An epidemiological MSEIR model described by the Caputo fractional derivative

  • Ricardo AlmeidaEmail author
  • Artur M. C. Brito da Cruz
  • Natália Martins
  • M. Teresa T. Monteiro


A fractional MSEIR model is presented, involving the Caputo fractional derivative. The equilibrium points and the basic reproduction number are computed. An analysis of the local asymptotic stability at the disease free equilibrium is given. Finally a numerical simulation, using Matlab based on optimization techniques, of the varicella outbreak among Shenzhen school children, China, is carried out.


Fractional calculus Fractional differential equations Epidemics MSEIR model Local stability 

Mathematics Subject Classification

26A33 92D30 37N25 



R. Almeida, A.M.C. Brito da Cruz and N. Martins were supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2019; M. T. Monteiro by COMPETE: POCI-01-0145-FEDER-007043 and FCT-Fundação para a Ciência e a Tecnologia within the Project Scope: UID/CEC/00319/2013. The authors are grateful to two anonymous referees for valuable comments and suggestions. ©2017 The MathWorks, Inc. MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders.


  1. 1.
    Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, OxfordGoogle Scholar
  2. 2.
    Bailey NTJ (1975) The mathematical theory of infectious diseases and its application. Griffin, LondonzbMATHGoogle Scholar
  3. 3.
    Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A 115:700–721zbMATHGoogle Scholar
  4. 4.
    Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics, II—the problem of endemicity. Proc R Soc Lond A 138:55–83zbMATHGoogle Scholar
  5. 5.
    Kermack WO, McKendrick AG (1933) Contributions to the mathematical theory of epidemics, III—further studies of the problem of endemicity. Proc R Soc Lond A 141:94–122zbMATHGoogle Scholar
  6. 6.
    Ackerman E, Elveback LR, Fox JP (1984) Simulation of infectious disease epidemics. Charles C. Thomas Publisher, SpringfieldGoogle Scholar
  7. 7.
    West RW, Thompson JR (1997) Models for the simple epidemic. Math Biosci 141:29–39zbMATHGoogle Scholar
  8. 8.
    Esteva L, Vargas C (1998) Analysis of a dengue disease transmission model. Math Biosci 150:131–151zbMATHGoogle Scholar
  9. 9.
    MacDonald G (1957) The epidemiology and control of malaria. Oxford University Press, LondonGoogle Scholar
  10. 10.
    Boots M, Norman R (2000) Sublethal infection and the population dynamics of host–microparasite interactions. J Anim Ecol 69:517–524Google Scholar
  11. 11.
    Feng Z, Towers S, Yang Y (2011) Modeling the effects of vaccination and treatment on pandemic influenza. Am Assoc Pharm Sci J 13(3):427–437Google Scholar
  12. 12.
    Glasser J, Taneri D, Feng Z, Chuang J-H, Tüll P et al (2010) Evaluation of targeted influenza vaccination strategies via population modeling. PLoS ONE 5(9):e12777Google Scholar
  13. 13.
    Lipsitch M, Cohen T, Muray M, Levin BR (2007) Antiviral resistance and the control of pandemic influenza. PLoS Med 4:01110120Google Scholar
  14. 14.
    Gray A, Greenhalgh D, Hu L, Mao X, Pan J (2011) A stochastic differential equation SIS epidemic model. SIAM J Appl Math 71(3):876–902MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zhang X, Liu X (2009) Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment. Nonlinear Anal 2:565–575MathSciNetzbMATHGoogle Scholar
  16. 16.
    Korobeinikov A (2006) Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull Math Biol 68(3):615–626MathSciNetzbMATHGoogle Scholar
  17. 17.
    Liu W, Levin SA, Iwasa Y (1986) Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J Math Biol 23(2):187–204MathSciNetzbMATHGoogle Scholar
  18. 18.
    Korobeinikov A, Maini PK (2004) A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math Biosci Eng 1(1):57–60MathSciNetzbMATHGoogle Scholar
  19. 19.
    Li MY, Graef JR, Wang L, Karsai J (1999) Global dynamics of a SEIR model with varying total population size. Math Biosci 160(2):191–213MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hisashi I (2007) Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model. J Math Biol 54:101–146MathSciNetzbMATHGoogle Scholar
  21. 21.
    Pinto CMA, Carvalho ARN (2017) A latency fractional order model for HIV dynamics. J Comput Appl Math 312:240–256MathSciNetzbMATHGoogle Scholar
  22. 22.
    Zafar ZUA, Rehan K, Mushtaq M (2017) HIV/AIDS epidemic fractional-order model. J Differ Equ Appl 23(7):1298–1315MathSciNetzbMATHGoogle Scholar
  23. 23.
    Latha VP, Rihan FA, Rakkiyappan R, Velmurugan G (2017) A fractional-order delay differential model for Ebola infection and CD8 T-cells response: stability analysis and Hopf bifurcation Int. J Biomath 10:1750111MathSciNetzbMATHGoogle Scholar
  24. 24.
    Latha VP, Rihan FA, Rakkiyappan R, Velmurugan G (2018) A fractional-order model for Ebola virus infection with delayed immune response on heterogeneous complex networks. J Comput Appl Math 339:134–146MathSciNetzbMATHGoogle Scholar
  25. 25.
    El-Shahed M, Alsaedi A (2011) The fractional SIRC model and influenza A. Math Probl Eng 3:378–387MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ahmed E, Hashish A, Rihan FA (2012) On fractional order cancer model. Fract Calc Appl Anal 3:1–6Google Scholar
  27. 27.
    Pooseh S, Rodrigues HS, Torres DFM (2011) Fractional derivatives in dengue epidemics. In: Simos TE, Psihoyios G, Tsitouras C, Anastassi Z (eds) Numerical analysis and applied mathematics, ICNAAM. American Institute of Physics, Melville, pp 739–742Google Scholar
  28. 28.
    Al-Sulami H, El-Shahed M, Nieto JJ, Shammak W (2014) On fractional order dengue epidemic model. Math Probl Eng. Article ID: 456537Google Scholar
  29. 29.
    Pinto CMA, Machado JAT (2013) Fractional model for malaria disease. In: Proceedings of the ASME 2013 international design engineering technical conferences and computers and information in engineering conference IDETC/CIE 01/2013Google Scholar
  30. 30.
    Rihan F. A, Baleanu D, Lakshmanan S, Rakkiyappan R (2014) On fractional SIRC model with salmonella bacterial infection. Abst Appl Anal. Article ID: 136263Google Scholar
  31. 31.
    Ndaïrou F, Area I, Nieto JJ, Silva CJ, Torres DFM (2017) Mathematical modeling of Zika disease in pregnant women and newborns with microcephaly in Brazil. Math Meth Appl Sci.
  32. 32.
    Angstmann CN, Henry BI, McGann AV (2016) A fractional order recovery SIR model from a stochastic process. Bull Math Biol 78(3):468–499MathSciNetzbMATHGoogle Scholar
  33. 33.
    Arenas AJ, González-Parrab G, Chen-Charpentierc BM (2016) Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order. Math Comput Simul 121:48–63MathSciNetGoogle Scholar
  34. 34.
    Huo J, Zhao H (2016) Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks. Phys A 448:41–56MathSciNetzbMATHGoogle Scholar
  35. 35.
    Goufo EFD, Maritz R, Munganga J (2014) Some properties of Kermack–McKendrick epidemic model with fractional derivative and nonlinear incidence. Adv Differ Equ. No 1. Article ID 278Google Scholar
  36. 36.
    Mouaouine A, Boukhouima A, Hattaf K, Yousfi N (2018) A fractional order SIR epidemic model with nonlinear incidence rate. Adv Differ Equ 2018(1):160MathSciNetzbMATHGoogle Scholar
  37. 37.
    Okyere E, Oduro FT, Amponsah SK, Dontwi IK, Frempong NK (2016) Fractional order SIR model with constant population. Br J Math Comput Sci 14(2):1–12Google Scholar
  38. 38.
    Santos JPC, Cardoso LCEM, Lemes NHT (2015) A fractional-order epidemic model for Bovine Babesiosis disease and tick populations. Abst Appl Anal. Article ID 729894Google Scholar
  39. 39.
    Sardar T, Rana S, Bhattacharya S, Al-Khaled K, Chattopadhyay J (2015) A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector. Math Biosci 263(Supplement C):18–36MathSciNetzbMATHGoogle Scholar
  40. 40.
    El-Shahed M, El-Naby FA (2014) Fractional calculus model for childhood diseases and vaccines. Appl Math Sci 8(98):4859–4866Google Scholar
  41. 41.
    El-Shahed M, Alsaedi A (2011) The fractional SIRC model and Influenza A. Math Probl Eng. Article ID 480378Google Scholar
  42. 42.
    Pinto CMA, Machado JAT (2013) Fractional model for malaria transmission under control strategies. Comput Math Appl 66:908–916MathSciNetGoogle Scholar
  43. 43.
    Kumar R, Kumar S (2013) A new fractional modelling on susceptible-infected-recovered equations with constant vaccination rate. Nonlinear Eng 3(1):11–19Google Scholar
  44. 44.
    Rostamy D, Mottaghi E (2016) Stability analysis of a fractional-order epidemic model with multiple equilibriums. Adv Differ Equ 2016:170MathSciNetzbMATHGoogle Scholar
  45. 45.
    González-Parra G, Arenas AJ, Chen-Charpentier BM (2014) A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1). Math Methods Appl Sci 37(15):2218–2226MathSciNetzbMATHGoogle Scholar
  46. 46.
    Özalp N, Demirci E (2011) A fractional order SEIR model with vertical transmission. Math Comput Model 54(1):1–6MathSciNetzbMATHGoogle Scholar
  47. 47.
    Asfour HA, Ibrahim M (2015) On the differential fractional transformation method of MSEIR epidemic model. Int J Comput Appl 113(3):10–16Google Scholar
  48. 48.
    Gómez-Aguilar JF, López-López MG, Alvarado-Martínez VM, Baleanu D, Khan H (2016) Chaos in a cancer model via fractional derivatives with exponential decay and Mittag–Leffler law. Entropy 19(2):1–19MathSciNetGoogle Scholar
  49. 49.
    Copot D, Ionescu CM, De Keyser R (2014) Relation between fractional order models and diffusion in the body. IFAC Proc 47(3):9277–9282Google Scholar
  50. 50.
    Ionescu CM, Lopes A, Copot D, Machado JAT, Bates JHT (2017) The role of fractional calculus in modelling biological phenomena: a review. Commun Nonlinear Sci Numer Simul 51:141–159MathSciNetGoogle Scholar
  51. 51.
    Magin RL (2006) Fractional calculus in bioengineering. Begell House, ConnecticutGoogle Scholar
  52. 52.
    Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59(5):1586–1593MathSciNetzbMATHGoogle Scholar
  53. 53.
    Navarro-Guerrero G, Tang Y (2017) Fractional order model reference adaptive control for anesthesia. Int J Adapt Control Signal Process 31(9):1350–1360MathSciNetzbMATHGoogle Scholar
  54. 54.
    Diethelm K (2010) The analysis of fractional differential equations. Springer, BerlinzbMATHGoogle Scholar
  55. 55.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., AmsterdamGoogle Scholar
  56. 56.
    Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, translated from the 1987 Russian original. Gordon and Breach, YverdonzbMATHGoogle Scholar
  57. 57.
    Diethelm K (2012) The mean value theorems and a Nagumo-type uniqueness theorem for Caputo’s fractional calculus. Fract Calc Appl Anal 15(2):304–313MathSciNetzbMATHGoogle Scholar
  58. 58.
    Ye H, Gao J, Ding Y (2007) A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl 328(2):1075–1081MathSciNetzbMATHGoogle Scholar
  59. 59.
    Almeida R (2017) A Gronwall inequality for a general Caputo fractional operator. Math Inequal Appl 20(4):1089–1105MathSciNetzbMATHGoogle Scholar
  60. 60.
    Schuette MC, Hethcote HW (1999) Modeling the effects of varicella vaccination programs on the incidence of chickenpox and shingles. Bull Math Biol 61:1031–1064zbMATHGoogle Scholar
  61. 61.
    Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653MathSciNetzbMATHGoogle Scholar
  62. 62.
    Diethelm K (2013) A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn 71(4):613–619MathSciNetGoogle Scholar
  63. 63.
    Dokoumetzidis A, Magin R, Macheras P (2010) A commentary on fractionalization of multi-compartmental models. J Pharmacokinet Pharmacodyn 37:203–207Google Scholar
  64. 64.
    Popovic JK, Atanackovic MT, Pilipovic AS, Rapaic MR, Pilipovic S (2010) A new approach to the compartmental analysis in pharmacokinetics: fractional time evolution of diclofenac. J Phamacokinet Pharmacodyn 37:119–134Google Scholar
  65. 65.
    Lin W (2007) Global existence theory and chaos control of fractional differential equations. J Math Anal Appl 332(1):709–726MathSciNetzbMATHGoogle Scholar
  66. 66.
    Ahmed E, El-Sayed AMA, El-Saka HAA (2007) Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J Math Anal Appl 325(1):542–553MathSciNetzbMATHGoogle Scholar
  67. 67.
    Matignon D (1996) Stability results for fractional differential equations with applications to control processing. Comput Eng Syst Appl (Lille, France) 2:963–968Google Scholar
  68. 68.
    Tang X, Zhao S, Chiu APY, Ma H, Xie X, Mei S, Kong D, Qin Y, Chen Z, Wang X, He D (2017) Modelling the transmission and control strategies of varicella among school children in Shenzhen China. PLoS ONE 12(5):1–17Google Scholar
  69. 69.
    Diethelm K, Ford NJ, Freed AD (2002) A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(3):3–22MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA)University of AveiroAveiroPortugal
  2. 2.Escola Superior de Tecnologia de SetúbalSetúbalPortugal
  3. 3.Department of Production and Systems, Algoritmi R&D CenterUniversity of MinhoBragaPortugal

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