Approximate analytical solution of the fractional epidemic model

Abstract

In this paper an analytical expression for the solution of the fractional order epidemic model of a non-fatal disease in a population which is assumed to have a constant size over the period of the epidemic is presented. Homotopy analysis method (HAM) is implemented to give approximate and analytical solutions of the presented problem.

The solution of fractional order epidemic model by implicit Adams methods

“…In this paper we discussed numerical methods to obtain the solution of fractional epidemic model (3) over a long time period where HAM [2,11] is not effective. Increasing the initial conditions the problem becomes difficult to solve.…”

“…The fractional order extension of this model have been first studied in [2] , where the authors replace the first derivatives in (1) by Caputo's fractional derivative of order 0 < α ≤ 1, defined by (see e.g. [3] ), where f is a given function, and ( •) denotes the gamma function.…”

“…Some of the recent analytic methods for solving nonlinear problems like (1) and (3) include the adomian decomposition method (ADM [7] ), homotopy-perturbation method (HPM [8] ), variational iteration method (VIM [9] ) and homotopy analysis method (HAM [10] and [2,11] ). They are relatively new approaches to provide an analytical approximate solution to linear and nonlinear problems and they provide immediate and visible symbolic terms of analytic solutions.…”

“…Moreover, using real-life values for the initial conditions (2) , problem (3) may become highly stiff and then it will be necessary to employ an implicit timestepping scheme. In this situation, the use of analytic approximation by polynomials, attainable for instance by homotopy analysis method (HAM [2,11] ), seems unreliable. Here, we consider the use of the implicit fractional Adams methods of order 2, which reduces to the classical trapezoidal rule in the case of α = 1 .…”

Analysis of stability and Hopf bifurcation in a fractional Gauss-type predator–prey model with Allee effect and Holling type-III functional response

“…There are many numerical methods for solving nonlinear fractional differential equations such as the predictor-corrector method [15,88], Adomian decomposition method [16], variational iteration method [17], and Adams method [89]. However, the Adams method is often used for solving nonlinear fractional differential equations [90][91][92] and is useful for studying the dynamic behavior (especially long time behavior) of the solutions [45]. Thus, in this study, the Adams method is used to solve model (3) by the Matlab software.…”