https://orcid.org/0000-0003-2382-6553
Abstract
Measles is a highly contagious illness that can spread throughout a population based on the number of susceptible or infected individuals as well as their social dynamics within the society. The measles epidemic is thought to be controlled for the suffering population using the susceptible-exposed-infectious-recovered (SEIR) epidemic model, which depicts the direct transmission of infectious diseases. To better explain the measles epidemics, we provided a nonlinear time fractional model of the disease. The solution of SEIR is obtained by using the Caputo fractional derivative operator of order . The Homotopy perturbation transform method (HPTM) and Yang transform decomposition methodology (YTDM) have been employed to obtain the numerical solution of the time fractional model. Obtaining numerical findings in the form of a fast-convergent series significantly improves the proposed techniques accuracy. The behaviour of the approximate series solution for several fractional orders is shown graphically which are derived through Maple. A graphic representation of the behaviours of susceptible, exposed, infected, and recovered individuals are shown at different fractional order values. Figures that depict the behaviour of the projected model are used to illustrate the developed results. Finally, the present work may help you predict the behaviour of the real-world models in the wild class with respect to the model parameters. It was found that the majority of patients who receive therapy join the recovered class when various epidemiological classes were simulated at the effect of fractional parameter
. These approaches shows to be one of the most efficient methods to solve epidemic models and control infectious diseases.
Figures
Citation: Alshammari NA, Alharthi NS, Mohammed Saeed A, Khan A, Ganie AH (2025) Numerical solutions of a fractional order SEIR epidemic model of measles under Caputo fractional derivative. PLoS One 20(5): e0321089. https://doi.org/10.1371/journal.pone.0321089
Editor: Rajnesh Lal, Fiji National University, FIJI
Received: October 9, 2024; Accepted: March 2, 2025; Published: May 28, 2025
Copyright: © 2025 Alshammari et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The numerical data used to support the fundings of this study are included within the article.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Medical research, public health, and health care evaluation are all grounded within the incredible field of epidemiology. These days, epidemiologists are interested in researching disease models using anonymous parameters. Mathematical epidemiology has been studying the transmission of infectious diseases since the early 20th century with the use of established mathematical models [1–4]. The deterministic and stochastic models of infectious diseases provide the researchers with important new insights. The individual populations for the deterministic models represent the various compartment-specific stages. The population transition rate stated mathematically as a derivative from one part to another depending on various parts and population transition amounts. The population as a function of time is represented by the differential equation system. In 1927, Kermack and McKendrick built a basic deterministic model that was helpful in creating more complex mathematical models of epidemics, and it is still regarded as a foundational model [5]. After the initial stage of infection, a latent period often occurs in various infectious diseases. This interval cannot be skipped while assessing the infectious stage. For this reason, it becomes sense to add an early stage into the epidemiological model. Four classifications make up the current SEIR model of disease, which corresponds to infectious diseases based on their condition. The model is made up of the following individuals: the susceptible ; the exposed
; the infective
; and the recovered
. In certain cases, it is discovered that a member of the infectious class does not exhibit symptoms for some time. SEIR models are employed to model these diseases [6, 7]. Comprising, proposing, planning, executing, testing, and assessing different detection, therapy, and control programmes are among the applications of mathematical epidemiological models [8–13].
The calculus of integrals and derivatives of any arbitrary real or complex order is the focus of fractional calculus [10, 14–17]. This makes it possible to think of it as a modification of classical calculus, which is one specific example that is covered in the theory. The de l’Hospital letter to Leibniz in 1695, which asked, "What does the derivative of order or
of a function mean?" was the moment the notion of derivatives of non-integer order arose. The study of this field attracted the interest of mathematicians in the eighteenth and nineteenth century. Famous scientist Abel became the first to investigate tautochrone problems implicitly using fractional calculus in 1823 [19]. After that, a number of fundamental papers have been published on different facets of fractional calculus [18,20–22]. A comprehensive collection of relevant findings, including case studies and examples, can be found in [23]. Additionally, there exists an extensive amount of work in the background literature pertaining to the exact and approximate solutions of fractional differential equations (FDEs) of the Riemann Liouville and Caputo types, as well as non-integer derivatives concerning polynomial products, non-integer derivatives and non-integer powers of operators, and boundary value problems (see, for example, [24–28]). Researchers have recently focused a lot of attention on fractional calculus, and several facets of the topic are being explored for research. This is an illustration of the fractional derivative’s importance as a tool for understanding the dynamic behaviour of many physical systems.
The use of fractional calculus in a number of scientific and engineering fields is noteworthy and highly significant [29–31]. More specifically, the biological mechanisms underlying a number of diseases have been studied using fractional framework [32–35]. It has been shown that fractional calculus is useful for modeling a wide range of processes and has many applications. The nonlocal properties of this differential operator are its strongest strength, which are absent within integer order differential operators. FDEs are unique in that they may describe the memory and transmission properties of many different mathematical models. It is a known reality that models with fractional orders are more useful and realistic than those with integer orders. In these models, the fractional order derivative yields a higher degree of freedom. Strong tools for controlling the dynamic behaviour of different biomaterials and systems are arbitrary order derivatives [27]. These models most important trait is their global (nonlocal) properties, which are absent within classical order models [36]. However, there is also an increasing amount of work on dynamic fractional differential systems that is related to modern mathematics and is focused on several scientific domains such as control theory, chemistry, and physics. Fractional calculus may be interesting because it provides a more detailed description of potential uncertainties in the dynamic model due to the numerical value of the fraction parameter. In particular, fractional calculus has been applied in a number of real-world scenarios recently [37–40. However, there is currently an expanding collection of mathematical literature on fractional differ-integral calculus that can successfully confirm support for studies conducted in other relevant fields [41–42].
In reality, the analytical analysis of a fractional derivative model is one of its difficulties. In fact, this restricts attention to the fractional model’s dynamics. To the best of our knowledge, there are no pertinent publications in the available literature that address the process of deriving analytical solutions for the measles model. Furthermore, not much research has been done to examine the dynamics of the SEIR model using non-integer derivatives. Overall, the literature suggests that more investigation and study are needed to better understand and study the SEIR model phenomenon. Therefore, we present a comprehensive analysis of the intricate phenomena of SEIR model to understand the transmission route of this infectious disease. Powerful techniques such as YTDM and HPTM are produced by combining the adomian decomposition method (ADM), homotopy perturbation method (HPM), and Yang transform. Adomain polynomials and He’s polynomials are used to decompose the nonlinear terms after the differential equations are transformed into algebraic equations with the aid of Yang transform. Both deterministic and stochastic differential equation systems can be solved effectively with these computational methods. More specifically, it can be applied to a system of fractional order, classical, linear, and nonlinear ordinary and partial differential equations. The structure of the article is as follows: The standard SEIR model is described in Sect 2. We give some fundamental definitions of FC in Sect 3. We created the HPTM for differential equations of any order in Sect 4. We created the YTDM for differential equations of any order in Sect 5. We discussed the convergence analysis of the suggested techniques in Sect 6. We solved the SEIR epidemic model in Sect 7 using the suggested methods. Discussion and numerical simulation are provided in Sect 8. Sect 9 concludes with some final remarks.
2 Model descriptions
To explain the spread dynamics of measles, we design a deterministic, compartmental mathematical model. The population is constantly interacting and mimics the demographics of a normal developing nation while exploring dynamics that are growing exponentially. The model equations [10] are described by classifying the total population into four categories:
, and
, which stand for susceptible, exposed, infected, and recovered population, respectively. The flow chart of these four classes are shown in Fig 1.
The class of susceptible is decreased with infected individuals at a rate
; the rate at which new immigrants or births occur is
; and the rate at which natural death occurs is
. The exposed individuals in
are formed by interaction with infected persons at a rate of
; these individuals break into the infected class at a rate of
, become diminished at a rate of
, and are decreased at a rate of
through testing and measles therapy. At a speed
, the class of infected individual
is formed from the exposed individual. It gets weaker at a rate
and reduced by infection recovery at a rate
. This creates a class
that is fully protected from individual disease. According to [10], the natural death at a rate
is diminished by
recovered individual.
The model’s fractional order expansion was first investigated in [43]. Although at a slower rate, they exhibit the realistic biphasic decrease behaviour of disease infection. The fractional SEIR system is the new differential equation system that is represented as follows.
The above model becomes the classical epidemic model for .
3 Basic concept
To provide more insight and elucidate the methods of solution, we will draw attention to a helpful concept pertaining to fractional calculus.
3.2 Definition
The yang transform (YT) of the function below is as [45]
with demonstrate the transform variable.
Some basic properties of YT are as
having inverse YT as
4 Analysis of the HPTM
Assume the general nonlinear fractional differential equation as
having
By using the YT
After we get
On utilizing the inverse YT
In terms of HPM, we may get
with .
Let
with
where
By switching Eq (14) and Eq (15) in Eq (13), we may get
Similarly,
Finally, the approximate solution is derived as
5 Analysis of the YTDM
Assume the general nonlinear fractional differential equation as
having
By using the YT
After we get
On utilizing the inverse YT
We may get the series form solution as
The nonlinear term is broken down as
with
By switching Eq (24) and Eq (25) in Eq (23), we may get
Similarly,
Finally, the approximate solution is derived as
6 Convergence analysis
Here we discuss the convergence analysis of the proposed approaches.
6.1 Theorem
Let us assume that the precise solution of (9) is and let
,
and
, where H symbolizes the Hilbert space. The solution achieved
will converge
if
, i.e., for any
, such that
,
Proof. We take a sequence of
We must demonstrate that forms a “Cauchy sequence” in order to achieve the desired outcome. Additionally, let’s take
For , we have
As , and
are bound, so take
, and we get
Hence, makes a “Cauchy sequence” in H. It proves that the sequence
is a convergent sequence with the limit
for
which complete the proof.
6.2 Theorem
Assuming that is finite and
reflect the series solution that was establish. Considering
such that
, the maximum absolute error is presumed by the resultant relation.
Proof. Assume is finite which indicates that
.
Let us consider
which complete the proof of theorem.
6.3 Theorem
The result of (20) is unique when
Proof: Let with the norm
is Banach space,
continuous function on J. Let
is a non-linear mapping, where
Suppose that and
, where
and
are are two different function values and
,
are Lipschitz constants.
I is contraction as . From Banach fixed point theorem the result of (19) is unique.
6.4 Theorem
The result of (20) is convergent.
Proof: Let . To show that
is a Cauchy sequence in H. Let,
Let m = n + 1, then
where . Similarly, we have
As , we get
. Therefore,
Since when
. Thus,
is a Cauchy sequence in H, indicating that the series
is convergent.
7 Applications
7.1 Example
Assume the fractional epidemic model (2) with below initial conditions.
By using the YT
After we get
On utilizing the inverse YT
In terms of HPM, we may get
For numerical results, we used the following (Table 1) values of parameters are considered from [10]. By comparing the coefficients, we may get
Finally, the approximate solution is derived as
Application of the YTDM
By using the YT
After we get
On utilizing the inverse YT
We may get the series form solution as
The nonlinear term is broken down as
For numerical results, we used the following Table (1) values of parameters are considered from [10]. By comparing both sides, we may get
On m = 0
On m = 1
Finally, the approximate solution is derived as
8 Results and discussion
The current paper presents the numerical solution of the measles epidemic model using a nonlinear differential equation system. The behaviour of the model is represented by looking at solutions up to a third-order series. From the plot, we see that when the order is smaller faster the decay of susceptible population , this behavior can be observed from the Fig 2 and Fig 3. From Fig 4 and Fig 5, we see that the exposed class significantly grows at smaller order with the passage of time. Similarly from Fig 6 and Fig 7, one can observes that smaller the fractional order fastest the decaying process of the infected class with the passage of time. In the Fig 8 and Fig 9, we may observe that the recovered class grows more rapidly on the smaller order of the differentiation over time. As demonstrated in all figures, evaluation is conducted for a range of
values in order to conduct a reliable research. In comparison to ordinary derivatives, we find that the fractional order SEIR epidemic model has a greater degree of freedom. HPTM and YTDM are used to establish the numerical results of SEIR population for various values of
in Tables 2–9. Outstanding responses from the compartments in the suggested model are obtained by using non-integer values for the fractional parameter. Another noteworthy thing to keep in mind is that we choose a short time interval because we assumed relatively small initial values. The initial values of the data are taken large for long intervals of time to prevent negative patterns in the population. The solution converges to a steady state for a range of
values. It provides faster convergence by decreasing the fractional values of
. The figures provided demonstrate that the anticipated model has a higher degree of flexibility and is significantly dependent on the order. The behaviour that has been described serves as a case study of the capability and efficacy of the suggested solution strategies. Moreover, the fractional operator under investigation provides more interesting outcomes for examining and projecting the future of the model under consideration. The current study may contribute to our understanding of the deadly virus because epidemic models depend heavily on genetic characteristics.
9 Conclusion
Finding the numerical solutions for the fractional SIER model of disease is necessary due to the increasing number of disease models. A system of coupled, non-linear ordinary differential equations describes the population dynamics during the illness in the model. To the best of the author’s knowledge, no precise solution for this model can be found in the literature. This work suggests comparing two unique approaches for the fractional SEIR epidemic model namely the YTDM and HPTM. The results are found as a quickly convergent series of solutions. Additionally, numerical simulations are displayed along with the compression for a variety of values of a. Tables and graphs show how the fractional parameter affected our found solutions. As compared to regular derivatives, it is important to note that fractional derivatives exhibit notable modifications and memory effects. By incorporating fractional calculus into mathematical models, researchers can gain deeper insights into the complex behaviors of biological systems and develop innovative approaches to address key challenges in biomedical research and health care. According to the authors, biologists will find this study to be more beneficial and efficient. Moreover, nonlinear fractional-order mathematical models of infectious diseases such as hepatitis, TB, and Ebola can be studied using the same techniques. Readers can employ hybrid transforms merging with our proposed schemes as a future study direction to attain better outcomes. The addition of more operators will therefore be highly desired in the future, especially in light of the benefits of the current operator.
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