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March  2021, 29(1): 1661-1679. doi: 10.3934/era.2020085

Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China

Zhimin Li 1, , Tailei Zhang 1,, and  Xiuqing Li 2,

1. 

School of Science, Chang'an University, Xi'an 710064, China
2. 

College of Economics and Management, Shanxi Normal University, Linfen 041004, China

* Corresponding author: Tailei Zhang

Received  February 2020 Revised  July 2020 Published  August 2020

Fund Project: Supported by the Fundamental Research Funds for the Central Universities, CHD (Grant No. 300102129202), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JM1011) and the National Natural Science Foundation of China (Grant No. 11701041)

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.

Keywords: Stochastic delay differential equation, threshold, permanence, extinction.
Mathematics Subject Classification: 34F05, 60H10, 92B05.
Citation: Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

T. Britton, Stochastic epidemic models: A survey, Math. Biosci., 225 (2010), 24-35.  doi: 10.1016/j.mbs.2010.01.006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The model (2.2.2) is simulated by the parameters values in Table 2, and compared with the HIV/AIDS and Gonorrhea data in Yunnan Province from 2007 to 2016
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Figure 2.  Partial rank correlation coefficients(PRCCs) results for the dependence of $ \mathcal R_i^* $ on each parameter
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Figure 3.  When $ \mathcal R_1 = 1.2729 > 1 > \mathcal R_1^* = 0.9672, $ model (2.2.1) describes HIV/AIDS infection $ I_1 $ will be persistent, but stochastic differential equation with time delay model (2.2.2) describes HIV/AIDS infection $ I_1 $ will be extinct
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Figure 4.  When $ \mathcal R_2 = 1.7050 > 1 > \mathcal R_2^* = 0.4189, $ model (2.2.1) describes Gonorrhea infection $ I_2 $ will be persistent, but stochastic differential equation with time delay model (2.2.2) describes Gonorrhea infection $ I_2 $ will be extinct
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Table 1.  Cumulative total of reported HIV/AIDS cases and the number of Gonorrhea infections increased annually from 2007 to 2016 in Yunnan Province, China (see [26,18])
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
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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
Table 2.  Parameters and numerical values chosen for the simulation
Parameters Definition Value Source
A Recruitment rate for the susceptible population 92136 Estimated
d Natural mortality rate 0.0149 [6]
$ {\alpha _1 } $ Death rate for HIV/AIDS 0.7114 [26]
$ {\alpha _2 } $ Death rate for Gonorrhea 0.3 Estimated
$ {r_1 } $ Cure rate for HIV/AIDS 0.79 Estimated
$ {r_2 } $ Cure rate for Gonorrhea 0.99994 Estimated
$ {\beta _1 } $ Infection rate for HIV/AIDS 0.9 Estimated
$ {\beta _2 } $ Infection rate for Gonorrhea 0.25 Estimated
$ {a_1 } $ Inhibition rate of HIV/AIDS on transmission 0.9 Estimated
$ {a_2 } $ Inhibition rate of Gonorrhea on transmission 1 Estimated
$ {\tau_1 } $ Incubation period of AIDS 8 year [26]
$ {\tau_2 } $ Incubation period of Gonorrhea 0 [18]
$ {S(0)} $ Initial value of susceptible population 80000 Estimated
$ {I_1 (0)} $ Initial value of HIV/AIDS patients 57325 [26]
$ {I_2 (0)} $ Initial value of Gonorrhea patients 12358 Estimated
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Parameters Definition Value Source
A Recruitment rate for the susceptible population 92136 Estimated
d Natural mortality rate 0.0149 [6]
$ {\alpha _1 } $ Death rate for HIV/AIDS 0.7114 [26]
$ {\alpha _2 } $ Death rate for Gonorrhea 0.3 Estimated
$ {r_1 } $ Cure rate for HIV/AIDS 0.79 Estimated
$ {r_2 } $ Cure rate for Gonorrhea 0.99994 Estimated
$ {\beta _1 } $ Infection rate for HIV/AIDS 0.9 Estimated
$ {\beta _2 } $ Infection rate for Gonorrhea 0.25 Estimated
$ {a_1 } $ Inhibition rate of HIV/AIDS on transmission 0.9 Estimated
$ {a_2 } $ Inhibition rate of Gonorrhea on transmission 1 Estimated
$ {\tau_1 } $ Incubation period of AIDS 8 year [26]
$ {\tau_2 } $ Incubation period of Gonorrhea 0 [18]
$ {S(0)} $ Initial value of susceptible population 80000 Estimated
$ {I_1 (0)} $ Initial value of HIV/AIDS patients 57325 [26]
$ {I_2 (0)} $ Initial value of Gonorrhea patients 12358 Estimated
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