Palabras clave
HIV/AIDS
fractional derivatives in the Caputo sense
optimal control problem
tuberculosis
diabetes
VIH/SIDA
derivadas fraccionarias en el sentido de Caputo
problema de control óptimo
tuberculosis
Cómo citar
Resumen
En este trabajo, aprovechando las ventajas del uso de las derivadas de orden fraccionario en el sentido de Caputo, presentamos un estudio de control óptimo de la eficacia del tratamiento de la tuberculosis (TB) en presencia del VIH/SIDA y la diabetes. El modelo matemático que es controlado se encuentra en [17] y estudia la relación entre la tuberculosis, el VIH/SIDA y la diabetes con respecto a la eficacia del tratamiento y diferencia en tuberculosis multirresistente (TB-MDR), y tuberculosis extremadamente resistente (TB-XDR). La definición de los controles se centra en evitar la reinfección/reactivación, la TB-MDR y la TB-XDR en las diferentes subpoblaciones (TB únicamente, TB-VIH/SIDA y TB-Diabetes). El modelo dividido en subpoblaciones nos permite diferenciar los comportamientos de la transmisión y la resistencia y evaluar los diferentes costos en la aplicación de los controles. Realizamos simulaciones computacionales con datos de la literatura para estudiar nuestro problema de control en un escenario específico. Exploramos el comportamiento del número básico de reproducción variando la tasa de contacto efectivo y los parámetros asociados a la resistencia para diferentes valores de _ (orden fraccional). Estudiamos diferentes estrategias de control basadas en la activación de los controles y encontramos que la más efectiva es cuando activamos todos los controles. Con esta estrategia, reducimos el número de casos resistentes, principalmente en la TB-XDR en diabéticos que tiene un fuerte impacto en la dinámica de resistencia y transmisión de la tuberculosis. Además, esta estrategia evita el crecimiento futuro del número de casos resistentes.
Citas
F. Abd-elal, N.H. Sweilam, A.M. Nagy, Y.S. Almaghrebi, Computational methods for the fractional optimal control HIV infection, Journal of Fractional Calculus and Applications, 7 (2016), no 2, 121–131. Link
F.B. Agusto, A.I. Adekunle, Optimal control of a two-strain tuberculosis- HIV/AIDS co-infection model, Biosystems 119 (2014) 20–44. Doi: 10.1016/j.biosystems.2014.03.006
B.A. Baba, B. Bilgehan, Optimal control of a fractional order model for the COVID-19 pandemic, Chaos, Solitons and Fractals 144 (2021) 110678. Doi: 10.1016/j.chaos.2021.110678
P. Baghaei et al., Diabetes mellitus and tuberculosis facts and controversies, J Diabetes Metab Disord 12 (2013), no 1. Doi: 10.1186/2251-6581- 12-58
J. Biazar, Solution of the epidemic model by Adomian decomposition method, Applied Mathematics and Computation 173 (2006), no 2, 1101–1106. Doi: 10.1016/j.amc.2005.04.036
O.K. Basoglu, F. Bacakoglu, G. Cok, Sayiner, M. Ate¸s, The oral glucose tolerance test in patients with respiratory infections, Monaldi Arch Chest Dis 54 (1999), no 4, 307-310.
F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical Models in Epidemiology, Springer, New York NY, 2019. Doi: 10.1007/978-1-4939-9828-9
S.M. Blower, A.R. Mclean, T.C. Porco, P.M. Small, P.C. Hopewell, M.A. Sanchez, A.R. Moss, The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine 1 (1995), no 8, 815–821. Doi: 1038/nm0895-815
R.F. Camargo, E. Capelas de Oliveira, Cálculo fracionário, Livraria da Física, São Paulo, 2015.
A.R.M. Carvalho, C.M.A. Pinto, Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection, Communications in Nonlinear Science and Numerical Simulation 61 (2018), 104–126. Doi: 10.1016/J.CNSNS.2018.01.012
A.R.M. Carvalho, C.M.A. Pinto, D. Baleanu, HIV/HCV coinfection model: A fractional-order perspective for the effect of the HIV viral load, Advances in Difference Equations 2018 (2018). Doi: 10.1186/s13662-017-1456-z
L. Carvalho de Barros, F. Santo Pedro, E. Esmi, J.P. Carvalho dos Santos, D.E. Sánchez The memory effect on fractional calculus: an application in the spread of COVID-19, Computational and Applied Mathematics 40 (2021), no 3, 72. Doi: 10.1007/s40314-021-01456-z
A.J.F. Cassetone, Alterações metabólicas associadas ao uso de medicamento antirretroviral em pessoas vivendo com HIV/AIDS: caracterização e desenvolvimento de algoritmos inteligentes aplicados à sua identificação e previsão. Doctorate Thesis, FMUSP, University of São Paulo, Brazil, 2017.
C. Castillo-Chavez, Z. Feng, To treat or not to treat: the case of tuberculosis, Journal of Mathematical Biology 35 (1997), no 6, 629–656. Doi: 10.1007/s002850050069
T. Cohen, C. Colijn, B. Finklea, M. Murray, Exogenous re-infection and the dynamics of tuberculosis epidemics: local effects in a network model of transmission, J. R. Soc. Interface 4 (2007), no 14, 523–531. Doi: 10.1098/rsif.2006.0193
E. Delgado Moya, A. Pietrus, S. Muniz Oliva, A mathematical model for the study of effectiveness in therapy in Tuberculosis taking into account associated diseases, Contemporary Mathematics 2 (2021), no 1, 77–102. Doi: 10.37256/cm.212021694
E. Delgado Moya, A. Pietrus, S. Muniz Oliva, Mathematical model with fractional order derivatives for Tuberculosis taking into account its relationship with HIV/AIDS and Diabetes, Jambura Journal of Biomathematics 2 (2021), no 2, 80–95. Doi: 10.34312/jjbm.v2i2.11553
G. Delogu, M. Sali, G. Fadda, The Biology of mycobacterium tuberculosis infection, Mediterr J Hematol Infect Dis. 5 (2013), no 1. Doi: 10.4084/MJHID.2013.070
O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the definition and the computation of the basic reproduction ratio <0 in model for infectious diseases in heterogeneous populations, Journal of Mathematical Biology 28 (1990), 365–382. Doi: 10.1007/BF00178324
O. Diekmann, J.A.P. Heesterbeek, M.G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface 7 (2010), no 47, 873–885. Doi: 10.1098/rsif.2009.0386
K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynamics 71 (2013) 613–619. Doi: 10.1007/s11071-012-0475-2
C. Dye, G.P. Garnett, K. Sleema, B.G. Williams, Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Directly observed short-course therapy, Lancet 352 (1998), no 9144, 1886–1891. Doi: 10.1016/s0140-6736(98)03199-7
Fatmawati, H. Tasman, An optimal treatment control of TB-HIV coinfection, Cogent Mathematics and Statistics 2016 (2016), article ID 8261208. Doi: 10.1155/2016/8261208
Fatmawati, M.A. Khan, E. Bonyah, Z. Hammouch, E.M. Shaiful, A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model, AIMS Mathematics 5 (2020), no 4, 2813–2842. Doi: 10.3934/math.2020181
F. Haq, K. Shah, G. Rahman, M. Shahzada, Numerical solution of fractional order smoking model via Laplace Adomian decomposition method, Alexandria Engineering Journal 57 (2018), no 2, 1061–1069. Doi: 10.1016/j.aej.2017.02.015
R.I. Hickson, G. N. Mercer, K.M. Lokuge, A metapopulation model of tuberculosis transmission with a case study from high to low burden areas, PloS ONE 7 (2012), no 4, e34411. Doi: 10.1371/journal.pone.0034411
C.Y. Jeon, M.B. Murray, Diabetes mellitus increases the risk of active tuberculosis: a systematic review of 13 observational studies, PLoS Med 5(2008) e152. Doi: 10.1371/journal.pmed.0050152
E. Jung, S. Lenhart, Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Mathematical Biosciences 2 (2002), no 4, 473–482. Doi: 10.3934/dcdsb.2002.2.473
H. Kheiri, M. Jafari, Optimal control of a fractional-order model for the HIV/AIDS epidemic, International Journal of Biomathematics 11(2008), no 7, 1850086. Doi: 10.1142/S1793524518500869
S. Kim, A.A de Los Reyes, E. Jung, Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines, Journal of Theoretical Biology 443 (2018), 100–112. Doi: 10.1016/j.jtbi.2018.01.026
W. Lin, Global existence theory and chaos control of fractional differential equations, Journal of Mathematical Analysis and Applications 332 (2007), no 1, 709–726. Doi: 10.1016/j.jmaa.2006.10.040
A. McIvor, H. Koornhof, B.D. Kana, Relapse, re-infection and mixed infections in tuberculosis disease, Pathog Dis. 75(2017), no 3. Doi: 10.1093/femspd/ftx020
D.P. Moualeu, S. Bowong, J.J. Tewa, Y. Emvudu, Analysis of the impact of diabetes on the dynamical transmission of tuberculosis, Mathematical and Computer Modelling: An International Journal 7 (2012), no 3, 117–146. Doi: 10.1051/mmnp/20127309
A.K. Niazi, S. Kalra, Diabetes and tuberculosis: a review of the role of optimal glycemic control, J Diabetes Metab Disord. 11 (2012) 28. Doi: 10.1186/2251-6581-11-28
D. Okuonghae, B.O. Ikhimwin, Dynamics of a mathematical model for tuberculosis with variability in susceptibility and disease progressions due to difference in awareness level, Frontiers in Microbiology 6 (2016), 1530. Doi: 10.3389/fmicb.2015.01530
L. Pealing, K. Wing, R. Mathur, D. Prieto-Merino, L. Smeeth, D.A.J. Moore, Risk of tuberculosis in patients with diabetes: population based cohort study using the UK Clinical Practice Research Datalink, BMC Med (2015). Doi: 10.1186/s12916-015-0381-9
C.M.A. Pinto, A.R.M. Carvalho, A latency fractional order model for HIV dynamics, Journal of Computational and Applied Mathematics 312 (2020) 240–256. Doi: 10.1016/j.cam.2016.05.019
S. Rosa, D.F.M. Torres, Optimal control and sensitivity analysis of a fractional order TB model, Statistics, Optimization and Information Computing 7 (2019), no 3, 617–625. Doi: 10.19139/soic.v7i3.836
M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G.R. Jafari, M. Ausloos, Memory effects on epidemic evolution: The susceptible-infectedrecovered epidemic model, Physical Review E 95 (2017), no 2, 022409. Doi: 10.1103/PhysRevE.95.022409
C.J. Silva, H. Maurer, D.F.M. Torres, Optimal control of a tuberculosis model with state and control delays, Mathematical Biosciences 14 (2017), no 1, 321–337. Doi: 10.3934/mbe.2017021
C.J. Silva, D.F.M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Mathematical Biosciences and Engineering 244 (2013), no 2, 154–164. Doi: 10.1016/j.mbs.2013.05.005
C.J. Silva, D.F.M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, SIAM Journal on Mathematical Analysis 35 (2015), no 9, 4639–4663. Doi: 10.3934/dcds.2015.35.4639
C.R. Stevenson, , J. A. Critchley, N.G. Forouhi, G. Roglic, B.G. Williams, C. Dye, N.C. Unwin, Diabetes and the risk of tuberculosis: a neglected threat to public health?, Chronic Illness 3 (2007), no 3, 228-245.Doi: 10.1177/1742395307081502
S. Suchindran, E.S. Brouwer, A. Van Rie, Is HIV infection a risk factor for multi-drug resistant tuberculosis? A systematic review, PLoS One 4 (2009), no 5. Doi: 10.1371/journal.pone.0005561
N.H. Sweilam, S.M. Al-Mekhlafi, D.Baleanu, Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains, Journal of Advanced Research 17 (2019) 125–137. Doi: 10.1016/j.jare.2019.01.007
N.H. Sweilam, S.M. AL-Mekhlafi, D.Baleanu, A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model, Journal of Advanced Research 32 (2021) 149–160. Doi: 1016/j.jare.2020.08.006
N.H. Sweilam, S.M. Al-Mekhlafi, Z.N. Mohammed, D. Baleanu, Optimal control for variable order fractional HIV/AIDS and malaria mathematical models with multi-time delay, Alexandria Engineering Journal 59 (2020), no 5, 3149–3162. Doi: 10.1016/j.aej.2020.07.021
M. Tahir, S.I.A. Shah, G.Z.Y. Rogovchenko, Prevention strategy for superinfection mathematical model tuberculosis and HIV associated with AIDS, Cogent Mathematics and Statistics 6 (2019), no 1, art: 1637166. Doi: 10.1080/25742558.2019.1637166
H. Wang, H. Jahanshahi, M.-K. Wang, S. Bekiros, J. Liu, A.A. Aly, A Caputo-Fabrizio Fractional-Order Model of HIV/AIDS with a treatment compartment: Sensitivity analysis and optimal control strategies, Entropy (Basel) 23 (2021), no 5, 610. Doi: 10.3390/e23050610
World Health Organization (WHO), Global tuberculosis report 2017. Disponible en: https://apps.who.int/iris/handle/10665/259366. Consultado el 05-enero-2020, 06:30 p.m.
World Health Organization (WHO), Global tuberculosis report 2021. Disponible en https://www.who.int/publications/i/item/9789240037021. Consultado el 23-abril-2021, 02:35 p.m.
W. Wojtak, C.J. Silva, D.F.M. Torres, Uniform asymptotic stability of a fractional tuberculosis model, Mathematical Modelling of Natural Phenomena 13 (2018), no 1, art:9. Doi: 10.1051/mmnp/2018015
M. Zamir, G. Zaman, A.S. Alshomrani, Sensitivity analysis and optimal control of anthroponotic cutaneous Leishmania, PLoS ONE 11 (2016), no 8, e0160513. Doi: 10.1371/journal.pone.0160513
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