2013, 10(4): 959-977. doi: 10.3934/mbe.2013.10.959

The ratio of hidden HIV infection in Cuba

1. 

Dept. of Applied Mathematics, University of Málaga, 29071 Málaga, Spain

2. 

Dept. of Electronics Technology, University of Málaga, 29071 Málaga, Spain, Spain

Received  August 2012 Revised  April 2013 Published  June 2013

In this work we propose the definition of the ratio of hidden infection of HIV/AIDS epidemics, as the division of the unknown infected population by the known one. The merit of the definition lies in allowing for an indirect estimation of the whole of the infected population. A dynamical model for the ratio is derived from a previous HIV/AIDS model, which was proposed for the Cuban case, where active search for infected individuals is carried out through a contact tracing program. The stability analysis proves that the model for the ratio possesses a single positive equilibrium, which turns out to be globally asymptotically stable. The sensitivity analysis provides an insight into the relative performance of various methods for detection of infected individuals. An exponential regression has been performed to fit the known infected population, owing to actual epidemiological data of HIV/AIDS epidemics in Cuba. The goodness of the obtained fit provides additional support to the proposed model.
Citation: Miguel Atencia, Esther García-Garaluz, Gonzalo Joya. The ratio of hidden HIV infection in Cuba. Mathematical Biosciences & Engineering, 2013, 10 (4) : 959-977. doi: 10.3934/mbe.2013.10.959
References:
[1]

, "WHO Case Definitions of HIV for Surveillance and Revised Clinical Staging and Immunological Classification of HIV-related Disease in Adults and Children,", World Health Organization, (2007). 

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, 1991.

[3]

M. Atencia, G. Joya, E. García-Garaluz, H. de Arazoza and F. Sandoval, Estimation of the rate of detection of infected individuals in an epidemiological model, in "Computational and Ambient Intelligence" (eds. F. Sandoval, A. Prieto, J. Cabestany and M. Graña), 4507 of Lecture Notes in Computer Science, 948-955. Springer, (2007). doi: 10.1007/978-3-540-73007-1_114.

[4]

M. Atencia, G. Joya and F. Sandoval, Modelling the HIV-AIDS Cuban epidemics with Hopfield neural networks, in "Artificial Neural Nets Problem Solving Methods" (eds. J. Mira and J. Álvarez), 2687 of Lecture Notes in Computer Science, 1053-1053. Springer, (2003). doi: 10.1007/3-540-44869-1_57.

[5]

M. Atencia, G. Joya and F. Sandoval, Robustness of the Hopfield estimator for identification of dynamical systems, in "Advances in Computational Intelligence" (eds. J. Cabestany, I. Rojas and G. Joya), Springer, 6692 (2011), 516-523. doi: 10.1007/978-3-642-21498-1_65.

[6]

N. Bailey, "The Mathematical Theory of Infectious Diseases and Its Applications," Oxford University Press, 1975.

[7]

F. Berezovskaya, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152.

[8]

H. de Arazoza and R. Lounes, A non-linear model for a sexually transmitted disease with contact tracing, Mathematical Medicine and Biology, 19 (2002), 221-234.

[9]

H. de Arazoza, R. Lounes, J. Pérez and T. Hoang, What percentage of the cuban HIV-AIDS epidemic is known?, Revista Cubana de Medicina Tropical, 55 (2003), 30-37.

[10]

R. P. Dickinson and R. J. Gelinas, Sensitivity analysis of ordinary differential equation systems-A direct method, Journal of Computational Physics, 21 (1976), 123-143. doi: 10.1016/0021-9991(76)90007-3.

[11]

O. Diekmann and J. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases," John Wiley, 2000.

[12]

E. García-Garaluz, M. Atencia, G. Joya, F. García-Lagos and F. Sandoval, Hopfield networks for identification of delay differential equations with an application to dengue fever epidemics in Cuba, Neurocomputing, 74 (2011), 2691-2697. doi: 10.1016/j.neucom.2011.03.022.

[13]

J. Gielen, A framework for epidemic models, Journal of Biological Systems, 11 (2003), 377-405. doi: 10.1142/S0218339003000919.

[14]

E. Hairer, S. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I. Non-Stiff Problems," Springer, 1993.

[15]

M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, 1974.

[16]

Y.-H. Hsieh, H. de Arazoza, R. Lounes and J. Joanes, A class of methods for HIV contact tracing in Cuba: Implications for intervention and treatment, in "Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention," 77-92. World Scientific, (2005). doi: 10.1142/9789812569264_0004.

[17]

Y.-H. Hsieh, H.-C. Wang, H. de Arazoza, R. Lounes, S.-J. Twu and H.-M. Hsu, Ascertaining HIV underreporting in low prevalence countries using the approximate ratio of underreporting, Journal of Biological Systems, 13 (2005), 441-454. doi: 10.1142/S0218339005001616.

[18]

P. A. Ioannou and J. Sun, "Robust Adaptive Control," Prentice-Hall, 1996.

[19]

R. Isermann and M. Münchhof, "Identification of Dynamic Systems," Springer, 2011. doi: 10.1007/978-3-540-78879-9.

[20]

H. Khalil, "Nonlinear Systems," Macmillan Publishing Company, New York, 1992.

[21]

L. Ljung, "System Identification: Theory for the User," Prentice Hall, 1999.

[22]

J. Murray, "Mathematical Biology," Springer, 2002.

[23]

R. Naresh, A. Tripathi and D. Sharma, A nonlinear HIV/AIDS model with contact tracing, Applied Mathematics and Computation, 217 (2011), 9575-9591. doi: 10.1016/j.amc.2011.04.033.

[24]

B. Rapatski, P. Klepac, S. Dueck, M. Liu and L. I. Weiss, Mathematical epidemiology of HIV/AIDS in Cuba during the period 1986-2000, Mathematical Biosciences and Engineering, 3 (2006), 545-556. doi: 10.3934/mbe.2006.3.545.

[25]

K. Schittkowski, "Numerical Data Fitting in Dynamical Systems," Kluwer Academic Publishers, 2002.

[26]

M. Vidyasagar, "Nonlinear Systems Analysis," Society for Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898719185.

show all references

References:
[1]

, "WHO Case Definitions of HIV for Surveillance and Revised Clinical Staging and Immunological Classification of HIV-related Disease in Adults and Children,", World Health Organization, (2007). 

[2]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, 1991.

[3]

M. Atencia, G. Joya, E. García-Garaluz, H. de Arazoza and F. Sandoval, Estimation of the rate of detection of infected individuals in an epidemiological model, in "Computational and Ambient Intelligence" (eds. F. Sandoval, A. Prieto, J. Cabestany and M. Graña), 4507 of Lecture Notes in Computer Science, 948-955. Springer, (2007). doi: 10.1007/978-3-540-73007-1_114.

[4]

M. Atencia, G. Joya and F. Sandoval, Modelling the HIV-AIDS Cuban epidemics with Hopfield neural networks, in "Artificial Neural Nets Problem Solving Methods" (eds. J. Mira and J. Álvarez), 2687 of Lecture Notes in Computer Science, 1053-1053. Springer, (2003). doi: 10.1007/3-540-44869-1_57.

[5]

M. Atencia, G. Joya and F. Sandoval, Robustness of the Hopfield estimator for identification of dynamical systems, in "Advances in Computational Intelligence" (eds. J. Cabestany, I. Rojas and G. Joya), Springer, 6692 (2011), 516-523. doi: 10.1007/978-3-642-21498-1_65.

[6]

N. Bailey, "The Mathematical Theory of Infectious Diseases and Its Applications," Oxford University Press, 1975.

[7]

F. Berezovskaya, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152.

[8]

H. de Arazoza and R. Lounes, A non-linear model for a sexually transmitted disease with contact tracing, Mathematical Medicine and Biology, 19 (2002), 221-234.

[9]

H. de Arazoza, R. Lounes, J. Pérez and T. Hoang, What percentage of the cuban HIV-AIDS epidemic is known?, Revista Cubana de Medicina Tropical, 55 (2003), 30-37.

[10]

R. P. Dickinson and R. J. Gelinas, Sensitivity analysis of ordinary differential equation systems-A direct method, Journal of Computational Physics, 21 (1976), 123-143. doi: 10.1016/0021-9991(76)90007-3.

[11]

O. Diekmann and J. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases," John Wiley, 2000.

[12]

E. García-Garaluz, M. Atencia, G. Joya, F. García-Lagos and F. Sandoval, Hopfield networks for identification of delay differential equations with an application to dengue fever epidemics in Cuba, Neurocomputing, 74 (2011), 2691-2697. doi: 10.1016/j.neucom.2011.03.022.

[13]

J. Gielen, A framework for epidemic models, Journal of Biological Systems, 11 (2003), 377-405. doi: 10.1142/S0218339003000919.

[14]

E. Hairer, S. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I. Non-Stiff Problems," Springer, 1993.

[15]

M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, 1974.

[16]

Y.-H. Hsieh, H. de Arazoza, R. Lounes and J. Joanes, A class of methods for HIV contact tracing in Cuba: Implications for intervention and treatment, in "Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention," 77-92. World Scientific, (2005). doi: 10.1142/9789812569264_0004.

[17]

Y.-H. Hsieh, H.-C. Wang, H. de Arazoza, R. Lounes, S.-J. Twu and H.-M. Hsu, Ascertaining HIV underreporting in low prevalence countries using the approximate ratio of underreporting, Journal of Biological Systems, 13 (2005), 441-454. doi: 10.1142/S0218339005001616.

[18]

P. A. Ioannou and J. Sun, "Robust Adaptive Control," Prentice-Hall, 1996.

[19]

R. Isermann and M. Münchhof, "Identification of Dynamic Systems," Springer, 2011. doi: 10.1007/978-3-540-78879-9.

[20]

H. Khalil, "Nonlinear Systems," Macmillan Publishing Company, New York, 1992.

[21]

L. Ljung, "System Identification: Theory for the User," Prentice Hall, 1999.

[22]

J. Murray, "Mathematical Biology," Springer, 2002.

[23]

R. Naresh, A. Tripathi and D. Sharma, A nonlinear HIV/AIDS model with contact tracing, Applied Mathematics and Computation, 217 (2011), 9575-9591. doi: 10.1016/j.amc.2011.04.033.

[24]

B. Rapatski, P. Klepac, S. Dueck, M. Liu and L. I. Weiss, Mathematical epidemiology of HIV/AIDS in Cuba during the period 1986-2000, Mathematical Biosciences and Engineering, 3 (2006), 545-556. doi: 10.3934/mbe.2006.3.545.

[25]

K. Schittkowski, "Numerical Data Fitting in Dynamical Systems," Kluwer Academic Publishers, 2002.

[26]

M. Vidyasagar, "Nonlinear Systems Analysis," Society for Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898719185.

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