In this paper, the fractional order model was adopted to describe the dynamics of measles and to establish how the virus that causes measles is transmitted as well as how to mitigate the conditions that cause the spread. We showed the existence of the equilibrium states. The threshold parameter of the model was evaluated in terms of parameters in the model using the next generation matrix approach. We provided the conditions for the stability of the disease free and the endemic equilibrium points. Also, the stability of the various equilibrium points was studied. Numerical simulations of the model are presented graphically using Adam-Bashforth Method and the results were interpreted. The result also shows that the use of vaccination is the best way to prevent measles outbreak.
Cite this paper
Aguegboh, N. S. , Nwokoye, N. R. , Onyiaji, N. E. , Amanso, O. R. and Oranugo, D. O. (2020). A Fractional Order Model for the Transmission Dynamics of Measles with Vaccination. Open Access Library Journal, 7, e6670. doi: http://dx.doi.org/10.4236/oalib.1106670.
Haq, F., Shahzad, M., Muhammad, S., Wahab, H.A. and Rahman, G. (2017) Numerical Analysis of Fractional Order Epidemic Model of Childhood Diseases. Discrete Dynamics in Nature and Society, 2017, Article ID: 4057089.
https://doi.org/10.1155/2017/4057089
Goufo, E.F.D., Noutchie, S.C.O. and Mugisha, S. (2014) A Fractional SEIR Epidemic Model for Spatial and Temporal Spread of Measles in Metapopulations. Hindawi Publishing Corporation. Abstract and Applied Analysis, 2014, Article ID: 781028.
https://doi.org/10.1155/2014/781028
Nazir, G., Shah, K., Alrabaiah, H., Khalil, H. and Khan, R.A. (2020) Fractional Dynamical Analysis of Measles Spread Model under Vaccination Corresponding to Nonsingular Fractional Order Derivative. Advances in Difference Equations, No. 1, 1-15. https://doi.org/10.1186/s13662-020-02628-7
Farman, M., Saleem, M.U., Ahmad, A.L. and Ahmad, M.O. (2018) Analysis and Numerical Solution of SEIR Epidemic Model of Measles with Non-Integer Time Fractional Derivatives by Using Laplace Adomian Decomposition Method. Ain Shams Engineering Journal, 9, 3391-3397. https://doi.org/10.1016/j.asej.2017.11.010
Ahmed, E., EI-Sayed, A.M.A. and EI-Saka, H.A.A. (2006) On Some Routh-Hurwitz conditions for Fractional Order Differential Equations and Their Applications in Lorenz, Rossler, Chua and Chen Systems. Physics Letters A, No. 358, 1-4.
https://doi.org/10.1016/j.physleta.2006.04.087
El-Sayed, A.M.A., Arafa, A.M.A., Kahlil, M. and Hassan, A. (2016) A Mathematical Model with Memory for Propagation of Computer Virus under Human Intervention. Progress in Fractional Differentiation and Applications, No. 2, 105-113.
https://doi.org/10.18576/pfda/020203
Van Den Driessche, P. and Watmough, J. (2002) Reproduction Numbers and Subthreshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48.
https://doi.org/10.1016/S0025-5564(02)00108-6
Diekmann, J.A., Heesterbeek, P. and Metz, J.A.J. (1990) On the Definition and the Computation of the Basic Reproduction Ratio in Models for Infectious Diseases in Heterogeneous Populations. Mathematical Biosciences, 28, 363-382.
https://doi.org/10.1007/BF00178324
Heffernan, J.M., Smith, R.J. and Wahl, L.M. (2005) Perspectives on the Basic Reproduction Ratio. Journal of the Royal Society Interface, 2, 281-293.
https://doi.org/10.1098/rsif.2005.0042
El-Saka, H.A.A. (2014) The Fractional-Order SIS Epidemic Model with Variable Population Size. Journal of the Egyptian Mathematical Society, 22, 50-54.
https://doi.org/10.1016/j.joems.2013.06.006
Kitengeso, R.E. (2016) Stochastic Modelling of the Transmission Dynamics of Measles with Vaccination Control. International Journal of Theoretical and Applied Mathematics, 2, 60-73.
