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 COVID-19, 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 policy-making. 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 time-dependent 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 anti-malarial 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 non-pharmaceutical 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 mask-wearing, 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 malaria-dengue ODE model to investigate transmission dynamics. They have used various control strategies to prevent infection but they have not applied any fractional-order optimal control technique. To get better result, we will include fractional-order 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 long-term 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, cost-effectiveness 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 fractional-order 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 COVID-19. The authors have also studied the complex dynamics of fractional-order epidemic models in the presence of vaccination and media control [41, 42]. Veersha et al. [59] investigated numerical techniques to analyze fractional-order SIR models for childhood and vector-borne diseases. Arqub et al. also investigated the advanced solution techniques for epidemic models [4, 5]. There are different definitions of fractional-order 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 fractional-order 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 fractional-order 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 fractional-order 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 fractional-order 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 fractional-ordered 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:

  1. I.

    How does the “memory effect” of the population alter the dynamics of a SEIR-type epidemic model with saturated media awareness?

  2. II.

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

  3. III.

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

  4. IV.

    What are the most economically effective strategies?

We have employed a fractional-order differential equation framework to address the “memory effect”. Furthermore, to investigate question (II), an optimal control problem minimizing the infectious cases constrained to the fractional-order system is developed. The sensitivity indices are obtained following the PRCC approach. The cost-effective 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, fractional-order optimal control and efficacy function with its impact are discussed. Cost-effectiveness 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 SEIR-type 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 bi-linear 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.

$$\begin{aligned} \begin{aligned} \frac{dS}{dt}&=A-\beta SI-dS+\sigma R-\frac{PSM}{1+qSM},\\ \frac{dE}{dt}&=\beta SI-(d+\eta )E,\\ \frac{dI}{dt}&=\eta E-(d+\delta )I-(m+bu)I,\\ \frac{dR}{dt}&=(m+bu)I-(d+\sigma )R+\frac{PSM}{1+qSM}. \end{aligned} \end{aligned}$$
(1)

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.

Fig. 1
figure 1

Flow diagram of the disease system

Full size image

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 cutting-edge method in our model system (3) because, as we have already explained, fractional-order differential equations can access many hereditary features and memory. Furthermore, the idea of fractional-order differential equations, as proposed by Caputo, would be better suitable for epidemic models because of its starting condition feature.

$$\begin{aligned} \begin{aligned} D^\zeta S&= \tilde{A}-\tilde{\beta } SI-\tilde{d}S\\&\quad +\tilde{\sigma } R-\frac{\tilde{P}SM}{1+qSM},\\ D^\zeta E&=\tilde{\beta } SI-(\tilde{d}+\tilde{\eta })E,\\ D^\zeta I&=\tilde{\eta } E-(\tilde{d}+\tilde{\delta })I-(\tilde{m}+\tilde{b}u)I,\\ D^\zeta R&=(\tilde{m}+\tilde{b}u)I-(\tilde{d}+\tilde{\sigma })R+\frac{\tilde{P}SM}{1+qSM}, \end{aligned} \end{aligned}$$
(2)

Due to the simplicity of calculation, we have arranged the system’s parameters (2) in such a way that. \(\tilde{A}=A,~\tilde{\beta }=\beta ,~\tilde{d}=d,~\tilde{\alpha }=\alpha ,~\tilde{P}=P,~\tilde{\eta }=\eta ,~\tilde{\delta }=\delta ,~\tilde{m}=m,~\tilde{b}=b.\)

Table 1 Description of system parameters
Full size table

Therefore, the new system becomes

$$\begin{aligned} \begin{aligned} D^\zeta S&=A-\beta SI-dS+\sigma R-\frac{PSM}{1+qSM},\\ D^\zeta E&=\beta SI-(d+\eta )E,\\ D^\zeta I&=\eta E-(d+\delta )I-(m+bu)I,\\ D^\zeta R&=(m+bu)I-(d+\sigma )R+\frac{PSM}{1+qSM}. \end{aligned} \end{aligned}$$
(3)

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 non-negativity 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:

$$\begin{aligned} D^\zeta z(x)=h(x,z),~z(x_0)=z_0\text { and }x_0>0 \end{aligned}$$
(4)

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

$$\begin{aligned} \begin{aligned} H_1(X)&=A-\beta SI-dS+\sigma R-\frac{PSM}{1+qSM},\\ H_2(X)&=\beta SI-(d+\eta )E,\\ H_3(X)&=\eta E-(d+\delta )I-(m+bu)I,\\ H_4(X)&=(m+bu)I-(d+\sigma )R+\frac{PSM}{1+qSM}. \end{aligned} \end{aligned}$$
(5)

For any Y, Z in \(\Omega \), we have

$$\begin{aligned}&||H(Y)-H(Z)||\nonumber \\&\quad =\left| A-\beta SI-dS+\sigma R-\frac{PSM}{1+q S M}-A\right. \nonumber \\&\qquad \left. +\beta S_1 I_1+d S_1-\sigma R_1+\frac{PS_1 M}{1+q S_1 M}\right| \nonumber \\&\qquad +\left| \beta SI-(d+\eta )E-\beta S_1 I_1+(d+\eta )E_1\right| \nonumber \\&\qquad +|\eta E-(d+\delta )I-(m+bu)I\nonumber \\&\qquad -\eta E_1+(d+\delta )I_1+(m+bu)I_1|\nonumber \\&\qquad +|(m+bu)I-(d+\sigma )R+\frac{PSM}{1+qSM}\nonumber \\&\qquad -(m+bu)I_1+(d+\sigma )R_1-\frac{PS_1M}{1+qS_1M}|\nonumber \\&\quad = \left| -d(S-S_1)-\beta (S I-S_1 I_1)+\sigma (R-R_1)\right. \nonumber \\&\qquad \left. -PM(\frac{S}{1+qSM}-\frac{S_1}{1+qS_1M})\right| \nonumber \\&\qquad +\left| \beta (SI-S_1 I_1)-(\eta +d)(E-E_1)\right| \nonumber \\&\qquad +\left| \eta (E-E_1)-(d+\delta +m+bu)(I-I_1)\right| \nonumber \\&\qquad \left| (m+bu)(I-I_1)-(d+\sigma )(R-R_1)\right. \nonumber \\&\qquad \left. +PM(\frac{S}{1+qSM}-\frac{S_1}{1+qS_1M})\right| \nonumber \\&\quad \le d|S-S_1|+2\beta |SI-S_1I_1|+(d+2\sigma )|R-R_1|\nonumber \\&\qquad +2PM|\frac{S}{1+qSM}-\frac{S_1}{1+qS_1M}|\nonumber \\&\qquad +(d+2\eta )|E-E_1|+(d+\delta +2(m+bu))|I-I_1|\nonumber \\&\quad \le d|S-S_1|+2\beta G_2(|S-S_1|+|I-I_1|)\nonumber \\&\qquad +(d+2\sigma )|R-R_1|+2PML|S-S_1|\nonumber \\&\qquad +(d+2\eta )|E-E_1|+(d+\delta +2(m+bu))|I-I_1|\nonumber \\&\quad = (d+2G_2\beta +2MPL)|S-S_1|+(d+2\eta )|E-E_1|\nonumber \\&\qquad +(d+\delta +2G_2\beta +2(m+bu))|I\nonumber \\&\qquad -I_1|+(d+2\sigma )|R-R_1|\nonumber \\&\quad \le M_1||Y-Z||. \end{aligned}$$
(6)

where

$$\begin{aligned} M_1= & {} max \left\{ (d+2G_2\beta +2MPL), (d+2\eta ),\right. \nonumber \\{} & {} \quad \left. (d+\delta +2G_2\beta +2(m+bu)), (d+2\sigma )\right\} ,\nonumber \\ L \le{} & {} \frac{1}{(1+qSM)(1+qS_1M)}. \end{aligned}$$
(7)

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 non-negative. 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 non-negative 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 non-negative 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

$$\begin{aligned} \begin{aligned} D^\zeta S|_{S_{t_0}=0}&=A+\sigma R,\\ D^\zeta E|_{E_{t_0}=0}&=\beta S I \ge 0, \\ D^\zeta I|_{E_{t_0}=0}&=\eta E \ge 0,\\ D^\zeta R|_{R_{t_0}=0}&=(m+bu)I+\frac{PSM}{1+qSM} \ge 0. \end{aligned} \end{aligned}$$
(8)

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 order-derivative of order \(\zeta \), we get

$$\begin{aligned} \begin{aligned}&D^\zeta N = D^\zeta S+D^\zeta E+D^\zeta I+D^\zeta R\\ i.e.,&D^\zeta N= A-dN-\delta I\\ i.e.,&D^\zeta N+dN \le A. \end{aligned} \end{aligned}$$
(9)

Using lemma 3.3 in Ref. [42], we have \(N(t)\le (N(t_0)-\frac{A}{d})E_\zeta [-d(t-t_0)^\zeta ]+\frac{A}{d} \rightarrow \frac{A}{d}\) 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 infection-free equilibrium point (IFE) \(P^0(S^0,0,0,R^0)\), where

$$\begin{aligned} \begin{aligned} S^0=&\frac{((d+\sigma )(AqM-d)-dPM)+\sqrt{(}((d+\sigma )(AqM-d)-dPM)^2+4AdqM(d+\sigma )^2)}{2d(d+\sigma )qM},\\ R^0 =&\frac{PS^0M}{(d+\sigma )(1+qS^0M)}. \end{aligned} \end{aligned}$$
(10)

where \(S^0\) always feasible.

and (ii) a unique endemic equilibrium \(P_1(S_1,E_1,I_1,R_1)\), where

$$\begin{aligned}&S_1= \frac{(\eta +d)(d+\delta +m+bu)}{\beta \eta },\nonumber \\&E_1= \frac{(d+\delta +m+bu)I_1}{\beta \eta },\nonumber \\&I_1= \frac{\eta (d{+}\sigma )(A-B)}{(\eta {+}d)(d{+}\sigma )(d{+}\delta {+}m{+}bu)-\sigma \eta (m+bu)},\nonumber \\&R_1= \frac{(m+bu)I_1}{d+\sigma }+\frac{PS_1M}{(1+qS_1M)(d+\sigma )}, \end{aligned}$$
(11)

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 infection-free 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 infinite-dimensional 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 next-generation 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:

$$\begin{aligned} \phi =\begin{pmatrix} \beta S I\\ \eta E \end{pmatrix},~ \psi =\begin{pmatrix} (\eta +d)E\\ (d+\delta +m+bu)I \end{pmatrix}. \end{aligned}$$

We have calculated the Jacobian matrices of \(\phi \) and \(\psi \) at \(P^0\) as follows:

$$\begin{aligned}{} & {} F=\begin{pmatrix} 0 &{} \beta S\\ \eta &{} 0 \end{pmatrix} \text { and } \\{} & {} V=\begin{pmatrix} (\eta +d) &{} 0\\ 0 &{} (d+\delta +m+bu) \end{pmatrix}. \end{aligned}$$

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:

$$\begin{aligned} {\mathscr {R}}_0=\sqrt{\frac{\beta S^0 \eta }{(\eta +d)(d+\delta +m+bu)}} \end{aligned}$$

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:

$$\begin{aligned} J(S,E,I,R)=\begin{pmatrix} -\beta I-d-\frac{PM}{(1+qSM)^2} &{}0 &{} -\beta S &{} \sigma \\ \beta I &{} -(\eta +d) &{} \beta S &{} 0\\ 0 &{} \eta &{} -(d+\delta +m+bu) &{} 0\\ \frac{PM}{(1+qSM)^2} &{} 0 &{} (m+bu) &{} -(d+\sigma ) \end{pmatrix}. \end{aligned}$$

Theorem 3.4

The infection-free 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

$$\begin{aligned} J(P^0)=\begin{pmatrix} -d-T_1 &{}0 &{} -\beta S_0 &{} \sigma \\ 0 &{} -(\eta +d) &{} \beta S_0 &{} 0\\ 0 &{} \eta &{} -(d+\delta +m+bu) &{} 0\\ T_1 &{} 0 &{} (m+bu) &{} -(d+\sigma ) \end{pmatrix}. \end{aligned}$$

The fourth-degree characteristic equation of the Jacobian matrix \(J(P^0)\) is

$$\begin{aligned} x^4+a_1 x^3+a_2 x^2+a_3 x+a_4=0, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} a_1&= (4d+\delta +m+bu+\eta +T_1+\sigma ),\\ a_2&= (1-{\mathscr {R}}_0^2)+(d^2+d\sigma +dT_1)\\&\quad +(2d+T_1+\sigma )(2d+\delta +m+bu),\\ a_3&= (2d+T_1+\sigma )(1-{\mathscr {R}}_0^2)\\&\quad +(2d+\delta +m+bu)(d^2+d\sigma +dT_1),\\ a_4&= (d^2+d\sigma +dT_1)(1-{\mathscr {R}}_0^2),\\ T_1&= \frac{PM}{1+qS^0M}. \end{aligned} \end{aligned}$$
(12)

Here \(a_1,a_2,a_3,a_4\) all are positive and \((a_1a_2-a_3)>0,~a_3(a_1a_2-a_3)-a_4a_1^2>0\) when \({\mathscr {R}}_0<1\). Therefore, by Routh-Hurwitz 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

$$\begin{aligned} J(P_1)=\begin{pmatrix} -\beta I_1-d-\frac{PM}{(1+qS_1M)^2} &{}0 &{} -\beta S_1 &{} \sigma \\ \beta I_1 &{} -(\eta +d) &{} \beta S_1 &{} 0\\ 0 &{} \eta &{} -(d+\delta +m+bu) &{} 0\\ \frac{PM}{(1+qS_1M)^2} &{} 0 &{} (m+bu) &{} -(d+\sigma ) \end{pmatrix}. \end{aligned}$$

The fourth degree characteristic equation of above Jacobian matrix \(J(P_1)\) is

$$\begin{aligned} x^4+b_1x^3+b_2x^2+b_3x+b_4=0, \end{aligned}$$

where, the coefficients are

$$\begin{aligned} \begin{aligned} b_1&= 4d+\beta I_1+T_2+\sigma +\delta +m+bu+\eta (>0),\\ b_2&= d(\beta I_1+d+T_2)+\sigma (\beta I_1+d)\\&\quad +(2d+\beta I_1+T_2+\sigma )\\&\quad \times (2d+\delta +m+bu+\eta )(>0),\\ b_3&= (2d+\delta +m+bu+\eta )(d(\beta I_1+d+T_2)\\&\quad +\sigma (\beta I_1+d))+\beta ^2 S_1 I_1 \eta (>0),\\ b_4&= (d(\beta I_1+d+T_2)+\sigma (\beta I_1+d))\\&\quad \times ((\eta +d)(d+\delta +m+bu)-\beta S_1 \eta )+\\&\quad \times \beta I_1 \eta (\beta S_1 (d+\sigma )-\sigma (m+bu)),\\ T_2&= \frac{PM}{(1+qS_1 M)^2}. \end{aligned} \end{aligned}$$
(13)

We observe that the coefficients \(b_1,b_2~b_3~\) are always positive. \(b_4\) is also positive and \((b_1b_2-b_3)>0,~b_3(b_1b_2-b_3)-b_4b_1^2>0\) if \(\beta \) satisfy the following inequality:

$$\begin{aligned} \beta >\frac{\sigma (m+bu)}{S_1(d+\sigma )}. \end{aligned}$$

Therefore, the system (3) satisfies all properties of Routh-Hurwitz 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:

$$\begin{aligned} {\mathscr {L}}=E+I. \end{aligned}$$

Applying fractional-order derivative (\(D^\zeta \)) on both sides, we get

$$\begin{aligned} \begin{aligned} D^\zeta {\mathscr {L}}&= D^\zeta E+D^\zeta I,\\ D^\zeta {\mathscr {L}}&= \beta S I-(\eta +d)E+\eta E-(d+\delta )I\\&\quad -(m+bu)I,\\ D^\zeta {\mathscr {L}}&\le \frac{\beta A}{d} I-d(E+I)-(\delta +m+bu)I,\\ D^\zeta {\mathscr {L}}&\le -d(E+I)-(\delta +m+bu-\frac{\beta A}{d})I. \end{aligned} \end{aligned}$$
(14)

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 \(\sigma <\min \{(6d+3PM+\delta +m+bu),(8d+4\delta +3\,m+3bu+PM),(8d+PM+2\delta +2\,m+2bu),(2d-PM-m-bu)\}\) 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:

$$\begin{aligned} {\mathscr {L}}_1= & {} (S+E+I)-(S_1+E_1+I_1)\\{} & {} \times \{1+\ln {\frac{S+E+I}{S_1+E_1+I_1}}\}+R-R_1 \{1+\ln {\frac{R}{R_1}}\}. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} A&=dS_1-\sigma R_1+\frac{PS_1M}{1+qS_1M}+dE_1\\&\quad +(d+\delta +m+bu)I_1,\\&\quad \times (m+bu)I_1-(d+\sigma )R_1+\frac{PS_1M}{1+qS_1M}=0. \end{aligned} \end{aligned}$$
(15)

After using lemma 4.3 of [42], we get

$$\begin{aligned} D^\zeta {\mathscr {L}}_1&= \left[ 1-\frac{S_1+E_1+I_1}{S+E+I}\right] D^\zeta [S+E+I]\nonumber \\&\quad +\left[ 1-\frac{R_1}{R}\right] D^\zeta R,\nonumber \\&= (1-\frac{S_1+E_1+I_1}{S+E+I})\nonumber \\&\quad \times \left\{ A-dS+\sigma R-\frac{PSM}{1+qSM}\right. \nonumber \\&\quad \left. -dE-(d+\delta +m+bu)I\right\} \nonumber \\&\quad + (1-\frac{R_1}{R})\left\{ (m+bu)I\right. \nonumber \nonumber \\&\quad \left. -(d+\sigma )R+\frac{PSM}{1+qSM}\right\} ,\nonumber \\&= (1-\frac{S_1+E_1+I_1}{S+E+I})\left[ -d(S-S_1)\right. \nonumber \\&\quad +\sigma (R-R_1)-d(E-E_1)\nonumber \\&\quad -(d+\delta +m+bu)(I-I_1)\nonumber \\&\quad +\left. \frac{PM}{(1+qSM)(1+qS_1M)}(S-S_1)\right] \nonumber \\&\quad +(1-\frac{R_1}{R})\left[ (m+bu)(I-I_1)-(d+\sigma )\right. \nonumber \\&\quad \left. \times (R-R_1) +\frac{PM}{(1{+}qSM)(1{+}qS_1M)}(S{-}S_1)\right] , \end{aligned}$$
(16)

By theorem 3.2, we have

$$\begin{aligned} D^\zeta {\mathscr {L}}_1&\le \frac{A}{d}\left\{ (S-S_1)+(E-E_1)+(I-I_1)\right\} \nonumber \\&\quad \times \left[ -(d+PM)(S-S_1)+\sigma (R-R_1)\right. \nonumber \\&\quad \left. -d(E-E_1)-(d+\delta +m+bu)(I-I_1) \right] \nonumber \\&\quad +\frac{A}{d}(R-R_1)\left[ (m+bu)(I-I_1)\right. \nonumber \\&\quad \left. -(d+\sigma )(R-R_1)+PM(S-S_1)\right] ,\nonumber \\&\le \frac{A(-6d-3PM-\delta -m-bu+\sigma )}{2d}(S-S_1)^2\nonumber \\&\quad +\frac{A(-8d{-}PM{-}2\delta {-}2m{-}2bu{+}\sigma )}{2d}(E-E_1)^2\nonumber \\&\quad + \frac{A(-6d{-}4\delta {-}3m{-}3bu{-}PM{+}\sigma )}{2d}(I-I_1)^2\nonumber \\&\quad +\frac{A(-2d+\sigma +PM+m+bu)}{2d}(R-R_1)^2. \end{aligned}$$
(17)

Therefore, \(D^\zeta {\mathscr {L}}_1 \le 0\),

$$\begin{aligned} \sigma&< \min \{(6d+3PM+\delta +m+bu),\nonumber \\&\quad (8d+4\delta +3m+3bu+PM),\nonumber \\&\quad \times (8d+PM+2\delta +2m+2bu),\nonumber \\&\quad \times (2d-PM-m-bu)\}. \end{aligned}$$
(18)

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 infection-free 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 Fractional-order 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 model-based 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

$$\begin{aligned} T(u,M)&=\min _{u,M} \int _{0}^{t_1} L(E,I,u,M)\nonumber \\&=\min _{u,M} \int _{0}^{t_1}\! (w_1 E+w_2 I+w_3 u^2+w_4 M^2)dt, \end{aligned}$$
(19)

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

$$\begin{aligned} T(u^*,M^*)=\min _{{u,M}\in \Gamma } T(u,M) \end{aligned}$$
(20)

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

$$\begin{aligned} T(u^*,M^*)=\min _{{u,M}\in \Gamma } T(u,M), \end{aligned}$$

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. (1)

    Both the sets (\(\Gamma \) and state variable’s set) are non-empty.

  2. (2)

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

  3. (3)

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

  4. (4)

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

  5. (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 non-empty. Hence, property 1 is satisfied. The non-negativity 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

$$\begin{aligned}{} & {} B(\overrightarrow{x_1})-B(\overrightarrow{x_2})\\{} & {} \quad =\begin{pmatrix} -\beta S_1 I_1-\frac{PS_1M}{1+qS_1M}+\beta S_2 I_2+\frac{PS_2M}{1+qS_2M}\\ \beta S_1 I_1-\beta S_2 I_2\\ 0\\ \frac{PS_1M}{1+qS_1M}-\frac{PS_2M}{1+qS_2M} \end{pmatrix}. \end{aligned}$$

So,

$$\begin{aligned}&|B(\overrightarrow{x_1})-B(\overrightarrow{x_2})|\nonumber \\&\quad = |-\beta S_1 I_1-\frac{PS_1M}{1+qS_1M}+\beta S_2 I_2\nonumber \\&\qquad +\frac{PS_2M}{1+qS_2M}|+|\beta S_1 I_1-\beta S_2 I_2|+\nonumber \\&\qquad \times | \frac{PS_1M}{1+qS_1M}-\frac{PS_2M}{1+qS_2M}|\nonumber \\&\quad \le 2\beta |S_1I_1-S_2I_2|+2PM\nonumber \\&\qquad \times |\frac{S_1}{1+qS_1M}-\frac{S_2}{1+qS_2M}|\nonumber \\&\quad \le 2\beta [|S_1||I_1-I_2|+|I_2||S_1-S_2|]\nonumber \\&\qquad +2PM|S_1-S_2|\nonumber \\&\quad \le 2\beta [\frac{A}{d}|I_1-I_2|+\frac{A}{d}|S_1-S_2|]\nonumber \\&\qquad +2PM|S_1-S_2|\nonumber \\&\quad = (\frac{2\beta A}{d}+2PM)|S_1-S_2|+\frac{2\beta A}{d}|I_1-I_2|\nonumber \\&\quad \le (\frac{2\beta A}{d}+2PM)|S_1-S_2|\nonumber \\&\qquad +\frac{2\beta A}{d}|I_1-I_2|+|R_1-R_2|\nonumber \\&\quad \le c|\overrightarrow{x_1}-\overrightarrow{x_2}|. \end{aligned}$$
(21)

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

$$\begin{aligned}{} & {} L(E,I,u,M)=w_1 E+w_2 I+w_3 u^2+w_4 M^2 \\{} & {} \quad \ge p_1(|u|^2+|M|^2)^\frac{p_3}{2}-p_2. \end{aligned}$$

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:

$$\begin{aligned} \begin{aligned}&L(E,I,u,M)= w_1E+w_2I+w_3u^2+w_4M^2.\\&H(E,I,u,M,\lambda _1,\lambda _2,\lambda _3,\lambda _4)\\&\quad = L(E,I,u,M)+\lambda _1 D^\zeta S+\lambda _2 D^\zeta E\\&\qquad +\lambda _3 D^\zeta I+\lambda _4 D^\zeta R. \end{aligned} \end{aligned}$$
(22)

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:

$$\begin{aligned} D^\zeta \lambda _1(t)&=-\frac{\partial H}{\partial S}=(\lambda _1-\lambda _2)\beta I+\lambda _1 d\nonumber \\&\quad +(\lambda _1-\lambda _4)\frac{PM}{1+qSM},\nonumber \\ D^\zeta \lambda _2(t)&=-\frac{\partial H}{\partial E}=-w_1+(\lambda _2-\lambda _3)\eta +\lambda _2 d,\nonumber \\ D^\zeta \lambda _3(t)&=-\frac{\partial H}{\partial I}=-w_2+(\lambda _1-\lambda _2)\beta S\nonumber \\&\quad +\lambda _3(d+\delta +m+bu)-\lambda _4(m+bu),\nonumber \\ D^\zeta \lambda _4(t)&=-\frac{\partial H}{\partial R}=(\lambda _4-\lambda _1)\sigma +\lambda _4 d. \end{aligned}$$
(23)

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

$$\begin{aligned} q^2\bar{S} ^2M^3+2q\bar{S} M^2+M- \frac{(\bar{\lambda }_1- \bar{\lambda } _4)P\bar{S}}{2w_4}=0 \end{aligned}$$

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:

$$\begin{aligned} E(t)=1-\frac{I(0)}{\bar{I}(t)}. \end{aligned}$$

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.

Table 2 Sensitivity indices of the parameters defining \({\mathscr {R}}_0\) and the parameter set
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Table 3 Infected averted ratio
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5 Cost-effectiveness 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 cost-effectiveness 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, cost-effectiveness 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 cost-effectiveness 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 cost-effectiveness 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:

  1. (a)

    Infection Averted Ratio (IAR)

  2. (b)

    Incremental Cost-Effectiveness 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

$$\begin{aligned} \text {IAR}=\frac{\text {Total infection averted}}{\text {Total number of recovered}}, \end{aligned}$$

where “Total infection averted” is the difference between the infected population with control and the infected population without control.

Table 4 Incremental cost-effectiveness ratio
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Table 5 Incremental cost-effectiveness ratio
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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 policy-III (treatment control) is the most cost-effective, which is followed by policy-I (treatment and media control).

5.2 Incremental cost-effectiveness ratio

In order to calculate ICER, we consider the subsequent formulae

$$\begin{aligned} \text {ICER}=\frac{\text {Difference in total cost of two competing policies}}{\text {Difference in total infection averted}} \end{aligned}$$

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 policy-II (media control) is lower than policy-III (treatment control). So, policy-III (treatment control) is more costly and less effective than policy-II (media control). Therefore, we can exclude policy-III from the list. In the ICER Table 5, we have calculated ICER for policy-I (both control) and policy-II (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 cost-effective. 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 real-world data is that it is simpler to identify the effects of interactions across classes. When using real-world 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 fractional-order \(\zeta (=1,~0.95,~0.9,~0.85)\) 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 fractional-order (\(\zeta \)) 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.

Fig. 2
figure 2

Local asymptotic stability of IFE for different \(\zeta \)

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Fig. 3
figure 3

Local asymptotic stability of endemic equilibrium for different \(\zeta \)

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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 \(t=60\).

Fig. 4
figure 4

Occurance of transcritical bifurcation at \({\mathscr {R}}_0=1\)

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Fig. 5
figure 5

Control parameter plot: a Treatment control and b Media control

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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.

Fig. 6
figure 6

Variation of state variables with and without controls

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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.

Fig. 7
figure 7

Variation of efficacy function

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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 decision-making 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 decision-makers 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:

$$\begin{aligned} \Gamma _T^{{\mathscr {R}}_0}=\frac{\partial {\mathscr {R}}_0}{\partial T}\times \frac{T}{{\mathscr {R}}_0}. \end{aligned}$$

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.

Fig. 8
figure 8

Bar plot of sensitivity indices

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Fig. 9
figure 9

Effect of \(\beta \) and \(\eta \) on \({\mathscr {R}}_0\)

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Fig. 10
figure 10

Effect of u and M on \({\mathscr {R}}_0\)

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Fig. 11
figure 11

Effect of P and q on \({\mathscr {R}}_0\)

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Fig. 12
figure 12

Impact of system parameters on exposed population (E)

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Fig. 13
figure 13

Impact of system parameters on infected population (I)

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Fig. 14
figure 14

Impact of system parameters on basic reproduction number (\({\mathscr {R}}_0\))

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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 high-order 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 non-linear and monotonic relationship among inputs and outcomes. Latin Hypercube Sampling (LHS), the methodology used to accomplish this strategy, is the source of the term “sampling-based method.” Both the methods (LHS and PRCC) are in the reference [63].

For the non-influential parameters, modest but non-zero PRCCs may be constructed by using numerical estimates rather than analytical solutions [63]. In this case, the Kolmogorov-Smirnov [29] test is also used to statistically identify non-influential features. The null hypothesis states that the parameter under consideration has no bearing on this situation. The null hypothesis is rejected if the p-value is less than or equal to the selected significance threshold \(\alpha \), which is often set to \(5\%\). A thorough explanation of the p-value 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 p-value that impacts exposed individual (E) is depicted in Fig. 12. The parameters \(A,d,\eta \) are most influential by investigating the PRCC values and p-values 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 p-values. 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 p-value 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 p-value. 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 fractional-order 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 four-compartmental model involving fractional-order 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 infection-free 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 infection-free population. In this model (3), the stability switch occurs at \({\mathscr {R}}_0=1\) through a transcritical bifurcation. Next, a fractional-order 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 cost-effective 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 p-values, 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 fractional-order 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 fractional-order 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 fractional-order (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 cost-effectiveness 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 cost-effective 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 dengue-malaria ODE epidemic model involving ODE systems. Comparing the results of the infected population for the ODE model and our fractional-order 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 SEIR-type 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 fractional-order 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 fractional-order optimal control. In literature, there are some articles on the SEIR-type epidemic model, but to our best knowledge, this is the first SEIR-type epidemic model utilizing a system of fractional-order differential equations that incorporates saturated media effects and treatment control. In addition to deriving fractional-order optimal control policies, the study includes both Partial Rank Correlation Coefficient (PRCC) and cost-effectiveness 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:

  1. I.

    Compared to the ODE-based model, our fractional-order model’s disease prevalence is more realistic.

  2. II.

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

  3. III.

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

  4. IV.

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

  5. V.

    A more suitable policy for infection control has been found from cost-effectiveness 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.