Research Article Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control KumamaRegassaCheneke ,KoyaPurnachandraRao ,andGeremewKenassaEdessa Department of Mathematics, Wollega University, Nekemte, Ethiopia Correspondence should be addressed to Kumama Regassa Cheneke; kumamaregassa@gmail.com Received 30 June 2021; Revised 4 August 2021; Accepted 26 November 2021; Published 20 December 2021 Academic Editor: Kenan Yildirim Copyright © 2021 Kumama Regassa Cheneke et al. is is an open access article distributed under the Creative Commons AttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkis properly cited. AmathematicalmodelofHIVtransmissionisbuiltandstudiedinthispaper.esystem’sequilibriumiscalculated.Anext- generationmatrixisusedtocalculatethereproductionnumber.enovelmethodisusedtoexaminethedevelopedmodel’s bifurcation and equilibrium stability. e stability analysis result shows that the disease-free equilibrium is locally as- ymptotically stable if 0 < R 0 < 1 but unstable if R 0 > 1. However, the endemic equilibrium is locally and globally asymp- totically stable if R 0 > 1 and unstable otherwise. e sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate (β 2 ) of HIV transmission from HIV in- dividuals 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 from0.1to1,thereproductionnumbervaluedecreasesfrom1.205to1.189,wheretheconstantvalueof β 2 � 0.1. Ontheotherhand,aswechangethevalueof β 2 from0.1to1,thevalueofthereproductionnumberincreasesfrom0.205to 1.347,wheretheconstantvalueof η � 0.1.Further,thedevelopedmodelisextendedtotheoptimalcontrolmodelofHIV/ AIDStransmission,andthecost-effectivenessofthecontrolstrategyisanalyzed.Pontraygin’sMaximumPrinciple(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. e numerical simulation depicts the number of newly infected HIV in- dividuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. e cost-ef- fectivenessstudydemonstratesthatwhencombinedandused,thepreventativeandtreatmentcontrolmeasuresareeffective. MATLAB is used to run numerical simulations. 1. Introduction Human immunodeficiency virus (HIV) is a retrovirus that attacks the human immune system and causes a highly killing disease called acquired immunodeficiency syndrome (AIDS) [1–6]. HIV was discovered in the early 1980, and it hasbeenpersistinginthepopulation[2].HIVistransmitted through unsafe sex, blood transfusion, breast feeding, ma- terials exposed to the virus, and mother to child during pregnancy [3]. Currently, there is no curing treatment for HIV-infectedindividuals[4].In2018,thenumberofhuman individuals living with HIV is estimated to be 37.9 million and the number of dead individuals with AIDS-related disease is 1.2 million. Among HIV-infected individuals, about62%aretestedandtakingantiretrotherapy(ART)[4]. e data indicate that Africa is the continent that is highly exposedtothehumanimmunodeficiencyvirusintheworld. Particularly in 2018, Ethiopia has about 690,000 peoples living with HIV, 23,000 new people are infected with HIV, and 11,000 individuals dead with AIDS-related disease [5]. To control the transmission of the human immunodefi- ciencyvirus,differentprotectiveandtreatmentstrategiesare used [6]. Some of the strategies used in controlling the transmission and progression of HIV are using condom, be faithful, abstaining, and ART [7]. Even though, different control strategies are used to eradicate and combat the Hindawi Journal of Mathematics Volume 2021, Article ID 7471290, 14 pages https://doi.org/10.1155/2021/7471290