| Citation: |
Lin Hu, Lin-Fei Nie. THE EFFECTS OF DELAY AND IMPULSIVE DRUG THERAPY IN AN HIV MODEL WITH CTLS IMMUNE RESPONSE[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 333-350. doi: 10.11948/20190418
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THE EFFECTS OF DELAY AND IMPULSIVE DRUG THERAPY IN AN HIV MODEL WITH CTLS IMMUNE RESPONSE
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Abstract
Considering the effects of immune response and drug therapy on HIV treatment, an HIV mathematical control model with CTLs immune response is therefore proposed, where the delay of virus invasion and impulsive drug therapy are introduced. By utilizing the comparison theorem, differential inequality theories and analytic method, the threshold values for the existence and global stability of the virus-free periodic solution, and the uniform persistence of disease without CTLs immune response are studied. Numerical simulations are performed to illustrate the main theoretical results and the feasibility of drug therapy. Our theoretical results suggest that long-term and standardized medication can prolong the infection process and spread of the virus, or suppress the virus concentration below the detectable level. -
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References
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Lin Hu, Lin-Fei Nie. THE EFFECTS OF DELAY AND IMPULSIVE DRUG THERAPY IN AN HIV MODEL WITH CTLS IMMUNE RESPONSE[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 333-350. doi: 10.11948/20190418
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Figure 1. The effect of delay and drug for HIV models (2.3) and (2.4) with
$ \lambda = 10 $ ,$ d = 0.02 $ ,$ r = 0.1 $ ,$ T_{max} $ ,$ \delta = 0.25 $ ,$ N = 1000 $ ,$ \beta = 2.5\times 10^{-5} $ ,$ c = 0.24 $ ,$ k = 1 $ ,$ \omega = 3.6 $ ,$ m = 0.36 $ ,$ p = 0.02 $ ,$ p = 0.12 $ ,$ d_E = 0.2 $ ,$ \theta = 10 $ ,$ \tau = 1 $ and the drug dosage$ D(0) = D_{in} = 10(mg) $ , where blue curves, green curves and red curves represent (2.3) and (2.4) without delay and drug therapy, with delay and without drug therapy, and with delay and drug therapy, respectively -
Figure 2. The existence and stability of the virus-free periodic solution of HIV models (2.3) and (2.4) with
$ \lambda = 1 $ ,$ d = 0.02 $ ,$ T_{max} $ ,$ \delta = 0.24 $ ,$ N = 1000 $ ,$ \beta = 2.4\times10^{-5} $ ,$ c = 0.24 $ ,$ \omega = 3.6 $ ,$ k = 1 $ ,$ m = 0.36 $ and$ \theta = 2 $ ,$ D(0) = D_{in} = 16(mg) $ and$ \tau = 0.5 $ , where the blue curves and red curves represent models (2.3) and (2.4) with drug therapy and without drug therapy, respectively -
Figure 3. The existence and stability of the endemic periodic solution of HIV models (2.3) and (2.4) with
$ \lambda = 5 $ ,$ d = 0.02 $ ,$ T_{max} = 250 $ ,$ \delta = 0.24 $ ,$ N = 1000 $ ,$ \beta = 3.6\times10^{-5} $ ,$ c = 0.24 $ ,$ k = 1 $ ,$ \omega = 3.6 $ ,$ m = 0.36 $ and$ \theta = 2 $ ,$ \tau = 1.5 $ and$ D(0) = D_{in} = 16(mg) $ -
Figure 4. The permanence and extinction of HIV models (2.3) and (2.4) with different drug therapy periodic,
$ \lambda = 4 $ ,$ d = 0.02 $ ,$ T_{max} = 200 $ ,$ \delta = 0.24 $ ,$ N = 1000 $ ,$ \beta = 2.4\times10^{-5} $ ,$ c = 0.24 $ ,$ \omega = 2.3 $ ,$ m = 0.36 $ and$ \theta = 2 $ and$ D(0) = D_{in} = 16(mg) $ , where, (a)-(c): the blue lines are$ \tau = 1.5 $ and the red lines are$ \tau = 0.4 $ ; (d)$ \tau = 2.5 $ ,$ 1.5 $ ,$ 0.5 $ ,$ 0.45 $ and$ 0.4 $ , respectively