Influence of Fractional Order on the Behavior of a Normalized Time-Fractional SIR Model

Abstract

In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized by the property that the sum of its weight function equals one. This unity property is crucial because it helps with evaluating how the fractional order influences the behavior of time-fractional differential equations over time. The normalized time-fractional derivative, with its unity property, provides an intuitive understanding of how fractional orders influence the SIR model’s dynamics and enables systematic exploration of how changes in the fractional order affect the model’s behavior. We numerically investigate how these variations impact the epidemiological dynamics of our normalized time-fractional SIR model and highlight the role of fractional order in improving the accuracy of infectious disease predictions. The appendix provides the program code for the model.

1. Introduction

Time-fractional derivatives, unlike classical derivatives, are important for developing more accurate epidemic mathematical models for transmission dynamics because they more effectively capture the history of disease dynamics, which is common in real-world disease spread [1]. For example, the well-known time-fractional Caputo derivative of order $\alpha$ for a given function $u\left(t\right)$ is defined as follows:

$$\begin{array}{c}\hfill D_t^{\alpha }u\left(t\right)=\frac{1}{\Gamma (1-\alpha )}{\int }_0^t\frac{ds\left(s\right)}{ds}\frac{ds}{{(t-s)}^{\alpha }},0<\alpha <1,\end{array}$$

(1)

where $\Gamma \left(\alpha \right)$ represents the Gamma function $\Gamma \left(z\right)={\int }_0^{\infty }{s}^{z-1}{e}^{-s}ds$ [2]. The time-fractional Caputo derivative is a generalization of the classical derivative that accounts for memory effects. In the Caputo derivative, the fractional derivative is computed on the function’s integer-order derivative rather than directly on the function itself. This definition is particularly useful in time-fractional differential equations, where it helps to describe systems exhibiting non-local and history-dependent memory processes [3]. Aghayan et al. [2] presented stability and stabilization criteria for uncertain fractional descriptor systems with neutral-type delay, using Lyapunov–Krasovskii functional and linear matrix inequalities, and designed control laws to ensure robust stability, which were illustrated by numerical examples that showed less conservative results than previous studies. Aghayan et al. [4] investigated the delay-dependent robust stability of uncertain fractional-order neutral-type systems with distributed delays and input saturation, and they proposed state-feedback controller gains via linear matrix inequalities and validated the results through numerical simulations. Their mathematical epidemic models used fractional derivatives results for the more accurate modeling of disease dynamics, particularly for diseases with long incubation periods, or complex interactions among individuals. Fractional models improve the understanding and prediction of the epidemic’s progression over time. This allows researchers to more effectively study long-term trends, short-term fluctuations, and the influence of past conditions on present disease dynamics [5].

Traditionally, susceptible–infected–removed (SIR) models have been studied using ordinary differential equations with uniform population distribution. However, recent research has focused on reaction–diffusion models to account for spatial movement and transportation networks, which significantly influence the spread of infectious diseases. Fractional derivatives provide a more accurate representation of complex systems by allowing non-integer orders that better fit the available data [6]. He et al. [7] developed a novel fractional-order discrete-time SIR epidemic mathematical model with vaccination. Sene [8] analyzed the SIR epidemic model by incorporating a delay within the framework of the fractional derivative using the Mittag–Leffler kernel. Zhang et al. [9] investigated the use of fractional-derivative equations (FDEs) to model the dynamics and reduction strategies of COVID-19 by applying a time-dependent susceptible–exposed–infectious–recovered (SEIR) model to predict the evolution of the virus in various countries, and they suggested that while FDEs capture memory effects in the death toll, strict social distancing may be more effective than self-quarantine in reducing the spread. Georgiev and Vulkov [10] presented a fractional temporal SIR model and extended the classical SIR framework by incorporating fractional derivatives to capture subdiffusion and memory effects, resolved model reconstruction through a coefficient identification inverse problem, reconstructed time-dependent parameters using an iterative algorithm, and proposed an approach for assessing the model’s economic impact. Djenina et al. [11] proposed a new fractional-order discrete model for predicting COVID-19, demonstrating its adaptability to periodic infection changes, with existence, uniqueness, and stability analyzed and supported by numerical simulations. Khan et al. [12] presented the dynamics of a fractional SIR model with a generalized incidence rate using Caputo and Atangana–Baleanu derivatives, conducted mathematical analysis of stability for disease-free and endemic equilibria, proved the existence and uniqueness of solutions, and demonstrated through numerical results that the Atangana–Baleanu derivative offers more biologically feasible outcomes than the Caputo operator. Gao et al. [13] presented a novel computational method for solving a nonlinear fractional dynamical system modeling the spread of coronavirus (2019-nCoV) using a compartmental model, modified with fractional operators, and investigated the impact of these operators on infection dynamics through a modified predictor–corrector approach. Majee et al. [14] introduced and analyzed a novel fractional-order SIR epidemic model with a saturated treatment function and examined the system’s dynamics, equilibrium points, stability, basic reproduction number, backward bifurcation conditions, and Hopf bifurcations, with numerical validation of the analytical results. Chen et al. [15] reviewed the impact of COVID-19 through epidemiological modeling, focusing on fractional epidemic models such as fractional SIR, SEIR, and SEIAR models; proposed a general fractional SEIAR model using single-term and multi-term fractional differential equations; and introduced an effective parameter estimation algorithm combining the Nelder–Mead simplex search and particle swarm optimization to predict and manage infectious disease outbreaks.

However, while traditional time-fractional derivatives effectively model memory effects, their definitions inherently introduce scaling issues when comparing the impacts of different fractional orders because the integral of the weight function depends not only on the fractional order but also on time, as noted in a recent study [16]. The main contribution and novelty of this paper is to introduce a new normalized time-fractional susceptible–infected–removed model based on a recently developed normalized time-fractional derivative [16], which preserves the property that the integral of the weight function equals one, regardless of both the fractional order and time. This characteristic enables a more consistent comparison of the fractional order’s influence on the behavior of time-fractional SIR models. We then numerically study the impact of fractional order on the dynamics of the proposed normalized time-fractional SIR model.

The structure of this paper is outlined as follows: In Section 2, we present the proposed normalized time-fractional SIR model. In Section 3, a numerical solution algorithm is described in detail. Computational experiments are conducted in Section 4. Finally, the conclusions are given in Section 5. The Appendix A includes the program code for the proposed normalized time-fractional SIR model, available for interested readers.

2. Proposed Normalized Time-Fractional SIR Model

The standard SIR model is a compartmental model used to describe the spread of infectious diseases. It divides the population into three compartments: susceptible (S), those who can contract the disease; infected (I), those who have the disease and can spread it; and recovered (R), those who have recovered and are immune. The model is governed by a system of ordinary differential equations (ODEs), where transitions between compartments depend on the infection and recovery rates. It provides a simplified but effective framework for analyzing disease transmission dynamics in a population. The standard SIR model [17] is expressed as follows:

$$\begin{array}{ccc}\hfill \frac{dS\left(t\right)}{dt}& =& -\beta S\left(t\right)I\left(t\right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill \frac{dI\left(t\right)}{dt}& =& \beta S\left(t\right)I\left(t\right)-\gamma I\left(t\right),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \frac{dR\left(t\right)}{dt}& =& \gamma I\left(t\right),\hfill \end{array}$$

(4)

where $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ are the susceptible, infected, and removed individuals who are not currently infected, currently infected, and recovered at time t, respectively.

In this paper, we present a normalized time-fractional SIR model based on a recently developed normalized time-fractional diffusion equation [16]:

$$\begin{array}{ccc}\hfill \frac{d^{\alpha }S\left(t\right)}{d{t}^{\alpha }}& =& -\beta S\left(t\right)I\left(t\right),\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill \frac{d^{\alpha }I\left(t\right)}{d{t}^{\alpha }}& =& \beta S\left(t\right)I\left(t\right)-\gamma I\left(t\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \frac{d^{\alpha }R\left(t\right)}{d{t}^{\alpha }}& =& \gamma I\left(t\right),\hfill \end{array}$$

(7)

where

$$\begin{array}{c}\hfill \frac{d^{\alpha }S\left(t\right)}{d{t}^{\alpha }}=\frac{1-\alpha }{t^{1-\alpha }}{\int }_0^t\frac{dS\left(s\right)}{ds}\frac{ds}{{(t-s)}^{\alpha }},0<\alpha <1,\end{array}$$

(8)

where $(1-\alpha )/t^{1-\alpha }$ is the normalizing factor, which ensures that the right-hand side term in Equation (8) becomes $dS/ds$ when $dS/ds$ is constant. That is,

$$\begin{array}{c}\hfill \frac{1-\alpha }{t^{1-\alpha }}{\int }_0^t\frac{ds}{{(t-s)}^{\alpha }}=1,0<\alpha <1.\end{array}$$

(9)

The other two normalized time-fractional derivatives, $d^{\alpha }I\left(t\right)/d{t}^{\alpha }$ and $d^{\alpha }R\left(t\right)/d{t}^{\alpha }$, are defined in a similar manner. Specifically, these derivatives characterize the fractional rate of change in the infected and removed populations over time, consistent with the normalization approach used for the susceptible individuals. Let

$$\begin{array}{c}\hfill w_{\alpha }^t\left(s\right)=\frac{1-\alpha }{t^{1-\alpha }{(t-s)}^{\alpha }}.\end{array}$$

(10)

Figure 1a,b show $w_{\alpha }^{30}\left(s\right)$ for various values of $\alpha =0.1,0.5$ and $0.9$ with $t=30$, and $w_{0.5}^t\left(s\right)$ for various time values $t=10,20$, and 30 with $\alpha =0.5$, respectively. Furthermore, from Equation (9), we obtain

$$\begin{array}{c}\hfill W_{\alpha }\left(t\right)={\int }_0^t{w}_{\alpha }^t\left(s\right)ds=1,\end{array}$$

(11)

which does not depend on both the fractional order $\alpha$ and the time t. We note that the normalized time-fractional derivative, Equation (8), shares the same integrand as the Caputo derivative, Equation (1), with the only difference being the coefficient terms. The former includes the factor $(1-\alpha )/t^{1-\alpha }$, which depends on both $\alpha$ and time t, whereas the latter involves $1/\Gamma (1-\alpha )$, a term dependent only on $\alpha$.

3. Numerical Solution Algorithm

Let $S_n=S\left(t_n\right)$, $I_n=I\left(t_n\right)$, and $R_n=R\left(t_n\right)$ for $n=1,\dots ,$ where $t_n=(n-1)\Delta t$. The normalized time-fractional derivative, Equation (8), can be discretized as follows:

$$\begin{array}{ccc}\hfill \frac{d^{\alpha }S(t_n+1)}{d{t}^{\alpha }}& =& \frac{1-\alpha }{t_{n+1}^{1-\alpha }}\sum _{p=1}^n{\int }_{t_p}^{t_{p+1}}\frac{dS\left(s\right)}{ds}\frac{ds}{{(t_{n+1}-s)}^{\alpha }}\approx \sum _{p=1}^n\frac{1-\alpha }{t_{n+1}^{1-\alpha }}{\int }_{t_p}^{t_{p+1}}\frac{ds}{{(t_{n+1}-s)}^{\alpha }}\frac{S_{p+1}-S_p}{\Delta t}\hfill \\ & =& \sum _{p=1}^n\frac{{(n+1-p)}^{1-\alpha }-{(n-p)}^{1-\alpha }}{n^{1-\alpha }}\frac{S_{p+1}-S_p}{\Delta t}.\hfill \end{array}$$

(12)

Hence, we obtain

$$\begin{array}{ccc}\hfill \sum _{p=1}^n{w}_p^n\frac{S_{p+1}-S_p}{\Delta t}& =& -\beta S_{n+1}{I}_n,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \sum _{p=1}^n{w}_p^n\frac{I_{p+1}-I_p}{\Delta t}& =& \beta S_{n+1}{I}_n-\gamma I_{n+1},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \sum _{p=1}^n{w}_p^n\frac{R_{p+1}-R_p}{\Delta t}& =& \gamma I_{n+1},\hfill \end{array}$$

(15)

where $w_p^n=[{(n+1-p)}^{1-\alpha }-{(n-p)}^{1-\alpha }]/n^{1-\alpha }$, which satisfies

$$\begin{array}{c}\hfill \sum _{p=1}^n{w}_p^n=1.\end{array}$$

(16)

We rewrite the discrete system of Equations (13)–(15) as follows:

$$\begin{array}{ccc}\hfill S_{n+1}& =& S_n+\frac{\Delta t}{w_n^n}\left(-\beta S_{n+1}{I}_n-\sum _{p=1}^{n-1}{w}_p^n\frac{S_{p+1}-S_p}{\Delta t}\right),\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill I_{n+1}& =& I_n+\frac{\Delta t}{w_n^n}\left(\beta S_{n+1}{I}_n-\gamma I_{n+1}-\sum _{p=1}^{n-1}{w}_p^n\frac{I_{p+1}-I_p}{\Delta t}\right),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill R_{n+1}& =& R_n+\frac{\Delta t}{w_n^n}\left(\gamma I_{n+1}-\sum _{p=1}^{n-1}{w}_p^n\frac{R_{p+1}-R_p}{\Delta t}\right),\hfill \end{array}$$

(19)

From Equations (17)–(19), it is clear that $S_{n+1}+I_{n+1}+R_{n+1}=S_n+I_n+R_n=N$. We reformulate the discrete system given by Equations (17)–(19) in the following manner:

$$\begin{array}{ccc}\hfill S_{n+1}& =& \left(S_n-\frac{\Delta t}{w_n^n}\sum _{p=1}^{n-1}{w}_p^n\frac{S_{p+1}-S_p}{\Delta t}\right)/\left(1+\frac{\Delta t\beta I_n}{w_n^n}\right),\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill I_{n+1}& =& \left[I_n+\frac{\Delta t}{w_n^n}\left(\beta S_{n+1}{I}_n-\gamma I_{n+1}-\sum _{p=1}^{n-1}{w}_p^n\frac{I_{p+1}-I_p}{\Delta t}\right)\right]/\left(1+\frac{\Delta t\gamma }{w_n^n}\right),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill R_{n+1}& =& R_n+\frac{\Delta t}{w_n^n}\left(\gamma I_{n+1}-\sum _{p=1}^{n-1}{w}_p^n\frac{R_{p+1}-R_p}{\Delta t}\right).\hfill \end{array}$$

(22)

4. Computational Experiments

Numerical solutions are obtained by solving Equations (20)–(22) using appropriate initial conditions and parameter values. All computations are performed using a time step of $\Delta t=0.01$.

Figure 2a–d show the numerical solutions of the normalized time-fractional SIR model for $\alpha =0.1,0.5,0.9$, and 1, respectively. Here, the parameters $\beta =0.3$, $\gamma =0.1$, $S_1=0.99$, $I_1=0.01$, $R_1=0$, and $T=90$ are used. The basic reproduction number is ${\mathcal{R}}_0=\beta /\gamma =3$. For smaller values of $\alpha$, the solutions exhibit faster temporal evolution during the early stages. As $\alpha$ decreases, the system responds more quickly, leading to rapid changes in the solution over time. This behavior suggests a strong influence of $\alpha$ on the dynamics of the system, where lower values accelerate the progression of the solution. Conversely, larger values of $\alpha$ result in slower temporal evolution, which indicates a more gradual transition in the system’s state during the early stages. The peak value of the infected population rises with an increase in $\alpha$. Therefore, $\alpha$ plays a crucial role in determining the rate at which the system evolves temporally. The computational result shown in Figure 2c, which is a case of $\alpha =0.9$, is particularly noteworthy. In contrast to the conventional SIR model, the susceptible population increases at later times.

Figure 3a–d display the numerical solutions for the normalized time-fractional SIR model with $\alpha$ values of 0.1, 0.5, 0.9, and 1, respectively. The parameters $\beta =0.6$, $\gamma =0.1$, $S_1=0.99$, $I_1=0.01$, $R_1=0$, and $T=90$ are applied. The basic reproduction number is ${\mathcal{R}}_0=\beta /\gamma =6$. For smaller values of $\alpha$, the solutions demonstrate more rapid temporal changes in the early stages. As $\alpha$ decreases, the system exhibits a quicker response and results in accelerated changes in the solution over time. This indicates that $\alpha$ significantly affects the system’s dynamics, with lower values accelerating the evolution of the solution. Conversely, higher values of $\alpha$ lead to slower temporal progression and reflect a more gradual change in the system’s state during the initial period. The peak value of the infected population rises with an increase in $\alpha$. Hence, $\alpha$ is crucial for determining the temporal rate of system evolution. Notably, the result shown in Figure 2c for $\alpha =0.9$ is of particular interest, as it reveals an increase in the susceptible population at later times, unlike in the conventional SIR model.

Finally, we consider a comparison test with a conventional time-fractional SIR model [18]:

$$\begin{array}{ccc}\hfill D_t^{\alpha }S\left(t\right)& =& -\beta S\left(t\right)I\left(t\right),\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill D_t^{\alpha }I\left(t\right)& =& \beta S\left(t\right)I\left(t\right)-\gamma I\left(t\right),\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill D_t^{\alpha }R\left(t\right)& =& \gamma I\left(t\right).\hfill \end{array}$$

(25)

The numerical solutions for the system of Equations (23)–(25) are similar to those in Equations (20)–(22), with the only difference being the weight term

$$\begin{array}{c}\hfill w_p^n=\frac{{(n+1-p)}^{1-\alpha }-{(n-p)}^{1-\alpha }}{{(\Delta t)}^{\alpha -1}\Gamma (2-\alpha )}\end{array}$$

(26)

instead of

$$\begin{array}{c}\hfill w_p^n=\frac{{(n+1-p)}^{1-\alpha }-{(n-p)}^{1-\alpha }}{n^{1-\alpha }}.\end{array}$$

(27)

Figure 4 shows a comparison of the computational results for the normalized and Caputo time-fractional SIR models. In the normalized model, the susceptible population decreases rapidly, and the infected population peaks early (around $t=8$), followed by a sharp decline. Recovery is fast, with $R\left(t\right)$ approaching 0.8 by $t=90$. In contrast, the Caputo model exhibits a more gradual progression, with infections peaking later (around $t=43$) and smoother falls in both infected and susceptible populations occurring. Recovery also occurs at a slower pace, which indicates a longer-lasting epidemic. The normalized model suggests faster epidemic dynamics, while the Caputo model implies a more prolonged outbreak with delayed peaks. These differences highlight the importance of model choice for predicting the spread and control of an epidemic.

5. Conclusions

In this article, we introduced a normalized time-fractional SIR model, which was based on a recently developed normalized time-fractional derivative characterized by the unity property of the weight function’s sum. This new derivative framework provided a robust means of investigating the influence of the fractional order on the temporal evolution of epidemiological dynamics. By systematically exploring how variations in the fractional-order parameter affect the behavior of the SIR model, we demonstrated that fractional calculus provides valuable insights into the complex nature of infectious disease progression. Our numerical investigation revealed that the normalized time-fractional derivative enables a more intuitive and interpretable approach for understanding the fractional dynamics within the model. This interpretability is very important for better assessing how changes in the fractional order influence epidemiological outcomes, and it makes it a promising tool for improving the accuracy and predictive capacity of epidemic models. The proposed methodology, therefore, represents a significant contribution not only to fractional calculus but also to its practical application in the field of epidemiology. In conclusion, the normalized time-fractional SIR model has the potential to enhance our understanding of infectious disease dynamics and provides a pathway toward more precise modeling approaches. In this paper, we focused on presenting a normalized time-fractional SIR model and performed numerical experiments to investigate the effects of the fractional order on the evolution of the system’s dynamics. Future work may extend this approach to other complex epidemic model systems, further demonstrating the utility of fractional calculus in diverse scientific domains. Furthermore, analyses such as deriving formulas for the equilibrium points and proving the existence, uniqueness, and global convergence of the solutions of the model would be valuable topics for future research.