# Successive Approximation, Variational Iteration, and Multistage-Analytical Methods for a SEIR Model of Infectious Disease Involving Vaccination Strategy

## DOI:

https://doi.org/10.5614/cbms.2020.3.2.3## Keywords:

infectious disease, multistage method, SEIR model, successive approximations, variational iterations## Abstract

We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

## References

Worldometers, COVID-19 Coronavirus Pandemic, https://www.worldometers.info/coronavirus/, accessed on 7 October 2020 at 01:30 pm GMT.

Aldila, D., Khoshnaw, S.H.A., Safitri, E., Anwar, Y.R., Bakry, A.R.Q., Samiadji, B.M., Anugerah, D.A., Alfarizi GH, M.F., Ayulani, I.D. and Salim, S.N., A mathematical study on the spread of COVID-19 considering social distancing and rapid assessment: The case of Jakarta, Indonesia, Chaos, Solitons and Fractals, 139(110042), 2020. DOI:10.1016/j.chaos.2020.110042

Asyary, A. and Veruswati, M., Sunlight exposure increased Covid-19 recovery rates: A study in the central pandemic area of Indonesia, The Science of the Total Environment, 729(139016), 2020. DOI:10.1016/j.scitotenv.2020.139016

Qureshi, S., Periodic dynamics of rubella epidemic under standard and fractional Caputo operator with real data from Pakistan, Mathematics and Computers in Simulation, 178, pp. 151–165, 2020. DOI:10.1016/j.matcom.2020.06.002

Carcione, J.M., Santos, J.E., Bagaini, C. and Ba, J., A simulation of a COVID-19 epidemic on a deterministic SEIR model, Frontiers in Public Health, 8(230), 2020. DOI:10.3389/fpubh.2020.00230

Arcede, J.P., Caga-Anan, R.L., Mentuda, C.Q. and Mammeri, Y., Accounting for symptomatic and asymptomatic in a SEIR-type model of COVID-19, Mathematical Modelling of Natural Phenomena, 15(2020021), 2020. DOI:10.1051/mmnp/2020021

He, S., Peng, Y. and Sun, K., SEIR modeling of the COVID-19 and its dynamics, Nonlinear Dynamics, 2020. DOI:10.1007/s11071-020-05743-y

Wei, F. and Xue, R., Stability and extinction of SEIR epidemic models with generalized nonlinear incidence, Mathematics and Computers in Simulation, 170, pp. 1–15, 2020. DOI:10.1016/j.matcom.2018.09.029

Sun, C. and Hsieh, Y.-H., Global analysis of an SEIR model with varying population size and vaccination, Applied Mathematical Modelling, 34(10), pp. 2685–2697, 2010. DOI:10.1016/j.apm.2009.12.005

Hou, J. and Teng, Z., Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates, Mathematics and Computers in Simulation, 79(10), pp. 3038–3054, 2009. DOI:10.1016/j.matcom.2009.02.001

Zhao, Z., Chen, L. and Song, X., Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate, Mathematics and Computers in Simulation, 79(3), pp. 500–510, 2008. DOI:10.1016/j.matcom.2008.02.007

Jansen, H. and Twizell, E.H., An unconditionally convergent discretization of the SEIR model, Mathematics and Computers in Simulation, 58(2), pp. 147–158, 2002. DOI:10.1016/S0378-4754(01)00356-1

Biswas, M.H.A., Paiva, L.T. and De Pinho, M., A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11(4), pp. 761–784, 2014. DOI:10.3934/mbe.2014.11.761

Neilan, R.M. and Lenhart, S., An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 75, pp. 67–81, 2010. DOI:10.1090/dimacs/075

Lenhart, S. and Workman, J.T., Optimal Control Applied to Biological Models, Chapman & Hall/CRC Press, Boca Raton, 2007. DOI:10.1201/9781420011418

Joshi, H.R., Lenhart, S., Li, M.Y. and Wang, L., Optimal control methods applied to disease models, Contemporary Mathematics, 410, pp. 187–207, 2006. DOI:10.1090/conm/410

Mungkasi, S., Improved variational iteration solutions to the SIR model of dengue fever disease for the case of South Sulawesi, Journal of Mathematical and Fundamental Sciences, accepted, 2020.

Mungkasi, S., Variational iteration and successive approximation methods for a SIR epidemic model with constant vaccination strategy, Applied Mathematical Modelling, 90, pp. 1–10, 2021. DOI:10.1016/j.apm.2020.08.058

Agarwal, R.P. and O’Regan, D., An Introduction to Ordinary Differential Equations, Springer, New York, 2008. DOI:10.1007/978-0-387-71276-5

He, J.-H., Variational iteration method – A kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34(4), pp. 699–708, 1999. DOI:10.1016/S0020-7462(98)00048-1

He, J.-H., Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation, 114(2–3), pp. 115–123, 2000. DOI:10.1016/S0096-3003(99)00104-6

Harir, A., Melliani, S., El Harfi, H. and Chadli, L.S., Variational iteration method and differential transformation method for solving the SEIR epidemic model, International Journal of Differential Equations, 2020(3521936), 2020. DOI:10.1155/2020/3521936