Document Type : Research Paper

Authors

1 Department of Mathematics, Bhabha Institute of Technology, Kanpur-209204, India

2 Department of Mathematics, Harcourt Butler Technological Institute, Kanpur-208002, India

Abstract

The most urgent public health problem today is to devise effective strategies to minimize the destruction caused by the AIDS epidemic. Mathematical models based on the underlying transmission mechanisms of the AIDS virus can help the medical/scientific community understand and anticipate its spread in different populations and evaluate the potential effectiveness of different approaches for bringing the epidemic under control. In this paper, we present the framework of conventional compartmental models for the spread of HIV infection to investigate the effect of various types of growths of host population. The model presented has been studied qualitatively using stability theory of differential equations. The equilibrium and stability analysis have been carried out by establishing local and global stability results and some inferences have been drawn to understand the spread of the disease. A numerical study in each case is also performed to see the influence of certain parameters on the disease spread and to support the analytical results. The model analysis has also been applied to compare the theoretical results with the known Indian HIV data.

Keywords

Main Subjects

References
[1] Anderson R. M., and May R. M., Population biology of infectious diseases I, Nature,180, 361-379, 1979.
[2] Anderson R. M., Jackson H. C., May R.M., and Smith A. D. M., Population dynamics of fox rabies in Europe. Nature, 289, 765-777, 1981.
[3] Bailey N. T.J., The mathematical theory of infectious diseases (2nd ed.). Macmillan, New York, 1975.
[4] Birkhoff G., and Rota G. C., Ordinary differential equations, Ginn (1982).
[5] Brauer F., and van den Driessche P., Models for transmission of disease with immigration of infectives. Math. Biosci., 171, 143-154, 2001.
[6] Brauer F., Models for the spread of universally fatal diseases. J. Math. Biol., 28, 451-462, 1990.
[7] Bremermann H. J., and Thieme H. R., A competitive exclusion principle for pathogen virulence. J. Math. Biol., 27, 179-190, 1989.
[8] Busenberg S. N., and Hadeler K.P., Demography and epidemics. Math. Biosci., 101, 41-62, 1990.
[9] Busenberg S. N., and van den Driessche P., Analysis of a disease transmission model in a population with varying size. J. Math. Biol., 28, 257-270, 1990.
[10] Fan M., Li M. Y., and Wang K., Global stability of an SEIS epidemic model with recruitment and varying total population size. Math. Biosci., 170, 199-208, 2001.
[11] Gao L. Q., and Hethcote H. W., Disease transmission models with density dependent demographics. J. Math. Biol., 30, 717-731, 1992.
[12] Hyman J. M., and Stanley E. A., Using mathematical models to understand the AIDS epidemic. Math. Biosci., 90, 415-473, 1988.
[13] Jacquez J. A., Simon C. P., Koopman J., Sattenspiel L., and Perry T., Modeling and analyzing HIV transmission: The effect of contact patterns. Math. Biosci., 92, 119-199, 1988.
[14] Kribs-Zaleta C. M., Lee M., Roman C., Wiley S., and Hernandez-Suarez C. M., The effect of the HIV/AIDS epidemic on Africa’s truck drivers. Math. Biosci. Eng., 2, 771-788, 2005.
[15] Massad E., A homogeneously mixing population model for the AIDS epidemic. Math. Comp. Model., 12, 89-96, 1989.
[16] May R. M., and Anderson R. M., Population biology of infectious diseases II. Nature, 280, 455-461, 1979.
[17] May R. M., and Anderson R. M., Transmission dynamics of HIV infection. Nature, 3426, 137-142, 1987.
[18] May R. M., Anderson R. M., and McLean A. R., Possible demographic consequences of HIV/AIDS epidemics. Math. Biosci., (1988), 90, 475-505.
[19] Mena-Lorca J., and Hethcote H. W., Dynamic models of infectious diseases as regulators of population size. J. Math. Biol., 30, 693-716, 1992.
[20] Naresh R., and Tripathi A., Modeling and analysis of HIV-TB co-infection in a variable size population. Math. Model. Anal., 10, 275-286, 2005.
[21]  Naresh R., Omar S., and Tripathi A., Modelling and analysis of HIV/AIDS in a variable size population. Far East J. Appl. Math., 18, 345-360, 2005.
[22] Naresh R., Tripathi A., and Omar S., Modelling the spread of AIDS epidemic with vertical transmission. Appl. Math. Comp., 178, 262-272, 2006.
[23] National AIDS Control Organisation: Country Scenario AIDS, Published by NACO, Ministry of Health, Govt. of India, New Delhi, 2006.
[24] Pugliese A., Population models for disease with no recovery. J. Math. Biol., 28, 65-82, 1990.
[25] Tripathi A., Naresh R., and Sharma D., Modelling the effect of screening of unaware infectives on the spread of HIV infection. Appl. Math. Comp., 184, 1053-1068, 2007.