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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

 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  March 2021 Early access  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.

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
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##### References:
, and compared with the HIV/AIDS and Gonorrhea data in Yunnan Province from 2007 to 2016">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
Partial rank correlation coefficients(PRCCs) results for the dependence of $\mathcal R_i^*$ on each parameter
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
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
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
 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 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
 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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