Stability analysis and numerical simulation of nonlocal extended epidemic models using positivity-preserving scheme

Abstract

In this paper we introduce a robust numerical framework for simulating the nonlocal extended epidemic models that incorporate the fractional diffusion to capture the complex spatial–temporal dynamics of disease spread. The presented numerical scheme uses the Fourier spectral approach for spatial discretization and a positivity-preserving exponential time differencing approach for temporal integration. Two fractional epidemic models—the SIR model with a modified saturated incidence rate and the SEIR model—are analyzed to investigate the impact of fractional diffusion on disease transmission. The transmission dynamics are explored using the developed L-stable, positivity-preserving numerical method. Numerical experiments demonstrate the influences of the fractional diffusion on the spatial distribution and temporal evolution of infectious diseases. These experiments also highlight the importance of fractional diffusion in epidemic models and provide a rigorous computational framework that can be used in future research in public health policy and epidemic control strategies. A benchmark test problem with a known exact solution is considered, which confirms that the proposed L-stable scheme achieves second-order accuracy for different values of .

Introduction

Fractional partial differential equations (FPDEs) have emerged as powerful tools for modeling phenomena with non-local effects and spatial heterogeneity. The main task in the study of fractional differential equations lies in obtaining the analytical solution of the nonlinear reaction-diffusion systems. For most such systems, exact solutions are not available, thereby necessitating the development of reliable and efficient numerical methods. However, solving numerically the nonlinear systems of fractional differential equations remains a formidable task due to the inherent complexity of fractional operators and their nonlocal behavior, which significantly increases computational and analytical challenges.

Most epidemic models are traditionally formulated as systems of nonlinear ordinary differential equations, an approach that often overlooks spatial heterogeneity and population mobility. However, recent epidemiological studies highlight the importance of nonlocal transmission mechanisms arising from heterogeneous landscapes, transportation networks, and movement between nonadjacent communities^1,2. Classical diffusion models, based on Brownian motion, restrict interactions to local neighborhoods and are therefore limited in their ability to represent such mobility patterns.

Empirical studies of human and animal mobility indicate that dispersal often follows Lévy flight–type or power-law patterns rather than purely local diffusion processes^3–7. This evidence motivates the use of fractional diffusion operators, which provide a natural framework for modeling long-range movement, heterogeneous connectivity, and heavy-tailed dispersal. Replacing the classical Laplacian with the fractional Laplacian enables epidemic models to represent sporadic long-distance transmission and patchy population structures. Such nonlocal transport fundamentally modifies epidemic dynamics by accelerating spatial spread, advancing epidemic peaks, and influencing convergence to equilibrium, potentially sustaining endemic states even when local transmission would decline^8,9. Consequently, while classical diffusion-based epidemic models have been successfully applied in homogeneous settings^10–13, their inherently local nature limits applicability in heterogeneous environments characterized by sparse populations and long-range interactions. In these settings, nonlocal formulations—including fractional and integro-differential models—offer a more realistic description of disease propagation across extended spatial scales^14–16.

We adopt the spectral definition of the fractional Laplacian on a bounded domain, constructed from the eigenvalues and eigenfunctions of the classical Laplacian under homogeneous Dirichlet or Neumann boundary conditions^17–19. The availability of explicit spectral representations enables an efficient numerical treatment of the fractional operator. Based on this formulation, we develop a unified numerical framework for nonlinear epidemic reaction–diffusion systems that couples a spectral approximation in space with a second-order exponential time-differencing (ETD) scheme.

For time integration of stiff fractional reaction–diffusion systems, exponential time differencing (ETD) schemes based on rational Padé approximations of the matrix exponential have been shown to deliver improved stability and accuracy compared to classical schemes. Dai et al.²⁰ used Padé approximants to construct ETD3-Padé and ETD4-Padé schemes with unconditional stability and rigorous error analysis. Yousuf et al.¹⁹ employ restricted Padé rational approximations within Fourier spectral ETD frameworks for space-fractional systems, demonstrating stability and spectral accuracy.

Diagonal Padé approximants, such as Padé(1,1) and positivity preserving Padé approximants, such as Padé(0,2) are commonly used to approximate the action of matrix exponentials in exponential integrators, balancing accuracy and stability. These rational forms reduce to solving one or more linear systems per timestep rather than explicitly computing matrix exponential. While Padé(1,1) yields an A-stable scheme analogous to ETD-Crank–Nicolson, its lack of L-stability may result in insufficient damping of stiff fractional modes. In contrast, Padé(0,2) provides stronger damping and L-stability, which is particularly advantageous for stiff fractional epidemic models. The ETD–Padé(0,2) scheme preserves positivity due to its completely monotone amplification factor on , maps non-negative vectors to non-negative vectors and yields a non-negative resolvent for diagonalizable fractional diffusion operators with nonpositive spectra. The associated ETD also inherits the same property. The enhanced stability improves long-time accuracy and supports the preservation of positivity and population balance, leading to substantial efficiency gains when combined with spectral or finite-difference spatial discretizations.

Comparison with existing numerical schemes

A comprehensive comparison of the proposed ETD-Padé(0,2) scheme with finite difference, finite element, and ETD-Padé(1,1) is conducted and is given in table 1. Some key observations are stated below.

• Finite Difference: These methods are easy to implement but suffer from conditional stability, loss of positivity, and severe timestep restrictions in fractional and stiff regimes.

• Finite Element: These methods provide geometric flexibility and higher spatial accuracy but do not inherently guarantee positivity and are computationally expensive for nonlocal fractional operators.

• ETD–Padé(0,2): The strong damping of stiff modes and preservation of positivity make them particularly well-suited for fractional reaction–diffusion and epidemic systems. These schemes produce smooth solutions even with non-smooth initial data.

• ETD–Padé(1,1): These schemes may lead to reduced robustness for highly stiff reactions or long-time simulations. These schemes may produce unwanted oscillations due to non-smooth initial data.

Table 1.

Comparison of numerical schemes for fractional SIR/SEIR reaction–diffusion models.

Criterion	FD	FEM	ETD-Padé(1,1)	ETD-Padé(0,2)
Stability	Conditionally stable (CFL-type)	Unconditionally stable (implicit)	Unconditionally stable (A-stable)	Unconditionally stable (L-stable)
Temporal convergence order			Second	Second
Spatial convergence order			Depends on spatial discretization	Depends on spatial discretization
Positivity of compartments	Not guaranteed	Conditional	Conditional	Guaranteed under standard assumptions
Population conservation	Generally violated	Weakly preserved	Approximately preserved	Preserved
Long-time error accumulation	High	Moderate	Moderate	Low
Damping of stiff modes	Weak	Moderate	Moderate	Strong
Accuracy of endemic equilibrium	Low	Moderate	Moderate	High
Implementation complexity	Simple	Complex	Moderate	Moderate
Oscillation suppression	Poor	Moderate	Moderate	Strong
Suitability for epidemic/biological models	Limited	Moderate	Highly suitable	Highly Suitable

Methodology

We begin with the two epidemic models given as coupled systems of nonlinear fractional differential equations. The biological meaning of the parameters used in these models is described. Next, we perform the stability analysis of both systems and compute the basic reproduction number of each model. Space fractional Laplacian operators are discretized using the Fourier spectral approach, and an exponential Runge-Kutta type time-stepping scheme is developed using positivity preserving L-stable Padé rational approximation of the matrix exponential function.

In addition to the methodological advances, we introduce an algorithm that explains a complete implementation strategy that leverages discrete sine and cosine transforms, explicit eigenvalue formulas, and a Runge-Kutta-type ETD update that avoids ill-conditioned matrix inverses. The proposed framework is applied to fractional SIR and SEIR models with saturated incidence, enabling systematic investigation of how nonlocal diffusion affects spatial spread, equilibrium behavior, and transient epidemic dynamics. Numerical experiments are performed to verify stability, accuracy, and consistency with classical diffusion when approaching 2, establishing the proposed algorithm as a robust computational tool for studying long-range spatial effects in epidemiological modeling.

Limitations

This study focuses on methodology and does not contain data-driven calibration or validation against real epidemiology datasets. As a result, model parameters such as fractional order and diffusion coefficients should be viewed as phenomenological descriptors of nonlocal movement rather than experimentally estimated values. Instead of making quantitative forecasts for individual outbreaks, the numerical results provide qualitative insights into how fractional diffusion may influence epidemic peaks, persistence, and spatial spread. While the findings have significant implications for spatial control tactics, turning these insights into policy-relevant conclusions would necessitate integration with mobility data, parameter estimates, and uncertainty analysis, which are all left to future research.

The remainder of the paper is organized as follows. Section "The description of diffusive epidemic models" introduces the two epidemic models. Section "Stability analysis of the epidemic models" presents the stability analysis. Spatial discretization of the reaction–diffusion system is described in Section "Reaction diffusion equation and spatial discretization". The derivation of the time-stepping scheme and its algorithmic implementation are given in Section "Time-stepping scheme". Linear error analysis and stability regions are discussed in Section "Stability and linear error analysis of the numerical scheme". Numerical experiments illustrating the performance of the proposed method are reported in Section “Numerical experiments”, followed by concluding remarks in Section “Conclusions”.

The description of diffusive epidemic models

We shall consider two epidemic models with fractional diffusion, SIR and SEIR models.

The following SIR model with modified saturated incidence rate and without delay factor is considered¹¹,

1

where S(x, t), I(x, t), and R(x, t) are the number of susceptible, infected, and recovered individuals, respectively, at location x and time t. The parameter is the recruitment rate, the rate of natural death, is the infection transmission rate, is the rate of death due to disease, and r is the rate of recovery. The modified incidence rate is,

with inhibitory effects measures, and represent limited contact capacity, crowding, or behavioral fatigue.

Nauman et al.^12,13 extended this model by including diffusion terms,

2

where the diffusion coefficients correspond to mobility rates of different compartments (e.g., movement of infectious individuals between communities). In this work, we shall consider an extension of this model by replacing the classical Laplacian with the fractional Laplacian,

3

with initial conditions,

and homogeneous Neumann boundary conditions.

The fractional order controls the degree of nonlocality or super-diffusion. Lower represents long-range jumps observed in air/road mobility networks.

A diffusive SEIR model for the transmission dynamics of measles considered by Al-Showaikh and Twizel¹⁰ determines the geographic spread of the disease in a population that has not been vaccinated against it. We have also considered an extension of this model by combining the fractional Laplacian instead of the classical Laplacian.

4

with some given initial conditions,

and homogeneous Neumann boundary conditions. Since R is not present in the first three equations, we shall solve the system only for S, E, and I and compute , where N is the total population under consideration. The parameter denotes the infection transmission rate, the average life expectancy per year, the average disease incubation, and the infection period.

The complexity of the fractional operators makes the task of finding numerical solutions of nonlinear systems more challenging. The numerical schemes must approximate the solution that preserves all the characteristics that hold for the solution of the epidemic model, such as stability properties of steady states, positivity, boundedness, etc.

Stability analysis of the epidemic models

The stability analysis of the SEIR model provides insights into the dynamics of disease spread and helps in understanding the impact of interventions. By adjusting parameters like in the model, public health officials can make informed decisions to control outbreaks.

Equilibrium analysis of the SIR model

Equilibrium points provide crucial information on epidemic behavior, leading public health policies, and ways to successfully minimize the spread of the disease. Considering these principles is critical for creating strong epidemic models and planning for prospective outbreaks. The equilibrium points of the epidemic models provide information on the various outcomes of an epidemic, such as whether the disease will die out or persist in the population.

Definition 1

A constant is said to be the equilibrium point of the following system of FDEs:

5

if and only if satisfies .

Definition 2

The equilibrium point is said to be a disease-free equilibrium (DFE) when there are no infected individuals in the population.

Definition 3

The equilibrium point is said to be an endemic equilibrium (EE) when a disease remains in the population at a steady average rate, neither increasing nor decreasing over time.

According to the above definitions, the disease-free equilibrium for the SIR model (3) occurs when . Hence, . while the endemic equilibrium is represented as

Theorem 4

Let J be the Jacobian matrix for a nonlinear system (1). The stability of the equilibrium has five different considerations based on the value of the eigenvalues of the Jacobian^21,22.

• The equilibrium will be stable when the real part of all eigenvalues is negative.

• If at least one of the eigenvalues has a negative real part and at least one has a positive real part, the equilibrium will represent a saddle point.

• The equilibrium is called a node point if all eigenvalues are real.

• The equilibrium is called a spiral point if we have at least a complex conjugate pair of eigenvalues.

• The equilibrium will be a linear center when having a zero real part on a pair of complex eigenvalues.

The Jacobian matrix of the SIR model:

For the SIR model (3), we let

6

The Jacobian matrix for the SIR model (3) is as follows:

At the disease-free equilibrium , we obtain,

The eigenvalues of the Jacobian matrix at (DFE) are as follows,

According to the theorem (4), the trajectory behavior of the (DFE) will be stable when,

The basic reproduction ratio of the SIR model:

The basic reproduction ratio () is a crucial factor in determining whether infectious diseases spread in the population or disappear. This ratio refers to the average number of secondary cases caused by an infected individual in a susceptible class²³. We rewrite the basic epidemic model as follows:

7

where represents the rate at which infections emerge. We let . Where refers to the rate of transfer of individuals out of the compartment and refers to the rate of transfer of individuals into the compartment. In the epidemic model, we let i be the number of compartments and j refer to the infected one. Then, we evaluate the following Jacobian matrices at (DFE) as follows:

8

and

9

The matrix represents the next generation matrix.

Definition 5

Based on the next-generation method²⁴, the reproduction ratio is the largest eigenvalue of the next-generation matrix .

Following the next generation method, the next generation matrix at (DFE) is represented as:

Hence, the basic reproduction ratio , which concludes that the infection will die out if . However, the disease will spread through the population when . This in fact confirms the stability behavior in the Jacobian analysis mentioned above.

Stability analysis of the SEIR model

The stability of the equilibrium point can be assessed using the eigenvalues of the Jacobian matrix. Referring to definition 2, the disease-free equilibrium is denoted as , where the zeros indicate the absence of both infected and recovered individuals. Moreover, referring to definition 3, the endemic equilibrium is expressed as follows:

This endemic equilibrium determines the state of the system (4) where the disease persists in the population.

The Jacobian matrix of the SEIR model:

Similarly, for the spatial-fractional SEIR model represented in (4), we let

10

Hence, the Jacobian matrix at the equilibrium point is computed as,

To analyze the stability of the disease-free equilibrium (DFE), we compute the eigenvalues of the following Jacobian matrix evaluated at ,

It is found that and . Referring to theorem 4, the trajectory behavior of the (DFE) will be stable when,

The basic reproduction ratio of the SEIR model:

Based on the next-generation method, the next-generation matrix in (DFE) for the spatial-fractional model SEIR (4) is represented as:

Thus,

Reaction diffusion equation and spatial discretization

We consider the following abstract initial boundary value problem,

11

with initial condition,

and having either Dirichlet or Neumann boundary conditions. The non-linear term F(u, t) satisfies the Lipschitz condition with respect to u,

for some . The initial condition function can be non-smooth, or we can have mismatched initial-boundary conditions.

The spectral definition of the fractional Laplacian is a very useful tool to develop numerical schemes. Using this approach, we need to compute the eigenvalues and eigenfunctions of the classical Laplacian using explicit analytical formulas.

Definition 6

Assume that the classical Laplacian operator is defined on a bounded domain and let is a complete set of eigenfunctions with corresponding eigenvalues , then,

Let

Then

where are the Fourier coefficients. The choice of the boundary conditions determines the appropriate choice of the formula to be used to compute the eigenvalues and eigenfunctions of the classical Laplace operator. The explicit formulas used to compute the eigenvalues and eigenfunctions of the classical Laplacian defined on with homogeneous Dirichlet boundary conditions are,

where the length of the interval is . For problems with homogeneous Neumann boundary conditions, the formulas used to compute the eigenvalues and eigenfunctions are,

respectively. We approximate the fractional Laplacian operator by the truncated series using a finite number of the orthonormal trigonometric functions ,

12

The Fourier coefficients are computed using “Discrete Fourier Transforms.” Highly efficient algorithms based on Fast Fourier Transform (FFT) are used to compute matrix vector products. Please see the reference²⁵ pages 23–46, for periodic boundary conditions.

For the one-dimensional problems, we represent equation (11) in Fourier space by applying the Fourier transform^18,26,

13

where is the discretization matrix. Denoting , , we can write (13) as,

14

We compute using Fourier transforms and use the inverse Fourier transforms to reconstruct the solution u(x, t) in the physical space. Based on the specified homogeneous boundary conditions, discrete sine or cosine transforms and their inverses are applied²⁷.

Time-stepping scheme

We develop a second order L-stable numerical method by considering the following abstract initial value problem

15

Using the integrating factor, we write the exact solution of (15) as:

16

For some fixed , let be some fixed time step and . Evaluating (16) at , we get,

17

Similarly evaluating (16) at , we get,

which is simplified as,

Now we apply the transformation or and and get the recurrence formula,

18

where denotes the numerical solution computed at . The computational challenges associated with approximating the recurrence formula (18) include an accurate and efficient approximation of matrix exponential functions and integral terms. Approximating the integral is quite challenging because it involves a non-linear function of the unknown function u(x, t). The computational accuracy of the solution depends on how well we approximate these matrix exponential functions and the integral terms in (18). We employ the exponential time-differencing (ETD) approach to approximate the integral.

A second order scheme

A second-order scheme is developed using the following steps.

Step:1

Approximate by a constant for all and compute the integral in the recurrence formula (18) and label it as ,

19

Step:2

Next we approximate using the linear interpolation,

for , substitute in the integral and evaluate the integral. Some simplification and rearrangement of the terms yield the following formula,

20

where , , and .

The formulas (19) and (20) constitute a Runge-Kutta type time-stepping scheme. We can easily implement this scheme for the cases when A is a scalar matrix or when it is a nonsingular diagonal matrix. When A is a non-diagonal matrix, the computation of the inverses and may become numerically problematic, particularly when the eigenvalues of A are close to zero. Moreover, evaluating the matrix exponential for a non-diagonal dense matrix is computationally expensive due to its high computational cost. These difficulties are effectively addressed by approximating the matrix exponential using the positivity-preserving second-order Padé(0, 2) approximation²⁸. Substituting with its Padé approximation not only alleviates the computational cost but also leads to the cancellation of the matrix inverses and , and thereby improving numerical stability.

A second order L-stable scheme

The Padé(0, 2) rational approximation of denoted by is defined as,

21

where is the identity matrix of the size of A.

Definition 7

Let be a rational approximation of the exponential . It is called A-acceptable if for all z with negative real part . If is A-acceptable and also satisfies as , then it is called L-acceptable.

For with ,

Thus as , and the method is L-stable. The scheme (19) and (20) is modified using Padé(0, 2) rational approximation as,

22

where,

Let be the exact solution computed at and be the numerical solution at computed by using the stable scheme (22). Let X be a finite-dimensional subspace of , with a bounded domain in .

Theorem 8

Let F be Lipschitz on . If , then the following error bound holds for the numerical solution ,

uniformly on . The constant C depends on T, but is independent of n and k.

This theorem can be proved in the same way as theorem 4.7 in²⁹ by replacing Padé(0, 2) approximant instead of Padé(1, 1).

Algorithm

The following algorithm is constructed for easy implementation and coding for the L-stable scheme (22) for problems with homogeneous Dirichlet boundary conditions.

1. Input: The initial point a end point b of the spatial domain, value of the order of fractional derivative , spatial mesh size N, the diffusion coefficient , the final time T, and the initial condition u

2. Compute and use the formula to compute the vector of eigenvalues

3. Compute and the spatial mesh as

4. Define , and

5. For do the following

   

Remark

The Fourier transforms dst (discrete sine transform) and idst (inverse discrete sine transform) are replaced by dct (discrete cosine transform) and idct (inverse discrete cosine transform) for the problems having homogeneous Neumann boundary conditions. The eigenvalues for this case are computed as,

and with . The amplification symbol of the rational approximation used is shown in Fig. 1. The approximation is L-acceptable and approaches to zero asymptotically without crossing the x-axis.

Fig. 1.

Amplification symbol of the rational approximations .

Stability and linear error analysis of the numerical scheme

In this section we shall compute the stability regions of the proposed scheme. For a Runge-Kutta type numerical scheme, the stability regions are the regions in the complex plane for which the numerical solution remains bounded and stable. Accordingly, we analyze the stability properties of the proposed method and subsequently present a linear error analysis.

Stability regions

We shall find the stability regions of the schemes (22) by considering the following non-linear autonomous ODE³⁰,

23

with a non-linear function F of the complex-valued function u. Assume that be a fixed point of (23) such that Let u be a perturbation of the fixed point and let , then linearizing about about this fixed point we obtain the following equation,

24

If , then the fixed point is said to be stable, see Cox and Matthews³⁰. In order to cover the dynamics of the system as much as possible, it is naturally crucial for a numerical scheme to satisfy this property. To determine the regions of stability, we apply the numerical scheme to equation (24) and and compute the ratio . Assuming and , with k as the time step size and applying the scheme, we get the ratio,

The stability regions computed for are plotted and given in Fig. 2.

Fig. 2.

Stability regions of the second-order L-stable method.

Stability analysis of the scheme

Stability of the proposed numerical scheme (22) is analyzed in this subsection. Within the framework of ordinary differential equations²⁸, various approaches have been developed for the stability analysis of numerical methods for linear and nonlinear systems. Accordingly, the semi-discretized problem (15) can be written as the following nonlinear autonomous ODE:

25

where f(y) is a nonlinear complex-valued function. Let be a fixed point of the function such that . Let y be a perturbation of the fixed point and then linearizing about this fixed point, we have

26

The solution of (26) is . If the initial condition is slightly perturbed as , then the solution of the perturbed initial value problem is

If with , and t is sufficiently large, then the following error,

becomes arbitrarily large for and for each and the problem (26) is unstable. If , then the error for can be arbitrarily small, which makes the initial value problem (26) stable. For , the difference is at least bounded.

Numerical stability plays a crucial role in the solution of systems of ordinary differential equations—particularly stiff systems—since it permits the step size to be governed primarily by accuracy requirements rather than stability constraints^31,32. According to Dahlquist³³, a numerical method is said to be A-stable if, for the test problem (26), all numerical solutions tend to zero as for any fixed step size. When (25) represents a system of differential equations, denotes a diagonal or block-diagonal matrix whose entries are the eigenvalues of f. Applying the proposed numerical scheme (19)–(20) to the test problem (26) yields the following result:

with the step size k and According to the definition 2 of²⁸, the rational function is called the stability function. Noted that Padé approximation has the property that for . Therefore our proposed numerical scheme (19)-(20) is L-stable with the stability domain of the method,

and according to the famous Dahlquist³³ test equation.

Linear error analysis

Linear error analysis is a methodology for analyzing and quantifying errors induced by numerical methods when solving differential equations. This analysis provides information on the stability and accuracy of the numerical method.

We prove the second-order convergence of the numerical scheme using a linearization of F(u). The linear error analysis of the numerical scheme (22) is conducted using the exact solution written in the recurrence relation form as,

27

of the linearized semi-discretized system,

28

Applying (22) to (28), we get

29

where

30

Now putting (30) into (29), we can write

31

Using the Taylor series expansion

32

equation (31) can be written as:

33

34

and the exact solution (27) is written as,

35

It clearly follows from (34) and (35) the method has second-order local truncation error as,

36

Numerical experiments

A test problem with a known exact solution is considered to validate the accuracy and efficiency of the proposed numerical scheme. Two space-fractional diffusive epidemic models, SIR and SEIR are numerically solved using the proposed numerical scheme. Hat-like non-smooth initials are considered for both SIR and SEIR models. The effects of space-fractional diffusion on the solution profile are examined; see Figures 5,6, and 8,9,10,11,12. For , the numerical solution of problem (7.2), and problem (7.3) matches the known results from Al-Showaikh and Twizel¹⁰ and Nauman et al.^12,13 respectively. These results also validate that all the results remain positivity-preserving for all relevant parameter regimes. Steady-state values from the Jacobian and reproduction number analysis match long-time numerical simulations, see Fig. 6

Fig. 5.

Problem 7.2: SIR Model for disease-free equilibrium: Susceptible, Infected, and Recovered with and .

Fig. 6.

Problem 7.2: SIR Model for endemic equilibrium: Susceptible, Infected, and Recovered with and .

Fig. 8.

SEIR model dynamics at time with and different values of diffusion coefficients.

Fig. 9.

SEIR Model: Susceptible for (Left) and (Right).

Fig. 10.

SEIR Model: Exposed for (Left) and (Right).

Fig. 11.

SEIR Model: Infected for (Left) and (Right).

Fig. 12.

SEIR Model: Recovered for (Left) and (Right).

A reaction-diffusion test problem

The following problem from Zhai et al.³⁴ with a known analytical solution is being considered to validate the expected rate of convergence of the methods.

37

The exact solution of the problem (7.1) is given as,

38

Accuracy of the proposed scheme is demonstrated by solving the test problem (7.1) for different values of . Figure 3 (LEFT) shows the graphs of the exact and the numerical solution at the final time . Since the exact solution involves the factor , therefore the exact solution (38) remains the same at for all values of . For , it will be different for different values of . The exact and numerical solutions computed for different values of at are shown in Fig. 3 (RIGHT). These graphs highlight the accuracy of our numerical scheme for different values of . The time evolution graph shown in Figure 4 validates the reliability of our scheme over the whole time domain . Convergence of the numerical scheme is validated by computing spatial convergence, temporal convergence, and space-time convergence. Third-order spatial accuracy is obtained for different values of , see Table 2. Second-order temporal as well as space-time combined accuracy obtained for different values of is shown in Tables 3 and 4. Computational efficiency of the scheme is demonstrated by recording the central processing unit (CPU) time for each iteration while computing these convergence tables.

Fig. 3.

Problem 7.1: Exact and numerical solutions at (LEFT) and at for different values of (RIGHT).

Fig. 4.

Problem 7.1: Solution profile for .

Table 2.

Problem 7.1: Convergence in space of the proposed scheme using time step size .

Values of	h	Error	Order-	CPU Time
	0.10000	3.0431e-04	0.0000	0.0699
	0.05000	4.4229e-05	2.7824	0.0485
	0.02500	5.7438e-06	2.9449	0.0528
	0.01250	7.2755e-07	2.9809	0.0641
	0.00625	9.2607e-08	2.9738	0.0824
	0.00313	1.2309e-08	2.9114	0.1140
	0.10000	3.5492e-04	0.0000	0.0708
	0.05000	5.1581e-05	2.7826	0.0511
	0.02500	6.6959e-06	2.9455	0.0539
	0.01250	8.4687e-07	2.9831	0.0628
	0.00625	1.0715e-07	2.9824	0.0813
	0.00313	1.3928e-08	2.9437	0.1091
	0.10000	3.9538e-04	0.0000	0.0719
	0.05000	5.7447e-05	2.7829	0.0524
	0.02500	7.4503e-06	2.9469	0.0533
	0.01250	9.3870e-07	2.9886	0.0671
	0.00625	1.1698e-07	3.0044	0.0869
	0.00313	1.4318e-08	3.0304	0.1081

Table 3.

Problem 7.1: Convergence in time of the proposed scheme using space step size .

	k	Error	Order-	CPU Time
	0.10000	3.8425e-03	0.0000	0.0277
	0.05000	1.1641e-03	1.7229	0.0068
	0.02500	3.2140e-04	1.8567	0.0086
	0.01250	8.4108e-05	1.9340	0.0152
	0.00625	2.1311e-05	1.9806	0.0459
	0.00313	5.3231e-06	2.0013	0.0837
	0.10000	3.9838e-03	0.0000	0.0271
	0.05000	1.3283e-03	1.5846	0.0073
	0.02500	3.9174e-04	1.7616	0.0084
	0.01250	1.0761e-04	1.8641	0.0154
	0.00625	2.8337e-05	1.9250	0.0402
	0.00313	7.2416e-06	1.9683	0.0660
	0.10000	3.8251e-03	0.0000	0.0272
	0.05000	1.3816e-03	1.4692	0.0062
	0.02500	4.3342e-04	1.6725	0.0076
	0.01250	1.2362e-04	1.8098	0.0147
	0.00625	3.3561e-05	1.8811	0.0428
	0.00313	8.8089e-06	1.9297	0.0791

Table 4.

Problem 7.1: Convergence in space-time of the proposed scheme.

	h	k	Error	Order-	CPU Time
	0.01000	0.01000	5.4203e-05	0.0000	0.0760
	0.00500	0.00500	1.3647e-05	1.9898	0.0483
	0.00250	0.00250	3.4027e-06	2.0038	0.1443
	0.00125	0.00125	8.4743e-07	2.0055	0.2999
	0.00063	0.00063	2.1129e-07	2.0039	0.9134
	0.00031	0.00031	5.2741e-08	2.0022	3.1301
	0.01000	0.01000	7.0309e-05	0.0000	0.0664
	0.00500	0.00500	1.8313e-05	1.9408	0.0592
	0.00250	0.00250	4.6484e-06	1.9781	0.1426
	0.00125	0.00125	1.1658e-06	1.9954	0.3245
	0.00063	0.00063	2.9135e-07	2.0005	0.9415
	0.00031	0.00031	7.2783e-08	2.0011	3.4588
	0.01000	0.01000	8.4122e-05	0.0000	0.0701
	0.00500	0.00500	2.2729e-05	1.8880	0.0515
	0.00250	0.00250	5.9481e-06	1.9340	0.1135
	0.00125	0.00125	1.5179e-06	1.9704	0.2865
	0.00063	0.00063	3.8216e-07	1.9898	0.9559
	0.00031	0.00031	9.5735e-08	1.9971	3.2679

Fractional diffusive SIR epidemic model

Here we consider the fractional diffusive SIR system (3) with the initial conditions,

and homogeneous Neumann boundary conditions. The parameter values taken for this experiment are: 

The spatial domain we consider is: and two temporal domains , and, are taken to show the disease-free equilibrium and endemic equilibrium.

Parameters Sensitivity Analysis: The numerical experiments are performed using and . The influence of fractional order and infection transmission rate is shown in Figs. 5 and 6. Figure 5 shows the graphs of susceptible, infected, and recovered for the disease-free equilibrium with , and corresponding . Figure 6 shows the graphs of susceptible, infected, and recovered populations for the endemic equilibrium with , and corresponding . The transmission rate determines epidemic intensity and the transition between disease-free and endemic regimes. These findings highlight the combined roles of mobility structure and transmission strength in shaping epidemic outcomes.

Fractional diffusive SEIR epidemic model

The fractional diffusive SEIR model (4) is also numerically solved. This model determines the geographic spread of the disease in a population that has not been vaccinated against it. Since R is not present in the first three equations of (4), we shall solve the system only for S, E, and I and compute , where is the total population under consideration. The total population taken for the numerical experiments is . The hat-shaped initial distributions considered are,

and homogeneous Neumann boundary conditions are considered. The parameter values taken for this experiment are: . The domain we consider is: . The numerical experiments are performed using and .

Graphs of the hat-shaped initial conditions are shown in Fig. 7. The maximum value of each class of individuals, which initially is concentrated around the midpoint of the interval, decreases linearly to zero at the boundaries and .

Fig. 7.

SEIR Model: Initial distributions of susceptible, exposed, infected, and recovered populations.

Parameters Sensitivity Analysis: The sensitivity analysis highlights how key parameters correspond to distinct epidemiological mechanisms. Changes in the diffusion coefficients modulate the spatial homogenization rate of each compartment, with higher diffusion accelerating the redistribution of infected individuals and smoothing spatial gradients, which may weaken the effectiveness of localized containment strategies. The number of susceptible people decreases with the passage of time, whereas the numbers of exposed and infectious individuals increase. The changes in the dynamical behavior of the model with respect to the changes in the diffusion coefficients are shown in Fig. 8. It can be seen that as rates increase, the number of susceptible individuals becomes larger than both the number of exposed and the number of infected individuals, and both the exposed and infected individuals spread on the x-axis. Figures 9,10,11,12 respectively, give three-dimensional plots of susceptible, exposed, infectious, and recovered individuals for and with diffusion coefficients . The profiles in Fig. 8 can clearly be seen in Figs. 9,10,11 at in each figure.

Variations in the fractional order primarily regulate the degree of long-range population movement: lower values of (stronger nonlocal diffusion) consistently lead to faster spatial spread and earlier epidemic peaks, reflecting mobility patterns with long-distance jumps; see the left-side and right-side graphs in Figs. 9,10,11,12.

Conclusions

By synthesizing different ideas, we have developed an implementation strategy is to numerically solve the fractional reaction-diffusion systems. The scheme is used for numerical simulation of two space fractional epidemic models. The Fourier spectral approach is applied to discretize the fractional Laplacian, and discrete Fourier transforms are used to efficiently implement the scheme for numerical experiments. Matrix exponential functions are approximated using positivity-preserving rational functions. An algorithm is derived from the scheme for straightforward implementation, accommodating both homogeneous Dirichlet and homogeneous Neumann boundary conditions.

Our future work plan is to extend these models in multi-spatial dimensions and apply optimal control strategies. We also have a plan to work on data-driven parameter estimation, stochastic mobility, and multi-region coupling.

Author contributions

M.Y. conceived the ideas and the experiment(s), and N.A. conducted the experiment(s) and wrote the manuscript. All authors reviewed the manuscript.

Funding

Funding for this research is provided through the scholarly support by the Deanship of Research, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia.

Data availability

All data generated or analyzed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.