Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control
A mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability a...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2021-01-01
|
| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/7471290 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832566418942459904 |
|---|---|
| author | Kumama Regassa Cheneke Koya Purnachandra Rao Geremew Kenassa Edessa |
| author_facet | Kumama Regassa Cheneke Koya Purnachandra Rao Geremew Kenassa Edessa |
| author_sort | Kumama Regassa Cheneke |
| collection | DOAJ |
| description | A mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability analysis result shows that the disease-free equilibrium is locally asymptotically stable if 0<R0 <1 but unstable if R0>1. However, the endemic equilibrium is locally and globally asymptotically stable if R0>1 and unstable otherwise. The sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate β2 of HIV transmission from HIV individuals to susceptible individuals and the parameter that contributes to the decreasing of the reproduction number is identified as progression rate η of HIV-infected individuals to AIDS individuals. Furthermore, it is observed that as we change η from 0.1 to 1, the reproduction number value decreases from 1.205 to 1.189, where the constant value of β2=0.1. On the other hand, as we change the value of β2 from 0.1 to 1, the value of the reproduction number increases from 0.205 to 1.347, where the constant value of η=0.1. Further, the developed model is extended to the optimal control model of HIV/AIDS transmission, and the cost-effectiveness of the control strategy is analyzed. Pontraygin’s Maximum Principle (PMP) is applied in the construction of the Hamiltonian function. Moreover, the optimal system is solved using forward and backward Runge–Kutta fourth-order methods. The numerical simulation depicts the number of newly infected HIV individuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. The cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective. MATLAB is used to run numerical simulations. |
| format | Article |
| id | doaj-art-8c6547e2f2b24966911b175b25e1bfa1 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-8c6547e2f2b24966911b175b25e1bfa12025-02-03T01:04:12ZengWileyJournal of Mathematics2314-47852021-01-01202110.1155/2021/7471290Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal ControlKumama Regassa Cheneke0Koya Purnachandra Rao1Geremew Kenassa Edessa2Department of MathematicsDepartment of MathematicsDepartment of MathematicsA mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability analysis result shows that the disease-free equilibrium is locally asymptotically stable if 0<R0 <1 but unstable if R0>1. However, the endemic equilibrium is locally and globally asymptotically stable if R0>1 and unstable otherwise. The sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate β2 of HIV transmission from HIV individuals to susceptible individuals and the parameter that contributes to the decreasing of the reproduction number is identified as progression rate η of HIV-infected individuals to AIDS individuals. Furthermore, it is observed that as we change η from 0.1 to 1, the reproduction number value decreases from 1.205 to 1.189, where the constant value of β2=0.1. On the other hand, as we change the value of β2 from 0.1 to 1, the value of the reproduction number increases from 0.205 to 1.347, where the constant value of η=0.1. Further, the developed model is extended to the optimal control model of HIV/AIDS transmission, and the cost-effectiveness of the control strategy is analyzed. Pontraygin’s Maximum Principle (PMP) is applied in the construction of the Hamiltonian function. Moreover, the optimal system is solved using forward and backward Runge–Kutta fourth-order methods. The numerical simulation depicts the number of newly infected HIV individuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. The cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective. MATLAB is used to run numerical simulations.http://dx.doi.org/10.1155/2021/7471290 |
| spellingShingle | Kumama Regassa Cheneke Koya Purnachandra Rao Geremew Kenassa Edessa Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control Journal of Mathematics |
| title | Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control |
| title_full | Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control |
| title_fullStr | Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control |
| title_full_unstemmed | Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control |
| title_short | Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control |
| title_sort | bifurcation and stabillity analysis of hiv transmission model with optimal control |
| url | http://dx.doi.org/10.1155/2021/7471290 |
| work_keys_str_mv | AT kumamaregassacheneke bifurcationandstabillityanalysisofhivtransmissionmodelwithoptimalcontrol AT koyapurnachandrarao bifurcationandstabillityanalysisofhivtransmissionmodelwithoptimalcontrol AT geremewkenassaedessa bifurcationandstabillityanalysisofhivtransmissionmodelwithoptimalcontrol |