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Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting
Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China
1. | School of Science, Chang'an University, Xi'an 710064, China |
2. | College of Economics and Management, Shanxi Normal University, Linfen 041004, China |
In this paper, we provide an effective method for estimating the thresholds of the stochastic models with time delays by using of the nonnegative semimartingale convergence theorem. Firstly, we establish the stochastic delay differential equation models for two diseases, and obtain two thresholds of two diseases and the sufficient conditions for the persistence and extinction of two diseases. Then, numerical simulations for co-infection of HIV/AIDS and Gonorrhea in Yunnan Province, China, are carried out. Finally, we discuss some biological implications and focus on the impact of some key model parameters. One of the most interesting findings is that the stochastic fluctuation and time delays introduced into the deterministic models can suppress the outbreak of the diseases, which can provide some useful control strategies to regulate the dynamics of the diseases, and the numerical simulations verify this phenomenon.
References:
[1] |
L. J. S. Allen,
An introduction to stochastic epidemic models, Mathematical epidemiology, Lecture Notes in Mathematics, 1945 (2008), 81-130.
doi: 10.1007/978-3-540-78911-6_3. |
[2] |
B. Berrhazi, M. El Fatini and A. Laaribi,
A stochastic threshold for an epidemic model with Beddington-DeAngelis incidence, delayed loss of immunity and Lévy noise perturbation, Phys. A, 507 (2018), 312-320.
doi: 10.1016/j.physa.2018.05.096. |
[3] |
T. Britton,
Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35.
doi: 10.1016/j.mbs.2010.01.006. |
[4] |
Y. Chen, B. Wen and Z. Teng,
The global dynamics for a stochastic SIS epidemic model with isolation, Phys. A, 492 (2018), 1604-1624.
doi: 10.1016/j.physa.2017.11.085. |
[5] |
K. Fan, Y. Zhang, S. Gao and et al,
A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Phys. A, 481 (2017), 198-208.
doi: 10.1016/j.physa.2017.04.055. |
[6] |
Z. Guo, M. Zhao and et al, Correction for life expectation of population and mortality of infant in Yunnan, Maternal and Child Health Care of China, 20 (2005), 681-685. Google Scholar |
[7] |
M. Y. Li and H. Shu,
Joint effects of mitosis and intracellular delay on viral dynamics: Two-parameter bifurcation analysis, J. Math. Biol., 64 (2012), 1005-1020.
doi: 10.1007/s00285-011-0436-2. |
[8] |
M. Liu,
Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.
doi: 10.1093/imamat/hxv002. |
[9] |
Z. Liu,
Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real World Appl., 14 (2013), 1286-1299.
doi: 10.1016/j.nonrwa.2012.09.016. |
[10] |
M. Liu, C. Bai and K. Wang,
Asymptotic stability of a two-group stochastic SEIR model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3444-3453.
doi: 10.1016/j.cnsns.2014.02.025. |
[11] |
Q. Liu and Q. Chen,
Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Phys. A, 428 (2015), 140-153.
doi: 10.1016/j.physa.2015.01.075. |
[12] |
Q. Liu, D. Jiang, T. Hayat and et al,
Stationary distribution of a stochastic delayed SVEIR epidemic model with vaccination and saturation incidence, Phys. A, 512 (2018), 849-863.
doi: 10.1016/j.physa.2018.08.054. |
[13] |
X. Mao, G. Marion and E. Renshaw,
Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110.
doi: 10.1016/S0304-4149(01)00126-0. |
[14] |
X. Meng, S. Zhao, T. Feng and et al,
Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227-242.
doi: 10.1016/j.jmaa.2015.07.056. |
[15] |
B. Monel, E. Beaumont, D. Vendrame and et al, HIV cell-to-cell transmission requires the production of infectious virus particles, and does not proceed through Env-mediated fusion pores, Journal of Virology, 86 (2012), 3924-3933. Google Scholar |
[16] |
P. W. Nelson, M. A. Gilchrist, D. Coombs and et al,
An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.
doi: 10.3934/mbe.2004.1.267. |
[17] |
Yunnan Bureau of Satistic, The Sixth National Census Data Bulletin in Yunnan Province 2010, Available from: http://www.stats.gov.cn/tjsj/tjgb/rkpcgb/dfrkpcgb/201202/t20120228_30408.html. Google Scholar |
[18] |
The Data-center of China Public Health, Available from: http://www.phsciencedata.cn/. Google Scholar |
[19] |
E. Tornatore and S. M. Buccellato,
Parasite population delay model of malaria type with stochastic perturbation and environmental criterion for limitation of disease, J. Math. Anal. Appl., 360 (2009), 624-630.
doi: 10.1016/j.jmaa.2009.06.078. |
[20] |
J. E. Truscott and C. A. Gilligan,
Response of a deterministic epidemiological system to a stochastically varying environment, Proceedings of the National Academy of Sciences of the United States of America, 100 (2003), 9067-9072.
doi: 10.1073/pnas.1436273100. |
[21] |
D. Xiao and W. H. Bossert,
An intra-host mathematical model on interaction between HIV and malaria, Bull. Math. Biol., 72 (2010), 1892-1911.
doi: 10.1007/s11538-010-9515-6. |
[22] |
L. Zhang, W. T. Li, Z. Wang and et al,
Entire solutions for nonlocal dispersal equations with bistable nonlinearity: Asymmetric case, Acta Math. Sin. (Engl. Ser.), 35 (2019), 1771-1794.
doi: 10.1007/s10114-019-8294-8. |
[23] |
T. Zhang, H. Li, N. Xie and et al, Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China, Mathematical Biosciences and Engineering, 17 (2019), 1757-1775. Google Scholar |
[24] |
S. Zhang, X. Meng and X. Wang,
Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function, Adv. Difference Equ., 2018 (2018), 50-71.
doi: 10.1186/s13662-018-1508-z. |
[25] |
T. Zhang and X.-Q. Zhao,
Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China, SIAM J. Appl. Dyn. Syst., 19 (2020), 1438-1471.
doi: 10.1137/19M1280259. |
[26] |
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. |
[27] |
D. Zhao,
Study on the threshold of a stochastic SIR epidemic model and its extensions, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 172-177.
doi: 10.1016/j.cnsns.2016.02.014. |
show all references
References:
[1] |
L. J. S. Allen,
An introduction to stochastic epidemic models, Mathematical epidemiology, Lecture Notes in Mathematics, 1945 (2008), 81-130.
doi: 10.1007/978-3-540-78911-6_3. |
[2] |
B. Berrhazi, M. El Fatini and A. Laaribi,
A stochastic threshold for an epidemic model with Beddington-DeAngelis incidence, delayed loss of immunity and Lévy noise perturbation, Phys. A, 507 (2018), 312-320.
doi: 10.1016/j.physa.2018.05.096. |
[3] |
T. Britton,
Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35.
doi: 10.1016/j.mbs.2010.01.006. |
[4] |
Y. Chen, B. Wen and Z. Teng,
The global dynamics for a stochastic SIS epidemic model with isolation, Phys. A, 492 (2018), 1604-1624.
doi: 10.1016/j.physa.2017.11.085. |
[5] |
K. Fan, Y. Zhang, S. Gao and et al,
A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Phys. A, 481 (2017), 198-208.
doi: 10.1016/j.physa.2017.04.055. |
[6] |
Z. Guo, M. Zhao and et al, Correction for life expectation of population and mortality of infant in Yunnan, Maternal and Child Health Care of China, 20 (2005), 681-685. Google Scholar |
[7] |
M. Y. Li and H. Shu,
Joint effects of mitosis and intracellular delay on viral dynamics: Two-parameter bifurcation analysis, J. Math. Biol., 64 (2012), 1005-1020.
doi: 10.1007/s00285-011-0436-2. |
[8] |
M. Liu,
Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays, IMA J. Appl. Math., 80 (2015), 1431-1453.
doi: 10.1093/imamat/hxv002. |
[9] |
Z. Liu,
Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates, Nonlinear Anal. Real World Appl., 14 (2013), 1286-1299.
doi: 10.1016/j.nonrwa.2012.09.016. |
[10] |
M. Liu, C. Bai and K. Wang,
Asymptotic stability of a two-group stochastic SEIR model with infinite delays, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3444-3453.
doi: 10.1016/j.cnsns.2014.02.025. |
[11] |
Q. Liu and Q. Chen,
Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Phys. A, 428 (2015), 140-153.
doi: 10.1016/j.physa.2015.01.075. |
[12] |
Q. Liu, D. Jiang, T. Hayat and et al,
Stationary distribution of a stochastic delayed SVEIR epidemic model with vaccination and saturation incidence, Phys. A, 512 (2018), 849-863.
doi: 10.1016/j.physa.2018.08.054. |
[13] |
X. Mao, G. Marion and E. Renshaw,
Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl., 97 (2002), 95-110.
doi: 10.1016/S0304-4149(01)00126-0. |
[14] |
X. Meng, S. Zhao, T. Feng and et al,
Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl., 433 (2016), 227-242.
doi: 10.1016/j.jmaa.2015.07.056. |
[15] |
B. Monel, E. Beaumont, D. Vendrame and et al, HIV cell-to-cell transmission requires the production of infectious virus particles, and does not proceed through Env-mediated fusion pores, Journal of Virology, 86 (2012), 3924-3933. Google Scholar |
[16] |
P. W. Nelson, M. A. Gilchrist, D. Coombs and et al,
An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.
doi: 10.3934/mbe.2004.1.267. |
[17] |
Yunnan Bureau of Satistic, The Sixth National Census Data Bulletin in Yunnan Province 2010, Available from: http://www.stats.gov.cn/tjsj/tjgb/rkpcgb/dfrkpcgb/201202/t20120228_30408.html. Google Scholar |
[18] |
The Data-center of China Public Health, Available from: http://www.phsciencedata.cn/. Google Scholar |
[19] |
E. Tornatore and S. M. Buccellato,
Parasite population delay model of malaria type with stochastic perturbation and environmental criterion for limitation of disease, J. Math. Anal. Appl., 360 (2009), 624-630.
doi: 10.1016/j.jmaa.2009.06.078. |
[20] |
J. E. Truscott and C. A. Gilligan,
Response of a deterministic epidemiological system to a stochastically varying environment, Proceedings of the National Academy of Sciences of the United States of America, 100 (2003), 9067-9072.
doi: 10.1073/pnas.1436273100. |
[21] |
D. Xiao and W. H. Bossert,
An intra-host mathematical model on interaction between HIV and malaria, Bull. Math. Biol., 72 (2010), 1892-1911.
doi: 10.1007/s11538-010-9515-6. |
[22] |
L. Zhang, W. T. Li, Z. Wang and et al,
Entire solutions for nonlocal dispersal equations with bistable nonlinearity: Asymmetric case, Acta Math. Sin. (Engl. Ser.), 35 (2019), 1771-1794.
doi: 10.1007/s10114-019-8294-8. |
[23] |
T. Zhang, H. Li, N. Xie and et al, Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China, Mathematical Biosciences and Engineering, 17 (2019), 1757-1775. Google Scholar |
[24] |
S. Zhang, X. Meng and X. Wang,
Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function, Adv. Difference Equ., 2018 (2018), 50-71.
doi: 10.1186/s13662-018-1508-z. |
[25] |
T. Zhang and X.-Q. Zhao,
Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China, SIAM J. Appl. Dyn. Syst., 19 (2020), 1438-1471.
doi: 10.1137/19M1280259. |
[26] |
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. |
[27] |
D. Zhao,
Study on the threshold of a stochastic SIR epidemic model and its extensions, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 172-177.
doi: 10.1016/j.cnsns.2016.02.014. |



Year | 2007 | 2008 | 2009 | 2010 | 2011 |
HIV/AIDS | 57325 | 64460 | 71852 | 78613 | 85999 |
Gonorrhea | 2358 | 2230 | 1818 | 1819 | 1720 |
Year | 2012 | 2013 | 2014 | 2015 | 2016 |
HIV/AIDS | 92666 | 98555 | 104903 | 111351 | 117817 |
Gonorrhea | 1893 | 1643 | 2104 | 3028 | 4098 |
Year | 2007 | 2008 | 2009 | 2010 | 2011 |
HIV/AIDS | 57325 | 64460 | 71852 | 78613 | 85999 |
Gonorrhea | 2358 | 2230 | 1818 | 1819 | 1720 |
Year | 2012 | 2013 | 2014 | 2015 | 2016 |
HIV/AIDS | 92666 | 98555 | 104903 | 111351 | 117817 |
Gonorrhea | 1893 | 1643 | 2104 | 3028 | 4098 |
Parameters | Definition | Value | Source |
A | Recruitment rate for the susceptible population | 92136 | Estimated |
d | Natural mortality rate | 0.0149 | [6] |
Death rate for HIV/AIDS | 0.7114 | [26] | |
Death rate for Gonorrhea | 0.3 | Estimated | |
Cure rate for HIV/AIDS | 0.79 | Estimated | |
Cure rate for Gonorrhea | 0.99994 | Estimated | |
Infection rate for HIV/AIDS | 0.9 | Estimated | |
Infection rate for Gonorrhea | 0.25 | Estimated | |
Inhibition rate of HIV/AIDS on transmission | 0.9 | Estimated | |
Inhibition rate of Gonorrhea on transmission | 1 | Estimated | |
Incubation period of AIDS | 8 year | [26] | |
Incubation period of Gonorrhea | 0 | [18] | |
Initial value of susceptible population | 80000 | Estimated | |
Initial value of HIV/AIDS patients | 57325 | [26] | |
Initial value of Gonorrhea patients | 12358 | Estimated |
Parameters | Definition | Value | Source |
A | Recruitment rate for the susceptible population | 92136 | Estimated |
d | Natural mortality rate | 0.0149 | [6] |
Death rate for HIV/AIDS | 0.7114 | [26] | |
Death rate for Gonorrhea | 0.3 | Estimated | |
Cure rate for HIV/AIDS | 0.79 | Estimated | |
Cure rate for Gonorrhea | 0.99994 | Estimated | |
Infection rate for HIV/AIDS | 0.9 | Estimated | |
Infection rate for Gonorrhea | 0.25 | Estimated | |
Inhibition rate of HIV/AIDS on transmission | 0.9 | Estimated | |
Inhibition rate of Gonorrhea on transmission | 1 | Estimated | |
Incubation period of AIDS | 8 year | [26] | |
Incubation period of Gonorrhea | 0 | [18] | |
Initial value of susceptible population | 80000 | Estimated | |
Initial value of HIV/AIDS patients | 57325 | [26] | |
Initial value of Gonorrhea patients | 12358 | Estimated |
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