Abstract
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 < R 0 < 1 but unstable if R 0 > 1 . However, the endemic equilibrium is locally and globally asymptotically stable if R 0 > 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 β 2 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 β 2 = 0.1 . On the other hand, as we change the value of β 2 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.
Highlights
Human immunodeficiency virus (HIV) is a retrovirus that attacks the human immune system and causes a highly killing disease called acquired immunodeficiency syndrome (AIDS) [1,2,3,4,5,6]
The endemic equilibrium is locally and globally asymptotically stable if R0 > 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 (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
The optimal system is solved using forward and backward Runge–Kutta fourth-order methods. e 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. e cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective
Summary
Human immunodeficiency virus (HIV) is a retrovirus that attacks the human immune system and causes a highly killing disease called acquired immunodeficiency syndrome (AIDS) [1,2,3,4,5,6]. 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 immunodeficiency virus, different protective and treatment strategies are used [6]. Different control strategies are used to eradicate and combat the Journal of Mathematics transmission of HIV in human population, the viruses still become one of the global issues that need attention to save human population from this merciless killing disease. We extended the classical SIA model of HIV to the SWIA model of HIV with optimal control problem to identify the best control measures that reduces the transmission and progression of the human immunodeficiency virus along with minimum cost. Mathematical models are efficient tools to describe and predict the transmission dynamics of disease in the population [8, 9]
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