Fractional order models of infectious diseases: a review

  • Abstract
  • Keywords
  • References
  • Untitled
  • PDF
  • Abstract

    The aim of this paper is to present a succinct review on fractional order models of infectious diseases. Fractional order derivative is a potential tool which gives a better understanding of the impact of memory on spread of infectious diseases. This paper reviews different infectious diseases models with constant, variable or complex fractional order. Fractional order models with time delay are presented in this paper as well. We argue that, such models are essential for decision makers in health organizations.




  • Keywords

    Constant/Variable Fractional Order Models-Models with Complex Fractional Order -Fractional Order Models with Time Delay-Infectious Diseases Models with Memory.

  • References

      [1] Cho E. Ahmed, A.M.A. El-Sayed and H.A.A. El-Saka, on some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and Chen systems, Phys. Lett. A358, 1–4 (2006).

      [2] E. Ahmed and H.A. El-Saka, on fractional order models for Hepatitis C, Nonlinear Biomed. Phys. 4 (2010).

      [3] E. Ahmed, A.M.A. El-Sayed, H.A.A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models, J. Math. Anal. Appl. 325 (2007) 542–553.

      [4] E. Ahmed and A.S. Elgazzar, on fractional order differential equations model for nonlocal epidemics, Physica A 379 (2007) 607–614.

      [5] C. N. Angstmann, B. I. Henry, A. V. McGann, A Fractional Order Recovery SIR Model from a Stochastic Process, Bull Math Biol (2016) 78:468–499.

      [6] A.A.M. Arafa, S.Z. Rida, M. Khalil, A Fractional-Order Model of HIV Infection: Numerical solution and Comparisons with Data of Patients, Int. J. Biomath, 7 (4) (2014).

      [7] A.A.M. Arafa, S.Z. Rida and M. Khalil, Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection, Nonlinear Biomed. Phys 6(2012).

      [8] A.A.M. Arafa, S.Z. Rida and M. Khalil, The effect of anti-viral drug treatment of human immunodeficiency, Appl. Math. Model, 37 (2013) 2189–2196.

      [9] A.A.M. Arafa, S.Z. Rida and M. Khalil, Fractional Order Model of Human T-cell Lymphotropic Virus I (HTLV-I) Infection of CD4+T-cells, Advanced Studies in Biology, 3(2011) 347 – 353.

      [10] A.A.M. Arafa, S.Z. Rida, M.Khalil, Solutions of Fractional Order Model of Childhood diseases with Constant Vaccination Strategy, Math. Sci. Lett. No. 1 1(2012) 17-23.

      [11] A.A.M. Arafa, S.Z. Rida, M.Khalil, A fractional-order model of HIV infection with drug therapy effect, J. Egyptian Math. Soc., 22(2014) 538–543.

      [12] I. Area, H. Batarfi, J. Losada, J. J Nieto, W. Shammakh, A. Torres, On a fractional order Ebola epidemic model, Adv. Difference Equ. 278(2015).

      [13] J. Arino, I.A. Soliman, A model for the spread of tuberculosis with drug-sensitive and emerging multidrug-resistant and extensively drug resistant strains, in Mathematical and computational modelling with applications in natural and social sciences, engineering and arts, Wiley; 2014.

      [14] S. Crotty, R. Ahmed, Immunological memory in humans, Seminars in Immunology 16 (2004) 197–203.

      [15] R.V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci.165 (2000) 27-39.

      [16] H. Dahari, A. Lo A, R.M. Ribeiro, A.S. Perelson: Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy. J Theoret Biol 2007, 247:371-381.

      [17] E. Demirci, A. Unal and N. Ozalp, A fractional order SEIR model with density dependent death rate, Hacet. J. Math. Stat. 40 (2011), 287–295.

      [18] K. Diethelm, A fractional calculus-based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn, 71(2013), 613–619.

      [19] Y. Ding, H. Ye, A fractional-order differential equation model of HIV infection of CD4+ T-cells, Math. Comput. Modelling, 50 (2009) 386-392.

      [20] A. O’Dowd, Infectious diseases are spreading more rapidly than ever before, WHO warns. BMJ 335 (2007)418.

      [21] M. Du, Z. Wang and H. Hu, Measuring memory with the order of fractional derivative. Sci. Rep. 3(2013).

      [22] M. Glomski, E. Ohanian, Eradicating a Disease: Lessons from Mathematical Epidemiology. The College Mathematics Journal, 43.2 (2012): 123-132.

      [23] N.C. Grassly, C. Fraser, Mathematical models of infectious disease transmission. Nat Rev Microbiol 2008; 6(6):477-487.

      [24] A.H. Hashish, E. Ahmed, towards understanding the immune system, Theor. Biosci. 126 (2–3) (2007) 61–64.

      [25] H.W. Hethcote, the Mathematics of Infectious Diseases, SIAM Rev. 42 (2000) 599.

      [26] G. M. Hwang, P.J. Mahoney, J.H. James, G.C. Lin, A.D. Berro, M.A. Keyb, D.M. Goedecke, J.J. Mathieu, T. Wilson: A model-based tool to predict the propagation of infectious disease via airports. Travel Med Infect Dis. 10(1) (2012)32-42.

      [27] R. M. G. J. Houben, D. W. Dowdy, A. Vassall, T. Cohen, M. P. Nicol, R. M. Granich, J. E. Shea, P. Eckhoff, C. Dye, M. E. Kimerling, R. G. White, How can mathematical models advance tuberculosis control in high HIV prevalence settings?, International Journal of Tuberculosis and Lung Disease 18 (2014) 509–514.

      [28] J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Anal. Real World Appl. 26 (2015) 289–305

      [29] A. Huppert A, G. Katriel, Mathematical modelling and prediction in infectious disease epidemiology. Clin Microbiol Infect. 2013; 19:999–1005.

      [30] M. Jit, M. Brisson, Modelling the epidemiology of infectious diseases for decision analysis: a primer. Pharmaco Economics 29(5), 371–386 (2011).

      [31] J. CA Jr, et al: The immune system in health and disease. In Immunobiology. Garland Pub, NewYork: 2001.

      [32] P. S. Kim, D. Levy, and P. P. Lee, “Modeling and simulation of the immune system as a self-regulating network,” Methods in Enzymology, vol. 467, no. C, pp. 79–109, 2009.

      [33] I.K. Kouadio, S. Aljunid, T. Kamigaki, K. Hammad, H. Oshitani, Infectious disease following natural disasters: prevention and control measures. Expert Rev Anti Infect Ther 10(2012); 95–104.

      [34] Z. Liu, P. Lu, Stability analysis for HIV infection of CD4+T-cells by a fractional differential time-delay model with cure rate, Advances in Difference Equations 2014, 2014:298.

      [35] Y. Liu, P. Lu, and I. Szanto, Numerical Analysis for a Fractional Differential Time-Delay Model of HIV Infection of CD4+ T-Cell Proliferation under Antiretroviral Therapy.

      [36] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review, Malar J, 10 (2011) 10:202.

      [37] A.K. Misra, Anupama Sharma, J.B. Shukla, Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Mathematical and Computer Modelling 53 (2011) 1221–1228.

      [38] K. Moaddy, A.G. Radwan, K.N. Salama, S. Momani, I. Hashim, the fractional-order modeling and synchronization of electrically coupled neuron systems, Comput. Math. Appl., 64 (2012) 3329–3339.

      [39] P. A. Morel, S. Ta’asan, B. F. Morel, D. E. Kirschner, and J.L. Flynn, “New insights into mathematical modeling of the immune system,” Immunologic Research, vol. 36, no. 1–3, pp.157–165, 2006.

      [40] Z. Mukandavire,W. Garira, J.M. Tchuenche, Modelling effects of public health educational campaigns on HIV/AIDS transmission dynamics, Applied Mathematical Modelling, 33(2009) 2084–2095.

      [41] H. Nishiura, Mathematical and statistical analyses of the spread of dengue. In: Dengue Bulletin, 30(2006) 51–67.

      [42] D. Okuonghae and B. O. Ikhimwin, Dynamics of a Mathematical Model for Tuberculosis with Variability in Susceptibility and Disease Progressions Due to Difference in Awareness Level, Front. Microbiol., (2016).

      [43] C.M.A. Pinto, J.A. Tenreiro Machado, Fractional model for malaria transmission under control strategies, Comput. Math. Appl. 66 (5) (2013) 908–916.

      [44] C.M.A. Pinto, A.R.M. Carvalho, Effect of drug-resistance in a fractional complex-order model for HIV infection, IFAC, 48(2015) 188–189.

      [45] D.A. Relman, R.R. Choffnes, A. Mack, Infectious disease movement in a borderless world: workshop summary. The National Academies Press, Washington, DC; 2010.

      [46] S.Z. Rida, A.S. Abdel Rady, A.A.M. Arafa, M. Khalil, Approximate solution of a fractional order model of HCV infection with drug therapy effect, International Journal of Applied Mathematical Research, 1 (2) (2012) 108-116.

      [47] L. Rong, Z. Feng, A.S. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment. Bull Math Biol 2007; 69:2027–2060.

      [48] M. Salah, Abdelouahab and N.E Hamri, The Grunwald-Letnikov Fractional-Order Derivative with Fixed Memory Length, Mediterr. J. Math., (2015) 1-16.

      [49] T. Sardar, S. Rana, S. Bhattacharya, K. Al-Khaled, J. Chattopadhyay, A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector, Math. Biosci. 263 (2015) 18–36.

      [50] A. Scherer and A. R. McLean, Mathematical models of vaccination, Brit. Med. Bull., 62 (2002) 187–199.

      [51] H.G. Sun, W. Chen, H. Wei, and Y.Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Special Topics 193, (2011) 185–192.

      [52] N.H. Sweilam, S. M. AL-Mekhlafi, Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives, J. Adv. Res. 7(2016) 271–283.

      [53] N.H. Sweilam, S.M. Al-Mekhlafi, and T.A.R Assiri, “Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order,” Complexity, 2017.

      [54] J.K. Taubenberger, D.M. Morens, 1918 Influenza: the mother of all pandemics. Emerg. Infect. Dis. 12 (2006) 15–22.

      [55] J. M. Tchuenche and C. T. Bauch, Dynamics of an Infectious Disease Where Media Coverage Influences Transmission, ISRN Biomathematics, (2012).

      [56] Mm C. Vinauger, E.K. Lutz and J. A. Riffell, Olfactory learning and memory in the disease vector mosquito Aedes aegypti, The Journal of Experimental Biology 217(2014) 2321-2330.

      [57] Y. Xu, Z. He, Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations, J Appl Math Comput, 43(1) (2013) 295-306.

      [58] M. Xu, J. Yang, D. Zhao and H. Zhao, an image-enhancement method based on variable-order fractional differential operators, Bio-Medical Materials and Engineering 26 (2015) S1325–S1333.

      [59] Y. Yan and C. Kou, “Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay,” Mathematics and Computers in Simulation, vol. 82, no. 9, pp. 1572–1585, 2012.

      [60] Africa Renewal: Ebola threatens economic gains in affected countries, 2014.

      [61] Global Health Policy: The U.S. Government & Global Emerging Infectious Disease Preparedness and Response, (2014).

      [62] London school of Hygine & Tropical medicine: Introduction to Infectious Disease Modelling and Its Applications, 2015.

      [63] Virginia Department of Health: Vector-borne Disease Control (2015):

      [64] World Health Organization, 2011. Dengue Net database and geographic information system. [Online; accessed 1-October-2013].

      [65] World Health Organization: Vector-borne diseases (2014).

      [66] World Health Organization: Ebola Situation Reports, 2015.

      [67] World Health Organization: HIV (2015).

      [68] World Health Organization: What is TB? How is it treated? (2015).

      [69] World Health Organization: Tuberculosis (2016).

      [70] Who organization, Zika virus (2016).

      [71] World Health Organization: Malaria (2017).

  • The PDF file you selected should load here if your Web browser has a PDF reader plug-in installed (for example, a recent version of Adobe Acrobat Reader).

    If you would like more information about how to print, save, and work with PDFs, Highwire Press provides a helpful Frequently Asked Questions about PDFs.

    Alternatively, you can download the PDF file directly to your computer, from where it can be opened using a PDF reader. To download the PDF, click the Download link above.





Article ID: 19436
DOI: 10.14419/jes.v1i1.19436

Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.