Abstract
A fourcompartmental fractionalorder epidemic model has been investigated to understand the transmission mechanism of infectious diseases with the population’s memory effect. The existence and uniqueness criterion of the model solution of the proposed fractionalorder model is verified. Utilizing the nextgeneration matrix method, a threshold quantity called, the basic reproduction number (\({\mathscr {R}}_0\)) is obtained. The model possesses two equilibrium points, infectionfree and endemic. The asymptotic stability (local and global) of the proposed system at the equilibrium points has been analyzed thoroughly. It is observed that the total number of infections during the disease is influenced by the fractionalorder of the model which represents the population’s memory. A transcritical bifurcation is exhibited around the infectionfree equilibrium point when the basic reproduction number crosses unity. Additionally, a fractionalorder optimal control problem has been studied by considering two disease interventions: media awareness and treatment. The policy containing infectious disease spread has been determined based on a costeffectiveness analysis. Sensitivity indices are computed to determine which parameters significantly impact \({\mathscr {R}}_0\) and hence may used in controlling the disease. Some numerical simulations have been performed to verify analytical results by using MATLAB2022a.
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1 Introduction
Throughout human history, diseases such as the plague, measles, smallpox, HIV, and the flu have negatively impacted our societies and economies [30]. In recent decades, increased human mobility has facilitated the rapid spread of infectious diseases like COVID19, Tuberculosis, HIV, etc. The causative agents behind these illnesses, such as viruses, bacteria, parasites, etc., are ubiquitous and can be transmitted to the human body through direct and indirect paths [14]. Mathematical modeling of infectious disease transmission plays a crucial role in identifying the key parameters regulating the disease dynamics in a population. Therefore, it is also essential in public health policymaking. The most paradigmatic framework was provided by KerMack and Mackendric in 1927 [31]. Thereafter, mathematicians develop a keen interest in studying various aspects of disease transmission and its control strategies [30, 34, 41, 42].
Based on the disease transmission patterns, prevention, and control strategies, several compartmental epidemic models [2, 7, 11, 17, 18, 55] have been proposed and analyzed for their asymptotic dynamics. The most common framework is called the SIR model, which compartmentalizes the whole population into three groups: susceptible, infected, and recovered, according to their clinical condition [28]. Infectious diseases with prolonged noninfectious incubation or latent period, like tuberculosis, influenza, ebola, etc., can be modeled with the inclusion of an extra compartment called exposed or latent period [13].
Many preventive measures like, quarantine, travel restrictions, and school closures, may be implemented to control the spread of diseases. These measures can cause economic hardships, educational disruptions, and emotional stress for individuals and communities. Public health policies focus on disease surveillance, prevention, and control strategies in response to the impact of infectious diseases. These include vaccination programs, promoting good hygiene practices, developing effective treatment options, public health campaigns, and strengthening healthcare systems to enhance preparedness and response capabilities [30, 38, 41]. Among these control strategies, effective treatment [15] is one of the most primary controls implemented for disease elimination [27]. Rowthorn and Toxvaerd [54] studied an epidemic model and found that treatments are typically socially optimal and prevention is socially suboptimal. Olaniyi et al. [49] studied a timedependent deterministic mathematical model to prevent and manage the dynamics of malaria transmission. The authors considered treatment management and other available controls in their model. They have demonstrated that vector control is the most effective single intervention, whereas antimalarial medicine (i.e., treatment) is the most economically advantageous for adoption. Therefore, treatment becomes an essential control policy for disease transmission, and hence, it becomes a major point of interest in this study.
Among numerous nonpharmaceutical control measures, the media profoundly influences public perceptions, awareness, and behaviors concerning the transmission of infectious diseases [21]. The media serves as a primary source of information about communicable diseases, providing updates on outbreaks, transmission patterns, prevention strategies, and treatment options. Its pivotal role lies in educating the populace and enhancing awareness regarding disease risks and precautionary steps, such as maskwearing, social distancing, and adherence to quarantine protocols. Multiple studies [41, 42] have observed that awareness programs facilitated by media significantly mitigate the risk of disease transmission. The behavioral change due to the awareness campaigns is incorporated into the epidemic model to investigate the transmission dynamics [46]. Das et al. [12] investigated a TB model considering the awareness effect on the disease incidence rate. It is observed that the awareness program can avoid the critical occurrence of backward bifurcation for the TB model dynamics. Kar et al. [30] investigated the stability and bifurcation scenario of an epidemic model considering the media effect on the transmission pattern of the disease. Similarly, multiple studies have shown that media awareness programs can significantly alter disease dynamics and help in mitigating the adverse effects of epidemics [14, 47]. Therefore, we consider this an essential control policy in our study. Butt et al. [10] have proposed a malariadengue ODE model to investigate transmission dynamics. They have used various control strategies to prevent infection but they have not applied any fractionalorder optimal control technique. To get better result, we will include fractionalorder optimal control technique in our present work.
Infectious diseases impose a substantial economic burden on societies [56]. The costs include healthcare expenditures, loss of productivity due to illness or death, costs of outbreak response and control measures, and the longterm healthcare needs of individuals affected by infectious diseases. Some factors, such as socioeconomic status, access to healthcare, living conditions, and social determinants of health, can influence disease susceptibility and outcomes [43]. Addressing health inequities is crucial for effective infectious disease control and improving public health. So, it is essential to find out which control (or group of controls) is most economically influential among the available controls for infectious disease. In this contest, costeffectiveness analysis is a suitable mathematical tool [43], which we will apply to find our present work’s most economically effective strategy.
Although systems of ordinary differential equations are utilized to construct the mathematical model describing disease transmission [30, 62], they do not adequately address the memory effect of such events. Over the past few decades, there has been considerable interest in employing fractionalorder differential equations for modeling biological phenomena [8, 9, 24], as they can effectively account for the memory effect [3, 32, 33]. Recently, Majee et al. [39] studied the transmission dynamics of a SIR epidemic model considering the saturated disease incidence rate in the context of COVID19. The authors have also studied the complex dynamics of fractionalorder epidemic models in the presence of vaccination and media control [41, 42]. Veersha et al. [59] investigated numerical techniques to analyze fractionalorder SIR models for childhood and vectorborne diseases. Arqub et al. also investigated the advanced solution techniques for epidemic models [4, 5]. There are different definitions of fractionalorder derivatives. Among these, Caputo’s definition is suitable for biological models as it defines a flow in the phase space of the system. It is observed that the inclusion of fractionalorder gives a wider range of stability compared to the corresponding classical part [39, 51]. Apart from the transmission dynamics of these models, the implementation of optimal control to mitigate the adverse effects of contagions is another important aspect. In a recent study, Majee et al. [40] studied a fractionalorder SEIR optimal control problem considering both treatment and vaccination controls. The author implemented a forward and backward sweep method to solve the problem. In recent times, some researchers have used the fractionalorder derivative to model infectious diseases, and they have proven to be more accurate in some cases when compared to the classical ones. Therefore, one of the main novelties of this work is to use past infection memory through the fractionalorder differential equation, and to introduce media awareness at the time of the disease outbreak.
A significant aspect of disease dynamics is sensitivity analysis [6, 10, 25, 44]. It illustrates how the threshold quantity, the basic reproduction number, is affected by the system parameter. By adjusting a few system factors, we can regulate and even stop the spread of illness. In their epidemic model, some researchers [39, 42] have spoken about several control tactics to lower infection, but they haven’t yet included sensitivity analysis as a control measure. Therefore, our epidemic system’s control mechanisms are improved by our sensitivity analysis.
Finally, our aim in this paper is to explore the complex dynamics of a fractionalordered SEIR epidemic model considering both treatment and media control. Though certain previous studies investigated media awareness control [30, 57] within their epidemic models, this study focuses on incorporating a saturated type of media control, which is more realistic. To the best of our knowledge, this is the first attempt to investigate the fractional optimal control epidemic model considering both treatment and media controls. The research questions we shall investigate in this study are the following:

I.
How does the “memory effect” of the population alter the dynamics of a SEIRtype epidemic model with saturated media awareness?

II.
What are the influences of treatment and media awareness control policies on mitigating disease prevalence?

III.
Indexing the model parameters according to their local and global sensitivity.

IV.
What are the most economically effective strategies?
We have employed a fractionalorder differential equation framework to address the “memory effect”. Furthermore, to investigate question (II), an optimal control problem minimizing the infectious cases constrained to the fractionalorder system is developed. The sensitivity indices are obtained following the PRCC approach. The costeffective analysis of the control policies are obtained by calculating IAR and ICER.
The remaining part of our work is arranged as follows: In the subsequent section, there are detailed descriptions of the model equations. In Sect. 3, we discuss asymptotic dynamics encompassing the existence, uniqueness, and uniform boundedness of the solution of the proposed model, stability criterion of the equilibrium points, stability criterion of the equilibrium points, and bifurcation analysis of the proposed model. In Sect. 4, fractionalorder optimal control and efficacy function with its impact are discussed. Costeffectiveness is addressed in Sect. 5. Numerical simulation and sensitivity analysis have been performed in Sect. 6. Finally, all key results of this script are briefly discussed in the Sect. 7.
2 Model formulation
Formulation of mathematical models based on some assumptions is crucial for scientific investigation and addressing particular research questions. Based on the disease characteristics, several mathematical models are studied. We have considered an SEIRtype epidemic model for our investigation.
This paper has separated the entire population into four compartments to investigate the contagious behavior of infectious diseases. These are: Susceptible (S(t)), who are at risk of contracting the virus. Exposed (E(t)), who have been exposed to the virus but have not yet developed any symptoms. Infected (I(t)), who suffers after contracting the virus. Recovered (R(t)), who have finished the infection course, where t denotes the time. Among all the individuals, only exposed and infected individuals can spread the disease. Now, let us review the factors influencing infection and the demographic makeup.

We assume that only the susceptible population has the recruitment rate A where all individuals drop by the natural mortality rate d.

Disease transmission is a key part of the epidemic model. Incidence function takes part in disease transmission. There are various kinds of incidence functions. We have taken the bilinear incidence function (\(\beta S I\)) among them. So, the quantity (\(\beta S I\)) goes to the exposed class from susceptible class. Here, \(\beta \) is the contact rate of the susceptible individual with the exposed individual. The exposed individual goes to the infected class at a rate \(\eta \). When the disease spreads in the population, some people die due to the infection. Here, we consider \(\delta \) as the death rate due to the infection.

Due to the good immunity power, some people recover naturally with a recovery rate of m. After applying treatment control, some infected individuals become cured. The portion, bu, of the infected individual becomes cured and stays in the recovered compartment. Here, u is the treatment control, and b is the effectiveness of the treatment. So, the quantity \((m+bu)I\) goes to the recovered class from infected class. In addition, we have considered that recovery is not permanent; hence, some recovered individuals return to the susceptible class at a rate \(\sigma \).

In this work, media is taken as the key part of infection control. Here, we have considered the media as a saturation factor \(\frac{PSM}{1+qSM}\), where the parameters \(P \text {~and~} q\) are the effectiveness of media and saturation constant, respectively. Also, the parameter M is the media effect. In this saturated type of media function, the rate of media awareness programs rises as the amount of infectious agents grows, reaches maximum, and then stays constant despite future increases in contagious agents. When few options for awareness programs are available, this kind of dynamics can undoubtedly be seen. Due to the effectiveness of media, some portion of the susceptible population becomes aware, and they directly go to the recovered compartment.
The above assumptions can be graphically shown in Fig. 1. Depending on the above assumptions, the following model can be formed.
with the initial conditions are \(S(0)> 0\), \(E(0)\ge 0\), \(I(0)\ge 0\), \(R(0)\ge 0\), and the details of each parameter is given in Table 1.
Fractional calculus is widely used in many different sectors as a result of the recent, fast progress of computer technology. Biological, engineering, and physical systems have a lot of temporal memory. Ordinary differential equations cannot access the memory characteristic. Therefore, adopting a method that can resemble this kind of quality is ideal. We use this cuttingedge method in our model system (3) because, as we have already explained, fractionalorder differential equations can access many hereditary features and memory. Furthermore, the idea of fractionalorder differential equations, as proposed by Caputo, would be better suitable for epidemic models because of its starting condition feature.
Due to the simplicity of calculation, we have arranged the system’s parameters (2) in such a way that. \(\tilde{A}=A,~\tilde{\beta }=\beta ,~\tilde{d}=d,~\tilde{\alpha }=\alpha ,~\tilde{P}=P,~\tilde{\eta }=\eta ,~\tilde{\delta }=\delta ,~\tilde{m}=m,~\tilde{b}=b.\)
Therefore, the new system becomes
with initial conditions \(S(0)> 0\), \(E(0)\ge 0\), \(I(0)\ge 0\), \(R(0)\ge 0.\)
3 Dynamical analysis
Analysis of stability is a component of any dynamical system. The stability of a system can be understood or predicted via mathematical modeling. It highlights how the model reacts to changes and disturbances. Consequently, the present needs of the model are covered in this part. These requirements include the nonnegativity property, equilibrium point categorization, stability analysis, boundedness of solutions, and basic reproduction number calculation.
3.1 Existence and uniqueness of solution
The subsequent lemma helps to prove the existence and uniqueness criteria of system (3).
Lemma 1
[36] Consider the fractional order arrangement shown below:
Where \(\zeta \in (0,1],~h:[t_0,\infty )\times \Delta \rightarrow {\mathbb {R}}^n\), the prerequisite for finding individualized solutions to the system (4) on \([t_0,\infty )\times \Delta \) is that the function h(x, z) satisfies the criteria of Lipschitz condition concerning z.
We have taken the region \([t_0, G_1]\times \Omega \) to make sure the uniqueness of solution and existence criteria of the system (3), where \(\Omega =\{(x,y,z,w)\in R^4:\max \{S,E,I,R\}\le G_2\}\) and \(0<G_1<\infty \) and \(0<G_2<\infty \).
Theorem 3.1
For an arbitrary point \(X_{t_0}=(S(t_0),\) \( E(t_0),I(t_0),R(t_0))\in \Omega \), there exist a unique solution \(X(t)=(S(t),E(t),I(t),R(t))\in \Omega \) of the system (3) for any time \(t>t_0\).
Proof
To prove this theorem, first we take two points \(Y=(S,E,I,R), ~Z=(S_1,E_1,I_1,R_1)\) in \(\Omega \). Then take a map \(H:\Omega \rightarrow {\mathbb {R}}^4\), where \(H(X)=(H_1(X),H_2(X),H_3(X),H_4(X))\) with
For any Y, Z in \(\Omega \), we have
where
Therefore, the function H(X) satisfies the Lipschitz condition. Hence, by lemma 1, the system (3) attains a unique solution in \(\Omega \). Hence, the proof. \(\square \)
3.2 Positivity and boundedness of solution
We know that population is finite and nonnegative. To show the positivity and boundedness of solutions of model (3), we consider a domain \(\Omega ^+=\{(S, E, I, R)\in \Omega : S, E, I, R\in {\mathbb {R}}^+\}\), where \({\mathbb {R}}^+\) is the set of nonnegative real numbers. Now, we are in a position to state and prove the following theorem.
Theorem 3.2
Every solution of system (3) beginning in \(\Omega ^{+}\) are nonnegative and uniformly bounded for all \(t>0\).
Proof
Let \(X_{t_0}=(S_{t_0},E_{t_0},I_{t_0},R_{t_0})\in \Omega ^+\) be initial solution of system (3). Then from the system (3), we get
By lemma 3.1 in Ref. [42], we obtain \(S(t),~E(t),~I(t),R(t) \ge 0\) for any \(t\ge 0\). Therefore the solution to system (3) beginning in \(\Omega ^{+}\) with its initial condition will remain in \(\Omega ^{+}\).
Let us take \(N(t)=S(t)+E(t)+I(t)+R(t)\). Then, after applying the fractional orderderivative of order \(\zeta \), we get
Using lemma 3.3 in Ref. [42], we have \(N(t)\le (N(t_0)\frac{A}{d})E_\zeta [d(tt_0)^\zeta ]+\frac{A}{d} \rightarrow \frac{A}{d}\) as \(t \rightarrow \infty \).
Hence, the solution of system (3) beginning from \(\Omega ^+\) lie in the region \(\{(S,E,I,R)\in \Omega : 0\le N(t) \le \frac{A}{d}\}.\) Hence, the proof. \(\square \)
3.3 Feasible equilibria
Since we have assured about the existence and uniqueness of the solution of the system (3) from the previous Sect. 3.1, we, therefore, aim to find equilibrium points. As a result, we have the following theorem.
Theorem 3.3
The system (3) exhibits (i) an infectionfree equilibrium point (IFE) \(P^0(S^0,0,0,R^0)\), where
where \(S^0\) always feasible.
and (ii) a unique endemic equilibrium \(P_1(S_1,E_1,I_1,R_1)\), where
where \(B=\) \( \frac{d(d{+}\eta )(d{+}\delta {+}m{+}bu)(d{+}\sigma )(\beta \eta {+}qM(\eta {+}d)(d{+}\delta {+}m{+}bu)){+}\beta \eta PM)}{\beta \eta (d{+}\sigma )[\beta \eta {+}qM(\eta {+}d)(d{+}\delta {+}m{+}bu)]}\) \((>0)\) and \(I_1\) exists if \(A>B\).
3.4 Basic reproduction number
The predicted number of secondary cases a typical infected person should cause in a community that is entirely susceptible is known as the basic reproduction number \({\mathscr {R}}_0\). A disease often disappears from the population if the basic reproduction number is less than unity and stays present if it is greater than unity. Basic reproduction number \({\mathscr {R}}_0\) was approached here via the next generation matrix (operator) method by Diekmann et al. [16], while compartmental epidemic models were given a computational formula by van den Driessche and Watmough [52]. \({\mathscr {R}}_0\) is a threshold value for the local stability of the infectionfree solution, according to Wang and Zhao’s [60] theory of \({\mathscr {R}}_0\) for periodic compartmental epidemic models. Additionally, Thieme [58] established the spectral bound and reproduction number theory for infinitedimensional population structure and time heterogeneity. \({\mathscr {R}}_0\) in heterogeneous contexts has recently been given a new description by Inaba [26] that is based on an operator for generational development. Ultimately, the necessity of computing \({\mathscr {R}}_0\) is to understand how the disease spreads. Here, we have applied the nextgeneration matrix method to compute \({\mathscr {R}}_0\). In this model (3), E and I are the infected compartments. The form of the infected compartment is represented as \(\phi \)\(\psi \). Here \(\phi \) and \(\psi \) are as follows:
We have calculated the Jacobian matrices of \(\phi \) and \(\psi \) at \(P^0\) as follows:
The basic reproduction number of system (3) is the spectral radius of \(FV^{1}\). Therefore, the explicit form of the basic reproduction number is as follows:
3.5 Stability of equilibrium points
We have obtained the basic reproduction number (\({\mathscr {R}}_0\)) in the previous Sect. 3.4. Now, our aim is to discuss the stability at the equilibrium points \(P^0\) and \(P_1\) depending on the threshold \({\mathscr {R}}_0\). For this purpose, the Jacobian matrix at any point (S, E, I, R) is computed as follows:
Theorem 3.4
The infectionfree equilibrium point \(P^0(S^0,0,0,R^0)\) of system (3) is locally asymptotically stable if \({\mathscr {R}}_0<1\) and unstable for \({\mathscr {R}}_0>1.\)
Proof
Jacobian matrix at \(P^0(S^0,0,0,R^0)\) is
The fourthdegree characteristic equation of the Jacobian matrix \(J(P^0)\) is
where
Here \(a_1,a_2,a_3,a_4\) all are positive and \((a_1a_2a_3)>0,~a_3(a_1a_2a_3)a_4a_1^2>0\) when \({\mathscr {R}}_0<1\). Therefore, by RouthHurwitz criteria, the system (3) is locally asymptotically stable. \(\square \)
Theorem 3.5
The EE (\(S_1,E_1,I_1,R_1\)) is locally asymptotically stable, if \(\beta>\frac{\sigma (m+bu)}{S_1(d+\sigma )},~~ {\mathscr {R}}_0>1\).
Proof
To discuss the local asymptotic stability at endemic equilibrium point \(P_1(S_1,E_1,I_1,R_1)\), we have obtained the jacobian matrix \(J(P_1)\) below
The fourth degree characteristic equation of above Jacobian matrix \(J(P_1)\) is
where, the coefficients are
We observe that the coefficients \(b_1,b_2~b_3~\) are always positive. \(b_4\) is also positive and \((b_1b_2b_3)>0,~b_3(b_1b_2b_3)b_4b_1^2>0\) if \(\beta \) satisfy the following inequality:
Therefore, the system (3) satisfies all properties of RouthHurwitz criteria. Hence, EE (\(P_1\)) of the system (3) is locally asymptotically stable. Hence, the proof. \(\square \)
The local stability of both equilibrium points has been discussed in this section. Now, we shall proceed with the global stability of both equilibrium points.
Theorem 3.6
The IFE (\(P^0\)) is globally asymptotically stable if \(\beta <\frac{d(\delta +m+bu)}{A}\) and \({\mathscr {R}}_0<1.\)
Proof
To check global asymptotic stability of the system (3), we choose a Lyapunov function (\({\mathscr {L}}\)) as follows:
Applying fractionalorder derivative (\(D^\zeta \)) on both sides, we get
If \(\beta <\frac{d(\delta +m+bu)}{A}\), we get \( D^\zeta {\mathscr {L}} \le 0\). Therefore, by lemma 3.2 of Ref. [42], we can conclude that the IFE (\(P^0\)) of the system (3) is globally asymptotically stable. \(\square \)
Theorem 3.7
If \(\sigma <\min \{(6d+3PM+\delta +m+bu),(8d+4\delta +3\,m+3bu+PM),(8d+PM+2\delta +2\,m+2bu),(2dPMmbu)\}\) and \({\mathscr {R}}_0>1\), then the EE (\(P_1\)) of the system (3) is globally asymptotically stable.
Proof
We have constructed a Lyapunov function (\({\mathscr {L}}_1\)) to establish global stability criteria as follows:
Since \(P_1(S_1,E_1,I_1,R_1)\) is equilibrium point of the system (3), we can write the model system as
After using lemma 4.3 of [42], we get
By theorem 3.2, we have
Therefore, \(D^\zeta {\mathscr {L}}_1 \le 0\),
and consequently, the system (3) is globally asymptotically stable, if above conditions hold good. \(\square \)
3.6 Bifurcation analysis
French mathematician Henri Poincare first described the bifurcation of a dynamical system. How the structural change takes place as the parameter(s) changes is the subject of bifurcation research. Dynamic evolution is primarily characterized by structural change and transitional behavior in a system. The bifurcation point is the place where the two parts of a system split. At bifurcation points, trajectories’ characteristics and the behavior of fixed points can both drastically alter. In general, when bifurcation happens, the characteristics of the attractor and repellor are changed.
Theorem 3.8
The model system (3) undergoes a transcritical bifurcation at \({\mathscr {R}}_0=1\) around the infectionfree equilibrium point \(P^0\).
Proof
In the stability Sect. (3.5), we have shown the local stability of model system (3) at IFE (\(P^0\)) and EE (\(P_1\)) due to the variation of \({\mathscr {R}}_0\). The theorem (3.4) says that IFE is stable for \({\mathscr {R}}_0<1\) and otherwise unstable also the theorem (3.5) says that EE is stable for \({\mathscr {R}}_0>1\) otherwise unstable. Therefore the stability of model system (3) alters after crossing (from left to right or from right to left) \({\mathscr {R}}_0=1\). Therefore, from the publication [23], we can say that the model system (3) has transcritical bifurcation at \({\mathscr {R}}_0=1\). \(\square \)
4 Fractionalorder optimal control and efficacy function
Minimizing or maximizing the objective functional while taking certain restrictions into consideration is what is meant by optimal control. Biological modelbased issues are where optimal control is most commonly used. The primary goal of optimum control is to eradicate the virus from the afflicted area. As a control parameter, we have taken the treatment (u) and media awareness (M) into consideration. One of the key methods for controlling the spread of the disease is treatment control. People take necessary precautions due to media awareness programs. The reduction of infection, the expense of applying control measures, and the eradication of the disease are our main priorities. We apply Pontryagin’s maximal concept in order to obtain the ideal control value. At this point, we wish to create an objective functional. The constructed objective functional is
subject to the system (3). The positive weights corresponding to exposed individual E(t), infected individual I(t), treatment control u(t) and media control M(t) are \(w_1,~w_2,~w_3,w_4\), respectively, where the weights \(w_3, w_4\) measure the side effect of the controls and consequently in objective functional both the controls are taken in quadratic forms. For obvious reasons, the control parameters u(t), M(t) take the maximum value 1 and minimum value 0.
4.1 Optimal control’s existency
Our intention is to find \(u^*,M^*\) such that
where \(\Gamma =\{u,M:~u,M\text { are measurable and }0\le u(t),M(t)\le 1\text { for }t\in [t_0,t_1]\}\) is the set of control parameters.
Theorem 4.1
There are optimal controls (\(u^*,M^*\in \Gamma \)) such that
subject to the system (3).
Proof
Fleming and Rishel, in their publication [20], provide some features of the existence of two optimal controls. If the following features are satisfied, then two optimal controls, \(u^*\) and \(M^*\), exist.

(1)
Both the sets (\(\Gamma \) and state variable’s set) are nonempty.

(2)
\(\Gamma \) is both closed and convex.

(3)
The R.H.S of the system (3) is bounded by state and control variables.

(4)
The integrand L(E, I, u, M) is convex on \(\Gamma \).

(5)
There are constants \(p_1,p_2\) and \(p_3>1\) such that L(E, I, u, M) holds
$$\begin{aligned} L(E,I,u,M) \ge p_1(u^2+M^2)^{\frac{p_3}{2}}p_2 \end{aligned}$$Luke’s result assures the existence of the solution of the system (3), which proves that both the sets (\(\Gamma \) and state variable’s set) are nonempty. Hence, property 1 is satisfied. The nonnegativity of the state variables and control parameters proves the convexity of control parameters. Therefore, \(\Gamma \) is both closed and convex. Clearly, we can say from theorem 3.2 that property 3 is satisfied. To show the property 4, let us write the system (3) as:
$$\begin{aligned} D^\zeta \overrightarrow{x}=G(\overrightarrow{x})=A\overrightarrow{x}+B(\overrightarrow{x}). \end{aligned}$$where,
$$\begin{aligned}{} & {} A=\begin{pmatrix} d &{} 0 &{} 0 &{} \sigma \\ 0 &{} (\eta +d) &{} 0 &{} 0\\ 0 &{} \eta &{} (d+\delta +m+bu) &{} 0\\ 0 &{} 0 &{} (m+bu) &{} (d+\sigma ) \end{pmatrix}\\{} & {} \text { and } B=\begin{pmatrix} A\beta S I\frac{PSM}{1+qSM}\\ \beta S I\\ 0\\ \frac{PSM}{1+qSM} \end{pmatrix}. \end{aligned}$$
Let us take \(\overrightarrow{x_1}=(S_1,E_1,I_1,R_1)^T\) and \(\overrightarrow{x_2}=(S_2,E_2,I_2,R_2)^T\), then
So,
where, \(c=max \left\{ (\frac{2\beta A}{d}+2PM),\frac{2\beta A}{d},1\right\} \). Therefore, \(G(\overrightarrow{x})\) is Lipschitz continuous. Hence, the property 4.
Again we have
If we take \(p_1=min \{w_3,w_4\}\), \(p_3=2,p_1=1\), then property 5 is satisfied. Hence, the theorem. \(\square \)
4.2 Optimality system
From the previous theorem, we are certain that there is an optimal solution. Now, we are supposed to discuss the dynamic features of the optimal solution. For this purpose, we have formed Lagrangian (L) and Hamiltonian (H) as follows:
Here the variables \(\lambda _1,\lambda _2,\lambda _3,\lambda _4\) are known as the adjoint variables. We have obtained the differential equations for adjoint variables by Pontryagin’s Maximum Principal [53] as:
These adjoint variables satisfy the transversality conditions \(\lambda _i(t_1)=0\). Next, we have chosen \(\bar{S},~\bar{E},~\bar{I},~\bar{R},~\bar{\lambda }_1,\bar{\lambda }_2,~\bar{\lambda }_3,~\bar{\lambda }_4\) as the optimal value of \(S,~E,~I,~R,~\lambda _1,~\lambda _2,\lambda _3,~\lambda _4\) respectively. We shall show that the next theorem will optimize objective functional (T(u, M)).
Theorem 4.2
If \(u^*\) and \(M^*\) be the optimal value of u and M respectively that minimizes T over the region \(\Gamma \) then \(u^*=\frac{(\bar{\lambda } _3\bar{\lambda } _4)b\bar{I}}{2*w_3}\) and \(M^*\) is the positive real root of the equation
Proof
We know that the stationary conditions \(\frac{\partial H}{\partial u}=0\) and \(\frac{\partial H}{\partial M}=0\) give the optimal value of u and M respectively. Now, from \(\frac{\partial H}{\partial u}=0\), we obtain \(u^*=\frac{(\bar{\lambda } _3\bar{\lambda } _4)b\bar{I}}{2*w_3}\). Again, from \(\frac{\partial H}{\partial M}=0\), we obtain a cubic equation of M which gives \(M^*\). Previous discussion says that control parameters u and M are bounded with lower bound 0 and upper bound 1.i.e. \(u=0,M=0\) if \(u^*<0, M^*<0\) and \(u=1,M=1\) if \(u^*>1, M^*>1\) and otherwise \(u=u^*, M=M^*\). Finally, for \(u=u^*=max \{0,min \{\frac{(\bar{\lambda }_3\bar{\lambda }_4)b\bar{I}}{2*w_3},1\}\}\) and \(M=M^*=max \{0,min \{\text {positive real root of} ~q^2\bar{S} ^2M^3+2q\bar{S} M^2+M \frac{(\bar{\lambda }_1 \bar{\lambda } _4)P\bar{S}}{2w_4} =0,1\}\}\), T is optimum. \(\square \)
4.3 Efficacy function
In the context of the study of infectious diseases, an efficacy function refers to a mathematical function used to determine the effectiveness or efficacy of a particular treatment or intervention. It is commonly used in clinical research to assess the impact of a drug, therapy, or medical intervention on the outcome of interest. It quantifies the relationship between the applied controls and the observed outcomes, allowing researchers to evaluate the applied control’s efficacy.
The efficacy function can be defined based on specific endpoints or measures, such as survival rates, disease progression, symptom improvement, or other relevant clinical outcomes. Here, the efficacy function E(t) can be defined as:
Definition 1
Let I(0) be the initial value of the infected population and \(\bar{I}(t)\) be the optimal value of the infected population after applying both the controls (treatment control and media control). Then, the efficacy function can be defined in the following way:
When we use both controls, the number of infected people decreases proportionately (treatment control and media control). This proportional decline is measured by the efficacy function. The efficacy function [41] is constructed on the control parameter. The implementation of treatment control has proven to be successful, as seen by the increasing graph of efficacy function. The upper and lower bound of efficacy function are 1 and 0, respectively. Figure 7 shows the decline in the number of infected people.
5 Costeffectiveness analysis (CEA)
The process of analyzing the anticipated or predicted costs and benefits connected with a project choice to assess whether it makes sense from a business standpoint is known as a costeffectiveness analysis. The demand for CEA in control measures of disease dynamics has a crucial role in public health. It is one type of economic analysis. As it may not be suitable for monetizing health effects, costeffectiveness analysis is frequently utilized in the field of health care. The costs and health effects of one or more interventions can be analyzed using a costeffectiveness analysis. By calculating the price to achieve a unit of a health result, such as an additional year of life or death averted, it compares one intervention to another. Prioritizing the funding of healthcare programs can be done with the aid of costeffectiveness analysis. Different policies are compared on the basis of cost and outcomes with other policies. There are various techniques to represent CEA [30, 41, 45, 48]. We shall discuss here two techniques:

(a)
Infection Averted Ratio (IAR)

(b)
Incremental CostEffectiveness Ratio(ICER).
In this paper, we have considered three policies: i) policy I (treatment and media control) ii). policy II (media control) iii). policy III (treatment control). We shall discuss which policy is more suitable for both IAR and ICER.
5.1 Infection averted ratio
In order to calculate IAR, we have used the following formulae
where “Total infection averted” is the difference between the infected population with control and the infected population without control.
We have calculated IAR [30] for three policies, which is shown in Table 3. After observation of the IAR result in Table 3, we arrive at the conclusion that policyIII (treatment control) is the most costeffective, which is followed by policyI (treatment and media control).
5.2 Incremental costeffectiveness ratio
In order to calculate ICER, we consider the subsequent formulae
At first, we arranged the policies in increasing order in terms of “Total infection averted”. Then, we have calculated ICER [45] for all the policies in Table 4. We see that policyII (media control) is lower than policyIII (treatment control). So, policyIII (treatment control) is more costly and less effective than policyII (media control). Therefore, we can exclude policyIII from the list. In the ICER Table 5, we have calculated ICER for policyI (both control) and policyII (media control). Here, policy II (media control) is lower than policy I (treatment and media control). So, policy II (media control) is less costly and more effective, which is followed by policy I (treatment and media control).
Note We have observed both IAR and ICER for three policies. The results of IAR and ICER are not the same. Both IAR and ICER say that policy I (treatment and media control) is not costeffective. Hence, the policymaker will decide which policy is more suitable for disease control.
6 Numerical simulation and sensitivity analysis
All analytical results can be verified through numerical simulation. We have used MATLAB2022a to do numerical simulation. A compelling argument for utilizing numerical analysis rather than realworld data is that it is simpler to identify the effects of interactions across classes. When using realworld data, it would be difficult to determine the reasons for disparate outcomes since pricing, expenses, and technology aspects would probably change from epidemic system to epidemic system. Also, it is important to remember that the simulations in this work are qualitative in nature rather than quantitative. The findings show the range of dynamical outcomes from all the scenarios, which were done across a wide range of physiologically plausible parameter spaces. Throughout this manuscript, we have used the parameter set in the Table 2. We have taken the unit of time as day.
To show the dynamic behavior of system 3, we have computed basic reproduction number \({\mathscr {R}}_0\). Depending on the range of \({\mathscr {R}}_0\), we have demonstrated the stability conditions. Figure 2 shows the local stability of model 3 at IFE (\(P^0\)) for the parameter set in the Table 2 with different fractionalorder \(\zeta (=1,~0.95,~0.9,~0.85)\) and \(\beta =0.005,~m=0.03405,~u=0.299,~\eta =0.01,~M=0.5\). Where \({\mathscr {R}}_0=0.5326\). Except for susceptible and recovered individuals, the infected and exposed individuals converge to zero. i.e., the disease dies out.
The analytical result of the theorem 3.5 is established numerically in Fig. 3. We have depicted this figure for different fractional order (\(\zeta =1,0.95,0.9,0.85\)) with a parameter set given in the Table 2, where \({\mathscr {R}}_0=1.1217\). In this figure, fractional order is impacted. When the fractionalorder (\(\zeta \)) decreases (i.e., memory increases), the effect of the infected population and the exposed population gradually decreases. Also, the susceptible and recovered population gradually increases the effect of memory.
The alteration of \({\mathscr {R}}_0\) affects the dynamical behavior of system 3. For this cause transcritical bifurcation occurs at \({\mathscr {R}}_0=1\). It is shown in Fig. 4. This figure says that when the range of \({\mathscr {R}}_0\) is less than unity, IFE (\(P^0\)) is stable (green line indicates the stability of IFE), and when the range of \({\mathscr {R}}_0\) is greater than unity, IFE (\(P^0\)) is unstable (red line indicates unstable IFE). Also, when the range of \({\mathscr {R}}_0\) is greater than unity, EE (\(P_1\)) is stable (the blue line indicates the stability of EE).
The treatment control (u(t)) is represented in Fig. 5a. Treatment control takes its value 1 from the time \(t=0\) to \(t=54\). After time \(t=54\), the treatment control curve strictly decreases, and it goes to 0 at time \(t=60\). Media control is plotted in Fig. 5b. Similar to treatment control, it attains value 1 from the time \(t=0\) to \(t=48\). After time \(t=48\), the curve of media control decreases, and it decreases to 0 when \(t=60\).
We have plotted the state variables after applying both the controls (treatment control and media control) in Fig. 6. Here, the red, black, cyan, and blue lines indicate both control, media control, treatment control, and without control, respectively. After observing the Fig. 6, we conclude that both control (treatment control and media control) are more fruitful than single control (treatment control or media control) in decreasing infected and exposed individuals and in enlarging recovered individuals. When a media awareness program happens, the susceptible population becomes aware of the disease, and they begin to take necessary precautions to control the disease. So, some portion of the susceptible population directly transfers to the recovered compartment. For this reason, media control is fruitful in susceptible populations. For the same reason, both controls is also fruitful in the susceptible population. In this compartment, there are no infected people, so treatment control has a slight effect in the susceptible compartment. Both control and media control have great contributions to reduce the exposed population. Also, treatment control has a slight effect on reducing exposed individuals. When the disease spreads in the population, people become infected, so media has a slight effect in reducing infected individuals. Hence, after applying treatment control, the infected population decreases. So, both control and treatment control have a great effect on reducing the infected population. In the recovered compartment, media control has a great effect in enlarging the recovered population because some portion of the susceptible population comes to the recovered compartment. Also, both control has a great impact on enlarging the recovered population. In Fig. 6, we have taken 60 days as unit time. In this phase, after applying control, the curves of both control, treatment control, and media control, without control in the S, E, I, R compartments, take different positions. When we extend the time, the initial condition will be the position that was acquired by the curves at time \(t=60\) days. So, after 60 days, the curves in S, E, I, R compartments start from the initial position at time \(t=60\) days. These curves will continue after starting from the initial position.
Figure 7 represents the efficacy function. This figure shows that the curve of efficacy function is strictly increasing. The curve starts from 0 and attains the highest value of 0.715 (at time \(t=60\)). The line that is strictly growing shows that the number of people who are infected is declining.
6.1 Sensitivity analysis
In this section, we have talked about the local and global sensitivity analysis. Sensitivity analysis (SA) is a method used in finance, engineering, and decisionmaking among other domains to evaluate how changes in input variables affect a model or system’s output. It aids in comprehending the connections between a system’s inputs and outputs and in determining which factors have the most effects on the outcomes.
There may be some errors in the parameter values when collecting data on various model parameters. So, uncertainty may arise in the output. Sensitivity analysis helps decisionmakers understand the uncertainty associated with their decisions and the potential risks or opportunities that may arise due to changes in input values. Here, we will discuss local and global sensitivity analysis.
6.1.1 Local sensitivity analysis (LSA)
In epidemiology, the basic reproduction number \({\mathscr {R}}_0\) is an important parameter, as based on the values of \({\mathscr {R}}_0\), one can say for an infectious disease that the disease will persist in the population or not. So, it is crucial to find the most influential parameter on \({\mathscr {R}}_0\). For this reason, we will use here the sensitivity index approach [19, 30, 42].
Definition 2
The sensitivity index of a variable T in terms of a partial derivative is:
By the observation from Table 2, the parameters \(~d,\delta ,m,b,u,M,P~\) have negative impact and \(~\beta ,\eta ,A,q,\sigma ~\) have positive impact on basic reproduction number (\({\mathscr {R}}_0\)). A parameter’s positive impact (\(\alpha \)) implies that \(10\%\) decrease (or increase) of the value of the parameter effects on the value of (\({\mathscr {R}}_0\)) by \(\alpha \times 10\% \) decrease (or increase) of (\({\mathscr {R}}_0\)). Also, a parameter’s negative impact (\(\tau \)) implies that \(10\%\) rise (or reduction) of the value of the parameter effects on the value of (\({\mathscr {R}}_0\)) by \(\tau \times 10\% \) reduction (or rise) of the corresponding value of (\({\mathscr {R}}_0\)). From the sensitivity Table 2, the parameters \(A,\sigma \) and \(\beta \) have great positive impact on (\({\mathscr {R}}_0\)). Also, the parameters d and P have a great negative impact on (\({\mathscr {R}}_0\)). We cannot control the natural death rate (d) for the population. So, the natural death rate (d) cannot be taken into account by the control measure program. The positive impact parameters \(A, \sigma \), and \(\beta \) can be included in the control measure program. The infection can be controlled by reducing the recruitment rate (A), contact rate (\(\beta \)), and transfer rate of the recovered individual to the susceptible individual (\(\sigma \)) and by increasing the value of effectiveness of media (P). Therefore, sensitivity analysis can be treated as a disease control method.
The parameters that have an impact on the threshold parameter (\({\mathscr {R}}_0\)) are depicted in the Fig. 8 by bar diagram. The parameters \(\beta , \eta , A, q, \sigma \) have a positive impact, and the parameters d, m, b, u, M, P have a negative impact. (\({\mathscr {R}}_0\)) is greatly positively impacted by the parameters \(\beta , A,\sigma \), and negatively impacted by the parameters d, M, P.
To show the impact of \(\beta \) and \(\eta \) on the threshold parameter \({\mathscr {R}}_0\), we have represented the surface and contour plot in Fig. 9. In this figure, the density of \({\mathscr {R}}_0\) is high due to the high value of \(\beta \) and \(\eta \), i.e., when the value of \(\beta \) and \(\eta \) increases, the density of \({\mathscr {R}}_0\) is high. In Fig. 9, the yellow area signifies the high density of \({\mathscr {R}}_0\). For the lower value of \(\beta \) and \(\eta \), there is a low density of \({\mathscr {R}}_0\). \({\mathscr {R}}_0\) has a lower density, as seen by the blue area in Fig. 9. From this figure, we conclude that the parameters \(\beta \) and \(\eta \) can control the infection.
The effect of control measures (treatment control (u) and media control (M)) on reducing infection is represented by the surface and contour plot in Fig. 10. This figure shows that the control measures significantly reduce the infectious population by lessening the threshold value (\({\mathscr {R}}_0\)). When the control measures (treatment control (u) and media control (M)) are applied, the threshold value (\({\mathscr {R}}_0\)) decreases. The figure’s blue area shows that the control measures have mostly been used to lower the value of \({\mathscr {R}}_0\). For the little use of control measures, the threshold parameter \({\mathscr {R}}_0\) attains a high value. In this picture, the high value of \({\mathscr {R}}_0\) is shown by the yellow section.
In Fig. 11, \({\mathscr {R}}_0\) is strongly effected by the effectiveness of media (P) and the saturation constant (q). These two parameters have different impacts on \({\mathscr {R}}_0\). When the value of P increases, there is a decrease of the value of \({\mathscr {R}}_0\). The blue portion of the Fig. 11 demonstrates this. Also, when the value of q increases, there is a decrease of the value of \({\mathscr {R}}_0\). The yellow portion of the Fig. 11 indicates this.
6.1.2 Global sensitivity analysis
In the local sensitivity Sect. 6.1.1, we have investigated the most sensitive parameters on \({\mathscr {R}}_0\). The main limitation of the method is that one parameter in LSA changes within a range while all other parameters remain constant. Thus, nonlinear interactions between different variables are ignored, and their influence is ignored. Furthermore, only certain preassigned values for parameters that remain constant (unchanged variables) are included in the sensitivity findings. So, to get more accurate results, we need global sensitivity analysis (GSA). GSA, which takes into account a wide range of uncertainty and potential nonlinear and highorder interconnections among model parameters, is an alternate strategy to solve these limitations. In this regrade, we calculate the PRCC [22], which measures the partial rank correlation between the parameters and the basic reproduction number. The most popular method of global sensitivity analysis, which provides monotonicity between the system parameters and output, can be used to quantify a nonlinear and monotonic relationship among inputs and outcomes. Latin Hypercube Sampling (LHS), the methodology used to accomplish this strategy, is the source of the term “samplingbased method.” Both the methods (LHS and PRCC) are in the reference [63].
For the noninfluential parameters, modest but nonzero PRCCs may be constructed by using numerical estimates rather than analytical solutions [63]. In this case, the KolmogorovSmirnov [29] test is also used to statistically identify noninfluential features. The null hypothesis states that the parameter under consideration has no bearing on this situation. The null hypothesis is rejected if the pvalue is less than or equal to the selected significance threshold \(\alpha \), which is often set to \(5\%\). A thorough explanation of the pvalue may be found in the [29, 64] reference.
In this instance, we also use this GSA approach to determine the parameter that is most sensitive to the infected, exposed population, as well as to the basic reproduction number. It is possible to choose physiologically suitable parameter areas for sampling using Latin Hypercube Sampling (LHS). We have considered 2000 samples. It is believed that the parameters have uniform random distributions. Two thousand sets of values for the infected, exposed population and the fundamental reproduction number will be produced using these 2000 samples.
According to the reference [1], a parameter is called most influential if the PRCC value is less than \(0.5\) or greater than \(+0.5\). The scatter plot of each system parameter with PRCC value and pvalue that impacts exposed individual (E) is depicted in Fig. 12. The parameters \(A,d,\eta \) are most influential by investigating the PRCC values and pvalues given in each subfigure of the plot 12. The parameter A has a positive PRCC value, and the parameters \(d,\eta \) have a negative PRCC value. We cannot control the natural death rate (d). By controlling the recruitment rate (A) and the transfer rate of exposed individuals to infected individuals (\(\eta \)), we can control the exposed population (E).
Figure 13 is each system parameter’s scatter plot that impacts an infected individual (I). In Fig. 13, we have observed that the parameters \(A,d,\eta \) are most influential by analyzing the PRCC and pvalues. Here, the parameters \(A,\eta \) have positive PRCC value, and the parameter d has negative PRCC value. The control program does not accept the argument d, but the parameters \(A,\eta \) can be regarded as in the control program. We can handle the infected population by reducing A and \(\eta \).
Figure 14 is the scatter plot of each system parameter that impacts basic reproduction number (\({\mathscr {R}}_0\)). The PRCC value and the pvalue for each parameter are shown in each subfigure of the plot 14. The parameters \(A,\eta ,\sigma \) are most influential by observing PRCC value and pvalue. The control program can consider the parameters \(A,\eta ,\sigma \). We can reduce the reproduction number by controlling the parameters \(A,\eta ,\sigma \).
We have used FDE12 to do numerical simulation. The main advantage of this numerical method is that the FDE12 tackles the integration of initial value issues for fractionalorder systems based on Caputo’s definition. Moreover, the use of FDE12 is beneficial since it is easy to use and does not need to calculate the Jacobian matrix as needed in FLMM2.
The main drawback of this numerical method is that if we take the fractional value \(\zeta >1\), we will not get any solution.
7 Conclusion
In this study, a fourcompartmental model involving fractionalorder differential equations has been launched to analyze the transmission dynamics of the proposed SEIR model. We have obtained the threshold parameter (\({\mathscr {R}}_0\)), through which the stability criterion of the system (3) at the equilibrium points (namely infectionfree equilibrium, endemic equilibrium) are analyzed. It is observed that the system (3) is locally asymptotically stable at IFE (\(P^0\)) for \({\mathscr {R}}_0<1\) and globally asymptotically stable conditionally. Similarly, we have discovered that the system (3) is locally asymptotically stable at (\(P_1\)) for \({\mathscr {R}}_0>1\), and globally asymptotically stable for a threshold parametric condition of \(\sigma \) (transfer rate of recovered individual to susceptible individual) and \({\mathscr {R}}_0>1\). Therefore, \({\mathscr {R}}_0>1\) indicates the persistence of infection and \({\mathscr {R}}_0<1\) indicates the infectionfree population. In this model (3), the stability switch occurs at \({\mathscr {R}}_0=1\) through a transcritical bifurcation. Next, a fractionalorder control problem has been formulated considering treatment and media awareness controls to reduce infection. We have applied Pontryagin’s maximum principle to solve control problems analytically and the solution is simulated numerically to visualize its effect on the disease prevalence. It is observed that the implementation of both these two controls gives a better result in reducing disease prevalence than applying any single one of them. Additionally, the costeffective analysis provides a better understanding of the most successful policy at a lesser cost. We have noticed that the proportional drop in the infected population is measured by the efficacy function. i.e., when the graph of the efficacy function increases, the infected population decreases significantly. This phenomenon is pointed out in the Fig. 7. Finally, the sensitivity analysis (local and global) finds the parameters that significantly impact the course of infected (I), exposed (E) populations, and the basic reproduction number (\({\mathscr {R}}_0\)). After verifying the PRCC values and pvalues, it is observed that the parameters \(A,d,\eta ,\sigma \) are more sensitive to the model outcomes and \({\mathscr {R}}_0\). Therefore, these parameters are key to control the spread of the disease.
Li et al. [37] have proposed an epidemic model with a saturated type of media coverage and recovery to investigate the complex dynamics of infectious disease transmission. However, the model ignores the memory effect of the population, which can significantly alter the disease dynamics affecting for a longer period. The temporal nonlocality of fractionalorder derivatives is used to represent this effect in our model (3). The ODE system in [37] is efficient in modeling a disease outbreak that occurred in a short period. Still, it fails to accumulate the ‘memory effect’ of the population for disease outbreaks affecting longer periods. The fractionalorder of the system (3) is inversely proportional to the memory of the population. It is observed that both exposed and infected cases significantly decrease with a decline in the fractionalorder (See Fig. 3). Moreover, the memory effect significantly reduces the susceptible population.
In addition to this, our current study also introduces an optimal control problem considering treatment and media awareness control variables. The costeffectiveness analysis, considering both IAR and ICER techniques, gives a clear understanding of the most economically efficient disease control strategies. Our findings indicate that implementing both controls simultaneously is not the most costeffective approach. Instead, either treatment control alone or media awareness control alone proves to be less costly yet more effective measures. The solution trajectories are sensitive to the model parameters, and hence, scaling their sensitivity is crucial to measure the uncertainty of the prediction of the disease prevalences and course. This study gives a detailed local and global sensitivity analysis of the model parameters. The results obtained in this study are new in the literature and significant from both mathematical and epidemiological points of view.
Butt et al. [10] have proposed a denguemalaria ODE epidemic model involving ODE systems. Comparing the results of the infected population for the ODE model and our fractionalorder SEIR model, we have observed that our model has given a better result than the ODE model. In the research works [35, 61], the authors propose mathematical models to investigate transmission dynamics in terms of ordinary differential equations. However, they have not considered any controls in their work, but we have shown that the treatment and media controls are more efficient in our SEIR model. The authors Zhang et al. [65] proposed an SEIRtype ODE model with treatment control; they observed that the infected population became less in quantity after increasing the rate of treatment. However, neither the expense of applying the controls nor the best degree of each control to use to reduce infection was studied. In this study, we used optimal control to find the theoretical and numerical maximum amount for each control. Peter et al. [50] have taken a fractionalorder model and predicted that the memory effect could be taken into account as a control parameter. However, they have not considered media as a control parameter, although in the present article, we have used media as a fractionalorder optimal control. In literature, there are some articles on the SEIRtype epidemic model, but to our best knowledge, this is the first SEIRtype epidemic model utilizing a system of fractionalorder differential equations that incorporates saturated media effects and treatment control. In addition to deriving fractionalorder optimal control policies, the study includes both Partial Rank Correlation Coefficient (PRCC) and costeffectiveness analyses. The results obtained in this study are new in the literature and significant from both mathematical and epidemiological points of view. The summary of key results:

I.
Compared to the ODEbased model, our fractionalorder model’s disease prevalence is more realistic.

II.
Sensitivity analysis confirms that the parameters \(A, \eta , \sigma \) have some substantial impact on the infection spread.

III.
The number of infections declines significantly as more people become conscious of the impact of media.

IV.
Optimal value of media and treatment control is derived to lower the infection rate.

V.
A more suitable policy for infection control has been found from costeffectiveness analysis.
Vaccination is a crucial factor in health intervention policy. It lessens the spread of infectious diseases. Also, quarantine is a crucial factor in epidemiology. So, our next research may be extended by including vaccination and quarantine in our present model. In future research, we may also use this type of SEIR model to investigate the transmission dynamics of diseases like TB, influenza, etc.
Data Availibility Statement
No data was used for the research described in the article.
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Funding
Research work of Snehasis Barman is financially supported by National Fellowship for Scheduled Caste Students (UGCNFSC, UGCRef. No.:191620004584, Dated:20/05/2020). Research work of Suvankar Majee is financially supported by Council of Scientific and Industrial Research (CSIR), India (File No. 08/003(0142)/2020EMRI, dated: 18th March 2020). The work of T. K. Kar is financially supported by Science and Engineering Research Board, Department of Science and Technology, Government of India (File No.MTR/2022/000734,dated:19/12/2022). Also, the authors are thankful to the anonymous reviewers and the Editorinchief, Prof. Walter Lacarbonara, for their comments and suggestions to substantially improve the revised manuscript.
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Barman, S., Jana, S., Majee, S. et al. Complex dynamics of a fractionalorder epidemic model with saturated media effect. Nonlinear Dyn 112, 18611–18637 (2024). https://doi.org/10.1007/s1107102409932x
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DOI: https://doi.org/10.1007/s1107102409932x