Numerical Modeling of SEIR Measles Dynamics with Diffusion

Nauman Ahmed, M. Rafiq, M. A. Rehman, Mubasher Ali, M. O. Ahmad

Abstract


A novel unconditionally positive finite difference (FD) scheme is developed to solve numerically SEIR measles epidemic model with diffusion. In population dynamics, positivity of subpopulations is an essential requirement. The proposed FD scheme preserves the positivity of the solution of the model. The consistency and unconditional stability is proved. The proposed FD scheme is explicit in nature which is an extra feature of this scheme. Comparisons are also made with forward Euler explicit FD scheme and Crank Nicolson implicit FD scheme. Simulations of a numerical test are also presented to verify all the attributes of the proposed scheme.

Keywords


SEIR Measles epidemic model with diffusion; Finite difference scheme; Positivity; Consistency; Stability

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References


L.J.S. Allen, B.M. Bolker, Y. Lou and A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. A 21 (2008), 1 – 20, DOI: 10.3934/dcds.2008.21.1.

F. Al-Showaikh and E. Twizell, One-dimensional measles dynamics, Appl. Math. Comput. 152: 169-1942004, DOI: 10.1016/S0096-3003(03)00554-X.

R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, DOI: 1992.10.1002/hep.1840150131.

A.R. Appadu, Jean M.-S. Lubuma and N. Mphephu, Computation study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems, Prog. Comput. Fluid Dyn. 17 (2017), DOI: 10.1504/PCFD.2017.082520.

R. Bhattacharyya, B. Mukhopadhyay and M. Bhattacharyya, Diffusion-driven stability analysis of a prey-predictor system with Holling type-IV functional response, Syst. Anal. Model. Smul. 43 (8) (2003), 1085 – 1093, DOI: 10.1080/0232929031000150409

H. Cai and X. Luo, Stochastic control of an epidemic process, Int. J. Syst. Sci. 25 (1994), 821 – 828, DOI: 10.1080/00207729408928999.

B.M. Chen-Charpentier and H.V. Kojouharov, Unconditionally positivity preserving scheme for advection-diffusion-reaction equations, Math. Comput. Modeling 57 (2013), 2177 – 1285, DOI: 10.1016/j.mcm.2011.05.005.

S. Chinviriyasit and W. Chinviriyasit, Numerical modeling of SIR epidemic model with diffusion, Appl. Math. Comp. 216 (2010), 395 – 409, DOI: 10.1016/j.amc.2010.01.028.

H. Jansen and E.H. Twizell, An unconditionally convergent discretization of the SEIR model, Math Comput. Simulat. 58 (2002), 147 – 158, DOI: 10.1016/S0378-4754(01)00356-1.

R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific (1994).

S.M. Moghadas, M.E. Alexander, B.D. Corbett and A.B. Gumel, A positivity-preserving Mickenstype discretization of an epidemic model, J. Differ. Equ. Appl. 9 (2003), 1037 – 1051, DOI: 10.1080/1023619031000146913.

M. Rafiq, M.O. Ahmad and S. Iqbal, Numerical modeling of internal transmission dynamics of dengue virus, 13th International Bhurban Conference on Applied Sciences and Technology (IBCAST), Islamabad, 2016, 85 – 91, DOI: 10.1109/IBCAST.2016.7429859.

A. Suryanto, W.M. Kusumawinahyu, I. Darti and I. Yanti, Dynamically consistent discrete epidemic model with modified saturated incidence rate, Comp. Appl. Math. 32 (2013), 373 – 383, DOI: 10.1007/s40314-013-0026-6.




DOI: http://dx.doi.org/10.26713%2Fcma.v9i3.794

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