A Minimal HIV-AIDS Infection Model with General Incidence Rate and Application to Morocco Data
AbstractWe study the global dynamics of a SICA infection model with general incidence rate. The proposed model is calibrated with cumulative cases of infection by HIV–AIDS in Morocco from 1986 to 2015. We first prove that our model is biologically and mathematically well-posed. Stability analysis of different steady states is performed and threshold parameters are identified where the model exhibits clearance of infection or maintenance of a chronic infection. Furthermore, we examine the robustness of the model to some parameter values by examining the sensitivity of the basic reproduction number. Finally, using numerical simulations with real data from Morocco, we show that the model predicts well such reality.
J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol 44 (1975),331–341.
C. P. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and tuberculosis coinfection, Bull. Math. Biol. 71 (2009), no. 7,1745–1780.
R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl. 257 (2001), no. 1, 206–222.
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci. 42 (1978),no. 1-2, 43–61.
N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria throughthe sensitivityanalysis of a mathematical model, Bull. Math. Biol. 70 (2008), no. 5, 1272–1296.
P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J.North. Am. Benth. Soc. 8 (1989), 211–221.
D. L. DeAngelis, R. A. Goldstein and R. V. O’Neill, A model for tropic interaction, Ecology 56 (1975), no. 4, 881–892.
J. Djordjevic, C. J. Silva and D. F. M. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett. 84 (2018),168–175. arXiv:1805.01425
K. Hattaf, A. A. Lashari, Y. Louartassi and N. Yousfi, A delayed SIR epidemic model with general incidence rate, Electron. J. Qual.Theory Differ. Equ. 2013 (2013), no. 3, 9 pp.
K. Hattaf, M. Mahrouf, J. Adnani and N. Yousfi, Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Phys. A 490 (2018), 591–600.
K. Hattaf and N. Yousfi, Global dynamics of a delay reaction -diffusion model for viral infection with specific functional response,
Comput. Appl. Math. 34 (2015), no. 3, 807–818.
K. Hattaf and N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response, Int. J.Dyn. Control 4 (2016), no. 3, 254–265.
K. Hattaf, N. Yousfi and A. Tridane, Mathematical analysis of a virus dynamics model with general incidence rate and cure rate,Nonlinear Anal. Real World Appl. 13 (2012), no. 4, 1866–1872.
K. Hattaf, N. Yousfi and A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Appl. Math. Comput. 221 (2013), 514–521.
K. Hattaf, N. Yousfi and A. Tridane, A delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst. 22 (2014),no. 2, 181–190.
C. Ji and D. Jiang, Threshold behaviour of a stochastic SIR model, Appl. Math. Model. 38 (2014), no. 21-22, 5067–5079.
W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part I, Proc. Roy. Soc. Edinburgh A 115 (1927), 700–721.
J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976.
X. Liu and L. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Anal. Real World Appl.13 (2012), no. 6, 2671–2679.
X.-Q. Liu, S.-M. Zhong, B.-D. Tian and F.-X. Zheng, Asymptotic properties of a stochastic predator-prey model with Crowley-Martin functional response, J. Appl. Math. Comput. 43 (2013), no. 1-2, 479-490.
E. M. Lotfi, M. Maziane, K. Hattaf and N. Yousfi, Partial differential equations of an epidemic model with spatial diffusion, Int. J.Partial Differ. Equ. 2014 (2014), Art. ID 186437, 6 pp.
E. M. Lotfi, M. Maziane, M. Mahrouf, K. Hattaf and N. Yousfi, Global stability of a diffused SIR epidemic model with general incidence rate and time delay, Int. J. Math. Anal. (Ruse) 10 (2016), no. 17, 807–816.
M. Mahrouf, K. Hattaf and N. Yousfi, Dynamics of a stochastic viral infection model with immune response, Math. Model. Nat.Phenom. 12 (2017), no. 5, 15–32.
J. P. Mateus, P. Rebelo, S. Rosa, C. M. Silva and D. F. M. Torres, Optimal control of non-autonomous SEIRS models with vaccination and treatment, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 6, 1179-1199. arXiv:1706.06843
M. Maziane, E. M. Lotfi, K. Hattaf and N. Yousfi, Dynamics of a class of HIV infection models with cure of infected cells in eclipse stage, Acta Biotheoretica 63 (2015), no. 4, 363–380.
Ministry of Health,Morocco,Department of Epidemiology and Disease Control, http://www.sante.gov.ma/Pages/Accueil.aspx
A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387(1997), 188–191.
Population Data, http://www.populationdata.net
Population Data, Maroc,http://www.populationdata.net/pays/maroc
H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Sensitivity analysis in a dengue epidemiological model, Conf. Papers in Math.2013 (2013), Art. ID 721406, 7 pp. arXiv:1307.0202
O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment, Math. Biosci. Eng. 5 (2008), no. 1, 145–174.
C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst. 35(2015), no. 9, 4639–4663. arXiv:1501.03322
C. J. Silva and D. F. M. Torres, A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecological Complexity 30 (2017), 70–75. arXiv:1612.00732
UNAIDS,Fast-Track-Ending the AIDS epidemic by 2030,2014. http://www.unaids.org/sites/default/files/media_asset/JC2686_WAD2014report_en.pdf
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.
J.-J. Wang, J.-Z. Zhang and Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonlinear Anal. Real World Appl. 11(2010), no. 4, 2390–2402.
World Bank Data, Morocco, http://data.worldbank.org/country/morocco
Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett. 34 (2014),90–93.
X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, Bull. Korean Math. Soc. 48(2011), no. 3, 555–574.
M. Zwahlen and M. Egger, Progression and mortality of untreated HIV-positive individuals living in resource-limited settings: update of literature review and evidence synthesis, Report on UNAIDS obligation no HQ/05/422204, Bern, 2006.
Ministère de la santé du Royaume du Maroc, Analyse des modes de tranmission du VIH au Maroc, 2010.
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