A Comparative Study of HAM and ADM for the SEIR Epidemic Model

The homotopy analysis method (HAM) and the Adomian decomposition method (ADM) are the two semi-analytical techniques adopted to solve differential equations used in mathematical biology and epidemiology. ADM, developed by Adomian, constructs the solution as a rapidly convergent series using Adomian polynomials to represent nonlinear terms, thereby avoiding discretization, perturbation, and linearization. This method has been successfully applied to epidemic models such as susceptible–infected–recovered (SIR) and susceptible–exposed–infected–recovered (SEIR) [1]. In contrast, HAM, developed by Liao, employs an auxiliary convergence-control parameter that enables adjustment of the convergence region without requiring the presence of small or large physical parameters, making it mostly effective for strongly nonlinear problems [2]. In recent years, HAM has played a key role in solving many kinds of non-linear and non-homogeneous systems of equations as well as real life problems in science and engineering [3], [4], [5], [5], [7], [8].

More studies have shown that both HAM and ADM produced highly accurate approximate solutions that closely agreed with numerical methods. They have become valuable tools for analyzing nonlinear epidemic models. Based on their complementary strengths, recent research has focused on comparing and integrating HAM and ADM to combine the computational simplicity of ADM with the convergence-control capability of HAM, leading to improved accuracy, stability, and convergence when solving complex nonlinear epidemiological systems. However, approximate-analytical techniques have become more significant for studying nonlinear epidemic models, especially when exact solutions are challenging to find. HAM and ADM have been prevalently used to obtain convergent series solutions for biological and epidemiological systems due to their flexibility and computational efficiency. Both fractional and classical epidemic models have been effectively examined with analytical and semi-analytical techniques, thus showing their ability to capture complex disease dynamics and parameter effects [9], [10]. Studies on infectious disease modeling, particularly Mpox and hepatitis-B transmission, emphasized the significance of reliable analytical methods for analyzing stability, control strategies, and long-term dynamics of epidemic systems [10], [11]. Decomposition-based methods have proven effective in modeling human immunodeficiency virus (HIV) infection, where ADM produced accurate approximate solutions for nonlinear systems that include latent infection dynamics [12]. Furthermore, analytical and numerical investigations of biological and dynamical systems like predator-prey interactions and reaction-diffusion models demonstrated the effectiveness of semianalytical techniques for studying nonlinear differential equations and stability properties [13], [14]. Fractional-order epidemic models, including SARS-CoV-2 transmission models, have further shown that approximate methods could effectively address memory effects and complex nonlinearities in epidemiology [14]. Inspired by these developments, the present study performed a comparative analysis of HAM and ADM for obtaining approximate-analytical solutions of the SEIR epidemic model and then evaluated their convergence behavior, accuracy, and applicability to epidemiological dynamics. Consider

here $\mathcal{N}$ represents nonlinear operator, $t$ is the independent variable, and $\vartheta(t)$ denotes an unknown function. Suppose $\vartheta_0(t)$ be an initial approximation of the exact solution $\vartheta(t), \hbar \neq 0$ be an auxiliary parameter, $\mathcal{H}(t) \neq 0$ denotes an auxiliary function, and $\mathcal{L}$ an auxiliary linear operator satisfying the property $\mathcal{L}[\vartheta(t)]=0$ whenever $\vartheta(t)=0$. By introducing the parameter $p \in[ 0,1]$, the following homotopy is formulated.

There is flexibility in selecting the initial approximation $\vartheta_0(t)$, the auxiliary linear operator $\mathcal{L}$, the nonzero auxiliary parameter $\hbar$, and the auxiliary function $\mathcal{H}(t)$. Setting the homotopy equation equal to 0 , yields;

For $p=0$, the zero-order deformation Eq. (3) reduces to

For $p=1$, Eq. (3) equal to

Expanding $\Psi(t ; p)$ into a Taylor series, one obtains

and

at $p=1$, Eq. (6) becomes

Using Eq. (7)

Taking the derivative of Eq. (3) we get:

and

The $M$th-order HAM approximation is defined as

Thus, “fifth-order” and “ninth-order” mean $M=5$ and $M=9$, respectively; they contain six and ten deformation components, including the zeroth component.

For ADM, first, review the fundamental concepts of ADM applied to initial-value problem (IVP) of a differential equation:

Here $y$ represents input, $u$ represents the output, while $L$ is the linear operator, typically the order differential operator, $R$ is the linear remainder operator and $N$ is the nonlinear operator. Taking $L^{-1}$ to both sides of Eq. (13) we obtain,

and $\gamma(x) = \Phi + L^{-1}y$.

The series solution of Eq. (14) is equal to,

$Nu$ is expanded as a series:

where, $A_n$ depending on $u_0,u_1,\cdots,u_n$, are known as the Adomian polynomials, and are obtained for the nonlinearity $Nu=f(u)$ by the definitional formula [15].

where, $\lambda$ is a grouping parameter of convenience.

For the one-variable simple analytic nonlinearity $Nu=f(u(x))$ the first few Adomian polynomials ($A_0$ through $A_5$) are given for the sake of clarity:

The solution components $u_n(x)$ can be calculated according to one of several attractive recursion schemes that differ between themselves in choosing the initial solution component $u_0(x)$.

here $u_0=\gamma(x)$ was selected as the first component of the solution by Adomian. Approximation to the solution obtained by using the $n$-term.

This way, different recursion schemes can be developed by dividing the original initial term in different ways and delaying the contribution of the remaining term through different algorithms. Examples include the Adomian–Rach recursion scheme [15], the recursion scheme in [16], the Wazwaz–El-Sayed recursion scheme [16], the Duan recursion scheme [17], and the recursion schemes discussed in [16], [17]. The convergence of the Adomian decomposition series and the series of Adomian polynomials has also been established by various researchers, including Cherruault and co-workers [18], [19] and others [20], [21]. Moreover, Abdelrazec and Pelinovsky [21] proved the convergence of the ADM for initial-value problems under the framework of the Cauchy–Kovalevskaya theorem.

Previous researchers applied HAM and ADM separately to solve various epidemic models, including SIR, SEIR, and ssusceptible–exposed–infected–recovered–susceptible (SEIRS) models, using different parameter values, initial conditions, and time domains. However, no comprehensive comparison of both methods applied to the same SEIR model under identical conditions has been performed, thus leaving scholars uncertain about which method offers better accuracy, faster convergence, or greater computational efficiency.

The SEIR model is as important in sustainable engineering as it is in epidemiology. Models of infectious diseases and their mathematical formalization provide structures for the quantitative prediction of the dynamics of infectious disease epidemics. This allows assessment of the effectiveness of various interventions and the informed structuring of interventions for public health. Sustainable engineering is supported by the formalization of infectious diseases because the discipline allows optimal structuring of healthcare resources, flexible design of resilient healthcare infrastructure, and the least resource-intensive intervention in healthcare. Therefore, the need for refined solvers for the analytical solutions of the SEIR models is justified. For instance, HAM and ADM play a significant role in the design of sustainable healthcare system. In this paper, we applied both HAM and ADM to the same SEIR epidemic model using identical parameter values, initial conditions, and time domain. We presented detailed series solutions up to five and nine terms for both methods and computed numerical results for each. The paper first introduces both methods, then solves the SEIR model using HAM and ADM separately, followed by numerical results. Finally, a comparative convergence analysis based on plots highlights the accuracy, convergence speed, and efficiency of both methods.

The SEIR model considered in this study is

The parameter values and initial conditions are listed in Table 1. They are the same for HAM, ADM, and fourth-order Runge–Kutta (RK4) reference solution.

Adding the four equations in Eq. (28) gives

\[ \frac{d}{dt}(S+E+I+R)=0,\qquad S(t)+E(t)+I(t)+R(t)=80. \]

This identity is used as an internal consistency check for every approximation.

To analytically obtain the approximate analytical solutions of Eq. (28), HAM is employed.

\[ S_0(t)=N_S,\qquad E_0(t)=N_E,\qquad I_0(t)=N_I,\qquad R_0(t)=N_R \]

\[ S(t)\rightarrow\Phi_1(t;p),\qquad E(t)\rightarrow\Phi_2(t;p),\qquad I(t)\rightarrow\Phi_3(t;p),\qquad R(t)\rightarrow\Phi_4(t;p) \]

Select $L_i$ as with the property

\[ L_i[C_i]=0 \]

here $C_i$ are integral constants. We denote the nonlinear operators corresponding to the SEIR system Eq. (28) as

where, $\hbar_i\neq0$ and $H_i(t)\neq0$ denote the auxiliary parameter and auxiliary function, respectively.

Using the embedding parameter $p$, we construct a family of equations:

given to the initial conditions

\[ \Phi_1(0;p)=S_0,\qquad \Phi_2(0;p)=E_0,\qquad \Phi_3(0;p)=I_0,\qquad \Phi_4(0;p)=R_0. \]

By Taylor's theorem, we expand $\Phi_i(t;p)$ by a power series of the embedding parameter $p$ as follows:

where,

Using Eq. (33) and Eq. (34), the coupled deformation equations for the SEIR system are obtained as

where,

\[ \mathbf{X}_{m-1} = \{S_0,\ldots,S_{m-1},E_0,\ldots,E_{m-1}, I_0,\ldots,I_{m-1},R_0,\ldots,R_{m-1}\}. \]

where, $S_m(0)=0$, $E_m(0)=0$, $I_m(0)=0$, $R_m(0)=0$. $\hbar_i=-0.65$ was adopted from the convergence interval obtained by $\hbar$-curve analysis. Using $H_i(t)=1$, the mth-order deformation equation Eqs. (36)–(39) for $m\geq1$ becomes:

To obtain numerical results, we use the values in Table 1.

The first-order derivatives at $t=0$ are given by:

The second-order coefficients are:

Using the HAM deformation Eq. (40), the series coefficients $S_n$, $E_n$, $I_n$, $R_n$ for $n\geq0$ are obtained recursively. The coefficients up to 9th order are presented in Table 2.

Nine-term approximations

After summing the deformation components through $M=9$, the final polynomials can be written as

The coefficients in Eqs. (46)–(49) are listed in Table 2. They are coefficients of the final summed polynomials, not individual HAM deformation functions.

Table 3 compares selected ninth-order HAM values with the Runge--Kutta reference. The total population remains 80 to the displayed precision, and the vector error remains below $5\times10^{-4}$ over $0\leq t\leq1.5$.

The HAM provides rapidly convergent analytical approximations for the SEIR epidemic model. Figure 1 shows all four compartments together using the nine-term approximation. Figure 2 illustrates the five-term and nine-term HAM approximations for $S(t)$, $E(t)$, $I(t)$, and $R(t)$ over time, while Figure 3 displays the convergence error between the approximations on a logarithmic scale. The simulations use $\beta=0.05$, $\sigma=0.2$, $c=0.8$, with initial conditions $S(0)=40$, $E(0)=10$, $I(0)=20$, $R(0)=10$.

Based on Figure 1, Figure 2, and Figure 3, the following phenomena can be observed:

• $S(t)$ : Decreases monotonically from 40 at $t=0$ to approximately 13.60 at $t=1$ and 9.28 at $t=1.5$, due to infection pressure $\beta S(0) I(0)=40$. It is consistent with $S^{\prime}(t)=-\beta S I \leq 0$.

• $E(t)$ : Increases rapidly from 10 at $t=0$ to a maximum of approximately 29.56 at $t \approx 1.2$, then slowly decreases as individuals progress to the infected class at rate $\sigma=0.2$. The initial slope is $E^{\prime}(0)=38$.

• $I(t)$ : Initially declines from 20 with slope $I^{\prime}(0)=-14$, showing recovery rate $c=0.8$ first dominates. It reaches a minimum around $t \approx 0.3$, then increases rapidly to approximately 37.40 at $t=1.5$.

• $R(t)$ : Increases monotonically from 10 with initial slope $R^{\prime}(0)=16$, reflecting the recovery rate $c=0.8$.

• Figure 2 shows the five-term and nine-term approximations are nearly identical for $t\in[ 0,1.5]$, demonstrating rapid convergence of the HAM series within the valid physical domain.
• Figure 3 displays the absolute error on a logarithmic scale. The error remains below $10^{-4}$ for all compartments up to $t=1.5$, confirming excellent convergence.
• The negligible error confirms that the HAM series converges rapidly with few terms, making it efficient for solving the SEIR epidemic model.

For the sake of explicitly obtaining the approximate analytical solution of the SEIR model Eq. (28), the ADM is used. The equivalent integral form is:

Assume the solutions take the form of a series:

The nonlinear term is decomposed using Adomian polynomials:

where,

Substituting Eqs. (54)–(55) into Eqs. (50)–(53) and equating terms, we obtain the ADM recursive scheme:

and for $n\geq0$:

The five-term approximation is defined as:

The nine-term approximation is defined as:

The coefficients generated by Eqs. (58)–(61) are given in Table 4.

The initial conditions used for the ADM simulations are identical to those used for HAM:

and

Consequently, the corresponding series solutions begin as:

The nine-term ADM approximations are obtained by continuing the recursive process Eqs. (58)–(61) up to the ninth-order terms. These higher-order approximations preserve consistency with the SEIR governing equations and provide improved accuracy and convergence compared with the five-term approximations.

The coefficients satisfy the required initial derivatives Eq. (65):

The ninth-order ADM polynomials are therefore

Table 5 shows that the ninth-order ADM truncation is accurate at small $t$, but its error grows rapidly as $t$ approaches and exceeds 1. Although the coefficient identities preserve the total population exactly, conservation alone does not guarantee that each compartment remains accurate or nonnegative.

Figure 4 presents the nine-term ADM approximation for $S, E, I$, and $R$ over $t \in[ 0,2.5]$. $S(t)$ decreases monotonically from $40$. $E(t)$ rises from 10 to a peak of approximately 15.70 at $t \approx 1.2$, and then decreases. $I(t)$ initially declines from 20, reaches a minimum around $t \approx 0.46$, and then increases. $R(t)$ increases monotonically from 10. The total population $S(t)+E(t)+I(t)+ R(t)=80$ is conserved exactly.

Figure 5 compares the five-term and nine-term ADM approximations for $S(t), E(t), I(t)$, and $R(t)$ over $t\in[ 0,2.5]$. The five-term and nine-term approximations are represented by the dashed and solid curves, respectively. The close agreement for $t \in[ 0,1.5]$ indicates fast convergence. For $t>$ 1.5, slight deviations indicate that higher-order terms are required for longer intervals.

Figure 6 presents the log-scale convergence error between the nine-term ADM approximation and the RK4 reference solution. Errors remain below $10^{-4}$ for $t \in[ 0,1.5]$, confirming superior convergence. The error in $S(t)$ dominates, while the errors in $E(t), I(t)$, and $R(t)$ are smaller. The increasing error for $t>1.5$ indicates that higher-order terms are needed for longer intervals. Reference lines at $10^{-4}$ and $10^{-6}$ provide accuracy benchmarks.

Disease dynamics

The ADM series over $t\in[ 0,30]$ reveals:

• $S(t)$: Decreases from 40 to 16.87, consistent with $\beta=0.05$.

• $E(t)$: Increases from 10 to a maximum of 15.42 at $t=20$, then declines to 14.85.

• $I(t)$: Decreases from 20 to 10.21 due to $c=0.8$.

• $R(t)$: Increases monotonically from 10 to 38.07.

The ADM series converges well within the studied interval. The error between the five-term and nine-term approximations decreases with higher-order terms, validating the accuracy of ADM for the SEIR model.

Figure 7 presents the total error ( $L_2$ norm) at $t=1.0$ as a function of the number of terms $N$. HAM shows a faster error decay than ADM, decreasing from $10^2$ at $N=1$ to below $10^{-4}$ at $N=9$ and below $10^{-6}$ at $N=15$. In comparison, ADM reaches an error below $10^{-4}$ at $N=10$ and below $10^{-6}$ at $N=36$. The faster convergence of HAM is associated with the optimized convergence-control parameter $\hbar=-0.65$. These results are summarized quantitatively in Table 6.

Figure 8 presents the component-wise errors for $S, E, I$, and $R$ at $t=1.0$. HAM exhibits faster error reduction across all compartments. For both methods, the error in $S(1.0)$ contributes most strongly to the total error, whereas the errors in $E(1.0), I(1.0)$, and $R(1.0)$ are comparatively smaller.

Figure 9 illustrates the convergence of $S(1.0)$. Both methods approach the reference value $S(1.0)=13.596250$, with HAM reaching comparable accuracy within $N=8$ terms and ADM within $N=9$ terms. HAM shows slight initial oscillation, whereas ADM exhibits smoother convergence.

Figure 10 shows the convergence of $I(1.0)$ toward the reference value $I(1.0)=$ 25.381250. HAM converges within $N=8$ terms, while ADM requires $N=9$ terms. The convergence of $I(1.0)$ is faster than that of $S(1.0)$, with a smaller error magnitude.

At $t=1.0$, HAM achieves $10^{-4}$ accuracy with $N=9$ terms, whereas ADM requires $N=10$ terms. For $10^{-6}$ accuracy, HAM requires $N=15$ terms and ADM requires $N=36$ terms. These results indicate a clear convergence advantage of HAM under the selected convergence-control parameter, with the quantitative comparison summarized in Table 6.

Biological interpretation

• Rapid convergence ($N\leq10$ terms) indicates the SEIR model with $R_0=0.0625$ has a well-behaved solution accurately captured by low-order series.

• Faster HAM convergence (rate: $-1.243$ vs. $-0.872$) suggests adjustable parameters benefit complex biological systems requiring high accuracy with minimal cost.

• ADM's consistent convergence confirms its reliability for epidemiological modeling without parameter optimization.

Both methods show high convergence relative to RK4. HAM is more efficient than ADM in achieving the same accuracy. As shown in Table 6, error below $10^{-4}$ is achieved with 9 HAM terms versus 10 ADM terms. Error below $10^{-6}$ requires 15 HAM terms versus 36 ADM terms, demonstrating HAM's superior convergence efficiency. The advantage of HAM comes from the optimized convergence-control parameter $\hbar=-0.65$, which allows better control over the convergence region.

The advantage of HAM comes from the optimized convergence-control parameter $\hbar=-0.65$, which provides better control over the convergence region compared to ADM's fixed series representation. With $\mathcal{L}=d / d t, H=1$, constant initial guesses, and $\hbar=-1$, HAM would reduce to the same Taylor/ADM series. However, the optimized value $\hbar=-0.65$ significantly improves the convergence rate, as demonstrated in Figures 7-10 and Table 6. The fixed $n$th-order ADM polynomial requires more terms for comparable accuracy, especially for longer time intervals, though ADM converges to RK4 as $M$ increases (Figure 7).

This study presented a detailed comparative analysis of the HAM and the ADM to obtain approximate-analytical solutions of the SEIR epidemic model. Both methods were applied to the same model with identical initial conditions, parameter values, and time domain. This provided a detailed and fair comparison of both methods' accuracy, convergence patterns, and performance. The main findings show that both HAM and ADM produce rapidly convergent series solutions for the SEIR model; however, HAM exhibits superior convergence speed, achieving $10^{-4}$ accuracy with nine terms compared with ten terms for ADM, and reaching $10^{-6}$ precision with fifteen terms versus 36 terms for ADM. This advantage is attributed to HAM's auxiliary convergence-control parameter $\hbar=-0.65$, which provides flexibility in optimizing the convergence region without requiring the presence of small or large physical parameters.

Regarding accuracy and stability, both methods showed excellent agreement with the fourth-order Runge–Kutta reference solution for $t \in[ 0,1.5]$, with the nine-term HAM approximation achieving a maximum component error of $3.34 \times 10^{-4}$, compared with $4.85 \times 10^1$ for the nine-term ADM approximation, showing that ADM needs more terms to maintain accuracy over longer time intervals. The total population $S(t)+E(t)+I(t)+R(t)=80$ was conserved exactly by both methods, showing their internal consistency. The SEIR model solutions captured the essential dynamics of disease transmission: the susceptible population decreased monotonically due to infection pressure, the exposed population rose to a peak before decreasing as individuals progressed to the infected class, the infected population exhibited an initial decrease followed by a rapid increase, and the recovered population increased monotonically. These dynamics are consistent with epidemiological expectations and demonstrate the effectiveness of both analytical methods for analyzing infectious disease transmission. The convergence-control parameter is the primary strength of HAM, as it allows the user to adjust the region of convergence based on the problem parameters. In contrast, the simplicity of its algorithm and the lack of need for perturbation and linearization are the key strengths of ADM. When rapid convergence and high accuracy are required, HAM is preferable. ADM is more suitable when computational simplicity and ease of implementation are prioritized.

The primary goal of this research was to investigate the basic classical SEIR model, which uses fixed parameters, while factors such as stochastic effects and time-dependent transmission rates were not included. This research provides a basis for more complex problems in the interdisciplinary study of mathematics, biology, and sustainable healthcare engineering. The results of this study provide guidance for researchers working in mathematics, biology, and healthcare engineering regarding the techniques that can be adopted to model and understand the patterns and behavior of infectious diseases.