On New High Order Iterative Schemes for Solving Initial Value Problems in Epidemiology

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 (S), who can catch the infection but are so far uninfected, the infectives (I), those who have the disease and can transmit it to the susceptibles, and the removed (R), who have either died or who have recovered and are therefore immune.

The governing equations for the problem are described [11] by

$$\frac{dS}{dt}=(1-P)\pi N-\beta \frac{SI}{N}-\mu S,$$E1

$$\frac{dI}{dt}=\beta \frac{SI}{N}-(\kappa +\mu )I,$$E2

$$\frac{dR}{dt}=P\pi N+\kappa I-\mu R,$$E3

whereS(t), I(t) and R(t) denote the susceptibles, infectives and the removed classes respectively.

The total population is denoted by N = S + I + R, μ is the death rate, P is the fraction of citizens vaccinated at birth each year, β is the average contact rate, π is the constant birth rate, and κ is the rate at which an individual recovers from the disease and enters the removed group which also contains vaccinated individuals. Equations (1 - 3) are solved using the new hybrid iteration schemes and the results are compared with results from the Runge-Kutta MATLAB in-built solver ode45.

2. Numerical solution

To simplify the formulation of the solution, equations (1) - (3) are scaled by dividing by N. We define new variables z₁ = S/N, z₂ = I/N and z₃ = R/N. This leads to z₁ + z₂ + z₃ = 1 and if we assume that π = μ, the scaled new system becomes

$${z^{\prime }}_1(t)=(1-P)\pi -\beta z_1(t)z_2(t)-\pi z_1(t),z_1(0)=s_0,$$E4

$${z^{\prime }}_2(t)=\beta z_1(t)z_2(t)-(\pi +\kappa )z_2(t),z_2(0)=i_0,$$E5

wheres₀ and i₀ are given constants. The solution for z₃(t) can be obtained from z₃ = 1−z₁ −z₂.

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 z₁(t) and z₂(t) it holds that the solution asymptotically approaches a disease free equilibrium (DFE) or the endemic equilibrium (EE) where

$$\underset{t\to +\infty }{\lim }(z_1(t),z_2(t))=\text{DFE}=(1-P,0),$$E6

$$\underset{t\to +\infty }{\lim }(z_1(t),z_2(t))=\text{EE}=\left(\frac{1-P}{R_v},\frac{\pi }{\beta }(R_v-1)\right).$$E7

Here R_v, the vaccination reproduction number, is the threshold that determines the stability of the equilibria and is defined by

$$R_v=\frac{\beta (1-P)}{\gamma +\pi }.$$E8

It was shown in [11] that the DFE is locally stable if R_v<1 and the EE is locally stable provided 1 <R_v≤ 4(κ + π)/π. In this work, we use develop new iteration schemes to solve the system (4) - (5) using parameters that yield both the DFE and EE.

3. Method of solution

To develop the method of solution, we assume that the true solution of (4 - 5) is z_s,α(s = 1, 2) and z_s,γare the initial approximations. We introduce the following coupled system,

$$L_j{z}_j+f_j(z_{1,\gamma },z_{2,\gamma })+{\displaystyle \sum _{s=1}^2(z_s-z_{s,\gamma })}\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })+g_j(z_1,z_2)={\Psi }_j,$$E9

$$g_j(z_1,z_2)=f_j(z_1,z_2)-f_j(z_{1,\gamma },z_{2,\gamma })-{\displaystyle \sum _{s=1}^2(z_s-z_{s,\gamma })}\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma }),$$E10

where

$$L_1{z}_1={z^{\prime }}_1+\pi z_1,L_2{z}_2={z^{\prime }}_2+(\pi +\kappa )z_2,$$E11

$$f_1(z_1,z_2)=\beta z_1{z}_2,f_2(z_1,z_2)=-\beta z_1{z}_2,{\Psi }_1=(1-P)\pi ,{\Psi }_2=0.$$E12

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 f (x) = 0.

We write equation (9) as

$$L_j{z}_j+{\displaystyle \sum _{s=1}^2{z}_s}\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })+g_j(z_1,z_2)={\Phi }_j(z_{1,\gamma },z_{2,\gamma }),$$E13

where

$${\Phi }_j(z_{1,\gamma },z_{2,\gamma })={\Psi }_j+{\displaystyle \sum _{s=1}^2{z}_{s,\gamma }}\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })-f_j(z_{1,\gamma },z_{2,\gamma }).$$E14

We use the quasilinearization method (QLM) of Bellman and Kalaba [1] to solve equation (13). The QLM determines the (i+ 1)th iterative approximation z_j,i+1 as the solution of the differential equation

$$L_j{z}_{j,i+1}+{\displaystyle \sum _{s=1}^2{z}_{s,i+1}}\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })+g_j(z_{1,i},z_{2,i})+{\displaystyle \sum _{s=1}^2(z_{s,i+1}-z_{s,i})}\frac{\partial g_j}{\partial z_s}(z_{1,i},z_{2,i})={\Phi }_j(z_{1,\gamma },z_{2,\gamma }),$$E15

which can be written as

$$\begin{array}{l}{L}_j{z}_{j,i+1}+{\displaystyle \sum _{s=1}^2\left[\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })+\frac{\partial g_j}{\partial z_s}(z_{1,i},z_{2,i})\right]}{z}_{s,i+1}=\hfill \\ {\displaystyle \sum _{s=1}^2{z}_{s,i}}\frac{\partial g_j}{\partial z_s}(z_{1,i},z_{2,i})-g_j(z_{1,i},z_{2,i})+{\Phi }_j(z_{1,\gamma },z_{2,\gamma }),\hfill \end{array}$$E16

subject to

$$z_{1,i+1}=s_0,z_{2,i+1}=i_0.$$E17

We assume that z_j,0is obtained as a solution of the linear part of equation (13) given by

$$L_j{z}_{j,0}+{\displaystyle \sum _{s=1}^2{z}_{s,0}}\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })={\Phi }_j(z_{1,\gamma },z_{2,\gamma }),$$E18

which yields the iteration scheme

$$L_j{z}_{j,r+1}+{\displaystyle \sum _{s=1}^2{z}_{s,r+1}}\frac{\partial f_j}{\partial z_s}(z_{1,r},z_{2,r})={\Phi }_j(z_{1,r},z_{2,r}).$$E19

We note that equation (19) is the standard QLM iteration scheme for solving (4 - 5).

When i= 0 in (16) we can approximate z_jas

$$z_j\approx z_{j,1}.$$E20

Thus, setting i= 0 in (16) we obtain

$$\begin{array}{ll}{L}_j{z}_{j,1}+{\displaystyle \sum _{s=1}^2\left[\frac{\partial f_j}{\partial z_s}(z_{1,\gamma },z_{2,\gamma })+\frac{\partial g_j}{\partial z_s}(z_{1,0},z_{2,0})\right]}{z}_{s,1}\hfill & ={\displaystyle \sum _{s=1}^2{z}_{s,0}}\frac{\partial g_j}{\partial z_s}(z_{1,0},z_{2,0})\hfill \\ \hfill & -g_j(z_{1,0},z_{2,0})+{\Phi }_j(z_{1,\gamma },z_{2,\gamma }),\hfill \end{array}$$E21

which yields the iteration scheme

$$\begin{array}{ll}{L}_j{z}_{j,r+1}+{\displaystyle \sum _{s=1}^2\left[\frac{\partial f_j}{\partial z_s}(z_{1,r},z_{2,r})+\frac{\partial g_j}{\partial z_s}(z_{1,r+1}^{(0)},z_{2,r+1}^{(0)})\right]}{z}_{s,r+1}\hfill & ={\displaystyle \sum _{s=1}^2{z}_{s,r+1}^{(0)}}\frac{\partial g_j}{\partial z_s}(z_{1,r+1}^{(0)},z_{2,r+1}^{(0)})\hfill \\ \hfill & -g_j(z_{1,r+1}^{(0)},z_{2,r+1}^{(0)})+{\Phi }_j(z_{1,r},z_{2,r}),\hfill \end{array}$$E22

where$z_{j,r+1}^{(0)}$is the solution of

$$L_j{z}_{j,r+1}^{(0)}+{\displaystyle \sum _{s=1}^2{z}_{s,r+1}^{(0)}}\frac{\partial f_j}{\partial z_s}(z_{1,r},z_{2,r})={\Phi }_j(z_{1,r},z_{2,r}).$$E23

The general iteration scheme obtained by setting i= m (m ≥ 2) in equation (16), hereinafter referred to as scheme-m is

$$\begin{array}{ll}{L}_j{z}_{j,r+1}+{\displaystyle \sum _{s=1}^2\left[\frac{\partial f_j}{\partial z_s}(z_{1,r},z_{2,r})+\frac{\partial g_j}{\partial z_s}(z_{1,r+1}^{(m-1)},z_{2,r+1}^{(m-1)})\right]}{z}_{s,r+1}\hfill & ={\displaystyle \sum _{s=1}^2{z}_{s,r+1}^{(m-1)}}\frac{\partial g_j}{\partial z_s}(z_{1,r+1}^{(m-1)},z_{2,r+1}^{(m-1)})\hfill \\ \hfill & -g_j(z_{1,r+1}^{(m-1)},z_{2,r+1}^{(m-1)})+{\Phi }_j(z_{1,r},z_{2,r}),\hfill \end{array}$$E24

where$z_{j,r+1}^{(m-1)}$is obtained as the solution of

$$\begin{array}{ll}{L}_j{z}_{j,r+1}^{(m-1)}+{\displaystyle \sum _{s=1}^2\left[\frac{\partial f_j}{\partial z_s}(z_{1,r},z_{2,r})+\frac{\partial g_j}{\partial z_s}(z_{1,r+1}^{(m-2)},z_{2,r+1}^{(m-2)})\right]}{z}_{s,r+1}^{(m-1)}\hfill & ={\displaystyle \sum _{s=1}^2{z}_{s,r+1}^{(m-2)}}\frac{\partial g_j}{\partial z_s}(z_{1,r+1}^{(m-2)},z_{2,r+1}^{(m-2)})\hfill \\ \hfill & -g_j(z_{1,r+1}^{(m-2)},z_{2,r+1}^{(m-2)})+{\Phi }_j(z_{1,r},z_{2,r}).\hfill \end{array}$$E25

The initial approximation for solving the iteration algorithms, scheme-m is obtained by solving the linear part of the governing equations (4 - 5). This gives

$$z_{1,0}=(1-P)(1-e^{-\pi t})+s_0{e}^{-\pi t},z_{2,0}=i_0{e}^{-(\pi +\kappa )t}.$$E26

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 t = t_F(τ + 1)/2 to map the interval [0, t_F] to [-1,1], where t_Fis a finite time. The basic idea behind the spectral collocation method is the introduction of a differentiation matrix D which is used to approximate the derivatives of the unknown variables z at the collocation points as the matrix vector product

$$\frac{dz}{dt}={\displaystyle \sum _{k=0}^N{D}_{jk}}z({\tau }_k)=DZ,j=0,1,\dots ,N,$$E27

whereN + 1 is the number of collocation points (grid points), D = 2D/t_F, and Z = [z(τ₀), z(τ₁),..., z(τ_N)]^Tis the vector function at the collocation points τ_j.

Applying the Chebyshev spectral method to (19), for instance, gives

where

$$a_{ji}=\frac{\partial f_i}{\partial z_i}$$E28

andI is an (N + 1) × (N + 1) identity matrix.

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 m = 0, 1, 2 to illustrate the effect of increasing m in the accuracy and convergence of the iteration schemes. The number of collocations points in all the results presented here is N = 40. In order to assess the accuracy of the proposed method, the present numerical results were compared against results generated using the MATLAB initial value solver ode45. In the numerical simulations presented here, following [11], the governing parameters were carefully selected in order to represent the cases which give rise to both the disease free equilibrium (DFE) and endemic equilibrium (EE).We consider the following cases

1. Case 1: s₀ = 1, i₀ = 0, β = 0.8, κ = 0.03, π = 0.4, P = 0.9.

In this case we observe that R_v= 0.186 <1, hence we expect the disease to be eradicated from the population after some time.

2. Case 2: s₀ = 0.8, i₀ = 0.2 β = 0.8, κ = 0.03, π = 0.4, P = 0.9.

In this case we observe that R_v= 0.186 <1 and as expected, using these parameters, the disease should be eradicated as t → ∞.

3. Case 3: s₀ = 0.8, i₀ = 0.2 β = 0.8, κ = 0.03, π = 0.4, P = 0.

In this caseR_v= 1.86 >1 which leads to the endemic equilibrium (no disease eradication).

4. Case 4: s₀ = 0.8, i₀ = 0.2 β = 0.8, κ = 0.03, π = 0.4, P = 0.3.

In this case R_v= 1.32 >1 which leads to the endemic equilibrium (no disease eradication).

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 s(t), r(t). We observe that s(t) decreases monotonically with time while r(t) increases with time. The graph of the profile for i(t) is not shown because i(t) = 0 in this particular case.

Figure 1.

Case 1: Comparison of the numerical solution of the population fractions s(t) against the results from the iteration schemes-0, 1 and 2

Figure 2.

Case 1: Comparison of the numerical solution of the population fractions r(t) against the results from the iteration schemes-0, 1 and 2

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 t → ∞.

Figure 3.

Case 2: Comparison of the numerical solution of the population fractions s(t) against the results from the iteration schemes-0, 1 and 2

Figure 4.

Case 2: Comparison of the numerical solution of the population fractions i(t) against the results from the iteration schemes-0, 1 and 2

Figure 5.

Case 2: Comparison of the numerical solution of the population fractions r(t) against the results from the iteration schemes-0, 1 and 2

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.

Figure 6.

Case 3: Comparison of the numerical solution of the population fractions s(t) against the results from the iteration schemes-0, 1 and 2

Figure 7.

Case 3: Comparison of the numerical solution of the population fractions i(t) against the results from the iteration schemes-0, 1 and 2

Figure 8.

Case 3: Comparison of the numerical solution of the population fractions r(t) against the results from the iteration schemes-0, 1 and 2

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.

Figure 9.

Case 4: Comparison of the numerical solution of the population fractions s(t) against the results from the iteration schemes-0, 1 and 2

Figure 10.

Case 4: Comparison of the numerical solution of the population fractions i(t) against the results from the iteration schemes-0, 1 and 2

Figure 11.

Case 4: Comparison of the numerical solution of the population fractions r(t) against the results from the iteration schemes-0, 1 and 2

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.