Modeling HIV In The Presence Of Infected Immigrants And Vertical Transmission: The Role Of Incidence Function
[Full Text]
AUTHOR(S)
Ochoche, Jeffrey M.
KEYWORDS
Keywords: Basic Reproduction number, Epidemiology, HIV, Incidence function, Infected immigrants, Stability, Vertical transmission
ABSTRACT
Abstract: We formulated three mathematical models for the transmission dynamics of HIV in the presence of infected immigrants and vertical transmission using a deterministic approach. Three forms of incidences popularly used in epidemiology were considered: The mass action, Standard and nonlinear/saturated incidences. A Basic Reproduction number was derived for each model and used to prove its local stability. All the models were found to be globally asymptotically stable at the diseasefree equilibrium. The mass action incidence and the non  linear incidence yielded similar analytical results e.g their Basic Reproduction numbers are identical and greater than the Basic Reproduction number of the Standard incidence model. Further, only the numerical simulation of the standard incidence model was biologically meaningful and we concluded that for sexually transmitted diseases the standard incidence is most appropriate.
REFERENCES
[1]. H.W. Hethcote “The mathematics of infectious diseases” Siam Review Vol. 42, No. 4, pp. 599–653, 2000.
[2]. F. Fenner, D. A. Henderson, I. Arita, Z. Jezek, and I. D. Ladnyi, “Smallpox and its Eradication” World Health Organization, Geneva, 1988.
[3]. A. S. Benenson , Control of Communicable Diseases in Man, 16th ed., American Public Health Association, Washington, DC, 1995.
[4]. Y. Shao and C. Williamson, “ The HIV1 Epidemic, Low to MiddleIncome Countries” Cold Spring Harb Perspect Med ; doi: 10.1101/cshperspect.a007187, 2012.
[5]. S.H. Vermund and A. J. LeighBrown, “The HIV Epidemic: HighIncome Countries” Cold Spring Harb Perspect Med ; doi: 10.1101/cshperspect.a007195,2012.
[6]. R.F Stengel, Mutation and Control of the Human Immunodeficiency Virus. Paper presented at the 13th Yale Workshop on Adaptive and Learning Systems, Yale University, New Haven, CT, May 30  June 1, 2005.
[7]. Q. Li, S. Cao, X. Chen, G. Sun, Y. Liu and Z. Jia, “Stability Analysis of an HIV/AIDS Dynamics Model with Drug Resistance”. Discrete Dynamics in Nature and Society Vol 2012, Article ID 162527, doi:10.1155/2012/162527, 2012.
[8]. United states embassy in Nigeria. Nigeria HIV fact sheet. 2011.
[9]. National Agency for the Control of AIDS, Federal Republic of Nigeria Global AIDS response: Country Progress report, 2012.
[10]. R. M. Anderson, G. F. Medly, R. M. May, A. M. Johnson, “A preliminary study of the transmission dynamics of the Human Immunodeficiency Virus (HIV), the causative agent of AIDS”, IMA J. Math. Appl. Med. Biol., vol 3 pp. 229263,1986.
[11]. R.M. Anderson, “The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS”, J. AIDS vol 1, pp. 241256, 1988.
[12]. R. M. Anderson and R. M. May, Infectious Diseases of Human Dynamics and Control. Oxford University press, Oxford, 1991.
[13]. C. C. McCluskey, “A model of HIV/AIDS with staged progression and amelioration”, Math. Biosci., 181 pp. 116, 2003.
[14]. Y.H. Hsieh, C.H. Chen, “Modeling the social dynamics of a sex industry: Its implications for spread of HIV/AIDS”, Bull. Math. Biol., 66: pp. 143166, 2004.
[15]. J.Y.T. Mugisha, L.S. Luboobi, “The Effect of Vaccinating Susceptible Adults against HIV/AIDS in a Two Age Groups Population”, Zim. J. Sc. Tech. Vol. 1 no. 2, pp. 91 – 103, 2000.
[16]. S.D HoveMusekwa and F. Nyabadza “The dynamics of an HIV/AIDS model with screened disease carriers” Computational and Mathematical Methods in Medicine Vol. 10, No. 4, pp. 287–305, 2009.
[17]. B.D. Agarwala, “On two ODE models for HIV/AIDS development in Canada and a logistic SEIR model”, Far East J, Appl. Math. Vol 6 no. 1, pp. 25–70, 2002.
[18]. O. Sharomi, C. Podder , A.BGumel, E.H. Elbasha, J. Watchmough, “Role of Incidence Function in Vaccine  Induced Backward Bifurcation in some HIV Models” Math Biosci, doi:10.1016/j.mbs.2007.05.012 pp. 436463, 2007.
[19]. R. Naresh , A. Tripath, J. Biazar and D. Sharma, “Analysis of the Effect of Vaccination on the Spread of AIDS Epidemics Using Adominian Decomposition Method” JNSST, Vol 2 no 1, ISSN 19330324, 2008.
[20]. C. M. KribsZaleta, M. Lee, C. Roman, S. Wiley and C. M. HernandezSuarez, “The effect of the HIV/AIDS epidemic on Africa’s truck drivers”, Math. Bios. Engg. Vol 2. No 4, pp. 771788 2005.
[21]. S.M. Moghadas, A.B. Gumel, R.G. Mcleod and R. Gordon “Could Condoms Stop the AIDS Epidemic?” Journal of Theoretical Medicine, Vol. 5 No 3–4, pp. 171–181, 2003
[22]. R.A Kimbir, M.J.I Udoo and T. Aboiyar “A Mathematical Model for the transmission dynamics of HIV/AIDS in a twosex population considering Counseling and antiretroviral therapy (ART)” , J. Math. Comput. Sci. Vol 2 , No. 6, pp. 16711684, 2012
[23]. A. B. Gumel, W. Z. Xue, P. N. Shivakumar, M. L. Garba and B. M. Sahai, “A New Mathematical Model for Assessing Therapeutic Strategies for HIV infection” Journal of Theoretical Medicine. Vol 4 No. 2, pp.147, 2002
[24]. S. Valle, A. M. Evangelista, M.C. Velasco, Effects of education, Vaccination and Treatment on HIV Transmission in Homosexuals with Genetic Heterogeneity, Cornell Univ., Dept. of Biometrics Technical Report BU1587M.,2001
[25]. C. M. KribsZaleta and J.X. VelascoHernandez, “A simple vaccination model with multiple endemic states”. Math. Biosci. 164 pp. 183 – 201, 2000.
[26]. O. Sharomi and A.B. Gumel “Dynamical analysis of a multistrain model of HIV in the presence of antiretroviral drugs”, Journal of Biological Dynamics, Vol 2 No.3, pp. 323345, 2008.
[27]. K.O. Okosun , O.D. Makinde, I. Takaidza, “Impact of optimal control on the treatment of HIV/AIDS and screening of unaware infectives”, Applied Mathematical Modelling 37,pp. 3802–3820, 2013.
[28]. H. Shim, S. Han, C. C. Chung, S. Nam and J.H. Seo, “Optimal Scheduling of Drug Treatment for HIV Infection: Continuous Dose Control and Receding Horizon Control”, International Journal of Control, Automation, and Systems Vol. 1, No. 3, 2003
[29]. L.S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, New York, Wiley, 1962.
[30]. O. Sharomi, C.N. Podder, A.B. Gumel and B. Song, “Mathematical Analysis of the Transmission Dynamics of HIV/TB Coinfection In The Presence Of Treatment” Mathematical Biosciences And Engineering Vol 5, No. 1, pp. 145–174,2008.
[31]. Z. Mukandavire, A.B. Gumel, W. Garira and J.M. Tchuenche, “Mathematical Analysis of a Model or HIVMalaria CoInfection” Mathematical Biosciences And Engineering Vol 6, No. 2, pp. 333–362, 2009.
[32]. S. Mushayabasa and C. P. Bhunu “Modeling Schistosomiasis and HIV/AIDS Codynamics”, Computational and Mathematical Methods in Medicine, doi:10.1155/2011/846174. Vol 2011, Article ID 846174, 2011.
[33]. P. van den Driessche and J. Watmough,” Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci. 180 pp. 29–48, 2002.
[34]. O. Diekmann, H. Heesterbeek, J.A.J. Metz, “On the definition and computation of the basic reproductive ratio in the models for infectious diseases in the heterogeneous populations”, J. Math. Biol. 28 ,pp. 365–382, 1990.
[35]. H. Hethcote, M. Zhien , L. Shengbing, “Effects of quarantine in six endemic models for infectious diseases”, Math. Biosci., 180 ,pp. 141–160, 2002.
[36]. J.O. LloydSmith, W.M. Getz and H.V. Westerhoff, “Frequencydependent incidence in models of sexually transmitted diseases: portrayal of pairbased transmission and effects of illness on contact behavior”, Proc. R. Soc. Lond. B, doi 10.1098/rspb.2003.2632, 2004
[37]. R.M. Anderson and R.M. May Population Biology of Infectious Diseases. SpringerVerlag, Berlin, Heilderberg, New York. 1982
[38]. S.M. Moghadas and A.B. Gumel, “Global stability of a twostage epidemic model with generalized nonlinear incidence”, Mathematics and Computers in Simulation 60, pp. 107–118, 2002.
[39]. A. Kaddar, “Stability analysis in a delayed SIR epidemic model with a saturated incidence rate”, Nonlinear Analysis: Modelling and Control, Vol. 15, No. 3, pp. 299–306, 2010.
[40]. P. Das, D. Mukherjee And Y. Hsieh, “An SI Epidemic Model With Saturation Incidence: Discrete And Stochastic Version”, Int. J. Nonlinear Anal. Appl. No. 19, 2011.
[41]. L. Cai, X. Li and J. Yu, “Analisis of a delayed HIV/AIDS epidemic model with saturation incidence”, J Appl Math Comp 27, pp. 365 – 377, 2008.
[42]. Z. Mukandavire, P. Das, C. Chiyaka, F. Nyabadza “Global analysis of an HIV/AIDS epidemic model”, World Journal of Modelling and Simulation Vol. 6 No. 3, pp. 231240, 2010.
[43]. S. AlSheikh, F. Musali, M. Alsolami, “Stability Analysis of an HIV/AIDS Epidemic Model with Screening”, International Mathematical Forum, Vol. 6 No. 66, pp. 3251 – 3273, 2011.
[44]. O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, “The construction of nextgeneration matrices for compartmental epidemic models”, J. R. Soc. Interface published online 5 November 2009, doi: 10.1098/rsif.2009.0386
[45]. C. CastilloChavez, Z. Feng, and W. Huang, On the computation of Ro and its role on global stability,in: CastilloChavez C., Blower S., van den Driessche P., Krirschner D. and Yakubu A.A.(Eds), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. The IMA Volumes in Mathematics and its Applications. SpringerVerlag, New York, 125(2002), pp. 229250.
[46]. CastilloChavez, C. and B. Song “Dynamical models of tuberculosis and their applications”, Mathematical Biosci. and Engr. Vol 1 No.2, pp. 361404, 2004.
[47]. V. Lakshmikantham, S. Leela and A.A. Martynyuk Stability Analysis of Nonlinear Systems. Marcel Dekker, Inc., New York and Basel, 1989.
[48]. H.L Smith and Waltman , The Theory of the Chemostat. Cambridge University Press, 1995.
[49]. A.B. Gumel, C. C. McCluskey, P. van den Driessche, “Mathematical Study of a StagedProgression HIV Model with Imperfect Vaccine”, Bull. Math Bio. 68 pp. 2105–2128 DOI 10.1007/s1153800690957, 2006
