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2024, Volume 29, Issue 2: 702-730. Doi: 10.3934/dcdsb.2023110

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A stochastic HIV/HTLV-I co-infection model incorporating the AIDS-related cancer cells

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China
2.
School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

^*Corresponding author: Weipeng Zhang
^*Corresponding author: Weipeng Zhang

Received: January 2023

Revised: May 2023

Early access: June 2023

Published: February 2024

The second author is supported by the National Natural Science Foundation of PR China (No.11971096), the National Key R & D Program of PR China (2020YFA0714102), the Natural Science Foundation of Jilin Province(No.YDZJ202101ZYTS154), and the Fundamental Research Funds for the Central Universities

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Abstract

This paper proposes a stochastic HIV/HTLV-I co-infection model incorporating the AIDS-related cancer cells and investigates its dynamical behaviours. The main methods are stochastic Lyapunov analysis and the ergodic theory. The existence and uniqueness of the global positive solution is proved as well as the stochastic ultimate boundedness. A unique stationary distribution is yielded. Furthermore, the sufficient conditions for the extinction of diseases are given. Finally, some numerical simulations are carried out to support the theoretical results.

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Mathematics Subject Classification: Primary: 60H10, 65C20.

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Figure 1. The action mechanism of model (3)

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Figure 2. Time evolution of $ C(t) $, $ H(t) $, $ I_{1}(t) $ and $ I_{2}(t) $ obtained from numerical simulation for the deterministic model (2)

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Figure 3. Ergodicity stationary distribution of model (3) with $ \sigma_{1} = 0.015 $, $ \sigma_{2} = 0.015 $, $ \sigma_{3} = 0.015 $ and $ \sigma_{4} = 0.015 $

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Figure 4. The sample path of $ C(t) $, $ \langle\langle C\rangle\rangle_t $, $ H(t) $, $ \langle\langle H\rangle\rangle_t $, $ I_{1}(t) $ and $ I_{2}(t) $ with $ \sigma_{1} = 0.025 $, $ \sigma_{2} = 0.02 $, $ \sigma_{3} = 0.2 $ and $ \sigma_{4} = 0.45 $, respectively

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Figure 5. The sample path of $ C(t) $, $ \langle\langle C\rangle\rangle_t $, $ H(t) $, $ \langle\langle H\rangle\rangle_t $, $ I_{1}(t) $ and $ I_{2}(t) $ with $ \sigma_{1} = \sqrt{0.2} $, $ \sigma_{2} = 0.02 $, $ \sigma_{3} = 0.3 $ and $ \sigma_{4} = 0.5 $, respectively

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Figure 6. The sample path of $ C(t) $, $ H(t) $, $ \langle\langle H\rangle\rangle_t $, $ I_{1}(t) $ and $ I_{2}(t) $ with $ \sigma_{1} = 0.5 $, $ \sigma_{2} = 0.02 $, $ \sigma_{3} = 0.2 $ and $ \sigma_{4} = 0.45 $, respectively. The distribution of $ H(t) $

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Table 1. The significance of parameters in model (2)

Parameter	Significance
$ r_{1} $	the intrinsic growth rate of cancer cells
$ r_{2} $	the intrinsic growth rate of healthy CD4+ T cells
$ e_{1} $	the killing rate of healthy CD4+ T cells
$ e_{2} $	the losing rate of healthy CD4+ T cells
$ k $	the HIV infection rate
$ q $	the HTLV-I infection rate
$ \frac{1}{h_{1}} $	the maximum carrying capacity of cancer cells
$ \frac{1}{h_{2}} $	the maximum carrying capacity of healthy CD4+ T cells
$ \mu_{1} $	the death rates of HIV-infected CD4+ T cells
$ \mu_{2} $	the death rates of HTLV-infected CD4+ T cells
$ \alpha_{1} $, $ \alpha_{2} $, $ \beta_{1} $, $ \beta_{2} $	positive constants

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Table 2. The parameters and their values in model (3)

Parameter	Value	Source
$ r_{1} $	$ 10^{-1} $	[47]
$ r_{2} $	$ 3\times 10^{-2} $	[44]
$ e1 $	$ 10^{-5} $	[44]
$ e2 $	$ 10^{-6} $	[44,47]
$ k $	$ 10^{-4} $	[53]
$ \alpha_{1} $	$ 5\times 10^{-3} $	Estimated
$ \beta_{1} $	$ 5\times10^{-2} $	Estimated
$ h_{1} $	$ \frac{1}{700} $	Estimated
$ h_{2} $	$ 1.25\times 10^{-3} $	Estimated
$ \mu_{1} $	$ 0.0015 $	Estimated
$ q $	$ 10^{-4} $	Estimated
$ \alpha_{2} $	$ 10^{-3} $	Estimated
$ \beta_{2} $	$ 10^{-2} $	[24]
$ \mu_{2} $	$ 2\times 10^{-2} $	[59]

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