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 "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.  Google Scholar

[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.  Google Scholar

[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.  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), 133-152.  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), 221-234. 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), 30-37. 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), 123-143. 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), 2691-2697. doi: 10.1016/j.neucom.2011.03.022.  Google Scholar

[13]

J. Gielen, A framework for epidemic models, Journal of Biological Systems, 11 (2003), 377-405. 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 "Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention," 77-92. World Scientific, (2005). 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), 441-454. 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, New York, 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), 9575-9591. 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), 545-556. 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 "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.  Google Scholar

[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.  Google Scholar

[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.  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), 133-152.  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), 221-234. 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), 30-37. 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), 123-143. 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), 2691-2697. doi: 10.1016/j.neucom.2011.03.022.  Google Scholar

[13]

J. Gielen, A framework for epidemic models, Journal of Biological Systems, 11 (2003), 377-405. 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 "Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention," 77-92. World Scientific, (2005). 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), 441-454. 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, New York, 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), 9575-9591. 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), 545-556. 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

[1]

Brandy Rapatski, Petra Klepac, Stephen Dueck, Maoxing Liu, Leda Ivic Weiss. Mathematical epidemiology of HIV/AIDS in cuba during the period 1986-2000. Mathematical Biosciences & Engineering, 2006, 3 (3) : 545-556. doi: 10.3934/mbe.2006.3.545

[2]

Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi, Glenn Webb. Preface: Population dynamics in epidemiology and ecology. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : i-ii. doi: 10.3934/dcdsb.2020125

[3]

Miaomiao Gao, Daqing Jiang, Tasawar Hayat, Ahmed Alsaedi, Bashir Ahmad. Dynamics of a stochastic HIV/AIDS model with treatment under regime switching. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021181

[4]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172

[5]

Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367

[6]

Praveen Kumar Gupta, Ajoy Dutta. Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 393-399. doi: 10.3934/naco.2019038

[7]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187

[8]

Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256

[9]

Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021005

[10]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[11]

Mahdi Khajeh Salehani. Identification of generic stable dynamical systems taking a nonlinear differential approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4541-4555. doi: 10.3934/dcdsb.2018175

[12]

Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423

[13]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[14]

MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777

[15]

Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233

[16]

Paul L. Salceanu. Robust uniform persistence in discrete and continuous dynamical systems using Lyapunov exponents. Mathematical Biosciences & Engineering, 2011, 8 (3) : 807-825. doi: 10.3934/mbe.2011.8.807

[17]

Doan Thai Son. On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3113-3126. doi: 10.3934/dcdsb.2017166

[18]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331

[19]

Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639

[20]

Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (1)

[Back to Top]