| Citation: |
Hai-Feng Huo, Li-Na Gu, Hong Xiang. MODELLING AND ANALYSIS OF AN HIV/AIDS MODEL WITH DIFFERENT WINDOW PERIOD AND TREATMENT[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1927-1950. doi: 10.11948/20200279
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MODELLING AND ANALYSIS OF AN HIV/AIDS MODEL WITH DIFFERENT WINDOW PERIOD AND TREATMENT
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Abstract
A novel HIV$ / $AIDS model with different window period and treatment is proposed. Window period individuals and latent individuals are divided into two types: treated and untreated in our model. The basic reproduction number $ R_{0} $ is obtained by using the next generation matrix. Stability of the disease-free equilibrium and existence of the endemic equilibrium is derived. Using the theory of central manifold, the generation of forward bifurcation is established. Existence of the optimal control pair is analyzed and the mathematical expression of the optimal control is also given by the Pontryagin maximum principle. The best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the AIDS data in Gansu province of China from 2004 to 2019. We also estimate that the basic reproduction number $ R_{0} $ is $ 2.1985 $ (95%CI: (1.3535-3.0435)). Numerical simulation and sensitivity analysis of several parameters are also presented. Our results suggest that individuals who are in the stage of window period for AIDS should receive treatment, and this plays a key role in the prevention and control of HIV$ / $AIDS.
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Keywords:
- Treatment /
- window period /
- forward bifurcation /
- optimal control
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References
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Hai-Feng Huo, Li-Na Gu, Hong Xiang. MODELLING AND ANALYSIS OF AN HIV/AIDS MODEL WITH DIFFERENT WINDOW PERIOD AND TREATMENT[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1927-1950. doi: 10.11948/20200279
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- Figure 1. Transfer diagram for the dynamics of HIV/AIDS.
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Figure 2. Disease free equilibrium
$ P_{0} $ is globally asymptotically stable. -
Figure 3. Endemic equilibrium
$ P^{*} $ is locally asymptotically stable. -
Figure 4. Illustration of forward bifurcation when one parameter
$ \beta_{1} $ in$ R_{0} $ is varied. - Figure 5. The actual number of people infected in Gansu Province from 2004 to 2019.
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Figure 6. (a) The number of Markov chain samples of
$ R_{0} $ . The curve represents the size of the$ R_{0} $ value. (b) The frequency distribution of$ R_{0} $ . The red curve is the probability density function curve of$ R_{0} $ . -
Figure 7. The Partial Rank Coefficients (PRCCs) of
$ R_{0} $ in system (2.1). -
Figure 8. The variation trend between four parameters and the basic regeneration number
$ R_{0} $ . - Figure 9. The sensitivity of several parameters changes on infected individuals.
- Figure 10. P values of each parameter in the 12th year.
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Figure 11. Number of people in different compartment when we select the weights on objective function are
$ c_{1} = 100; c_{2} = 1000 $ and under different optimal control strategies, that is, (1) without$ u_{1} = 0, u_{2} = 0 $ ; (2) single control$ u_{1} = 0.5, u_{2} = 0 $ ; (3) single control$ u_{1} = 0, u_{2} = 0.5 $ ; (4) middle control$ u_{1} = u_{2} = 0.5 $ ; (5) optimal control$ u_{1} = u_{1}^{*}, u_{2} = u_{2}^{*} $ . -
Figure 12. The impact of different weight coefficients on optimal control
$ u_{1}, u_{2} $ : (a)$ c_{1} = 100, c_{2} = 1000 $ ; (b)$ c_{1} = 1000, c_{2} = 100 $ .