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A Global Stability Analysis of a Susceptible-Infected-Removed-Prevented-Controlled Epidemic Model

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DOI: 10.4236/am.2014.510131    3,223 Downloads   4,323 Views   Citations
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Muhammad A. Yau, Hussaini S. Ndakwo, A. M. Umar

Affiliation(s)

Department of Mathematical Sciences, Nasarawa State University, Keffi, Nigeria.

ABSTRACT

A mathematical model of HIV transmission dynamics is proposed and analysed. The population is partitioned into five compartments of susceptible S(t), Infected I(t), Removed R(t), Prevented U(t) and the Controlled W(t). Each of the compartments comprises of cohort of individuals. Five systems of nonlinear equations are derived to represent each of the compartments. The general stability of the disease free equilibrium (DFE) and the endemic equilibrium states of the linearized model are established using the linear stability analysis (Routh-Hurwitz) method which is found to be locally asymptotically stable when the infected individuals receive ART and use the condom. The reproduction number is also derived using the idea of Diekmann and is found to be strictly less than one. This means that the epidemic will die out.

KEYWORDS

Epidemic Model, Stability Analysis, HIV/AIDS, Disease Free Equilibrium Points

Cite this paper

Yau, M. , Ndakwo, H. and Umar, A. (2014) A Global Stability Analysis of a Susceptible-Infected-Removed-Prevented-Controlled Epidemic Model. Applied Mathematics, 5, 1393-1399. doi: 10.4236/am.2014.510131.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Centers for Disease Control and Prevention (1989) Perspectives in Disease Prevention and Health Promotion Condoms for Prevention of Sexually Transmitted Diseases.
[2] Tinuola, F.R., Ayodele J.O. and Ola, T.M. (2006) HIV/AIDS in Africa and the Politics of the West. Nigerian Journal of Social Research, 1, 100-112.
[3] Kimbir A.R. and Oduwole, H.K (2008)A Mathematical Model of HIV/AIDS Transmission Dynamics Considering Counselling and Antiretroviral Therapy. Journal of Modern Mathematics and Statistics, 2, 166-169.
[4] Muhammad, A.Y. (2010) A Mathematical Model of HIV Transmission Dynamics Considering the Use of Antiretroviral Therapy and a Preventive Measure (Condom). M.Sc Thesis (Unpublished).
[5] Yang H.M. and Ferreira, W.C. (1999) A Population Model Applied to HIV Transmission, Considering Protection and Treatment. IMA Journal of Mathematics Applied in Medicine and Biology, 16, 237-259.
[6] Hethcote, H.W. (1989) Three Basic Epidemiological Models. Applied Mathematical Ecology. Springer-Verlag, Berlin, 119-144. http://dx.doi.org/10.1007/978-3-642-61317-3_5
[7] Diekmann, O., Heesterbeek, J.P. and Metz, J.A. (1990) On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations. Journal of Mathematical Biology, 28, 365-382. http://dx.doi.org/10.1007/BF00178324

  
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