity of individuals under treatment around 0.01 [10]
$h_{k}$ Infectivity of untreated individuals in stage $k$ of infection per year around 2.76 for $h_{1}$, around 0.106 for $h_{2}$ [13]
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