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). Google Scholar

[2]

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

[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, 4507 (2007), 948. doi: 10.1007/978-3-540-73007-1_114. Google Scholar

[4]

M. Atencia, G. Joya and F. Sandoval, Modelling the HIV-AIDS Cuban epidemics with Hopfield neural networks,, in, 2687 (2003), 1053. doi: 10.1007/3-540-44869-1_57. Google Scholar

[5]

M. Atencia, G. Joya and F. Sandoval, Robustness of the Hopfield estimator for identification of dynamical systems,, in, 6692 (2011), 516. doi: 10.1007/978-3-642-21498-1_65. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0021-9991(76)90007-3. Google Scholar

[11]

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

[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. doi: 10.1016/j.neucom.2011.03.022. Google Scholar

[13]

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

[14]

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

[15]

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

[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, (2005), 77. doi: 10.1142/9789812569264_0004. Google Scholar

[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. doi: 10.1142/S0218339005001616. Google Scholar

[18]

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

[19]

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

[20]

H. Khalil, "Nonlinear Systems,", Macmillan Publishing Company, (1992). Google Scholar

[21]

L. Ljung, "System Identification: Theory for the User,", Prentice Hall, (1999). Google Scholar

[22]

J. Murray, "Mathematical Biology,", Springer, (2002). Google Scholar

[23]

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

[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. doi: 10.3934/mbe.2006.3.545. Google Scholar

[25]

K. Schittkowski, "Numerical Data Fitting in Dynamical Systems,", Kluwer Academic Publishers, (2002). Google Scholar

[26]

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

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). Google Scholar

[2]

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

[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, 4507 (2007), 948. doi: 10.1007/978-3-540-73007-1_114. Google Scholar

[4]

M. Atencia, G. Joya and F. Sandoval, Modelling the HIV-AIDS Cuban epidemics with Hopfield neural networks,, in, 2687 (2003), 1053. doi: 10.1007/3-540-44869-1_57. Google Scholar

[5]

M. Atencia, G. Joya and F. Sandoval, Robustness of the Hopfield estimator for identification of dynamical systems,, in, 6692 (2011), 516. doi: 10.1007/978-3-642-21498-1_65. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/0021-9991(76)90007-3. Google Scholar

[11]

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

[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. doi: 10.1016/j.neucom.2011.03.022. Google Scholar

[13]

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

[14]

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

[15]

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

[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, (2005), 77. doi: 10.1142/9789812569264_0004. Google Scholar

[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. doi: 10.1142/S0218339005001616. Google Scholar

[18]

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

[19]

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

[20]

H. Khalil, "Nonlinear Systems,", Macmillan Publishing Company, (1992). Google Scholar

[21]

L. Ljung, "System Identification: Theory for the User,", Prentice Hall, (1999). Google Scholar

[22]

J. Murray, "Mathematical Biology,", Springer, (2002). Google Scholar

[23]

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

[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. doi: 10.3934/mbe.2006.3.545. Google Scholar

[25]

K. Schittkowski, "Numerical Data Fitting in Dynamical Systems,", Kluwer Academic Publishers, (2002). Google Scholar

[26]

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

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