Optimal Control of an HIV/AIDS Epidemic Model with Infective Immigration and Behavioral Change
- 1 College of Mathematics and System Sciences, Xinjiang University, Urumqi, China
- 2 College of Mathematics and System Sciences, Xinjiang University, Urumqi, China
Abstract
In order to find out the effect of human (sexual) behavior change and immigration in spreading the HIV/AIDS, a deterministic model of HIV/AIDS with infective immigration is formulated. First, basic properties of the model, including non-negativity and boundedness of the solutions, existence of the endemic equilibrium and the basic reproduction number, R 0 are analyzed. The geometrical approach is used to obtain the global asymptotic stability of endemic equilibrium. Then the basic model is extended to include several control efforts aimed at reducing infection and changing behavior. Pontryagin’s maximum principle is used to derive the optimality system and solve the system numerically. Our numerical findings are illustrated through simulations using MATLAB, which shows reliability of our model from the practical point of view.
- Seidu, B. and Makinde, O.D. (2014) Optimal Control of HIV/AIDS in the Workplace in the Presence of Careless Individuals. Computational & Mathematical Methods in Medicine, 2014, 119-128.
- Sharma, S. and Samanta, G.P. (2014) Dynamical Behaviour of an HIV/AIDS Epidemic Model. Differential Equations and Dynamical Systems, 22, 369-395. https://doi.org/10.1007/s12591-013-0173-7
- Okosun, K.O., Makinde, O.D. and Takaidza, I. (2013) Impact of Optimal Control on the Treatment of HIV/AIDS and Screening of Unaware Infectives. Applied Mathematical Modelling, 37, 3802-3820. https://doi.org/10.1016/j.apm.2012.08.004
- Yusuf, T.T. and Benyah, F. (1969) Optimal Strategy for Controlling the Spread of HIV/AIDS Disease: A Case Study of South Africa. Journal of Biological Dynamics, 6, 475-494. https://doi.org/10.1080/17513758.2011.628700
- Garnett, G.P. and Anderson, R.M. (1996) Antiviral Therapy and the Transmission Dynamics of HIV-1. Journal of Antimicrobial Chemotherapy, 37, 135-50. https://doi.org/10.1093/jac/37.suppl_B.135
- Castillo-Chavez, C. and Feng, Z. (1998) Global Stability of an Age-Structure Model for TB and Its Applications to Optimal Vaccination Strategies. Mathematical Biosciences, 151, 135-154. https://doi.org/10.1016/S0025-5564(98)10016-0
- Li, G.H., Wang, W.D. and Jin, Z. (2006) Global Stability of an SEIR Epidemic Model with Constant Immigration. Chaos, Solitons and Fractals, 30, 1012-1019. https://doi.org/10.1016/j.chaos.2005.09.024
- Naresh, R., Tripathi, A. and Sharma, D. (2009) Modelling and Analysis of the Spread of AIDS Epidemic with Immigration of HIV Infectives. Mathematical and Computer Modelling, 49, 880-892. https://doi.org/10.1016/j.mcm.2008.09.013
- Tripathi, A., Naresh, R., Tchuenche, J.M. and Sharma, D. (2013) Modeling the Spread of HIV-AIDS with Infective Immigrants and Time Delay. International Journal of Nonlinear Science, 16, 313-322.
- Coffeea, M., Lurieb, M.N. and Garnett, G.P. (2007) Modelling the Impact of Migration on the HIV Epidemic in South Africa. AIDS, 21, 343-350. https://doi.org/10.1097/QAD.0b013e328011dac9
- Brauer, F. and van den Driessche, P. (2001) Models for Tranmission of Disease with Immigration of Infectives. Mathematical Biosciences, 171, 143-154. https://doi.org/10.1016/S0025-5564(01)00057-8
- Tumwiine, J., Mugisha, J.Y.T. and Luboobi, L.S. (2010) A Host-Vector Model for Malaria with Infective Immigrants. Journal of Mathematical Analysis & Applications, 361, 139-149. https://doi.org/10.1016/j.jmaa.2009.09.005