1. Introduction
Most problems arising from mathematical epidemiology are often described in terms of differential equations. However, it is often very difficult to obtain closed form solutions of such equations, especially those that are nonlinear. In most cases, attempts are made to obtain only approximate or numerical solutions. In this work, we revisit the SIR epidemic model with constant vaccination strategy that was considered in [11], where the Adomian decomposition method was used to solve the governing system of nonlinear initial value differential equations.
In this work we develop new accurate iterative schemes which are based on extending Taylor series based linearization method to obtain accurate and fast converging sequence of hybrid iteration schemes. At first order, the hybrid iteration scheme reduces to quasilinearization method (QLM) which was originally developed in [1]. More recently Mandelzweig and his co-workers [8–10] have extended the application of the QLM to a wide variety of nonlinear BVPs and established that the method converges quadratically. In this work we demonstrate that the proposed hybrid iteration schemes are more accurate and converge faster than the QLM approach.
To implement the method we consider the SIR model that describes the temporal dynamics of a childhood disease in the presence of a preventive vaccine. In SIR models the population is assumed to be divided into the standard three classes namely, the susceptibles (
The governing equations for the problem are described [11] by
where
The total population is denoted by
2. Numerical solution
To simplify the formulation of the solution, equations (1) - (3) are scaled by dividing by
where
Previous studies [4–7, 12] have shown that the long term behaviour of systems like (4) - (5) can be classified into two categories namely, endemic or eradication. From the long term behaviour of
Here
It was shown in [11] that the DFE is locally stable if
3. Method of solution
To develop the method of solution, we assume that the true solution of (4 - 5) is
where
This idea of introducing the coupled equations of the form (9-10) have previously been used in [3] the construction of Newton-like iteration formulae for the computation of the solutions of nonlinear equations of the form
We write equation (9) as
where
We use the quasilinearization method (QLM) of Bellman and Kalaba [1] to solve equation (13). The QLM determines the (
which can be written as
subject to
We assume that
which yields the iteration scheme
We note that equation (19) is the standard QLM iteration scheme for solving (4 - 5).
When
Thus, setting
which yields the iteration scheme
where
The general iteration scheme obtained by setting
where
The initial approximation for solving the iteration algorithms, scheme-
The iteration schemes (19),(24 - 25) can be solved numerically using standard methods such as finite difference, finite elements, spline collocation methods,etc. In this study we use the Chebyshev spectral collocation method to solve the iteration schemes. For brevity, we omit the details of the spectralmethods, and refer interested readers to ([2, 13]). Before applying the spectral method, it is convenient to transform the domain on which the governing equation is defined to the interval [-1,1] on which the spectral method can be implemented. We use the transformation
where
Applying the Chebyshev spectral method to (19), for instance, gives
where
and
4. Results and discussion
In this section we present the results of solving the governing equations (4-5) using the iteration scheme-m. For illustration purposeswe present the results for
1. Case 1:
In this case we observe that
2. Case 2:
In this case we observe that
3. Case 3:
In this case
4. Case 4:
In this case
The results for Case 1 are shown on Figs. 1 - 2. In this case, the initial guess and the first few iterations match the numerical solution all the iterative schemes in the plots of
Figs. 3 - 5 show the numerical approximation of the profiles of the different classes for Case 2. Again, all the iterative schemes rapidly converge to the numerical solution. The population of the susceptibles decreases with time and that of the removed (those recovered with immunity) increases with time. The infected population initially increases and reaches a maximum, then gradually decreases to zero as
Figs. 6 - 8 show the numerical approximation of the profiles of the different classes for Case 3. It can be noted from the graphs that the Scheme-2 converges fastest towards the numerical results. Only 10 iterations are required for full convergence in Scheme-2 compared to 14 iterations in Scheme-1 and 28 iterations in Scheme-1.
Figs. 8 - 11 shows the variation all the population groups with time for Case 4. Again, we observe that Scheme-2 converges fastest towards the numerical results. Only 5 iterations are required for full convergence in Scheme-2 compared to 6 iterations in Scheme-1 and 12 iterations in Scheme-1.
5. Conclusion
In this work, a sequence of new iteration schemes for solving nonlinear differential equations is used to solve the SIR epidemic model with constant vaccination strategy. The proposed iteration schemes are derived as an extension to the quasi-linearization method to obtain hybrid iteration schemes which converge very rapidly. The accuracy and validity of the proposed schemes is confirmed by comparing with the ode45 MATLAB routine for solving initial value problems. It is hoped that the proposed method of solution will spawn further interest in computational analysis of differential equations in epidemiology and other areas of science.
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